Paper Title: An Earth-sized Planet on the Verge of Tidal Disruption.




















































Fei Dai
fdai@hawaii.edu

0000-0002-8958-0683]Fei Dai
Institute for Astronomy, University of Hawai`i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
Division of Geological and Planetary Sciences,
1200 E California Blvd, Pasadena, CA, 91125, USA
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA


0000-0001-8638-0320]Andrew W. Howard
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

0000-0003-1312-9391]Samuel Halverson
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA

0000-0003-2066-8959]Jaume Orell-Miquel
Instituto de Astrofísica de Canarias (IAC), Calle Vía Láctea s/n, 38205 La Laguna, Tenerife, Spain 

0000-0003-0987-1593]Enric Pallé
Instituto de Astrofísica de Canarias (IAC), Calle Vía Láctea s/n, 38205 La Laguna, Tenerife, Spain 

0000-0002-0531-1073]Howard Isaacson
Department of Astronomy,  University of California Berkeley, Berkeley CA 94720, USA
Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD, Australia

0000-0003-3504-5316]Benjamin Fulton
NASA Exoplanet Science Institute/Caltech-IPAC, MC 314-6, 1200 E California Blvd, Pasadena, CA 91125, USA

0000-0002-3286-3543]Ellen M. Price
Heising-Simons Foundation 51 Pegasi b Postdoctoral Fellow
Department of the Geophysical Sciences, University of Chicago, 5734 South Ellis Ave., Chicago, IL 60637, USA

0000-0002-9479-2744]Mykhaylo Plotnykov
Centre for Planetary Sciences, University of Toronto, 1265 Military Trail, Toronto, ON, M1C 1A4, Canada

0000-0003-0638-3455]Leslie A. Rogers
Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA

0000-0003-3993-4030]Diana Valencia
Centre for Planetary Sciences, University of Toronto, 1265 Military Trail, Toronto, ON, M1C 1A4, Canada

0000-0003-0062-1168]Kimberly Paragas
Division of Geological and Planetary Sciences,
1200 E California Blvd, Pasadena, CA, 91125, USA

0000-0002-0371-1647]Michael Greklek-McKeon
Division of Geological and Planetary Sciences,
1200 E California Blvd, Pasadena, CA, 91125, USA

0000-0002-0672-9658]Jonathan Gomez Barrientos
Division of Geological and Planetary Sciences,
1200 E California Blvd, Pasadena, CA, 91125, USA


0000-0002-5375-4725]Heather A. Knutson
Division of Geological and Planetary Sciences,
1200 E California Blvd, Pasadena, CA, 91125, USA


0000-0003-0967-2893]Erik A. Petigura
Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA


0000-0002-3725-3058]Lauren M. Weiss
Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN 46556, USA

0000-0001-7058-4134]Rena Lee
Institute for Astronomy, University of Hawai`i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0002-4480-310X]Casey L. Brinkman
Institute for Astronomy, University of Hawai`i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0001-8832-4488]Daniel Huber
Institute for Astronomy, University of Hawai`i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006, Australia


0000-0001-7409-5688]Guðmundur Stefánsson
Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

0000-0003-1298-9699]Kento Masuda
Department of Earth and Space Science, Osaka University, Osaka 560-0043, Japan


0000-0002-8965-3969]Steven Giacalone
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

0000-0001-9352-0248]Cicero X. Lu
Gemini Observatory/NSF’s NOIRLab, 670 N. A’ohoku Place, Hilo, HI 96720, USA
Science Fellow, Gemini North


0000-0002-1426-1186]Edwin S. Kite
Department of Geophysical Sciences, University of Chicago, Chicago, IL 60637, USA


0000-0003-2215-8485]Renyu Hu
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
Division of Geological and Planetary Sciences, California Institute of Technology


0000-0002-5258-6846]Eric Gaidos
Department of Earth Sciences, University of Hawai'i at Mänoa, 1680 East-West Road, Honolulu, HI 96822 USA


0000-0002-0659-1783]Michael Zhang
Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637


0000-0003-3856-3143]Ryan A. Rubenzahl
NSF Graduate Research Fellow
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

0000-0002-4265-047X]Joshua N. Winn
Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA

0000-0002-7127-7643]Te Han
Department of Physics & Astronomy, The University of California, Irvine, Irvine, CA 92697, USA

0000-0001-7708-2364]Corey Beard
NASA FINESST Fellow
Department of Physics & Astronomy, The University of California, Irvine, Irvine, CA 92697, USA

0000-0002-5034-9476]Rae Holcomb
Department of Physics & Astronomy, University of California Irvine, Irvine, CA 92697, USA

0000-0002-5812-3236]Aaron Householder
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA


0000-0003-0742-1660]Gregory J. Gilbert
Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA


0000-0001-8342-7736]Jack Lubin
Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA


0000-0001-7664-648X]J. M. Joel Ong
Institute for Astronomy, University of Hawai`i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
NASA Hubble Fellow

0000-0001-7047-8681]Alex S. Polanski
Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA


0000-0003-2657-3889]Nicholas Saunders
NSF Graduate Research Fellow
Institute for Astronomy, University of Hawai`i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA


0000-0002-4290-6826]Judah Van Zandt
Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA

0000-0001-7961-3907]Samuel W. Yee
Center for Astrophysics  Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
Heising-Simons Foundation 51 Pegasi b Postdoctoral Fellow


0000-0002-2696-2406]Jingwen Zhang
NASA FINESST Fellow
Institute for Astronomy, University of Hawai`i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA






Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA



0000-0002-6153-3076]Bradford Holden
University of California Observatories
University of California, Santa Cruz
1156 High St, Santa Cruz, CA 95064



Caltech Optical Observatories, Pasadena, CA, 91125, USA

0009-0008-9808-0411]Max Brodheim
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Waimea, HI 96743


0000-0002-1835-1891]Ian J. M. Crossfield
Department of Physics & Astronomy, University of Kansas, 1082 Malott,1251 Wescoe Hall Dr., Lawrence, KS 66045, USA


0009-0000-3624-1330]William Deich
University of California Observatories
University of California, Santa Cruz
1156 High St, Santa Cruz, CA 95064


Space Sciences Laboratory, University of California Berkeley, Berkeley, CA 94720, USA




0009-0004-4454-6053]Steven R. Gibson
Caltech Optical Observatories, Pasadena, CA, 91125, USA

0000-0002-7648-9119]Grant M. Hill
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Waimea, HI 96743

0009-0006-2900-0401]Sharon R Jelinsky
Space Sciences Laboratory, University of California Berkeley, Berkeley, CA 94720, USA


0000-0001-8414-8771]Marc Kassis
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Waimea, HI 96743


0000-0003-2451-5482]Russ R. Laher
NASA Exoplanet Science Institute/Caltech-IPAC, MC 314-6, 1200 E California Blvd, Pasadena, CA 91125, USA

0009-0004-0592-1850]Kyle Lanclos
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Waimea, HI 96743

0000-0001-7323-7277]Scott Lilley
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Waimea, HI 96743


0009-0008-4293-0341]Joel N. Payne
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Waimea, HI 96743


Space Sciences Laboratory, University of California Berkeley, Berkeley, CA 94720, USA

0000-0003-0149-9678]Paul Robertson
Department of Physics & Astronomy, University of California Irvine, Irvine, CA 92697, USA

0000-0001-8127-5775]Arpita Roy
Astrophysics & Space Institute, Schmidt Sciences, New York, NY 10011, USA



0000-0002-4046-987X]Christian Schwab
School of Mathematical and Physical Sciences, Macquarie University, Balaclava Road, North Ryde, NSW 2109, Australia

0000-0003-3133-6837]Abby P. Shaum
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA



0009-0007-8555-8060]Martin M. Sirk
Space Sciences Laboratory, University of California Berkeley, Berkeley, CA 94720, USA


Space Sciences Laboratory, University of California Berkeley, Berkeley, CA 94720, USA


0009-0003-6534-0428]Adam Vandenberg
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Waimea, HI 96743


0000-0002-6092-8295]Josh Walawender
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Waimea, HI 96743

0000-0002-6937-9034]Sharon X. Wang
Department of Astronomy, Tsinghua University, Beijing 100084, People's Republic of China

0009-0002-4850-5377]Shin-Ywan (Cindy) Wang
NASA Exoplanet Science Institute/Caltech-IPAC, MC 314-6, 1200 E California Blvd, Pasadena, CA 91125, USA


Space Sciences Laboratory, University of California Berkeley, Berkeley, CA 94720, USA

0000-0001-6160-5888]Jason T. Wright
Department of Astronomy & Astrophysics, 525 Davey Laboratory, Penn State, University Park, PA, 16802, USA
Center for Exoplanets and Habitable Worlds, 525 Davey Laboratory, Penn State, University Park, PA, 16802, USA
Penn State Extraterrestrial Intelligence Center, 525 Davey Laboratory, Penn State, University Park, PA, 16802, USA




0000-0002-4037-3114]Sherry Yeh
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Waimea, HI 96743







0000-0002-7349-1387]José A. Caballero 
Centro de Astrobiología (CSIC-INTA), ESAC campus, Camino Bajo del Castillo s/n, 28692 Villanueva de la Cañada, Madrid, Spain

0000-0003-0061-518X]Juan C. Morales 
Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB, Can Magrans sn, 08193 Bellaterra, Barcelona, Spain 
Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

0000-0001-9087-1245]Felipe Murgas
Instituto de Astrofísica de Canarias (IAC), Calle Vía Láctea s/n, 38205 La Laguna, Tenerife, Spain 

Departamento de Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain

0000-0002-4019-3631]Evangelos Nagel
Institut für Astrophysik und Geophysik, Georg-August-Universität  Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

0000-0003-1242-5922]Ansgar Reiners 
Institut für Astrophysik und Geophysik, Georg-August-Universität  Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

0000-0002-1624-0389]Andreas Schweitzer
Hamburger Sternwarte, Gojenbergsweg 112, D-21029 Hamburg, Germany

0000-0002-8087-4298]Hugo M. Tabernero
Departamento de Física de la Tierra y Astrofísica & IPARCOS-UCM (Instituto de Física de Partículas y del Cosmos de la UCM), Facultad de Ciencias Físicas, Universidad Complutense de Madrid, 28040 Madrid, Spain

0000-0002-6532-4378]Mathias Zechmeister 
Institut für Astrophysik und Geophysik, Georg-August-Universität  Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany



0000-0001-9263-6775]Alton Spencer
Western Connecticut State University, Danbury, CT, 06810


0000-0002-5741-3047]David R. Ciardi
NASA Exoplanet Science Institute/Caltech-IPAC, 1200 E. California Blvd., Pasadena, CA 91125, USA

0000-0002-2361-5812]Catherine A. Clark
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
NASA Exoplanet Science Institute/Caltech-IPAC, 1200 E. California Blvd., Pasadena, CA 91125, USA


0000-0003-2527-1598]Michael B. Lund
NASA Exoplanet Science Institute/Caltech-IPAC, 1200 E. California Blvd., Pasadena, CA 91125, USA

0000-0003-1963-9616]Douglas A. Caldwell
SETI Institute, Mountain View, CA 94043 USA/NASA Ames Research Center, Moffett Field, CA 94035 USA




0000-0001-6588-9574]Karen A. Collins
Center for Astrophysics  Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA


0000-0001-8227-1020]Richard P. Schwarz
Center for Astrophysics  Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA


0000-0003-1464-9276]Khalid Barkaoui
Astrobiology Research Unit, Université de Liège, 19C Allée du 6 Août, 4000 Liège, Belgium
Department of Earth, Atmospheric and Planetary Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Instituto de Astrofísica de Canarias (IAC), Calle Vía Láctea s/n, 38200, La Laguna, Tenerife, Spain


0000-0001-8621-6731]Cristilyn N. Watkins
Center for Astrophysics  Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA


0000-0002-1836-3120]Avi Shporer
Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA


0000-0001-8511-2981]Norio Narita
Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan
Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Instituto de Astrofísica de Canarias (IAC), Calle Vía Láctea s/n, 38205 La Laguna, Tenerife, Spain 


0000-0002-4909-5763]Akihiko Fukui
Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan
Instituto de Astrofísica de Canarias (IAC), Calle Vía Láctea s/n, 38205 La Laguna, Tenerife, Spain 


Kotizarovci Observatory, Sarsoni 90, 51216 Viskovo, Croatia


0000-0001-9911-7388]David W. Latham
Center for Astrophysics Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA

0000-0002-4715-9460]Jon M. Jenkins
NASA Ames Research Center, Moffett Field, CA 94035, USA

0000-0003-2058-6662]George R. Ricker
Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

0000-0002-6892-6948]Sara Seager
Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA
02139, USA
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA


0000-0001-6763-6562]Roland Vanderspek
Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
















§ ABSTRACT 
TOI-6255 b (GJ 4256) is an Earth-sized planet (1.079±0.065 R_⊕) with an orbital period of only 5.7 hours. With the newly commissioned Keck Planet Finder (KPF) and CARMENES spectrographs, we determined the planet's mass to be 1.44±0.14 M_⊕. The planet is just outside the Roche limit, with P_ orb/P_ Roche = 1.13 ±0.10.  The strong tidal force likely deforms the planet into a triaxial ellipsoid with a long axis that is ∼10% longer than the short axis.  Assuming a reduced stellar tidal quality factor Q_⋆^'≈10^7, we predict that tidal orbital decay will cause TOI-6255 to reach the Roche limit in roughly 400 Myr. Such tidal disruptions may produce the possible signatures of planet engulfment that have been on stars with anomalously high refractory elemental abundances compared to its conatal binary companion. TOI-6255 b is also a favorable target for searching for star-planet magnetic interactions, which might cause interior melting and hasten orbital decay. TOI-6255 b is a top target (Emission Spectroscopy Metric of about 24) for phase
curve observations with the James Webb Space Telescope. 








§ INTRODUCTION

The “ultra-short-period” (USP) planets are generally terrestrial planets (R<2 R_⊕) that orbit their host stars in less than one day <cit.>. More than a hundred USPs have been described in the literature. Their occurrence rate around sun-like stars is about 0.5% <cit.>. USPs provide favorable opportunities to study the composition of terrestrial planets <cit.>. A true Earth analog on a yearly orbit induces a Doppler wobble of 9 cm s^-1, which is beyond the capabilities of current-generation spectrographs. The same planet on a sub-day orbit would induce a Doppler wobble of a few m s^-1, which is within the reach of current spectrographs. Moreover, the irradiation of a USP by its host star is more than a thousand times the Earth's insolation. There is both theoretical and empirical evidence that any primordial H/He atmosphere on these planets would have been eroded by intensive mass loss <cit.>. Thus, one can directly make use of the mass and radius
measurements to probe the planet's interior composition. USP planets are also amenable to observations of thermal phase curves and secondary eclipses with the James Webb Space Telescope (JWST). The albedo, phase offset, day-night temperature contrast, and emission spectra might reveal the presence or lack of an outgassed secondary atmosphere. In the absence of an atmosphere, phase curve observation may reveal the dominant surface mineralogy on these rocky planets <cit.>. 


Several recent works suggested that the photospheric composition of a star can be significantly altered by the engulfment of rocky planets and their refractory elements <cit.>. Specifically, there are examples of stars for which the refractory elemental abundances are enhanced by about 0.1 dex relative to a co-moving and presumably co-natal binary companion. The USPs are the natural candidates for the planets that are engulfed and produce such signatures. Tidal interactions with the host star gradually shrink the orbit of a USP, and eventually bring them within the tidal disruption limit <cit.>. The resulting debris would fall onto the surface convective layer of the host star and alter the photospheric abundances. The strength of tidal interaction is steeply dependent on the orbital separation (Eqn <ref>); thus, only the shortest-period planets may be tidally disrupted or engulfed by the host star within the age of the universe <cit.>. In this work, we present the discovery of perhaps the most compelling planet that is doomed for engulfment. TOI-6255 b is an Earth-sized planet with an extraordinarily short orbital period of merely 5.7 hours. Tidal disruption could happen in the next 400 Myr if the host star has a reduced tidal quality factor Q_⋆^'=10^7.

In Section 2, we characterize the host star TOI-6255 in detail. We measure the radius of TOI-6255 b in Section 3 using transits. We extract the mass of the planet from Doppler observations in Section 4. In Section 5, we discuss the composition, tidal decay, tidal distortion, and phase curve of TOI-6255 b, as well as the suspected star-planet magnetic interaction. Section 6 is a summary of the paper.



§ HOST STAR PROPERTIES



 §.§ Spectroscopic Analysis


We obtained a high-resolution, high-signal-to-noise-ratio (SNR), iodine-free spectrum of TOI-6255 
with the High Resolution
Echelle Spectrometer on the 10-meter Keck Telescope <cit.>. The observation was taken on UT Nov 26th, 2023. We exposed for 900 seconds and reached a signal-to-noise ratio (SNR) of about 80 per reduced pixel near 550 nm.  We utilized the SpecMatch-Emp pipeline <cit.> to extract the spectroscopic parameters (T_ eff and [Fe/H]). In short, SpecMatch-Emp cross-matches the observed spectrum of a star with a library of hundreds of well-calibrated stellar spectra observed by the California Planet Search collaboration. This empirical approach circumvents many known systematic effects that plague the direct spectral modeling of low-mass stars. Some of these systematic effects include the poor definition of a continuum and the imperfect molecular line list. SpecMatch-Emp gives an effective temperature T_ eff=3421±70 K and a metallicity [Fe/H] = –0.14±0.09.  

We then included the Gaia parallax
<cit.> in our analysis. One can obtain a direct constraint on the stellar radius using the Stefan-Boltzmann law. The effective temperature, the K-band magnitude (minimal extinction), and the parallax (distance) of a star together constrain its radius. We used the Python package Isoclassify <cit.> to carry out this calculation. We adopted the MESA Isochrones & Stellar Tracks <cit.>. We used default settings recommended by Isoclassify. Tab. 1 summarizes the posterior distribution of the stellar parameters of TOI-6255. We note that Isoclassify results do not include any systematic errors that may be present in different theoretical model grids. <cit.> estimated a ∼2% error on T_ eff, ∼4% on M_⋆, and ∼5% on R_⋆.

To check on the systematic errors in stellar parameters, we performed an independent analysis using the CARMENES template spectrum of TOI-6255 (see Section <ref> for details of the CARMENES observations). In short, the CARMENES data were analyzed with the publicly available SteParSyn package <cit.> in combination with the procedures described in <cit.> and <cit.>. While,  Isoclassify uses the MIST models, the SteParSyn parameter determination used a synthetic model grid computed using BT-Settl <cit.> models as described in <cit.>.  Isoclassify and SteParSyn results agree well within 1σ: T_ eff = 3421±70 K (Isoclassify) vs. 3455±70 K (SteParSyn); logg = 4.850±0.044  vs. 4.78±0.05; [Fe/H]= -0.14±0.09  vs. -0.16±0.06; M_⋆ = 0.353±0.015 M_⊙  vs. 0.357±0.019 M_⊙; R_⋆  = 0.370±0.011 R_⊙  vs. 0.361±0.015 R_⊙. 
All these values are also consistent with previous determinations by 
<cit.>
and <cit.>.

We also checked whether there is infrared excess due to the presence of a debris disk using the stellar photosphere models with stellar parameters derived from the CARMENES spectrum of TOI-6255. We plotted 2MASS JHK_s bands, WISE W1, W2, W3, and W4 photometry, and found no evidence for infrared excess out to 22 μm against a BT-Settl model, in agreement with the results of <cit.>. However, the WISE bands are not sensitive to colder debris disks.
Finally, with the K_s magnitude and the Gaia parallax as input, the widely used empirical relation by <cit.> also gives a mass of 0.3591±0.0086 M_⊙, which is consistent with the Isoclassify within 1σ. We chose to use the Isoclassify results for further analysis. 






 §.§ Nearby Stellar Companion

We checked if TOI-6255 has a nearby stellar companion that may produce a false positive transit signal. We observed TOI-6255 with Palomar/PHARO <cit.> on UT Jun 30th, 2023.  No stellar companion was detected in the narrow-band Brγ filter (λ_o = 2.1686 μm; Δλ = 0.0326 μm). Each dither position was observed three times, offset in position from each other by 0.5 for a total of 15 frames; with an integration time of 5.7 seconds per frame, the total on-source time was 85.5 seconds. Fig. <ref> shows the resultant contrast limit as a function of angular separation. 


Using Gaia DR3 data <cit.>, no comoving companion was identified within a radius of 10 arcmin. The Gaia Renormalised Unit Weight Error (RUWE) can be considered as a reduced χ^2 of their single-star astrometric solution. TOI-6255 has a RUWE of 1.39 just below the threshold of 1.4 below which the astrometric solution is considered consistent with a single star. 
This is consistent with the previous high-resolution imaging survey for close companions performed by <cit.>.

The proper motion of TOI-6255 reported by Gaia does not match with any known comoving associations reported in Banyan-Σ <cit.> and in <cit.>. Using the framework of <cit.>, TOI-6255 has about 4% chance of being in the thick disk based on its Galactic UVW velocity (U, V, W = 32.0±1.0, -15.0±2.1, -27.2±0.5 km s^-1) after correcting for the Local Standard of Rest. In this calculation, we used the values: U_⊙,V_⊙,W_⊙ = 10.0±1.0, 11.0±2.0, 7.0±0.5 km s^-1 from <cit.>.



 §.§ Host Star Age


While many methods of estimating stellar ages do not apply to M dwarfs, gyrochronology (relating declining rotation rate to increasing age through temperature-dependent relations) may still be applicable.  No periodic signal is obvious in either a Lomb-Scargle periodogram <cit.> or an auto-correlation analysis <cit.> of TESS lightcurves of TOI-6255, probably because the rotation periods of middle-aged mid-type M dwarfs are longer than the TESS orbit (and lightcurve systematics timescale) of 13.7 days.  Each TESS sector is 27-days in duration, and only two sectors have been observed for TOI-6255. Therefore, measuring the stellar rotation of TOI-6255 using TESS data can be challenging.  A periodogram of a lightcurve from the longer-duration WASP survey <cit.> contains two peaks at ∼68 and 85 days: application of the <cit.> gyrochronology to the former yields an age of 6 ± 2 Gyr, consistent with main sequence status.  However, we caution that this gyrochronology is poorly calibrated at the cool end, and stars like TOI-6255 near or at the fully convective boundary can have markedly different spin-down rates <cit.>. 



lcc
Stellar Parameters of TOI-6255 <cit.> 


Parameters     Value and 68.3% Confidence Interval     Reference
TIC ID      261135533     A

R.A.      22:06:00.76     A

Dec.      +39:17:55.8      A
 
V (mag)     12.747 ± 0.052    A

K_s (mag)     8.071 ± 0.02    A

Effective Temperature T_eff (K)     3421±70     B 

Surface Gravity log g (cm s^-2)    4.850 ± 0.044    B 

Iron Abundance [Fe/H] (dex)    -0.14 ± 0.09    B 


Stellar Mass M_⋆ (M_⊙)    0.353±0.015    B 

Stellar Radius R_⋆ (R_⊙)    0.370±0.011    B 

Stellar Density ρ_⋆ (g cm^-3)    9.8±1.0   B 

Limb Darkening q_1 <cit.>     0.50±0.28    B

Limb Darkening q_2 <cit.>     0.35±0.18    B

Parallax π (mas)   49.0544±0.0236    C 

A:TICv8; B: this work; C: <cit.>







§ PHOTOMETRIC OBSERVATIONS




 §.§ TESS Observations

TESS <cit.> observed TOI-6255 during Sectors 16 and 56 in September 2019 and in September 2022. We downloaded the 2-min cadence light curve produced by the TESS Science Processing Operations Center <cit.>. The data is available on the Mikulski Archive for Space
Telescopes website[<https://archive.stsci.edu>] or the following [DOI]doi:10.17909/t9-nmc8-f686<cit.>. Our analysis made use of both the Presearch Data Conditioning Simple Aperture Photometry <cit.> light curve and the Simple Aperture Photometry <cit.>. PDC-SAP light curve underwent more extensive systematic mitigation and is hence used for modeling the transit signal. On the other hand, SAP light curve better preserves any long-term stellar variability, it was used in our phase curve analysis and the measurement of the rotation period for the host star. We discarded all data points that had a non-zero Quality Flag.






 §.§ Additional Transiting Planets?

The TESS Science Processing Operations Center <cit.> initially detected the transit of TOI-6255.01 (0.238-day planet). We searched for other transiting signals in the light curve. We first removed any stellar activity or instrumental variation by fitting a cubic spline in time of a width of 0.75 days. We searched the detrended light curve with our own Box-Least-Square code <cit.> that was used in previous works <cit.>. We recovered TOI-6255.01, as well as another 14.48-day transit-like signal TOI-6255.02 with a signal detection efficiency <cit.> of 14.3. This signal was independently reported on the ExoFOP website [<https://exofop.ipac.caltech.edu/tess/>] by citizen scientist Alton Spencer. SPOC analysis of TOI-6255.02 showed a substantial flux centroid shift of 27±3 arcsecond towards a nearby background star. This suggests that TOI-6255.02 is most likely a false positive. We are unable to detect any other prominent transit signal from the current data set. 



 §.§ Transit Modeling


Our transit model is based on the Python package Batman <cit.>. The stellar density of the host star is a parameter for which a prior can be set by our spectroscopic analysis ρ = 9.8±1.0 g cm^-3. The stellar density is crucial for breaking the degeneracy between the semi-major axis and impact parameter of the planet <cit.>. For limb darkening coefficients, we adopted the formulation of <cit.> (q_1 and q_2). The transit parameters further include the orbital period P_orb, the time of conjunction T_c, the planet-to-star radius ratio R_p/R_⋆, the scaled orbital distance a/R_⋆, the cosine of the orbital inclination cosi, the orbital eccentricity e, and the argument of pericenter ω. e and ω are recombined to √(e) cosω and √(e) sinω to give rise to a uniform prior on eccentricity <cit.>. We initially allowed for non-zero eccentricities. However, the resultant constraint on eccentricities using the current dataset is weak (even after incorporating the radial velocity data in Section <ref>). To reduce model complexity, we adopted circular orbits.



We first fitted all transits observed by TESS assuming a linear ephemeris (i.e. constant period, no transit timing variations). The best-fit constant-period model was found by maximizing the likelihood with the Levenberg-Marquardt method implemented in Python package lmfit <cit.>. The best-fit constant-period model then served as a template model. Each individual transit was fitted with this template model. The only free parameters are the mid-transit times and three parameters for a quadratic function of time for any out-of-transit flux variation. This process yielded a list of individual transit times. We were not able to identify any statistically meaningful transit timing variations. To sample the posterior distribution of the transit parameters, we performed a Markov Chain Monte Carlo analysis using the emcee package <cit.>. We initialized 128 walkers in the vicinity of the maximum-likelihood model from  lmfit. We ran emcee for 50000 steps which is much longer than the autocorrelation length of each parameter (hundreds of steps). We summarize the posterior distribution in Tab. 2. Fig. <ref> is the phase-folded and binned transit of TOI-6255 b. TOI-6255 b has been confirmed by our strong radial velocity (RV) detection and the well-modeled transit light curve (Section <ref>). 
 Appendix <ref> shows a tentative detection of phase curve variation of TOI-6255 b in TESS light curves. Appendix <ref> shows additional transit modeling results using ground-based light curves. 
lll

Planetary Parameters of TOI-6255 


Parameter      Symbol      Posterior Distribution 
TOI-6255 b

Planet/Star Radius Ratio     R_p/R_⋆      0.0267±0.0014  

Time of Conjunction (BJD-2457000)     T_c      1738.71248±0.00043  

Impact Parameter     b      0.84±0.05 

Scaled Semi-major Axis     a/R_⋆      3.10± 0.12  

Orbital Inclination (deg)     i      74.4±1.2  

Orbital Eccentricity      e      0 (fixed)  

Orbital Period (days)     P_ orb       0.23818244±0.00000012  

RV Semi-amplitude (m/s)      K    2.98±0.28 

Planetary Radius (R_⊕)      R_ p    1.079±0.065 

Planetary Mass (M_⊕)      M_ p    1.44±0.14 

KPF RV Jitter (m s^-1)      σ_ KPF    0.7 ± 0.3

CARMENES RV Jitter (m s^-1)      σ_ CARMENES    <0.4 (95% confidence)

Kernel Amplitude  (m s^-1)     h    8.9 ± 1.1

Correlation Timescale (days)     τ    1.2_-0.1^+0.4





§ SPECTROSCOPIC OBSERVATIONS




 §.§ KPF Observations

The Keck Planet Finder <cit.> is an echelle spectrometer that was commissioned at the Keck Observatory in March 2023.  KPF covers the wavelength range of 445–870 nm with a resolving power of 98,000. Moreover, KPF's unique combination of high efficiency, 10-m aperture, and scheduling flexibility makes it well suited for studying ultra-short period planets. This paper is one of the first science results based on KPF data. 

We observed TOI-6255 with KPF from May 22nd to Nov 24 of 2023 for a total of 91 exposures. The exposure time was 10 min. We achieved a typical SNR of 87 at 550 nm.  The spectra were reduced with the KPF Data Reduction Pipeline (DRP) which is publicly available on Github[<https://github.com/Keck-DataReductionPipelines/KPF-Pipeline>]. The KPF DRP performs quadrant stitching, flat-fielding, order tracing, and optimal extraction. The host star is an M dwarf for which traditional cross-correlation function RV extraction is sometimes inferior to template-matching methods. The cross-correlation method is limited by the lack of a good line list for the many molecular absorption features that characterize the spectra of low-mass stars.
The continuum level and a list of isolated lines are also hard to define. We used the publicly available template-matching code  <cit.> pipeline to extract the RVs from our KPF spectra.

KPF's wavelength calibration sources include Th-Ar and U-Ne lamps, a Laser Frequency Comb, a Fabry-Pérot Etalon, and the Solar Calibrator <cit.>. Wavelength calibration sources and the wavelength calibration algorithm in the KPF DRP are still subject to fine-tuning at the time of writing this paper. Some of the apparent radial velocity variations in our data for TOI-6255 are due to night-to-night instrumental drifts. Whenever possible, we tried to obtain several exposures of the star per calendar night such that night-to-night instrumental drifts can be successfully mitigated either with a correlated noise model or a floating chunk offset technique (see Section <ref>). Such an observational strategy has proven useful in the RV characterization of many USP planets <cit.>. We implemented this observation strategy using the KPF-Community Cadence algorithm described in <cit.>. The KPF RVs are shown in Tab. 3 and 4.




 §.§ CARMENES Observations


TOI-6255 was also observed with CARMENES (Calar Alto high-Resolution search for M dwarfs with Exoearths with Near-infrared and optical Échelle Spectrographs; ), located at the 3.5-m telescope of the Calar Alto Observatory, Almería, Spain. CARMENES is a dual-channel spectrograph: the optical channel (VIS) covers the wavelength range from 0.52 to 0.96 μm (ℛ = 94,600), and the near-infrared channel (NIR) covers from 0.96 to 1.71 μm (ℛ = 80,400).

We obtained 33 high-resolution spectra between Jun 24th and Aug 8th, 2023. The exposure time was 30 min, and the typical median SNR achieved in the visible channel was about 68. The spectra were reduced using the  <cit.> pipeline.
We corrected the spectra for telluric absorption () and derived relative RVs with the  pipeline <cit.>. The RVs were further adjusted using measured nightly zero point corrections (). The CARMENES RVs are shown in Tab. 5. 





 §.§ Gaussian Process Model


The measured RV variations of TOI-6255 (Fig. <ref>)  display a correlated noise component that does not phase up with the orbital period of TOI-6255 b. As mentioned in Section <ref>, the correlated noise could be due to a combination of stellar activity and instrumental drifts. We employed a Gaussian Process (GP) model <cit.> to disentangle the planetary signals from the correlated noise component.

We used the same GP framework as in our previous works <cit.>. However, instead of using a quasi-periodic kernel which is widely used in exoplanet radial velocity analyses <cit.>, we used a simpler squared-exponential kernel. In other words, we dropped the periodic component of the quasi-periodic kernel. The quasi-periodic kernel is designed to model the quasi-periodic variation of stellar activity produced by the host star rotation coupled with its surface magnetic features. For TOI-6255, we were not able to robustly identify the quasi-periodic rotational modulation of TOI-6255 with TESS observations (Section <ref>). Moreover, we suspect at least some of the correlated noise is due to instrumental drift which is unrelated to the stellar rotation period. Using the squared-exponential kernel, we are agnostic to the source of the radial velocity jitter. Our covariance matrix takes the following form:


    C_i,j = h^2 exp[-(t_i-t_j)^2/2τ^2]+[σ_i^2+σ_jit^2]δ_i,j


where C_i,j is the covariance matrix.
h is the amplitude of the squared-exponential kernel; τ is the correlation timescale. t_i is the time of each observation. δ_i,j is the Kronecker delta function i.e. the white noise component.  σ_i are the nominal uncertainty produced by the RV pipelines and σ_jit is a jitter term in case we have underestimated the white noise component. With this covariance matrix, our likelihood function is:


    logℒ =  -N/2log2π-1/2log|C|-1/2r^TC ^-1r

where ℒ is the likelihood function; N is the total number of RV measurements. r is the residual vector: r≡ RV(t_i)-M(t_i) measured RV minus the model RV. The RV models are circular Keplerian described by the RV semi-amplitude K, the orbital period P_ orb, and the time of conjunction T_c. We imposed Gaussian priors on  P_ orb and T_c using the results of the transit modeling in Section <ref>.

We allowed all of the hyperparameters of the GP to float freely in our modeling of the combined KPF and CARMENES RV dataset. The only exception is that we placed an ad hoc lower boundary on the correlation timescale τ >1 day so that the GP does not subsume the planetary signal of TOI-6255 b on 5.7 hours. We again sampled the posterior distribution using an MCMC analysis with emcee. The procedure is similar to that presented in Section <ref>. The results are summarized in Tab. 2. The RV variation of TOI-6255 b is robustly detected with K=2.98±0.28m s^-1. 



For readers who are concerned that a GP model may produce a spurious detection of planetary signals, we highlight the KPF observations taken on Nov 24 2023 (see the inset panel of Fig. <ref>). The continuous set of 21 exposures obtained that night show the expected radial-velocity variation over the 5.7-hour period of TOI-6255 b. The correlated noise component modeled by GP (blue dotted line) is essentially constant on such a short timescale. We are unable to detect another RV planet in the presence of the correlated noise.



 §.§ Floating Chunk Offset Method

Another commonly used strategy for dealing with correlated noise in RV datasets is the Floating Chunk Offset method <cit.>. If the planet of interest has an orbital period of less than 1 day, one can rely exclusively on radial-velocity variations observed during a single night, and discard any information about radial-velocity variations between nights. 
In essence, any long-term stellar activity or instrumental effects are removed by allowing each night to have an independent additive RV offset.

This method naturally requires multiple observations taken within a single night. We trimmed our RV datasets by removing the isolated exposures. We were left with 106 data points taken on 29 different calendar nights, each given an offset γ_1 to γ_29. In this model, the only major change to the covariance function (Eqn. <ref>) is that we have dropped the squared-exponential kernel and only retained the white noise component. The likelihood is unchanged other than the implicit inclusion of the nightly offsets in RV(t_i). 

We again used emcee to sample the posterior distribution following a similar procedure as in the previous section. The Floating Chunk Offset method was able to robustly detect TOI-6255 b as well, yielding the RV semi-amplitude K=2.93±0.38m s^-1. This is fully consistent with the value determined in the GP model. Fig. <ref> compares the detection of TOI-6255 b using the GP and Floating Chunk Offset models. The GP model has far fewer parameters than the Floating Chunk Offset model and makes use of more data points. We adopted the GP model for all further analyses in this paper.



§ DISCUSSION




 §.§ Core Compositions


With precise mass and radius measurements in hand, we can now investigate the possible compositions of TOI-6255 b. To put TOI-6255 b into context, we queried the NASA Exoplanet Archive [<https://exoplanetarchive.ipac.caltech.edu>] for the latest sample of terrestrial planets (<1.8R_⊕) with well-measured masses and radii (both with <20% uncertainties). The sample is shown in Fig. <ref>. We further identified a sub-sample of ultra-short-period planets (USP, R_p<1.8R_⊕, P_ orb<1 day, blue symbols in Fig. <ref>) to compare with TOI-6255 b. Previous studies <cit.> have suggested that USP planets are so strongly irradiated by their host stars that any primordial H/He envelope should have been completely stripped away by intensive atmospheric erosion <cit.>. As such, one can directly probe the planet composition of these planets without worrying about a low-mean-molecular-weight atmosphere.  Indeed, <cit.> have shown that the measured masses and radii of USPs are consistent with expectations for atmosphere-free rocky planets with an Earth-like composition of roughly 30%Fe-70% silicate rocks. 



To fully characterize the composition of TOI-6255 b, we explore two models. The first model is a 2-layer interior model with an iron core and a silicate mantle <cit.>. The composition can be parameterized by a single number: the iron core mass fraction (CMF). We found that the CMF of TOI-6255 b is 0.45±0.32 using this model. Fig. <ref> shows TOI-6255 b in comparison with other rocky exoplanets. We also employed the more sophisticated model  superearth <cit.>.  superearth further distinguishes between the true iron mass fraction (Fe-MF, Plotnykov & Valencia, 2024) and the core mass fraction (CMF). This is to account for the fact that the silicate mantle may contain iron, while the iron core may contain other elements. Following the method in <cit.>, we included different degrees of core differentiation (expressed as the amount of iron in the mantle, x_Fe), and different values for the Si alloy in the core (x_Si). We use priors on these values based on the Terrestrial Planets x_Fe∼ U(0,20) % by mol and x_Si∼ U(0,20) % by mol. Using a Markov Chain Monte Carlo scheme, we found that TOI-6255 b has a CMF=0.41±0.20 and Fe-MF=0.38±0.15. These results are in agreement with the simpler model of <cit.>. A magma ocean may be present on TOI-6255 (see later discussion); <cit.> showed that the presence of a magma ocean imposes negligible change on the inferred composition of a rocky planet from mass and radius measurement. On the other hand, the significant tidal distortion of TOI-6255 may affect the inference on its composition as will be explained in the next section.































 §.§ Tidal Distortion


Given the extremely short orbital period of TOI-6255 b, it may be significantly tidally deformed. To demonstrate that TOI-6255 b is under extreme tidal stress, we first compared its orbital period to the Roche limit of an incompressible fluid:

    P_Roche≈ 12.6 h(ρ_p/1 g cm^-3)^-1/2,

as in <cit.>.  P_Roche is the orbital period of the Roche Limit where tidal forces due to the star are stronger than the gravity of the planet i.e. the planet starts to disintegrate if material strength is negligible. The orbital period of TOI-6255 b is dangerously close to the Roche limit with P_orb/P_Roche = 1.13±0.10. Fig. <ref> shows that although dozens of ultra-short period planets have been discovered, TOI-6255 b stands out as the planet that is closest to tidal disruption.

 One can estimate the extent of tidal distortion of the planet using the Love number h_2 <cit.>. For a homogeneous solid planet:




    δ R_p=h_2ζ



    h_2 = 5/2/1+μ̃



    μ̃ = 19μ/2ρ g R_p



    ζ= M_⋆/M_p(R_p/a)^3R_p


where R_p and δ R_p are the nominal radius and tidal distortion of the planet M_p and M_⋆ are the masses of the planet and the host star. μ the tensile strength of the planet. ρ is the mean density of the planet.  g is the surface gravity of the planet.  a is the semi-major axis.  These quantities have been directly measured with the exceptions of h_2 and μ. For a centrally concentrated planet, h_2 is further reduced from 5/2/1+μ̃. <cit.> showed that increasing the iron core mass fraction of an Earth-sized planet from 0 to Earth-like (30%) leads to roughly a factor 2 reduction in the tidal Love number. As for the tensile strength μ, <cit.> recommend a mean tensile strength of 50 GPa for rocky bodies within the solar system; this would further reduce h_2 by a factor of (1+μ̃)≈ 1.9. However, for TOI-6255 b the high equilibrium temperature (∼1300K) likely leads to at least partial melting and hence significant weakening of the tensile strength of the surface rocks <cit.>. <cit.> showed that from room temperature to 1000 K, the tensile strength of common rocks can easily drop by an order of magnitude. Moreover, since the planet is likely in synchronous rotation and on a circular orbit (see Section <ref> for tidal despin and circularization timescale), the tidal bulge on the planet is static in the corotating frame centered on the planet. Solid body thermal creep (again equilibrium temperature is high enough) might have adjusted the planet to a hydrostatic equilibrium by slowly releasing any internal stress. Thus, μ̃ can be negligible on TOI-6255 b. Empirically, estimates of Earth's Love number h_2 range between 0.6 and 0.9 <cit.>. For TOI-6255 b, h_2 is likely higher than Earth given the weakened material strength and larger planetary size.  We adopt a fiducial h_2=1 for further analysis. 

The 5.7-hour rotation period of TOI-6255 also leads to substantial rotational deformation:


    q = Ω^2 R_p^3/G M_p

where Ω is rotational angular velocity of the planet, G is the gravitational constant. q quantifies surface gravity against the centrifugal force at the surface of the planet.  q amounts to about 4% on TOI-6255 b. For a synchronously rotating planet, it can be shown that the effective potential of the rotation is -1/3 of the tidal potential <cit.>. The tidal bulge is along the line connecting the host star and the planet (assuming a negligible tidal lag), while the rotational deformation bulges out the equator of the planet. The addition of these two effects results in an ellipsoidal shape (with semi-major axes [R_1,R_2,R_3]) as illustrated in Fig. <ref>. The long axis R_1 points towards the host star. The intermediate axis R_2 points along the direction of orbital motion. The short axis R_3 is the polar axis parallel to the rotation axis of the planet. The three semi-axes of the ellipsoids are given by <cit.>:


    R_1=R_ vol(1+7/6δ R_p)


    R_2=R_ vol(1-1/3δ R_p)


    R_3=R_ vol(1-5/6δ R_p)



where R_ vol≡ (R_1R_2R_3)^1/3 is the volumetric radius of the planet. A tidally distorted planet should also produce a distinct transit signal compared to a spherical model <cit.>. In particular, if the planet is in synchronous rotation, the long axis of the planet always points towards the star. Hence, the shadow of the planet projected on the star changes as a function of the orbital phase and orbital inclination: 


    A(ϕ) = π√(R_1^2R_2^2cos^2i+R_3^2sin^2i(R_1^2sin^2ϕ+R_2^2cos^2ϕ))


for a transiting planet during transit, the orbital inclination i is close to 90^∘ and the  orbital phase ϕ is close to 0.  R_1 is mostly hidden from the observer. Therefore, the transit radius of the planet is roughly given by R_ tran≡ (R_2R_3)^1/2. As such, the volumetric radius of the planet is larger than the transit radius of the planet: R_ vol = R_ tran(1+7/12δ R_p).  




We now revisit the composition of TOI-6255 b if we account for both tidal and rotational deformation. We can obtain the volumetric radius of the planet from its transit radius and the expected distortion. We then used the volumetric radius to infer the composition, primarily the iron CMF, of TOI-6255 b. Assuming h_2 = 1, this resulted in roughly a 3% increase in the transit radius. The inferred CMF of the planet dropped from 45±32% to 31±30%. Such a correction does not change the qualitative result presented in Section <ref>. 

In principle, the ellipsoidal shape of the planet can be directly measured in the transit light curve and phase curve variation. An ellipsoidal planet produces two major effects in the transit light curve: 1) it gives rise to an extension of the ingress/egress timescale, the tidal bulge will occult the star before the nominal ingress of spherical planet, and 2) the tidal bulge is hidden from observer's view at mid-transit, but away from mid-transit it slowly rotates into view and causes extra absorption compared to a spherical planet. We looked for both effects in the TESS light curve by explicitly computing the transit light curve of an ellipsoidal planet (lower panel of Fig. <ref>). The light curve correction due to an ellipsoidal planet is <10ppm. The binned uncertainty of the TESS light curve on a 2-min timescale (comparable to the ingress/egress timescale) is about 120ppm. We could not robustly detect the signature of tidal distortion in the TESS light curve.  <cit.> performed a detail analysis on detectability of tidal distortion in TESS data, they found that such a detection is challenging even for giant planets.




The calculation of the tidal and rotational deformation presented so far assumes small deformation and linear responses. In other words, the deformations are small compared to the size of the planet, and the different effects can be simply summed linearly. However, the tidal distortion on TOI-6255 (∼10%) is probably no longer linear i.e. beyond the yield point of most solids <cit.>. In comparison, the lunar tide on Earth is of order 10^-7. 

Previously, <cit.> expanded on the self-consistent field method of <cit.> to compute the tidal distortion of small, rocky planets in three dimensions, taking into account gravitational forces from the star and planet and the centrifugal force from the planet's rapid orbital motion. The model allows the planet to have an arbitrary number of layers. The <cit.> method is a relaxation procedure for solving the integro-differential equation governing the planet's equilibrium shape. In practice, the method rapidly converges to the equilibrium solution, within a specified tolerance, in just a few iterations.

We applied the model of <cit.> on TOI-6255 b. A potential limitation of the model by <cit.>  is that parameters of scientific interest such as orbital period, planet mass, and core mass fraction are outputs and cannot be specified upfront. For our purposes, we generated a library of models spanning a wide range of core-mantle boundary pressures, core pressures, aspect ratios, and planet-star distances. Within this model library, we drew and interpolated models that are broadly consistent with the measured stellar mass, planet mass, orbital period, and transit radius of TOI-6255 b. The results suggest that TOI-6255 b may indeed be more significantly distorted than what the Love theory predicts. The long axis may be 15% longer than the short axis. The model by <cit.> should be valid in the non-linear regime of tidal distortion; however, it does not include material strength yet. In Love's linear theory of tidal distortion, material strength is included (see Eqn. <ref>). It is not clear which model offers a better description of the tidal distortion of TOI-6255 b; nonetheless, the two models agree that TOI-6255 b is one of the most tidally distorted terrestrial planets discovered so far. We defer a detailed modeling of the tidal distortion when it can be empirically constrained.













 §.§ Tidal Decay and Dissipation
 


We have seen that TOI-6255 b may be tidally distorted due to its proximity to the Roche limit. Is it in any danger of being tidally disrupted in the near future? Using the constant-lag-angle model of equilibrium tides <cit.>, the tidal decay timescale due to tides raised on the slowly rotating host star is given by 


    P/P≈ 30  Gyr(Q_⋆^'/10^6)(M_⋆/M_p/M_⊙/M_⊕)(ρ_⋆/ρ_⊙)^5/3(P_ orb/1  day)^13/3

where Q_⋆^' is the tidal quality factor for the host star. M_p and M_⋆ are the masses of the planet and the host star. ρ_⋆ is the mean stellar density. Using a nominal Q_⋆^'=10^7 for the host star, TOI-6255 b has a tidal decay timescale of τ_P≡ P/P ≈3.7 Gyr. However, the orbital period only needs to change by ∼ 13% to reach the Roche Limit, which takes about τ_ Roche≈ 400 Myr if integrating Eqn. <ref> (note the difference between τ_P and  τ_ Roche). The major source of uncertainty of this calculation comes from the tidal quality factor of the host star Q_⋆^' which may vary by more than an order of magnitude and may be frequency-dependent as suggested by recent theoretical and empirical results <cit.>. Q_⋆^'=10^7 represents a relatively slow stellar dissipation rate <cit.>.  TOI-6255 is likely a mature M-star with an age of several Gyr as evidenced by the slow rotation. Tidal perturbation by a USP around a slowly spinning host stars is outside the inertial 
range rendering the suppression of dynamical tide <cit.>.

A priori, it is unlikely that a planet would be discovered right before its destruction. As shown in Section <ref>,
400 Myr represents about 5-10% of the system's age. More than a hundred USPs have been confirmed in the literature. In this light, it seems possible that the sample contains a planet whose tidal decay timescale is 5-10% of the system's age. Fig. <ref> shows the time required for TOI-6255 b and other shortest-period planets to reach the tidal disruption limit τ_ Roche. Among the USPs (P_ orb<1 day), the timescales can be shorter than the age of the universe or the age of the host star. Once the planet is disrupted, the resultant debris is likely to fall onto the host star. Depending on how permeable the radiative-convective boundary is, the refractory elements may linger in the surface convective layer. The photospheric composition of a star would thus be enhanced in refractory elements found in rocky planets. By comparing the abundance of refractory elements on the photospheres of comoving, co-natal binaries, it has been suggested that the star that is preferentially enriched in refractory elements experienced engulfment of rocky planets  <cit.>. TOI-6255 b may be a progenitor of such a catastrophic planet engulfment event. 


If TOI-6255 b still has a non-zero orbital eccentricity, tidal dissipation may take place in the planet. If so, the orbital decay rate can be sped up by including an additional term due to the tidal dissipation in the planet:


    P/P≈ 2  Gyr(Q_p^'/10^3)(M_p/M_⊕)(M_⋆/M_⊙)^2/3
         (R_p/R_⊕)^-5(P_ orb/1  day)^13/3(e/0.01)^-2


 Using a nominal planetary tidal quality factor Q_p^'=1000 similar to Mars <cit.> and an e of 0.001 level <cit.>, the tidal decay timescale for TOI-6255 b is about τ_P ≈200 Myr. The tidal dissipation amounts to ∼7% of the insolation the planet receives. Again the major source of uncertainty is in Q_p^' whose uncertainty can be more than an order of magnitude. <cit.> pointed out that as the planet's core gets molten, the solid mantle gets increasingly thinner. The mantle experiences stronger deformation and thus stronger dissipation. On the other hand, if the planet's interior is completely molten (by tidal heating or by star-planet magnetic interaction Section <ref>), the fluid-like interior is much less dissipative. Q_p^' can be orders of magnitude higher, similar to the giant planets.  
 
 Equally important, can TOI-6255 b maintain a non-zero eccentricity? The tidal circularization time is given by: 

    τ_e≡e/e≈ 0.7  Myr(Q_p^'/10^3)(M_p/M_⋆/M_⊕/M_⊙)
       (R_⋆/R_p/R_⊙/R_⊕)^5(ρ_⋆/ρ_⊙)^5/3(P_ orb/1  day)^13/3

 For TOI-6255 b, this is only τ_e≈ 400 yr if Q_p^' is 10^3. We briefly note that the tidal despin timescale for TOI-6255 b to become tidally locked is even shorter by a factor (R_p/a)^2, and can be shorter than one year for TOI-6255 b at its current orbital period. Such a short timescale justifies our earlier assumption that the planet is tidally locked with a constant day side and night side.

Given the short tidal circularization timescale, the planet should have a circular orbit unless the other planets constantly pump up the eccentricity through resonant or secular interaction <cit.>. Previous works have shown that USP planets like TOI-6255 b are almost always found in multi-planet systems <cit.>. <cit.> proposed a scenario for 55 Cnc e in which the planet maintains a non-zero eccentricity due to the perturbation of other planetary companions. We are currently unable to confirm any additional planets around TOI-6255 that could pump up the eccentricity of TOI-6255 b. Without an orbital eccentricity, most of the tidal dissipation happens within the host star as described by Eqn <ref>.

