Paper Title: Early Planet Formation in Embedded Disks (eDisk) XVI: An asymmetric dust disk driving a multi-component molecular outflow in the young Class 0 protostar GSS30 IRS3.



   
   

   European Southern Observatory, 3107, Alonso de Córdova, Santiago de Chile

              alejandrosantamariamiranda@gmail.com
         Academia Sinica Institute of Astronomy & Astrophysics, 11F of Astronomy-Mathematics Building, AS/NTU, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan, R.O.C.
        National Radio Astronomy Observatory, 520 Edgemont Rd, Charlottesville, VA, 22903, USA
        Niels Bohr Institute, University of Copenhagen, Øster Voldgade 5-7, 1350 Copenhagen K., Denmark
        Korea Astronomy and Space Science Institute, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea
        University of Virginia, 530 McCormick Rd., Charlottesville, Virginia 22904, USA
        Department of Physics and Astronomy, Graduate School of Science and Engineering, Kagoshima University, 1-21-35 Korimoto, Kagoshima,Kagoshima 890-0065, Japan
        Department of Earth Science Education, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea
        SNU Astronomy Research Center, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea
        Division of Astronomy and Space Science, University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea
        Department of Astronomy, University of Illinois, 1002 West Green St, Urbana, IL 61801, USA
        Department of Astronomy, University of Michigan, 1085 S. University Ave., Ann Arbor, MI 48109-1107, USA
        Institute for Astronomy, University of Hawai‘i at Mānoa, 2680 Woodlawn Dr., Honolulu, HI 96822, USA
        



   

 
We present the results of the ALMA Large Program Early Planet Formation in Embedded disks observations of the Class 0 protostar GSS30 IRS3. Our observations included 1.3 mm continuum with a resolution of 005 (7.8 au) and several molecular species including  ^12CO, ^13CO, C^18O, H_2CO and c-C_3H_2. The dust continuum analysis unveiled a disk-shaped structure with a major axis size of ∼200 au. We observed an asymmetry in the minor axis of the continuum emission suggesting that the emission is optically thick and the disk is flared. On the other hand, we identified two prominent bumps along the major axis located at distances of 26 and 50 au from the central protostar. The origin of the bumps remains uncertain and might be due to an embedded substructure within the disk or the result of the temperature distribution instead of surface density due to optically thick continuum emission. The ^12CO emission reveals a molecular outflow consisting of three distinct components: a collimated one, an intermediate velocity component exhibiting an hourglass shape, and a wider angle low-velocity component. We associate these components with the coexistence of a jet and a disk-wind. The C^18O emission traces both a Keplerian rotating circumstellar disk and the infall of the rotating envelope. We measured a stellar dynamical mass of 0.35±0.09 M_⊙. 

 
Early Planet Formation in Embedded Disks (eDisk) XVI: An asymmetric dust disk driving a multi-component molecular outflow in the young Class 0 protostar GSS30 IRS3
    Alejandro Santamaría-Miranda1
          Itziar de Gregorio-Monsalvo1
          Nagayoshi Ohashi2  
          John J. Tobin3
          Jinshi Sai 2
          Jes K. Jørgensen4
          Yusuke Aso5
          Zhe-Yu Daniel Lin6
          Christian Flores2
          Miyu Kido7
          Patrick M. Koch2
          Woojin Kwon8,9
          Chang Won Lee5,10
          Zhi-Yun Li6
          Leslie W. Looney3,11
          Adele L. Plunkett3
          Shigehisa Takakuwa2,7
          Merel L.R  van 't Hoff12
          Jonathan P. Williams13
          Hsi-Wei Yen2
        
    Received 14 March 2024; accepted 29 July 2024
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§ INTRODUCTION


Circumstellar disks arise as a direct consequence of the angular momentum conservation of rotating material from an infalling envelope  <cit.>. In these disks, planets are expected to form at some point during the star formation process. In the last decade, the efforts to understand planet formation have been focused mainly on Class II disks which have dispersed their envelopes. The advent of the Atacama Large Millimeter/submifllimeter Array (ALMA) provided the needed sensitivity and spatial resolution to resolve structures inside disks such as rings, gaps, and dust traps <cit.>. These structures are ubiquitous in the star-forming regions studied in different ALMA programs (e.g., ) and may reflect the presence of planets <cit.>.

If those substructures are the consequence of the presence of planets, we should investigate the origin of planets in earlier evolutionary stages. The lack of sufficient mass to form the observed population of exoplanets in Class II disks <cit.> and the fact that Class 0/I disks are likely more massive and solid-abundant <cit.> support the scenario where planets are formed at earlier stages. 


Dedicated high angular resolution observations millimeter/sub-millimeter wavelengths have revealed the presence of substructures in the dust disks surrounding some young Class 0/I sources <cit.> but it remains a question whether these are ubiquitous and when they arise. To address this question, the ALMA Large Program 'Early Planet Formation in Embedded disks' <cit.> aims to study the origin of substructures systematically in Class 0/I disks with ALMA at 1.3 mm (Band 6) and a spatial resolution of 004. The sample comprises twelve Class 0 and seven Class I protostars in nearby (d < 200 pc) star-forming regions. The program aimed at detecting hints of disk substructures that may indicate the presence of planets or the presence of giant spirals as a byproduct of gravitational instabilities. 

The source GSS30 IRS3 was observed as part of the eDisk Large Program. It is part of the  L1688 molecular cloud, located at a distance of ∼138.4 ± 2.6 pc <cit.> based on the recent Gaia measurements. GSS30 IRS3 is a deeply embedded protostar <cit.>, not detected at optical wavelengths, preventing a direct measurement of its distance from Gaia. Therefore, we assume that the distance to the source is similar to its host cloud. GSS30 IRS3 is classified as a Class 0 protostar, with a bolometric luminosity (L_bol) of  1.7 L_⊙, and the bolometric temperature is 50 K <cit.>. GSS30 IRS 3 was first detected by <cit.>  and later observed at 2.7 mm at higher-angular resolution in a focused paper on the three sources at GSS30 <cit.>. The existence of a molecular outflow has been reported with single-dish observations taken at the James Clerk Maxwell Telescope <cit.> and later with ALMA <cit.> in CO (2-1) at a resolution of ∼15.

In this paper, we present observations of GSS30 IRS3 at a very high angular resolution (0056) in the continuum (at 1.3 mm) and (∼015) in line emission (CO isotopologues, H_2CO and c-C_3H_2) as part of the eDisk ALMA Large Program. The paper is structured as follows: we describe the observations in Section <ref>. Then, we present the continuum, spectral line maps, and dust mass estimation in Section <ref>. Section <ref> presents the molecular outflow analysis and the estimation of the protostar's dynamical mass. We discuss the origin of the continuum asymmetries, twisted feature observed in the C^18O velocity map and the coexistence of a jet and a wide-angle outflow in Section <ref>. Finally, we summarize the main results in Section <ref>. 