 




 §.§ Star-Planet Magnetic Interaction


It has long been suspected that ultra-short-period planets like TOI-6255 b may be within the stellar Alfvénic sphere and experience a strong star-planet magnetic interaction <cit.>. The interaction is much analogous to our Io-Jupiter system <cit.>. The transfer of ions from the planet back to the star, through a structure often referred to as the Alfvén wings, may lead to radio emission from the system <cit.>. The associated Ohmic dissipation may also give rise to orbital decay and melting of the interior of the planet <cit.>. TOI-6255 b is a top candidate for detecting such star-planet magnetic interactions.  

The deca-meter radio emission from Jupiter is likely due to the electron cyclotron maser (ECM) instability <cit.>. Star-planet magnetic interactions are capable of generating auroral radio emission from both the star and the planet. This non-thermal emission would have a characteristic gyro-frequency of ν_G = 2.8   B MHz, where B is the local magnetic field (in Gauss). The emission would be coherent, broad-band(Δ ν∼ν_G/2), and highly circularly polarized (probably reaching close to 100% in some cases).  Moreover, the stellar radio emission may be modulated by the orbital phase of the planet just like Jupiter's decameter emission.

We note that any potential emission coming from the planet would most likely be undetectable, as the magnetic field strengths of rocky planets in the solar system do not exceed a few Gauss. The resulting ECM frequency is below Earth's ionosphere cutoff. On the other hand, the stellar magnetic field could reach a few hundred G or even a few kG <cit.>. Spectropolarimetry has been used to infer a surface magnetic field strength for many nearby M dwarfs using Zeeman Doppler Imaging <cit.>. Magnetic fields of hundreds of gauss would make the stellar ECM emission detectable with current radio observatories such as LOFAR <cit.> and FAST <cit.>. We encourage follow-up observation of TOI-6255 using these radio telescopes.







 §.§ Is TOI-6255 b already disintegrating?


Given TOI-6255 b's proximity to the Roche limit (see Section <ref>), one may wonder whether TOI-6255 b is already losing mass, making it similar to the “disintegrating planets” KOI-2700 b <cit.>, KIC 12557548 b <cit.>, and K2-22b <cit.>. All three of those disintegrating planets have longer orbital periods than TOI-6255 b (Fig. <ref>). The transits of all three disintegrating planets are notably asymmetric ( egress can be several times longer than ingress) about the mid-transit times, and there can be an order of unity change in the transit depths over time. The prevailing explanation is that we are observing a tail of dusty materials episodically emanating from these planets <cit.>. See also upcoming JWST observations of K2-22 b by Tusay et al. (Program 3315). In addition, due to their material strength, 
up-to-km-size and less conspicuous debris from an ongoing 
disrupting planet may be preserved despite the intense tidal
tearing in the proximity of the host star's surface<cit.>. 

We show that TOI-6255 b is likely not undergoing catastrophic disintegration. 
We did not detect any asymmetry in the TESS transit signal. We fitted a simple trapezoidal model to the phase-folded light curve similar to <cit.>, and found no statistical significant difference between the ingress and egress of the light curve. Moreover, the transit depth of TOI-6255 b also appears to be constant. The standard deviation of in-transit and out-of-transit flux residuals of each TESS 2-min exposures are almost identical (1415 ppm v.s. 1429 ppm). Moreover, the transit depths in TESS Sectors 16 and 56 (separated by three years) are consistent within 1 σ of each other. <cit.> showed that catastrophic evaporation only takes place on small (0.1 M_⊕), strongly irradiated (2000 K) planets (i.e. high thermal sound speed and low escape velocity from the planet). TOI-6255 b (1.44 M_⊕) is likely too massive to undergo this mode of mass loss, although with its P_ orb/P_ Roche≃ 1.13, 
Roche-lobe overflow of the planet's atmosphere may significantly ease this stringent requirement<cit.>.  



 §.§ Why the shortest-period planets orbit late-type stars?


An intriguing observation is that TOI-6255 b and other top contenders for the shortest period planet – KOI-1843.03 <cit.>, K2-137 b <cit.>, K2-141 b <cit.>, GJ-367 b <cit.> – are all around late-type stars. The disintegrating planets are also all around K or M-type host stars. <cit.> showed that the occurrence rate for USPs is about 0.15±0.05 % for F stars, 0.51±0.07% for G stars, but increases to 1.1±0.4% for M dwarfs. Why are the shortest period planets found around late-type stars? What would be the implication for planet formation at the inner edge of the protoplanetary disks?

<cit.> noted that the inner edge of the protoplanetary disk plays a pivotal role in setting the lower boundary for the orbital periods of planets. Magnetospheric disk truncation is expected to happen at the corotation radius set by the rotation period of the host star. Rotation measurements of young clusters <cit.> have indeed indicated that late-type stars tend to maintain fast rotation throughout the timescale of planet formation (with periods often <3 days on 10 Myr timescales). On the other hand, sun-like stars tend to spin down faster even before the planetary disk dissipates. It could be the case that the smaller disk inner edge around late-type stars enables the formation of more short-period planets like TOI-6255 b before tides further shrink their orbits over Gyr timescale.








 §.§ JWST Phase Curve Observations


USP planets like TOI-6255 b are particularly amenable to phase curve and secondary eclipse observations with the James Webb Space Telescope <cit.>. The flux ratio between USPs and their host stars is at a favorable level of a few hundred parts per million in the infrared. Furthermore, the USPs are expected to be fully tidally locked with their host star  <cit.>, resulting in a constant day side and a constant night side. Phase curve observations map out the temperature distribution as a function of longitudes on the planet <cit.>, which in turn provides insights into the presence or absence of an outgassed, high-molecular-weight atmosphere <cit.>.  In the absence of an atmosphere, the same measurements may directly probe the surface mineralogy on these planets <cit.>. The stellar insolation TOI-6255 b receives is about 590±90F_⊕; the equilibrium temperature is 1340±60 K assuming a low albedo of 0.1 appropriate for dark basaltic surface <cit.>. The dayside may have a different extent of lava pool, depending on the detail of heat transport and if there is substantial tidal heating <cit.>. 

Ten USP planets (GJ 367 b, K2-141 b, 55 Cnc e, LHS 3844 b, TOI-561 b, WASP-47 e, TOI-1685 b, TOI-4481 b, and TOI-2445 b) have been selected as JWST targets. The majority of these are super-Earths (>1.2 R_⊕, Fig. <ref>). This preference for super-Earths is mainly driven by signal-to-noise considerations, as the Emission Spectroscopy Metric <cit.> is proportional to (R_p/R_⋆)^2. TOI-6255, on the other hand, is an Earth-sized planet (1.079±0.065 R_⊕) on a 5.7-hour orbit around a nearby M dwarf (K=8.1). It boasts an ESM of 24, making it a top ESM rocky planet for phase curve observations with MIRI/LRS, trailing only LHS 3844 b <cit.> and 55 Cnc e <cit.>. 




Finally, we examined how the tidal distortion of a planet can modify the phase curve variations. The obvious effect is the ellipsoidal light variation (ELV). Since TOI-6255 b is likely in synchronous rotation, the long axis of the ellipsoidal planet always points toward the star. Hence, at different orbital phases, we should see a different projection of the planet as described in Eqn <ref>, which roughly scales as (√(sin^2(ϕ)); ϕ is the orbital phase). The projection effect has two peaks at both quadratures of the orbit. To first order, phase curve variations of the planet will be modulated by this area projection effect. This effect is analogous to the ellipsoidal light variation of the binary stars <cit.>. For TOI-6255, we calculated that the stellar ELV is <1 ppm, whereas the planetary ELV is tens of ppm in amplitude depending on the tidal Love number and the strength of thermal emission. The planetary ELV is directly proportional to the change of the projected area of the tidal distorted planet (i.e. 10% fractional area change depending on the tidal Love number h_2).  In principle, the planetary ELV can be distinguished from the thermal phase curve.  The thermal emission of a bare rocky planet only peaks at the secondary eclipse, whereas the ELV peaks at quadratures (-cos(ϕ) v.s. √(sin^2(ϕ))). The tidal distortion of TOI-6255 may further modify the phase curve variation through geological effects. The long axis of the planet (i.e. at the substellar and antistellar points) is where the silicate mantle is the thickest, and the thicker mantle there may promote convection and volcanic activity. This effect would result in an even hotter sub-stellar point. However, a proper simulation of this effect and their observability with JWST requires a dedicated model; we defer that to future work.





§ SUMMARY

In this work, we present the discovery and confirmation of TOI-6255 b using transit observations and Doppler monitoring of the host star. We summarize the key findings as follows:


    
  * TOI-6255 b has an orbital period of 5.7 hours and a transit radius of 1.079±0.065 R_⊕.

      
  * By applying both Gaussian Process regression and the Floating Chunk Offset method on the KPF and CARMENES radial velocity measurements, we constrained the mass of TOI-6255 b to be 1.44±0.14 M_⊕.

      
  * TOI-6255 b has iron core mass fraction of 45±32% <cit.>, which is broadly consistent with an Earth-like composition.

      
  * TOI-6255 b experiences extreme tidal forces. It is marginally outside the Roche Limit with P_ orb/P_ Roche = 1.13 ±0.10 <cit.>. The planet would tidally decay to the Roche Limit within the next 400 Myr if the host star's reduced tidal quality factor is Q_⋆^'=10^7. The resultant tidal disruption may produce the long-suspected planet engulfment signature.
      
              
  * TOI-6255 b may experience star-planet magnetic interaction analogous to our Io-Jupiter interaction. Resultant radio emission may be orbital-phase-modulated and circularly polarized.
      
      
  * TOI-6255 b is likely ellipsoidal in shape. Assuming a tidal Love number h_2=1, the planet's long axis is about 10% longer than its short axis. Accounting for this effect, the CMF of the planet is reduced to 31±30%. The ellipsoidal shape produces a subtle distortion of the transit light curve (<10 ppm) that could not be detected with the existing TESS data.

       
  * With an ESM of 24, TOI-6255 b is a top-ranking target for JWST phase curve variations which may determine the presence/lack of a heavy-mean-molecular-weight atmosphere. If the planet is a bare rocky core, emission spectra may tell us the dominant surface mineralogy. The ellipsoidal shape of TOI-6255 further modifies the phase curve variation; the planetary ellipsoidal light variation should enhance planetary emission near quadratures of the orbit.



AstroImage <cit.>, Isoclassify <cit.>,isochrones <cit.> MIST <cit.>, SpecMatch-Syn <cit.>, Batman <cit.>, emcee <cit.>

Keck I: (KPF), 3.5 m Calar Alto, TESS, LCOGT, MuSCAT2, MuSCAT3, WASP, Palomar




ACKNOWLEDGEMENTS



We thank Henrique Reggiani for an independent investigation of the stellar spectrum. We thank Doug Lin, Luke Bouma, Eugene Chiang, Darryl Seligman, Saul Rappaport, Jennifer van Saders, Ji Wang, and Li Zeng for insightful discussions. We also thank Connie Rockosi for their contribution to successful construction of KPF. 

Support for this work was provided by NASA through the NASA Hubble Fellowship grant HST-HF2-51503.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.

A NASA Key Strategic Mission Support titled “Pinning Down Masses of JWST Ultra-short-period Planets with KPF” (PI: F. Dai) provided the telescope access and funding for the completion of this project.

This work was supported by a NASA Keck PI Data Award, administered by the NASA Exoplanet Science Institute. Data presented herein were obtained at the W. M. Keck Observatory from telescope time allocated to the National Aeronautics and Space Administration through the agency's scientific partnership with the California Institute of Technology and the University of California. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.

The data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.


The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community.  We are most fortunate to have the opportunity to conduct observations from this mountain.

We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products.

This paper made use of data collected by the TESS mission and are publicly available from the Mikulski Archive for Space Telescopes (MAST) operated by the Space Telescope Science Institute (STScI). 
 
Funding for the TESS mission is provided by NASA’s Science Mission Directorate. KAC and CNW acknowledge support from the TESS mission via subaward s3449 from MIT.

This research has made use of the Exoplanet Follow-up Observation Program (ExoFOP; DOI: 10.26134/ExoFOP5) website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.



This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by NSF. 


This work is partly supported by JSPS KAKENHI Grant Number JPJP24H00017 and JSPS Bilateral Program Number JPJSBP120249910.
This article is based on observations made with the MuSCAT2 instrument, developed by ABC, at Telescopio Carlos Sánchez operated on the island of Tenerife by the IAC in the Spanish Observatorio del Teide.
This paper is based on observations made with the MuSCAT3 instrument, developed by the Astrobiology Center and under financial supports by JSPS KAKENHI (JP18H05439) and JST PRESTO (JPMJPR1775), at Faulkes Telescope North on Maui, HI, operated by the Las Cumbres Observatory.

This research was carried out, in part, at the Jet Propulsion Laboratory and the California Institute of Technology under a contract with the National Aeronautics and Space Administration and funded through the President’s and Director’s Research & Development Fund Program.

We acknowledge financial support from the Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación MCIN/AEI/10.13039/501100011033 and the ERDF “A way of making Europe” through project PID2021-125627OB-C32, and from the Centre of Excellence “Severo Ochoa” award to the Instituto de Astrofisica de Canarias.

D.H. acknowledges support from the Alfred P. Sloan Foundation, the National Aeronautics and Space Administration (80NSSC21K0652), and the Australian Research Council (FT200100871).

DRC and CAC acknowledge support from NASA 18­2XRP18_2­0007









figuresection



§ SUPPLEMENTARY FIGURES


















ccccc

Keck/KPF Radial Velocities of TOI-6255


Time (BJD)     RV (m/s)     RV Unc. (m/s) 
2460087.06212686     7.91     0.95

2460090.08283583     2.73     0.91

2460091.04582747     -5.73     0.84

2460093.05386041     -11.03     1.06

2460093.11093392     -12.58     1.05

2460100.02074132     -10.36     1.00

2460100.07705751     -4.79     1.12

2460100.98823238     -2.14     0.95

2460101.02651958     -0.00     0.87

2460106.06587267     -16.29     0.89

2460106.10595727     -18.94     0.91

2460124.04069331     -11.19     0.89

2460124.11356605     -6.04     0.96

2460126.07766735     9.79     1.03

2460126.11463478     6.95     0.89

2460129.99777854     -9.10     0.97

2460130.99139985     -5.80     1.00

2460135.10930206     -0.66     1.01

2460136.98291478     -8.14     1.49

2460137.07709318     -11.15     1.21

2460146.02819387     -1.36     1.02

2460146.10394072     -3.63     0.98

2460150.02103948     15.30     0.86

2460150.09242876     21.43     1.00

2460153.01666907     13.54     1.06

2460153.06132063     10.06     0.93

2460154.90337595     14.78     1.00

2460155.91078596     3.31     0.94

2460157.02830366     -1.22     0.96

2460157.05650057     -2.94     0.94

2460157.06840406     -2.94     0.98

2460185.76215501     -7.86     1.41

2460185.91568898     -8.41     1.46

2460186.05100537     -6.90     1.41

2460187.84396189     -10.25     1.01

2460188.00378558     -2.40     1.01

2460188.10885611     -7.88     1.29

2460188.77511581     -6.39     0.93

2460188.86314294     -1.67     0.96

2460188.96253992     2.61     1.08

2460190.00555127     1.58     0.92

2460190.04392353     4.74     1.00

2460190.07231719     7.42     1.04

2460191.00661120     5.15     1.00

2460191.05434485     10.08     0.92

2460191.07363138     14.80     0.81




ccccc

Tab. 3 Continued


Time (BJD)     RV (m/s)     RV Unc. (m/s) 
2460192.02226241     15.08     0.92

2460192.03837202     14.70     0.92

2460192.09976076     14.96     0.97

2460192.76651900     3.67     1.11

2460193.77734790     3.34     1.35

2460193.82599660     -1.97     1.31

2460195.76304918     -5.14     0.94

2460195.80161774     -2.75     1.05

2460195.83124630     -2.24     0.97

2460196.76153753     -7.40     1.02

2460196.80613649     -3.91     1.02

2460196.85435330     -4.87     0.94

2460198.74800479     11.49     0.91

2460198.81742941     7.85     0.87

2460198.84542413     5.39     0.77

2460200.74046532     -7.21     1.07

2460200.78534718     -2.20     1.08

2460200.83814583     -1.26     1.07

2460201.77332723     6.47     0.73

2460201.79622377     9.70     0.81

2460201.85800751     7.00     0.81

2460204.77482518     -21.49     0.96

2460204.80641455     -20.91     0.88

2460204.91623978     -15.53     0.99

2460272.70896174     8.38     0.92

2460272.71643373     11.46     0.98

2460272.72410032     11.83     1.07

2460272.73387913     10.94     0.96

2460272.74149438     14.75     0.98

2460272.74894656     14.15     1.10

2460272.75645266     13.54     1.01

2460272.76405543     14.87     0.98

2460272.77146772     14.92     0.92

2460272.78168415     13.15     0.95

2460272.78911734     16.08     1.07

2460272.79657609     15.12     1.10

2460272.80441531     13.52     0.97

2460272.81143582     13.67     1.02

2460272.81907260     13.19     1.03

2460272.82888544     11.59     1.04

2460272.83632523     13.13     1.07

2460272.84397265     10.70     1.14

2460272.85134004     11.59     1.10

2460272.85912790     8.78     1.12

2460272.86647520     10.47     1.14



ccccc

CARMENES Radial Velocities of TOI-6255


Time (BJD)     RV (m/s)     RV Unc. (m/s) 
2460120.46393455     -3.06     1.62

 2460120.51080524     -0.05     1.52

 2460120.58599291     2.33     1.33

 2460120.64040814     0.25     1.31

 2460121.47968095     2.55     3.92

 2460122.46464087     11.23     4.30

 2460122.52944385     -0.20     2.53

 2460122.56474435     2.99     2.34

 2460122.60303551     -0.93     1.97

 2460123.46324037     6.29     3.24

 2460123.52151940     -3.84     1.94

 2460123.57083056     -3.55     1.81

 2460123.61803215     -3.06     1.74

 2460124.46376153     -4.43     2.35

 2460124.52255125     -5.01     1.82

 2460124.57117621     -5.00     2.85

 2460124.61768596     -0.29     1.90

 2460126.46060562     0.59     3.39

 2460126.49574863     4.84     2.23

 2460126.56197642     3.09     1.83

 2460126.60298822     5.36     1.55

 2460160.37799236     5.27     3.90

 2460160.42652334     1.76     2.57

 2460160.48450259     1.42     3.46

 2460160.52696557     3.37     3.23

 2460160.57725878     5.95     2.67

 2460160.62478288     3.84     2.38

 2460164.37810974     -1.80     2.52

 2460164.42647410     0.01     1.81

 2460164.48147214     -1.44     2.08

 2460164.53039479     -2.59     2.06

 2460164.57831531     -4.88     2.46

 2460164.63229627     -2.66     2.30







§ PHASE CURVE AND SECONDARY ECLIPSE


Given the extremely short orbital period of TOI-6255 b, we searched for any phase curve variation or secondary eclipse in the TESS light curve. We removed data taken during the transits of TOI-6255 b. We then detrended any long-term  stellar variability and instrumental effects following the method of <cit.>. Briefly, for each data point in the original light curve, we isolated a window of 1×, 2× or 3× P_ orb the orbital period around that data point. We removed any 5-σ outliers, and then fitted a linear function of time to the remaining data points within that window. The original data point is then divided by the local best fit linear function of time. We experimented with both the PDC-SAP and SAP TESS light curves as well as using different window width (1×, 2× or 3× P_ orb). The resultant phase curve variations did not show any substantial differences between these choices. We report results based on the SAP light curve and with window width of 2P_ orb).

Our occultation (secondary eclipse) model uses the best-fit system parameters (period, orbital inclination, etc.) from the much more precise transit (primary eclipse) model (Section <ref>). The only difference is that the limb darkening coefficients have been set to 0, and the secondary eclipse depth (δ_ sec) is allowed to vary freely to account for a combination of reflected stellar light and thermal emission from the planet in the TESS band (600-1000 nm). The out-of-eclipse phase curve variation is modeled as a Lambertian disk <cit.>. This model is characterized by the amplitude of illumination effect A and any phase offset of the peak of the phase curve θ. We initially let δ_ sec and A to vary independently, in essence allowing for a non-zero flux from the night side of the planet. However, TOI-6255 has only been observed by TESS for two sectors, the SNR of the existing data does not support a statistically significant detection of the nightside flux or a phase offset (see Fig. <ref>). We therefore report the results of the simplest phase curve model where: δ_ sec≡A (no night-side contribution) and  θ=0 (no phase offset). We sampled the posterior distribution using another MCMC analysis with emcee <cit.>. The procedure is similar to that described in the previous section. We found that δ_ sec≡A_ ill = 43±25 ppm. This is less than 2-σ detection of the phase curve/secondary eclipse.

In Fig. <ref>, we plot the ratio of planetary to stellar flux F_p/F_⋆ in the TESS band as a function of the planet's Bond Albedo. We assumed that the albedo in the TESS band is the same as the Bond Albedo. The higher the Bond albedo, the more the reflective the planet is, and the lower the equilibrium temperature will be. The planet reflects more stellar light but gives out weaker thermal emission. The blue-shaded area is the 1-σ confidence interval from TESS. The confidence interval is so wide that we cannot place a meaningful constraint on the planet's albedo. TOI-6255 b is a top-ranking target for future JWST phase curve characterization in the near and mid-infrared, any phase curve variation of TOI-6255 will be much more observable in the infrared (see Section <ref>). 




§ TRANSIT MODELING WITH OTHER LIGHT CURVES

TOI-6255 b was observed by a suite of ground-based facilities including the Las Cumbres Observatory Global Telescope <cit.> network nodes at the Teide Observatory (1-m telescope), the McDonald Observatory (1-m telescope), MuSCAT2 <cit.>, and LCOGT MuSCAT3 <cit.>. 

Due to the small size of the transit signal (∼0.7 mmag), only MuSCAT3 (the largest aperture) detected the transit signal robustly. We highlight the results from MuSCAT3 here. All other ground-based photometric observations can be downloaded from the ExoFOP website. MuSCAT3 is mounted on the 2-m Faulkes Telescope North at Haleakala Observatory on Maui, Hawai'i. MuSCAT3 has 4 simultaneous channels: g^' (400-550 nm), r^' (550-700 nm), i^' (700-820 nm), and z_s (820-920 nm). We combined the four channels together in our photometric analysis using their weighted mean and a single set of limb darkening coefficients. 

TOI-6255 was observed by MuSCAT3 on UT Aug 14th, Aug 18th, Sept 8th, Sept 28th, and Nov 24th of 2023.  The images  were calibrated using standard procedures by the BANZAI pipeline [<https://github.com/LCOGT/banzai>]. Differential photometric data were extracted using AstroImageJ <cit.> with circular photometric apertures having radius 6 or smaller that excluded flux from the nearest known neighbor (Gaia DR3 1956328333129802112) that is 9 west of TOI-6255. We followed the same procedure as our analysis of the TESS light curves. The phase-folded and binned MuSCAT3 light curve and the best-fit transit model are shown in Fig. <ref>. The posterior distribution of the transit parameters are consistent with that from TESS albeit with higher uncertainty. Notably, the MuSCAT3 observations also suggest a high impact parameter (b=0.77±0.09) for TOI-6255 b (c.f. b = 0.84±0.05 from TESS). MuSCAT3 suggests  R_p/R_⋆ = 0.0286±0.0031 which agrees with the  R_p/R_⋆ = 0.0267±0.0014 from TESS) well within 1-σ.  

We also modeled the TESS Gaia Light Curve <cit.> which used point-spread function (PSF) modeling to remove contamination from nearby stars in the TESS 200-sec Full-Frame-Image light curves. TGLC light curve is only available for Sector 56 for TOI-6255. We found that SPOC/PDC-SAP and TGLC produced consistent results on the radius of TOI-6255 b. TGLC <cit.> light curves also give a consistent R_p/R_⋆ = 0.0273±0.0014.




 Paper Title: An all-sky catalogue of stellar reddening values.

 


T_ eff
km s^-1
log g
ξ_ t
ζ_ RT
v_ r
vsin i
E(B-V)
Kepler

   


Department of Theoretical Physics and Astrophysics, Masaryk University, Kotlářská 2, 611 37 Brno, Czechia, epaunzen@physics.muni.cz
Kuffner Observatory, Johann-Staud-Straße 10, A-1160 Wien, Austria
Advanced Technologies Research Institute, Faculty of Materials Science and Technology in Trnava, Slovak University of Technology in Bratislava, Bottova 25, 917 24 Trnava, Slovakia





 
  
  
   When observing astronomical objects, we have to deal with extinction (i.e. the absorption and scattering of the emitted radiation by dust 
   and gas between the source and the observer). Interstellar extinction depends on the location of the object and the wavelength. The different extinction laws describing these effects are difficult to estimate for a small sample of stars.
  
   Many sophisticated and automatic methods have recently been developed for estimating astrophysical parameters (age and metallicity, for example) depending on the reddening, which is normally treated as a free parameter within the corresponding estimations. However, many reddening values for stars have been published over the last few decades, most of which include observations in the ultraviolet, which are essential for a good estimation but are essentially no longer taken into account. 
  
   We searched the literature through the end of 2022 for published independent reddening values of stellar objects based on various methods that exclude estimates from reddening maps. In addition, we present new reddening estimates based on the classical photometric indices in the Geneva, Johnson, and Strömgren-Crawford systems. These are based on well-established and reliable calibrations.
  
   After a careful identification procedure and quality assessment of the data, we calculated the mean reddening values of 157 631 individual available measurements for 97 826 objects. We compared our results with the ones from recent automatic pipeline values, including those from the Gaia consortium. In addition, we chose star cluster members to compare their mean
   values with estimates for the corresponding aggregates. Within the different references, we find several statistically significant offsets and trends and discuss possible explanations for them.
  
         Our new catalogue can serve as a starting point for calibrating and testing automatic tools such as isochrone and
  spectral energy distribution fitting. Our sample covers the whole sky, including the Galactic field, star clusters, and Magellanic Clouds, and so can be used for a variety of astrophysical studies.

   

   An all-sky catalogue of stellar reddening values
    E. Paunzen1
        M. Netopil2
        M. Prišegen3
                                N. Faltová1
        
    
=============================================================================================================








§ INTRODUCTION
 


Extinction and reddening caused by interstellar dust affect the detected radiation
from most observable astronomical sources. The problem worsens when objects lie close to the
Galactic plane due to the high dust density <cit.>. An error in determining the 
reddening will have severe consequences, for example for drawing colour-magnitude diagrams and 
determining astrophysical parameters such as the effective temperature, luminosity, mass, and age
<cit.>. 

However, <cit.> conducted multi-colour photometry of O- and B-type stars 
and found no unique reddening law. In addition, they concluded that there is a minor variation in the total-to-selective extinction ratio (R_ V) with Galactic longitude. This was later confirmed by <cit.> from the spatial variation in the wavelength of maximum polarisation. Finally, <cit.> summarised the knowledge
about the interstellar reddening law, R_ V, varying for different sight lines.
As a further complication, the interstellar extinction is variable depending on the wavelength <cit.>.
The combination of these two effects leads to different extinction coefficients for
different photometric filters for a specific reddening law. This must be considered 
for the ultraviolet (UV) and infrared region <cit.>, which is normally not accessible from the ground.
An excellent overview of the relevant processes and methods can be found in <cit.>. 

Reddening measurements for the interstellar medium in the past relied on three techniques: (1)
star counts <cit.>, (2) spectroscopic and photometric measurements with their calibrations
<cit.>, and (3) main sequence fitting of colour-colour and
colour-magnitude diagrams <cit.>. Each of these techniques has its own advantages and disadvantages.

With the availability of precise photometric data from the UV to infrared
region, spectral energy distribution (SED) fitting techniques <cit.> 
are now widely used to build 3D reddening maps. This technique needs, in addition to a good knowledge
of the stellar astrophysical parameters, a set of stellar flux models that introduces another
source of uncertainty <cit.>.   

The astrometric results from the Gaia satellite <cit.> open a new
window for the determination of astrophysical parameters based on
precise parallaxes. The corresponding final catalogue will include ultra-precise parallaxes
(and thus distances) and auxiliary measurements such as photometry and spectroscopy, which
makes the reddening determination a critical issue.  

In this paper we present a catalogue of mean reddening values compiled from the literature,
which, to our knowledge, does not exist elsewhere.
It provides a source of standard values for future calibration purposes, especially as a starting
point for the SED fitting procedures and new Gaia data releases. 






§ SAMPLE SELECTION AND DATA SOURCES
 



Besides the inclusion of references already known to us, the literature was searched until the end of the year 2022 for reddening estimates of stellar objects based on various methods, and no time limit was set for the past. Almost all larger catalogues are already available from the CDS/VizieR, and we used some reddening related keywords of the unified content descriptor – such as phot.color.excess or phys.absorption – to identify appropriate papers. A pre-selection based on the catalogue title was made because of the many returned references. Furthermore, we do not cover works that include reddening estimates for fewer than ten objects. This certainly results in an incomplete list, but a complete census cannot be gathered anyway. The remaining references were checked to see if these include independent reddening estimates – thus, we excluded works that adopt estimates of others that are, for example, already included in our list. The compiled references and the number of studied objects are listed in Table <ref>. We do not cover results of reddening maps or Gaia data, as we intend to provide a catalogue for their validation.

The reddening values are not always listed as E(B-V) but also in the Strömgren-Crawford 
<cit.> and Two Micron All Sky Survey <cit.> photometric
systems. For the transformation to E(B-V), we used the following transformations 
<cit.>:


    E(B-V)    =    1.35E(b-y) = 0.36E(V-K_ S) =  
       =    2.00E(J-K_ S) = 0.32A_ V = 2.78A_ K_ S.


The errors for the colour excess ratios of the different photometric bands 
listed in <cit.> are typically below 0.04 mag. We also
considered a non-standard reddening law if listed in the individual references.
In addition to the references from the literature, we also calibrated reddening values in the 
Geneva seven-colour, Johnson, and Strömgren-Crawford photometric systems. In the following,
we overview the corresponding calibrations and data sources.

The Geneva photometric sample: The Geneva photometric catalogue <cit.>
belongs to one of the most homogeneous datasets, based on a unique instrumentation 
and reduction procedure in both hemispheres <cit.>. The about 43 000 measured stars in the General Catalogue of 
Photometric Data <cit.> cover all spectral types and luminosity classes. 
For O/B type stars, the reddening free X/Y parameters <cit.> enable an estimate of the reddening in E[B-V] that can be transformed to the Johnson UBV equivalent E(B-V) = 0.842 × E[B-V].
In the first step we used the parameter space -0.3 ≤ X ≤ +1.4 and -0.07 ≤ X ≤ +0.15 to extract the most likely 
O- and B-type stars population <cit.>. The sample was then refined by a cross-match with SIMBAD and a check of available spectral classifications. This was done on the basis of coordinates (a
search radius of ten arcseconds was used) and apparent magnitudes.
The final cleaned sample includes 10 233 stars with a reddening based on Geneva photometric data.

The Johnson photometric sample: We employed the Q parameter, which is usable for early-type (i.e. hotter than B9) stars.
It utilises the colours U-B and B-V and their ratio <cit.>. It has been widely applied, especially for young open clusters <cit.>. For this work, the mean Johnson UBV 
measurements from the GCPD were taken. As the next step, the Q values for different spectral types were calculated, and all 
relevant hot objects were identified.
For this, the spectroscopic data compilation by <cit.> and further spectroscopic data from the SIMBAD database
were used. If no spectroscopic information was available, objects were deleted from the list rather than introducing wrong
reddening values for highly reddened cool-type objects, for example. The final list of reddening values based on the Q-method
includes 9878 stars.

The Strömgren-Crawford photometric sample: The uvbyβ photometric system has been widely and successfully used since its introduction almost 60 years ago <cit.>. Multiple calibrations for different stellar groups have been published since then.
We used the catalogue of mean uvbyβ values by <cit.>. It includes 60 668 stars with unweighted mean 
indices and their errors. From this compilation, only stars with the complete set of uvbyβ measurements were considered.
This results in a list of 32 529 stars in total.
We used the program TEMPLOGG <cit.> to calculate the reddening values. 
This program is a menu-driven application written in C-Shell script. It provides convenient access to the different Fortran
codes that perform the inversions from observed colour indices to fundamental
parameters, and it also enables the user to process tables of photometric data for a
list of stars in batch form. It applies the calibrations from
<cit.>, <cit.>, and <cit.> and covers the whole spectral type range.

The identifications in the references (Table <ref>) are manifold. They 
range from commonly known IDs (for example, HD and HIP numbers) to coordinates in several 
epochs. Especially the Lausanne code numbering system (LID) used in the GCPD, is sometimes
troublesome. However, a common object identification must be done before the final means can be calculated.
Our primary source for this task is the 2MASS catalogue. It is 
deep enough for our purpose but also covers the brightest stars. All objects were carefully
checked within the 2MASS survey. If no unambiguous identification was possible, we skipped the
measurements. In total, only about 1700 data points were skipped. This is a small number compared
to the over 120 000 measurements in the final catalogue.

In the last step, we calculated unweighted mean values and their standard deviations. Finding criteria
for introducing meaningful weights on a justified basis was impossible. As expected for many objects and references, some mean reddening values exhibit large errors up to 2 mag. 
The reasons for that could be wrong identifications or simply very divergent measurements.

The final catalogue includes 157 631 individual available measurements for 97 826 objects and will be available in electronic form, an excerpt is listed in Table <ref>.








§ ANALYSIS
 

Our primary source for common identification is the 2MASS catalogue. For the further analysis,
we cross-matched it with the Gaia Data Release 2 and 3 (DR2 and DR3) catalogues <cit.> using the coordinates
and J as well as G magnitudes. Again, we were conservative, rather than excluding a 
Gaia source instead of including a false identification. We could identify 96 549 from
the 97 826 objects (or 98.69%) of our catalogue.

The final catalogue represents a sample of stars distributed over the whole sky, as shown in Fig. <ref>.
Several important Galactic features, such as the disk, the central and Bulge region, and the north
and south poles, are seen. However, the Magellanic Clouds are also visible.

In Fig. <ref> the histogram of the G magnitudes is shown. Our sample includes objects with magnitudes of 1.7 to 21.2,  with maxima at 8.5 and 16.5 magnitudes. The 16.5 magnitude maximum is caused
by objects from the Galactic Bulge <cit.>. It shows a statistically
significant number of stars in all magnitude bins, which can be used for calibration.



 §.§ The references for assessing the results
 

We selected six recent and widely used sources
of reddening values to compare and analyse our values with the literature: <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>.
From these, the work by <cit.> presents a 3D map, whereas the other three
sources list reddening values for individual stars.

We included a comparison with results from the Gaia DR2 <cit.> because many papers devoted to star clusters and stellar astrophysics are based on it. Most of them use reddening values for the analysis.

For all six references, we have to emphasise that no filter in the UV region (bluewards of 400 nm)
was used for the reddening estimates. It is well known that indices such as Johnson (U-B) and 
Strömgren (u-b) are most efficient for such a purpose. This is a severe drawback. We note that
the G and G_ BP filters of the Gaia photometric system measure the flux down to about 330 nm, with a minimal efficiency compared to the other, redder regions. Furthermore, the filters are much too broad (blue cutoffs at
680 and 1000 nm, respectively) to be useful for the reddening estimate.

We also notice that the StarHorse <cit.> and 
StarHorse2021 <cit.> compilations include negative reddening values. It is
written in the papers that these values must be treated with caution, but no other comments
for the reasons are given.



<cit.>: They presented a 3D interstellar dust reddening map of the Galactic plane
based on Gaia DR2, the Two Micron All Sky Survey (2MASS), and the Wide-Field Infrared Survey Explorer <cit.> photometry. The filters range 
from 400 nm to 2400 nm.
It covers the whole
Galactic longitude range with Galactic latitudes | b | < 10. They applied a machine
learning algorithm called random forest regression to derive E(G-K_ S), 
E(G_ BP-G_ RP), and E(H-K_ S). In doing so, they built an empirical
training sample of stars selected from several large-scale spectroscopic
surveys, including the APO Galactic Evolution Experiment <cit.>
Large Sky Area Multi-Object Fiber Spectroscopic Telescope <cit.>, and
Sloan Extension for Galactic Understanding and Exploration <cit.> surveys.
Together with the photometry, reddening values were derived with
the star-pair technique. They state that a comparison with results in the literature
shows good agreement and that their results have typical uncertainties of about 0.07 mag
in E(B-V). We used the three listed reddening values for the individual Gaia DR2 stars and
computed a mean reddening and its standard deviation using the following relations 
<cit.>:

    E(B-V) = 0.45E(G-K_ S) = 6.09E(H-K_ S)                
    .

There are 37 045 stars that are also on our target list. 

Gaia DR2 catalogue <cit.>: It includes both the G-band extinction,
A_ G, and the E(G_ BP-G_ RP) reddening. They were first derived using the parallaxes and photometric 
data to get each passband's effective temperature, absolute magnitudes, and reddening
values. Then, an interpolation routine within the PARSEC evolutionary models <cit.>
for solar metallicity (Z_⊙ = 0.0152), and a standard reddening law was applied. The authors state that
the extinction estimates are inaccurate on a star-by-star level but mostly unbiased
and so are applicable at the ensemble level.  
We transformed the reddening of (G_ BP-G_ RP) according to <cit.>:

    E(B-V) = 0.77E(G_ BP-G_ RP)   
    .

The standard deviation was derived by the given upper and lower limits for E(G_ BP-G_ RP). 
In total, we have 51 356 stars in common.

Gaia DR3 catalogue <cit.>: In addition to more precise astrometric parameters, 
this data release also includes low-resolution BP/RP spectra for 219 million sources. They cover the wavelength range 330 to 1050 nm with a resolution between 13 and 85. This resolution does not permit the measurement of individual spectral lines but can be used similar to spectrophotometry. The overall method
has not changed <cit.>. We note that the filter curves have been altered
compared to the Gaia DR2. With these new data sources, the individual
reddening estimates should be expected to be closer to the already published ones, as compiled in this work.



StarHorse <cit.>: They combined parallaxes and photometry from the Gaia DR2 together 
with the photometric catalogues
of Pan-STARRS 1, 2MASS, and AllWISE <cit.> in order to derive
Bayesian stellar parameters, distances, and extinctions. For this purpose, the StarHorse code <cit.> was
applied. It is a Bayesian
parameter estimation code that compares many observed
quantities to stellar evolutionary
models. Given the set of observations plus several priors, it finds the posterior probability over a
grid of stellar models, distances, and extinctions. A mean precision of 0.20 mag for A_ V was achieved.
Our analysis used the listed line-of-sight extinction 50th percentile for 67 437 stars in common.

StarHorse2021 <cit.>: For this version, they significantly updated the
StarHorse algorithm in the context of the Gaia DR3. Furthermore, the photometric data from 
SkyMapper DR2 <cit.> without the u filter were included. It was
concluded that the systematic
errors of the astrophysical parameters are smaller than the nominal uncertainties for most objects.

Bayestar2019 <cit.>: This is a 3D map of interstellar dust 
reddening that covers three-quarters of
the sky (i.e. declinations of δ > -30). It is based on the 
Pan-STARRS 1
and
2MASS colours ranging from 400 nm to 2400 nm. Including the astrometric data from the
Gaia significantly improved the accuracy compared to the prior version of the 
maps <cit.>.
They group stars into small angular 
patches in the sky. Based on the available photometric measurements of each
star, its spectral type and the parallax from Gaia (if available), they compute a probability distribution over the star's
distance and foreground dust column. Each star puts a constraint
on the line-of-sight distance versus dust column relation.
Assuming a constant reddening law, a significant number of stars
is needed along a single sight line  
to put a strong constraint on the dust density as a function
of distance. We used the distances from <cit.> as well as Galactic
coordinates
to estimate mean reddening values. 



 §.§ Comparison with the references
 

We statistically analysed the differences in the reddening
values of the individual references among each other and our catalogue.
We expect a normal distribution centred at zero with a certain width
or standard deviation. Two other important quantities are the third and fourth
standardised moments, the skewness and kurtosis <cit.>. 



The skewness is a measure of symmetry
around the zero point or the lack of symmetry. In the ideal case,
the skewness for a normal distribution is zero (mean is equal to median), and any symmetric data should have a value near zero. 
Negative values indicate data that are skewed left in the sense that the left tail of the 
distribution is long relative to the right one. Values between -0.5 and +0.5 for a large
sample, as ours can be considered symmetric. 
The kurtosis measures whether the data are tailed relative to a normal distribution. The value for an
ideal normal distribution is three. Excess kurtosis is often used, subtracting a value of three. 

Table <ref> presents all statistical values for the
different references, whereas Fig. <ref> shows the data graphically. 
The histograms are shown in Figs. <ref> to
<ref>.
It has to be kept in mind that these characteristics are for
the extinction in (B-V).

All the investigated datasets' mean and median values are compiled 
around zero. There are no apparent outliers. 
The standard deviations for the datasets of the literature vary between 0.12 and 0.23 mag. 
These values also seem to limit the global statistical accuracy for any of these references.
The comparison with our datasets yields slightly higher standard deviations (up to 0.3 mag).
We think that the reason is the use of UV data within our compilation. As stated before, it is
essential to include this wavelength region, especially for early-type stars <cit.>.
Looking at the skewness and kurtosis values, we find that only two datasets <cit.>
qualify as being close to normal distributions. All others deviate significantly,
especially when it comes to kurtosis. We
also calculated all values for those 8500 objects common in all datasets to exclude selection effects. The results
are the same, with slightly lower standard deviations. 

Figure <ref> presents more detailed structures of the individual differences with our catalogue.
The asymmetry of the different distributions is visible, especially for the upper main sequence stars (which are, in general, at
more considerable distances from the Sun). For the cool-type stars, there is, in four cases, a similar effect in the other
direction evident. 

In general, we advise the users of the published reddening values to search all
available sources and to compare them. The object's location in the Milky Way needs to be checked for significant deviating values. 
De-reddening procedures using reddening-free indices, for instance those described in Sect.
<ref>, are always preferable.



 §.§ Reddening values of open clusters
 

As the next step, we investigated the members of star clusters in our sample. The releases of the Gaia data brought a flood of newly
discovered star clusters based on astrometric and photometric data. However, wrongly identified or already known aggregates make a
homogeneous analysis difficult. 
We used the star cluster membership lists from the catalogue of <cit.> to identify possible open cluster members among the studied stars.
This catalogue contains the parameters (age, reddening, and distance) of 7167 star clusters, with more than 700 newly discovered high-confidence star clusters. In addition to this,
they also include a list of cluster members with membership probabilities for each star. For example, they performed several cross-checks with already published
catalogues and found offsets for the extinction (see Fig. 7 therein). We must emphasise that determining the cluster parameters is 
still challenging, although, from the Gaia datasets, we already get a reasonable estimate of the distances <cit.>.
We cross-matched our sample with the lists by <cit.> and identified
4270 high-confidence (P>0.7) star cluster members. For the comparison, we only took star clusters for which more than two members are included in
our sample, leaving us with 786 aggregates. In Fig. <ref>, we present the comparison result with the published values by 
<cit.>. Plotting star clusters with more than five members does not change the results.
The skewness and
kurtosis of the corresponding distribution are -1.99 and +8.50.
We see a wide spread of differences with a clear trend for E(B-V) < 0.2 mag. This means that their Bayesian neural network technique generally 
overestimates the reddening. 
This might be a result of the applied binning by <cit.> of 
0.11 mag in (G_ BP-G_ RP).
For larger reddening values, the effect reverts. 
This conclusion is also valid using mean reddening values from the six sources
we used for our comparison.
Determining the cluster parameters is still very much limited by the choice of reddening. At the same time,
the distance can be accurately estimated using the Gaia data (which will improve further with forthcoming releases). We also now have very good
metallicities from the Gaia-ESO <cit.> and Galactic Archaeology with Hermes <cit.> surveys, for example, which could serve
as a starting point for fitting isochrones to get the age. But still, the fitting techniques need to consider all these starting values
to derive homogeneous cluster parameters <cit.>.





 §.§ Variable extinction of stellar objects
 

It has been well documented that several star groups show variable extinction. Most of
these objects, like the classical T Tauri and UX Ori stars show this effect because of a
varying circumstellar environment <cit.>.

On a larger scale, such effects have also been observed for Hii regions 
and the Galactic centre <cit.>. In general, it
can be said that reddening varies in regions with rapid variations in dust and gas content.

Intrinsically variable stars are at the centre of many scientific studies <cit.>. The periods, amplitudes, and light curve characteristics <cit.> are as manifold as the underlying physical mechanisms <cit.>. For example, Cepheid variables and their period-luminosity-relations <cit.> have allowed us to start constructing a distance ladder, which helped us explore large regions of the Universe. For such studies, the knowledge of a precise extinction
value is essential. Therefore, our catalogue can help identify possible outliers and offsets
due to incorrect reddening values.



§ CONCLUSIONS
 

Compiling mean reddening values of stellar objects is essential for calibrating and testing observations
and models. The interstellar and inter-cluster absorption must be considered for various aspects of astrophysical
studies. Its wavelength dependence and different reddening laws due to multiple mixtures of interstellar
gas and dust make it necessary to work with a statistically sound sample of objects. We present a catalogue
of mean reddening values for 97 826 objects distributed across the visible parts of the Milky Way and the Magellanic
Clouds.

A comparison with published reddening values derived via sophisticated automatic fitting procedures reveals
systematic offsets for both Galactic field objects and star clusters. Therefore, estimating the reddening using well-established techniques and datasets, especially observations
in the UV region, is still essential. Future satellite missions such as the Quick Ultra-VIolet Kilonova surveyor <cit.>
will help improve this situation significantly. Also, a (re-)analysis of photometric
UV observations available in scientific archives is very much needed <cit.>.


We are grateful to the referee for the constructive input.
This work was supported by the grant GAČR 23-07605S and the European Regional Development Fund, project No. ITMS2014+: 313011W085 (MP).
This research has made use of the SIMBAD database,
operated at CDS, Strasbourg, France and of the Two Micron All Sky Survey (2MASS), 
which is a joint project of the University of Massachusetts and the Infrared 
Processing and Analysis Center/California Institute of Technology, funded by 
the National Aeronautics and Space Administration and the National Science Foundation.
This work presents results from the European Space Agency (ESA) space mission 
Gaia. Gaia data are being processed by the Gaia Data Processing and Analysis 
Consortium (DPAC). Funding for the DPAC is provided by national institutions, 
in particular the institutions participating in the Gaia MultiLateral Agreement 
(MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia. 
The Gaia archive website is https://archives.esac.esa.int/gaia. 


aa





§ HISTOGRAMS OF THE REDDENING DIFFERENCES















 Paper Title: The effect of dynamical interactions in stellar birth environments on the orbits of young close-in planetary systems.



firstpage–lastpage
Temperature dependence of the electron and hole Landé g-factors in CsPbI_3 nanocrystals in a glass matrix
    Sergey R. Meliakov^1, Evgeny A. Zhukov^2,1, Vasilii V. Belykh^2, Mikhail O. Nestoklon^2,  Elena V. Kolobkova^3,4, Maria S. Kuznetsova^5, Manfred Bayer^2, Dmitri R. Yakovlev^2,1
    August 5, 2024
====================================================================================================================================================================================




§ ABSTRACT 
Stars do not form in isolation but together with other stars, and often in a clustered environment. Depending on the initial conditions in these environments, such as initial density and substructure, the distances of encounters between stars will differ. These encounters can also affect just-formed exoplanetary systems. Using N-body simulations, we show the effect of a single fly-by on a common type of exoplanetary system: close-in Super-Earths/sub-Neptunes with or without a distant Giant planet. Even a single encounter can significantly modify the architecture of these exoplanetary systems over their long lifetimes. We test fly-bys with different characteristics, such as distance and mass, and show how they perturb the inner planets long after the encounter, leading to collisions and mutual inclination excitation, which can significantly modify the observed architecture of these systems in transit. We find that our initially four-planet inner systems reduce to three or two inner planets depending on their initial separation and that the mutual inclinations of these remaining planets can be high enough to reduce the number of observable, transiting planets. In our 500 Myr simulations, we show that this reduction in the number of transiting planets due to stellar fly-bys can contribute to the observed excess of single-transit systems.







methods: numerical – planets and satellites: dynamical evolution and stability – planet–star interactions – planetary systems








§ INTRODUCTION
 

The birth environment of planets is intrinsically linked with that of their host stars, as planets form from the protoplanetary discs around these young protostars <cit.>. Many of the exoplanet host stars have likely formed with other stars in grouped or clustered environments. While these regions are still young, stars and any planets around these stars will be subject to a higher-density environment than what they will eventually experience after dispersal into the field <cit.>. As such, this birth environment has the potential to influence not only the stars and their protoplanetary discs but also any planets that have already formed around them. 

The effects of dynamical interactions caused by being in an evolving young star-forming region have been investigated for different types of single <cit.> and multi-planet architectures <cit.> as well as how planets still within their protoplanetary discs are affected <cit.>. Fly-bys have even been thought to have had an influence on the shape of our own outer Solar system <cit.>. A lot of this work has focused on more distant, colder Giant planets, however, observations indicate that they are rarer <cit.>. Close-in planets that can be observed via transits are more common <cit.>, which is why we focus on the influence of an encounter on these smaller planets here.

Our knowledge about planet systems around other stars has dramatically evolved over the past two decades, and the most productive planet-finding mission to date was Kepler, which launched in 2009 with its primary mission lasting ∼4 years. It used the transit method to discover thousands of planets, both in single-transit and multi-transit systems <cit.>. A majority of these discovered planets are smaller (1-4 R_⊕), relatively low-mass planets (< 20 M_⊕) in close-in orbits (period < 100 days). 


For a multi-planet system to be discovered by the transit method requires its planets to be orbitally aligned with respect to the observing direction, necessitating low mutual inclinations. However, even in a fully coplanar configuration, not all close-in planets are guaranteed to transit their host star together <cit.>. The Kepler mission discovered about three times as many single-transiting systems compared to the number of systems with more than one transiting planet. <cit.> showed that this difference in the numbers of transiting planets towards more single-transit systems was likely not the result of a single underlying planet population of low mutual inclination systems (∼1–2) with the same initial number of planets, as this would result in too few single-transit systems. Shortly thereafter, <cit.> referred to this observational excess as a dichotomy between single-transit and multi-transit systems and it has since been known as the “Kepler Dichotomy”. 

<cit.> investigated the Kepler detection pipeline of multiples to determine if the dichotomy was actually a real feature instead of a detection bias. They found that incompleteness caused the dichotomy to probably be exaggerated but failed to account for it fully. Regardless of this aspect, the dichotomy is now generally assumed to be due to differences in the inclination of planets in the same system. However, it is unclear if this is due to the existence of a population of planets with higher mutual inclinations in addition to a lower-inclination population or just a spread of inclinations with a high inclination tail from a single population <cit.>. In particular, <cit.> showed that the dichotomy was likely not due to a population of single-planet systems. <cit.> showed that mutual inclinations are distributed continuously and suggested that this result pointed to the oblateness of the host star or late assembly of the planets as the cause for the dichotomy. 

While the inclination-population model explains why we might observe the dichotomy, it does not provide an underlying cause for the different inclinations. The explanations brought forward for the origin of these differences can be split into two different avenues. The first one comprises theories based on the idea that differences in/during planet formation lead to the observation of a dichotomy. 

<cit.> suggested that giant planet-planet scattering or migration during the formation process prevented the subsequent formation of smaller, close-in planets, leaving highly inclined large planets behind. They noted that single-transiting systems did not seem to have a preferred planet size, whereas multi-transiting systems appeared to preferentially host smaller planets. Alternatively, <cit.> proposed that differences in the protoplanetary disc surface density and mass could result in differences in, e.g. multiplicity and mutual inclination. Another explanation is centred around a misalignment of the spin-orbit of the star with its planets as a consequence of excitation during the disc phase. This then caused subsequent instabilities in the planetary orbits and higher mutual inclination differences <cit.>. Simulations by <cit.> suggested that instabilities in the resonant chains created during the formation of the planets were another possible cause. 


<cit.> found that the dichotomy could be caused not by higher mutual inclinations of close-in planets but by fewer planets within the Kepler detection window. This was thought to be due to differences in migration and clustering of planets trapped outside the transit-observable inner au-region. In addition to this, the mass of the discs was found to also play a role in how many of these planet clusters would form.

The second explanation route for the dichotomy is based on the idea that inclination differences are caused by the long-term evolution of the planet systems in question. This evolution can either be investigated in isolation or with other stars impacting the planetary systems. There appears to be a correlation between the occurrence of inner Super-Earth planets and a more distant outer giant <cit.>. While the distant giant planet would not be detected in transit surveys, it could nevertheless influence the dynamical evolution of the inner planet system. This is the idea behind several studies investigating the dynamical evolution of a planet system with an outer giant. Perturbations of the inner planets' orbits can be caused by the presence of an outer giant planet (system), e.g. when it is on an inclined/misaligned orbit or  becomes dynamically unstable <cit.>. 



Interactions with other stars in the birth environment were suggested as a cause of disruption in a planet system, potentially explaining the dichotomy. <cit.> concluded from their simulations of the evolution of planet systems in a young star cluster that single-transit systems could originate from denser regions within the birth environment. <cit.> investigated the effect of a fly-by on different star-planet systems, each with two planets in different configurations. While these authors did not use their simulations to investigate the dichotomy directly, they found that fly-bys of their close-in SE systems around an M-dwarf could not disrupt them sufficiently to depart from their initial coplanar set-up. This result could be in part due to their relatively short integration time of only 100 Myr.

<cit.> also investigated if stellar fly-bys could cause excitation of initially coplanar planetary systems to explain the dichotomy. They found that the presence of one or more exterior larger companion planets decided the amount of disturbance in the inner planet system after a fly-by. Their idea was centred around the giant planets being disturbed by the fly-by, which then caused the inner planet system to destabilise. However, they only evolved their planetary systems for a short period of time (<1 Myr) after the fly-by occurred, not considering that the perturbations introduced by the fly-by could take several Myr to hundreds of Myr to have any measurable disturbing effect <cit.>.

In this paper, we take a similar approach of simulating the effect of a single stellar fly-by on a planetary system that could be observed using the transit method, but we follow the evolution of the planet systems for 500 Myr after the encounter. The fly-by characteristics are based on typical stellar encounters within a dynamically evolving young star-forming region that we extracted from N-body simulations of a typical region (based on the Orion Nebula Cluster - ONC). Our planetary systems either feature a distant giant planet around a close-in system of smaller planets or only of the close-in system. In section <ref>, we describe the close encounters that can occur in a typical young star-forming region during its early dynamical evolution and the type of fly-bys we investigate in our simulations. We then provide details about the planetary systems that are subjected to a fly-by and how we determine the number of transiting planets. In section <ref>, we provide the results of our planet system simulations and discuss them in section <ref>. Finally, section <ref> provides concluding remarks.




§ METHOD




 §.§ Determining typical interactions between stars


We simulate the effect of typical interactions of young star-planet systems with other stars in young star-forming regions. The input for these interactions (i.e. perturber mass, fly-by distance, and perturber velocity) are based on close encounter information from 20 simulations of the dynamical evolution of a typical (Orion Nebula Cluster - like) star-forming region used in <cit.>. These N-body simulations simulated the early dynamical evolution of stars (without planets) in young star-forming regions. They are set up using the box fractal method <cit.>, which can create different levels of initial kinematic and spatial substructure to mimic what has been found in observations. A detailed overview of how fractals can be used to create substructure in simulations and how to construct them can be found in <cit.> and also <cit.>. For the stellar interactions, we choose simulations that start with a fractal dimension of D = 2.0, which represents a high amount of initial spatial substructure and an initially subvirial ratio (α_vir = 0.3), which sets up the initial kinematic substructure. The virial ratio is the ratio of the total kinetic energy to the total potential energy of all stars in the simulation.  These substructures and subvirial initial conditions are representative of those that have been suggested for typical star-forming regions, like the Orion Nebula Cluster <cit.>.



These simulations include primordial binaries as well as stellar evolution. The number of systems is 2000, with the primary masses sampled randomly from a <cit.> IMF. This IMF is a combination of a Chabrier () lognormal IMF approximation for lower-mass stars with the power-law slope of Salpeter () for stars more massive than 1 M_. The presence of primordial binaries increases the number of stars to ∼2800 per simulation. The primaries and single stars have masses within a range from 0.1 M_ and 50 M_, whereas the secondary binary components can have masses down to 0.01 M_ (i.e. brown dwarfs). Further details, e.g. on setting up the stellar and binary populations in the simulations, can be found in <cit.>.


Fig. <ref> shows the distribution of the (non-primordial binary) close interaction distances plotted against simulation time (in Myr) from one of these 20 previously described simulations. Star-forming regions, such as the ONC-like ones simulated in <cit.> and used here, undergo “cool” collapse due to their initially substructured and subvirial nature <cit.>. This evolution can occur on very short timescales, which is evident in numerous close interactions within the first 1 Myr. 



The encounters considered here do not include those that occur between two stars that are in a primordial binary but do include those between the binary stars and other stars. More than half of the encounters occurring after 1 Myr (after core-collapse) have a close encounter distance <300 au. This encounter distribution is evident in Fig. <ref>, where we show a cumulative distribution of encounter distances between 1-4 Myr of simulation time from 20 simulations combined for two different initial condition sets. Differences between the physically more realistic ONC-like and the often used Plummer sphere <cit.> initial conditions are in the number of stars: ∼2800 stars including primordial binaries versus 1000 stars without primordial binaries. The Plummer sphere simulations were set up without stellar evolution, however, this is expected to have little effect on the encounter profiles over the 4 Myr evolution time considered here and they are initially virialised. The ONC-like simulations (initially subvirial and spatially substructured) have similar encounter profiles compared to those fitting other star-forming regions <cit.>. The Plummer sphere distribution is distinctly different, with encounters typically occurring at larger separations i.e. 50 per cent are larger than ∼500 au.
For our single fly-by analysis, we select three different distances (50 au, 150 au and 250 au) covering the distance below which half the encounters occur (300 au) in the star-forming region simulations. In our fly-by simulations, these close encounter distances are set in relation to the centre of mass of the star-planet system. 

In the 20 simulations (ONC-like initial conditions) after core-collapse during 1-4 Myr of the simulation time,  up to 23.4 per cent of the initially single, approximately solar-mass (0.9-1.1 M_) stars have at least one encounter with another star/system at a distance <1000 au and up to 12.5 per cent have at least one encounter with another star at <300 au. In contrast, up to 2.2 per cent of single stars with masses less than 0.9 M_ and up to 19.8 per cent of those with masses above 1.1 M_ have at least one encounter below <300 au. For the following star-planet simulations, we chose three different perturber masses (0.1, 0.5 and 1.0 M_), whereas the star hosting the planet systems is always a solar-mass star (1.0 M_).



Finally, we select two different perturber velocities based on the median velocities of the non-binary stars in the stellar simulations. The median stellar velocities across the 20 simulations are very similar at ∼2 km s^-1. We assume two different fly-by cases, first the extreme case of two stars passing each other in exactly opposite directions, each with this median velocity. In a reference frame centred on the host star, the velocity of the perturber would be double, i.e. ∼4 km s^-1. The second perturber velocity we test in the fly-by simulation uses half of this velocity, simulating a slower fly-by (∼2 km s^-1). The perturbers start their journey at distances between ∼8000–9000 au at different positions in relation to the simulated planetary systems. The perturbers take around ∼9,000–10,000 and ∼18,000–20,000 yr (depending on their velocity and close encounter distance) to reach the host star and interact with the planetary system located there.



 §.§ Setting-up and evolving the planetary systems


As we are interested in investigating whether close encounters in young star-forming regions can play a part in explaining the Kepler dichotomy, we set up two different exoplanet architectures that are commonly associated with those found via the transit method. These architectures all feature four close-in planets, each with a mass of 5 M_⊕ around the 1.0 M_ host star. This choice of using close-in planets with initially the same mass is motivated by the results in <cit.>, who showed intra-system uniformity in planet masses in multi-planet systems (similar to the intra-system uniformity seen in exoplanet radii ).

Planets with these masses are classified as Super-Earths (SE) or Sub-Neptunes (SN), depending on their composition. As we use fly-by information from the early dynamical evolution of a star-forming region, we assume that these planets are still SNs with a gaseous atmosphere, resulting in higher radii than if they were SEs. We chose a radius of three R_⊕ as planets at this young age are more likely to have a large H/He atmosphere/envelope <cit.>. 

The radii of the stars (host and perturber) are chosen based on their mass using a main-sequence mass-radius relationship. We disregard that at this early stage of the dynamical evolution of a star-forming region, they might still be pre-main sequence stars with radii larger than what they would have once they reached the main sequence <cit.>. The only effect that a larger solar radius would have in our simulations is to cause collisions of planets with the host star (which do not occur in our simulations). All solar-mass stars in our simulation have a radius of 1 R_ and the two sub-solar mass perturbers (0.5 M_ and 0.1 M_) have radii following the mass-radius-relationship R ∝ M^0.8. 



We do not evolve the planets' radii unless they are involved in a collision. If two SNs collide with one another, the gaseous envelope is likely removed, leaving a rocky SE behind, as suggested by the result of giant impacts in <cit.>. To calculate the new radius, we use the mass-radius relationship for rock/iron planets based on Eq. 8 in <cit.> with a rock to iron fraction rmf = 2/3, i.e. Earth-like composition. Our collisions result in a perfect merging of the two involved planets, with their masses combining to form a larger planet. 



The inner-most planet is always placed at 0.1 au from the host star, which is a location commonly associated with Kepler close-in planets <cit.>. The other three planets are placed at a separation based on their mutual Hill radius R_MH. The mutual Hill sphere can be calculated using

    R_MH = a_1 + a_22( m_1 + m_23 M)^1/3,

where m_i and a_i are the masses and semi-major axes of two adjacent planets and the central host stellar mass is M. The above calculates a radius, which can be put in relation to the separation between these two planets to calculate a dimensionless number K:

    K = a_2 - a_1R_MH

<cit.> suggested that planetary systems with similar masses set up in circular and coplanar orbits and a minimum of 10 R_MH are stable for at least 10^9 yr. <cit.> also found that 93 per cent of their observed planet sample had separations with at least this K-value. 

We set up our initially circular and coplanar planetary systems with separations of the inner planets of 10 and 14 R_MH. The former is just at the edge of stability, whereas the latter is suggested to be stable even if eccentricities are non-zero <cit.>. We run simulations of these systems in isolation (without perturber), and all of them remain virtually coplanar (mutual inclination < 10^-7) and circular (eccentricity < 10^-3) up to at least 500 Myr, which is the age which we run all of our fly-by simulations to in this analysis. This simulation time is much shorter than the average ages of many known Kepler planet systems, but it is not feasible to run our systems for this long, e.g. 5 Gyr or more.










As mentioned in section <ref>, there is an apparent connection between the occurrence of close-in planets with a distant Giant <cit.>. In our second set-up, we add simulations where the already described inner planetary systems feature a more distant outer Giant with a mass of 5 M_J at three different distances to the host star (5, 10 and 20 au). We set the radius of this Giant planet to 1 R_J <cit.>.

For all of the planetary architecture combinations, we simulate two different orbital starting positions for the inner planets. They are either lined up at the start or separated by a quarter of their orbital period. The inclinations of the planetary systems as a whole are chosen at random; mutual inclinations are zero. For each of our 18 perturbers (differing in mass [3 options], velocity [2 options] and close encounter distance [3 options]), we use two different starting positions, resulting in a total of 36 different fly-bys. In addition to the random choice of planet system inclinations, this last choice results in different interaction angles between the perturbers and planets for otherwise identical simulations.

We use the REBOUND N-body code <cit.> to integrate the systems. We run a total of 576 fly-by simulations using different planetary orbit and fly-by configurations. The input parameters for these simulations are summarised in Table <ref>.





We run these simulations up to a simulation time of 500 Myr. The fly-by occurs early (first 20 kyr) in the simulation, but we continue the integration with the perturber up to a simulation time of 1 Myr in the first instance. The simulation snapshots are stored in the REBOUND SimulationArchive with timesteps of 10^3 yr during this period using the IAS15 15th order Gauss-Radau integrator <cit.>. To speed up the simulation, the perturber is removed at this time (1 Myr). We also change the integrator to the hybrid MERCURIUS integrator <cit.> and increase the time between stored snapshots to 10^4 yr. 

The MERCURIUS integrator is a combination of two different integrators. When the particles (e.g. planets) are far apart from each other, it uses the symplectic Wisdom-Holman integrator WHFast with a fixed integration time step calculated from the period of our inner-most planet multiplied by 10^-2 <cit.>. For most of our simulations, this results in an internal fixed integration timestep dt≈0.12 days based on the period of ∼11.6 days of the inner planet located at 0.1 au. 

The hybrid MERCURIUS integrator automatically switches over to the high-order IAS15 integrator during close encounters. We set a generous switch-over distance at a value of 5 R_MH, which is half the smaller initial separation distance of the IP systems. In this way, IAS15 is not triggered while the planets remain at their original separations. To prevent IAS15 from stalling during very close encounters, we set a minimum timestep of 10^-4 dt <cit.>.





§ RESULTS


There are two main reasons why we would not be able to observe all planets transiting that formed around a star at a later point in time. One, the number of planets at the time of observation is lower than the number of planets when the system formed. Two, the mutual inclinations between the planets are too large for all of these to be within the same observation window. We first analyse the evolution of the average number of planets caused by collisions, after the planetary orbits start to overlap due to changes in eccentricity. This analysis is followed by the evolution of the maximum mutual inclinations. Then, we show how the combination of collisions and mutual inclination differences will affect the number of transiting planets that could be observed from randomly sampled viewing angles, which will be described in the following sections.



 §.§ Number of remaining planets


The number of potentially observable inner planets can be reduced through collisions after orbits start to overlap. We find that none of these close-in planets get ejected in our simulations, which is in agreement with the theory that in the inner region of these systems, collisions happen before ejections <cit.>. All collisions are between these planets themselves. We do not see any collisions with either the distant outer Giant, the host or the perturber star. In the left-hand graph in Fig. <ref>, we show the evolution of the average number of inner planets over the simulation time separated for each of the four different planetary architectures with 10 R_MH. The error in the number of planets in the different simulations is calculated using the standard error of the mean. The first collision occurs after ∼53 Myr in a simulation with a Giant placed at 5 au. The average number of inner planets drops off continuously for all four-planet system architectures and reaches values of around ∼2.2–2.3 planets per system at the end of our simulations (500 Myr) compared to 4 initial inner planets. The simulations without a Giant and one at 5 au show collisions first (55.6 and 53 Myr), followed by the other two architectures (Giant planet at 10 and 20 au) over the next 15 Myr. At 500 Myr, we see that any of the simulations with a Giant have a similar average (∼2.2 planets), whereas the simulations without one have a slightly higher average of remaining planets (∼2.3 planets). However, this difference is not statistically significant, as can be seen from the overlapping standard error regions. On the right-hand graph of Fig. <ref>, we show a histogram of the number of remaining close-in planets at 500 Myr separated by architecture and show the fraction of simulations on the y-axis. We see that ∼80 per cent of all initially four inner planet systems reduce to systems with only two inner planets at 500 Myr for the 10 R_MH initial separation systems. Of these 2-planet systems, ∼38 per cent feature one 15 M_⊕ (a result of two collisions) and one 5 M_⊕ planet, whereas the other ∼62 per cent feature two 10 M_⊕ planets (two planets with one collision each). We also find a single 20 M_⊕ Super-Earth after three subsequent collisions in a simulation with a Giant planet at 5 au. 

Fig. <ref> shows that for the fly-bys with the closest encounter distance of 50 au the planet-planet collisions start the earliest (at ∼53 Myr) in the 10 R_MH simulations, followed by collisions in the 150 au and the 250 au simulations (at ∼61 Myr and 78 Myr). None of the systems with a 150 au fly-by retain their initial four inner planets at 500 Myr, whereas some of the 50 au and 250 au keep their initial four inner planet configuration intact until this simulation time. The 50 au and 150 au simulations show a similar drop-off and final value of the average number of inner planets with slightly different starting times. We plot the standard error of the mean which shows that for most of the simulation time, the average number of remaining planets is significantly different for the three different fly-by distances.

Compared to the 10 R_MH separation simulations, the 14 R_MH simulations show the first planet-planet collisions later (at ∼150 Myr) leading to a higher average number of planets at the end of our simulations. In Fig. <ref> on the right, we see that about half of the systems retain their four planets until 500 Myr. In the left plot of that figure, we see that the simulations with a Giant at 20 au show collisions first, followed a few ten Myr later by the simulations without a Giant planet and the other initial Giant placements. However, the simulations with a Giant planet at 5 and 10 au show a lower average number of remaining inner planets after 500 Myr, having gone through a slightly steeper decline in planet numbers, i.e. more collisions. The shaded regions in this graph represent the standard error of the mean and indicate that there is no significant difference in the average number of remaining planets between the four planet architectures.

When analysing the effect of the encounter distance in Fig. <ref> for the 14 R_MH simulations, we find a much clearer and significant separation in the start of collisions by fly-by distance compared to the 10 R_MH simulations. Collisions for the 50 au fly-bys start ∼90 Myr earlier than those in the 150 au fly-bys, which are ahead of the 250 au fly-bys by another ∼40 Myr. The average number of planets at 500 Myr shows clear, significant differences between the fly-by distances with most of the 250 au simulations only going through one or no planet-planet collision. However, the averages in the 50 and 150 au simulations towards the end of the simulation cannot be clearly distinguished from each other due to their overlap in the standard error of the mean. When we look at the histogram on the right of that figure, we find that in the 250 au fly-by simulations only ∼10 per cent go through two collisions, making two-planet systems the exception amongst mostly three- and four-planet systems.































  §.§.§ Orbital excitation of the inner planets



Comparing the left and right graphs in Fig. <ref>, which show the semi-major axes of all remaining inner planets after 500 Myr, we find that the majority of planets in the 14 R_MH simulations can still be found at their original semi-major axis (right). Whereas virtually all of the planets in the 10 R_MH simulations (left) have moved to positions either closer or further away from the host star. Based on this orbital evolution of the inner planet system, larger excitation in the orbits of the planets in the initially 10 R_MH simulations are the consequence. Due to their initially closer spacing, the planets are more easily able to collide with each other. In these plots, we only show the surviving planets and scale them to 1.0, so the number of planets in the right graph of Fig. <ref> is much higher than the one on the right.





In Fig. <ref>, we show the eccentricities of the remaining planets after 500 Myr of simulation as a cumulative distribution. We plot both initial separation set-ups together and find that the planet systems with an initial separation of 10 R_MH show higher eccentricities for all remaining inner planets than for those simulations set up with an initial separation of 14 R_MH. This further illustrates the higher level of excitation that is the result of their initial set-up.



 §.§ Mutual inclination and simulated transits


The probability of a planet transiting its host star on a circular orbit can be calculated using R_∗/a <cit.>, with R_∗ being the stellar radius and a the semi-major axis of the planet's orbit. Based on this relation, we use the approach as shown in <cit.> to calculate the largest angle θ at which a planet on a circular orbit just transits its host star, i.e. its impact parameter is still below 1:

    θ = arcsin( R_∗a).

The above shows that the smaller the semi-major axis of the planet in question, the higher its transit probability. To account for non-circular cases that occur when the systems have been disrupted, we modify Eq. <ref> to R_∗/r, where the semi-major axis a is replaced by the actual distance r between the star and the planet along the line of sight in eccentric orbits <cit.>:

    r = a(1-e^2)1 + e cos f,

Here, e is the eccentricity of the planet's orbit and f is the true anomaly. A planet will transit its host star if its inclination i is located within ±θ of the observing direction. By definition, i is expressed in reference to the plane of the sky. For an observer on Earth, a transit will be visible when viewing the star-planet system roughly “edge-on”, i.e. (90 - θ) ≤ i ≤ (90 + θ) <cit.>. After evaluating the mutual inclination evoluiont, we then use simulated transit observations to determine the number of transiting planets. Instead of using a 90 viewing angle, we use a random sample of 1000 different viewing inclinations located within ±90 of the highest/lowest planet inclination from our simulations. Even with this favouring of viewing angles located towards the actual transit plane of the planets, for most of the sampled viewing inclinations, no planets will be within the transit zone indicated by ±θ. 

We then evaluate if our simulated transits could be recovered as shown in several injection/recovery experiments using the Kepler pipeline <cit.>. We start by calculating an approximate Signal-to-Noise Ratio (SNR) m_i following Eq. 2 in <cit.>:

    m_i = ( R_PR_∗)^2 √(T_obs,iP) ( 1CDPP_dur,i),

where R_P and P are an injected planet's radius and period, and R_∗ is as previously defined. T_obs,i is the total observation time of the star and we chose four years for this value, which is the total lifetime of the primary Kepler mission. The final variable in the above equation is the CDPP_dur,i, which is the Combined Differential Photometric Precision <cit.> related to the transit duration for each of the planets. The approximate transit durations for our four inner planets are between ∼4.5–6 hours. We download the “Robust RMS CDPP for a searched transit” for these durations (4.5, 5.0 and 6.0 hours) from the latest data release in the Kepler database [DR25, https://exoplanetarchive.ipac.caltech.edu/] restricting the stellar radii for the host stars to 0.8–1.2 R_. We then calculate median values, which are between 130–115 ppm (decreasing precision with increasing transit duration) and use four, evenly spread values in this range for the approximation of the SNRs. 

Fig. 5 in <cit.> provides an approximation for the fraction of recovered signals depending on the SNR-value. Based on their analysis, ∼100 per cent of the injected signals are expected to be recovered if m_i≳ 15, whereas the probability drops to zero for m_i ≲ 5. We calculate the SNR values for all planets at times between 0 and 500 Myr and compare them to these two critical values. For all inner planets in the 576 simulations at 0 Myr, m_i is larger than 38, which means all of these transit signals would likely be recovered. The signal recovery changes throughout the simulation time, mainly caused by the reduction in individual planetary radii after collisions, as the SNR depends partly on the planet's radius. When calculating if a planet transits, we multiply an injected transit by the recovery fraction calculated using the Γ cumulative distribution function given in Eq. 3 in <cit.>. As shown in there, smaller SNR values result in a not full recovery of the injected transits, which would lead to non-integer values in the recovered transits. For any injected transit that cannot be fully recovered, we therefore randomly sample a number 10 times between 0 and 1 and count a planet transit for this planet system as recovered if this number is smaller than the calculated recovery fraction. If it is larger this transit is not recovered, reducing the total number of transiting planets for this system. For most of our injected transits, the recovery fraction is above 0.95, which leads to most of them being fully counted.

In addition to calculating if a planet will transit when viewed from different observation angles, we also calculate the mutual inclinations between any of our inner planets throughout the simulation time and analyse them based on different characteristics. In particular, we are interested in the maximum mutual inclination in a simulation and use this as a proxy to track the level of excitation in our simulations and how this could then affect their observability. We choose a boundary value of ∼4. This value is derived by taking the sum of the θ-values of the inner-most (2.65) and outer-most close-in planet (1.39) on a circular orbit. The idea behind this approach is that two planets with this minimum difference in mutual inclination would not transit together.



  §.§.§ Evolution of maximum mutual inclination


All our planets start in coplanar orbits with mutual inclinations of 0. We track the change in mutual inclinations between the four inner planets over the 500 Myr simulation time. In Fig. <ref> and <ref>, we show the fraction of simulations that have a maximum mutual inclination between any two of the inner planets above 4 for both initial separation cases.

In the left graph of Fig. <ref>, we see the maximum mutual inclination values for simulations with initial inner-planet separation of 10 R_MH. We show the evolution of the four-inner-planets-only system separately and combine all four-inner-planets with a Giant at any of the three distances cases. For the first ∼50 Myr, none of the simulations have mutual inclinations above the boundary value defined above. Simulations with an outer Giant present start to show higher mutual inclinations first, followed shortly by the first four inner-planet-only simulation a few Myr later at ∼58 Myr. In these excited systems, we see collisions following shortly thereafter. The four inner-planet-only plots show a larger spread of fluctuations between individual snapshots due to the lower number of simulations (72) used in the calculation compared to the three Giant planet set-ups (total of 216). While starting to show mutual inclinations departing the coplanar set-up at different times, both reach a value of ∼25 per cent after 500 Myr. In the graph on the right in Fig. <ref>, we show the maximum mutual inclination value depending on the encounter distance over the simulation time. Mutual inclinations above the boundary are reached for more simulations with the lower encounter distance of 50 au, which is apparently able to excite the planetary systems more. The 150 au and 250 au fly-by simulations do not show large differences in the time evolution of the mutual inclinations across our sample.






The fraction of simulations showing high maximum mutual inclination in the 14 R_MH set-up is shown in Fig. <ref>. The fly-bys happen at the same time as in the 10 R_MH simulations, but it takes much longer for perturbations to show a measurable effect. The larger initial separations allow these systems to remain coplanar for longer. While the curves start increasing at much later times (∼100 Myr), they reach higher values much quicker than the 10 R_MH simulations, indicating that more simulations are excited long after the fly-by. The inner planet-only simulations show slightly higher excitation compared to the systems with a Giant planet present and no flattening off as the simulations in Fig. <ref> did. Regarding the fly-by distance, the closer the fly-by, the sooner we see simulations with higher mutual inclinations.  This is in line with the staggered start of collisions that we have shown earlier in Fig. <ref>. The final fraction of excited simulations by fly-by difference is very similar at 500 Myr.




  §.§.§ Disruptive effect of other perturber characteristics


In addition to the effect of the distance of the fly-by and the planet system architecture, we also evaluate the effect of other perturber characteristics on the maximum mutual inclination between any two of the four inner planets. We find that this proxy can be used to estimate the level of excitation as collisions between planets occur shortly after a simulated planet system passes the 4 boundary. For the 10 R_MH simulations, the first collisions occur at ∼53 Myr, which is just one Myr after the first max. mutual inclination simulations passed our 4 mark. For the 14 R_MH simulations, the first collision occurred at ∼151 Myr, which is also just a few Myr after passing the same mutual inclination mark. This suggests that our planetary systems remain in an excited, but pre-collision state only for a brief amount of the covered evolution period.

We test three different perturber masses. The simulations with higher-mass perturbers show higher and earlier excitation indicated by an increase in their simulations passing our mutual inclination boundary. For the 10 R_MH simulations, all three different perturbers disrupt 15–30 per cent of the simulations to show higher mutual inclinations at 500 Myr. In the 14 R_MH, only the 1.0 and 0.5 M_ perturbers can disrupt the planet systems during our 500 Myr simulation time, whereas the 0.1 M_ perturber causes very little excitation during the simulation time. We also use two different fly-by velocities, doubling the amount of time any of the perturbers spends in the proximity of the planetary system for the slower velocity fly-by. For both R_MH sets of simulation, the faster fly-bys lead to an earlier start of the mutual inclinations to surpass our 4 boundary (see Fig. <ref> in the appendix). However, the fraction of excited systems for the lower velocity fly-bys is offset just by a later start of the disruption, reaching similar values just at later times.

We also change the angle between the perturber and the planetary system orbit. The excitation after 500 Myr as indicated by the maximum mutual inclination in any of our simulations is not correlated with this angle in either of the two R_MH simulation set-ups. Finally, we analyse if the orbital starting position of the inner planets, either lined up or with 1/4 of an orbit separation, makes a difference in the amount of excitation. We find that the orbital starting position does not affect the amount of disruption within the planetary systems in either of the two R_MH simulation sets at all, as would be expected from a secular interaction.





  §.§.§ Transiting planets


















Above, we describe how the mutual inclination between the inner planets and the number of transiting planets are connected. The larger the mutual inclinations between the inner planets, the fewer planets will transit together, but even at low or zero mutual inclination, not all planets will transit together for all viewing angles <cit.>. In Fig. <ref>, we show the number of transiting planets at different simulation times (0, 100, 200, 300, 400 and 500 Myr) for the two R_MH simulations set-ups. We show all architecture combinations of inner planets and Giants together at different times. These histograms have been adjusted to consider the likelihood of a transit signal being successfully recovered in the Kepler pipeline as described above.

Due to their close orbital spacing for the 10 R_MH systems, all four inner planets will transit for a larger range of viewing angles at 0 Myr (initial set-up), followed by only the inner-most planet, the inner-most two and the inner-most three planets. The difference between the four-planet transit and inner-most-planet-only transit counts is in favour of the single transit for the 14 R_MH simulations due to the slightly larger separation between them, directly affecting the angle at which they will transit together.

The mutual inclinations in the 10 R_MH simulations start to increase much earlier than for the 14 R_MH simulations, which results in the number of simultaneously transiting inner planets decreasing quickly in the case of these initially tighter spaced inner planets. For the initial separation systems with wider spacing, the decrease of transiting planets becomes more noticeable between 200-300 Myr, when more of the systems become excited. This goes along with a decrease in the average number of planets as well as an increase in the number of simulations with higher mutual inclinations (as evident in Fig. <ref> and the left-hand graph in Fig. <ref>). At 500 Myr, ∼70 per cent of the simulations in the 10 R_MH separation case show only single transits, with a slow-down of two-planet to single transits evolution between 400–500 Myr. The 14 R_MH simulations show a similar evolution but at a slower pace and a later starting time. The movement towards fewer transiting planets is a mix of increased mutual inclinations and decreased number of planets due to collisions, which is also evident in the earlier figures showing both of these aspects.

We analyse the distribution of the number of remaining planets depending on their transit observability and fly-by distance in Fig. <ref> at 500 Myr. Comparing the left and right plots, we immediately notice that the initially closer separation of the inner planets (left) leads to a larger fraction of single-transiting planets at all three fly-by distances than for the initially larger inner planet separation (right). Also, the number of planets within these transiting systems differ, i.e. for the 10 R_MH systems, most of the single (and double-) transits are caused by two-planet systems, with smaller fractions of three-planet systems. For the 14 R_MH systems, two-/three-/four-planet systems make up similar proportions of the single and double-transiting systems. We can see in both graphs that there is a drop in the fraction of single-transiting systems for the 250 au fly-bys, which are either counted as part of the double-transits (left) or triple-transits (right).

Finally, we have a look at how collisions are distributed amongst the different numbers of transiting planets in Fig. <ref>. For the closer separation case (left), we find that single (and double-) transiting systems are predominantly made up of systems having undergone 2 collisions, whereas there is a mix of collision histories for the initially more distant separation case (right). Starting with initially closer inner planets, three transit systems are almost exclusively made up of three planet systems, whereas mutually inclined four-planet systems dominate for the more distant initial separation case.







§ DISCUSSION


In this analysis, we set out to investigate if a typical fly-by that can occur in a young star-forming region can disrupt a system of four close-in planets so that only a single planet transits at any given time. This is done to test if the “Kepler Dichotomy” - the apparent excess of single-transiting planets - can, at least partly, be explained as a result of interactions within the birth environment. We do not attempt to fully explain the dichotomy due to several limitations in our approach. One, our simulated planetary systems are still young compared to the planet systems that form the basis for the observed Kepler distribution. Running our simulations, e.g. to 5 Gyr to make them more comparable to the ages of the observed planet systems is not feasible. A second limitation is that we only evolve two example initially coplanar and circular planet system architectures under the influence of a perturbing fly-by whereas the Kepler data is the result of a multitude of different initial planet system architectures, which we are unable to observe close to their initial formation state. Finally, less than 12.5 per cent of the approximately solar-mass initially single stars in our simulations of young star-forming regions have an encounter <300 au after the denser initial core-collapse phase, thereby leaving a large number of planet systems without this perturbing effect.

The evolution of the number of transits over time in both graphs in Fig. <ref> shows a clear trend to a lower number of transits for more evolved, older systems. For the 10 R_MH simulations, most planets are either single or double-transits, which is caused by the reduction in the total number of planets in addition to higher mutual inclinations. These simulations show a slowdown of the increase in the number of single-transit systems from 400 Myr year onwards. Simulating these further in time is required to evaluate if the double-transit systems will move towards single-transit over longer periods of time. This slow-down is also apparent from the flattening of the curve showing the fraction of simulations with mutual inclinations > 4, which appear to be caused by the two more distant fly-bys (150 and 250 au).

The 14 R_MH simulations do not show such a slowdown in the movement toward single transit systems during our simulation time. This is likely due to the later start of the disruption in the planetary systems as the larger initial separations apparently make these systems stable for longer after the fly-by. In these simulations, more than half of the systems still have four planets and it remains to be seen if collisions are as numerous as in the 10 R_MH simulations. Fig. <ref> shows that there appears not to be a slowing of the fraction of simulations that show mutual inclinations > 4 leading to more and more systems with high enough mutual inclinations to only show single-transits. This suggests that these planetary systems might become more excited over time and move towards fewer transiting planets. 

In the 10 R_MH simulations, we see a large number of collisions (∼97 per cent of our simulations undergo at least one collision amongst the planets in the inner system) leading to an average number of only ∼2.2–2.3 remaining planets. As all of the collisions in our simulations happened between the inner planets, most of these systems now feature either two 10 M_⊕ or one even more massive 15 M_⊕ Super-Earth. 

Based on our simulations, we suggest that we should find higher-mass SE planets in systems with lower multiplicity and that single-transiting systems would likely have a mix of different planet masses and sizes (either Super-Earths or Sub-Neptunes). The work by <cit.> ('peas-in-the-pod') found that planets in a multi-planet system are more likely to show similar or correlated sizes, which is similar to the outcome of our simulations, where some of our planetary systems survive with four inner planets of the same size. 


All of the above are results from initially coplanar and circular planetary systems. We have also run a smaller set of initially slightly eccentric (randomly set for each planet at <0.05) and mutually inclined (randomly set for each of the planets 2–4 at <1 mutual inclination compared to the inner-most planet, which has the same initial inclination as in the coplanar simulations to ensure a similar angle between the planet system and the fly-by star). These initial values are motivated by results, e.g. in <cit.>, <cit.> and <cit.>.  The evolution of these simulations (which remained stable in the absence of a flyby) shows similar perturbations leading to increasing mutual inclinations, collisions and the reduction of the number of transiting planets. However, this evolution is accelerated compared to the initially coplanar, circular ones, with an earlier start of the perturbations resulting in a quicker reduction of the number of transiting planets.

We now compare our findings to the simulations of <cit.>, who are one of only a few who directly investigated fly-bys as a potential cause for the “Kepler Dichotomy”. A key difference to our analysis is that these authors only continue to run their simulations for a short period of time after the fly-by, i.e. the time until the perturber is the same distance apart as at the start of the simulation (maximum of 100 times the Giant planet semi-major axis). This was done under the assumption that the largest perturbance is caused by the fly-by itself, even though <cit.> showed in their simulations that it can take tens to hundreds Myr for the perturbance introduced by the fly-by to show a measurable effect. We remove the perturber after 1 Myr, at which point most of them have already reached distances more than ∼10^4-times that of the Giant's semi-major axis and continue the planet system evolution for another 499 Myr.

<cit.> simulated close-in inner-planet-systems with one or two Giant planets and found that only the presence of two outer Giants in a fly-by simulation could induce “sufficient” inclination misalignment (Δ i ≳ 0.5) to cause the inner planets to no longer transit together. Our simulations only include a single Giant planet at three different distances or none at all around a close-in planet system. We show that more than sufficient inclination misalignments (our criteria is Δ i ≳ 4) is achieved even in simulations without an outer Giant (as well as those with one) 500 Myr after a fly-by. Our simulations suggest that some level of perturbance appears unavoidable after a fly-by but that it can take a long time before the perturbation causes significant, measurable excitation to the system. 

In the future it would be interesting to investigate how/if the presence, initial location and mass of the Giant planet impacts the evolution of the inner planet system and how much it is perturbed by the fly-by itself, e.g. testing the hypothesis of an outside-in cascade of inclination perturbance suggested by <cit.>. This next analysis will also shed light on how the Giant might become inclined or dynamically unstable to cause disruption/perturbation in the inner system <cit.>.

<cit.> simulated the fly-by of two star-planet systems with two planets with different configurations. While their simulations are not directly comparable to ours (different number, semi-major axes of planets and mass of their central star), their findings for their close-in Super-Earth systems can be compared to ours. These authors found that their Super-Earths are resistant to the fly-by effect and would require a very close fly-by to destabilise (within a few au). They also suggested that their systems (lower mass than ours) are stable against planet-planet disruption long after the encounter. Our findings agree with the first but disagree with their second conclusion. All our inner planet systems continue to be stable for several tens to hundreds of Myr after the encounter, and none of our fly-by distances are close enough to destabilise them directly, even though the planets are placed close to the stability limit. However, we find that many of our IP systems become unstable enough over time to cause planet-planet collisions, but we see no inner planet ejections.



§ CONCLUSIONS


In this paper, we use N-body simulations to investigate the effect of close encounters that can occur during the early dynamical evolution of a typical young star-forming region on planetary systems containing close-in planets (with and without a distant Giant planet) around a solar-mass host star.



  
  * A single fly-by at <300 au can excite the orbits of close-in inner planets in such a way that initially, coplanar and circular systems evolve into systems with higher eccentricities and mutual inclinations. These orbital changes can lead to collisions between the inner planets. Both effects then result in a reduction in the number of transiting planets over time.
  
  * We find that many of the simulated planet systems have long periods of apparent stability before they become unstable over a very short period of time, ending in collisions and/or with high mutual inclinations. At the end of our simulations (500 Myr), we see a considerable amount of single-transiting systems but do not reproduce the Kepler transit distribution. We did not set out to re-produce this due to our choice of only two basic planetary architectures; however, we show that fly-bys can contribute to the production of single-transit systems by themselves. We also find that a longer simulation time is required to evaluate the evolution, especially of the systems with higher initial orbital separation.
  
  * The distances of the fly-bys result in differences in the maximum mutual inclinations for the initially closer inner planet separations but not for the more distant set-up. There is a time-lag in when the systems start to get excited but the result after 500 Myr is roughly similar. The fly-by distance difference has a small effect on the number of collisions in the systems, but this difference is small and not statistically significant in all cases. Other perturber characteristics have similar delay effects to when the inner planet systems get excited, i.e. velocity or perturber mass. But neither the orbital starting position nor the angle between the orbital plane of the inner planets and the perturber produces any difference in the resulting maximum mutual inclinations.
  
  * In addition to the initial inner planet separation, we also test four different architectures, where three feature a Giant planet at different locations. This difference does not affect the behaviour of the inner planet systems with regard to collisions or mutual inclinations, as we find no statistical difference between the architecture setups.




§ ACKNOWLEDGEMENTS


This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Framework Programme (grant agreement no. 853022, PEVAP). JEO is supported by a Royal Society University Research Fellowship.

Simulations in this paper made use of the REBOUND N-body code <cit.>. The simulations were integrated using the hybrid symplectic MERCURIUS integrator <cit.> and IAS15, a 15th order Gauss-Radau integrator <cit.>. The SimulationArchive format was used to store fully reproducible simulation data <cit.>. 

This work was performed using the DiRAC Data Intensive service at Leicester, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant ST/R001014/1. DiRAC is part of the National e-Infrastructure. This work was performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (www.csd3.cam.ac.uk), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/T022159/1), and DiRAC funding from the Science and Technology Facilities Council (www.dirac.ac.uk).

This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.





§ DATA AVAILABILITY


The data underlying this article will be shared on reasonable request
to the corresponding author.





mnras
 









§ EFFECT OF FLY-BY VELOCITY ON MUTUAL INCLINATION EVOLUTION OF INNER PLANETS











 Paper Title: X-ray Emission of Nearby Low-mass and Sun-like Stars with Directly Imageable Habitable Zones.



X-ray Emission of Nearby Stars]X-ray Emission of Nearby Low-mass and Sun-like Stars with Directly Imageable Habitable Zones


Breanna A. Binder
babinder@cpp.edu

0000-0002-4955-0471]Breanna A. Binder
Department of Physics and Astronomy, California State Polytechnic University, Pomona, CA, USA

0000-0002-1046-025X]Sarah Peacock
University of Maryland, Baltimore County, Baltimore, MD, 21250, USA
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

0000-0002-2949-2163]Edward W. Schwieterman
Department of Earth and Planetary Sciences, University of California, Riverside, CA, USA
Blue Marble Space Institute of Science, Seattle, WA, USA

0000-0002-0569-1643]Margaret C. Turnbull
SETI Institute, Carl Sagan Center for the Study of Life in the Universe, Off-Site: Madison, WI 53713, USA


Department of Physics and Astronomy, California State Polytechnic University, Pomona, CA, USA

0000-0002-7084-0529]Stephen R. Kane
Department of Earth and Planetary Sciences, University of California, Riverside, CA, USA

0000-0003-0944-2334]Alison Farrish
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

0000-0001-6398-8755]Katherine Garcia-Sage
Heliophysics Division, NASA Goddard Space Flight Center, Greenbelt, MD, USA


§ ABSTRACT 
Stellar X-ray and UV radiation can significantly affect the survival, composition, and long-term evolution of the atmospheres of planets in or near their host star's habitable zone (HZ). Especially interesting are planetary systems in the solar neighborhood that may host temperate and potentially habitable surface conditions, which may be analyzed by future ground and space-based direct-imaging surveys for signatures of habitability and life. To advance our understanding of the radiation environment in these systems, we leverage ∼3 Msec of  and  observations in order to measure three fundamental stellar properties at X-ray energies for 57 nearby FGKM stellar systems: the shape of the stellar X-ray spectrum, the luminosity, and the timescales over which the stars vary (e.g., due to flares). These systems possess HZs that will be directly imageable to next-generation telescopes such as the Habitable Worlds Observatory and ground-based Extremely Large Telescopes (ELTs). We identify 29 stellar systems with / ratios similar to (or less than) that of the Sun; any potential planets in the habitable zones of these stars therefore reside in present day X-ray radiation environments similar to (or less hostile than) modern Earth, though a broader set of these targets could host habitable planets. An additional 19 stellar systems have been observed with the  X-ray Telescope; in total, only ∼30% of potential direct imaging target stars has been observed with , , or . The data products from this work (X-ray light curves and spectra) are available via a public Zenodo repository (doi: 10.5281/zenodo.11490574).








§ INTRODUCTION
 

In the coming decades, ground- and space-based direct imaging will afford the best potential for detecting and spectrally characterizing terrestrial planets in the habitable zones of our nearest neighbors <cit.>. While the James Webb Space Telescope (JWST) has and will continue to characterize some rocky exoplanets <cit.>, its ability to spectrally examine temperate terrestrial—and therefore potentially habitable—exoplanets is limited primarily to target systems with transiting planets and late M dwarf host stars. This is due to the lack of coronagraphic instrumentation on JWST to image rocky planets around more massive, Sun-like stars, and because the characterization of terrestrial planets through transit observations is most favorable for M dwarf hosts, which have a combination of favorable transit probabilities, planet-star size ratios, and planet-star separations that allow for the probing of low atmospheric altitudes when accounting for the effects of refraction <cit.>. Moreover, some key candidate biosignature molecules like O_2 cannot be detected directly at Earth-like abundances with JWST via transit spectroscopy even for the most favorable targets <cit.>, but could be more easily detected by reflected light observations of planets in nearby systems <cit.>.

The occurrence rates derived from Kepler indicate that terrestrial planets are likely present within the habitable zones (HZs) of a large fraction of nearby stars <cit.>. The Astro2020 Decadal Survey has identified the discovery and characterization of these worlds as the highest priority for NASA's investments <cit.>, and the Habitable Worlds Observatory[<https://habitableworldsobservatory.org/home>] will build on the LUVOIR and Habitable Exoplanets Observatory concepts <cit.> to realize this goal. Due to detection biases, the vast majority of nearby temperate rocky planets amenable for direct imaging have not yet been discovered, but will be revealed by HWO in its survey phase <cit.>, precursor science with extreme precision radial velocity (EPRV) measurements <cit.>, or direct imaging with upcoming 30-m class telescopes <cit.>.

A detailed understanding of the target stars themselves is a strong prerequisite for maximizing the science yield of exoplanet imaging missions, including extracting the statistical features of the overall planet population <cit.> as well as characterizing the climates, atmospheric composition and stability <cit.>, and potential biosignatures of individual planets <cit.>. Evaluating planetary habitability, specifically, must consider myriad factors beyond circumstellar location including the planet’s bulk composition <cit.> and stellar X-ray and extreme ultraviolet (XUV; 1-912 Å; corresponding to an energy range of 0.01-12 keV) emission <cit.>. Planets orbiting in or near their star’s HZ can experience high levels of XUV radiation, which directly influences atmospheric survival, composition, and long-term evolution <cit.>. In some cases, planetary atmospheres and surface volatiles can be entirely driven away by XUV irradiation <cit.>. Further, the atmospheric states of planets orbiting M dwarf stars may be profoundly different than for those orbiting Sun-like FGK stars due their to extended super-luminous pre-main sequence phase <cit.> during which XUV radiation could ablate away planetary atmospheres or lead to atmospheric oxidation via massive H escape <cit.>. Outgassing could replenish these atmospheres <cit.> though the retention and photochemistry of the atmosphere would continue to be controlled by the host star spectrum and activity <cit.>. 

While X-ray detections have been made for some HWO target stars <cit.> and more detailed studies of the X-ray properties of low-mass nearby stars have been performed <cit.>, no comprehensive and uniform X-ray analysis of potential direct imaging target stars has yet been done. In this paper, we present a comprehensive analysis of archival X-ray observations for all FGKM stellar systems with HZs that will be directly-imageable with upcoming telescopes such as HWO and ground-based Extremely Large Telescopes (ELTs). A follow-up paper yielding panchromatic XUV–IR spectra (including modeled EUV wavelengths) for each one of these systems is forthcoming (Peacock et al., in prep). We describe our sample and the archival data in Sections <ref> and <ref>. Our analysis procedure is detailed in Section <ref>. We extract light curves for stars detected at high significance and identify periods of count rate variability (such as flaring events or epochs of elevated count rates). We then extract X-ray spectra for stars, separating variable- from non-variable epochs, and fit the resulting spectra with one-, two-, or three-component thermal plasma models. Finally, the average best-fit spectral models are used to convert observed count rates (or count rate upper limits, in the case of non-X-ray-detected stars) to luminosities. In Section <ref>, we discuss the evolution of the / ratio with effective temperature and stellar age, and present an extensive appendix describing the X-ray observations and a summary of the known stellar physical parameters and exoplanet systems from the literature for each stellar system in our sample.



§ THE SAMPLE


To identify the highest priority targets for “deep dive” investigations of nearby stars and their planetary systems, we have used the predicted imaging and spectroscopy capability of (1) NASA’s next generation space-based Great Observatories such as HWO, and (2) ground-based ELTs. To first order, the performance of these facilities is described in terms of two specifications: the inner working angle (IWA) — the smallest planet-star angular separation that can be imaged (expressed in milliarcseconds, or mas), and the minimum planet to star flux ratio (or fractional planet brightness, FPB) — where starlight suppression technology and postprocessing algorithms can separate the planet signal from stellar photons and systematic noise. Intriguingly (and somewhat vexingly), for HZ planets the IWA requirement favors stars that are more luminous and therefore have wider HZs, while the limiting flux ratio favors target stars that are less luminous and therefore outshine their HZ planets by a smaller factor <cit.>. A third specification, planet limiting magnitude (V_ lim), determines which of these targets planets would be bright enough to obtain spectra in a reasonable amount of observing time – and this requirement simply favors the nearest stars.

To form our preliminary target list, for space-based missions we chose stars for which the HZ (0.95 - 1.7 AU scaled by √(L_ bol)) falls at least partially outside an inner working angle of 58 mas, and for which Earth-sized planets of albedo ∼0.3 would be brighter than V_ lim < 30, with a fractional planet brightness (FPB) > 4×10^-11. For ground-based observatories, we examined the 25 stars within 10 pc having the most favorable FPB whose HZs are at least partially visible outside an IWA of ∼30 mas, consistent with the literature <cit.>. This results in a well-defined list of 175 target stars whose HZs fall at least partially within the detection space of upcoming exoplanet direct imaging observatories. We compared our target list with the potential target stars for HWO <cit.>, who used slightly different cut-off requirements for V_ lim, FPB and IWA. Our list contains more M-dwarfs (more favorable targets for ELTs) while the <cit.> list includes more F stars (more favorable for space-based observations). <cit.> also excludes stars in close binaries or higher-order multiple systems, while we produce a more inclusive X-ray catalog of nearby stars, assuming technical challenges posed by binaries may eventually be surmounted. There are 229 unique stars when combining the two lists, which we summarize in Table <ref>. 

cccccc[!htbp]
Potential Target Stars for Future HZ Imaging Surveys


    
    HD     Common     Distance     Spectral     From     Potential 

    ID     Name        (pc)         Type       This Work?        HWO Targeta 
    
    166     V439 And         13.77     G8V           

    693     6 Cet 	       18.89    	F8V	   		    

    739     θ Scl     21.72    	F5V	   		    

    1326     GX And		    3.57    	M1.5V	   	     

    1581     ζ Tuc	    8.61    	F9.5V	   	    

    ⋮    ⋮    ⋮    ⋮    ⋮    ⋮

    ^aFrom <cit.>. Select entries are shown to illustrate the table form and content. The full machine-readable table (229 rows) is available online from the journal. 




Using this input list of 229 viable targets for direct imaging campaigns, we searched the  archives of the XMM-Newton Observatory and the Chandra X-ray Observatory for publicly available data. Many M stars (and some K stars) in this initial target list were previously studied as part of the Measurements of the Ultraviolet Spectral Characteristics of Low-mass Exoplanetary systems (MUSCLES) and Mega-MUSCLES Hubble Space Telescope Treasury programs, which characterized the X-ray/UV emission from K and M exoplanet host stars. We incorporate these results, presented in <cit.>, into our work here (see Section <ref>). We retrieve 192  and  observations for 57 stellar systems containing Sun-like and low-mass stars from the X-ray archives, amounting to ∼3 Msec of observing time (not including the MUSCLES/Mega-MUSCLES observations). An additional 19 stellar systems have been observed by . Figure <ref> shows a map of the 3D spatial distribution of these nearby stars and indicates which stars have X-ray observations available.