 








§ OBSERVATION AND DATA REDUCTION


GSS30 IRS3 was observed with the ALMA 12-m array in Band 6 as part of the eDisk ALMA Large Programme (project code: 2019.1.00261.L: PI N. Ohashi). A separate DDT program (2019.A.00034.S: PI J. Tobin) complements the eDisk data with shorter baselines for the same molecular lines for some sources. Observations toward GSS30 IRS3 from the Large Program were executed four times in October 2021 in configuration C43-8, while two executions were completed for the DDT program in June 2022 with configuration C43-5. Detailed information about the baseline length, number of antennas, precipitable water vapor, and calibrators are provided in <cit.>. 

The correlator was set up in dual-polarization mode using four basebands. The first baseband was divided into four spectral windows to detect C^18O (2-1), ^13CO (2-1), H_2CO (3_2,1-2_2,0), and SO (6_5-5_4) with a spectral resolution of ∼0.17 km s^-1 and a bandwidth of 58.59 MHz. Basebands two and three were set up with a bandwidth of 1875 MHz to detect continuum. A spectral resolution of ∼1.3 km s^-1 was selected for the detection of several molecular lines (CH_3OH, SiO, DCN, H_2CO, and c-C_3H_2) and those lines were identified and masked when measuring the continuum. The fourth baseband has a bandwidth of 937.5 MHz (0.64 km s^-1 spectral resolution) to detect the continuum and the ^12CO (2-1) line. The time on source for data from the Large Program was 2.75 h, and for the DDT program the time on source was 0.5 h. 

Calibration of the ALMA data was performed using the standard ALMA calibration pipeline. Then, we self-calibrated the data using the Common Astronomy Software Applications package <cit.> version 6.2.1 and 6.4.1 for 2019.1.00261.L and 2019.A.00034.S programs, respectively. The pipeline version numbers should be supplied in addition to just the CASA number. Before performing the self-calibration, we first imaged the six executions and aligned the emission peaks using fixvis and fixplanets. After the alignment, we rescaled the flux in the UV plane to the execution on the 27 October. Then we proceeded with the self-calibration but using the rescaled non-aligned data, first with four phase-only cycles and two phase and amplitude cycles of the short-baseline data. Next, we performed joint self-calibration of the long and short baseline data and proceeded with four iterations of self-calibration in phase-only mode. Finally, the self-calibration solutions were applied to both the continuum and the spectral line data, and the task TCLEAN was used to produce continuum and spectral line images after continuum subtraction. The final image is a combination of the short and long configurations. We adopted a Briggs robust parameter equal to 0 for the continuum image and 0.5 for the spectral line data as a compromise between sensitivity and resolution. We also adopted a Briggs robust parameter equal to 2 for the H_2CO and c-C_3H_2 spectral to enhance the sensitivity. The spectral line data were tapered at 2000 kλ to bring out larger-scale structures. Primary beam correction was applied before inferring physical parameters from the images. The angular resolution of the continuum maps is 0056 (7.8 au at a distance of 138 PC), with a maximum recoverable angular scale of 29, which is also the nominal value for Configuration 5 and a field of view of ∼22”. The absolute flux calibration uncertainty in Band 6 is estimated to be 10%. A more detailed explanation of the reduction procedure can be found in <cit.>. 



§ RESULTS













 §.§ Dust emission
 
Figure <ref> shows the dust emission at 1.3 mm of GSS30 IRS3, at an angular resolution of 0068×0045 (Position angle of 76.61^o), obtained using a robust parameter equal to 0. The continuum emission traces a disk-like structure around the central protostar with a high inclination angle. The deconvolved disk size is  055×017 (76×23 au) estimated from a two-dimensional Gaussian fitting (see Table <ref>). The best fit peak intensity is 3.87 ± 0.04 mJy beam^-1 at the Gaussian central position R.A.=16h26m21.715s, Dec= -24^o 22'5109. The flux density obtained from the Gaussian fitting is 123.6 ± 1.2 mJy. The continuum emission measured above a 5σ contour has major and minor axes of 143×062 (∼198 ×∼86 au), while the peak intensity and the flux density obtained by this method are 6.38 mJy beam^-1 and 133.7 mJy, respectively. Since the residuals from the Gaussian fitting show that the fit is far from perfect (see model and residual in Figure <ref>), and also a fitting attempt using two (2-D) Gaussian components did not converge, we will adopt the values of the flux density and size inferred above the 5σ contour in the analysis section. 
Assuming a geometrically flat and axisymmetric disk, we can estimate the disk inclination as cos i = b / a, where a and b  are the major and minor axes. The inclination derived using the deconvolved Gaussian fitting is 71.7 ± 0.3^o while using the 5σ contour, we obtained 64.3 ± 1.5^o, with the same position angle in both cases (109^o). Given the poor quality of the Gaussian fitting, we will adopt a value of 64.3^o as the most reliable inclination estimation. We note that if the disk is geometrically thick this value should be treated as a lower limit.  Additionally, the dust disk exhibits a 'boxy' shape (see Section <ref> for further discussions). 

Moreover, we also detected the dust emission of another source (GSS30 IRS1) in the field of view. See Section <ref> for further information.

  




 §.§ Brightness asymmetries

There are no apparent substructures in the disk surrounding GSS30 IRS3 (Figure <ref>), such as clear spirals, gaps, or sharp rings, similar to those found in many more evolved Class II sources <cit.>. Instead, the image shows a striking asymmetry along the minor axis. To show that asymmetry more clearly, we performed a one-dimensional cut along the minor axis centered on the peak derived from the Gaussian fitting (Figure <ref>). 
Additionally, along the major axis, there are two bumps on both sides of the disk, located at 018  (∼25 au) and 036 (∼50 au) East and West from the disk center, respectively. There is only one bump along the minor axis, which could be due to a flared structure of the disk.  The center of the bump is located at a deprojected distance of ∼20 au from the center of the disk. The radial brightness profile of the continuum emission is discussed in Sections <ref> and <ref>. 