There are three M-dwarfs highlighted in <cit.> as strong potential HWO targets (Lacaille 8760, Lacaille 9352, and Lalande 21185). Both Lacaille 8760 and Lacaille 9352 were observed by  and are included in this work. A detailed study of the panchromatic spectrum and habitability potential of Lalande 21185 is in preparation (Schwieterman et al., in prep) and is not included in the sample here. Roughly half of the target stars contained in our study are F- and G-type stars. Table <ref> provides a summary of the stellar physical parameters for all stars included in our study, including the effective temperature (), spectral type <cit.>, stellar mass (M), bolometric luminosity (), and approximate age (in Gyr). All data were compiled using the NASA Exoplanet Archive[See <https://exoplanetarchive.ipac.caltech.edu/>]. Right ascension (R.A.), declination (Dec.), and distances were taken from Gaia DR3 <cit.>. More detailed discussion of each stellar system is provided in Appendix <ref>.







§ X-RAY DATA
 
We retrieved all publicly available imaging observations for our target stars from the [See <https://nxsa.esac.esa.int/nxsa-web/#home>] and [See <https://cda.harvard.edu/chaser/>] archives. Table <ref> summarizes the number of observations and the approximate exposure times utilized in this work. Below, we provide a more detail description of how the observations were processed and the methods used to extract light curves and spectra for each facility.



 §.§ XMM-Newton

We retrieved  observations from the  Science Archive and reprocessed the data from the raw  files using SAS 18[<https://www.cosmos.esa.int/web/xmm-newton/sas>]. The European Photon Imaging Camera (EPIC) on board  carries a set of three X-ray CCD arrays: one array uses pn-CCDs (referred to as PN) and two utilize metal oxide semi-conductor CCDs (referred to as MOS1 and MOS2). The PN data were reprocessed with  and the MOS data were reprocessed using  and filtered on an energy range of 0.2-15 keV. Due to the brightness of many of the stars, pile-up (when two or more photons fall on a pixel in less than the readout time) is a significant concern. The PATTERN parameter records the number and pattern of CCD pixels that are triggered for a given event; we use the recommended PATTERN=0 to mitigate pile-up effects[See the  ABC Guide at <https://heasarc.gsfc.nasa.gov/docs/xmm/sl/epic/image/sas_cl.html>]. Background light curves were extracted using  and inspected for background flares, and good time intervals (GTIs) were generated to reject observation times exhibiting strong background flaring events. 

We inspected the reprocessed, cleaned images for evidence of an X-ray source at the location of the target star. If the target star was detected at ≳500 counts in the PN image, we extracted source light curves (binned to 100 s) to search for count-rate variability. We used  to extract both PN and MOS1/2 spectra for stars with ≳2000 net counts,  to generate redistribution matrix files (RMFs), and  to generate ancillary response functions (ARFs) for all available observations. All spectra were binned to contain at least 25 counts per bin.

The majority (∼83%) of the observations analyzed in this work are from the  archive, containing ∼75% of the total exposure time. However, with a spatial resolution of ∼15^'', only stellar binaries with angular separations larger than ∼30^'' can be resolved with the PN camera. Figure <ref> (left panel) shows a 10 ks /PN image of the binary system GL 412, where the two components (and a third, unidentified X-ray source) are resolved.

ccccccccccc
Properties of Sample Stars

    
    
    
    Star     Alternate     R.A.     Dec.     Spectral     Distance     Mass     T_ eff          Age^*      
 3-4
    Name     Name(s)        2c(J2000)          Type^*       (pc)        ()     (K)          ()     (Gyr)       References
    
    η Crv     HIP 61174     12:32:03.75     -16:11:46.55     F2V     17.96±0.17     1.48^+0.29_-0.19     6854^+124_-99     4.62^+0.22_-0.25     1.5^+0.2_-0.4     1,2,50 

    ξ Oph     HIP 84893     17:21:00.68     -21:06:49.81     F2V     17.06±0.10     1.44^+0.30_-0.20     6756^+110_-127     4.13^+0.17_-0.18         1,2 

    υ And A     HD 9826     01:36:47.60     41:24:13.54     F8V     13.41±0.06     1.29±0.04     6157±112     3.35^+0.10_-0.14     3.12     1,2,29 

    ι Hor     HR 810, HD 17051          02:42:34.03     -50:47:57.54     F9V          17.32±0.02     1.17^+0.17_-0.15     6146^+106_-184     1.74^+0.04_-0.06     0.47; 2.72     1-4,9 

    ν Phe       HIP 5862           01:15:12.13     -45:31:51.14       F9V     15.17±0.04     1.15^+0.18_-0.14     6119^+130_-154     2.00±0.06     2.88     1,2,3,9 

    γ Pav     HIP 105858     21:26:26.81     -65:21:45.50       F9V     9.27±0.04     1.15^+0.18_-0.14     6109^+110_-176     1.47±0.05     1,7.25     1,3,44,45 

    β Vir     HIP 57757     11:50:42.51     01:45:48.67     F9V     11.12^+0.07_-0.06     1.13^+0.16_-0.14     6071^+120_-90     3.77^+0.11_-0.14     2.96     1,2 

    GJ 1095     HIP 35136     07:15:50.19     +47:14:20.89     F9V     16.86^+0.19_-0.20     1.12     5866     1.54±0.06     1, 7     1,2,10 

    LHS 237     GJ 288, HIP 37853     07:45:34.63     -34:09:53.86     F9V     15.20±0.13      1.11     5744     1.51±0.06         1,2 

    ι Per     HIP 14632     03:09:06.10     +49:36:46.35     F9.5V     10.51±0.03     1.08^+0.18_-0.10     5938^+141_-169     2.23^+0.04_-0.10     4.20±0.48     1,2,9 

    β Hyi     HIP 2021     00:25:55.80     -77:15:10.06     G0V     7.47±0.03     1.14^+0.01_-0.02     5839     3.70±0.05     6.32     1,2,9 

    β Com     HIP 64394                 13:11:51.43     +27:52:55.58     G0V          9.18±0.03     1.10^+0.14_-0.13     5969^+188_-100     1.41^+0.05_-0.04      <1.12, 1.7     1,2,9 

    LHS 208         HD 39091, π Men       05:37:11.89     -80:27:52.05     G0V          18.27±0.02     1.094±0.039     6037±45     1.44±0.02     2.98^+1.40_-1.30     5 

    ρ CrB     GJ 9537, HIP 78459      16:01:02.41     +33:18:00.28     G0V     17.47±0.02     1.05^+0.12_-0.15     5833^+141_-124     1.81^+0.06_-0.04     11.04     1,2,9 

    GL 672     HIP 84862     17:20:39.74     +32:27:47.22     G0V     14.54±0.02     1.05^+0.16_-0.10     5817^+120_-84     1.29^+0.04_-0.03     12.04     1,2,9 

    χ^1 Ori     HIP 27913, GJ 222        05:54:22.76     +20:16:33.07     G0V          8.66±0.08      1.028^+0.03_-0.028     5822     1.17±0.06     0.35; 4.32     1,2,6,9 

    GL 788     HIP 100017     20:17:32.60     +66:51:18.04     G0V     17.57±0.52     1.02±0.03     5886      1.18±0.10     3.76^+1.88_-1.92     2,9 

    47 UMa          GJ 407, HD 95128     10:59:27.53     +40:25:49.80     G0V          13.80±0.03     1.005±0.047              5829±95               1.58±0.04     6.48^+1.44_-1.04     9,29 

    GL 620.1A       HD 147513                 16:24:01.39     -39:11:34.68     G1V          12.903±0.02     1.06^+0.12_-0.14     5873^+110_-103     1.032^+0.003_-0.004     2; <1; 0.4     1-4,9,19 
 
    GL 311          π^1 UMa, HD 72905      08:39:11.64     +65:01:16.67     G1.5V        14.27^+0.14_-0.15     1.01     5920     1.03±0.05     0.2     2,6,8 

    α Cen A     GJ 559A, LHS 50        14:39:36.49     -60:50:02.37     G2V          1.347±0.003     1.06^+0.05_-0.04     5801     1.61±0.07     7.84^+1.08_-1.28     2,9 

    44 Boo A     ι Boo, HIP 73695 A     15:03:46.74     +47:39:15.89     G2V     12.8±0.2     1.04±0.1     5877     1.552     1.5     46,47,48 

    18 Sco          GJ 616, HIP 79672         16:15:37.52     -08:22:17.91     G2V          14.13±0.02     1.04^+0.12_-0.14     5791^+117_-93     1.09^+0.04_-0.02     5.84^+1.88_-1.96     1,9,10 

    51 Peg     HIP 113357, GJ 882     22:57:28.22     +20:46:08.78     G2V     15.46±0.03     1.03^+0.17_-0.09     5758^+102_-120     1.37^+0.02_-0.05     6.76     1,2 

    ζ^1 Ret     LHS 171, HIP 15330     03:17:49.26     -62:34:20.76     G2V     12.12±0.08     0.96^+0.03_-0.06     5699     0.80±0.05     1.56±0.44     2,9 

    HD 136352       GJ 582, ν^2 Lupi      15:21:45.55     -48:19:07.88     G2V          14.68±0.02     0.87±0.04        5664±61     1.04±0.06     12.3^+1.2_-2.9     11,12 
 
    GL 327     HIP 43726     08:54:17.50     -05:26:03.56     G3V     17.42^+0.44_-0.47     1.09±0.04     5790     1.09^+0.05_-0.04     1.32     2,9 

    μ Ara       HD 160691                 17:44:08.68     -51:50:05.65     G3IV/V       15.60±0.03      1.03^+0.15_-0.11     5772^+119_-84     1.90^+0.04_-0.06     5.7±0.6     1,2,7 

    κ^1 Cet     HD 20630               03:19:21.98     +03:22:14.22     G5V          9.14±0.03      1.02^+0.16_-0.10     5712^+161_-124     0.86^+0.02_-0.03     0.35; 0.7; 2.2     1,2,6,20,21 

    HD 140901      GL 599     15:47:28.54     -37:55:02.15     G7IV/V     15.25±0.01     1.06±0.05     5587^+153_-125     1.22±0.03     3.2±2.8     1,49 

    GJ 777A     LHS 3510, HD 190360     20:03:38.25     +29:53:40.08     G7IV/V     16.01±0.02     0.99±0.08     5552±11     1.20±0.06     13.4     1,2,9 

    ξ Boo A     37 Boo, HD 131156     14:51:23.53     +19:06:00.97     G7V     6.71±0.16     0.90±0.04     5551±21     0.55±0.04     <4     38 

    δ Pav     GJ 780                   20:08:46.81     -66:11:13.53     G8IV         6.09±0.01       0.99^+0.14_-0.10     5590^+156_-140     1.25^+0.02_-0.04     6.2     1,2
 
    LHS 2156     HIP 47080     09:35:38.55     +35:48:32.33     G8IV/V     11.20±0.02     0.96^+0.16_-0.09     5499^+156_-136     0.80^+0.02_-0.04         1,2 

    82 Eri          GJ 139, HD 20794          03:20:00.08     -43:03:59.59     G8V          6.003±0.009      0.94±0.12      5413^+91_-118     0.65±0.01     5.76; >12     1,2,9,13 
 
    τ Ceti     HD 10700, GJ 71       01:44:02.17     -15:56:01.25     G8.5V       3.603±0.007          0.92^+0.12_-0.10     5333^+124_-75     0.51±0.01     >12     1,2,9 

    44 Boo Ba         15:03:46.74     +47:39:15.89     K0V     12.8±0.2     0.98     5300     0.51         46,47,48 

    55 Cnc A     ρ^1 Cnc, HIP 43587     08:52:35.22     +28:19:47.22     K0IV/V     12.59±0.01     0.90^+0.14_-0.09     5252^+123_-172     0.64±0.02     9.5     1,2 

    70 Oph A        LHS 458, HIP 88601A       18:05:27.47     +02:29:42.81     K0V          5.15±0.03      0.89±0.02      5300±50     0.65±0.02     6.2±1.0     1,22 
 
    40 Eri A        GJ 166A, HD 26965         04:15:13.91     -07:40:05.08     K0.5V        5.04±0.01      0.85^+0.11_-0.10     5092^+140_-149     0.43±0.01     6.9±4.7     1,14,23 
 
    δ Eri     HIP 17378     03:43:14.80     -09:45:36.30     K1III/IV     9.04±0.07     1.19     5095     3.37^+0.07_-0.06     6.28     2 

    GL 451A         Groombridge 1830      11:53:04.16     +37:41:34.18     K1IV         9.16^+0.16_-0.17       0.66     4950     0.23±0.04     5.0±0.3     2,6 

    GL 117          HIP 13402, HD 17925        02:52:32.56     -12:46:14.00     K1.5V        10.355±0.005     0.89^+0.09_-0.13     5225^+91_-136     0.40±0.01     1.5;<1.2;0.1     1,2,6,9 
 
    α Cen B     GJ 59 B, LHS 51       14:39:35.06     -60:50:15.10     K2IV          1.347±0.03     0.87±0.07     5234±63     0.52±0.05     8, >11     2,9,30 

    GL 783 A         HD 191408A, LHS 486       20:11:12.54     -36:06:29.58     K2.5V        6.05±0.03      0.74     4922     0.28±0.10     7.05±0.65     1,2,6,25 
 
    GL 183          HD 32147, HIP 23311       05:00:49.59     -05:45:30.96     K3V          8.845±0.003      0.81^+0.11_-0.09     4931^+133_-134     0.29±0.01     2     1,3,6 
 
    GL 892     HD 219134     23:13:21.06     +57:10:10.80     K3V     6.531±0.004     0.79±0.03     4817±62     0.29±0.01     11.0±2.2,12.46     1,2,36 

    GJ 667 A        HIP 84709                  17:18:56.53     -34:59:25.40           K3V         6.97±0.83     0.59              4810              0.13^+0.17_-0.09         2,3,31 

    LHS 1875     GL 250, HIP 32984     06:52:17.47     -05:10:25.42     K3.5V     8.747±0.004     0.76^+0.09_-0.10     4719^+157_-129     0.22^+0.02_-0.01     3.1,<0.48     1,2,9 

    GL 570 A     HIP 73184      14:57:29.18     -21:25:23.31     K4V     5.881±0.003     0.75^+0.08_-0.10     4681^+165_-126     0.22^+0.01_-0.02     <0.6;3     1,2,9 

    ξ Boo B          14:51:23.19     +19:06:03.97    K4V     6.71±0.16     0.66±0.07     4350±150     0.10±0.02     <4     38 

    44 Boo Bb         15:03:46.74     +47:39:15.89     K4V     12.8±0.2     0.55     5035     0.24         46,47,48 

    GJ 667 B        WDS J17190-3459 B                   K4V                             31 

    70 Oph B        LHS 459       18:05:27.82     +02:29:39.12     K5V     5.15±0.03     0.73±0.01     4390±200     0.15±0.02     6.2±1.0     1,22 

    61 Cygni A       HIP 104214               21:06:59.64     +38:45:49.90     K5V          3.497±0.001       0.67^+0.08_-0.09      4304^+161_-128     0.15±0.01     6     1,2,26  
 
    61 Cygni B      HIP 104217                 21:07:00.88     +38:45:21.86     K7V          3.497±0.001       0.61^+0.08_-0.09     3949^+156_-137     0.104^+0.012_-0.009     6     1,2,26 
 
    GL 412A        HIP 54211                 11:05:22.09     +43:31:51.41     M1V          4.83±0.03       0.38     3688     0.020^+0.090_-0.014     3     2,15,28 

    GL 570 B     (HIP 73182)     14:57:27.70     -21:25:07.94     M1.5V     5.86±0.06     0.55±0.05     3345     0.13±0.09         1,2,37 

    GJ 832      LHS 3685, HD 204961       21:33:33.90     -49:00:45.47     M1.5V     4.964±0.001     0.44±0.02     3601     0.03^+0.09_-0.02     9.24     1,2,4 

    GJ 667 C                                    17:19:00.30     -34:59:52.06           M1.5V      7.244±0.005     0.33±0.02     3350±50     0.0137±0.0009     >2     31 

    Kapteyn's Star     GJ 191                 05:11:50.38     -45:02:37.73     M2V        3.9331±0.0004      0.28±0.02     3742±157     0.016±0.004     11.5^+0.5_-1.5     1,2,16 
 
    Wolf 1055     HD 180617, GL 752 A     19:16:55.26     +05:10:08.04     M2.5V     5.912±0.02     0.48±0.02     3534±51     0.032±0.001         40 

    AD Leo       GJ 388     10:19:35.70     19:52:11.30     M3V     4.964±0.001     0.423±0.012     3477±23     0.0236±0.0001     0.4±0.1     39 

    GL 570 C     HIP 73182     14:57:27.70     -21:25:07.94     M3V     5.86±0.06     0.39±0.03         0.018         37 

    Wolf 1061           GJ 628, LHS 419     16:30:17.96     -12:40:04.26     M3.5V     4.306±0.001     0.30±0.02     3309±157     0.011^+0.003_-0.002         1,2 

    Luyten's Star     GJ 273     07:27:25.11     +05:12:33.78     M3.5V     5.92±0.02     0.29±0.02     3382±49     0.0087^+0.0063_-0.0065         1,2,43 

    55 Cnc B           08:52:40.28     +28:18:54.91     M4.5V     12.48±0.02     0.26±0.02     3187±157     0.008±0.002         1,35 

    GL 783B         HD 191408B, LHS 487       20:11:12.80     -36:06:32.17     M4.5V     6.05±0.03     0.24         0.0009     7.05±0.65     25 

    GJ 777B     LHS 3509     20:03:37.57     +29:53:45.49    M4.5V     15.97±0.01     0.21±0.02     3099±157     0.0047±0.0012         1,34 

    40 Eri C        LHS 25          04:15:19.12     -07:40:15.25     M4.5V     5.04±0.01     0.204±0.006     3100     0.008      6.9±4.7     1,14,23,24 
 
    υ And B          01:36:50.16     41:23:25.98     M4.5V     13.47±0.02     0.19±0.02     3159±157     0.0044^+0.0012_-0.0011         29 
 
    Proxima Centauri     GJ 551               14:29:34.16     -62:40:33.87     M5.5V        1.3012±0.0003       0.13±0.02     2900±100     0.0016^+0.0004_-0.0006     4.85     1,17,18 

    GL 412B         WX UMa, LHS 39        11:05:24.50     +43:31:33.27     M6.6V     4.83±0.03     0.095     2700     0.00095     3     27,28 

    VB 10     GL 752 B     19:16:57.61     +05:09:01.59     M8V     5.912±0.02     0.0881^+0.0026_-0.0024     2508^+63_-60     0.000499±0.000004     1     41,42 

    1: <cit.>, 2: https://nexsci.caltech.edu/missions/EXEP/EXEPstarlist.htmlExoCat-1, <cit.>, 3: <cit.>, 4: <cit.>, 5: <cit.>, 6: <cit.>, 7: <cit.>, 8: <cit.>, 9: <cit.>, 10: <cit.>, 11: <cit.>, 12: <cit.>, 13: <cit.>, 14: <cit.>, 15: <cit.>, 16: <cit.>, 17: <cit.>, 18: <cit.>, 19: <cit.>, 20: <cit.>, 21: <cit.>, 22: <cit.>, 23: <cit.>, 24: <cit.>, 25: <cit.>, 26: <cit.>, 27: <cit.>, 28: <cit.>, 29: <cit.>, 30: <cit.>, 31: <cit.>, 32: <cit.>, 33: <cit.>, 34: <cit.>, 35: <cit.>, 36: <cit.>, 37: <cit.>, 38: <cit.>, 39: <cit.>, 40: <cit.>, 41: <cit.>, 42: <cit.>, 43: <cit.>, 44: <cit.>, 45: <cit.>, 46: <cit.>, 47: <cit.>, 48: <cit.>, 49: <cit.>, 50: <cit.> 

    ^*Significant disagreements in literature are discussed in Appendix <ref>.


cccccc[!htbp]
    Summary of X-ray Observations
    
    
    
                          2c        2c 
 2-3 5-6
    Star     # Obs.     Exp. Time (ks)         # Obs.     Exp. Time (ks) 
    
    
    η Crv       0            4     38.5 

    ξ Oph        0            1     19.8 

    υ And     1     5.3           4     58.9 

    ι Hor      32     211.3          0     

    ν Phe        1     16.0            0     

    γ Pav     1     15.9            0     

    β Vir      1     41.1            0     

    GJ 1095          0            1     96.2  

    LHS 237          1      15.6           0     

    ι Per      0            1     4.9  
















































    ⋮    ⋮    ⋮       ⋮    ⋮

    Total           151     2235.7        41     815.0 

    Select entries are shown to illustrate the table form and content. The full machine-readable table (57 rows) is available online from the journal.







 §.§ Chandra

We retrieved  ACIS imaging observations from the  archive[<https://cda.harvard.edu/chaser/>] and reprocessed all  data using the CIAO v4.15 <cit.> task  and standard reduction procedures. We ran the point source detection algorithm  on each individual exposure to generate a list of X-ray source positions and error ellipses. The major and minor axes of these error ellipses were increased by a factor of five, and all X-ray sources were masked so that a background light curve could be extracted for each observation. We inspected these background light curves for background flaring events. We created good time intervals (GTIs) using  and filtered all event data on these GTIs. Finally, we restricted the energy range of our resulting “clean”  files to 0.5-7 keV.



We re-ran  on the cleaned image to identify significant X-ray sources and obtain first-pass estimates of their source properties. Due to its exquisite spatial resolution (0.496^'' pixel size) and a typical on-axis PSF size less than 1^'' for the ACIS instruments,  is the instrument of choice for resolving crowded fields. For stellar systems with multiple components and angular separations of less than a few arcseconds,  is currently the only telescope that can be used to definitively associate X-ray emission with a given stellar component. Figure <ref> (right panel) illustrates the spatial resolving power of  with a 5 ks exposure of the triple star system 40 Eri, where the brightest X-ray source is clearly associated with the position of 40 Eri C, a fainter X-ray source is found at the location of 40 Eri A, and 40 Eri B is undetected.

If the target star was detected in the cleaned  image with ≳50 net counts, we extracted source light curves (binned to 100 s) to search for evidence of X-ray variability. For sources with ≳500 net counts, the CIAO tool  was used to extract spectra and generate RMFs and ARFs. All spectra were binned to contain at least 10 counts per bin. 



 §.§ Swift

The X-Ray Telescope (XRT) onboard the Neil Gehrels  Observatory additionally observed some nearby stars.  is optimized to quickly slew to observe transient events, and typically obtains short “snapshot” X-ray observations (∼2 ks or less) with XRT of relatively bright X-ray sources. These observations are useful for providing flux estimates (or flux upper limits) for stars of interest, but often cannot provide the longer-duration light curves or higher signal-to-noise spectra that can be obtained from  or . We therefore use the High Energy Astrophysics Science Archive Research Center (HEASARC)[<https://heasarc.gsfc.nasa.gov/cgi-bin/W3Browse/swift.pl>] to determine which stars were observed by  but not  or , and we use the  data products generator[<https://www.swift.ac.uk/user_objects/index.php>] to create stacked images of these additional stars and measure count rates (or count rate upper limits). The 19 stars that were observed only by /XRT, but not by  or , are not considered part of our primary stellar sample, but we still provide updated provide X-ray luminosity estimates in Appendix <ref>.




§ ANALYSIS PROCEDURE

We first inspected all X-ray observations for evidence of X-ray emission at the location of the target stars. For stars that were detected, we first extract light curves (binned to 100 s) to search for rapid variability within the observation. For brighter stars (i.e., those with ≳2000 net counts with  or ≳500 net counts with ), changes in observed count rate are readily apparent by visual inspection; for fainter stars, a more quantitative assessment of variability is needed in addition to visual inspection. 



 §.§ Assessing Count Rate Variability

We use the Anderson-Darling test, which is a non-parametric statistical test that is more sensitive to short duration variations than the more commonly used Kolmogorov-Smirnov (KS) test <cit.>. The Anderson-Darling statistic, A^2, is computed as <cit.>:


    A^2 = -N-1/N∑_i=1^N [(2i-1) ln(t_i)+(2N+1-2i)ln(1-t_i)],


where N is the number of photons observed and t_i is the arrival time of the i-th bin, scaled to the interval (0,1). We use 's  routine to compute A^2 and the critical values for the 5% and 1% significance levels when compared to a normal distribution. An A^2 value that is larger than a critical value for a given significance level is evidence of variability; the typical 5% critical value is ∼0.75 and the typical 1% critical value is ∼1.05 for the observations considered here. We additionally compare the observed light curve to a constant count rate and compute a standard reduced χ^2_r (χ^2 divided by the degrees of freedom) statistic.

Variability in observed count rate is, to first order, driven by a change in X-ray flux incident on the detector. In our case, this change in flux is due to changes in the intrinsic X-ray luminosity of the target star. However, changes in the underlying spectra of the star can additionally impart more subtle changes on the light curve, as the sensitivity of  and  to X-ray photons is energy-dependent. We therefore “flag” segments of each light curve according to the type of variability observed, hereafter referred to as the VarFlag, so that the spectral properties can be measured and an accurate count rate-to-luminosity conversion factor can be established as a function of time during the observation. The VarFlags are defined as follows:


    
  * Q: quiescent periods, when the count rate is low and constant within the uncertainties
    
  * F: strong flaring events, where the count rate is above ∼80% the quiescent maximum count rate
    
  * D: decaying periods following flares or descending count rates that follow highly elevated count rates; these periods are roughly defined as immediately following an obvious flare where the count rate is below 80% the maximum flare count rate but still above ∼150% of the quiescent count rate
    
  * E: elevated count rate periods, when count rates are approximately 50% higher than the quiescent count rate but not obviously associated with a strong flaring event
    
  * R: rising count rate periods that precede a strong flare


Dividing a single observation into a series of sub-exposures is a subjective process: we wish to define sub-exposures that are short enough to capture unique variations seen in the light curve (such as rapid flaring events, some lasting only 200-300 s) but long enough so that a sufficient number of X-ray counts are available for adequate spectral modeling (the ∼2000 and ∼500 count limits for  and , respectively). Our main goal is to accurately measure the X-ray luminosity of the star as a function of time.

Figure <ref> shows an example light curve of Proxima Centauri ( observation number 0801880501) and a cumulative distribution function (CDF) of photon arrival times compared to a constant count rate. The A^2 statistic, critical values, and χ^2_r are shown, and the light curve has been color-coded according to the VarFlag assigned to each segment of the light curve. Table <ref> summarizes the variability metrics for every X-ray observation of every star available in the archives (the full machine-readable table is available online from the journal). While some stars in our sample have been the targets of extensive observing campaigns by  and/or  for many years, observational data is quite sparse for other stars. For stars with many observations, long-term periodic X-ray variability has been detected and is likely the result of coronal activity cycles <cit.>. Similar long-term X-ray variability cycles may exist for sparsely observed stars in our sample, but will not be detectable in single snapshot observations. Additional observations are required to constrain the long-term coronal activity cycle and/or flaring activity for many stars in the present sample.



clcccc
    X-ray Variability Metrics
    
    
                              Exp.     Net         

     Star       Observation ID     Time (ks)      Counts     χ^2_r     A^2 
    
    
    





    ι Hor        XMM/0673610201     5.3      4230±70     1.5     0.49  
































    β Vir       XMM/0044740201     41.1     18450±160     3.1     0.73 

    υ And A     XMM/0722030101     5.3     940±40     2.1     0.36 








    44 Boo           XMM/0100650101     18.8     287110±550     3.1     0.64 

    GL 620.1A        XMM/0822070201     7.8     12220±110     1.9     0.41  



































    κ^1 Cet     Chandra/22327      14.0     6600±80     1.7     3.26 
 






































































































    Wolf 1061        Chandra/20163      38.3     250±40     0.9     29.78 


 







    Prox Cen         XMM/0801880501     20.0     66790±260      35.2     17.55  



    GL 412B           XMM/0742230101     10.3     32030±210     243.1     3.43 

    VB 10        XMM/0504010101     24.2     660±110     1.1     11.01 

    ⋮    ⋮    ⋮    ⋮    ⋮    ⋮

    Select entries are shown to illustrate the table form and content. The full machine-readable table (200 rows) is available online from the journal. Net counts are reported for 0.2-15 keV for  observations and 0.5-7 keV for  observations.






 §.§ Spectral Modeling

All X-ray spectral modeling was performed with XSPEC v12.11.1 <cit.> using standard χ^2 statistics. Depending on the number of counts and the overall complexity of the spectra, we used one-, two-, or three-temperature apec models to parameterize the X-ray spectra. Since all of the stars are located very close to the Sun (within ∼20 pc), the intervening hydrogen column density is expected to be low; we adopt a fixed absorbing column N_ H (using the tbabs model) of 10^19 cm^-2 for all stars <cit.>. We assume abundances from <cit.> for both the plasma emission and interstellar absorption models. The plasma temperature and normalizations were left as free parameters during the fit. The normalization of each apec component is defined as (neglecting cosmological terms):


    𝒩=10^-14/4π d^2∫ n_e n_ H dV,


where d is the distance to the source, n_e and n_ H are the electron and hydrogen column densities (in units of cm^-3), respectively, and dV is the volume element (in units of cm^3). The integral ∫ n_e n_ H dV is the volume emission measure (VEM). Once the best-fit model is obtained, we use the cflux convolution model to measure the flux on each plasma component. 

While the spectra of some stars required only one or two apec components to achieve a statistically acceptable fit, most spectra required a third apec component. We initialized all spectral models with the default (solar) abundance. For some stars (e.g., ι Hor, κ^1 Cet) the best fit model was insensitive to the exact choice of abundance; for these stars the abundance was kept fixed at the solar value. If a statistically acceptable fit was not obtained with solar abundances, we allowed the abundance to be a free parameter of the fit, subject to the constraint that the abundance be the same for all plasma components.

For stars that exhibited count rate variability, spectra were extracted and modeled for each sub-exposure. Table <ref> provides full best-fit parameters for all stars with enough counts to perform spectral modeling for all observations and sub-exposures (the full machine-readable table is available online from the journal). For stars with multiple observations or multiple sub-exposures with the same VarFlag, we present the average best-fit model for a given VarFlag in Table <ref>. For uniformity, we refer to the coolest component of a three-component model as “APEC #1," the hottest component as “APEC #3," and the intermediate-temperature component as “APEC #2." The two-temperature thermal plasma models have their individual components matched to the three-temperature component with the most similar temperature.  Figure <ref> shows an example of the averaged spectrum for ι Hor, which did not exhibit significant count rate variability in any of the publicly available 38  observations and was therefore assigned a VarFlag of Q (quiescent). The best-fit models for the individual exposures are shown in black, with the average best-fit spectrum superimposed in red.



cccccccccccccc
All Best-Fit Parameters from X-ray Spectral Modeling

    
    
                  2cAPEC #1        2cAPEC #2        2cAPEC #3 
 4-5 7-8 10-11
        Star     ObsID     VarFlag     kT_1     𝒩_1        kT_2     𝒩_2        kT_3     𝒩_3     logL_ X     abund     <χ^2/ dof> 

                   (keV)     (10^-4)        (keV)     (10^-4)         (keV)     (10^-4)     [erg s^-1]         
    
    
    η Crv     14474     Q     0.16^+0.07_-0.14     3.3^+10.0_-1.6        0.50±0.08     4.6^+5.0_-2.9                28.43±0.06     0.32^+0.56_-0.16     1.14 






































    β Vir     0044740201     Q     0.09±0.01     4.7^+1.2_-0.9        0.37±0.01     2.4±0.1                28.08±0.02     1 (fixed)     2.48 

    χ^1 Ori     0111500101/0-10ks     E     0.10±0.02     39.7^+55.7_-19.9                0.49±0.03     52.9^+10.5_-9.4                  0.84^+0.10_-0.09     12.1^+7.8_-4.8      28.88^+0.03_-0.04     0.31^+0.07_-0.05     1.31 




    β Com     0148680101/0-5ks     Q     0.07±0.02     23.5^+114.8_-15.1                0.39±0.02     3.4±0.3                          28.11±0.02     1 (fixed)     1.55 











    44 Boo     0100650101/0-6ks     D     0.25±0.02     23.8^+4.4_-6.6        0.50^+0.13_-0.12     9.8^+5.4_-3.6        0.91^+0.03_-0.02     275.3^+13.8_-15.7     28.67±0.01     0.09±0.01     1.45 




    GL 620.1A     0822070301     Q     0.11^+0.02_-0.01     9.5^+5.3_-3.6                0.46±0.03     1.4^+2.5_-2.3                  0.77^+0.05_-0.04     6.3^+1.9_-1.7      28.82±0.03     0.43^+0.09_-0.06     1.60 














































































































































































































































































































    ⋮    ⋮    ⋮    ⋮    ⋮        ⋮    ⋮       ⋮    ⋮    ⋮    ⋮    ⋮

    Select entries are shown to illustrate the table form and content. The full machine-readable table (360 rows) is available online from the journal.



Figure <ref> shows the VEM for each star (calculated from the best-fit normalization, defined in Equation <ref> above) as a function of temperature for the coolest APEC component (circles), the intermediate APEC component (squares), and the hottest APEC component (diamonds) using the measurements summarized in Table <ref>. The points are color-coded by stellar spectral type: F and G stars are shown in yellow, K stars in orange, and M stars in red. In general, the temperatures and VEMs of M stars exhibit the largest spread in coronal temperatures relative to more massive stars. The coronal temperatures of FG and K stars are similar for the coolest and hottest APEC components, although VEMs of FG stars are roughly an order of magnitude higher than those of K stars at similar temperatures. The VEMs do not significantly change within a given spectral type group, but the intermediate-temperature APEC component of FG stars is on average hotter (by ∼2 MK) than the intermediate APEC component of K stars.

The three-component thermal plasma model that generally describes the X-ray emission from F and G stars is described by plasma temperatures of ∼1.3 MK, 5.1 MK, and 9.4 MK, with corresponding (log) VEMs of 55.2 [cm^3], 55.4 [cm^3], and 55.0 [cm^3], respectively. For K stars, the temperatures become 1.2 MK, 3.3 MK, 10 MK, with (log) VEM values of 54.1 [cm^3], 53.9 [cm^3], and 53.6 [cm^3], respectively. For these hotter stars, periods of elevated count rates or flaring events different from the quiescent periods in that VEMs increased, while the plasma temperatures remained mostly constant. This result is in sharp contrast to the model fits for M stars, which showed more significant variations in both temperature and VEM during flaring events. 


ccccccccccccc
Average Best-Fit Parameters from X-ray Spectral Modeling

    
    
               2cAPEC #1        2cAPEC #2        2cAPEC #3 
 3-4 6-7 9-10
        Star     VarFlag     kT_1     𝒩_1        kT_2     𝒩_2        kT_3     𝒩_3     logL_ X     abund     <χ^2/ dof> 

                (keV)     (10^-4)        (keV)     (10^-4)         (keV)     (10^-4)     [erg s^-1]         
    
    
    η Crv          Q     0.20^+0.12_-0.09     2.3^+4.3_-1.1        0.55^+0.09_-0.07     3.1^+2.2_-1.7                28.41^+0.05_-0.10     0.27^+0.61_-0.18     1.31 

    υ And A     Q     0.29±0.03     0.8±0.1                           27.57±0.03     1 (fixed)     0.91 

    ι Hor         Q     0.09±0.03     7.7±1.1                            0.42±0.08     2.9±0.2                  0.71±0.14     1.6±0.2      28.75^+0.10_-0.14     1 (fixed)     1.27 

    β Vir     Q     0.09±0.01     4.7^+1.2_-0.9        0.37±0.01     2.4±0.1                28.08±0.02     1 (fixed)     2.48 

    χ^1 Ori        Q     0.08±0.03     258±228                            0.42±0.07     28.6±9.2                 0.69±0.06     21.3±8.2     28.94^+0.07_-0.10     0.20±0.12     1.55 

    ⋮    ⋮    ⋮    ⋮       ⋮    ⋮       ⋮    ⋮    ⋮    ⋮    ⋮








































































    Select entries are shown to illustrate the table form and content. The full machine-readable table (76 rows) is available online from the journal.









 §.§ X-ray Luminosity

In addition to insights about the temperatures and VEMs of coronal structures associated with our target stars, the best-fit X-ray spectral models provide us with a count rate-to-luminosity conversion factor that mitigates against instrumental effects from the extracted light curves and gives us a clearer view of the luminosity evolution of the target star. Furthermore, we can “unfold” our X-ray spectra (i.e., de-convolve the observed spectrum and the energy-dependent detector response function) to assess the intrinsic energy emission from the target star as a function of energy (or wavelength). 

Figure <ref> shows the continuum luminosity of each star (in units of ) as a function of wavelength during quiescent times. All spectra have been corrected for minor foreground absorption, and we use the XSPEC command  to extend our best-fit model to a wavelength range of ∼1-200 Å. For nearly all stars, flaring events are driven by (sometimes dramatic) enhancements in X-ray luminosity at wavelengths ≲10 Å (energies ≳1.2 keV). Figure <ref> shows quiescent and flaring spectra for three representative stars (κ^1 Cet, GJ 570A, and Prox Cen) from 1-1000 Å (top panel), as well as the ratio of the flaring and quiescent spectra (bottom panel). For both κ^1 Cet (G5V) and GJ 570A (K4V), the flux at ∼1 Å (∼12 keV) during flaring times is a factor of ∼30 larger than during quiescent times, but the fluxes at longer wavelengths (≳100 Å; energies ≲0.12 keV) are similar. In the case of Prox Cen (M5.5V), however, the flaring spectrum is brighter than the quiescent spectrum by at least an order of magnitude at all wavelengths, with short-wavelength (≲ 3 Å) emission enhanced by more than a factor of 10^3. This finding is unlikely to be the result of the reduced soft energy responses of  or , as the loss of effective area due to contamination build-up occurs most significantly at energies below 1 keV <cit.>. Nevertheless, we use detector response functions specific to the orbital cycle in which the individual observations were obtained to de-convolve the detector response as accurately as possible. Plots of individual stellar X-ray spectra, color-coded by VarFlag, are discussed in Appendix <ref>.





A question that our analysis can elucidate is: For what fraction of time is a star observed to be emitting at a given X-ray luminosity? While this is related to the flaring frequency and the typical peak luminosity and duration of flares, our goal in this work is not to directly characterize flaring events or the physical mechanisms that cause them. Rather, we wish to predict the X-ray environments in which HZ planets reside. 

To do this, we use our best fit spectral models to first convert the observed count rates to luminosities. For stars that are too faint to have their spectra directly modeled, we use the spectral model of the star with the closest spectral type and physical parameters to convert the observed count rates into luminosities. For stars that are detected but too faint for reliable light curve extraction, we simply estimate their time-averaged X-ray luminosity; for un-detected stars we calculate 3σ luminosity upper limits. We then apply Astropy's <cit.>  algorithm to the light curves; the “fitness” parameter is used to define the measurement uncertainties. The Bayesian block algorithm <cit.> is a segmentation technique, which aims to optimally split (in this case) time series data into “blocks” such that each block is statistically different from its neighboring block. An example Bayesian blocked light curve for κ^1 Cet is shown in Figure <ref>.



The output of the  routine is a list of time bins and the luminosity the star was emitting at during each bin. From these lists, we measure the fraction of the time that each star was observed to be fainter than a given  or / value. The results are shown in Figure <ref>. For comparison,  of the modern-day Sun is 6.31×10^26  (log/ = -6.78) when quiet and 8.04×10^27  (log/ = -5.68) when active <cit.>, while ∼4 Gyr ago the  of the Sun was ∼75% the modern value and its  was a factor of ∼2-3 higher <cit.>. On this plot, a perfectly non-variable star would be a straight vertical line, while stars that experience significant changes in X-ray luminosity (i.e., Proxima Centauri) will show a pronounced horizontal extent. We measure the luminosities at which the stars spend less than 25%, 50%, 70%, and 90% of the observed time (which we refer to as L_25, L_50, L_75, and L_90, respectively). These values are summarized in Table <ref>; we also include the time-averaged luminosities (with uncertainties) of faintly detected stars and the 3σ upper limits for non-detected stars in the L_50 column. 







§ DISCUSSION

We compared the 25th and 90th percentile / ratios for stars in our sample to the current minimum and maximum ratios for the Sun <cit.>, and find that eight stars that were detected in the X-ray observations at high significance exhibit / ratios similar to that of the modern Sun: ξ Oph (F2V), β Vir (F9V), 47 UMa (G0V), β Com (G0V), 55 Cnc A (G8V), GL 892 (K3V), GL 783 (K3V), and GL 183 (K3V). Three of these stars (β Vir, β Com, and GL 183) are younger than the Sun, while four are older (47 UMa, 55 Cnc A, GL 892, and GL 783) and one (ξ Oph) does not have an age estimate available in the literature. Twelve stars have / ratios consistent with those of the early Sun: η Crv (F2V), β Com, GL 311 (G1.5V), ζ^1 Ret (G2V), GL 327 (G3V), HD 140901 (G7IV), the unresolved binary 70 Oph (K0V+K5V), LHS 1875 (K3V), GL 570A (K4V), 61 Cyg A (K5V) and B (K7V), and the unresolved GL 570 BC pair (M1.5V+M3V). We note that one star, β Com, exhibits a range in / that fits within the narrow overlap region between the modern and early Sun (log/≈ -6.4 to -5.7). An additional 11 stars were not detected or only marginally detected and have / upper limits lower than the modern day minimum solar /: ν Phe, LHS 237, LHS 208, ι Per, ρ CrB, GL 672, 51 Peg, GJ 777A, 82 Eri, GL 451A, and Kapteyn's Star. For roughly a dozen stars in our sample, the observed  (or  upper limit) translates into X-ray surface fluxes below the ∼10^4  minimum observed in the quiet Sun and other “coronal hole” stars <cit.>. A closer look at these very X-ray faint stars will be the subject of an upcoming study (Binder et al., in preparation). In Table <ref>, we summarize these X-ray-detected Solar-like stars (their spectral types and approximate ages are taken from Table <ref>) and the relevant solar / comparison (modern or early Sun). We additionally provide the “tier” each star was assigned in <cit.>, indicating the priority which each star should be considered for an HWO direct imaging survey (with A indicating the highest priority group, and tiers B and C requiring further study).



Any HZ planets that may exist around the stars listed in Table <ref> are currently experiencing high energy radiation environments that are similar to or less hostile than the modern Earth. Out of the 229 unique stars we identified as potential targets for future direct imaging surveys, only ∼25% had ever been imaged with either  or  (this fraction increases to ∼30% when  imaging is included), and out of the 57 stellar systems with high quality X-ray observation we find that 28 systems (∼50%) have / ratios similar to or less than that of either the modern or early Sun. These stars span the full range of spectral types and ages considered here, and represent all three tiers of the <cit.> ranking system. It is therefore likely that there are an additional ∼100 potential direct imaging target stars with HZs with current X-ray radiation environments similar to or even less hostile than that of the Sun. We note that threshold /ratios below that of the modern or early Sun could be a conservative condition for finding habitable planets. Although, the past evolution of the star would also have to be taken into account. Detailed coupled geological and atmospheric evolution analyses would be most informative for predicting the targets with the best chances of maintaining potentially habitable conditions <cit.>. A systematic X-ray survey of these as-of-yet unobserved nearby stars is needed to inform the selection of the best targets for direct imaging surveys.

In general, M stars exhibit significantly more X-ray variability than earlier type stars and spend a significant amount of time at / ratios dramatically larger than that of the Sun. We can supplement our sample of Sun-like and low mass stars with X-ray observations of an 23 additional M- and K-type stars from the MUSCLES/Mega-MUSCLES survey <cit.>. <cit.> found a major decrease in the X-ray-to-bolometric luminosity ratio (/) with increasing effective temperature. We compute / (or upper limits, in the case of X-ray non-detections) for all stars in our sample using L_50 from Table <ref> and  from Table <ref>. In Figure <ref>, we plot the (logarithmic) / ratio as a function of T_ eff (also from Table <ref>) for our full sample (colored points), with the MUSCLES/Mega-MUSCLES sample shown in gray. Our sample extends this inverse relationship between / and T_ eff by ∼1000 K in T_ eff, with / ratios that are about an order of magnitude lower than observed for K and M stars. 

There is a small clustering of stars with T_ eff∼5200-6200 K with significantly higher / ratios than the majority of comparable stars, likely due to young stellar ages. Chromospheric activity in solar-type stars is observed to decrease rapidly after ∼2-3 Gyr <cit.>, after which activity levels remain mostly constant. We indicate the approximate age of each star in Figure <ref>: young stars (≲2 Gyr) are shown in red, older stars (≳2 Gyr) are shown in purple, and stars without age estimates or ambiguous ages are shown in dark blue. Most of these high-/, high-T_ eff stars are indeed believed to be consistent with the younger ages where each spectral type emits elevated high energy emission <cit.>. There are three stars with ambiguous ages that lie within this region of the diagram (χ^1 Ori, 55 Cnc A, and GL 451A); the observed X-ray emission of these stars suggests that younger age estimates may be preferred for these stars.



In Figure <ref> we again show the / ratio as a function of T_ eff for stars in our sample, but we use L_25, L_50, L_75 and L_90 to compute the / ratio. This allows us to assess the degree to which intrinsic X-ray variability contributes to the scatter in / observed in Figure <ref>. The ratio of L_75/L_25 provides an estimate of how much a star varies “normal” about the median . We find that the variability of the youngest stars in our sample exhibit the smallest dynamic range, with L_75/L_25∼1.3 for stars with T_ eff≳5000 K (for the coolest young star in the plot, AD Leo, L_75/L_25∼1.8). The L_75/L_25 ratio for older stars shows more dispersion, varying by up to a factor of ∼3.5.



 


§ CONCLUSIONS

We have analyzed ∼3 Msec of high quality  and  observations of 57 nearby stellar systems, as well as snapshot  images of an additional 19 stellar systems, to provide a uniform catalog of the X-ray properties of potential target stars for future direct imaging surveys. All X-ray light curves and spectroscopic data is publicly available on Zenodo (DOI: 10.5281/zenodo.11490574). The X-ray environments in the HZs around F, G, and early K stars are more likely to be similar to those that the Earth has evolved in, and we identify 29 stars with current / ratios similar to or less than that of the Sun. Our results are consistent with <cit.>, who found that the X-ray fluxes of late M stars can be a factor of ∼3-15 times larger than for K stars. In addition to high average / ratios, M stars exhibit more frequent and dramatic flares than earlier-type stars. We note that, of the 229 candidate target stars for direct imaging, only ∼30% have been observed by , , or . 




This work is supported by NASA Exoplanets Research Program (XRP) award #80NSSC23K0039 (PI Turnbull). This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. B.B. is grateful for the warm hospitality of the University of California, Santa Barbara Kavli Institute for Theoretical Physics, where the majority of this manuscript was written; this research was supported in part by grant no. NSF PHY-2309135 to the Kavli Institute for Theoretical Physics. S.P. acknowledges support from NASA under award number 80GSFC21M0002. E.S. and S.P. acknowledge support from the CHAMPs (Consortium on Habitability and Atmospheres of M-dwarf Planets) team, supported by the National Aeronautics and Space Administration (NASA) under grant nos. 80NSSC21K0905 and 80NSSC23K1399 issued through the Interdisciplinary Consortia for Astrobiology Research (ICAR) program. This work has made use of data from the European Space Agency (ESA) mission Gaia (<https://www.cosmos.esa.int/gaia>), processed by the Gaia Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. We would like to thank Eric Mamajek for useful discussions related to this work, and the anonymous reviewer for helpful comments that improved our paper. 



CXO, XMM, Swift (XRT), Exoplanet Archive

astropy <cit.>
          




§ DISCUSSION OF INDIVIDUAL STARS

In this appendix we provide additional context for each stellar system (e.g, when there is a notable discrepancy in the published literature with regards to a given physical parameter) and summarize the known or candidate exoplanets associated with each star. We then present the results of our X-ray analysis for each stellar system. Table <ref> summarizes the observations of faint/non-detected stars, including if the star was detected in a given observation or not, the count rate (or upper limit on the count rate) and detection significance, and the “match star” that was used to convert count rates to luminosities.

Figure <ref> shows X-ray images of resolved and unresolved binary star systems included in our study. Individual images are discussed further for each star below. Figure Set <ref> shows the light curves extracted for all stars from all observations in which they were detected, with sub-exposures colored by VarFlag. Figure Set <ref> shows the average best-fit spectra for all sufficiently bright stars. For stars that exhibit flaring events, we also show the ratio of the flaring-to-quiescent spectra as a function of wavelength.














 §.§ η Crv

There are no significant discrepancies in the stellar physical parameters of η Crv. Although there are no known or candidate exoplanets, the system is known to host two debris disks: one at a distance of ∼160-180 AU, and a hotter disk at ∼3-7 AU <cit.>.

η Crv was observed four times with . All observations were taken with ACIS-S in 1/4 subarray mode for ∼10 ks. In all observations, η Crv was detected with ∼1000 net counts (0.5-7 keV), but no significant variability was observed in the X-ray light curves (see Figure Set <ref>). The spectrum of η Crv is well-described by a two-component thermal plasma model, with no significant changes in the temperatures or normalizations across the different observations.


 §.§ ξ Oph

There are no significant discrepancies in the stellar physical parameters of ξ Oph, and the star does not host any no known or candidate exoplanets. ξ Oph was observed once with /ACIS-S for 19.8 ks and robustly detected with ∼160 counts (8σ significance). There are hints of low-level variability in the X-ray emission of ξ Oph, but the relatively low number of net counts prohibits spectral modeling.


 §.§ υ And

υ And A is an F8V dwarf in a presumed binary system with a proper motion companion υ And B, an M4.5V dwarf (also referred to as υ And D). The two stars have an angular separation of ∼0.9^' (corresponding to ∼750 AU). There are no significant discrepancies in the physical parameters of either star. υ And A hosts three confirmed exoplanets (υ And b, c, and d) which were discovered via RV measurements. The masses of υ And b, c, and d are  ∼0.67 M_ Jup, ∼0.67 M_ Jup, ∼2 M_ Jup, and ∼4 M_ Jup, respectively, with orbital periods (orbital distances) of ∼4.6 days (0.06 AU), ∼241 days (0.82 AU), and ∼1282 days (2.5 AU), respectively <cit.>.

The υ And system was observed four times with /ACIS-S in 1/8 subarray mode (for ∼15 ks per observation) and once with  for 5.3 ks. υ And A was detected in all five observations with sufficient counts to extract light curves (see Figure Set <ref>) and with enough counts in  observations 10976 and 10977 to enable spectral modeling. There is no evidence for strong X-ray variability in any observation, and both  spectra were consistent with a single-temperature thermal plasma model with a temperature of ∼0.3 keV (∼3.4 MK; see Table <ref>). υ And B was out of the field of view of the  observations. However, the star was within the field of view of the  PN detector but fell just on the gap between two CCDs and was not detected (see Figure <ref>). 