 §.§ Dust disk mass

The dust disk mass surrounding GSS30 IRS3 is estimated under the assumption that the emission at 1.3 mm is optically thin and comes entirely from isothermal dust. We estimated the dust mass using the relationship from <cit.>:


    M =  S_ν D^2/κ_νB_ν(T_d),

where D is the distance to the source (138 pc),  B_ν(T_d) is the Planck function at the observed frequency (225 GHz) at a dust temperature T_d, and κ_ν(2.3 cm^2 g^-1) is the absorption coefficient adopted from <cit.>. S_ν is the flux density. We assumed  two different values of the dust temperature to estimate the dust disk mass. The first temperature value we used was a fixed value of 20 K, commonly assumed for Class II sources <cit.>. For the second temperature, we took into account the typical temperature of the disk based on the source's luminosity. We derived this temperature using the formula T= 43(L_bol/L_sun)^0.25 from <cit.>, which was obtained from a grid of radiative transfer models. For our specific case of GSS30 IRS3, this formula provides a temperature of 49 K.
Using these two temperature values, we calculated the dust disk mass using the flux density derived from the Gaussian fitting and the one calculated above the 5σ contour level. For the Gaussian fitting, we obtained a dust disk mass of 69.5 ± 2.6 M_ and 23.87 ± 0.90 M_, respectively, for a temperature of 20 K and 49 K. The dust mass estimation when using the flux density above the 5σ contour level provides values of  75.1 ± 2.9 M_ and 25.80 ± 0.99 M_ considering a temperature of 20 K and 49 K, respectively. The values that we derived are between 4.9 and 1.5 times more massive than the ones that were derived by <cit.>, however, their κ_ν value is much larger (7.4 cm^2 g^-1)  than the value used in this paper, so we associate the differences to the adopted absorption coefficient. Comparing to other Class 0 sources in the eDISK sample we find relatively similar values to R CrA IRS 5N (protostellar mass of 0.3 M_⊙) with a mass dust disk between 23.3 and 66.6 M_ <cit.> or L1527 (protostellar mass of 0.5 M_⊙) with a mass dust disk between 38.9 and 97.4 M_ <cit.>. Finally, we note that the derived dust mass might be a lower limit than the actual mass because the inclined disk could be optically thick.



 §.§ Gas emission

We report the detection of several molecular species and transitions: ^12CO (2-1), ^13 CO (2-1), C^18O (2-1), c-C_3H_2 (blended lines 6_0,6-5_1,5 and 6_1,6-5_0,5), c-C_3H_2 (5_1,4-4_2,3),   c-C_3H_2 (5_2,4-4_1,3),  H_2CO (3_0,3-2_0,2) and H_2CO (3_2,2-2_2,1). The DCN (3-2), H_2CO  3_2,1-2_2,0, SO (6_5-5_4) and SiO (5-4), lines are not detected. The systemic velocity is 2.84 km s^-1 (which is derived from the C^18O (2-1) diagram fitting discussed in Section <ref>). Moment maps of the detected molecular transitions are presented in Figures <ref>, <ref> and <ref>, where we show integrated intensity (moment 0), mean velocity (moment 1), and peak intensity (moment 8) maps. A summary of the main properties of the detected lines including the angular resolution for each molecule is shown in Table <ref>, and an extensive description is provided in the following subsections.










  §.§.§ ^12CO
 
The ^12CO (2-1) emission line traces primarily a bipolar molecular outflow centered at the position of GSS30 IRS3 and perpendicular to the dust disk, as was also previously reported by <cit.> using lower angular resolution and less sensitive observations. Our ALMA data reveal three different outflow components in morphology and velocity:  a collimated inner high-velocity component,  an intermediate velocity component with an hourglass or conical shape,  and a wider angle lower velocity component that is more prominent in the northern lobe which has a parabolic shape (see Figure <ref>). The velocity channel maps of the  ^12CO (2-1) emission are presented in  Figure <ref> and the intensity-weighted velocity map in Figure <ref>.  In Figure <ref>, we plot the ^12CO contours overlaid on a recent JWST image of the ρ Ophiuchus region (P.I. Klaus Pontoppidan) where the H_2(0-0 S9) emission is tracing the outflow and delimiting very clearly the cavity wall beyond the molecular outflow. 

The collimated high-velocity component shows blueshifted emission to the northeast from the central source (spanning velocities between -19.9 and -18.7 km s^-1 in local standard rest velocity) and redshifted emission to the southwest (23.2 to 23.8 km s^-1; top left panel Figure <ref>). The JWST image reveals the presence of knots located at 49 and 67 from the center of the GSS30 IRS3 disk in the same direction of the blueshifted high-velocity component (right panel in Figure <ref>). Correcting the outflow velocities from the inclination of the system, and considering that the inclination angle is a lower limit of the actual angle, the maximum deprojected velocity in the inner collimated high-velocity component is 52 km s^-1. It is important to note that the deprojected velocity derived is also a lower limit because  the inclination angle is a lower limit of the actual angle. This deprojected velocity falls within the expected velocity range observed in jets associated with Class 0 sources (100 ± 50 km s^-1, ).

The intermediate velocity component of the outflow delineates an hourglass shape, and spanning velocities between  -18.0 and -2.2 km s^-1 for the blueshifted lobe and between 8.6 to 22.6 km s^-1 for the redshifted lobe. Both redshifted and blueshifted emission are present in both sides, as expected in outflows with a non-narrow opening angle and close to the plane of the sky  (see the top right panel in Figure <ref>). 
The wider angle low-velocity component of the molecular outflow shows blueshifted emission between -1.6 and 1.6 km s^-1m and redshifted emission between 5.4 and 7.9 km s^-1. It is formed by an extended but well-defined northeast arc mostly seen at blueshifted velocities and less extended southwestern lobe where both blueshifted and redshifted components overlay in the same area of the sky (see the bottom left panel in Figure <ref>). 


We also report the detection of ^12CO (2-1) in two other sources (GSS30 IRS1 and IRS2) that are in the ALMA field of view. GSS30 IRS1 observations will be presented in a future paper while the IRS2 detection is in Appendix <ref>.



  §.§.§ ^13CO
 
The ^13CO (2-1) emission line, an optically thinner line due to its lower abundance than the ^12CO (2-1) line, is detected at a peak S/N of 10. The velocity channel maps of the ^13CO (2-1) emission line are presented in Figure <ref>.
The blueshifted emission is found at velocities between -0.66 km s^-1 and  2.18 km s^-1, and it is stronger to the west of the central protostar. The redshifted emission is much fainter and found mainly to the east of the protostar. It spans velocities between 4.35  km s^-1 and 6.19  km s^-1. 

Two different faint structures are observed in the channel maps, tracing both the base of the northeast and southwest components of the molecular outflow and a gaseous disk-like structure surrounding the central protostar. The emission associated with the molecular outflow is only detectable between velocities of 0.35 km s^-1 and 2.35 km s^-1. The disk-like structure is detected between -0.66 and 2.18 km s^-1 (blueshifted component) and between 4.35 km s^-1 to 6.19 km s^-1 (redshifted component) velocities. The blueshifted component is more intense than the redshifted one. We plot the position-velocity diagram (PV) in Figure <ref> to discriminate between redshifted emission associated with cloud or disk. Although it is a very faint emission, it is compatible with a rotating disk, which will be further discussed in Section <ref>.




  §.§.§ C^18O
  
The C^18O (2-1) emission line is often considered optically thin and trace envelopes and disks, although it is only sometimes optically thick (e.g. ). Blueshifted emission is detected at velocities ranging from 0.01 km s^-1 to 2.68 km s^-1 (see Figure <ref>). Redshifted emission spans velocities between 3.02 km s^-1 and 5.18 km s^-1. The size of the emission traced by the C^18O (2-1) emission above the 5σ contour is 23×093, with a position angle similar to the 109^o of the dust disk.  

In GSS30 IRS3, C^18O (2-1) clearly traces a disk-like structure with an obvious redshifted and blueshifted pattern compatible with a Keplerian rotating disk (Figure <ref>). When studying the velocity map, we note a weak S-shape close to the systemic velocity (Figure <ref>). Its nature will be discussed in Section <ref>. 
The spectrum of C^18O (2-1) shows a double-peak profile typical of an inclined disk in Keplerian rotation.



  §.§.§ c-C_3H_2
 
 In Class 0 sources, c-C_3H_2 is usually located in the outflow cavity walls due to exposure to UV radiation from the central protostar <cit.>. On the other hand, in Class I sources, hydrocarbons such as c-C_3H_2 are located in the disk <cit.>. We detected the three targeted transitions of c-C_3H_2 at 217.822 GHz, 217.94 GHz, and 218.160 GHz. The first one has two lines blended (6_0,6-5_1,5 and  6_1,6-5_0,5) that are very close (<<m/s) for the kinematic analysis, and it is more intense than the other two detections (see Figure <ref>). c-C_3H_2 at 217.822 GHz and 217.94 GHz shows emission between V_lsr of 0.7 and 4.7 km s^-1 while the c-C_3H_2 at 218.16 GHz is only detected between 0.7 and 3.5 km s^-1. In GSS30 IRS3, the c-C_3H_2 emission is located in the same position as the dust disk Figure <ref>). The association of the c-C_3H_2 detection is consistent with both the disk and the disk wind based on its positional information. Regarding kinematics, its velocities align with a rotation pattern. Therefore we tentatively attribute the association of c-C_3H_2 with the disk-like structure.




  §.§.§ H_2CO
 
Two transitions, H_2CO (3_0,3-2_0,2) and H_2CO (3_2,2-2_2,1), out of three, are detected. The transition H_2CO (3_0,3-2_0,2) at 218.22 GHz seems to be located in the same position as the external part of the dust disk and also exhibits a rotation pattern (Figure <ref>). H_2CO (3_0,3-2_0,2) emission is detected at Vlsr values of 0.7 and 4.7 km s^-1. The transition H_2CO (3_2,2-2_2,1) is tentatively detected at the 3σ level in one channel at a V_lsr of 0.7 km s^-1.



§ ANALYSIS












 §.§ Disk rotation

The C^18O (2-1) emission line traces a disk-like structure exhibiting indications of rotation (Section <ref>). It is important to investigate the nature of the rotation, in particularly whether it is Keplerian. If it is Keplerian rotation, the dynamical mass of the central star can be derived. In order to investigate the rotation curve of the C^18O (2-1) emission, we will use a PV diagram. The C^18O (2-1) line is the most suitable tracer for this analysis because the disk-like structure is clearly detected with good S/N. This is consistent with being optically thinner than the ^13CO (2-1) line, which partially traces the gaseous disk in GSS30 IRS3. In any case, we analyze the rotation curve of both molecular transitions to compare the obtained results. For computing the position-velocity diagram, we use a PV diagram cut of 017 width, that corresponds to the size of the major axis of the synthesized beam, along the major axis of the C^18O (2-1) and  ^13CO (2-1) disk. We adopt a disk position angle value of 109^o estimated in Section <ref>. 

The SLAM code[<https://github.com/jinshisai/SLAM>] <cit.> is used to analyze the rotation curve of GSS30 IRS3. The process is based on two main steps; the first step is to obtain a 1-D intensity profile to derive the edge and ridge data points. Ridge points are defined as the intensity-weighted mean position or the center in the 1-D intensity profile from the Gaussian fitting. The edge points or radius is the outermost position at a certain S/N level. The second step is fitting a power-law (single or double) function, that is defined as a positive value, to the derived representative data as a function of velocity or radius using the Markov-Chain Monte Carlo method. Whether it is a function of radius or velocity depends on along which axis the 1D profile for ridge/edge calculations is obtained see <cit.> criteria to use the v_i-r_i data points as a function of radius or velocity. Note that the edge and ridge data points are independently fitted. Thus, the double power law implies two components having different rotation laws, such as a Keplerian disk and an infalling envelope conserving its specific angular momentum, where the break radius (r_b) is the boundary radius between a Keplerian disk and a rotation/infalling envelope, and where the break velocity (v_b) is the velocity at (r_b). We also derived the reduced chi-squared (χ^2) for each fit in order to compare all the results. A more detailed explanation of the SLAM methodology for the eDisk papers can be found in <cit.>.

The SLAM code can fit the systemic velocity as a free parameter. We ran a first test to derive the systemic velocity as 2.84 km s^-1 for the C^18O (2-1) emission, then we checked this value with the C^18O (2-1) spectra and fixed this value for the following test until we derived the final dynamical mass. 

We tested a single-power law (bottom-left panel in Figure <ref>) and double-power law fit (bottom-right panel in Figure <ref>) to the C^18O (2-1) PV diagram. The single power-law fittings give power-law indices (p_in) of ∼0.71 ± 0.03 for the edge method and ∼0.65±0.02 for the ridge method as shown in Table <ref>. These values are between 0.5 (Keplerian rotation) and 1 (constant specific angular momentum), suggesting that the observed rotation may not be explained by only Keplerian rotation and may also include rotation in an infalling envelope. To investigate such possibility, double power-law fittings were also performed; the fitting results with the edge method give a power-law index (p_in) of ∼0.45±0.08 and dp is the difference of the index between the inner and outer parts that is ∼0.65±0.15 with r_b of 107^+26_-19 au as shown in Table <ref>, meaning that the power-law index for the outside of r_b is ∼1.1. These results suggest that the observed rotation may be explained as a combination of Keplerian rotation and rotation conserving its specific angular momentum. This kind of transition of the rotation between a Keplerian disk and an infalling envelope was observed before in some other protostars as well <cit.>. The ridge method was not converging using p_in as a free parameter, so we fixed it at 0.5 under the assumption that there is a Keplerian disk and we derived a mass of  0.26  ± 0.07 M_⊙. Then, the mass derived from the single power-law fit is 0.27 ± 0.07 M_⊙ and  0.35 ± 0.09 M_⊙ for the double power law fit as the average of the ridge and edge methods because the edge and the ridge methods over- and under-estimate the stellar mass by a factor that depends on the resolution <cit.>.

For the ^13CO (2-1) line, we only considered the blueshifted component of the emission for the fit, since the redshifted component is very faint. The single-power law fit (top-left panel in Figure <ref>) shows a lower value of p_in than the C^18O (2-1) single-power fit. The double-power law fitting is not converging. The single-power fit is returning a mass at the break radius (M_b) of 0.24 ± 0.01 M_⊙. The edge method results were discarded because we are not obtaining stable solutions after several iterations. Additional details about the fitting are described in Appendix <ref>.