 §.§ ι Hor

ι Hor is characterized either as a late F-type <cit.> or early G-type <cit.> star. There is additional disagreement on the age of the system: <cit.> quotes an age of 2.72 Gyr, while <cit.> suggest a much younger age of ∼0.47 Gyr <cit.>. We adopt a spectral type of F9V and an age of 0.47 Gyr in this work. ι Hor hosts one confirmed exoplanet that was first detected via radial velocity (RV) measurements by <cit.>. It is a giant planet with ∼2.3  and an orbital period of ∼300 days <cit.>.

ι Hor is one of the most thoroughly observed stars in our sample, with 32 publicly available  observations taken between 2011 and 2018. A detailed study of the coronal activity and structure of the star is presented in <cit.>, who combined the  monitoring with observations from TESS and the STIS instrument on the Hubble Space Telescope. They corroborate a 1.6 year X-ray activity cycle for the star, although ι Hor does not exhibit the same type of dramatic changes in luminosity as seen in other stars in our sample (see Figure <ref>). Our X-ray analysis of ι Hor is consistent with <cit.>, although we do not undertake a more extensive study of the abundances of the star. Light curves of ι Hor are shown in Figure Set <ref>, and the best-fit spectrum is described in Table <ref>.


 §.§ ν Phe

ν Phe is characterized either as a late F-type <cit.> or early G-type <cit.> star. There are no confirmed or candidate exoplanets in the system, and the star was undetected in a 16 ks  observation.


 §.§ γ Pav

γ Pav is a known F-type star, with the specific spectral type ranging from F9V <cit.> to F7V <cit.>. There are significant discrepancies in the estimated age of γ Pav: <cit.> quote an age of ∼1 Gyr based on photometric observations and comparison to stellar evolution models, while and asteroseismic analysis by <cit.> yields an age of 7.25 Gyr. There are no known or candidate exoplanets orbiting γ Pav. The star was only marginally detected (2.8σ) in a 25.9 ks  observation. The low X-ray luminosity of γ Pav is consistent with the older age estimate of <cit.>.


 §.§ β Vir

There are no significant discrepancies in the stellar physical parameters for β Vir, and the star does not host any currently known or candidate exoplanets. β Vir was observed once with  (observation ID 0044740201) and was detected with ∼18,450 net counts on the PN camera (thick filter). The X-ray light curve does not show evidence of significant variability (see Figure Set <ref>). The spectrum is well described by a two-temperature thermal plasma model (see Table <ref>).


 §.§ GJ 1095

The spectral type of GJ 1095 is given as F9V by <cit.> or G0V by <cit.>. There are no known or candidate exoplanets orbiting GJ 1095. The star was only marginally detected (2.9σ) in a 96.2 ks  observation.


 §.§ LHS 237

There are no significant discrepancies in the stellar physical parameters of LHS 237, and no age estimate is currently available in the literature. LHS 237 is a triple-star system: the inner spectroscopic binary has an orbital period of ∼10 years <cit.>, and the third component is one of the coolest known white dwarfs (NLTT 18141 = GJ 288B) located ∼14.5^' from the central binary <cit.>. There are no known or candidate exoplanets in the system.

LHS 237 was serendipitously imaged by  for 15.6 ks in observation 0840210501 (the intended target for the observation was a Wolf-Rayet star). The inner spectroscopic binary is not detected, and the white dwarf is outside the  field of view.


 §.§ ι Per

There are no significant discrepancies in the stellar physical parameters for ι Per. The spectral type is either F9.5V <cit.> or G0V <cit.>. There are no known or candidate exoplanets in the system. ι Per was observed for 4.9 ks with  in ACIS-S 1/4 subarray mode, but the star was not detected.


 §.§ β Hyi

There are no significant disagreements in the stellar physical parameters of β Hyi, and the star does not host any currently known or candidate exoplanets. β Hyi was observed once with /ACIS-S in 1/4 subarray mode and once with . The star was detected in both observations with sufficient counts to extract light curves, but the star is too faint for accurate spectral modeling to be performed. There is no strong evidence for X-ray variability in either the  or  light curves of β Hyi (see Figure Set <ref>).


 §.§ β Com

There are only minor differences in the reported age of β Com: <cit.> report an age <1.12 Gyr, while <cit.> report an age of 1.7 Gyr. All other spectral parameters are taken from <cit.>. There are currently no known or candidate exoplanets orbiting β Com.

β Com was robustly detected (with ∼38,000 net counts) in a 57.2 ks observation by . We extracted a light curve from this observation and found marginal evidence for count rate variability (see Figure Set <ref>). Given the large number of counts available, we split the observation into 5 ks-long sub-exposures and extracted spectra from each sub-exposure to search for evidence of underlying spectroscopic variability. Throughout the observation, a two-temperature thermal plasma model was sufficient to describe the X-ray spectrum of β Com, with the best fit temperatures and normalizations remaining constant within the uncertainties. All best-fit sub-exposure spectral parameters for β Com are provided in Table <ref>, and the average best-fit spectral parameters are listed in Table <ref>. The best-fit spectral models are shown in Figure Set <ref>.



 §.§ LHS 208 (π Men)

All spectral parameters for LHS 208 are taken from <cit.>. LHS 208 hosts three confirmed exoplanets. HD 390931 b is a giant planet that was detected via RV measurements <cit.>; it has ∼12  and an orbital period of ∼2090 days <cit.>. π Men c is a super-Earth that was discovered via transits <cit.>; it has an ∼3.5  with an orbital period of ∼6 days <cit.>. π Men d was detected via RV measurements with ∼13  and an orbital period of ∼125 days <cit.>. LHS 208 not detected in a 36.9 ks observation with  or a 19.6 ks observation with .


 §.§ ρ CrB

Some minor discrepancies, mass ranges from 0.95 <cit.> to 1.05  <cit.> and spectral type from G0V <cit.> to G2V <cit.>. ρ CrB hosts three confirmed exoplanets (ρ CrB c, d, and e), with the existence of a fourth exoplanet controversial <cit.>. The masses of ρ CrB c, d, and e are  ∼28 , ∼22 , and 3.8 , respectively, with orbital periods of ∼102 days (∼0.4 AU), ∼282 days (∼0.8 AU), and ∼13 days (∼0.11 AU), respectively. ρ CrB is also approaching the end of its main sequence lifetime, with a prognosis of planetary engulfment for most of the known planets in the system <cit.>. ρ CrB was observed once with /ACIS-S for 9.8 ks, but was not detected.


 §.§ GL 672

There are no significant discrepancies in the stellar physical parameters for GL 672, and there are no known or candidate exoplanets in the system. GL 672 was observed for 9.9 ks by /ACIS-S but was not detected.


 §.§ χ^1 Ori

χ^1 Ori is a G0V-type star in a ∼14 year elliptical (e∼0.45) orbit with a low-mass companion. RV measurements compiled by <cit.> were used to infer a secondary mass of ∼0.15  with a likely spectral type of M4V or M5V. The estimated angular semimajor axis of the orbit is ∼0.668^'', unresolvable by  but potentially spatially resolvable by  (see Section <ref>). The inferred spectral parameters may be biased due to the presence of this low-mass companion. There is significant disagreement in the literature about the age of the system. <cit.> and <cit.> estimate an age of 4.32 Gyr, while <cit.> use activity-rotation diagnostics to estimate an age of 300-400 Myr. There are no confirmed or candidate exoplanets in the system.

Since only one  observation is available for χ^1 Ori, we are unable to identify the source of the X-ray emission – whether it be from the G0V star, the low-mass companion, or both. There is one potential, minor flaring event during the observation, starting ∼21 ks after the beginning of the observation and lasting for ∼4 ks, where the peak count rate exceeds the median count rate by ∼40% (see Figure Set <ref>). While this may not be a bona fide flare event, we classify this period as flaring so that its spectrum can be compared to the remainder of the observation. The light curve otherwise shows a count rate that is either constant within the uncertainties (which we classify as quiescent) or somewhat elevated above the median count rate level by ∼20% (which we classify as elevated).

Figure Set <ref> shows the quiescent, elevated, and potential flare spectra for the χ^1 Ori system. The elevated and flare spectra are nearly indistinguishable, and both show excess emission over the quiescent spectrum at wavelengths ≲10Å. This difference in short-wavelength emission is driven primarily by the hottest thermal plasma component in our model, which has a temperature of ∼0.8 keV for the elevated and flaring spectra but ∼0.7 keV in the quiescent spectrum (see also Table <ref>). The integrated X-ray luminosity of χ^1 Ori is constant within the uncertainties across the three spectra.



 §.§ GL 788

There are no significant discrepancies in reported stellar parameters of GL 788. The system does not host any known or candidate exoplanets. GL 788 was detected in a 9.6 ks  observation with ∼4470 net counts. There was no evidence of count rate variability apparent in the X-ray light curve (see Figure Set <ref>). The spectrum was well-described with a single, cool thermal plasma (kT∼0.2 keV).


 §.§ 47 UMa

There are no significant discrepancies in the stellar physical parameters reported in the literature for 47 UMa. The star hosts three confirmed giant exoplanets that were detected via radial velocity measurements. 47 UMa b, c, and d have masses of ∼2.4 M_ Jup, ∼0.5 M_ Jup, and 1.5 ∼2.4 M_ Jup, respectively, and have orbital periods (distances) of ∼1077 days (∼2 AU), ∼2287 days (∼7.8 AU), and ∼19000 days (∼12.4 AU), respectively. All planetary parameters were taken from <cit.>.

47 UMa was detected at ∼12σ significance in a 6.2 ks  observation. We extracted a light curve from the observation but found no evidence of variability (see Figure Set <ref>), and the star was too faint to enable spectroscopic modeling.


 §.§ GL 620.1A

GL 620.1A is either co-moving with <cit.> or in a wide binary with <cit.> the DA-type white dwarf WD 1620-391. The angular distance between the two components is ∼5.8^', which would correspond to a physical separation of ∼4500 AU at the distance of GL 620.1A. GL 620.1A is classified as a barium star, and its over-abundance in s-process elements has been attributed to pollution due mass transfer during the white dwarf progenitor's AGB phase <cit.>. There is some disagreement in the literature over the spectral type and age of GL 620.1A. Spectral types range from G1V <cit.> to a later G3/5V <cit.>; we adopt a spectral type G1V. While <cit.> quote an age of 2 Gyr, other estimates of the age of GL 620.1A are lower: <cit.> and <cit.> derive upper limits of <1 Gyr and <0.68 Gyr, respectively, while <cit.> adopt an age of 0.4 Gyr. One giant exoplanet is confirmed via RV measurements of GL 620.1A <cit.>; the planet has ∼1.2  and an orbital period of ∼528 days (orbital distance of ∼1.3 AU).

GL 620.1A is detected in both publicly available  observations (Figure <ref>) with sufficient counts to enable spectral modeling (see Figure Set <ref> and Table <ref>). There is no significant variability present in the count rate light curves of GL 620.1A (Table <ref>, see Figure Set <ref>), so we do not attempt to divide the observations into sub-exposures.




  §.§.§ The White Dwarf WD 1620-391

While GL 620.1A is robustly detected in both publicly available  observations, WD 1620-391 is not (see Figure <ref>a). The 3σ upper limits on the count rates of 0.014 ct s^-1 (observation ID 0822070201) and 0.0236 ct s^-1 (observation ID 0822070301). We use WebPIMMs[See <https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl>] to convert the observed /PN count rates (both observations were taken with the thick filter) to a flux assuming an APEC thermal plasma model with kT≈0.5 keV <cit.>. We find flux (luminosity) upper limits of 2.0×10^-14  (4×10^26 ) and 3.4×10^-14  (6.8×10^26 ) for observations 0822070201 and 0822070301, respectively. These luminosity limits are a factor of ∼5-7 higher than the observed luminosity of G 29-38; targeted observations with a more sensitive X-ray instrument are needed to determine if WD 1620-391 is emitting X-rays at a levels comparable to other X-ray-detected white dwarfs.



 §.§ GL 311 (π^1 UMa)

There are no significant discrepancies in the stellar physical parameters reported in the literature for GL 311. The system does not host any currently known or candidate exoplanets.

The variability metrics for GL 311 (see Table <ref>) suggest that count rate variability is observed in the available  observation (observation number 0111400101; see Figure Set <ref>). There are two minor flare-like events in the light curve. The first begins at ∼21 ks after the beginning of the observation, and reaches a peak count rate that is a factor of ∼2 higher than the median count rate. This event lasts for approximately 8 ks and it immediately followed by a second count rate increase, which exceeds the median count rate by a factor of ∼1.6 and lasts for approximately 5 ks. The post-flare count rate is elevated by ∼15% relative to the pre-flare count rate. Spectroscopic modeling before, during, and after the flaring event show that the temperatures of all three thermal components increased in temperature during the flaring event, although a modest decrease in the normalization (related to the VEM) during the flare led to a constant (within the uncertainties) luminosity (Figure Set <ref>; see also Table <ref>). 

Thus, although GL 311 clearly exhibits a degree of rapid X-ray variability (on a timescale of a few hours), this activity does not have a significant affect on the overall X-ray luminosity output of the star. The relative stability of the star's X-ray luminosity is also apparent in Figure <ref>, where the luminosity curve for GL 311 is near-vertical.


 §.§ α Cen

α Centauri A (G2V) and B (K1V) are members of the extensively studied triple system that also includes Proxima Centauri (M5.5V; see Section <ref>). α Cen A and B are in a ∼80 year elliptical orbit <cit.>. One candidate exoplanet, α Cen B b, was reported <cit.> but was later found to be a false positive signal <cit.>. There are currently no other known or candidate exoplanets in the α Cen AB system. 

The semi-major axis of the α Cen system is ∼18^'', but due to the brightness of the stars there is considerable blending in the  images (see Figure <ref>). We therefore do not attempt to extract X-ray products (light curves, spectra) for each star individually. The α Cen system exhibits significant X-ray variability (see Figure Set <ref>). The system is well-described by a three-component thermal plasma model, and increases in the observed count rates are driven largely by increases in the hottest plasma temperature (from ∼0.6 keV during quiescent periods to ∼0.8-0.9 keV during times associated with flaring events) and normalization (which increases by a factor of ∼12). The spectra of α Cen are shown in Figure Set <ref> (see also Table <ref>). Due to the significant blending of the two components, we do not use the quiescent α Cen spectrum as a baseline for inferring other stellar luminosities.


 §.§ 44 Boo

44 Boo (also called ι Boo) is a triple star system with a G0V primary (44 Boo A) in ∼210 year orbit with a spectroscopic binary composed of a K0V (44 Boo Ba) dwarf and a K4V (44 Boo Bb) dwarf <cit.>. 44 Boo A is separated from the spectroscopic binary by ∼0.2^''. The 44 Boo B binary has an orbital period of only ∼6.4 hours and is a possible contact binary <cit.>. There are no known or candidate exoplanets in the 44 Boo system. 

44 Boo is robustly detected with  in an 18.8 ks observation, with count rate variations clearly seen in the light curve (see Figure Set <ref>). We do not include the final ∼5 ks in our spectral analysis due to a significant background flaring event during the observation. Our spectral modeling revealed no significant differences between the minor flaring event from ∼6-10 ks and the relatively constant count rate period from 15-20 ks. The spectrum of 44 Boo did change during periods of declining count rate (the first 6 ks and from 10-15 ks) as the result changes in the hottest thermal plasma component: kT_3 decreased from ∼2.1-2.2 keV during quiescent/flaring times to ∼1.5 keV during periods of secular decline, while the normalization of this component (related to the VEM) more than doubled. These changes to APEC #3 result in a decrease in X-ray flux at the shortest wavelengths (≲10 Å); see Table <ref> and Figure Set <ref>.


 §.§ 18 Sco

There are minor differences in the reported spectral type of 18 Sco: <cit.> quote a spectral type of G5V, but <cit.> classifies the star as a G2V. We adopt a spectral type of G2V. 18 Sco hosts one candidate super-Earth with  ∼6.8 M_⊕ and an orbital period of ∼20 days <cit.>.  The star is detected in both a 20.1 ks  observation and a 62.2 ks observation with /ACIS-S. However, the 18 Sco is not bright enough in either observation for timing or spectroscopic analysis to be performed. 


 §.§ 51 Peg

There are no significant discrepancies reported for the stellar parameters of 51 Peg. The star is famous for hosting the first discovered exoplanet <cit.>, the hot Jupiter 51 Peg b with  ∼0.46 M_ Jup and an orbital period of ∼4.2 days <cit.>. 51 Peg was observed with both  and , but was not detected in either observation.


 §.§ ζ Ret

The wide binary ζ Ret contains ζ^1 Ret, which is listed as a G3V/G5V by <cit.> but G2V as <cit.>, and ζ^2 Ret, a G2V star <cit.>. The two stars are separated by ∼5.2^'. The system does not hot any known or candidate exoplanets.

Although this separation would be easily resolvable by , only ζ^1 Ret is contained with the fields of view of the  cameras. The star is robustly detected (with ∼7700 net counts) in a 20.9 ks  observation (ID 0404920101). No significant variability is observed in the X-ray light curve of ζ^1 Ret (see Figure Set <ref>), and the spectrum is well-described by a two-component thermal plasma model (with temperatures of ∼0.1 keV and ∼0.4 keV; see Figure Set <ref> and Table <ref>).


 §.§ HD 136352

There are no significant discrepancies in the stellar physical parameters reported in the literature for HD 136352. The star hosts three confirmed exoplanets, all of which have masses in the super-Earth/sub-Neptune regime. HD 136352 b, c, and d have  of ∼4.7 , ∼11 , and ∼8.8 , respectively, with orbital periods (distances) of ∼11.6 days (∼0.1 AU), ∼28 days (∼0.17 AU), and ∼107 days (∼0.43 AU), respectively. The inner two planets were found to transit the host star by <cit.>, providing bulk densities for the planets. Subsequent observations by <cit.> revealed that the third, outer planet also transits the host star. HD 136352 is marginally detected (∼2.7σ) in a 16.9 ks  observation with a count rate of 0.0041±0.0015 ct s^-1 with the PN camera and medium filter (Table <ref>).


 §.§ GL 327

There are no significant discrepancies in reported stellar parameters of GL 327 in the literature. The system does not host any currently known or candidate exoplanets. GL 327 was robustly detected with ∼10,500 net counts in a 23.5 ks exposure with  (observaton ID 0404920201). Count rate variability was observed in the X-ray spectrum as shown in Figure Set <ref>, so we split the observation into three sub-exposures and modeled the three spectra independently. In all three sub-exposures, a single-temperature thermal plasma component was sufficient to describe the observations. During the quiescent period from ∼6-21 ks, the plasma temperature was ∼0.4 keV, which increased to ∼0.5 keV during the descending and flaring periods. The normalization during periods of increased X-ray count rate additionally increased by ∼20%. The spectra of GL 327 are shown in Figure Set <ref> (see also Table <ref>.


 §.§ μ Ara

No significant discrepancies in stellar parameters are present in the literature for μ Ara, although there is some uncertainty about the evolutionary stage; we adopt a spectral type of G3IV/V <cit.>. μ Ara hosts four known (giant) exoplanets, all detected via RV measurements. HD 160691 b, c, d, and e have masses of  ∼1.7 , ∼2.4 , ∼10.2 , and ∼0.5 , respectively, and the planets have orbital periods (orbital distances) of ∼646 days (∼1.5 AU), ∼4470 days (∼5.5 AU), ∼10 days (∼0.09 AU), and ∼307 days (∼0.9 AU), respectively. All planetary data were taken from <cit.>. μ Ara is not detected in either of the two publicly available  observations.


 §.§ κ^1 Cet

There is some discrepancy in the reported age estimates of κ^1 Cet: <cit.> lists an age of 2.2 Gyr and <cit.> derive an upper limit of <2.76 Gyr, but many studies report significantly younger ages. For example, <cit.> derive an age of 300-400 Myr, while <cit.> estimated an age of 750 Myr and and <cit.> estimate an age of 650 Myr. There are no currently known or candidate exoplanets orbiting κ^1 Cet.

κ^1 Cet is relatively well-observed in X-rays, with five  observations and two  observations available in the archives, and the star is easily detected in all exposures. It was also observed twice with  using ACIS-S7 in 1/8 subarray mode with a 0.44 s readout time in both observations. While there are clear indications of variability in the light curves of κ^1 Cet the maximum observed count rate only ever exceeds the median count rate by a factor of ∼2 (see Figure Set <ref>). However, despite the only modest changes in count rate, the spectra during these minor flare-like events is quite distinct from the lower count rate quiescent periods, with significant excess emission a wavelengths shortward of 10 Å compared to quiescent times. The change in count rate is being driven by a significant increase in the temperature (by ∼0.2 keV, or ∼2.3 MK) and normalization (a factor of ∼2.2 increase) of the hottest component in a three-thermal plasma component model (Figure Set <ref>; see also Table <ref>).


 §.§ HD 140901

There are no significant discrepancies in the published stellar physical parameters of HD 140901. Two planets have been identified via RV measurements. HD 140901 c has  ∼1.8 M_ Jup with an orbital period of ∼14390 days <cit.>. The existence of HD 140901 b is controversial; if the planet exists, it has a mass of  of ∼16M_⊕ with an orbital period of ∼9 days (orbital distance 0.085 AU), but the RV data of HD 140901 can be equally well described by a retrograde orbital inclination of ∼138^∘ <cit.>. 

HD 140901 was observed once by /ACIS-I for 24.7 ks. The star was robustly detected with ∼220 net counts. The light curve shows clear evidence for variability (see Figure Set <ref>), but the star is not bright enough to enable spectral modeling.


 §.§ GJ 777

GJ 777 is a binary star system composed of a G6IV primary (GJ 777A) and an M4.5V secondary (LHS 2509), which are separated by ∼3000 AU (∼3^').
GJ 777A hosts two currently known exoplanets, detected via RV measurements: HD 190360 b has  ∼1.8 M_ Jup and an orbital period of ∼2854 days <cit.>, and HD 190360 c has  ∼19 M_⊕ and an orbital period of ∼17 days <cit.>. No X-ray emission is detected coincident with the GJ 777 binary system in either of two snapshot  observations.


 §.§ ξ Boo

ξ Boo is a visual binary <cit.> with an angular separation of ∼4.9^'' <cit.>, the stars are unresolvable by  (see Figure <ref>). No discrepancies in stellar physical parameters in the literature. No known or candidate exoplanets. ξ Boo A is a known BY Draconis variable <cit.>.

The ξ Boo binary is robustly detected by , although the two components cannot be resolved. The system exhibits dramatic X-ray variability, as shown in the light curve in Figure Set <ref>. The X-ray spectra of ξ Boo are described by three-component thermal plasma models in all variability states. The flaring spectrum requires hotter temperatures for all three plasma components than the quiescent spectrum, with the normalization of the hottest component increasing by a factor of ∼3.9 during flaring times compared to quiescent times. Figure Set <ref> shows the best fit spectral models for ξ Boo (see also Table <ref>).


 §.§ δ Pav

There are no significant discrepancies in the stellar physical parameters reported in the literature for δ Pav. The star does not host any currently known or candidate exoplanets.

δ Pav was detected in a 32.6 ks  observation (ID number 0780510401) ∼1000 net counts. The A^2 statistic provides evidence that the star may be variable on short timescales, however there are not a sufficient number of counts to split the observation into sub-exposures (see Figure Set <ref>). We attempted to model the spectrum of δ Pav despite the low number of counts; the spectrum required two thermal plasma components to achieve a statistically acceptable fit (see Table <ref>). The best-fit spectrum in shown in Figure Set <ref>. 



 §.§ LHS 2156

There are no significant discrepancies in the stellar physical parameters reported in the literature for LHS 2156. There are no known or candidate exoplanets in the system.

LHS 2156 was detected with ∼60,000 net counts in a 68.4 ks  observation with the PN camera (thick filter). Figure Set <ref> shows the X-ray light curve, which exhibits count rate variability over the duration of the observation. The spectra of LHS 2156 are well described by three-component thermal plasma models. The best-fit temperatures and normalizations of two cooler plasma components remain constant within the uncertainties in all sub-exposures. The hottest thermal plasma component increases in temperature by ∼0.1 keV during flaring periods compared to quiescent periods, an the normalization increases by ∼25%. The best-fit spectra for LHS 2156 are shown in Figure Set <ref> (see also Table <ref>).


 §.§ 82 Eri

There is some discrepancy in the literature about the age of 82 Eri: <cit.> and <cit.> quote ages >12 Gyr, but a much younger age of 5.76±0.66 Gyr was found by <cit.>. 82 Eri hosts four confirmed exoplanets, all of which were discovered via RV measurements <cit.>. HD 20794 b, c, d, and e have  values of ∼2.8 , ∼2.5 , ∼3.5 , and ∼4.8 , respectively, with orbital periods (distances) of ∼18 days (∼0.13 AU), ∼43 days (∼0.23 AU), ∼89 days (∼0.26 AU), and ∼147 days (∼0.5 AU), respectively. All planetary parameters were taken from <cit.>. 82 Eri has been observed by both  and  (see Table <ref>), but was not detected in either observation. 


 §.§ τ Ceti

There are no significant discrepancies in the stellar physical parameters of τ Ceti reported in the literature. τ Ceti hosts four exoplanets, all confirmed via radial velocity measurements <cit.>, with some evidence of additional planets in or near the τ Cet HZ <cit.>. τ Ceti e and f both have masses  ∼3.9 M_⊕ with orbital periods of ∼163 days (∼0.55 AU) and ∼636 days (∼1.3 AU), respectively, while τ Ceti g and h both have masses of  ∼1.8 M_⊕ and orbital periods of ∼20 days (∼0.13 AU) and ∼49 days (∼0.24 AU), respectively.

τ Ceti was observed by  ACIS-S for 45.1 ks in 1/8 subarray mode and detected at ∼7σ significance with ∼110 net counts. We extracted a light curve from this observation and find an A^2 statistic that is strongly suggestive of variability (see Table <ref> and Figure Set <ref>), but due to the relatively low number of counts we did not attempt to extract a spectrum. τ Ceti was marginally detected in an 11.9 ks  observation, but with insufficient counts for light curve extraction or spectral modeling to be performed. The luminosity estimates for τ Ceti suggest that the star's X-ray luminosity changes by a factor of ∼2.


 §.§ 55 Cnc

55 Cnc is a binary star system containing a G8V star (55 Cnc A) and a M4.5V dwarf (55 Cnc B). No significant discrepancies in stellar physical parameters. The two components are separated by ∼1,065 AU <cit.>. Five exoplanets are known to orbit 55 Cnc A. 55 Cnc b was the first to be discovered <cit.> with a mass of  ∼0.8 M_ Jup and an orbital period ∼14.7 days <cit.>. 55 Cnc c, d, e, and f has masses of  ∼0.16 M_ Jup, ∼3.9 M_ Jup, ∼8 M_⊕, and ∼0.15 M_ Jup, respectively, with orbital periods of ∼44 days (∼0.24 AU), ∼4870 days (∼5.5 AU), ∼0.7 days (∼0.02 AU), and ∼260 days (∼0.8 AU), respectively.

Cnc 55 was observed twice by /ACIS-S in 1/8-subarry mode (for 10.7 ks and 18 ks) and once for 10.4 ks by . 55 Cnc A was not detected by , but was detected (at ∼5.6σ) significance by . No evidence for strong variability was observed in the  light curve of 55 Cnc A (Table <ref>, see also Figure Set <ref>). Despite the relatively low number of net counts, we attempted to model the X-ray spectrum of 55 Cnc A. We find that a single-component thermal plasma model is sufficient to describe the X-ray spectrum (see Table <ref> and Figure Set <ref>). 55 Cnc B was not detected in the  observation or one of the  observations (the star was not within the field of view of the second  observation).


 §.§ 70 Oph

70 Oph is a visual and spectroscopic binary containing a K0V (70 Oph A) star and a K5V (70 Oph B) star with an orbital period of ∼88 years <cit.>. The orbit is highly elliptical, with an eccentricity e=0.50 that causes the orbital separation of the stars to vary from ∼11.4 AU to ∼34.8 AU <cit.>. These orbital separations correspond to angular separations of ∼2.3-6.8^'', unresolvable by  but potentially spatially resolvable by  (see Section <ref>). <cit.> find a best-fit age of 6.2±1.0 Gyr, in broad agreement with the 6.8-7.0 Gyr found by <cit.>. These ages (which where derived using astroseismic observations) are in contrast to the significantly younger age (∼680 Myr) predicted by <cit.>. There are no currently known or candidate exoplanets in the system.

There are three  observations available for the 70 Oph system. The individual stars are not resolvable in the  observations (see Figure <ref>); we therefore cannot separate the X-ray emission from the K0V star (70 Oph A) from the K5V star (70 Oph B) with the currently available observations. 

Despite the changes in X-ray count rate observed in the light curves (see Figure Set <ref>), the best-fit spectral models of 70 Oph across different VarFlags are nearly indistinguishable (Figure Set <ref>; see also Table <ref>), and the X-ray luminosity of the two stars changes only by ∼40% between the flaring and quiescent periods. Since we cannot separate the X-ray emission of the binary components individually, we do not use the best-fit spectrum of the 70 Oph system as template for inferring luminosities of other stars in our sample.


 §.§ 40 Eri

The 40 Eri triple system contains a K0V star (40 Eri A) that is either in a wide orbit or co-moving with a white dwarf (40 Eri B)-M4.5V dwarf (40 Eri C) pair. The projected angular distance between 40 Eri A and 40 Eri BC is ∼1.3^'. While the orbital period of the wider A-BC pair is very long, the orbital period of the B-C pair is measured to be ∼230 years <cit.>. The angular separation between 40 Eri B and 40 Eri C is ∼6.9^'', easily resolvable by  (see Figure <ref>). 40 Eri C has a visual magnitude of ∼11, which we use to estimate , and <cit.> estimate  to be ∼3,100 K. There is one confirmed planet orbiting 40 Eri A, which was discovered via RV measurements. HD 26965 b has  of ∼8.5  with an orbital period of ∼42 days <cit.>. No currently known or candidate planets are associated with the 40 Eri BC pair.

The first soft (0.1-2.0 keV) X-ray detection of the 40 Eri system was obtained by HEAO-1, which measured a luminosity of (9.6±3.2)×10^28  <cit.>. Due to the poor spatial resolution of HEAO-1, the binary components could not be resolved. Different scenarios for X-ray production are explored in <cit.>, and the X-ray emission was attributed to 40 Eri C. 

The  image of the system clearly shows X-ray sources coincident with both 40 Eri A and C (although not the white dwarf, 40 Eri B; see discussion below). 40 Eri C is the brighter of the two sources, with roughly an order of magnitude more net counts than observed for 40 Eri A. There is no evidence for strong variability in either star (see Figure Set <ref>). While the coolest thermal plasma component in the 40 Eri C spectrum is similar in temperature to other late-type stars in our sample, the hotter two thermal plasmas are considerably hotter (see Table <ref> and Figure Set <ref>). Despite the lack of obvious variability in 40 Eri C, the best-fit X-ray spectral parameters most closely resemble those of WX UMa in its flaring and post-flaring states. It is possible this star indeed exhibits significant coronal variability, but it was not detectable in the short snapshot  observation available in the archive.




  §.§.§ The White Dwarf 40 Eri B

The white dwarf 40 Eri B is not detected in the 5 ks /ACIS-S observation of the system. The 3σ upper limit on the count rate is 0.0033 ct s^-1. We again use WebPIMMs to convert the observed count rate upper limit to a flux (and luminosity) upper limit, assuming the same spectral model as for WD 1620-391 (Section <ref>). We find a flux upper limit of 1.6×10^-13 , which corresponds to a luminosity of ∼5×10^25 . This luminosity upper limit is below the measured X-ray luminosity of G 29-38 <cit.>, strongly suggesting a lack of a cold accretion disk around this white dwarf.



 §.§ δ Eri

There are no significant discrepancies in the reported physical parameters for δ Eri, and the star does not host any known or candidate exoplanets. δ Eri was observed once with  (observation observation) for 56.2 ks. There was marginal evidence for significant variability in the X-ray light curve of δ Eri (see Figure Set <ref>), but the star was too faint for spectral modeling.


 §.§ GL 451A (Groombridge 1830)

There are no significant discrepancies in the stellar physical parameters reported in the literature for GL 451A. The system does not host any currently known or candidate exoplanets. GL 451A was observed once with /ACIS-S (observation ID 9931) for 32.8 ks in 1/8-subarray mode across ACIS detectors 5-7 with a readout time of 0.54 s. No X-ray source is found coincident with the star's position.


 §.§ GL 117

<cit.> report an age of 1.5 Gyr for GL 117 and <cit.> derive an upper limit of <1.2 Gyr. These ages are in contrast with the very young age (∼100 Myr) predicted by <cit.>. There are no currently known or candidate exoplanets orbiting GL 117.

GL 117 is robustly detected in  observation 0203060501, and there are low-level count rate variations observable in the light curve (see Figure Set <ref>). The largest of these variations reaches a count rate level that is ∼30% above the median count rate. Given the ample number of X-ray counts recorded by , we divide the light curve into sub-exposures but we find no significant differences in the best fit spectral parameters nor the predicted luminosities (see Figure Set <ref> and Table <ref>). The bright X-ray luminosity of the star, particularly when compared to other K dwarfs in our sample, are consistent with a young age for GL 117.


 §.§ GL 783

GL 783 is a binary system containing a K3V star (A) and an M4.5V star <cit.>. The two stars were stated to be separated by ∼8^'' by <cit.> but Gaia DR3 positions indicate a separation of ∼2.6^'' <cit.>, both of which are unresolvable by  but potentially spatially resolvable by  (see Section <ref>). GL 783B has an apparent visual magnitude of ∼12.5 <cit.>, which we use to estimate . At the time of this writing there were no published values of  for GL 783B. The system does not host any currently known or candidate exoplanets.

There is one  observation available for the GL 783 system, and the individual stars are not resolvable (see Figure <ref>). The therefore cannot separate the X-ray emission from the K3V star (GL 783A) from the M4.5V star (GL 783B) with the currently available observations. The A^2 statistic provides evidence that one or both stars may be variable on short timescales (see Figure Set <ref>), however there are not a sufficient number of counts to split the observation into sub-exposures. We attempted to model the spectrum of the GL 783 system despite the low number of counts; the spectrum required three thermal plasma components to achieve a statistically acceptable fit (see Figure Set <ref> and Table <ref>). The coolest thermal plasma component (∼0.1 keV) is similar to the coolest components of the other FGK stars in our sample, while the temperatures of the hottest (∼1 keV) and intermediate (∼0.3 keV) components more closely resemble components found in the later-K and M dwarfs in our sample. 

Since we cannot separate the X-ray emission of the binary components individually, we do not use the best-fit spectrum of the GL 783 system as template for inferring luminosities of other stars in our sample. Follow-up observations with higher spatial resolution are needed to associate the observed X-ray emission with one or both binary components.



 §.§ GL 183

There is some discrepancy in the reported spectral type of GL 183: <cit.> report a spectral type K3V, while K4III is reported by <cit.>. We adopt a spectral type K3V. There is additional tension in the age estimates, with a younger age (∼2 Gyr) reported by <cit.> and older ages reported by <cit.> and <cit.>. There are no currently known or candidate exoplanets orbiting GL 183.

GL 183 was detected in a 18.4 ks  observation (ID number 0780510301) ∼650 net counts. The A^2 statistic provides evidence that the star may be variable on short timescales (see Figure Set <ref>), however there are not a sufficient number of counts to split the observation into sub-exposures. We attempted to model the spectrum of GL 183 despite the low number of counts and found the spectrum required two thermal plasma components to achieve a statistically acceptable fit (Figure Set <ref>; see also Table <ref>), and the fit parameters were reasonably well-constrained.


 §.§ GJ 667

The GJ 667 is a triple star system wherein the two more massive components, A and B (both of which are early/mid K dwarfs) orbit one another in a highly elliptical orbit (e≈0.58). The average angular separation of the two stars is ∼1.8^'', possibly resolvable by . The third star in the system, the M1.5V dwarf GJ 667 C, is separated from the AB pair by ∼0.5^' (which would be easily resolvable with both  and ). There is no published age estimate for the system. GJ 667C has been studied previously in X-rays by <cit.>, and is known to host two confirmed super-Earth exoplanets: GJ 667C b, with  ∼5.6  and an orbital period of ∼7.2 days <cit.>, and GJ 667C c, with with  ∼3.8  and an orbital period of ∼28 days <cit.>. An additional three planets in wider orbits were reported by <cit.> but were unable to be confirmed by <cit.>; their existence remains controversial.

GJ 667 was observed twice with /ACIS-S in 1/8-subarray mode. GJ 667 A and B were not within the field of view of either observation. GJ 667 C was detected in both observations. We extracted an X-ray light curve from  ObsID 17317 (with an exposure time of 18.2 ks, see Figure Set <ref>), but the star was too faint for spectral modeling to be performed. There were insufficient counts in the second  observation (ObsID 17318, with an exposure time of 9.1 ks) for light curve or spectral analysis.


 §.§ LHS 1875

There is some discrepancy in the literature about the age of LHS 1875: <cit.> quote an age of ∼3.1 Gyr, while <cit.> provide a significantly younger upper limit of <0.48 Gyr. There are no known or candidate exoplanets orbiting LHS 1875. LHS 1875 was detected with ∼9260 net counts in a 21.4 ks /PN observation. No evidence for significant X-ray variability was observed (see Figure Set <ref>), and the spectrum was well-described by a three-component thermal plasma model (see Table <ref>).


 §.§ GL 570

There is a significant age discrepancy in the literature for the GL 570 quadruple-star system: <cit.> report a very young age of <0.6 Gyr, while <cit.> adopt an ade of ∼3 Gyr. There are no other significant discrepancies in the stellar physical parameters for the stars in this system. There are no currently known or candidate exoplanets in the system.

GL 570A (K4V) orbits a binary red dwarf pair (GL 570B, an M1.5V dwarf, and GL 570C, an M3V dwarf) with a semi-major axis of ∼190 AU (corresponding to an angular separation of ∼0.5^'). GL 570A and the GL 570BC pair are both detected in X-rays and are easily resolvable with  (see Figure <ref>). The red dwarfs the make up the BC pair are separated by ∼0.2^'' <cit.> and cannot be resolved. The fourth component, GL 570D, is a brown dwarf in a very wide orbit (∼4^') from the GL 570 ABC triple system <cit.>. GL 570D is not detected in X-rays.

We extract X-ray light curves for both GL 570A and the GL 570BC pair. There is clear evidence for variability in both light curves (see Figure Set <ref>), although there are not enough net counts from GL 570BC to enable spectroscopic modeling on sub-exposures. We therefore divide observation of GL 570A into three distinct sub-exposures (the first ∼20 ks, 20-30 ks, and 30-40 ks) and independently model the spectra, but only extract one spectrum of GL 570BC from the entire observation. All spectra are well described by a two-component thermal plasma model. The flaring event seen from 20-30 ks in the GL 570A light curve is characterized by changes in plasma temperatures (∼0.1 keV for the cooler component and ∼0.2 keV for the hotter component) and an increase in the hot-component normalization by ∼170%. The different best-fit spectral models for GL 570A, as well as a comparison of the flaring-to-quiescent spectra, are shown in Figure Set <ref>. The GL 570A model is overall hotter than the GL 570BC model (see Table <ref>).



 §.§ 61 Cyg

The 61 Cygni AB system is a wide binary containing a K5V star (A) and a K7V star (B), with an angular separation of ∼12^'' (the orbital period of the system is ∼659 years). Most age estimates suggest the system is old <cit.>. The system is resolved by  <cit.>. The system does not host any currently known or candidate exoplanets.

61 Cyg has been observed 38 times with , making it the best X-ray-observed star system in our sample. The observations are spaced ∼6 months apart and span ∼18 years, and the two stars are well-resolved in all available images (see Figure <ref>). Both stars are observed to flare (see Figure Set <ref>), and all the spectra extracted from both stars are well describe by a three-component thermal plasma model. The three best-fit temperatures in the models of 61 Cyg A remain constant within the uncertainties during different variability epochs. The changes in count rate observed during flaring events in 61 Cyg A are driven by changes in the normalizations of the two hotter thermal plasma components (and hence the VEMs): the hottest component normalization increases by an order of magnitude, and the intermediate temperature component normalization increases by a factor of ∼2.5. The spectra of 61 Cyg B show similar changes in normalization during flaring events, but the two cooler plasma components also increase in temperature during flaring events (by ∼0.1 keV). The best-fit spectra for 61 Cyg A and B are shown in Figure Set <ref> (see also Table <ref>). 


 §.§ GL 412

The GL 412 AB system contains a M1V primary (A) and a UV Ceti-type flaring M6.6V secondary <cit.>. X-ray flares were previously observed from the system with ROSAT <cit.> and attributed to GL 412B; GL 412A was not believed to be a significant source of X-rays. The system does not host any currently known or candidate exoplanets.

The two stars have an angular separation of ∼31^'' and are resolved in the  image (observation number 0742230101; see Figure <ref>). The image clearly suggests that the observation caught a significant WX UMa flaring event and that the M1V primary, while fainter, emits a detectable quantity of X-ray radiation. A third X-ray source is detected ∼37^'' from WX UMa, but is not obviously associated with the GL 412 binary. We measure a count rate of 0.057±0.002 ct s^-1 in the PN image (medium filter) for GL 412A. 

The light curve of GL 412B is shown in Figure Set <ref>. Spectral modeling shows that the X-ray luminosity increases by a factor of ∼30 between quiescent and flaring times. The quiescent spectrum (the first ∼2 ks of the observation) is well-described with only two thermal plasma components. The steep, ∼0.5 ks flare, the elevated count rate (from ∼3-7 ks), and subsequent declining period all require three components to adequately describe the X-ray spectra. The flaring spectrum exhibits a very hot thermal plasma component, with a temperature of ∼3.5 keV (∼40 MK), which cools as the flare subsides (it has a temperature of ∼2.2 keV during the “elevated” period and ∼1.4 keV during the “descending” count rate period; see Figure Set <ref> and Table <ref>).


 §.§ GJ 832

No significant discrepancies in stellar physical parameters. One confirmed exoplanet, GJ 832 b with  ∼0.6 M_ Jup and an orbital period of 3853 days <cit.>.

GJ 832 was observed twice by , once during a 8.9 ks (observation 0748010201) exposure during which the star was detected but too faint to enable timing or spectroscopic modeling, and once during a 28.6 ks exposure (observation 0860303301) when the star was overall brighter and underwent a short flaring event (see Figure Set <ref>). The quiescent spectrum is well-described by two thermal plasma components, while the lower signal-to-noise during the short flaring period allows the flare spectrum to be modeled with a single thermal plasma component. Overall, the luminosity of GJ 832 increases by a factor of ∼2.5 during the flare. The best-fit spectral models and a comparison of the flaring-to-quiescent spectrum are shown in Figure Set <ref> (see also Table <ref>). The spectrum extracted during the short flaring time during the observation shows an excess of X-ray flux at ∼2-30 Å, but shows weaker X-ray emission (compared to the quiescent spectrum) at wavelengths ≲2 Å.



 §.§ Kapteyn's Star

There are no significant discrepancies in the stellar physical parameters reported in the literature for Kapteyn's Star. Although the detection of sub-Neptune mass planet was reported by <cit.>, this was later refuted by <cit.> who argued that the observed RV variations were artifacts of stellar activity and rotation. There are not currently any additional candidate exoplanets around Kapteyn's Star.

Kapteyn's Star has been observed numerous times by , , and  (see Table <ref>), but is not detected in any of archival X-ray observations. We explored whether Kapteyn's Star would be detected in a stacked  X-ray image (the  background is lower and significantly less affected by spurious flaring events than ). The stacked  image has an effective exposure time of 72.5 ks. Kapteyn's Star is still not detected in the stacked image, with a count rate upper limit of <1.24×10^-4 ct s^-1. Assuming the best fit quiescent spectrum of Wolf 1055, this count rate corresponds to a luminosity upper limit of <1.1×10^24 .


 §.§ Wolf 1055

There are no significant discrepancies in the stellar physical parameters of Wolf 1055 or its binary companion VB 10 <cit.>. There is one confirmed exoplanet orbiting Wolf 1055: HD 180617 b (also identified as GL 572 Ab), with  ∼12 M_⊕ and an orbital period of ∼106 days (∼0.34 AU). 

Wolf 1055 and VB 10 are separated by ∼1.2^' and are easily resolvable with both  and . Both Wolf 1055 was observed once with  (for 24.2 ks) and twice with  (with exposure times of 12.2 ks and 29.2 ks). Both Wolf 1055 and VB 10 are detected in all three observations. A comparison of the  observation and the longer  observation (ObsID 7428) is shown in Figure <ref>. Wolf 1055 is bright enough in two out of these three observations to enable spectral modeling. Although VB 10 is bright enough to be significantly detected in all three observations and we are able to extract light curves from all three observations. The light curves of both stars obtained during  observation 0504010101 are shown in Figure Set <ref>.

The quiescent spectrum of Wolf 1055 is well-described by a three-component thermal plasma model, while the lower signal-to-noise during the flaring period in the second half of the  observation 0504010101 allows the flare spectrum to be modeled with a single thermal plasma component. Overall, the luminosity of Wolf 1055 increases by ∼80% during the flare. The best-fit spectral models and a comparison of the flaring-to-quiescent spectrum are shown in Figure Set <ref> (see also Table <ref>). The spectrum extracted during the flaring period during the second half of observation shows an excess of X-ray flux at ∼2 Å, and shows weaker X-ray emission (compared to the quiescent spectrum) at wavelengths ≲2 Å.


 §.§ AD Leo

There are no significant discrepancies in the stellar physical parameters reported for AD Leo; all stellar parameters are taken from <cit.> and references therein. There are no confirmed or candidate exoplanets orbiting AD Leo.

AD Leo is a known flaring star, and exhibits strong X-ray flaring both times the star was observed by . Both light curves are shown in Figure Set <ref>. High count rates enable spectroscopic modeling over multiple sub-exposures. The X-ray spectra during quiescent periods are generally well-described with a two-component thermal plasma model, while periods of flaring and enhanced variability require three thermal plasma components to adequately describe the observed spectra. Increases in X-ray count rate are driven almost entirely by the appearance of this third, hot (∼1.5 keV) thermal plasma component, which drives a significant increase (by three- to five-orders of magnitude) in X-ray emission below ∼2 Å (see Figure Set <ref>).


 §.§ Wolf 1061

There are no significant discrepancies in stellar physical parameters of Wolf 1061. No age estimate is available for the star. Wolf 1061 hosts three currently known exoplanets <cit.>: Wolf 1061 b, c, and d have masses of  ∼1.9 , ∼3.4 , and ∼7.7 , respectively, with orbital periods of ∼4.9 days (∼0.04 AU), ∼17.9 days (∼0.09 AU), and ∼217 days (∼0.47 AU), respectively. All planetary data taken from <cit.>.

Wolf 1061 was detected in one 38.3 ks observation with Chandra/ACIS-S with ∼240 net counts. The light curve shows evidence for variability (see Figure Set <ref>), but the star is not bright enough to enable spectroscopic modeling.


 §.§ Luyten's Star

There are no significant discrepancies in the reported stellar parameters for Luyten's Star in the literature. No age estimate is available. There are two confirmed exoplanets orbiting Luyten's Star: GJ 273b and GJ 273c, which were detected via RV measurements by <cit.>. GJ 273b and GJ 273c have masses of ∼2.9 M_⊕ and ∼1.2 M_⊕, respectively, and the planets have orbial periods of ∼18.6 days (∼0.09 AU) and ∼4.7 days (∼0.03 AU), respectively.

Luyten's star is detected at ∼5σ significance in a 28.4 ks observation with /ACIS-S. The light curve shows evidence of X-ray variability (see Figure Set <ref>), but are are unable to extract X-ray spectra due to the low number of net counts.


 §.§ Proxima Centauri

Proxima Centauri is a member of an extensively studied triple system along with α Cen A and α Cen B. There is currently one confirmed Earth-like exoplanet, Proxima Centauri b, which has  ∼1  and an orbital period of ∼11 days <cit.>. Two candidate exoplanets, Proxima Centauri c and d, are awaiting follow-up observations. If present, Prox Cen c <cit.> and d <cit.> are estimated to have  values of ∼6  and ∼0.26 , respectively, with orbital periods (distances) of ∼1900 days (∼1.5 AU) and ∼5 days (∼0.029 AU).

Proxima Centauri exhibits significant flaring events, such as those illustrated in Figure <ref> (see also Figure Set <ref>). The X-ray spectrum of Prox Cen is well-described by a two-component thermal plasma model (with temperatures of ∼0.25 keV and ∼0.85 keV) during quiescent times and periods of elevated or rising count rates. A third thermal plasma component is required to describe the spectra during flaring periods and during periods of descending count rates following a flare. During the flare, this third component is hot (∼1.5 keV), and cools to ∼0.6 keV as the count rate decreases (see Figure Set <ref> and Table <ref>). The flaring spectrum drives dramatic increases in both X-ray luminosity. Although the flaring spectrum of Prox Cen exhibits a significant increase in the X-ray flux at wavelengths <10 Å, the spectrum remains elevated (by an order of magnitude) compared to the quiescent spectrum out to long wavelengths (see Figure <ref>). This is in contrast to earlier-type stars, which generally show a similar X-ray emission between quiescent and flaring periods at wavelengths >100 Å.


cccccccccc
Stars with Available /XRT Imaging



    
    Star     Alternate     R.A.     Dec.     Spectral     Distance     Mass     T_ eff          log 
 3-4
    Name     Name(s)        2c(J2000)          Type^*       (pc)        ()     (K)          ()     ([]) 

    

    71 Ori     HIP 29650     06:14:50.77     +19:09:20.29     F6V     20.87     1.35     6533     2.94     <27.59 

    γ Lep A     GL 216A     05:44:27.45     -22:27:00.08     F6.5V     8.88     1.22     6258     2.34     27.53 

    ϕ^2 Cet     GL 37     00:50:07.34     -10:38:43.26     F7V     15.77     1.17     6250     1.85     <26.94 

    η CrB     GL 584A     15:23:12.23     +30:17:17.7     G2V     18.82     1.14     6029     1.64     <27.75 

    61 Vir     GL 506     13:18:24.97     -18:18:31.0     G7V     8.50     0.91     5585     0.84     <26.78 

    GL 567     LHS 5279     14:53:23.27     +19:09:13.54     K0V     11.54     0.93     5258     0.54     28.28 

    GL 68     LHS 1287     01:42:29.41     +20:15:55.87     K1V     7.60     0.88     5190     0.45     <26.29 

    μ Cas     GL 53A     01:08:22.74     +54:54:47.53     K1V     7.55     0.82     5316     0.46     <26.77 

    GL 785     LHS 488     20:15:18.88     -27:02:01.61     K2V     8.79     0.85     5071     0.4     <26.50 

    AK Lep     GL 216B     05:44:26.19     -22:25:24.26     K2V     8.89     0.80     4869     0.3     28.00 

    GL 688     HD 160346     17:39:16.72     +03:33:17.32     K2V     10.71     0.76     4982     0.34     <26.47 

    Lacaille 8760     AX Mic     21:17:10.80     -38:52:20.84     K9V     3.97     0.56     3599     0.1     26.89 

    Wolf 1453     GL 205     05:31:28.21     -03:41:11.50     M1V     5.70     0.56     3690     0.07     27.53 

    Lacaille 9352     GL 887     23:06:00.94     -35:50:49.79     M1V     3.29     0.54     3676     0.04     26.71 

    BR Pis     GL 908     23:49:13.59     +02:23:48.90     M1V     5.90     0.41     3685     0.029     <26.80 

    GL 229Ab     LHS 1827     06:10:34.46     -21:52:04.16     M1.5V     5.76     0.54     3912     0.06     27.55 

    GL 1     HD 225213     00:05:31.99     -37:22:03.88     M1.5     4.36     0.38     3696     0.02     26.75 

    Kruger 60 A     GJ 860A     22:27:58.11     +57:41:38.52     M3V     4.01     0.34     3344     0.01     27.45 

    Struve 2398Ba     GL 725B     18:42:43.94     +59:38:18.09     M3V     3.52     0.25     3345     0.021     27.57 

    Struve 2398Aa     GL 725A     18:42:43.94     +59:38:06.52     M3.5V     3.52     0.33     3401     0.015     27.57 

    ^aUnresolved binary. ^bUnresolved binary with a T7 brown dwarf.