The best χ^2, ignoring the value of ridge double-power law because p_in is fixed, is the one obtained from  C^18O (2-1) single power fit ridge method, however the p_in is not close to the expected value of a Keplerian disk, so we should discard this value. Then, the second best  χ^2 value is the one obtained from C^18O (2-1) edge double-power law fit. Therefore, we adopt a mass of GSS30 IRS3 is 0.35±0.09 M_⊙.  




 §.§ Characterization of the molecular outflow

In this subsection, we analyze deeper the three different components of the molecular outflow described in Section <ref>. 

The most collimated inner high-velocity outflow component is extended to a maximum deprojected length of 23 (1.5×10^-3pc). The maximum deprojected extension of the intermediate velocity component with an hourglass shape is 33, while the maximum deprojected length of the wider angle lower velocity component is 83 and 27  for the northeast and southwest arc-like structures, respectively. The size of the different components was measured using a bisecting line from the center of the disk continuum emission to the maximum extension of each component, considering the 5σ contour. 
 

The most collimated inner high-velocity outflow component exhibits an opening angle of ∼34 degrees. On the other hand, the intermediate velocity component with an hourglass shape shows an opening angle of ∼70 degrees. For the wider angle low-velocity outflow component, we estimate values of ∼120 degrees and ∼100 degrees for the northeastern and southwestern lobes, respectively. These values were measured by hand using a protractor over the flux-integrated map (Figure <ref>). Table <ref> summarizes all these values.  

Given that our observations were not designed nor optimized for studying extended molecular outflows and that we are likely filtering structures larger than ∼20, we derived the dynamical parameters of the outflow as a first characterization in Appendix <ref>, however, the results in Table <ref> must be considered cautiously and valid only for the most compact emission of the outflow. 


















§ DISCUSSION
 


 §.§ Asymmetric Intensity Distribution along the minor axis of the Dust Disk


The eDisk results has shown the presence of asymmetric intensity distribution along the disk minor axis in several sources <cit.>. The asymmetries found in the minor axis in these eDisk subsample are generally explained due to a combination of optically-thick emission and disk flaring, as proved in the radiative transfer studies of <cit.> and <cit.>. As a result, dust is not settled onto the midplane but is distributed vertically, which, in some cases, can create a dark lane. This dark lane observed in edge-on disks has been described as a “hamburger" shape <cit.> .

In the case of the orientation of the GSS30 IRS3 disk, the northeastern region corresponds to the far side of the disk. This conclusion is based on the peak intensity position, which exhibits a slight skew from the geometric center and towards the northeast, as determined by fitting an ellipse to a 5σ contour level. Moreover, this is supported by the orientation of the collimated high-velocity molecular outflow, where the blueshifted emission points towards the northeast. This alignment provides a rationale for the observed brightness asymmetry in the minor axis between the two sides of the disk. Additionally,  the dust disk 'boxy' shape (Figure <ref>) can be potentially explained by the truncated edge of a flared disk. 

Consequently, the asymmetry in the minor axis can be attributed to a combination of disk flaring and optically thick emission.




 §.§ The origin of the bumps along the major axis

A subset of eDisk sources reported the presence of brightness asymmetry or bumps along the major axis of the protostellar disks <cit.>. In the case of GSS30 IRS3, two bumps are detected along the disk major axis. 
To confirm the authenticity of the bumps and eliminate the possibility of them being spurious artifacts from the cleaning process, we fit the radial intensity profile in the uv plane using Frank (see Appendix <ref>). Frank <cit.> first deproject the visibilities to correct the disk inclination and uses the Fourier-Bessel series to reconstruct the radial brightness profile. The bumps are located at a radius of ∼019 (∼26 au) and the other located at ∼036 (∼50 au) from the center of the disk. The positions of the identified bumps, closely coincide with those obtained from the radial intensity profile in the image plane. This alignment strongly rules out the possibility of being artifacts. However, we emphasize that our interpretation is only limited to discarding spurious signals in the cleaning process given the Frank limitations when dealing with a flared disk. 

We discuss two main possible explanations for the origin of the bumps. First, there could be a real substructure within the dusty disk that will likely evolve into a more well-defined structure over time. Alternatively, the bumps might be the result of the temperature distribution due to optically thick continuum emission. To test the second possibility, we compare the dust brightness temperatures at the bumps' locations with the expected midplane temperature at the same location in a passively heated and flared disk.

To obtain the midplane temperature, we employ the equation T_mid(r) = (ϕ L_*/8πr^2σ_SB)^0.25  <cit.>, where r is the radius (019 and 036), σ_SB denotes the Stefan-Boltzmann constant, L_* is the bolometric luminosity of the central protostar (1.7 L_⊙) and ϕ corresponds to the assumed flaring angle of 0.02 radian <cit.>. The predicted midplane temperature is 27.8 K at a radius of 019 and 20.2 K at a radius of 036. However, the observed brightness temperature at  019  and 036 are 18.9 K and 9.6 K, respectively. The observed brightness distribution along the major axis of the disk is compared with the estimated midplane temperature distribution in Figure <ref>, which shows that the observed brightness temperature is lower than the estimated midplane temperature. This suggests that the 1.3 mm continuum emission is not completely optically thick, as we would expect the observed brightness temperature to be similar to the actual physical temperature. It is, however, advisable to exercise caution in interpreting the observed brightness temperature distribution as an accurate representation of the actual radial brightness temperature distribution of the disk. This is due to the fact that the disk is inclined at a significant angle, which introduces a number of complications. In an edge-on disk, the brightness temperature is measured at a point where the optical depth along the line of sight becomes unity. Consequently, the distance of the measurement point from the center is not necessarily equivalent to the corresponding project distance from the center.



Future continuum observations at lower frequencies with ALMA that trace optically thinner dust emission could help to validate our findings. Also, a more detailed radiative transfer modeling, which is beyond the scope of this paper, might further elucidate the true nature of these bumps and their relationship with the substructures observed in more evolved sources.











 §.§ Kinematics and the origin of the S-shape in the C^18O line emission


In this subsection, we discuss the origin of the twisted feature observed in the C^18O (2-1) velocity map (Section <ref>). Similar S-shape or twisted velocity map close to the systemic velocity are usually associated with warped disks in young stellar objects <cit.>. There are several possible explanations for the origin of these features: the existence of a misaligned binary <cit.>, a fast radial infall attributed to the presence of a giant planet, brown dwarf, or low-mass companion <cit.>, the effect of a stellar fly-by <cit.>, or the transition from a Keplerian disk to an infalling envelope <cit.>. The first two theories are usually applied to more evolved young stellar objects. 

In the eDisk sample there is another source with a twisted kinematic map<cit.>. The C^18O (2-1) and SO (6-5) moment 1 maps showed an S-shaped feature which is explained by infalling motion. In the case of GSS30 IRS3 observations we are also detecting an infalling envelope according to the double-power law fit from SLAM (see Section <ref>)
Although the most likely explanation is an infalling envelope onto the disk, the twisted kinematic feature alone does not provide sufficient evidence for further analysis. On the other hand,  we are unable to confirm that the origin of the S-shape is due to the presence of an undetected close binary (within ∼8 au) or an embedded companion. Also, we do not observe indications of a recent star flyby. Future modeling, incorporating hydrodynamic simulations may be able to explain the origin of the S-shape; however, such modeling is beyond the scope of this initial analysis.





 §.§ Coexistence of a jet and a wide-angle outflow

Accretion and ejection are fundamental processes in star formation, shaping the evolution of young stars and their environment. The relation between ejection and accretion processes is not completely understood, but the accretion of gas and dust that fall from the protoplanetary disk´s inner edge onto the young stellar object drives the ejection of material. Ejection of material removes the excess of angular momentum through two different types of structures: a collimated component at high velocity and a wider-angle component at lower velocities.  The collimated component or jet propagates into the surrounding molecular cloud, forming bow shocks that push the gas and produce outflow shells in the jet´s vicinity <cit.>. Different knots are usually detected along the jet direction as a signal of episodic material ejection. On the other side, the wide-angle component is a radial disk wind that blows the ambient cloud, forming expanding shells <cit.>. Therefore, the jet and wind angle disk-driven structures are able to explain the mass loss process in star formation with different theories. Interestingly, <cit.> presented simulations demonstrating that jet and wide-angle disk-wind driven ejections could coexist, expecting the simultaneous detection of collimated and wide-angle outflow components during the early stages of star formation

The ^12CO observations of GSS30 IRS3 reveal three distinct components: a collimated outflow which axis is aligned with several knots seen in molecular hydrogen  (Figure <ref>), a conical hourglass shape and a wide-angle, low-velocity component with a parabolic morphology (Figure <ref>). The analysis of the conical outflow, described in Section <ref>, suggests that it is produced by a jet. While the blueshifted lobe of the wider angle outflow clearly exhibits emission shifting away from the central object as velocity increases, this is consistent with the predictions of the wind-driven shell model <cit.>. 

The coexistence of collimated jets/outflows and slow wider-angle molecular outflows has been previously detected in young protostars at early stages, specifically in two Class I sources in HH 46/47 <cit.> and DG Tauri B <cit.> or more recently in several sources <cit.>. Compared with those sources, GSS30 IRS3 exhibits very distinguishable components, even with observations that were not initially designed to study molecular outflows. Moreover, this is one of the few sources that the coexistence of a jet and a wide-angle outflow has been clearly detected in a young Class 0 protostar in the submm regime, making this source one of the best objects for future detailed studies of the accretion-ejection processes at the early stages of star formation.



§ CONCLUSIONS


We present high-resolution ALMA observations at 1.3 mm of the Class 0 protostar GSS30 IRS3, conducted as part of the eDisk ALMA program at a spatial resolution of ∼8 au. Our observations targeted the continuum emission as well as several molecular species, including ^12CO (2-1), ^13CO (2-1), C^18O (2-1), and c-C_3H_2 (blended lines 6_0,6-5_1,5 and 6_1,6-5_0,5 217.822 GHz), c-C_3H_2 (5_1,4-4_2,3 217.940 GHz), c-C_3H_2 (5_2,4-4_1,3 218.16 GHz), H_2CO (3_0,3-2_0,2 218.22 GHz), and H_2CO (3_2,2-2_2,1 218.47 GHz). We summarize our findings as follows: 


1. We detect dust continuum emission tracing a disk-like structure. The dust disk size is 143×062 (∼198 ×∼86 au) with an inclination of 64.3 ± 1.5^o and a  position angle of 109^o. The mass of the dust disk in GSS30 IRS3 is estimated to range between 23.87 ± 0.90 M_ and 75.1 ± 2.9 M_ depending on the adopted T_dust. 

 
2. The dust disk of GSS30 IRS3 does not exhibit clear substructures such as spirals. We report the presence of an asymmetry in the minor axis, which can be attributed to an optically thick emission and disk flaring. Additionally, we note the detection of two bumps located at 26 au and 50 au from the disk center along the major axis. The origin of these bumps remains unclear, and we propose two main hypotheses: they could indicate embedded real substructures within the disk, or that we are tracing the temperature distribution. 
 

3. The C^18O (2-1) emission traces a Keplerian disk and the infalling rotating envelope, allowing us to derive a dynamic mass of 0.35±0.09 M_⊙ of the central protostar with a disk gas size radius between 47 and 107 au. 

4. Our ^12CO (2-1) high-resolution and sensitive observations reveal the coexistence of a jet and a wide-angle outflow. 



    We thank the referee, Tien-Hao Hsieh, for the helpful comments and suggestions on this manuscript. We would like to thank all the ALMA staff supporting this work. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2019.1.00261.L, 2019.A.00034.S ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. J.J.T. acknowledges support from NASA XRP 80NSSC22K1159.  The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
    N.O. acknowledges support from National Science and Technology Council (NSTC) in Taiwan through the grants NSTC 109-2112-M-001-051, 110-2112-M-001-031, 110-2124-M-001-007, and 111-2124-M-001-005.
    J.K.J. acknowledges support from the Independent Research Fund Denmark (grant No. 0135-00123B). C.W.L. is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2019R1A2C1010851), and by the Korea Astronomy and Space Science Institute grant funded by the Korea government (MSIT; Project No. 2022-1-840-05).
    Z.-Y.L. is supported in part by NASA NSSC20K0533 and NSF AST-2307199 and AST-1910106.
    L.W.L. acknowledges support from NSF AST-2108794.
    J.P.W. acknowledges support from NSF AST-2107841.
    W.K. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2021R1F1A1061794).
ZYDL acknowledges support from NASA 80NSSCK1095, the Jefferson Scholars Foundation, the NRAO ALMA Student Observing Support (SOS) SOSPA8-003, the Achievements Rewards for College Scientists (ARCS) Foundation Washington Chapter, the Virginia Space Grant Consortium (VSGC), and UVA research computing (RIVANNA). S.T.  is supported by JSPS KAKENHI grant Nos. 21H00048  and 21H04495, and by NAOJ ALMA Scientific Research grant No. 2022-20A.
    H.-W.Y. acknowledges support from the National Science and Technology Council (NSTC) in Taiwan through the grant NSTC 110-2628-M-001-003-MY3 and from the Academia Sinica Career Development Award (AS-CDA-111- M03).
    IdG acknowledges support from grant PID2020-114461GB-I00, funded by MCIN/AEI/10.13039/501100011033.
    ASM thanks Laura Pérez and Anibal Sierra for their valuable help and insights on using the Frank. 

aa.bst 






§ CHANNEL MAPS
 
This section shows the channel maps for ^12CO (Section <ref>), ^13CO (Section <ref>) and C^18O (Section <ref>)









§ JWST IMAGES

We compare the image of the ^12CO molecular outflow with recent JWST images obtained with NIRCAM (PI K. Pontoppidan) centered over GSS30 IRS3.