§ UPDATED X-RAY LUMINOSITIES FOR /XRT-IMAGED STARS

We searched the /XRT archive for all nearby stars that did not have available  or  imaging. Nineteen of these stars (mostly K and M dwarfs; one system, Struve 2398, is a binary that is unresolved in the  imaging) have fluxes or flux upper limits available from , which we summarize in Table <ref>. The R.A. and Decl. coordinates for each star is taken from <cit.> and the stellar physical parameters are from <cit.>. We use the quiescent spectrum from Table <ref> of the star that mostly closely matches each  star's stellar physical parameters and the  response files[See <https://swift.gsfc.nasa.gov/proposals/swift_responses.html>] to convert  0.3-10 keV count rates to luminosities. Figure <ref> shows the / ratio as a function of T_ eff (as in Figure <ref>) with the nineteen  stars added.




aasjournal




 Paper Title: SOLES XI. The Aligned Orbit of TOI-2533 b, a Transiting Brown Dwarf Orbiting an F8-type Star.



TOI-2533 b's Spin-Orbit Alignment
Ferreira et al.



Thiago Ferreira dos Santos
thiago.dossantos@yale.edu

0000-0003-2059-470X]Thiago Ferreira
Department of Astronomy, Yale University, 219 Prospect Street, New Haven, CT 06511, USA

0000-0002-7670-670X]Malena Rice
Department of Astronomy, Yale University, 219 Prospect Street, New Haven, CT 06511, USA

0000-0002-0376-6365]Xian-Yu Wang
Department of Astronomy, Indiana University, 727 East 3rd Street, Bloomington, IN 47405-7105, USA

0000-0002-7846-6981]Songhu Wang
Department of Astronomy, Indiana University, 727 East 3rd Street, Bloomington, IN 47405-7105, USA

§ ABSTRACT 

Brown dwarfs occupy a middle ground in mass space between gaseous giant planets and ultra-cool dwarf stars, and the characterisation of their orbital orientations may shed light on how these neighbouring objects form. We present an analysis of the Rossiter-McLaughlin (RM) effect across the transit of TOI-2533 b, a brown dwarf on a moderately eccentric (e_b = 0.2476±0.0090) and wide-separation (a_b/R_⋆ = 13.34±0.30) orbit around an F8-type star, using data from the NEID/WIYN spectrograph in combination with archival photometry and radial velocity observations. Spin-orbit analyses of brown dwarfs are relatively rare, and TOI-2533 stands out as the fifth brown dwarf system with a measured spin-orbit constraint. We derive a sky-projected stellar obliquity of λ = -7±14^∘ for TOI-2533 b, finding that the brown dwarf is consistent with spin-orbit alignment. Our joint model also indicates that TOI-2533 b falls near the lower bound of the hydrogen-burning minimum mass range (M_b = 74.9±5.3 M_ Jup). Ultimately, we find that TOI-2533 b is consistent with formation from disc fragmentation in a primordially spin-orbit aligned orientation, although we cannot rule out the possibility that the system has been tidally realigned during its lifetime. 
    






§ INTRODUCTION


The formation of eclipsing systems – including exoplanets, brown dwarfs, and stars – can be constrained at a population level, and in some cases at an individual-system level, through precise measurements of the masses and relative orbits of the system components <cit.>. The key formation channels of brown dwarfs, whether star-like or planet-like (or a mix), have been long contested <cit.>. While a rough mass cutoff is generally used for convenience to delineate the borders between planets (M≲13M_ Jup), brown dwarfs (M ∼ 13-80M_ Jup), and stars (M≳80M_ Jup), an improved understanding of the formation mechanisms that separate each regime, including what properties typically characterise each group, is needed to define physically motivated boundaries (e.g., ).

Divergent eccentricity distributions for sub-stellar companions suggest distinct formation mechanisms at the high- and low-mass end. For example, <cit.> showed that systems with low-mass (M_ BD < 42.5 M_ Jup) brown dwarf companions demonstrate an anticorrelation between mass and maximum eccentricity, whereas this trend is not apparent for systems with high-mass brown dwarf companions (M_ BD > 42.5 M_ Jup), which follow an eccentricity distribution more similar to that of stellar binaries. <cit.> more recently showed that high-mass-ratio systems (M_ BD/M_*>0.01) follow a broader eccentricity distribution than low-mass-ratio systems (M_ BD/M_*<0.01), which tend toward lower eccentricities. These findings suggest that low-mass (M_ BD < 42.5 M_ Jup) and low-mass-ratio (q<0.01) brown dwarf companions may form through gravitational instability within circumstellar discs <cit.>, while high-mass (M_ BD > 42.5 M_ Jup) and high-mass-ratio (M_ BD/M_*>0.01) brown dwarf companions instead form more like stars, through molecular cloud collapse <cit.>.

Improved characterisation of brown dwarf systems, particularly in light of a growing population of known transiting brown dwarfs <cit.> with the advent of the Transiting Exoplanet Survey Satellite <cit.>, offers a compelling path forward to further distinguish between planet-like and star-like formation histories. Each formation mechanism should have a distinct imprint on the final distribution of brown dwarf spin-orbit orientations – that is, the alignment of the brown dwarf's orbit normal with the spin axis of its host star. Turbulent fragmentation, for instance, may result in a diverse range of orientations <cit.>, while disc-based pathways are typically linked to primordial alignment (although see <cit.> for exceptions). 



Spin-orbit orientations can be observationally constrained for transiting systems (see <cit.> for a review of this topic). During the transit of a spin-orbit aligned occulter, the blue-shifted hemisphere of a star is first obscured, followed by an equal-duration obscuration of the red-shifted hemisphere <cit.>. The corresponding distortion in the net Doppler shift measured across the integrated light from the star during the transit event is known as the Rossiter-McLaughlin (RM) effect <cit.>. 

The RM effect can be measured to derive the sky-projected spin-orbit angle λ (a 2D proxy for the true, 3D spin-orbit angle ψ) between the stellar spin axis and the occulter's orbit normal. Asymmetries in the RM profile are indicative of orbits that are misaligned with the host star's equator, hinting that a particular system either did not emerge from a spin-orbit aligned circumstellar disc or that it was dynamically perturbed after disc dispersal <cit.>.

To date[As of May 2024; based on the Encyclopaedia of Exoplanetary Systems (<https://exoplanet.eu/home/>) and the Transiting Exoplanets Catalogue (TEPCat <cit.>; <https://astro.keele.ac.uk/jkt/tepcat/tepcat.html>)], orbital obliquity measurements have been reported for 211 transiting exoplanets and brown dwarfs, with most of the observations focused on short-period, Jovian-mass exoplanet companions. Among these systems, only four RM observations have been reported for transiting brown dwarfs: CoRoT-3b (λ = -37.6^+10.0_-22.3 deg, M = 21.23^+0.82_-0.59 M_ Jup; ), KELT-1b (λ = 2±16 deg, M = 27.38±0.93 M_ Jup; , WASP-30b (λ = 7^+19_-27 deg, M = 62.5±1.2 M_ Jup; ), and HATS-70b (λ = 8.9^+5.6_-4.5 deg, M = 12.9^+1.8_-1.6 M_ Jup; ). Brown dwarf spin-orbit observations remain rare, despite their utility to constrain the formation pathways of these systems.

In this paper, we present an RM analysis of TOI-2533 b, a brown dwarf with orbital period P∼6.68 days first confirmed by <cit.> that transits an F8-type star bordering the Kraft break <cit.>. Our study leverages observations obtained using the NEID Spectrograph <cit.> on the WIYN 3.5-meter Telescope. This is the eleventh result in the Stellar Obliquities in Long-period Exoplanet Systems (SOLES) survey <cit.>, which is designed to extend the set of spin-orbit constraints to relatively wide-orbiting sub-stellar companions. The observations of TOI-2533 obtained within this work are described in Section <ref>, and a description of the stellar fundamental parameters are presented in Section <ref>. The joint transit, radial velocity, and RM effect models are presented in Section <ref>, and we discuss dynamical timescales in Section <ref>. Insights into the formation and evolution of TOI-2533 b and its placement among other transiting brown dwarfs with measured spin-orbit orientations are presented in Section <ref>. Our conclusions are then given in Section <ref>.



§ OBSERVATIONS


We obtained 21 in-transit radial velocity measurements of TOI-2533 (TYC 2010-124-1; Gaia DR3 1259922196651254272; TIC 418012030) in high-resolution mode (R ∼ 110 000) with the NEID Spectrograph mounted on the 3.5–meter WIYN Telescope at Kitt Peak National Observatory in Arizona/US <cit.>. Observations were performed on April 15, 2023, from 06:37 to 11:42 UT, with exposure times of 900 seconds resulting in an average SNR = 20.8 px^-1 at 5500 Å. Air masses varied from 1.06 at the beginning of the observations to 1.47 at the end, and the collected radial velocities are provided in Table <ref>. The data was reduced using the NEID Data Reduction Pipeline <cit.>. 






§ STELLAR CHARACTERISATION




 §.§ Atmospheric Parameters

We performed a spectral fitting analysis on our co-added out-of-transit NEID spectra, employing the  Python package <cit.>, to derive stellar atmospheric parameters for TOI-2533. From this fit, we obtained T_ eff=6228±114 K, log g = 4.45±0.18 cm s^-2, [Fe/H] = -0.33±0.07, and vsin i_*=5.55±0.75 km/s. Our derived values agree with those from <cit.> within 2σ.




 §.§ SED Analysis

We also leveraged the empirical stellar spectra library of <cit.> to further characterise the TOI-2533 host star. Our SED fit drew archival data from the Two Micron All Sky Survey (2MASS; ), Gaia DR3 <cit.>, the Hubble Space Telescope (HST; ), the Javalambre Physics of the Accelerating Universe Astrophysical Survey (J-PAS; ), the Javalambre-Photometric Local Universe Survey (J-PLUS; ), the Sloan Digital Sky Survey (SDSS; , the Tycho-2 Catalogue <cit.>, and the Wide-field Infrared Survey Explorer (WISE; ).

By integrating the observed Spectral Energy Distribution (SED) from the optical (λ_ min = 3551.05 Å) to the infrared region (λ_ max = 115608 Å) within the Virtual Observatory SED Analyser (VOSA[<http://svo2.cab.inta-csic.es/theory/vosa/>]; ), we found that TOI-2533 is an F8-type star. The mass (1.10^+0.01_-0.04 M_⊙), radius (1.12±0.02 R_⊙), and effective temperature (6183^+16_-84 K) derived from the SED fitting also agree with the values reported in <cit.>.

We note that this analysis was conducted independently of the joint fit described in Section <ref>. Parameters from <cit.>, obtained through spectroscopic analyses, were adopted as priors to derive the stellar host's physical and orbital parameters. 



 §.§ Additional System Properties


We searched for signatures of rotational variability within our TESS photometry of the TOI-2533 star using both the Generalised Lomb-Scargle periodogram (GLS; ) and an Autocorrelation Function analysis (ACF; ). No consistent, statistically significant rotational signal was detected, such that we were unable to derive the degree of line-of-sight alignment or the 3D spin-orbit orientation. We also found no evidence for bound stellar companions to the TOI-2533 system through a search leveraging both Gaia DR3 and existing adaptive optics constraints (see Appendix <ref>).





§ JOINT SPIN-ORBIT MODELLING




We used the allesfitter Python package <cit.> to jointly model the new RM measurement along with other publicly available data sets for TOI-2533. The star was observed during TESS Sectors 23 and 50 at 1800 s and 600 s cadence, respectively. Sector 23 data were processed by the Science Processing Operations Center (SPOC; ) standard aperture pipeline and retrieved using the lightkurve software package <cit.>, while Sector 50 data were processed using the TESS Quick-Look Pipeline (QLP; ) and retrieved from the Mikulski Archive for Space Telescopes (MAST)[<https://exo.mast.stsci.edu>]. Outliers in the data were removed with an upper sigma clipping of 2 away from the median. 

Archival radial velocity data, obtained with the Tillinghast Reflector Échelle Spectrograph (TRES) spectrograph <cit.>, were drawn from <cit.>. To remove systematic instrumental trends characterised by a gradual increase/decrease in the measured stellar flux over time in the TESS data, we applied a de-trending model following the method outlined in <cit.>, where a polynomial of either 1st, 2nd, or 3rd, order is selected based on a minimisation of the Bayesian Information Criterion of the data. 

The free parameters in our joint model include the companion-to-star radius ratio (R_b / R_⋆), sum of radii divided by the orbital semi-major axis ((R_⋆ + R_b) / a_b), cosine of the orbital inclination (cosi_b), transit epoch (T_0;b), orbital period (P_b; with initial guess as reported in ), radial velocity semi-amplitude (K_b), orbital eccentricity and argument of periapsis coupled as √(e)cos(ω) and √(e)sin(ω), limb-darkening coefficients in quadratic form for TESS (q_1; TESS and q_2; TESS) and NEID (q_1; NEID and q_2; NEID) data (see <cit.> and <cit.> for details), jitter terms for the NEID and TRES datasets that were added in quadrature, and a white noise flux error scaling term for the TESS photometry (see ). The sky-projected spin-orbit angle λ was allowed to freely vary between ±180^∘, and the projected stellar rotation velocity vsini_⋆ was also set as a free parameter. The priors for each hyper-parameter are summarised in Table <ref>, and the host star priors were drawn from <cit.> – specifically, R_⋆ = 1.11±0.01 R_⊙, M_⋆ = 1.02^+0.06_-0.07 M_⊙, and T_ eff, ⋆ = 6180±60 K.

The Markov hyper-parameter space was probed using the Dynamic Nested Sampling algorithm dynesty <cit.> over 1000 live-points and tolerance of the convergence criterion of 0.01 (see details in ). The resulting orbital and physical parameters derived for TOI-2533 b are listed in Table <ref>, and the RM best-fit model, along with the transit and RV models, is presented in Figure <ref>. 


We find that the TOI-2533 b brown dwarf is consistent with spin-orbit alignment, with λ = -7±14 degrees. This measurement is shown in context with the current census of exoplanet and brown dwarf spin-orbit constraints in Figure <ref>. All other derived system parameters – corresponding to a brown dwarf with a radius R_b=0.841±0.018 R_ Jup and mass 74.9±5.3 M_ Jup, on a short-period (P=6.685784±0.000013 days) and moderately eccentric (e = 0.2476 ± 0.0090) orbit – match with those of <cit.> within 1σ.











§ DYNAMICAL TIMESCALES


Interactions with the primary star in a two-body system may provide a damping mechanism to push a misaligned or eccentric orbit toward its lowest energy state <cit.>. In this section, we examine relevant dynamical timescales and discuss their implications for the evolutionary history of TOI-2533 b. 



 §.§ Tidal Realignment and Turbulent Friction


Cross-comparing with Figure 9 from <cit.>, we conclude that, based on the host star's effective temperature T_ eff = 6180±60 K and its metallicity [Fe/H] = -0.3±0.2 as reported in <cit.>, TOI-2533 most likely hosts a convective envelope and lies below the Kraft break <cit.>. We note that TOI-2533's projected stellar rotational velocity vsin i_*=8.4^+1.5_-1.4 km/s is consistent with the star lying either below or above the Kraft break (see e.g., ). Stars with convective envelopes have efficient magnetic dynamos at a young age – especially relevant for star-disc interactions over timescales of a few million years – which may facilitate orbital realignment <cit.>. 

Furthermore, star-companion tidal interactions push systems toward low obliquities over time. Stars above the Kraft break possess radiative envelopes and shallower convective zones, typically resulting in longer predicted tidal realignment timescales within the equilibrium tides framework <cit.>. For hot Jupiters, realignment may also be induced through a coupling of the companion's orbit and the host star's oscillation modes (i.e., gravity modes in the stellar interior) that preferentially realign systems with cool stellar hosts <cit.>; however, this mechanism does not extend to systems with companion masses greater than a few Jupiter masses.

Since TOI-2533 falls within a temperature range directly coinciding with the Kraft break transition, we consider both the case in which it does and does not have a substantive convective envelope. For stars with significant convective envelopes, i.e., those below the Kraft break, the empirical tidal realignment timescale can be estimated within the equilibrium tides framework as <cit.>


    τ_ CE = 10^10 q^-2(a/R_⋆/40)^6,

calibrated from observations made in binary star systems by <cit.>. Here, q ≡ M_ b/M_* represents the mass ratio of the companion to the host star, and a/R_⋆ denotes the companion's orbital separation scaled by the stellar radius. Hotter stars with radiative envelopes follow a separate empirical scaling given as


    τ_ RA = 6.25×10^9 q^-2(1 + q)^-5/6(a/R_⋆/6)^17/2.


Based on our derived properties for the TOI-2533 system, we find that τ_ CE∼ 10^9 years and τ_ RA∼10^15 years. If the host star possesses a substantial convective envelope, the system's age and τ_ CE coincide within a 3σ margin (Γ = 4.26^+2.18_-1.74 Gyr; ), such that the companion may have plausibly realigned over its lifetime if it formed in or was previously excited to a misaligned configuration. Alternatively, if the host star has a radiative envelope, the system would be difficult to realign within the system lifetime, implying that the observed alignment is likely primordial.



 §.§ Eccentricity Evolution Driven by Tidal Dissipation


For comparison with the system's tidal realignment timescale, we also consider the evolution timescale of TOI-2533 b's orbital eccentricity τ_e ∼ e/ė. Assuming that the brown dwarf is in pseudo-synchronous rotation – i.e., that its rotational frequency Ω is commensurate with its orbital angular frequency at periastron (although see <cit.>, which describes possible expected deviations from pseudo-synchronisation) – its energy dissipation rate can be expressed as


    ė = [21 k_2 GM_*^2 Ω R_*^5 ζ(e)/2𝒬a^6] a(1-e^2)/GM_*M_be,


where M_* and R_* correspond to the mass and radius of the star, and M_b refers to the companion's mass. Both Ω and the corrective factor ζ(e) depend on the companion's eccentricity, with each given as <cit.>


    Ω = 1/P[f_1(e)/f_2(e) β^3],



    ζ(e) = 2/7[f_0(e)/β^15 - 2 f_1(e)/β^12 + f_2(e)/β^9]


where


    f_0(e)    =    1 + 31/2e^2 + 255/8e^4 + 185/16e^6 + 25/64e^8,
    
        f_1(e)    =    1 + 15/2e^2 + 45/8e^4 + 5/16e^6, 
    
        f_2(e)    =    1 + 3e^2 + 3/8e^4, , 
    β   ≡   √(1-e^2).


Following <cit.>, we set fiducial values 𝒬 = 10^5 as the companion's effective tidal dissipation parameter and k_2 = 0.3 as the Love number, which represents the companion's rigidity and susceptibility of its shape to change in response to a tidal potential.

For TOI-2533 b, we estimate τ_e ∼ 10^12 years, which exceeds the proper system's age by three orders of magnitude. Therefore, we anticipate that, if TOI-2533 has a substantive convective envelope, the system should tidally realign before its orbital circularisation.



§ DISCUSSION




 §.§ Comparison of TOI-2533 b with Theoretical Models for Brown Dwarfs





Fully convective objects with Kelvin-Helmholtz timescales on the order of a few million years were postulated around sixty years ago by <cit.>, and despite the discovery of over 4000 brown dwarfs so far[As of February 2024; see the UltracoolSheet v2.0.0 catalogue <cit.>.], ongoing debate continues to refine our understanding of their formation mechanisms within the context of star and planet formation. Such objects fall short of initiating hydrogen nuclear fusion, with upper masses spanning M ∼ M_ HBMM = 0.07-0.09 M_⊙ <cit.>, but are massive enough for deuterium burning in their cores (M > M_ DBMM = 11-16.3 M_ Jup; ). Relatively high-mass brown dwarfs (⪆60 M_ Jup) have radii predominantly regulated by electron-degeneracy pressure, while those at the lower limit (∼ 14-20 M_ Jup) are mainly constrained by Coulomb pressure <cit.>.

The mass of TOI-2533 b (M_b = 74.9±5.3 M_ Jup = 0.0715±0.0051 M_⊙) falls marginally within the hydrogen-burning minimum mass regime (0.07-0.09 M_⊙), leaving its classification ambiguous between a brown dwarf and an ultra-cool dwarf star (see Figure <ref>). Notably, TOI-2533 b's mass falls above the lithium-burning minimum mass limit (M_ LBMM∼ 51.4 M_ Jup ). Nevertheless, lithium absorption features are not expected in the brown dwarf's spectrum, since lithium is expected to be completely depleted in brown dwarf atmospheres older than 100-250 Myr (e.g., ). 

Evolutionary models for field brown dwarfs suggest that objects with ages around 0.5 Gyr exhibit radii comparable to those of Jovian planets, while high-mass brown dwarfs (≥ 50 M_ Jup) tend to shrink over time, resulting in radii smaller than 1 R_ Jup after 1 Gyr <cit.>. This contrasts with the straightforward mass-based pattern observed in stars (R ∝ M^ξ, with ξ∈ℝ^+; ). The estimated radius of TOI-2533 b is 0.841±0.018 R_ Jup based on our TESS light curve modelling (see Section <ref>), consistent with theoretical predictions given the object's mass (M_b = 74.9±5.3 M_ Jup) and age (4.26^+2.18_-1.74 Gyr; see Figure <ref>).






 §.§ Possible Formation Scenarios for TOI-2533 b


At first glance, TOI-2533 b appears similar to a scaled-up hot Jupiter system, with a very short orbital period (P ∼ 6.68 days) but higher mass (M ∼ 75 M_ Jup). Short-period brown dwarfs around Sun-like stars also have a very low projected occurrence rate (≲2%) comparable to the ∼1% occurrence rate of hot Jupiters <cit.>. While the brown dwarf's aligned and moderately eccentric orbit may be suggestive of mechanisms such as co-planar high-eccentricity migration <cit.>, no outer planetary companion is known to date, and the brown dwarf's eccentricity evolution is slow compared to the system's age (see Section <ref>). Therefore, formation through high-eccentricity migration, as has been suggested for hot Jupiters based on their stellar obliquity distribution <cit.>, is unlikely for this system. 

The high mass of TOI-2533 b, together with the system's relatively high mass ratio (see Table <ref>), instead suggests a formation process reminiscent of stellar binary systems <cit.>. Several mechanisms have been proposed to explain the formation of close binary systems, including gravitational capture <cit.>, turbulent fragmentation <cit.>, and disc fragmentation <cit.>. 

Gravitational capture, in which a companion is captured into a bound orbit by the gravitational well of the primary star, typically occurs in dense environments <cit.>. In globular clusters, the capture rate of brown dwarfs by main-sequence stars has been estimated with an upper limit of 15% based on the absence of observed transits in the 47 Tucanæ globular cluster <cit.>, whereas the capture rate should be lower for brown dwarfs born in loose stellar clusters with a comparatively low number density  <cit.>. Thus, while somewhat rare, gravitational capture may be consistent with the low occurrence rate of short-period brown dwarfs around Sun-like stars <cit.> and remains as a possibility for the formation of TOI-2533 b.

In the framework of turbulent fragmentation, density perturbations within a molecular core already under collapse induce fragmentation into smaller cores or filaments when their Jeans mass is exceeded, leading to the formation of gravitationally bound binary or multiple body systems <cit.>. Turbulent fragmentation is typically expected to produce no preferential directionality of spin-orbit orientations due to the stochasticity of angular momenta throughout the molecular cloud. As a result, turbulent fragmentation would naturally produce an isotropic distribution of primordial spin-orbit orientations <cit.>. Since TOI-2533 most likely has a convective envelope, and, in that case, the system has a relatively short tidal realignment timescale τ_ CE, it is possible that the system may have formed through turbulent fragmentation and subsequently realigned to its currently observed orbital configuration.

Disc fragmentation during the second collapse phase of a molecular cloud[In the first collapse phase, also known as the isothermal phase, the number density around the centre of the cloud is on the order of n_c ≤ 10^11 cm^-3, while the second collapse occurs when n_c>10^16 cm^-3 and the cloud reaches a temperature of about 2000 K, causing molecular H_2 dissociation and a reduction of the adiabatic constant γ≡ C_P/C_V to γ_ critical < 4/3 <cit.>.] is a third potential pathway to form close stellar binaries <cit.>. In this scenario, a rotating molecular cloud flattens into a disc-like structure before its fragmentation (see Figure 1 in , and ), which takes place when there is a sufficiently high ratio between rotational and gravitational energies during the free-fall collapse phase <cit.>. These fragments break into smaller ones, increasing density, opacity, and temperature until further fragmentation ceases <cit.>. Accretion of material onto these proto-brown dwarfs continues until their surrounding gas dissipates and before they reach stellar masses <cit.>. Disc fragmentation naturally produces aligned and moderately eccentric close-in systems like TOI-2533 b <cit.>, with no need to invoke tidal realignment. 

While the formation mechanism for TOI-2533 b remains ambiguous across these three mechanisms, the system highlights the need for further brown dwarf spin-orbit constraints. In particular, brown dwarfs with long projected tidal realignment timescales may inform how the broader population of short-period brown dwarfs, including TOI-2533 b, likely reached their tight orbits. An extended sample of brown dwarf spin-orbit orientations has the potential to more generally inform the dynamical pathways that sculpt main-sequence/brown dwarf binaries.



§ CONCLUSIONS


TOI-2533 b stands out as the fifth transiting brown dwarf with a measured spin-orbit angle reported to date, as well as the eleventh result of the SOLES survey. We have leveraged in-transit radial velocity observations from the NEID/WIYN spectrograph to show that TOI-2533 b's orbit is consistent with alignment, with a low-sky-projected spin-orbit angle, λ = -7±14^∘, that falls within the range observed for quiescent near-resonant exoplanet systems <cit.> – a possible tracer of primordial circumstellar disc alignments. 

With its aligned and moderately eccentric orbit, TOI-2533 b's origin is consistent with stellar binary-like formation through disc fragmentation. This formation pathway is consistent with the brown dwarf's relatively high mass M_b = 74.9±5.3 M_ Jup and relatively high mass ratio q = 0.0702_-0.0019^+0.0021. However, we cannot rule out the possibility that the system initially had a larger misalignment that was corrected over its lifetime, given the short tidal realignment timescale τ_ CE in the case that the host star has a significant convective envelope. Further brown dwarf spin-orbit measurements across a wider parameter space, particularly for systems with long projected tidal realignment timescales, would offer useful insights to delineate the range of orbital architectures for these intermediate-mass eclipsing systems.







§ ACKNOWLEDGEMENTS


T.F. acknowledges support from Yale Graduate School of Arts and Sciences. M.R. acknowledges support from Heising-Simons Foundation Grant #2023-4478, Oracle for Research grant No. CPQ-3033929, and JPL/NASA Grant #1709150. S.W. acknowledges support from the Heising-Simons Foundation Grant #2023-4050. We acknowledge support from the NASA Exoplanets Research Program NNH23ZDA001N-XRP (Grant No. 80NSSC24K0153). 
This paper contains data taken with the NEID instrument, which was funded by the NASA-NSF Exoplanet Observational Research (NN-EXPLORE) partnership and built by Pennsylvania State University. NEID is installed on the WIYN telescope, which is operated by the National Optical Astronomy Observatory, and the NEID archive is operated by the NASA Exoplanet Science Institute at the California Institute of Technology. NN-EXPLORE is managed by the Jet Propulsion Laboratory, California Institute of Technology under contract with the National Aeronautics and Space Administration. All TESS data used in this paper can be downloaded from MAST: [10.17909/vpkg-8e10]http://dx.doi.org/10.17909/vpkg-8e10. 

NEID, TESS, TRES, The Encyclopædia of Exoplanetary Systems, TEPCat.

 <cit.>,  <cit.>,  <cit.>,  <cit.>,  <cit.>,  <cit.>, ,  <cit.>,  <cit.>,  <cit.>,  <cit.>.





§ NO BOUND STELLAR COMPANIONS TO TOI-2533


An inspection of the TESS Target Pixel Files (TPF) using the tpfplotter[<https://github.com/jlillo/tpfplotter>] <cit.> software initially revealed two nearby sources with Δ G ≤ 6 mag relative to TOI-2533: (i) Gaia DR3 1259922127931777664 (μ_α = -1.62±0.02 mas/yr, μ_δ = 2.91±0.02 mas/yr, π= 0.025±0.023 mas) and (ii) Gaia DR3 1259921754269906560 (μ_α = -12.50±0.02 mas/yr, μ_δ = 15.62±0.01 mas/yr, π= 1.71±0.02 mas).

The 3D velocity difference between Gaia DR3 1259921754269906560 and TOI-2533 is Δ v_ 3D∼ 13.77 km/s based on the two stars' parallax and proper motion measurements, surpassing the threshold of Δ v_ 3D≤ 2 km/s typically associated with gravitationally bound objects with a < 10^7 au <cit.>. Gaia DR3 1259921754269906560 maintains a projected separation of 53.07 from TOI-2533, corresponding to approximately 19600 AU. While the projected angular separation between TOI-2533 and Gaia DR3 1259922127931777664 is smaller, at 42.05, the projected distance between the two stars is much larger, at d ∼ 38910±35579 pc from inverting the parallax of Gaia DR3 (although we caution that this approach overestimates large distances; see e.g., ). By employing corrections from <cit.>, the median geometric distance is found to be d ∼ 11422_-1762^+2613 pc. Therefore, the likelihood that either star is gravitationally bound to TOI-2533 is low. 

Neither of these two sources nor TOI-2533 itself is present in the catalogue of binary systems presented by <cit.>. We also note that no close-separation companions have been identified within a ∼8 field of view from high-resolution adaptive optics imaging on the Exoplanet Follow-up Observing Program (ExoFOP) website[<https://exofop.ipac.caltech.edu/tess/>].


aasjournal



 Paper Title: Why Do Stars Turn Red?.


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0000-0003-1295-8235]Po-Sheng Ou （歐柏昇）
Institute of Astronomy and Astrophysics, Academia Sinica, No.1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan, R.O.C. 
Department of Physics, National Taiwan University, No.1, Sec. 4, Roosevelt Rd.,  Taipei 10617, Taiwan, R.O.C. 


Po-Sheng Ou
psou@asiaa.sinica.edu.tw

0000-0002-4848-5508]Ke-Jung Chen（陳科榮）
Institute of Astronomy and Astrophysics, Academia Sinica, No.1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan, R.O.C. 














§ ABSTRACT 

When a star exhausts the hydrogen fuel in its core, the core contracts, and the envelope may expand to ∼ 100 times its original size. Consequently, stars with ≲ 8 solar masses become red giants, while those with ≳ 8 solar masses evolve into red supergiants. However, the physics driving this extreme expansion is unclear. This letter presents a new mechanism grounded in the instability of stellar envelopes to explain the expansion. If the gravitational energy released by core contraction can trigger the initial expansion to a critical radius where the envelope becomes unstable, the star then continues expanding until stabilizing, thereby becoming a red giant or supergiant. We demonstrate that the envelope instability can explain several elusive features in the Hertzsprung–Russell diagram, such as the Hertzsprung gap between the main sequence and the red giant branch, as well as the dichotomy of the supergiant populations in red and blue.




























§ INTRODUCTION
 

The existence of red giants (RGs) and red supergiants (RSGs) as a theoretical solution in stellar astrophysics is well established; however, the question of why stars turn red, becoming RG/RSGs, remains in debate. Classical theories often refer to the “mirror effect", which describes reciprocal motion resulting from the release of gravitational energy as the helium core contracts <cit.>. While the “mirror effect" provides an explanation to the expansion of post-main-sequence stars, it does not fully explain why stars transition to certain radii with a red color.

Various theories have been proposed to explain why a star becomes an RG/RSG, attributing the key role to factors such as degree of core contraction <cit.>, thermal imbalance driven by opacity <cit.>, gradients of the opacity and mean molecular weight <cit.>, and the pressure and density contrast between the shell and the core <cit.>. Some other authors suggested that no single factor plays a major role <cit.>. Despite these varied theories, there has been no consensus on what precisely drives stars to turn red.

The observed distribution of giants and supergiants on the Hertzsprung–Russell (HR) diagram presents intriguing puzzles. A notable observed phenomenon is the “Hertzsprung gap" <cit.>, a region with very low stellar density that bridges the main sequence and the red giant branch (RGB). This gap is conventionally attributed to the rapid evolution of stars towards the RGB. However, the precise reason why stars rapidly pass through this region, instead of lingering, remains elusive. Another notable phenomena is division of supergiants into two distinct categories: RSGs and blue supergiants (BSGs). The scarcity of yellow supergiants <cit.>, which would exhibit intermediate effective temperatures (T_ eff), results in a gap between BSGs and RSGs. To explain the dichotomy between RSGs and BSGs would require a deeper understanding of stellar physics.

A clue to understanding the physics of stellar evolution into RG/RSG is the concept of “critical metallicity" required for RSG formation, as suggested by recent stellar models <cit.>. According to these models, not all stars successfully evolve into RSGs during their core-helium (He) burning phase. Only stars with metallicities above a specific threshold can transition into RSGs; those below this threshold remain as BSGs with smaller radii. Thus, the stable state of a supergiant can be either red or blue, typically not yellow. The transition between these two phases is characterized by a discontinuous change in radii and T_ eff across different metallicities. This discontinuous transition between the RSG and BSG phases should be governed by a specific physical criterion.



§ METHODS
 

To investigate the physical origin of RSGs, we construct stellar models using the Modules for Experiments in Stellar Astrophysics <cit.> No.10108. MESA is a one-dimensional stellar evolution code that includes microphysics such as nuclear reaction networks, equations of state, and opacities. 

We develop stellar models with initial masses of 3-60 M_⊙. For stars with masses of ≤ 10  M_⊙, we employ settings from the test suite “1M_pre_ms_to_wd" to evolve the stars from pre-main-sequence stage to the white dwarf phase. For stars with masses of ≥ 11  M_⊙, we first calculate pre-main-sequence evolution to create zero-age-main-sequence (ZAMS) models, then continue to evolve these models from the ZAMS to iron core collapse. We adpot the default settings for massive stars from the “inlist_massive_defaults” file within the  module for simulations after the ZAMS. 



§ ENERGY SOURCES FOR EXPANSION


First, we compare two models of stars, each with an initial mass of 25 M_⊙; one with an absolute metallicity Z=5× 10^-3 and the other with Z=5× 10^-4. This comparison is shown in Figure <ref> (details about the rate of heat flow are described in Appendix <ref>). The star with higher metallicity successfully evolves into an RSG, whereas the star with lower metallicity only reaches the BSG stage during its core-He burning phase. Despite of different metallicities, the heat released from core contraction into the envelope is comparable for both models. The higher metallicity star that evolves into an RSG does not receive more energy from its core for envelope expansion. Therefore, the “mirror effect" alone is insufficient to explain the formation of RSGs. 





Following core contraction, stellar nuclear reactions mainly take place in the hydrogen (H) shell and He core. As shown in Figure <ref>, it is not the difference in the energy generation from these nuclear reactions that drive the RSG formation. The energy generation rate by either shell-H burning or core-He burning is not significantly higher in the stars that eventually become RSGs. Instead, these stars expand into RSGs by trapping more energy within their envelopes, at the expense of their surface luminosities. A sharp decrease in surface luminosity is a distinctive characteristic in the transition to an RSG, a feature that is absent in stars that evolve only into BSGs. It is this luminosity drop, rather than an increase in energy generation, that provides the energy required for the envelope to expand into an RSG. 

A straightforward explanation for this luminosity drop could be that increased opacity within the stellar envelope traps more energy from outgoing radiation, thereby driving the expansion towards the RSG phase. However, as shown in Figure <ref>, the stellar models progressing to RSGs do not always exhibit higher opacities. In fact, even models with significantly reduced opacities can still reach the RSG extent as long as the nuclear reaction rates are high enough. This indicates that factors beyond simple opacity variations are responsible for RSG formation.







§ ENVELOPE INSTABILITY



 §.§ Stellar radius and envelope instability


Our study suggests that the determining factor for a star's evolution into an RSG is its size prior to expansion, which is the product of evolution with various stellar parameters such as opacity and nuclear reaction rates during the main-sequence stage. To evolve into an RSG, a star must achieve a threshold size at the end of the main-sequence stage, known as the termial-age main sequence (TAMS). Figure <ref> demonstrates this concept with various stellar models, all with the initial mass of 25M_⊙ but exhibiting different radii at the TAMS. The details of these models are described in Appendix <ref>. For these 25M_⊙ stars, only those with radii larger than ∼12 solar radii (R_⊙) at the TAMS successfully become RSGs, while the others remain as BSGs. This threshold radius for RSG formation varies with different stellar masses. 

Only stars that surpass a certain radius threshold at the TAMS can eventually become RSGs. We suggest that this is largely due to a radius range of envelope instability, as depicted in Figure <ref>, which shows how surface pressure changes with a growing radius in the post-main-sequence stage. As the radius expands, the surface pressure of the star first decreases, then increases, and finally decreases again. A decrease in surface pressure with an expanding radius indicates stability, as the star naturally reverts to its initial radius after a small perturbation. In contrast, in a region where the surface pressure increases as the radius expands, even a small perturbation in radius can lead to further expansion, indicating instability. For 25M_⊙ stars, there exists an unstable radius range at ∼200–600R_⊙, with small variations influenced by metallicity. This instability due to the change in surface pressure as a function of radius is similar to the instabilities involved in the derivation of the Jeans length <cit.> and the Schönberg–Chandrasekhar limit <cit.>.





We suggest that the formation of RSGs is driven by the envelope instability. In Figure <ref>, the star icons represent the maximum radii that stars reach during their core-He burning stage. Notably, no star remains within the unstable radius range, leading to a clear division into two distinct groups: RSGs and BSGs.  For the stars with smaller TAMS radii due to factors such as low metallicity, the “mirror effect" leads to expansion only within the first stable radius range, so these stars remain as BSGs. In contrast, stars that are sufficiently large at the TAMS are expanded into the unstable radius range by the “mirror effect". Once the stars enter this unstable range, they cannot maintain their size but continue expanding until their radii reach the next stable zone. This continued expansion in the instability zone is what causes them to turn red. This mechanism explains why the RSG phase is distinctly separate from BSGs in stellar evolution.



 §.§ Formation of blue and red supergiants


The evolution of supergiants encompasses two distinct stages, each caused by different physical mechanisms: the transformation from the main sequence to BSGs, followed by the expansion from BSGs to RSGs. Generally, all massive stars are capable of expanding into BSGs through the “mirror effect". However, advancing beyond this stage, from BSG to RSG, requires more than just the “mirror effect". The key determinant for this second stage of expansion is the presence of envelope instability. Only if the star is large enough at the TAMS, larger than ∼12R_⊙ for 25M_⊙ stars, the “mirror effect" can expand the star into the unstable radius range, causing further expansion into the RSG phase. Conversely, the 25M_⊙ stars with radii at the TAMS smaller than ∼12R_⊙ only expand into the BSG phase and do not further transition into the RSG phase. 

The energy source for the expansion from the main sequence to BSGs is simply the gravitational energy released by He-core contraction. The further expansion from BSGs to RSGs is trigger by envelope instability, rather than by any additional energy source. Therefore, this expansion requires a sacrifice the surface luminosity, leading to a sharp luminosity drop, as shown in Figure <ref>.



 §.§ Envelope instability criteria for 3 to 60 M_⊙ stars


The existence of envelope instability is not exclusive to massive stars such as those with 25M_⊙; it is generally applicable for stars of 3 to 60 M_⊙. In Figure <ref>, we show the relationship between T_ eff and stellar radii for stars within this mass range. The envelope instability zone is delineated by identifying the two turning points on the curve of surface pressure versus radius for each star, similar to the curves in Figure <ref>. These turning points define the lower and upper limits of the unstable radius range of each star. This range varies continuously with stellar mass, with lower-mass stars exhibiting instability at smaller radii. As a result, the radii after expansion are smaller for lower-mass stars compared to more massive stars. Nevertheless, T_ eff at these limits, ∼ 8000 K for the lower limits and ∼ 4500 K for the upper limits, are roughly consistent across different stellar masses, as shown in the boundary of the instability zone of Figure <ref>. 



Our results suggest that the formation of RGs with initial masses larger than ∼ 3M_⊙ and RSGs originate from the same physical mechanism of envelope instability. However, for RGs with initial masses less than ∼ 2M_⊙, we cannot identify instability zones or corresponding luminosity drops during their expansion phases. The formation of these RGs is probably associated with helium flashes rather than envelope instability, and therefore, they are beyond the scope of this study. 










§ THE ORIGIN OF HERTZSPRUNG GAP ON THE HR DIAGRAM


The envelope instability can explain some notable observational phenomena on the HR diagram. In Figure <ref>, we have converted the luminosities and T_ eff of theoretical stellar evolutionary tracks into observable magnitudes and color indices using the bolometric correction tables from the MESA Isochrones and Stellar Tracks (MIST) project <cit.>. This conversion allows us to directly map the envelope instability zone along with the observed stellar population in the Large Magellanic Cloud (LMC) on the HR diagram. Since our stellar models cover a wide range of stellar masses, we separately compare the instability zones for RGs from 3 to 8M_⊙ stars with the Magellanic Clouds Photometric Survey <cit.> data, and for RSGs from 10 to 60M_⊙ stars with Gaia Data Release 2 <cit.>. The details of the method for these color-magnitude diagrams are described in Appendix <ref>.

For our investigation of RGs, we selected a circular region with a radius of 0.5 degrees in the northwest arm of the LMC. We obtained the B and V band magnitudes of stars in this region from the MCPS. The color-magnitude diagram of these stars, presented in the top panel of Figure <ref>, shows the main sequence, the RGB, and the Hertzsprung gap between them. Notably, the envelope instability zone that we have identified matches well with the Hertzsprung gap. This result supports the existence of envelope instability zone and explains the physical origin behind the Hertzsprung gap between the main sequence and the RGB; because the limited number of stars within the Hertzsprung gap can be explained by the instability zone, which leads to a short transit phase of ∼ 10^5 years that stars swiftly evolve through.  


For evolved massive stars, we compare the theoretical instability zone with the Gaia data of the LMC bright stars, using the catalog compiled by ref. <cit.>. In the color-magnitude diagram shown in the bottom panel of Figure <ref>, these stars are distributed into one group of RSGs and another group of BSGs, separated by a sporadic distribution of YSGs between them. The population counts for each type of evolved stars are as follows: 2974 RSGs, 508 YSGs, and 4786 BSGs <cit.>. By overlaying the envelope instability zone on this color-magnitude diagram, we observe that this zone matches well with the sporadic distribution of YSGs. This result suggests that the instability zone clearly separate the populations between RSGs and BSGs in the color-magnitude diagram. We infer that supergiants predominantly appear as either blue or red because these colors correspond to stable zones, where stars stay for most of their time. In contrast, the YSGs are relatively rare because they are located within the instability zone. Massive stars rapidly evolve from BSGs to RSGs, passing through the YSG phase in a short period of ∼ 10^4 years. 


Furthermore, it is intriguing to note that the envelope instability zone coincides with the well-known instability strip in the HR diagram. For example, many observed YSGs are variable stars <cit.>. This overlap suggests a possible link between the envelope instability and the mechanisms driving stellar pulsation, which remains an area for further research.



§ CONCLUSIONS


In conclusion, we suggest that the instability of stellar envelope is key for a star to turn red. As the moderate expansion caused by the “mirror effect" propels a star into the unstable radius range, it naturally continues expanding until it reaches a stable radius, thus becoming an RG/RSG. The envelope instability zone on the HR diagram provides a self-consistent description of  the observed Hertzsprung gap and the bifurcation of supergiants into red and blue categories. 










This research is supported by the National Science and Technology Council, Taiwan under grant number MOST 110-2112-M-001-068-MY3 and by the Academia Sinica, Taiwan under the career development award AS-CDA-111-M04.  Our simulations were performed at the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231, and on the TIARA Cluster at the Academia Sinica Institute of Astronomy and Astrophysics (ASIAA).





















MESA <cit.>
          














§ CALCULATION OF THE RATE OF HEAT FLOW

From the MESA output, we obtain the ϵ_ grav term, which is the negative value of rate of heat flow per unit mass, expressed as

    ϵ_grav≡ -T ∂ s/∂ t,

where T is the temperature, s is the specific entropy, and t is the time. The ϵ_ grav term is usually used to diagnose stellar expansion (when ϵ_ grav<0) and contraction (when ϵ_ grav>0). To quantify the overall change in heat, we integrate the ϵ_ grav term across either the core or the envelope:

    -Q̇≡∫_M_1^M_2ϵ_grav dm,

where m is the mass coordinates, and M_1 and M_2 define the lower and upper limits of the mass range for calculation. Note that the physical meaning of ϵ_grav is the negative rate of heat flow per unit mass, rather than directly representing the rate of gravitational energy release <cit.>.




§ MODELS OF VARIOUS OPACITIES AND NUCLEAR REACTION RATES

As shown in Figure <ref>, we investigate how R_ TAMS influence supergiant size by developing a grid of MESA models for 25 M_⊙ and Z=0.001 stars with various opacities and CNO cycle rates. The details of these models are as follows.


The opacity in our models is adjusted by using different ζ_opacity, referred to as Z_ base in the MESA code, which is the input metallicity value used in the opacity table, not necessarily identical to the stellar metallicity used for evolution. For example, if we set ζ_opacity=0.02 for a star with Z=0.001, its opacity used for the radiation transport calculation is equivalent to a star of Z=0.02, but its metallicity for stellar evolution remains Z=0.001. The adjustment of ζ_opacity is intended only to examine the opacity effect on envelope expansion, instead of producing realistic stellar models. Generally, a higher ζ_opacity results in increased opacity.

The reaction rates of the CNO cycle are changed by applying a linear scaling factor η_ CNO. We change the rates for the five main reactions of the CNO cycle:

(1) ^12 C+^1 H→ ^13 N+γ, 

(2) ^14 N+^1 H→ ^15 O+γ, 

(3) ^16 O+^1 H→ ^17 F+γ, 

(4) ^15 N+^1 H→ ^16 O+γ, and 

(5) ^15 O+^1 H→ ^12 C+^4 He.

Changing the CNO reaction rates is for experiments on stellar evolution, rather than to construct realistic models.

From each models in this grid, which varies in opacity and nuclear reaction rates, we calculate the two radii: (1) R_ TAMS and (2) the maximum radius during the core-He burning stage (R_ max), representing the supergiant size. The relationship between R_ TAMS and R_ max is shown in Figure <ref>, which highlights the threshold radius required for RSG formation.




§ METHOD FOR THE COLOR-MAGNITUDE DIAGRAMS


We select the LMC, a nearby galaxy with low extinction, as a laboratory to compare observations with the theoretical envelope instability zone. We use two different observational dataset to create color-magnitude diagrams: one for massive stars (ranging from ∼ 10-60 M_⊙) and another for lower-mass stars (ranging from ∼ 3-8 M_⊙).

For the study of lower-mass stars, we use the Johnson B and V magnitudes of LMC stars from the Magellanic Clouds Photometric Survey <cit.>. This survey was conducted using the Las Campanas Swope Telescope (1 m) and the Great Circle
Camera, covering the central 8.5 by 7.5 degrees of the LMC.  We select a circular region with a radius of 0.5 degrees centered at RA=05:00:00, Dec=-66:00:00, located in the northwest arm of the LMC. We extract the MCPS data for stars within this region to create a color-magnitude diagram using the B magnitudes and B-V color indices. 

To investigate massive evolved stars, we utilize the multiwavelength catalog of bright sources in the LMC compiled by <cit.>. This catalog contains 197,004 sources and was constructed by crossmatching and deblending between the source list of Spitzer Enhanced Imaging Product and Gaia Data Release 2 <cit.>. Gaia astrometry is used to remove the foreground contamination, ensuring that the data specifically represents LMC stars. From this catalog, we extract the Gaia G band magnitude and Gaia BP-RP color to create a color-magnitude diagram.

To compare the theoretical stellar evolutionary tracks with observational data, we use the bolometric correction values to convert luminosities to magnitudes, and T_ eff to color indices for the stellar models of the LMC metallicity (Z=0.007). The bolometric correction tables are from the MESA Isochrones and Stellar Tracks (MIST) project <cit.>. We apply these bolometric corrections for each timestep of a theoretical evolutionary track through tri-linear interpolation of the values in the tables based on T_ eff, gravitational acceleration log g, and the LMC's average extinction value A_V=0.3 <cit.>. The theoretical quantities are converted into Johnson B, Johnson V, Gaia G, Gaia BP, and Gaia RP band magnitudes. The distance modulus of 18.5 is applied to transform the absolute magnitudes to apparent magnitudes for LMC objects. These conversion allows us to overlay the evolutionary tracks and the boundaries of instability zones on color-magnitude diagrams. In Figure <ref>, we mark the instability zones calculated from stellar models with the LMC metallicity on the color-magnitude diagrams.













aasjournal













 Paper Title: Investigating the crust of neutron stars with neural-network quantum states.




Lifelong Person Search
    Jae-Won Yang, Seungbin Hong, and Jae-Young Sim, Member, IEEE

    August 5, 2024
=================================================================





empty



§ INTRODUCTION

Low-density nuclear matter found in the crust of neutron stars exhibits a complex and fascinating structure that varies significantly with density. In the outer crust, nucleons are bound to fully ionized nuclei. As density increases within this region, these nuclei become increasingly neutron-rich, and consequently, terrestrial experiments can only directly determine the dominant nuclei species that are present at lower densities. When the neutron drip density is reached, about 2×10^-4 fm^-3, the neutron chemical potential becomes too large for nuclei to bind additional neutrons. This leads to the formation of an unbound neutron component, marking the boundary between the outer and inner crust of the star. As the density increases, the proportion of unbound neutrons grows significantly, and at the highest densities, the nuclei undergo phase transitions into various “pasta” phases where they deform into non-spherical shapes. Eventually, upon reaching nuclear saturation density, these nuclei coalesce, transitioning into a uniform distribution of nucleons <cit.>.