§ OTHER SOURCES DETECTED IN THE FIELD OF VIEW



 §.§ GSS30 IRS 1

We report the continuum detection of GSS30 IRS1, a Class I young stellar object <cit.>, located at a distance of approximately 14.7 arcseconds from GSS30 IRS3. Figure <ref> shows the 1.3 mm dust emission from GSS30 IRS1 with an angular resolution of 0052×0042 (position angle of 2.5^o), obtained using a robust parameter equal to 0.5. The continuum emission traces an inclined disk-like structure around the central protostar, with a small appendix similar to a spiral-like structure located at the west-south of the disk. The continuum emission measured above a 5σ contour for the disk-like structure (excluding the spiral-like structure) has major and minor axes of 034×022 (∼67 ×∼44 au), with a peak intensity and flux density of 2.72 mJy beam ^-1 and 3.92 mJy, respectively. Coordinate of the peak intensity is R.A.=16h26m21.354s, Dec= -24^o 23'0503. The rms is 17.1 µJy beam s^-1. The inclination derived from the disk-like structure is approximately 50^o.

Several emission lines, including ^12CO (2-1), ^13CO (2-1), C^18O (2-1), SO (6_5-5_4), and SiO (5-4), were detected surrounding GSS30 IRS1. A detailed analysis of the gas source and the spiral-like structure will be presented in a follow-up paper (Santamaría-Miranda et al. in prep).








 §.§ GSS30 IRS2

We report the detection of gas material surrounding the Weak-line T-Tauri star <cit.> GSS 30 IRS2 that is classified as M2 spectral type <cit.>. The ^12CO emission is detected between -2.2 km/s and 1.6 km/s with respect to the local standard of velocity rest, and it integrated map is shown in Figure <ref>. Redshifted emission is very contaminated by cloud emission, and it is impossible to distinguish between emission from the source and from the cloud. The emission size is 13×08 based on the 5σ contour. There is not also very clear molecular outflow emission. Furthermore, there is no clear rotation in the intensity-weighted velocity map (Right panel in Figure <ref>). There is no detection of the continuum at 3σ level (5.2 μJy/beam) or any other of the emission lines. A summary of the main properties is shown in Table <ref>.







§ SLAM FITTING
 
This section shows the result of corner plot (Figures <ref> and <ref>) of the SLAM fitting in Section <ref> for the C^18O and  ^13CO lines.










§ DYNAMICAL PARAMETERS OF THE MOLECULAR OUTFLOW COMPONENTS
 

In this subsection of the Appendix, we will analyze the dynamic properties of the three different components of the molecular outflow, separated in blueshited and redshifted emission with respect to the systemic velocity (2.84 km s^-1, as inferred in Section <ref>). We will consider the emission inside the 3σ contour level in Figure <ref> to perform the calculation in this subsection.  For inferring the dynamical properties of the molecular outflow, we will consider the ^12CO (2-1) molecular line  since this transition clearly traces the different components of the outflow with good S/N, and the ^13CO (2-1) line traces only the outflowing material closer to the central protostar together with a rotating disk.  
We calculate the dynamical time, the outflow mass, the mass-loss rate,  the outflow momentum, the kinetic energy, the mechanical luminosity,  and the outflow force.  

The dynamical time of the various outflow components was determined by considering the deprojected maximum velocity, the  deprojected length, and an outflow axis perpendicular to the dust disk, whose inclination is derived in Section <ref>.

The maximum inferred dynamical time is approximately 535 years for the wider angle lower velocity blueshifted component. When considering both the blueshifted and redshifted emission, the average dynamical time for this component is around 340 years. We used the maximum velocity to calculate the dynamical time of each of the components. Then we averaged the dynamical time between the blue and red component. The intermediate velocity hourglass shape component exhibits an average dynamical time of approximately 43 years when considering both blueshifted and redshifted emission. Similarly, the most collimated inner high-velocity component shows an average dynamical time of approximately 31 years. 

The ^12CO (2-1) emission line is optically thick at systemic velocities, since the brightness temperature of the line peak 57.9 K is similar to the kinetic temperature that we measured (58.7 K) from the linewidth. Therefore we determine the excitation temperature for the ^12CO (2-1) line using the peak intensity obtained close to the position of the star as 
    T_ex = hν_12/k/ln[1+ hν_12/k/T_o12+J_12(2.7 K)],
 where T_0 is the peak brightness temperature obtained close to the central continuum peak position. We derived a T_ex for the ^12CO (2-1) of 25.7 K, and we assumed this value for all the molecular outflow components. 
 
Then, we followed the prescription in <cit.> and <cit.> to calculate the column density and the mass of the outflow. We derived the mean optical depth in the wings of the ^12CO (2-1) emission line. We define the line wings as [-19.9 km/s, -18,7 km/s] and [23.2 km/s, 23.8 km/s] for the inner molecular outflow, [-18.0 km/s, -17.4 km/s] and [21.9 km/s, 22.6 km/s] for the intermediate hourglass shape and only two channels (-1.6 km/s and 7.9 km/s)  for the wider angle low velocity molecular outflow. The optical depth is obtained for each of the three molecular outflow components, obtaining  values between 0.04 and 0.51, in the optically thin regime. Then we assume that the emission is optically thin in the line wings. We infer a column density ∼2×10^16cm^2 for the wider angle and the inner molecular outflow components and a value one order of magnitude smaller for the highly collimated high-velocity molecular outflow component. Finally, we calculated the outflow mass using the inferred column density, the emission area, and a CO abundance of 10^-4. 


We derived the rest of the parameters following the formulas in <cit.> correcting for inclination. The total outflow mass of the molecular outflow is 6.1×10^-5 M_⊙, the momentum varies between 9.8×10^-6  M_⊙ km s^-1 and 4.5×10^-3 M_⊙ km s^-1, the kinetic energy varies between 4.7 ×10^39 erg and  4.7 ×10^41 erg, and the mechanical force varies between 4.8×10^-7 M_⊙ km s^-1 yr^-1 and 1.0×10^-5 M_⊙ km s^-1 yr^-1. All these parameters are summarized in Table  <ref>.