Modeling inner crust material poses significant challenges due to the inherent complexity of clustering phenomena, the emergence of superfluidity, and the existence of both free neutrons and neutron-rich nuclei. Various phenomenological approaches have been developed to address some of these challenges, including the compressible liquid-drop model <cit.> and self-consistent mean-field models <cit.>. In the former, the energy is parameterized as a function of global properties, and the nucleons within clusters are treated separately from the free neutrons, all of which are assumed to be uniformly distributed. While this treatment has a low computational cost, it neglects quantum mechanical shell effects, which are critical for determining the equilibrium composition of the crust. In contrast, self-consistent mean-field models are fully quantum mechanical, constructing the many-body state in terms of single-particle or quasi-particle wave functions. However, since they neglect correlations beyond the mean-field approximation, they have limited applicability for low-density nuclear matter, where strong correlations can lead to significant deviations from the mean-field behavior.

Ab initio methods offer a more fundamental approach, as they solve atomic nuclei and neutron-star matter starting from the individual interactions among nucleons <cit.>. Most of them, however, suffer from significant limitations when modeling the transition from neutron-rich nuclei to uniform matter <cit.>. Their applicability is presently limited to the uniform case—a notable exception in this regard is the lattice effective field theory approach <cit.>. Even continuum diffusion Monte Carlo methods, which are well known for their accuracy across different physical systems, are limited to the uniform phase of neutron-star matter because of the fermion-sign problem <cit.>. 

We overcome the limitation of diffusion Monte Carlo approaches by leveraging a variational Monte Carlo method based on neural-network quantum states (NQS) <cit.>. Owing to their flexibility, NQS enable an accurate description of the ground-state of quantum many-body systems. This feature is particularly relevant for modeling the different phases of matter comprising the crust of neutron stars, which is inaccessible by conventional ansätze. This approach also circumvents the fermion-sign problem, as Monte Carlo techniques are only employed to sample the modulus squared of the wave function, which is positive even for systems of fermions. 

In a recent application <cit.>, NQS based on the hidden-nucleon architecture have demonstrated success in modeling the normal and superfluid phases of dilute neutron matter within a unified framework. A more compact and flexible neural Pfaffian-Jastrow architecture, featuring a message-passing neural network to efficiently capture pairing and backflow correlations, was shown to outperform state-of-the-art diffusion Monte Carlo methods in modeling ground-state properties of ultracold Fermi gases <cit.>. Chiefly, the neural Pfaffian-Jastrow approach is designed to capture strong pairing effects, which characterize both cold atoms and the matter found in the crust of neutron stars. 

In this work, we put forward a generalization of the neural Pfaffian-Jastrow ansatz and adapt it to nuclear potentials, which strongly depend on the spin-isospin degrees of freedom of the interacting particles. We provide evidence that this ansatz is capable of modeling the different phases matter found in the crust of neutron stars, including the self-emergence of nuclear clusters and their dissolution. In particular, we find dramatically lower energies than those obtained with state-of-the-art auxiliary-field diffusion Monte method in the low-density regime. In addition to ground-state energies, we compute two-body distribution functions and the proton fraction of beta-equilibrated matter. Our findings underscore the viability of utilizing neural Pfaffians to achieve an ab initio description of the crust of neutron stars from the nuclear physics perspective. 




§ RESULTS




 §.§ Simulation setup


We model infinite nuclear matter using periodic boundary conditions and N=28 nucleons in a cubic simulation cell with side length L, where L is determined by the baryon density n_B=N/L^3. 
For each density investigated, we run three separate simulations with different proton fractions x ≡ n_p/n_B = N_p/N, where n_p is the proton density and N_p is the number of protons in the simulation. Specifically, we run simulations with N_p = 0, 4, and 14 at each density, allowing us to study pure neutron matter (PNM), isospin-symmetric nuclear matter (SNM), and an intermediate isospin-asymmetric case. The simulated nucleons are confined within the cell, with potential contributions from image particles considered up to a distance of 3L/2. Each nucleon has a continuous three-dimensional position vector and discrete spin and isospin projections on the z-axis. As common in quantum Monte Carlo calculations of nuclear systems <cit.>, we make the approximation that protons and neutrons have equal masses. 

The dynamics of neutrons and protons is modeled according to a non-relativistic Hamiltonian based on a leading-order pionless effective field theory expansion that contains two- and three-nucleon interactions. This “essential” Hamiltonian has been shown to reproduce properties of various nuclei across the nuclear chart, including ground-state energies, radii, and magnetic moments, within few-percent from their experimental values <cit.>. It also reproduces well the equation of state (EOS) of dilute neutron matter, as the predicted values agree with highly realistic phenomenological and chiral-EFT interactions <cit.>. 
Additionally, we assume the repulsive Coulomb interactions among finite-size protons to be completely screened by the electrons present in the system. 




 §.§ Energy per particle




We begin our analysis by comparing the energy per particle of SNM obtained from NQS with that obtained from state-of-the-art auxiliary-field diffusion Monte Carlo (AFDMC), as shown in Figure <ref>. Details on the AFDMC calculations are provided in the Auxiliary-field diffusion Monte Carlo subsection of the Methods section. For all simulated densities, n_B = 0.001 fm^-3, 0.01 fm^-3, 0.04 fm^-3, and 0.08 fm^-3, the NQS energies are lower than those predicted by AFDMC, especially at the lowest densities. Additionally, the curves exhibit distinctly different characters, with the NQS energies reaching a plateau at lower densities, while AFDMC energies continue to rise. This behavior highlights an important limitation of the AFDMC calculations, as they fail to capture the formation of nuclear clusters. This is ultimately due to the fermion sign problem, which is exacerbated by using a variational ansatz that is designed for the uniform phase. 
Therefore, the zero-density limit of the AFDMC results corresponds to an infinitely dilute gas of protons and neutrons, whose ground-state energy vanishes. As an additional point of reference for the zero-density ground state in the fully clustered system, we present the NQS results for ^28Si with no electromagnetic contribution to the potential, to be consistent with the SNM case. 
The similarity between the NQS energies of ^28Si and SNM at n_B=0.001 fm^-3 suggests that all nucleons in low-density SNM coalesce to form ^28Si. We stress that ^28Si self-emerges as the minimum of the quantum-mechanical energy of the system and is not a priori encoded in the Pfaffian ansatz. 

In the outer crust of neutron stars, nucleons are bound to fully ionized nuclei. At higher densities within the outer crust, these nuclei become increasingly neutron-rich. 
While terrestrial experiments can determine the dominant nuclei species at lower densities, at higher densities these nuclei are too neutron-rich to be stable and cannot be directly measured. Nonetheless, the dominant nuclei species in this regime can be reliably inferred through extrapolations from experiments using semiempirical nuclear mass formulae<cit.>.

When the neutron drip density is reached, about 2×10^-4 fm^-3, the neutron chemical potential becomes too large for nuclei to bind additional neutrons, leading to the formation of an unbound neutron component. This marks the boundary between the outer and inner crusts of the star. 
As the density increases, the proportion of unbound neutrons grows significantly, and at the highest densities, the nuclei undergo phase transitions into various “pasta” phases where they deform into non-spherical shapes. 
Modeling matter at these densities presents several challenges. Not only are the extremely neutron-rich nuclei impossible to measure directly, but they also coexist with a sea of unbound neutrons. A commonly used approach at this density is the compressible liquid drop model (CLDM). However, being a classical model, CLDM does not account for the shell effects within nuclei<cit.>. 

Check my understanding, they only mention this for outer crust material but CLDM seems obviously inferior there. However it seems to be commonly used for the inner crust and I would imagine the shell effects still matter... Probably need a more relevant citation too. The end of the inner crust and beginning of the core of the star is marked by the transition to uniform matter from the full dissociation of the nuclei, around half nuclear saturation density.

mention superfluid. mention core modeling is different and its difficult to have a consistent model over the crust-core boundary<cit.>, maybe mention this method can do that but we don't here? Even more thorough description of inner crust densities as that's what we have, especially as it relates to the plots I show.








Figure <ref> shows the ground-state energy of nuclear matter as a function of the baryon density n_B and proton fraction x. To generate this surface, the NQS energies for SNM and PNM, E(n_B, 1/2) and E(n_B, 0) respectively, were fitted to the following function based on a cluster expansion of the energy<cit.>,

    E(n_B, x) = a_0 
        + a_2/3( n_B/n_0)^2/3
        + a_1 ( n_B/n_0)
        + a_2 ( n_B/n_0)^2.

Here, n_0=0.16 fm^-3 represents the nuclear saturation density and the constant a_0 was fixed at zero for PNM. These fits enabled the computation of the symmetry energy S(n_B) as a function of baryon density, defined as the energy difference between PNM and SNM, 

    S(n_B)=E(n_B, 0)-E(n_B, 1/2).

The symmetry energy is an important quantity for characterizing nuclear matter, as it directly affects its neutron richness.
Finally, the energy surface between the two extremes of x=0 and x=1/2 was approximated through a well-known expansion in neutron richness, 1-2x, around symmetric matter<cit.>.
The energy computed from this expansion is given by

    E(n_B,x)=E(n_B, 1/2)+(1-2x)^2 S(n_B),

where we have already assumed that higher-order terms (which would come in as a fourth power of neutron richness) are negligible, hence the quadratic coefficient is simply the symmetry energy S(n_B) from Eq. (<ref>). Note that other expansions have been proposed <cit.>, and found to be more accurate in the low-density regime. We plan to employ them in future NQS calculations of isospin-asymmetric matter, as controlling finite size effects at small x requires using more than N=28 nucleons — see the Appendix on Finite-size effects for additional details. 

From Figure <ref>, we observe that the energy of nuclear matter decreases as the proton fraction increases, regardless of density. As expected, the change in energy from x=0 to x=1/2 is more pronounced in the high-density region compared to the low-density region. However, the onset of nuclear clustering makes the energy per particle of SNM at low-density negative by about -10 MeV per particle, as shown in Figure <ref>. Thus,
even at very low densities, the curvature of the energy per particle as a function of x is positive.
Finite-size effects are discussed in more detail in the Finite-size effect corrections subsection of the Methods section. 





 §.§ Beta-equilibrated nuclear matter


Cold nuclear matter at densities found in the inner crust of neutron stars is mostly composed of neutrons, protons, and electrons. Assuming the matter is neutrino transparent, the relative abundance of these particles at a specific density can be determined using a model for the energy as a function of proton fraction, under the well-supported assumptions of charge neutrality, n_p = n_e, and beta equilibrium, μ_e=μ_n-μ_p <cit.>. We will use this model of beta-equilibrated nuclear matter and our simulation results for SNM and PNM to construct an equation of state for our Hamiltonian. 



The proton fraction in beta-equilibrated matter as a function of baryon density is displayed in Figure <ref>. The NQS predictions align closely with the phenomenological Skryme parameterizations of the equation of state (Sly4 EOS) for the inner crust of neutron stars <cit.>. Figure <ref> underscores the flexibility of the NQS: by leveraging identical neural network architectures across all densities, we successfully capture the complexity of nuclear structures relevant to the inner crust of neutron stars. In contrast, the AFDMC yields proton fractions that tend towards those of non-interacting matter. This behavior is related to the inability of AFDMC to capture the onset of nuclear clusters at low density, making the presence of protons less energetically favorable compared to more accurate NQS calculations.

Due to the self-formation of nuclei in the low-density region, the NQS energy per particle of SNM obtained for N=28 would likely be even lower if we could carry out NQS calculations with N=40 particles or more. For example, the experimental energy per particle of ^28Si is 8.45 MeV, while that of ^40Ca is 8.55 MeV. These differences would be even larger if Coulomb interactions were turned off. 
On the other hand, the PNM simulations, once corrected for finite-size effects arising from the kinetic energy<cit.>, are expected to provide results close to the infinite case <cit.>. Thus, finite-size effects might underestimate the symmetry energy, yielding too small a proton fraction.




 §.§ Clustering

The Pfaffian NQS ansatz allows us to capture the self-emergence of nuclei and nuclear clusters from first principles, when their formation corresponds to a lower minimum of the variational energy than the uniform case. To better quantify the impact of clustering, we compute the two-body pair distribution functions, which provide the probability of finding two nucleons with defined isospin at a given distance from each other. The analytical expressions for the distributions are given by

    g_α(r) = 1/2π r^2 n_B N∑_i < j^N ⟨Ψ | δ (r_ij-r) P_α| Ψ⟩/⟨Ψ | Ψ⟩,

where P_α denotes the projector onto either neutron-neutron (nn), proton-neutron (pn), or proton-proton (pp) pairs. 





Instead of showing g_α(r) for SNM, we find it more informative to focus on an isospin-asymmetric case, specifically x = 4/28, as it yields simpler scenarios. The first possibility is the uniform case, in which the distributions should resemble those of the free Fermi Gas (FG), as the input Hamiltonian is relatively soft. A second possibility is the formation of four deuterons, which would yield a peak in the pn distribution but not in the pp one, as the small clusters would float in the simulation box. Finally, if at least one ^4He cluster is formed, the pp distribution should exhibit a peak at short distances. The two-body distributions displayed in Figure <ref> indeed seem to favor the latter case. In fact, compared to the non-interacting NQS case, obtained turning off the two- and three-body potentials in the input Hamiltonian, the distributions show a significant enhancement of pairs at short distances across all channels. For comparison, Figure <ref> also displays the FG distributions in the thermodynamic limit, whose expressions can be derived analytically. 

Unfortunately, the analysis of two-body distribution functions alone does not reveal whether heavier nuclei are present, including ^8Be or more neutron-rich isotopes. To overcome this limitation, we show a three-dimensional snapshot of Monte Carlo configurations for n_B =0.01 fm^-3 and x = 4/28 in the left panel of Figure <ref>. Clearly, the four protons (represented by solid orange circles) are always close to each other, indicating the presence of ^8Be. While at least four neutrons (represented by solid blue circles) are consistently close to the protons, there is also evidence of a free neutron component. This behavior is a microscopic indication of the neutron-drip, which is expected to occur in neutron-star matter at these densities. We note that these features are not specific to the Monte Carlo configurations displayed in Figure <ref>; rather, analogous structures are evident in all the samples we examined. 


In the right panel of Figure <ref>, we visualize Monte Carlo configurations for SNM at n_B = 0.01 fm^-3. Consistent with the energy per particle plotted in Figure <ref>, the 14 protons and 14 neutrons coalesce to form ^28Si. As a critical improvement compared to the state-of-the-art AFDMC method, the NQS effectively accommodates the self-emergence formation of a medium-mass nucleus in dilute isospin-symmetric matter. In addition, since the NQS does not require the a priori specification of free neutrons, it provides a more realistic and unbiased representation of the physical system as determined by our Hamiltonian.








The summation portion of this calculation is accomplished in our Monte Carlo simulation by a histogram over pair distances. This requires that the spherical shell area is replaced by the volume of a thin shell with thickness corresponding to the histogram bin size, 4π r^2 → (4/3)π (r_o^3-r_i^3).






§ DISCUSSION

In this work, we introduce a highly flexible Pfaffian NQS that leverages backflow transformations through a message-passing neural network to solve the nuclear quantum many-body Schrödinger equation with high accuracy. Notably, fermion anti-symmetry, parity, and translation invariance are respected by construction. As a crucial improvement over conventional variational wave functions used in state-of-the-art AFDMC calculations, a single Pfaffian ansatz is able to capture the different phases of matter across a wide range of densities corresponding to the crust of neutron stars. These phases, including extremely neutron-rich nuclei surrounded by superfluid neutrons and a uniform gas of protons and neutrons, are extremely difficult to model within conventional nuclear quantum many-body approaches, with the exception of lattice effective field theory <cit.>.

We demonstrate how the formation of nuclei dramatically decreases the energy of SNM in the low-density region, thereby lowering the value of the symmetry energy. This lowering has a tangible consequence on the proton fraction in beta-equilibrated, neutrino-transparent matter. The Pfaffian NQS yields a proton fraction that agrees well with the one computed using the phenomenological SLy4 functional. The predicted fraction is considerably larger than that obtained from AFDMC calculations in the low-density regime, while it agrees with this method in the uniform case, corresponding to n_B = 0.08 fm^-3 and beyond.

To probe the formation of nuclei heavier than the deuteron, we computed isospin-dependent two-body distribution functions at n_B = 0.04 fm^-3, simulating 4 protons and 24 neutrons with periodic-box boundary conditions. The appearance of a peak at short distances in all channels is a clear indication of the presence of nuclei heavier than the deuteron, which is corroborated by Monte Carlo samplings of the many-body wave functions at n_B = 0.01 fm^-3. The latter indicate the formation of neutron-rich Beryllium isotopes, surrounded by free neutrons. The flexibility of the ansatz is such that when simulating SNM at n_B = 0.01 fm^-3, the 14 protons and 14 neutrons self-organize to form ^28Si. 

In future calculations, we will leverage leadership-class GPU clusters to simulate a larger number of particles. Reducing finite-size effects will allow us to accurately simulate systems with proton fractions between SNM and PNM, which currently suffer from large systematic uncertainties. Concurrently, we plan to include screened Coulomb repulsion between protons, a fundamental component for investigating signatures of the “pasta” phase. A first approach in this direction involves treating the Coulomb lattice of nuclei using the Wigner-Seitz approximation <cit.>.

Finally, we are incorporating high-resolution Hamiltonians, both phenomenological and derived from chiral effective field theory, into our variational Monte Carlo method based on NQS. This step is crucial for achieving a unified description of neutron-star matter, from the crust to the inner core.



§ METHOD




 §.§ Neural-network quantum states

NQS have proven effective in solving the continuous-space Schrödinger equation for nuclear many-body systems, ranging from the deuteron to ^20Ne <cit.>. NQS based on the hidden-nucleon architecture have also successfully modeled both the normal and superfluid phases of dilute neutron matter using a single unified ansatz, demonstrating their flexibility in capturing a wide range of quantum many-body phenomena from first principles. Additionally, a more compact and flexible neural Pfaffian-Jastrow (PJ) architecture, which utilizes a message-passing neural network (MPNN) to efficiently encode pairing and backflow correlations, has been shown to outperform state-of-the-art diffusion Monte Carlo methods in modeling ultracold Fermi gases <cit.>. This ansatz is particularly effective at capturing strong pairing effects relevant to both cold atoms and neutron star crust matter.

In this work, we build upon the PJ+MPNN ansatz introduced in Ref. <cit.>,

    Ψ_PJ(X) = 
        e^J(X)× pf[           0 ϕ(x_1, x_2)           ⋯ ϕ(x_1, x_N); ϕ(x_2, x_1)           0           ⋯ ϕ(x_2, x_N);           ⋮           ⋮           ⋱           ⋮; ϕ(x_N, x_1) ϕ(x_N, x_2)           ⋯           0;             ].

To ensure skew-symmetry, the pairing orbital ϕ(x_i, x_j) is defined as 

    ϕ(x_i, x_j) 
    = ν(x_i, x_j) 
    - ν(x_j, x_i),

where ν(x_i, x_j) is a dense feedforward neural network (FNN). 
Consistent with Ref. <cit.>, we enforce translational invariance and parity in the ansatz. However, to encompass nuclear potentials, the MPNN is modified to take single-nucleon isospin coordinates through the one-body stream and isospin coordinates for each pair of nucleons through the two-body stream. Additionally, the pairwise spin inputs are modified to use the individual spins of each pair instead of their spin products.

We make a notable improvement to the PJ+MPNN ansatz by allowing the Pfaffian and Jastrow correlator to be complex, while keeping the MPNN real-valued. Generalizing the architecture proposed in Ref. <cit.> for spin systems, we make the Pfaffian complex-valued by feeding the outputs of the MPNN into two separate, real-valued FNNs, which represent the phase and amplitude of ν(x_i, x_j). Similarly, the MPNN outputs are fed into two real-valued, permutation-invariant Deep Sets <cit.>, representing the phase and amplitude of the Jastrow correlator J(X). 


To help mitigate runaway instabilities <cit.>, the logarithms of the real-valued amplitudes of all the neural networks, generally denoted as x, are regularized as

    x ↦ a tanh(x/a),

where a > 0 is a constant that sets the minimum and maximum allowed values of x. For all of our simulations, a was fixed to 8. All the individual FNNs in our ansatz have two hidden layers, each with a width of 16 nodes. The output dimension of the FNNs is either 1 or 16 nodes, depending on whether the FNN maps to a latent space or the output space. The activation function used throughout this ansatz is hyperbolic tangent and the pooling operation is logsumexp. In total, this NQS consists of 10,772 trainable parameters.



 §.§ Optimization

The optimization of the NQS is performed using a version of the stochastic reconfiguration (SR) algorithm, enhanced with RMSProp-like regularization <cit.>. For a complex wave function that depends on real parameters, the SR algorithm is modified such that only the real parts of the energy gradient g≡∇_pE and the quantum Fisher information matrix S are used to update the parameters p <cit.>, 

    p_t = p_t-1 - η(S_t-1)^-1(g_t-1).

Here, η is the learning rate and t is the optimization step. The RMSProp-inspired gradient accumulation is computed as

    v_t = βv_t-1 + (1-β) (g_t-1)^2,

with β = 0.99. Then, the quantum Fisher information matrix is regularized as

    S_t ↦ S_t + ε diag(√(v_t) + 10^-8),

where the small constant ε > 0 is typically set to 10^-3. 

All simulations are executed in parallel across 8 NVIDIA A100 GPUs, with a total of 16,000 configuration samples from 800 independent chains. Each optimization step, including Metropolis sampling and observable computation, takes approximately 80 seconds, with convergence generally requiring between 1,000 and 3,000 iterations. For the lowest baryon density examined, n_B = 0.001 fm^-3, transfer learning was employed to improve convergence and stability. Unlike the other densities, which were initialized with random parameters, the optimization for this density utilized parameters previously trained for n_B = 0.01 fm^-3 with the same number of protons and neutrons, significantly reducing the convergence time. 



 §.§ Finite-size effects

To mitigate finite-size effects, we adjust the NQS energies for a given n_B and x obtained with N=28 nucleons by subtracting the free kinetic energy of the finite system and adding the energy of the free FG in the thermodynamic limit <cit.>. These corrections, listed in Table <ref>, apply to the energies shown in Figure <ref>, Figure <ref>, and Table <ref>, as well as those used for calculating the proton fractions in Figure <ref>. 



It should be noted, however, that the above procedure is a rather crude one, as finite-size effects are not limited to the kinetic energy expectation value alone <cit.>. Furthermore, leveraging the free FG in the thermodynamic limits is only applicable when clustering is not as prominent as we found. Illuminating in this regard are simulations with low but nonzero proton fractions, as the number of simulated protons may be insufficient to form clusters large enough to significantly lower the energy. To better understand this effect, consider the hypothetical formation of stable nuclei in a sea of free neutrons at rest for a proton fraction of x=4/28 in the zero-density limit. Using experimental data for ground-state energies, we can infer that the binding energy per particle for the formation of a ^8Be cluster is 2.02 MeV, which increases to 2.28 MeV for ^16O, and further increases to 2.44 MeV for ^40Ca. This effect becomes even more amplified if the formed nucleus is neutron-rich. For instance, the binding energy is 2.45 MeV for a ^12Be cluster, 3.02 MeV for an ^24O cluster, and 3.21 MeV for a ^56Ca cluster. Therefore, while we can extract useful information to investigate clustering by simulating of N_p=4 protons, it is more reliable to employ Eqs. (<ref>) and (<ref>) to interpolate the NQS energies between PNM and SNM.




 §.§ Beta-equilibrated matter solution

To model neutrino-transparent nuclear matter in beta equilibrium, 
we apply the charge neutrality condition, n_p = n_e, and the beta equilibrium condition, μ_e = μ̂, where μ̂≡μ_n - μ_p represents the difference between neutron and proton chemical potentials. Using the well-known expansion of the energy per particle in neutron richness from Eq. (<ref>), the difference in chemical potentials can expressed in terms of the symmetry energy as

    μ̂ = - ∂ E(n_B, x)/∂ x = 4 (1-2x) S(n_B).

Next, by calculating the relativistic electron density at zero temperature and applying the charge neutrality condition, we obtain

    x≡n_p/n_B = n_e/n_B = (μ_e^2-m_e^2)^3/2/3π^2 n_B.

Finally, the beta equilibrium condition couples Equations (<ref>) and (<ref>), enabling the numerical solution for the proton fraction x. 




 §.§ Auxiliary-field diffusion Monte Carlo



The AFDMC solves the ground-state component of a given Hamiltonian by performing an imaginary-time propagation upon a starting variational ansatz <cit.>

    |Ψ_0⟩ = lim_τ→∞ |Ψ(τ)⟩ = lim_τ→∞ e^-(H-E_V)τ |Ψ_V⟩  .

In the above equation, E_V is a normalization constant, which is chosen to be close to the true ground-state energy E_0. The variational ansatz is assumed to be of the Slater-Jastrow form: |Ψ_V⟩ = F̂ |Φ⟩. To keep a polynomial cost in the number of nucleons, only linear terms in the spin-isospin correlation operators F̂ are included <cit.>. The optimal values of the variational parameters defining the correlation operator are found by employing the same stochastic reconfiguration method used to optimize NQS <cit.>. 



The mean-field part | Φ⟩ of the starting variational ansatz is the ground state of the uncorrelated Fermi gas, built from a Slater determinant of plane waves. The infinite system is simulated with N nucleons in a cubic periodic box of volume L^3. The momentum vectors in this box are discretized as 

    𝐤 = 2 π/L{ n_x, n_y, n_z}  ,

where n_x, n_y, and n_z are integer numbers. 
In this work, we carry out AFDMC calculations of SNM, simulating 14 protons and 14 neutrons as these “magic” numbers fill the discretized momentum shells for each species. AFDMC calculations of PNM with 28 particles in box would require either employing multiple Slater determinants in |Φ⟩ or twist–averaged boundary conditions <cit.>, which goes beyond the scope of this work. On the other hand, there is excellent agreement between NQS and AFDMC calculations of the energy per particle of PNM carried out with N=14 particles within a periodic box <cit.>. Thus, to obtain the symmetry energy results labeled as “AFDMC” in Fig. <ref>, we use the PNM energies obtained with NQS. 



 §.§ Data

The ground-state energies of obtained from NQS simulations at different n_B and x are listed in Table <ref>. For the isospin-symmetric case, we also report the corresponding AFDMC values. 



§ ACKNOWLEDGMENTS

We would like to thank Omar Benhar and Anthea Fantina for illuminating discussions. AL and BF are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contracts DE-AC02-06CH11357, by the 2020 DOE Early Career Award program, and by the NUCLEI SciDAC program. JK is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under grant No. DE-SC0021152. MHJ is supported by the U.S. National Science Foundation Grants No. PHY-1404159 and PHY-2013047.



§ AUTHOR CONTRIBUTIONS

B. F., J. K., and A. L. developed the code, designed the Pfaffian-Jastrow ansatz. B. F. and J. K. performed the numerical simulations. A. L. and M. H. J. provided overall supervision and guidance throughout the project. All authors contributed significantly to writing the manuscript.


 






 Paper Title: A study in scarlet -- I. Photometric properties of a sample of Intermediate Luminosity Red Transients.

 


   

   

   Valerin et al.
   

   Affiliations are listed after the acknowledgements
             

   



 
  
  
   
  
   We investigate the photometric characteristics of a sample of Intermediate Luminosity Red Transients (ILRTs), a class of elusive objects with peak luminosity between that of classical novae and standard supernovae. Our goal is to provide a stepping stone in the path to unveil the physical origin of such events, thanks to the analysis of the datasets collected.
  
   We present the multi-wavelength photometric follow-up of four ILRTs, namely NGC 300 2008OT-1, AT 2019abn, AT 2019ahd and AT 2019udc. Through the analysis and modelling of their spectral energy distribution and bolometric light curves we infer the physical parameters associated with these transients. 
  
   All four objects display a single peaked light curve which ends in a linear decline in magnitudes at late phases. A flux excess with respect to a single black body emission is detected in the infrared domain for three objects in our sample, a few months after maximum. This feature, commonly found in ILRTs, is interpreted as a sign of dust formation. Mid infrared monitoring of NGC 300 2008OT-1 761 days after maximum allows us to infer the presence of ∼10^-3-10^-5 M_⊙ of dust, depending on the chemical composition and the grain size adopted. The late time decline of the bolometric light curves of the considered ILRTs is shallower than expected for 56Ni decay, hence requiring an additional powering mechanism. James Webb Space Telescope observations of AT 2019abn prove that the object has faded below its progenitor luminosity in the mid-infrared domain, five years after its peak. Together with the disappearance of NGC 300 2008OT-1 in Spitzer images seven years after its discovery, this supports the terminal explosion scenario for ILRTs. With a simple semi-analytical model we try to reproduce the observed bolometric light curves in the context of few M_⊙ of material ejected at few 10^3 km s^-1 and enshrouded in an optically thick circumstellar medium.
  
   

   

   A study in scarlet - I. Photometric properties of a sample of Intermediate Luminosity Red Transients
    G. Valerin1E–mail: giorgio.valerin@inaf.it,
A. Pastorello1,
A. Reguitti1,2, 
S. Benetti1,
Y.-Z. Cai3,4,5, 
T.-W. Chen6,          
D. Eappachen7,8, 
N. Elias-Rosa1,9,
M. Fraser10, 
A. Gangopadhyay11,12,  
E. Y. Hsiao13,            
D. A. Howell14,15,
C. Inserra16,            
L. Izzo17,18,
J. Jencson19,         
E. Kankare20,
R. Kotak20, 
P. A. Mazzali21,22,         
K. Misra23, 
G. Pignata24,
S. J. Prentice25,           
D. J. Sand26,         
S. J. Smartt27,28,
M. D. Stritzinger29,    
L. Tartaglia30,
S. Valenti31,
J. P. Anderson32,33,
J. E. Andrews26,       
R. C. Amaro26,                           
S. Brennan11,
F. Bufano34,          
E. Callis11,
E. Cappellaro1,     
R. Dastidar35,33,
M. Della Valle17,36,
A. Fiore1,37,38,          
M. D. Fulton28,
L. Galbany9,39,        
T. Heikkilä20,            
D. Hiramatsu13,14,40,41,            
E. Karamehmetoglu11,29,    
H. Kuncarayakti20,42,
G. Leloudas43,             
M. Lundquist26,               
C. McCully14,
T. E. Müller-Bravo9,39,
M. Nicholl28,                
P. Ochner1,44,                
E. Padilla Gonzalez14,15,    
E. Paraskeva45,
C. Pellegrino46,          
D. E. Reichart47,          
T. M. Reynolds20,48,49,      
R. Roy50,                                     
I. Salmaso1,  
M. Singh51,
M. Turatto1,              
L. Tomasella1,              
S. Wyatt26,
D. R. Young28
          
    Received —-; accepted —-
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================





§ INTRODUCTION




It is well established that single stars with an initial mass below ∼8 M_⊙ will end their lives as white dwarfs, cooling down while supported by the electron degeneracy pressure in their cores. In contrast, stars with initial masses between ∼10 M_⊙ and 40 M_⊙ will complete all the nuclear burning cycles and will undergo a violent explosion as their core collapses <cit.>.
This apparently simple distinction raises a complicated question: what is the exact initial mass limit that separates the two opposite fates?
Stars with a zero-age-main-sequence mass between 8 M_⊙ and 10 M_⊙ are expected to form a degenerate O-Ne-Mg core during their lifetime <cit.>. Such stars are labelled Super-Asymptotic Giant Branch stars (SAGB), and the outcome of their evolution is uncertain. In fact, if the O-Ne-Mg core accretes enough material to approach the Chandrasekhar limit, the star will explode as an Electron Capture Supernova (ECSN), but if the core fails to reach this critical mass the star will end its evolution as an O-Ne-Mg white dwarf (e.g. ).
Whether such critical mass can be reached depends on the competing effects of mixing, convective overshooting and mass loss rates, which make the modelling of the core and its evolution a challenging endeavor <cit.>.
An additional complication, as pointed out by <cit.>, is that even small changes in the initial mass and metallicity of the progenitor star may give rise to a Fe core-collapse supernova (SN) instead of an ECSN, overall showing similar observables.

While stellar evolution theory predicts the existence of ECSNe, finding their observational counterparts is still an open issue. Proving that a transient originates from the core-collapse of an O-Ne-Mg core, rather than from a classical Fe core collapse, is not trivial. However, there has been no shortage of attempts: Low Luminosity Supernovae Type IIP (LL SNe IIP) (e.g. ) and also some interacting transients (e.g. ) have been proposed as ECSN candidates.
In order to be a reasonable ECSN candidate, an object should fulfill the key expectations for the explosion following the collapse of an O-Ne-Mg core.
First of all, the energy released by an ECSN should be significantly lower (∼ 10^50 erg) compared to classical SN explosions, therefore directing the investigation towards faint targets with low velocity ejecta <cit.>.
Secondly, the nucleosynthesis following the collapse of an O-Ne-Mg core yields limited amounts of 56Ni (few 10^-3 M_⊙), placing constraints on the luminosity of the late time decline of the candidate <cit.>.
Finally, the progenitor star of a candidate should be compatible with a luminous (∼10^5 L_⊙) SAGB star, since that is the only kind of star able to produce a degenerate O-Ne-Mg core massive enough to trigger an ECSN explosion <cit.>.

Intermediate Luminosity Red Transients (ILRTs) are a class of objects, populating the luminosity gap between classical novae and standard SNe <cit.>, which are appealing ECSN candidates.
Their physical origin is still debated, with some studies associating ILRTs to non–terminal eruptions of post–main sequence stars (e.g. ).
However, there are several indicators that favour the ECSN interpretation to explain the observed properties of these transients.
The low luminosity characterizing ILRTs, which show peak absolute magnitudes ranging between M_r ∼ –12 mag and –15 mag, is consistent with the expected weak explosion originating from the collapse of an O-Ne-Mg core <cit.>. Likewise, the late time decline in luminosity points towards small 56Ni masses synthesized, fulfilling the condition presented by <cit.> (see also ).
Furthermore, all of the progenitor stars associated with ILRTs have been consistent with a SAGB star, corroborating the ECSN scenario <cit.>. Recent estimates have shown that the rate of ILRTs is a few percent of all the local CC SNe events (≈8% according to , or ≈1–5%, according to ), compatible with the theoretical expectations for ECSNe <cit.>.
Finally, an important step towards understanding the nature of ILRTs was performed by <cit.>, who showed that a few years after their maximum luminosity the remnants of the two ILRTs SN 2008S <cit.> and NGC 300 2008OT-1 <cit.> had become fainter in the mid-infrared (MIR) than their progenitor stars.
Extreme dust extinctions would be needed to obscure a surviving star, therefore favouring a genuine terminal explosion over a non-terminal outburst.

While the considerations presented so far are certainly encouraging, the discussion on ILRTs as ECSN candidates is still ongoing. This is, after all, a relatively young class of transients, only established around a fifteen years ago. Their rates are not particularly low but their faintness makes their discovery occasional and their follow-up challenging. Since only a handful of ILRTs have been accurately characterized so far, additional data is key to improve our understanding of this poorly studied class of objects.
In this paper we present and analyse original photometric data of four ILRTs: NGC 300 2008OT-1, AT 2019abn, AT 2019ahd and AT 2019udc. This work is the first part of a series of two papers: in the second installment (“A study in scarlet II. Spectroscopic properties of a sample of Intermediate Luminosity Red Transients", Valerin et al. 2024, hereafter “Paper II") we will present and discuss the spectroscopic data collected for the same targets analysed here.
This paper is organized as follows: in Section <ref> we discuss the methodology used to obtain and reduce the data, while in Section <ref> the photometric data and the host properties are presented. In Section <ref> we discuss the reddening estimate, while in Section <ref> we discuss the physical parameters obtained through modelling of the spectral energy distribution fits. In Section <ref> we study the late time behaviour of NGC 300 2008OT-1 and in Section <ref> we present a simple model to reproduce the light curves of ILRTs. Finally, in Section <ref> we summarise the results obtained.





§ DATA REDUCTION
 
The objects presented in this paper were followed with several instruments at different facilities reported in Table [<ref>]. In particular, the majority of the private data was collected with the Nordic Optical Telescope (NOT) within the NOT Unbiased Transient Survey 2 (NUTS2) collaboration <cit.>, with the Liverpool Telescope <cit.>, with the GROND imager <cit.>, within the ePESSTO+ collaboration <cit.> as well as the Global Supernova Project (GSP, ).
Images obtained were reduced through standard iraf tasks <cit.>, removing the overscan, correcting them for bias and flat field. When multiple exposures were taken the same night, we combined them to improve the signal to noise ratio (S/N). To measure the magnitudes of the transients observed, we used a dedicated, python-based pipeline called ecsnoopy <cit.>. ecnoopy is a collection of python scripts that call iraf standard tasks like daophot through pyraf, and it was designed for Point Spread Function (PSF) fitting of multi-wavelenght data acquired from different instruments and telescopes. The PSF model was built from the profiles of isolated, unsaturated stars in the field. The instrumental magnitude of the transient was then retrieved by fitting this PSF model and accounting for the background contribution around the target position through a low-order polynomial fit. The error on this procedure was obtained through artificially placed stars close to the target, with magnitudes similar to that inferred for the object. The dispersion of the artificial stars instrumental magnitudes was combined in quadrature with the PSF fitting error given by daophot to obtain the total error associated with that measure. Zero Point (ZP) and Colour Terms (CT) corrections were computed for each instrument by observing standard fields: SDSS <cit.> was used as reference for Sloan filters, the <cit.> catalogue was used for Johnson filters and the 2MASS <cit.> catalogue was used for Near Infrared (NIR) filters. 

For the "Asteroid Terrestrial–impact Last Alert System" (ATLAS) data <cit.>, we combined the flux values obtained through forced photometry released from their archive[https://fallingstar–data.com/forcedphot/], and converted the result into magnitudes as prescribed in the ATLAS webpage.
While reducing WISE <cit.> and Spitzer <cit.> data we adopted the ZP provided on their respective websites[https://wise2.ipac.caltech.edu/docs/release/prelim/expsup/sec4_3g.html, https://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/iracinstrumenthand book/14/]. We also performed PSF fitting measurements on publicly available images taken with the James Webb Space Telescope (JWST, ). The pipeline-reduced JWST+NIRCam/MIRI `i2d' images (and Level-3 mosaics when available) were retrieved from the MAST archive[https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html].
It is worth noticing that in the NIR and Mid Infrared (MIR) we assumed negligible CT, so we only computed the ZP correction. 
Images from the Swift UV/Optical Telescope <cit.> were reduced using standard heasoft tasks, and the magnitudes were retrieved through aperture photometry, applying the aperture correction reported in the Swift
website[https://swift.gsfc.nasa.gov/analysis/uvotdigest/apercor.html] when needed.
In order to account for non-photometric nights, we selected a series of stars in the field of each observed transient: measuring the average magnitude variation of the reference stars, we computed the ZP correction for each night in each band. Applying ZP and CT corrections to the instrumental magnitudes of our targets, we obtained the apparent magnitudes which are reported in this paper. We adopted the AB magnitudes system for u, g, r, i, z, cyan, orange bands and Vega magnitudes for UVW2, UVM2, UVW1, U, B, V, R, I, J, H, K, W1, W2, [3.6] μ m, [4.5] μ m bands.
We resorted to template subtraction at late epochs, when the transients were too faint to be detected otherwise. The template subtraction procedure was performed on late time observations, again with ecsnoopy, with template images taken from SDSS <cit.>, Pan–STARRS1 <cit.> and Liverpool Telescope <cit.>.
The photometric measurements we obtained are reported in Appendix <ref>.







§ PHOTOMETRIC FOLLOW-UP
 



 §.§ AT 2019abn
 
AT 2019abn was discovered on 2019 January 22.6 UT by the Zwicky Transient Facility (ZTF, ) on a spiral arm of Messier 51 (M51) at the coordinates RA= 13^h29^m42^s.41, Dec = +47°11' 16”.6. The discovery and early observations are discussed by <cit.>, while the evolution of the transient up until 200 days from the discovery is covered by <cit.>. In this paper we provide additional optical data, especially at later stages of evolution, while also publishing original NIR and MIR observations obtained with IO:I and NOTCam, which put constraints on a critical section of the Spectral Energy Distribution (SED). 
By measuring the magnitude of the standard stars used as reference by <cit.> we integrate their dataset with our observations by applying the following magnitude corrections to their observations (likely due to a different choice of reference stars): Δ B = +0.07 mag, Δ r = +0.04 mag and Δ i = +0.05 mag. Discrepancies in V, z and NIR bands were within photometric uncertainty, and no correction was needed. Similarly, we incorporate the observations performed by <cit.> in our dataset after applying the following corrections: Δ J = -0.13 mag, Δ H = -0.05 mag and Δ K = -0.06 mag. 
We adopt a distance modulus of μ = 29.67 ± 0.02 mag to M51, obtained through the method of the tip of the red giant branch <cit.>. The Galactic absorption in the direction of M51 is A_V = 0.096 ± 0.006 mag, from <cit.>, under the assumption that R_V=3.1 <cit.>. The local absorption is more challenging to estimate and will be discussed, for all objects, in Section <ref>. 



The light curves of AT 2019abn are shown in Figure <ref>.
Thanks to the early discovery, it is possible to follow the evolution of AT 2019abn from very early stages. The rise in luminosity is observed in multiple bands, and lasts roughly 20 days before the transient reaches a peak apparent magnitude of m_r = 16.73 ± 0.01 mag on MJD = 58527.3. Thanks to this excellent coverage of the rise, it is possible to estimate the rising rates (γ_1) for the observed bands. To do so, we perform a linear regression of the measured magnitudes in each band between 25 and 11 days before maximum: such rising (and declining) rates are useful to quantitatively compare the behaviour of each band at different phases, as well as the differences between each transient (see  for similar measurements on other ILRTs). For sake of simplicity, we separate each light curve in multiple linear segments, and we display their declining (or rising) rates in Table <ref> and following tables. As a reference, here in the text we report the decline rates measured on the r band in units of [mag/100 days].
At 11 days before maximum, AT 2019abn starts departing from its initial linear increase in magnitude (γ_1=-21±1), forming a broad peak also described by <cit.>. The post maximum luminosity evolution is slow (γ_2=1.37±0.01) especially in the red bands, while the decline rate becomes more steep after 110 days (γ_3=2.49±0.04). Between 180 and 195 days after maximum, there is a sudden drop in luminosity of ∼ 0.9 magnitudes in all observed optical and NIR bands. After this abrupt change, the light curves settle on slow decline rates (γ_4=0.80±0.06). The MIR sampling of AT 2019abn, obtained through the Spitzer and WISE (survey NEOWISE) space telescopes, is unprecedented for an ILRT. After the first data point, at 50 days after maximum, the MIR light curves show at first a decline faster compared to the optical bands, but after ∼100 days the decline becomes more shallow. Interestingly, the luminosity drop at ∼180 days is not as evident in the MIR bands.








 §.§ AT 2019ahd
 
The discovery of AT 2019ahd was reported by the ATLAS survey <cit.> on 2019 January 29.0 UT. The coordinate of the transient are RA = 10^h 51^m 11^s.737 Dec = +05°50' 31”.03, which is 2”.6 north and 6”.9 south of the centre of its host, the spiral galaxy NGC 3423. As the distance modulus of the host galaxy, we chose to adopt an average of the different independent values reported on the NASA/IPAC Extra-galactic Database (NED) obtaining a distance modulus μ = 30.22 ± 0.14 mag, where the error comes from the standard deviation of the measurements <cit.>. We assumed a cosmology where H_0 = 73 km s^-1 Mpc^-1, Ω_Λ = 0.73 and Ω_M = 0.27 <cit.>, which will be used throughout this whole work. The Galactic absorption in the direction of NGC 3423 is A_V = 0.079 ± 0.003 mag <cit.>. The object was initially classified as a Luminous Blue Variable (LBV, ) due to its narrow hydrogen features and relatively red spectrum. The following photometric and spectroscopic evolution of the transient, in particular the prevalence of calcium features (Ca II H&K, [Ca II] and Ca NIR triplet; see Paper II), proved that it is an ILRT instead. 
The majority of the follow-up performed for this object is obtained through the GROND telescope, which yielded a remarkably homogeneous data set (Figure <ref>). The brightest magnitude is well constrained at m_r = 17.57 ± 0.07 mag, reached on MJD = 58525.0. Similar to AT 2019abn, AT 2019ahd displays a slow decline just after peak luminosity (γ_1=1.12±0.09), but this ”pseudo-plateau” only lasts for 45 days, followed by a steeper decline (γ_2=2.31±0.10) that ends 105 days after maximum. The subsequent decline rate is again slower (γ_3=1.05±0.03), in particular in the NIR bands (Table <ref>).








 §.§ AT 2019udc
 
The discovery of AT 2019udc was reported by the survey DLT40 <cit.> on 2019 November 4.1 UT. The transient lies on a spiral arm of the galaxy NGC 0718, at RA = 01^h 53^m 11^s.190 Dec = +04°11' 46”.96. We adopt a kinematic measure of distance for NGC 0718, that provides μ = 31.49 ± 0.15 mag, obtained through the redshift of the galaxy with respect to 3K CMB <cit.>. The Galactic absorption towards NGC 0718 is A_V = 0.100 ± 0.001 mag. Similar to AT 2019ahd, also AT 2019udc was originally classified as an LBV due to its spectral features <cit.> but its evolution proves that it is actually an ILRT.
AT 2019udc is the most distant object studied in this sample. The follow-up campaign was stopped 120 days after maximum, due to solar conjunction. The GSP multi-band monitoring yielded a high cadence follow up of the rise and peak of AT 2019udc, crucially contributing to the collection of a high quality data set for this object (Figure <ref>, left panel). During the first two days after discovery (-10 to -8 days from maximum), the rise in magnitudes can be approximated as linear, with an increase rate up to three time faster than the rise of AT 2019abn, with bluer bands showing a systematically faster evolution (γ_1=-47.5±4.8). After this phase the rise quickly slows down, departing from a linear behaviour and reaching peak luminosity in about 10 days (m_r=17.40 ± 0.06, MJD = 58801.28). Immediately after maximum, AT 2019udc displays a fast linear decline (γ_2=3.93±0.09), again around three times faster than the decline of AT 2019abn just after maximum. The luminosity drop settles on a gentler slope in all bands at around 35 days after maximum (γ_3=2.22±0.12). The resulting light curve is notably fast evolving for an ILRT: although peculiar compared to the other objects in the sample, these features are not unique among ILRTs, as shown in Section <ref>.

Thanks to the prompt discovery and high cadence monitoring of the DLT40 survey, the luminosity rise of AT 2019udc is exceptionally well sampled, as presented in the upper right panel of Figure <ref> (DLT40 data are unfiltered observations scaled to sloan r band observations).
The rise immediately after the discovery is remarkably fast, 0.8 magnitudes in less than one day, followed by a more gentle brightening of 0.6 mag in 2.5 days. At this point, 3 days after the first DLT40 detection, AT 2019udc is already close to maximum luminosity, and its magnitude will only marginally change (within ±0.15 mag) in the following two weeks. We try to fit the rise to maximum of AT 2019udc with a simple fireball model, where the flux increases as t^2 (as detailed by e.g. ). However, as shown by the black solid line in the upper right panel of Figure <ref>, the observed light curve quickly departs from the fireball model extrapolated from steep rise of the first two days. One possible explanation for this discrepancy is that the early light curves of ILRTs are not dominated by 56Ni decay, unlike the light curves of SNe Ia (for which the fireball model is a reasonable approximation).
DLT40 also provided several years of upper limits to the optical luminosity of the progenitor of AT 2019udc, which are displayed in the lower right panel of Figure <ref>. The absence of outbursts and the overall non-detection in the optical domain of the progenitor of AT 2019udc is well in line with the expectations for ILRTs, whose precursors have been identified as dust enshrouded stars, heavily extincted in the optical wavelengths but luminous in the MIR <cit.>.







 §.§ NGC 300 2008 OT-1
 
NGC 300 2008 OT-1 (hereafter NGC 300 OT) was discovered on 2008 May 14 during the SN search program at the Bronberg Observatory <cit.>. The event, located in the nearby NGC 300 at RA = 00^h 54^m 34^s.51 Dec = -37°38' 31”.4, was extensively studied in the subsequent years <cit.> and became a prototype for the class of ILRTs together with SN 2008S <cit.>. Here we present additional optical, NIR and MIR data, adding this object to our sample of ILRTs. 
For the distance of NGC 300, we adopt the results published by <cit.>, where a distance modulus μ = 26.43 ± 0.09 mag is obtained through the Red Clump method. The Galactic absorption towards NGC 300 is A_V = 0.034 ± 0.001 mag <cit.>.
NGC 300 OT is the closest ILRT ever observed, making it an extremely valuable which was monitored in detail for an extended period of time. In Figure <ref> we present the original optical and NIR photometric data we collected for this transient. The object was behind the sun during its rise and peak luminosity, so we lack the first part of its evolution. Our observed maximum is m_r = 14.38±0.02 mag on MJD = 54602.38. We note that there is a serendipitous optical detection of NGC 300 OT on MJD = 54580.65 when the transient is rising (m_R=16.30±0.04 mag, ), providing a rough estimate for the onset epoch of the event.
The first 35 days display a slow decline (γ_1=1.11±0.05), especially in the red bands. From 35 to 75 days, the transient falls from this "pseudo-plateau", and its luminosity starts to fade faster (γ_2=3.32±0.08). Between 75 and 120 days the decline in luminosity is particularly rapid (γ_3=5.24±0.06, calculated on the R band due to lack of r band coverage in this phase), comparable to the fast declining phase of AT 2019udc. From 120 and 255 days the fast decline stops, and a slow evolution ensues (γ_4=0.69±0.07) before the final phase (γ_5=1.56±0.05) that encompasses from 255 days onwards.
As done for AT 2019abn, we integrate the dataset provided by <cit.> with our observations by applying magnitude corrections to their observations, calculated by measuring the magnitude of the reference star chosen in their work: Δ B = +0.07 mag, Δ R = +0.05 mag. No correction was needed for V band, I band and NIR data, since they were perfectly matching. We also performed aperture photometry on the public images of NGC 300 OT taken with Swift UVOT which were analysed by <cit.>. We find a remarkable agreement in the U, B, V magnitudes reported there, while we measure overall fainter UVW2 and UVW1 magnitudes, possibly due to a different background selection: in particular, we restricted the aperture down to 3” to limit possible background contamination. The resulting UV fluxes from our measurements are in line with the behaviour expected from a black body emission, as shown in the top left panel of Figure <ref>.





 §.§ Comparison with other transients
 
In Figure <ref>, we compare the absolute r and H band evolution of our sample of ILRTs along with that of other transients of similar luminosity.
The first considerations can be made observing the sample of ILRTs, with the addition of the well studied SN 2008S <cit.> and AT 2017be <cit.>. We remark the large spread in the optical peak magnitudes, which span from –12 mag for AT 2017be to –15 mag for AT 2019abn. We note that AT 2019abn (M_r = –15.00±0.08 at maximum) is, to date, the brightest ILRT ever observed, with AT 2019udc (M_r = –14.17±0.16 at maximum) also falling on the brighter end of the ILRTs luminosity distribution. 
AT 2019ahd and NGC 300 OT, instead, display fainter peak magnitudes, –13.17±0.16 mag and –12.69±0.11 mag respectively. Among the least luminous ILRT we find AT 2017be, which is barely brighter than M_r = –12 mag.
There are also significant differences in light curve shapes within the class. AT 2019abn and AT 2019udc represent the two extreme cases, with the former being characterised by a long phase of slow decline after peak, almost a pseudo-plateau in the r band (1.37 ± 0.01 mag/100 days), while AT 2019udc undergoes a decline that is three times faster (3.93 ± 0.09 mag/100 days), after peak luminosity.
This variability is less pronounced in the NIR domain, where both the decline rates and the peak magnitudes span a smaller range of values. Regardless of their differences, ILRTs tend to settle on a linear decline at late times, which is compatible with the expected luminosity decline sustained by 56Ni radioactive decay. This can be visually evaluated by comparing the late time behaviour of ILRTs with that of SN 2005cs, one of the prototypes of low-luminosity SNe IIP <cit.>, whose late decline is known to be powered by 56Ni decay. We also remark that SN 2005cs is fainter than AT 2019abn at peak luminosity, revealing an overlap between the brightest ILRTs and the faintest core collapse SNe.
To summarize the features described above, ILRTs are characterized by their single-peaked, monotonically declining light curves which terminate in a linear decline at late phases.
A clearly different light curve shape is instead associated to Luminous Red Novae (LRNe, ), another class of transients populating the luminosity "Gap" which separates classical novae from standard SNe.
LRNe are non-degenerate stellar mergers, and typically display double peaked light curves, as shown for by AT 2017jfs <cit.> and to a lesser degree M31-LRN-2015 <cit.>, both reported in Figure <ref>. With its peak absolute magnitude fainter than –10 mag, M31-LRN-2015 appears detached from the other transients shown: indeed, LRNe can be much dimmer, with peak absolute magnitude even below ∼–4 mag (e.g. V1309Sco ). On the other hand, ILRTs discovered to date have been strictly confined within the luminosity Gap (–10 mag<M_V< –15 mag).






§ REDDENING ESTIMATE
 
Estimating the reddening affecting ILRTs is a challenging task, and different approaches can be found in the literature, depending on the available data. A first method consists in using the empirical relation between the Na ID Equivalent Width (EW) and the absorption along the line of sight <cit.>, as was done by <cit.>. This method has the advantage of not requiring any assumption on the intrinsic properties of the target, but it needs high S/N spectroscopy in order to be reliable. Furthermore, the EW of Na ID observed may saturate, preventing the application of empirical relations (e.g. ). Furtheremore, the EW of the Na ID varies with time in ILRTs (e.g. , and further discussed in Paper II), increasing the uncertainty associated with this method.
A second method is based on the observation that the spectral features of ILRTs resemble those of F-type stars <cit.>. 
It is then assumed that all ILRTs reach a temperature of ∼7500 K at maximum luminosity, and a reddening correction is applied to the Spectral Energy Distribution (SED) at peak in order to obtain a blackbody continuum corresponding to such temperature. 
Yet another strategy is adopted for SN 2008S by <cit.>, who estimate the extinction affecting the target based on an IR echo model of the SED at early stages.



In this work we adopt the same procedure carried out by <cit.>, who notes that the observed V–r and r–i colour curves of several ILRTs have a similar shape. Assuming that ILRTs should display a similar evolution of optical colours, a reddening estimate can be obtained by attempting to match the colour evolution of the bluest objects. While this is likely an approximation, it is not baseless. In fact, these transients present a marked homogeneity in their optical spectra (which will be further discussed in Paper II): since similar conditions in temperature and ionization state are required to produce the same spectral features, we can expect the optical colours of these transients to be at least compatible in their photospheric phases.
Following this reasoning, we consider the bluest object in our sample, AT 2019udc, together with AT 2017be, which shows remarkably similar colours, and we apply a reddening correction to all the other objects in our sample in order to superimpose their B-V, g-r and r-i colour curves through a least squares minimization procedure. Only the values measured within 100 days after maximum are considered. The absorption values for each transient inferred in this way are reported in the last column of Table <ref>. AT 2019abn is by far the most reddened object in our sample, with an estimated internal absorption of A_V=2.34 ± 0.06 mag. AT 2019udc, by construction, is assumed to be reddening-free, since it is among the bluest ILRT observed. 
In Figure <ref> we report the colour evolution for our sample of ILRTs after applying the reddening correction. The optical colours present an initial decrease (i.e. the objects become bluer) during the rise to maximum. This is particularly clear in AT 2019udc and AT 2019abn, thanks to their high cadence monitoring pre-maximum. After peak luminosity, ILRTs become steadily more red in the following months. This trend may invert as the flux becomes dominated by emission lines rather than continuum emission, as seen for NGC 300 OT r-i after 150 days post maximum.
We note that we are using a conservative estimate for the reddening affecting SN 2008S, for which we adopt just the Milky Way absorption of A_V=1.13 mag. This is a lower limit to the total reddening along the line of sight: adopting a light echo model, <cit.> infer an internal absorption of A_V ∼ 1 mag. This leads to a peak temperature of 8400±120 K, closer to an A type star, rather than an F type star.
In any case, even with minimal absorption, SN 2008S is among the bluest ILRTs to date: therefore if the colour evolution of ILRTs is indeed perfectly homogeneous, our adopted absorption is likely a lower limit of the actual value. On the other hand, it is reasonable to expect some scatter in the temperature and consequently in the colours of ILRTs, granted that all of them should fulfill the physical requirements for producing the spectrum of an F type star <cit.>. In this context, the absorption estimated so far can be seen as a first order correction, crucial for heavily extincted targets such as AT 2019abn.












For the sake of completeness we compare the reddening estimates just obtained (which will be adopted for this work) with a straightforward application of the linear relations between Na ID EW and E(B-V) presented by <cit.>. The detailed evolution of the Na ID EW for this sample of ILRTs will be shown and discussed in Paper II: here we simply consider the first measured value for each transient, specifically 0.7±0.1 Å for NGC 300 OT, 1.1±0.1 Å for AT 2019abn, 0.9±0.1 Å for AT 2019ahd and 5.4±0.2 Å for AT 2019udc. The two different linear relations provided by <cit.> allow to determine both a lower and upper estimates of E(B-V) for each object. The resulting A_V range obtained for NGC 300 OT spans between 0.3 and 0.9 mag, in good agreement with our previous estimation of 0.78 mag. In case of the lowest reddening inferred, the peak luminosity of NGC 300 OT in r band would be 0.4 mag fainter, leading to a peak absolute magnitude M_r= -12.4 mag. For AT 2019ahd we obtain 0.4 mag<A_V<1.3 mag, with the lower limit being compatible with our previous measurement of 0.37 mag. If we accepted the larger estimate of A_V∼1.3 mag, the peak r band absolute magnitude of AT 2019ahd would increase from -13.3 mag to -14.1 mag. For the other two ILRTs in the samples the two methods for estimating A_V yield more substantial differences.
According to the empirical relations obtained by <cit.>, the EW of the Na ID observed in the early phases of AT 2019abn would imply an extinction of 0.5 mag<A_V<1.6 mag, significantly lower than the value 2.34 mag previously obtained. The peak absolute magnitude of AT 2019abn would decrease from M_r= -15.1 mag to -14.4 mag, or even -13.5 mag in the case of lowest extinction. On the contrary, AT 2019udc initially displays a remarkably large Na ID EW of 5.4±0.2 Å (however, we note that the Na ID EW rapidly declines to 1.3 Å in just two days, and even below 1 Å after that phase): this would imply an extinction A_V of at least 2.6 mag. The resulting peak magnitude would be at least M_r= -16.5 mag, firmly within the luminosity realm of full-fledged SNe.
Such critical discrepancies in A_V would also lead remarkable variations in temperature, in particular a hotter AT 2019udc and a much cooler AT 2019abn, especially around peak luminosity (compared to the results discussed in the next section and displayed in Figure <ref>). This is an unlikely circumstance, given the already mentioned spectral homogeneity of ILRTs.





§ SED EVOLUTION
 
In order to extract physical quantities of our targets, we perform blackbody fits on the SED at different epochs.
This analysis is carried out through Monte Carlo simulations, using the python tool curve_fit[https://docs.scipy.org/doc/scipy/reference/generated/ scipy.optimize.curve_fit.html] to perform fits on 200 sets of fluxes randomly generated with a Gaussian distribution centered at the measured flux value, and σ equal to the error associated to the measurement. Such procedure was already adopted and described in <cit.>. The blackbody fit to the SED of the target yields the estimated temperature, and by integrating over the wavelength we obtain the total flux emitted. 
Adopting the distances discussed in Section <ref> and assuming spherical symmetry, we calculate the bolometric luminosity of the source. 
Finally, the radius is estimated through the Stefan–Boltzmann law. This whole procedure is repeated for each epoch with suitable photometric coverage, in order to study the evolution of the inferred physical parameters with time.
During the first phase of evolution all the objects are well fitted by a single black body, associated to the photosphere, throughout the optical and NIR domains. We refer to this component as the ”hot” blackbody, in contrast with the ”cold” blackbody which emerges at later stages. The phase in which only the hot blackbody is visible has a variable duration: up to 180 days for AT 2019abn, 110 days for AT 2019ahd and only 80 days for NGC 300 OT (Figure <ref>). The SED of AT 2019udc is well fit by a single black body at all epochs observed, but it is possible that the cold black body emerged after the end of our follow-up campaign.


The values of temperature, luminosity and radius obtained for this hot blackbody are displayed in Figure <ref>. The temperature evolution of our sample is especially homogeneous between 25 and 75 days after maximum, due to our reddening estimate through colour curves superposition (Section <ref>). However we still find interesting differences between the various targets in the pre–maximum phases. AT 2019abn displays an almost constant temperature of ∼4600 K for several days before slowly reaching the peak temperature of ∼5900K in ∼20 days. AT 2019udc qualitatively follows the same behaviour, but in a shorter timescale (less than 10 days) and reaching 7000 K at peak.
This initial increase in temperature could be explained with the injection of additional energy provided by the interaction between ejecta and CSM, as happens for interacting SNe (e.g. ).
AT 2019ahd, on the other hand, starts from a temperature of ∼7500 K, before quickly cooling to 6000 K at peak luminosity. From this point onward, AT 2019ahd closely follows the behaviour of AT 2019abn. In this respect, the early temperature evolution of AT 2019ahd is reminiscent of that of SN 2005cs (displayed in Figure <ref> for comparison) and LL SN IIP in general, where the temperature quickly declines during the rapid expansion of the ejecta. 
NGC 300 OT shows a very simple, monotonic temperature evolution, but in this case we miss the pre–maximum photometric coverage.







In the middle panel of Figure <ref> we present the bolometric luminosity obtained for the ”hot” blackbody. The bolometric luminosity behaves in a similar way to what is described in Sect <ref>, with AT 2019abn being the brightest ILRT, showing a peak luminosity of 1.9± 0.3 × 10^41 erg s^-1. As previously pointed out, its decline rate is slower compared to other objects of the same class, but it is not flat like the plateau of SNe IIP. AT 2019udc is characterized by a marked peak (1.4±0.5 × 10^41 erg s^-1) followed by a fast decline. AT 2019ahd and NGC 300 OT display more modest peak luminosities, respectively of 4.4±1.1 and 2.7±1.0 × 10^40 erg s^-1.
As for the evolution of the radius of the emitting source, shown in the bottom panel of Figure <ref>, it appears that all ILRTs in our sample follow a similar behaviour. After maximum, the transients show a roughly constant radius with time, although with different values. AT 2019abn again stands out from the group, showing a blackbody radius that at first quickly increases from 21±2 to 36±3 AU from discovery to peak luminosity, and then remains at 30 to 40 AU in the following 125 days. These values are roughly two to three times those obtained for the other three targets. AT 2019ahd shows qualitatively the same behaviour, with a radius growing from 7±1 to 15±1 AU from discovery to peak magnitude, and a subsequent slow evolution ranging from 14±1 to 18±1 AU within 110 days after maximum. AT 2019udc on the other hand does not show an increase in the radius during the pre–maximum phase, but rather a marginal decrease, from 23±3 to 19±2 AU. The following slow evolution of the blackbody radius between 16±2 and 21±2 AU over the course of 90 days is reminiscent of those of the other two ILRTs already presented. Finally, for NGC 300 OT we do not have pre–maximum data, but the evolution of the blackbody radius after the observed maximum is again slow, spanning from 11±1 to 16±1 AU in 90 days. This tendency of the hot blackbody radius to linger around a constant value is in stark contrast with the monotonic increase in radius for SNe IIP and Low Luminosity SNe IIP during the first 80 days, as shown by the radial evolution of SN 2005cs.



So far we described the physical properties of the hot blackbody, which is associated to the photosphere of the transient and is usually well visible for ∼100-150 days after maximum.
At later phases however, as the gas cools down and expands, an excess in the NIR flux is detected (extending to the MIR, when such measures are available). Such feature can be associated with the formation of dust, which contributes to the SED with a second emission component characterised by a low temperature compared to the hot continuum observed in the first months of evolution (T ≲ 1500 K, ).
In the bottom panels of Figure <ref> are shown blackbody fits performed to the SED of our targets at epochs when two components coexist. At the latest phases, as the objects shift from the photospheric to the nebular phase, the optical flux is entirely supported by emission lines (Hα, Hβ, [CaII], Ca NIR triplet), and there is no more evidence of a hot spectral continuum. At these epochs we still identify a contribution from the "cold" black body associated with dust emission. This feature is especially clear in AT 2019abn and NGC 300 OT (at 761 days after maximum, see Section <ref>) thanks to the MIR measurements. 
In Figure <ref> we present the physical parameters obtained for the “cold" blackbody component associated with dust emission. In principle this cold blackbody could be the result of a NIR light echo (e.g. ). However, the timing of the appearance of this NIR excess (80 days for NGC 300 OT being the shortest) would imply that the dust causing the light echo is located at least ∼6900 AU away from the transient. <cit.> show that the flash from SN 2008S is able to create a dust-free cavity of just ∼2000 AU: it appears unlikely that the energy radiated by our ILRTs can bring such a distant body of dust close to its sublimation temperature (∼1400 K for silicates and ∼1800 K for graphite, e.g. ). Therefore, we favour the interpretation that this cold blackbody component is emitted by newly condensed dust rather than by a light echo.
AT 2019abn and AT 2019ahd show very similar cold components, with temperatures cooling from ∼1600 to ∼1200 K over a timescale of hundreds of days. The luminosity of both components evolve slowly, starting from ∼3 × 10^39 erg s^-1. The radius of the emitting source is around 60 AU for AT 2019ahd and 75 AU for AT 2019abn.
The most relevant difference between the two transients is that AT 2019ahd develops the "cold" component sooner, at around 110 days. We stress that the blackbody fits at these epochs are affected by large errors, due to the uncertainty in disentangling the hot and cold components in the SED.

This is true also for NGC 300 OT, but in this case we have a higher cadence of NIR observations (also accounting for previously published observations), allowing us to better constrain the evolution of the cold component. A first, fast decline in temperature is observed between 71 days and 130 days after maximum, when the dust cools from 1550±170 K to 1150±90 K. The subsequent temperature decline is slower, reaching 570±10 K at 761 days. Such late estimate is possible thanks to publicly available WISE data. The other two late measurements at 398 and 582 days were performed on MIR data taken by <cit.>.
The luminosity of this "cold" component fades from 5.5±1.9 × 10^39 erg s^-1, just after its emergence, to 9.5±0.9 × 10^38 erg s^-1 at 761 days. The radius inferred for the dust emission at first shrinks from ∼100 AU to ∼40 AU, but around 180 days the trend inverts and the radius grows to 235±20 AU at 761 days.
The initial shrinking of the emitting source can be qualitatively explained as follows: we are observing dust formation far from the star, with dust condensing at progressively smaller radii as the transient becomes dimmer and the temperature decreases. Indeed, we know that the progenitor star was enshrouded in dust <cit.> before the transient event caused its sublimation, and we may be witnessing the rebirth of this dusty cocoon.
On the other hand, the increase in radius of emitting dust observed after 200 days can be found also in SNe, where the dust expands together with the ejecta (e.g. ). As a simple comparison, in the bottom panel of Figure <ref> we mark with a dashed line the position of matter expanding with a velocity of 550 km s^-1, and we note that the three latest measures of radius for NGC 300 OT are compatible with such trend. As will be shown in Paper II, the velocity of 550 km s^-1 is also well in line with the full width at half maximum velocity estimated from the emission lines of NGC 300 OT, corroborating the idea that the dust is moving jointly with the expanding gas seen during the photospheric phase. As previous observations displayed, part of the dusty cocoon surrounding ILRTs may survive the explosion, avoiding sublimation thanks to a large distance from the star <cit.>. However, in the present work we lack the mid and far-infrared coverage at early phases necessary to detect the emission from such dust component.



In order to obtain the bolometric luminosity of ILRTs, we combine the contribution of the “hot", photospheric component, and the "cold", dusty component. At late epochs, a blackbody continuum is no longer discernible in the optical domain, with only emission lines dominating the spectra. Therefore, to measure the bolometric luminosity in those cases we integrate the fluxes in the optical domain using the trapezoidal rule. On the other hand, at longer wavelengths it is still possible to perform a blackbody fit and integrate the flux even beyond the observed spectral region. In fact, while the optical domain only displays emission lines at late time, there still is still evidence of thermal continuum in the NIR and MIR domain.
The bolometric luminosity obtained for our sample of ILRTs is reported in Figure <ref>.
We analysed an unprecedented amount of NIR and MIR data for ILRTs, which allow us to better infer the luminosity contribution at longer wavelengths, crucial at late times.
The dotted lines in Figure <ref> show the expected behaviour of a light curve powered exclusively by 56Co decay: it is evident that AT 2019abn, AT 2019ahd and NGC 300 OT are characterised by a late time decline shallower than what is expected if only 56Co decay is supporting their late time luminosity.
Of course a more shallow slope in the light curve does not exclude the presence of 56Co, but rather it requires an additional mechanism to explain the late time luminosity of ILRTs. In order to estimate an upper limit to the 56Ni mass powering the late linear decline of ILRTs, we employ the analytical formulation provided by <cit.> for the luminosity of 56Ni and 56Co decay. By considering the first data points available during each linear decline, we obtain upper limits to the 56Ni mass of 6.7x10^-3 M_⊙ for AT 2019abn, 2.0x10^-3 M_⊙ for AT 2019ahd and 1.2x10^-3 M_⊙ for NGC 300 OT. AT 2019udc is the only object that shows a linear decline compatible with the one expected from 56Co decay, resulting in an estimate of 3.3±0.3 x10^-3 M_⊙ of 56Ni. 












We can also provide values of the total energy radiated by our ILRTs during the time they were monitored: NGC 300 OT emitted 2.5×10^47 erg over the course of 760 days, with 1.5×10^47 erg released in the first 120 days after maximum. Similarly, AT 2019ahd radiated 4.0×10^47 erg in 440 days, 2.3×10^47 erg in the first 100 days, while AT 2019udc emitted 4.0×10^47 erg in about 100 days. Finally, AT 2019abn radiated 1.7×10^48 erg in 480 days, the vast majority (1.6×10^48 erg) emitted in the first 240 days. Aside from the total energy released by these transients, the decline rate of their bolometric luminosity at late time appears particularly interesting.



§ LATE TIME MID-INFRARED MONITORING OF ILRTS
 



 §.§ Dust evolution and composition
 



<cit.> monitored the evolution of NGC 300 OT in the IR domain (2-5 μ m) at 398 and 582 days, finding that the hot dust component had progressively cooled down to 810 K and ultimately to 680 K. The probed wavelength range did not provide information on the cool, pre-existing dust component, which by that time could only contribute ∼1% of the observed flux. <cit.> also set lower limits for the optical depth of the dust: τ_ν > 12 at 398 days and τ_ν > 6 at 582 days, effectively stating that the dust remains optically thick even at very late phases. In this condition, the lower limit to the mass of emitting dust was estimated to be ∼10^-5 M_⊙.


Thanks to the WISE data collected on 2010-06-17 in the W1, W2, W3 and W4 filters, we are able to expand the analysis on the late time evolution of the dust in NGC 300 OT, 761 days after maximum. To correct these values for absorption, we extend the reddening law to 22 μ m by employing the tabulated values and prescriptions found in <cit.> and <cit.>.
The late time K band coverage with SOFI obtained 853 days after maximum allows us to linearly extrapolate the K-band flux at 761 days in order to have a more detailed SED of the transient.
We fit our flux measures through the same procedure detailed in Section <ref>. Following in the footsteps of <cit.>, instead of a simple Planck function we use f(ν) ∝ (1-e^-τ_ν) B_v(T_d) as a fitting function, where τ_ν is the optical depth, B_ν is the Planck function and T_d is the dust temperature. The multiplicative factor in front of the Planck function aims to account for the different opacity of dust at different wavelengths.
It is relevant to point out that this model is isothermal, while a cloud of optically thick dust would present a temperature gradient, so this is already an approximation.


To better differentiate the properties of the dust depending on the size of the dust grains and their chemical composition, we perform the fits using four different sets of opacity k_ν, tabulated by <cit.>. Two sets are opacity values for graphite dust, with grain size of 0.1 and 1 μ m respectively, while the other two sets of opacity assume the same grain size, but present a silicate composition. Since the values presented by <cit.> only cover up to 13 μ m while our data reach 22 μ m (thanks to the W4 channel), we have to extrapolate the behaviour of k_ν in that region. For the graphite dust we follow the prescription from <cit.>, using a λ^-2 scaling. The silicate opacity is instead characterised by an additional bump in opacity around 19 μ m: to reproduce it, we adopt the opacity behaviour reported by <cit.>. Beyond 9 μ m, graphite dust tends to have a lower opacity compared to graphite dust (see Figure <ref> in Appendix). Silicate dust stands out because of the "bumps" in opacity in the MIR domain, in contrasts with the smooth behaviour of the graphite opacity.







In Figure <ref> are reported the results of the SED fitting procedure for the different dust compositions and grain sizes considered. The parameters obtained are reported in Table <ref>. The parameters of the best fitting black body are similar regardless of dust particles size. There is, instead, a small variation on whether we adopt graphite dust or silicate dust. The dust temperature obtained for graphite is slightly lower than the temperature obtained for silicate dust, with a radius that is consequently larger.
All things considered, the differences between the black body parameters of silicate dust and graphite dust are negligible: the real difference lies in the optical depth inferred. We consider the optical depth at 5 μ m (τ_5), to perform a comparison between the different models. As shown by the dotted lines in Figure <ref>, at low optical depth (τ_5 <1), the models deviate significantly from the Planck function: silicate dust shows a double peaked emission, while the emission of graphite dust with grains of 1.0 μ m is characterised by a "shoulder" at around 3.2 μ m. Graphite dust with grains of 0.1 μ m do not show such clear features, but the SED model is systematically too narrow to properly fit the observed data in the optically thin case. Indeed, all the best fit yield optically thick dust at all wavelengths considered. However, in the case of silicate dust the best fit is obtained with an optical depth of τ_5=19.4 for 0.1 μ m grains and τ_5=16.5 for 1.0 μ m grains, basically returning to a Planck function. On the other hand, the optical depth inferred for graphite dust are τ_5=5.9 for 0.1 μ m grains and τ_5=4.9 for 1.0 μ m grains: optically thick, but still distinguishable from a pure Planck function. We notice that none of the models can accurately fit the point at 22 μ m: we speculate that this infrared excess is produced by the cold dust component that further cooled down since its detection by <cit.>. 
Finally, through our measures of the optical depth, it is possible to obtain an estimate of the dust mass using the equation (as done by ):




    M_dust = 4 π R_dust^2τ_ν/3 k_ν.




Here we are assuming a constant density profile in the dust cloud, which is of course an additional approximation. Depending on the composition and grain size considered, the inferred dust mass spans from 4.5×10^-5 M_⊙ up to 1.9×10^-3 M_⊙ (see Table <ref>). It is interesting to note that such values, obtained at ∼800 days after explosion, are perfectly in line with the amount of dust found in the ejecta of core-collapse SNe [https://nebulousresearch.org/dustmasses/] (e.g. ).
We remark that this fitting procedure was performed on the dust which was identified by <cit.> 93 days after discovery at a temperature of 1510 K (and named "warm" dust), and whose cooling is tracked in the upper panel of Figure <ref>. The initial temperature of this dust component is close to the dust sublimation temperature: this behaviour is compatible with newly formed dust which condensed few months after the peak luminosity of NGC 300 OT, and by the time of our fit has cooled down to ∼580 K. Similarly, also the third blackbody component with a temperature of 485 K detected by <cit.> at 93 days (identified as circumstellar dust that survived the explosion) must have cooled down, and the peak of its emission has shifted outside the monitored domain. As mentioned above, this component may be the origin of the exceeding flux observed at 22 μ m.



We replicate this SED analysis on AT 2019abn, thanks to publicly available images of M51 taken by the JWST on 2022-12-13 (proposal 1240, PI M.E. Ressler), 2023-06-08, 2023-06-14 (proposal 1783 PI A. Adamo) and 2024-05-31 (3435 PI K.M. Sandstrom). The object is still visible even five years after its peak luminosity: we report the magnitudes obtained with PSF fitting through ecsnoopy in Table <ref>.  
The wavelength coverage of the first three epochs is too scarce to provide a well sampled SED, therefore we focus on the measurements obtained on 2024-05-31, 1956 days after discovery. To fill the gap between the F430M and F1000W filters, we infer the behaviour of the F560W and F770W filters observed the previous year (2023-06-08). To do so, we assume a constant colour evolution using the filters F444W (observed on 2023-06-08) and F430M (observed on 2024-05-31) as anchors. The flux in these overlapping filters decreases by a factor of 1.5 between the two epochs, and we assume the same holds true for the F560W and F770W filters. Since cooling objects tend to become redder, this is likely an underestimation of the fluxes in F560W and F770W on 2024-05-31, yet this simple approximation allows us to constrain a crucial spectral region. The resulting SED is displayed in Figure <ref>.
Firstly, we point out that the Paα line is located within the F182M filter, and this could account for the relative brightness of the transient in this band, which is therefore excluded from the following discussion.



In Figure <ref> we show as solid lines the various fits to the observed SED of AT 2019abn with dust models characterised by different chemical composition and grain size. Parameters of each fit are shown in Table <ref>. The luminosity of the transient is consistently found to be (2.8±0.1)×10^38 erg s^-1 (in Figure <ref> this result is added to the bolometric light curve of AT 2019abn, updating Figure <ref>). There is instead a discrepancy in the resulting temperature and radius depending on the dust composition: in the case of graphite, the best fits yield a temperature of ∼350 K and a radius of ∼350 AU, while for the silicate dust the temperature found is ∼375 K, with a blackbody radius of ∼300 AU. Grain size only plays a marginal role in shaping these results. Graphite dust models yield high optical depth (τ_5>>1), leading to infer lower limits to the dust masses in the order of several 10^-3 M_⊙. On the other hand, the best fitting silicate dust models are closer to the optically thin regime (τ_5∼1), with inferred dust masses of the order of ∼2×10^-4 M_⊙.
These results, however, need to be interpreted with caution, since even the best fits struggle to accurately reproduce the observed data. Particularly puzzling is the flux “depression" in the F1000W band compared to the flux values found in the F770W and F1130W bands.
A possible explanation for this departure from a simple blackbody emission may be the presence of polycyclic aromatic hydrocarbon (PAH). In fact, PAH are characterised by emission features around 7 μ m and 11 μ m. In Figure <ref> we display a scaled emission spectrum of ionized PAH with size of 14 Å <cit.> to show the match between the PAH emission features and the peculiarities of the observed SED. The presence of PAH is also corroborated by MIR spectroscopy of AT 2019abn obtained on 2022 <cit.>. Accurate PAH modelling is beyond the scope of this study and is deferred to later works. Beyond these considerations on the shape of the SED, JWST observations provide valuable information on the MIR luminosity evolution of AT 2019abn five year after its discovery, as discussed in the following section. 







 §.§ Photometric decline below progenitor luminosity


<cit.>, monitored the two ILRTs NGC 300 OT and SN 2008S with the Spitzer telescope for several years, until they both faded below the luminosity of their respective progenitor.
We update this follow-up campaign by analysing the Spitzer images collected between 2015-10-20 and 2019-11-24 with the Spitzer [4.5] μ m channel. A visual inspection of the images shows that NGC 300 OT has faded below the detection threshold (Figure <ref>, left panel). We measure the flux on all public Spitzer [4.5] μ m images through PSF fitting employing ecsnoopy. Our results are in good agreement with the values measured with aperture photometry by <cit.>, with the exception of the very last detection, on 2015-02-09: we find a value of 600±300 L_⊙ against the 140±110 L_⊙ obtained with aperture photometry (the difference can be appreciated on the right panel of Figure <ref>). The complex background surrounding the target, as seen in the left panel of Figure  <ref>, contributes to the uncertainty of this measure. All the images collected from that moment only provide upper limits to the luminosity of NGC 300 OT: in particular, in the right panel of Figure  <ref> we show the 1σ limits obtained from PSF fitting. We therefore confirm that since 2015 NGC 300 OT has faded below the detection threshold, well below the progenitor luminosity.

We are also able to include AT 2019abn among the ILRTs with MIR coverage spanning several years. The light curve up to ∼500 days is provided by the Spitzer and WISE monitoring already shown in Figure <ref>, while the last two epochs are provided by the late JWST observations obtained 1598 and 1956 days after discovery and presented in the previous section. Despite the slight mismatch in the covered wavelength, the JWST filters F444W and F430M are sufficiently close to the [4.5] μ m and W2 filters to provide a coherent light curve, which is shown with red stars symbols in Figure <ref>. The behaviour of AT 2019abn is similar to that of SN 2008S, where the initial decline rate significantly slows down after ∼200 days. It is interesting to consider the evolution of the transient compared to its progenitor luminosity in the [4.5] μ m channel provided by , which is marked in Figure <ref> as a red horizontal dashed line. It is possible to notice that on 2023-06-08 AT 2019abn has already faded below its progenitor luminosity, with the latest measure obtained on 2024-05-31 being over two times dimmer than the luminosity of the star before the event. All three ILRTs that were monitored for multiple years in the MIR domain eventually became dimmer than their progenitor star, and so far none of them has shown sign of stopping their steady luminosity decline: this fact supports the interpretation that ILRTs are terminal events. 








§ TEMPLATE LIGHT CURVE
 
In order to better characterize this class of objects, we produce an r band template light curve for ILRTs. 
We are motivated also by the potential usefulness of such templates in our future attempts to classify new transients discovered by the Vera Rubin Observatory (formerly Large Synoptic Survey Telescope, ) and released by dedicated science brokers such as Lasair <cit.> and Alerce <cit.>. With the foreseen amount of data, it will be impossible to study in detail every single target, and template light curves for different classes of transient could prove to be useful tools for filtering the data stream, in addition to the classification algorithms and tools that are currently being developed (e.g. ). 
Apart from NGC 300 OT, AT 2019abn, AT 2019ahd and AT 2019udc, which were presented above, we broaden our sample including SN 2002bu <cit.>, SN 2008S <cit.>, AT 2017be <cit.>, SNHunt120 <cit.>, AT 2010dn, AT 2013jc, AT 2013la, AT 2013lb, AT 2018aes <cit.> and AT 2022fnm (Moran et al. 2024, submitted). We chose to focus on the r-band, since it is typically well-sampled and allows for a reliable comparison between objects.
The peak absolute magnitude of different ILRTs can span at least three magnitudes (from -12 mag to -15 mag, see Figure <ref>): however, here we are interested in recovering just the expected light curve shape of ILRTs. For this reason we normalised the peak magnitude of all objects considered to the same arbitrary value. Both the light curve shape and the absolute magnitude at peak can then be used as filters to identify future ILRTs candidates.



As a first step to produce a light curve template we make use of Gaussian Processes to interpolate the light curves of each object (see e.g.  for a more detailed description of the procedure).
In particular, we employ the Matern 3/2 kernel found in the python package george  <cit.>, and as for the mean function we adopt equation 1 given by <cit.>.
It is worth noticing that we performed the interpolation procedure in the ”luminosity - time" space.
Given the different sampling of each event, this first step is performed to ensure that each objects holds the same weight in the construction of the template, regardless of how many data points were collected at a given phase.
After interpolating the light curve of each object, we recover the mean value and the standard deviation of the ILRTs luminosity within temporal bins of 3 days, which we empirically choose as a good compromise that yields a rather smooth template which still has a reasonable time resolution. The resulting ”mean light curve" of an ILRT is shown as a black solid line in the upper panel of Figure <ref>, with the grey shaded area representing the 1 σ deviation from the mean.










Due to the low number of available ILRTs, the rising part of the template becomes reliable only at about -10 days with respect to maximum luminosity: before that phase the template is dominated by the widely different behaviours of AT 2019abn and AT 2018aes.
The overall template light curve rises of about 0.5 mag from -10 days to 0 days, and fades by 1 mag from maximum to 50 days. As time progresses, the 1 σ region gets wider, with objects evolving at different rates. Objects fading below the detection threshold can result in a jagged profile of the template, as happens at about 100 days due to the disappearing of AT 2018aes and AT 2022fnm.
There is a variety of properties in the light curves of the considered ILRTs, but we can roughly divide them in two subclasses, based on their behaviour in the 50 days after maximum. The first group of objects present slowly declining, convex light curves, which on average only fade by 0.8 mag from day 0 to day 50. The second group of objects display fast declining, concave light curves, which on average fade by more than 1.3 mag in the same time span (see Figure <ref>, lower panels). While this distinction is clear in some cases (AT 2019abn and AT 2019ahd are slow decliners, while AT 2019udc and AT 2022fnm are fast decliners), there are some objects with intermediate properties, such as AT 2013la, whose light curve quickly fades in the first two months, only to flatten after 75 d, ultimately becoming even brighter than AT 2019abn after 130 d.
Another outlier is AT 2018aes, which is characterised by the shallowest rise to maximum light, followed by a rather "standard" slow decline for 40 days before abruptly fading by 3.2 mag in 60 days.
The dichotomy between slow and fast declining ILRTs is reminiscent of the division between SNe IIP and IIL (e.g. ), with the progenitors of the slow declining ILRTs retaining larger hydrogen envelopes compared to the progenitors of fast declining ILRTs. Of course, a larger sample of ILRTs is needed to verify if this analogy holds true or if it is a mere coincidence.





§ A TOY MODEL FOR ILRTS LIGHT CURVES
 



In this section we present a simple model in the attempt to reproduce the shape of the bolometric light curves of ILRTs in the context of a weak SN explosion. The end goal is to obtain a rough estimate of parameters such as the amount of mass ejected and its velocity. The basic concepts are taken from <cit.>, which in turn expand the approach introduced by <cit.>. First of all, 56Ni radioactive decay is expected to be a relevant power source for SNe, and its luminosity over time is given by the equation:


    L_Ni(t) = 2M_Ni/t_de^-[t^2/t^2_d + 2R_0t/vt^2_d][(ϵ_NI-ϵ_Co)×
    ×∫^t_0[R_0/vt_d +t'/t_d]e^[t'^2/t^2_d + 2R_0t'/vt^2_d]e^-t'/t_Nidt' + 
     
        + ϵ_Co∫^t_0[R_0/vt_d +t'/t_d]e^[t'^2/t^2_d + 2R_0t'/vt^2_d]e^-t'/t_Codt'](1-e^-At^-2) ,


where M_Ni is the mass of synthesised 56Ni, t_Ni and t_Co are the half lives of 56Ni and 56Co, while ϵ_Ni and ϵ_Co are the energy generation rates of 56Ni and 56Co released during the radioactive decay. A is a constant that accounts for the opacity of the ejecta to γ-rays, with large values of A corresponding to complete trapping. The constant values are the same reported in <cit.>. This equation takes into account the diffusion time through an homologously expanding gas of mass M, initial radius R_0 and characteristic expansion velocity v. As discussed in Paper II, the velocity inferred from the width of the emission lines of our sample of ILRTs can reach only up to ∼10^3 km s^-1. However, the observed emission lines of ILRTs originate in a dense CSM which is obscuring the underlying material: as happens for SNe IIn, fast ejecta may be hiding below this layer of dense gas. Theoretical expectations for ECSNe suggest that their ejecta should be less energetic than a standard SN (v ≲ 10^4 km s^-1).
The characteristic timescale t_d of the light curve can be written as a combination of the diffusion timescale t_0 and the hydrodynamical timescale t_h:


    t_d = √(t_0 t_h)  t_0=kM/β c R_0 t_h = R_0/v ,


where β is a constant linked to the density profile of the mass, and k is the opacity of the ejecta, for which we adopt a value of 0.33 cm^2 g^-1. 

As an additional source of luminosity, we consider the radiation emitted by the expanding and cooling gas, which was heated by the blast wave that followed the explosion. This term was already introduced by <cit.> to reproduce the slowly declining light curves of SNe IIP. Using the formalism of <cit.>, such luminosity term is written as:


    L_blast(t) = E_th/t_0 e^-[t^2/t^2_d + 2R_0t/vt^2_d],


where E_th is the internal energy which was deposited in the ejecta during the explosion. To better reproduce the observed shape of the light curve, we consider two blast terms: one associated with the envelope, less massive and more extended, characterized by a short diffusion time, and another associated with the bulk of the ejecta, more massive and dense, with a much longer diffusion timescale. Dividing the ejecta in two regions is not a novelty: such approach has been successfully used, for example, by <cit.>.
The total time dependent luminosity is therefore the sum of these three contributions:


    L_tot = L_Ni + L_bulk + L_envelope .


The final piece of the model is the thick CSM surrounding the transient, which is a key feature observed in ILRTs spectra. 
This element does not provide additional energy, but it simply reprocesses the total luminosity emitted by the ejecta and 56Ni, delaying its appearance. To reproduce this effect, we use the fixed photosphere approximation presented by <cit.>:


    L_final(t) = 1/t_CSM e^-t/t_CSM∫^t_0 e^t'/t_0 L_tot(t')dt' ,


where t_CSM is the diffusion time through the thick CSM shell surrounding the transient. Its effect is crucial in the first phases, during which the blast terms L_bulk and L_envelope start abruptly from the maximum luminosity, without a rising phase. A CSM shell with a diffusion time of few days allow the model to reproduce the rise observed in ILRTs.
In Figure <ref> are shown the models and the bolometric light curves they are trying to emulate. The contribution of each component is shown with different colours: the parameters relative to each of them are reported in Table <ref>.
One weakness of the approach outlined above is the large number of free parameters, as well as the degeneracy between some of them: similar solutions can be obtained while inputting different parameters. Instead of blindly applying a fitting procedure, we tried to input reasonable parameters in the context of SNe, gradually modifying them to improve the agreement with the data.
An additional fragility of this approach is the assumption of spherical symmetry, which may be a crude approximation in the case of these transients (e.g. ).

AT 2019abn is definitely the most successful example within our sample. The emission from a rather massive envelope, combined with a CSM diffusion time of 10 days, seems to properly reproduce the rise and the maximum phases. The subsequent slow decline can be explained by slow ejecta which trap the photons for almost 200 days. The 56Ni contribution is not significant until the late phases, and it is definitely too faint to explain the evolution after 200 days: as mentioned in the previous section, an additional source of energy is likely needed to explain the late time luminosity of ILRTs. 

The parameters used to model AT 2019ahd do not differ significantly from the ones used for AT 2019abn: they are mainly scaled down in mass and internal energy of the ejecta. The blast term associated with the envelope is not as crucial, but the overall evolution appears to be reproduced sufficiently well, excluding of course the late shallow decline.
Since we miss the first evolutionary phases of NGC 300 OT, we only use a single blast (associated to the bulk of the ejecta) term plus 56Ni decay to reproduce the data: the blast term associated with the envelope is relevant during the early phases, so it is difficult to constrain in this case. Apart from this, the parameters used for NGC 300 OT are very similar to the ones used for AT 2019ahd.

The same conclusions cannot be reached for AT 2019udc, which is definitely an outlier in this sample. First of all, the fast decline hints at the presence of a single blast term. Most strikingly, the evolution of AT 2019udc is much faster compared to the other transients considered so far. For this reason the diffusion time within the ejecta must be remarkably lower: this leads to a low ejected mass and a high scale velocity, several times larger compared to the other ILRTs mentioned so far. Finally, it is worth noticing that in the case of AT 2019udc, 56Ni decay is a key element to reproduce the observed late time data points. 
All the masses mentioned so far are just relative to the ejected material: a compact remnant is expected to be left behind, in the scenario in which ILRTs arise from a core collapse event. Since the mass of the remnant can be estimated to be 1.3–2.0 M_⊙ <cit.>, the inferred mass of our sample of ILRTs right before the explosion adds up to 5.3–6.0 M_⊙ for AT 2019ahd,  5.5–6.2 M_⊙ for NGC 300 OT and 9.3–10 M_⊙ for AT 2019abn. Such values are compatible with the expectations for SAGB stars and consequently ECSN events, especially accounting for the fact that some mass was likely lost during the evolution of the star, therefore leading to a slightly higher ZAMS. AT 2019udc, with its estimated progenitor mass of just 2.2–2.9 M_⊙ at the time of explosion, sets itself apart from the rest of the sample, although its origin could be tentatively explained as an ECSN arising from a SAGB star that underwent extreme mass loss.
We remark that the toy model presented in this section is a simplified approach to the light curve modelling of ILRTs: more detailed models envisioning hydrodynamical modelling should be used to retrieve more robust estimates of the explosion parameters of these objects. Accounting for the possible luminosity contribution from ejecta-CSM interaction will also be an important step forward: while ILRTs never display spectral signatures typically associated with shocks, there may be hidden CSM interaction powering their light curve (Paper II).
The puzzling results obtained for AT 2019udc may be significantly revised by using a more refined approach.



§ SUMMARY AND CONCLUSIONS
 
In this work we present data sets for NGC 300 OT, AT 2019abn, AT 2019ahd and AT 2019udc. AT 2019abn and AT 2019udc stand out for their bright peak magnitude (M_r=–15 mag and –14.5 mag respectively). On the other hand, NGC 300 OT and AT 2019ahd are characterised by a more modest peak magnitude (M_r= –12.7 mag and –13.2 mag). We detect a NIR excess in the SED of three of our ILRTs, tentatively associated with dust formation, in three out of four targets. The only exception is AT 2019udc, characterised by the fastest evolution timescale among the ILRTs observed. At late phases, the contributions in the NIR and MIR domains to the bolometric light curve cause a decline shallower than the luminosity decline supported by the 56Ni decay, hinting at the presence of an additional powering mechanism.

The late time monitoring of NGC 300 OT with WISE allows us to build the SED of the transient 761 days after maximum. From its study, we infer a dust mass of the order of ×10^-5-×10^-3 M_⊙, depending on the composition and size of the grains adopted. Furthermore, the evolution of NGC 300 OT has been monitored with Spitzer for several years, showing a steady dimming of the flux at the transient position over 4 years. This strengthens the argument suggested by <cit.> that NGC 300 OT was a terminal event. Thanks to JWST observations we were able to show that also AT 2019abn faded below its progenitor luminosity in the MIR domain. Its SED appears to depart from a simple blackbody, possibly due to the presence of PAH emission features.
Finally, a simple model is presented in the context of a SN explosion, with the goal to estimate the parameters characterising the transients. For AT 2019abn, 2019ahd and NGC 300 OT, the low masses ejected (between 4 and 8 M_⊙) and low velocity of the material (up to 4500 km s^-1) are compatible with a weak explosion from a low mass star. AT 2019udc is an outlier in this sense, since its very fast decline rate leads to inferring high velocity (12000 km s^-1) and remarkably low ejected mass (0.9 M_⊙). 
Overall, due to their low energy and their apparent terminal nature, ILRTs remain solid candidates for being ECSNe.



§ ACKNOWLEDGEMENTS


We thank the staff of the various observatories where data were obtained for their assistance.
Based on observations made with the Nordic Optical Telescope, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and the University of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.
Observations from the Nordic Optical Telescope were obtained through the NUTS2 collaboration which are supported in part by the Instrument Centre for Danish Astrophysics (IDA). The data presented here were obtained in part with ALFOSC, which is provided by the Instituto de Astrofisica de Andalucia (IAA).
This work makes use of data from the Las Cumbres Observatory network. The LCO team is supported by NSF grants AST–1911225 and AST–1911151, and NASA SWIFT grant 80NSSC19K1639. 
Data were also obtained at the Liverpool Telescope, which is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos with financial support from the UK Science and Technology Facilities Council. Part of the observations were collected at Copernico and Schmidt telescopes (Asiago, Italy) of the INAF – Osservatorio Astronomico di Padova.
Based on observations made with the Gran Telescopio Canarias (GTC), installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the island of La Palma
This work has made use of data from the Asteroid Terrestrial-impact Last Alert System (ATLAS) project. 
Time domain research by the University of Arizona team and D.J.S.  is supported by NSF grants AST-1821987, 1813466, 1908972, & 2108032, and by the Heising-Simons Foundation under grant #20201864. 
G.V., N.E.R. and I.S. acknowledge financial support from PRIN-INAF 2022 “Shedding light on the nature of gap transients: from the observations to the models".
N.E.R. also acknowledge support from the Spanish MICINN grant PID2019-108709GB-I00 and FEDER funds, and from the program Unidad de Excelencia Maria de Maeztu CEX2020-001058-M.
A.R. acknowledges financial support from the GRAWITA Large Program Grant (PI P. D’Avanzo) and the PRIN-INAF 2022 "Shedding light on the nature of gap transients: from the observations to the models.
Y.-Z. Cai is supported by the National Natural Science Foundation of China (NSFC, Grant No. 12303054), the Yunnan Fundamental Research Projects (Grant No. 202401AU070063) and the International Centre of Supernovae, Yunnan Key Laboratory (No. 202302AN360001).
T.E.M.B. acknowledges financial support from the  
Spanish Ministerio de Ciencia e Innovacion (MCIN), the Agencia  
Estatal de Investigacion (AEI) 10.13039/501100011033, and the  
European Union Next Generation EU/PRTR funds under the 2021 Juan de la  
Cierva program FJC2021-047124-I and the PID2020-115253GA-I00 HOSTFLOWS  
project, from Centro Superior de Investigaciones Cientificas (CSIC)  
under the PIE project 20215AT016, and the program Unidad de Excelencia  
Maria de Maeztu CEX2020-001058-M.
MF is supported by a Royal Society - Science Foundation Ireland University Research Fellowship.
The Aarhus supernova group is funded by the Independent Research Fund Denmark (IRFD, grant numbers 8021-00170B, 10.46540/2032-00022B).
L.G. acknowledges financial support from the Spanish Ministerio de 
Ciencia e Innovación (MCIN), the Agencia Estatal de Investigación 
(AEI) 10.13039/501100011033, and the European Social Fund (ESF).
This work was funded by ANID, Millennium Science Initiative, ICN12_009
M.N. is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 948381). Part of the funding for GROND (both hardware as well as personnel) was generously granted from the Leibniz-Prize to Prof. G. Hasinger (DFG grant HA 1850/28-1).
H.K. was funded by the Academy of Finland projects 324504 and 328898. 
R.K. acknowledges support from the Research Council of Finland (340613).


aa




§ AFFILIATIONS

^1INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy

^2INAF – Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807, Merate (LC), Italy 

^3Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, P.R. China 

^4Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, P.R. China 

^5International Centre of Supernovae, Yunnan Key Laboratory, Kunming 650216, P.R. China 

^6 Graduate Institute of Astronomy, National Central University, 300 Jhongda Road, 32001 Jhongli, Taiwan 

^7SRON, Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA, Leiden, The Netherlands 

^8Department of Astrophysics/IMAPP, Radboud University Nĳmegen, P.O. Box 9010, 6500 GL, Nĳmegen, The Netherlands 

^9Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, E-08193 Barcelona, Spain 

^10School of Physics, O’Brien Centre for Science North, University College Dublin, Belfield, Dublin 4, Ireland 

^11The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, 10691 Stockholm, Sweden 

^12Hiroshima Astrophysical Science Center, Hiroshima University, Higashi-Hiroshima, Japan 

^13Department of Physics, Florida State University, 77 Chieftan Way, Tallahassee, FL 32306, USA 

^14Las Cumbres Observatory, 6740 Cortona Dr. Suite 102, Goleta, CA, 93117, USA

^15Department of Physics, University of California, Santa Barbara, CA, 93106, USA

^16School of Physics & Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK 

^17INAF, Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, I-80131 Napoli, Italy 

^18DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark 

^19Caltech, Mail Code 220-6, Pasadena, CA 91125 

^20Tuorla Observatory, Department of Physics and Astronomy, University of Turku, 20014, Turku, Finland 

^21Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK 

^22Max-Planck-Institut für Astrophysik, Karl-Schwarzschild Str. 1, D-85748 Garching, Germany 

^23Aryabhatta Research Institute of observational sciences, Manora Peak, Nainital, 263001, India 

^24Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile 

^25School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland 

^26Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA

^27Department of Physics, University of Oxford, Keble Road, Oxford, OX1 3RH 

^28Astrophysics Research Centre, School of Mathematics and Physics, Queens University Belfast, Belfast BT7 1NN, UK 

^29Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark 

^30INAF – Osservatorio Astronomico d’Abruzzo, via M. Maggini snc, Teramo, I-64100, Italy 

^31Department of Physics, University of California, Davis, CA 95616, USA 

^32 European Southern Observatory, Alonso de Córdova 3107, Casilla 19, Santiago, Chile 

^33Millennium Institute of Astrophysics, Nuncio Monsenor Sótero Sanz 100, Providencia, 8320000 Santiago, Chile 

^34INAF-Osservatorio Astrofisico di Catania, Via Santa Sofia 78, I-95123 Catania, Italy 

^35Instituto de Astrofísica, Universidad Andres Bello, Fernandez Concha 700, Las Condes, Santiago RM, Chile 

^36 ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy 

^37Institut für Theoretische Physik, Goethe Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany 

^38INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Via Sommarive 14, I-38123 Trento, Italy 

^39Institut d’Estudis Espacials de Catalunya (IEEC), E-08034 
Barcelona, Spain. 

^40Center for Astrophysics, Harvard & Smithsonian, Cambridge, Massachusetts, MA 02138, US 

^41The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, USA 

^42Finnish Centre for Astronomy with ESO (FINCA), University of Turku, Väisäläntie 20, 21500 Piikkiö, Finland 

^43DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, 2800Kgs. Lyngby, Denmark 

^44Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli studi di Padova Vicolo dell’Osservatorio 3, I-35122 Padova, Italy 

^45IAASARS, National Observatory of Athens, Metaxa & Vas. Pavlou St., 15236, Penteli, Athens, Greece 

^46Department of Astronomy, University of Virginia, Charlottesville, VA 22904, USA 

^47Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA 

^48Cosmic Dawn Center (DAWN), Denmark 

^49Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 København N, Denmark 

^50Manipal Centre for Natural Sciences, Manipal Academy of Higher Education, Manipal - 576104, Karnataka, India 

^51Indian Institute Of Astrophysics, 100 Feet Rd, Santhosapuram, 2nd Block, Koramangala, Bengaluru, Karnataka 560034, India































§ ADDITIONAL TABLES AND FIGURES








































§ MEASURED DATA
 















































 
 
 