These results are affected by several assumptions that may affect the estimated mass. Previous studies point out that the optical depth of molecular outflows is underestimated, and masses should be at least a factor two higher <cit.> or five <cit.>. The first source of uncertainty is T_ex; our value is in the expected range for molecular outflows (T_ex = 10-50 K ); if we use a T_ex=50 K, the mass will increase by a factor of 1.4 with respect to that with T_ex=25.7 K. Another source of uncertainties is related to the parental cloud material that contaminates the emission of the molecular outflow close to the systemic velocity, needing the exclusion of those channels for the dynamical parameters calculations.  <cit.>, using synthetic observations, found that the missed outflow mass might increase by a factor 5 to 10 within 2 km s^-1 from the rest velocity. However, the most significant uncertainty factor in our calculations is the interferometric filtering. It is noticeable when comparing our dynamical properties with the ones obtained in low-mass Class 0 molecular outflows observed with single-dish telescopes. For example, the outflow mass in Class 0 and I sources is in the range between ×10^-3 M_⊙ and ×10^-1 M_⊙  <cit.>, and our total outflow mass is below these values. Another example is the mechanical force  expected to be between  1.0×10^-5 M_⊙ year^-1 km  s^-1 and ×10^-2 M_⊙ yr^-1 km s^-1 for Class 0 sources <cit.> and the mechanical force of GSS30 IRS3 outflow is in the lower limit.
Therefore, we conclude that all derived parameters may be treated as lower limits. 




§ RADIAL BRIGHTNESS PROFILE FITTING


To further investigate if the bumps detected in the major axis of the radial intensity profile of the dust disk surrounding GSS30 IRS3 are real features and not a byproduct of the cleaning process, we fit a 1D profile to the ALMA visibilities with a nonparametric model using the Frank library <cit.>, which provides a super-resolution brightness profile reconstruction. A major advantage of fitting visibilities is avoiding false structures created during the CLEAN process. Data were averaged in frequency to get spectral windows of 8 channels to reduce data volume and speed up calculations, excluding baseband 1 and 4, which are dedicated to detecting molecular emission lines or has a strong detected line where the contribution to the continuum was negligible. Frank can estimate the inclination and position angle of the protoplanetary disk; however, <cit.> noted that Frank does not fit these two parameters accurately, so we fixed them to an inclination of 64.30 deg and a position angle of 109 deg, as inferred in Section <ref> considering the emission above the 5σ contour.

Before fitting a Bessel functions to the brightness profile, Frank applies a power spectrum estimate as a prior. There are several hyperparameters that affect this power spectrum, but the two most relevant ones are α and w_smooth.  α  works as an S/N threshold for the maximum baseline where visibilities are fitted (a higher value implies a stricter S/N threshold), while w_smooth affects the smoothness of the power spectrum, where a higher value implies a smoother power spectrum. We explore the fitting quality using different hyperparameters values (α between 1.05 and 1.30 and w_smooth between 10^-4 and 10^-1), not finding any substantial difference between the different fittings. We fixed the hyperparameters to α= 1.30 and w_smooth=10^-4 because they present the least noisy results. The best-fit model (Figure <ref>) showed the presence of two bumps. 
To obtain the bumps size and location, we fit an exponential to the brightness profile obtained from Frank to isolate the bumps. Then, we fitted Gaussians around the areas where the bumps are located. Their positions are ∼019 (∼26 au) and  ∼036 (∼50 au) from the center of the disk, 009  (∼13.6 au) and 014 (∼19 au) wide, respectively.



  





 Paper Title: The evolution of the $M_{\mathrm{d}}-M_{\star}$ and $\dot M-M_{\star}$ correlations traces protoplanetary disc dispersal.

 


   

   


   


   European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching bei München, Germany

              alice.somigliana@eso.org
         Fakultat für Physik, Ludwig-Maximilians-Universität München, Scheinersts. 1, 81679 München, Germany
         Dipartimento di Fisica e Astronomia, Universita' di Bologna, Via Gobetti 93/2, I-40122 Bologna, Italy
         INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy
        Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy
        Université Paris-Saclay, CNRS, Institut d'Astrophysique Spatiale, Orsay, France
        Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany
        Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany
        Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France
        Istituto Nazionale di Astrofisica-IAPS, Via Fosso del Cavaliere 100, I-00133 Roma, Italy
             

   



 
  
  Observational surveys of entire star-forming regions have provided evidence of power-law correlations between the disc integrated properties and the stellar mass, especially the disc mass (M_d∝M_⋆^λ_m) and the accretion rate (Ṁ∝M_⋆^λ_acc). Whether the secular disc evolution affects said correlations is still a matter of debate: while the purely viscous scenario has been investigated, other evolutionary mechanisms could have a different impact. In this paper, we study the time evolution of the slopes λ_m and λ_acc in the wind-driven and viscous-wind hybrid case and compare it to the purely viscous prediction. We use a combination of analytical calculations, where possible, and numerical simulations performed with the 1D population synthesis code , that we also present and release to the community. Assuming M_d(0) ∝M_⋆^λ_m, 0 and Ṁ(0) ∝M_⋆^λ_acc, 0 as initial conditions, we find that viscous and hybrid accretion preserve the power-law shape of the correlations, while evolving their slope; on the other hand, MHD winds change the shape of the correlations, bending them in the higher or lower end of the stellar mass spectrum depending on the scaling of the accretion timescale with the stellar mass. However, we show how a spread in the initial conditions conceals this behaviour, leading to power-law correlations with evolving slopes like in the viscous and hybrid case. We analyse the impact of disc dispersal, intrinsic in the wind model and due to internal photoevaporation in the viscous case: we find that the currently available sample sizes (∼ 30 discs at 5 Myr) introduce stochastic oscillations in the slopes evolution, which dominate over the physical signatures. We show that we could mitigate this issue by increasing the sample size: with ∼ 140 discs at 5 Myr, corresponding to the complete Upper Sco sample, we would obtain small enough error bars to use the evolution of the slopes as a proxy for the driving mechanism of disc evolution. Finally, from our theoretical arguments we discuss how the observational claim of steepening slopes necessarily leads to an initially steeper M_d - M_⋆ correlation with respect to Ṁ - M_⋆.

  
  
  
  


   


   The M_d-M_⋆ and Ṁ - M_⋆ correlations trace disc dispersal
   Somigliana, A. et al.
    
   The evolution of the Mdisc-Mstar and Mdot-Mstar correlations traces protoplanetary disc dispersal
    Alice Somigliana
          1,2email: alice.somigliana@eso.org,
          Leonardo Testi3,4,
          Giovanni Rosotti5,
          Claudia Toci1,
          Giuseppe Lodato5,
          Rossella Anania5,
          Benoît Tabone6,
          Marco Tazzari4,
          Ralf Klessen7, 8,
          Ugo Lebreuilly 9,
          Patrick Hennebelle 9,
          and Sergo Molinari 10
          
    Received XXX; accepted YYY
====================================================================================================================================================================================================================================================================================================================================================================================================









§ INTRODUCTION


The secular evolution of protoplanetary discs is deeply intertwined with both the planet formation process <cit.> and the accretion onto the central protostar <cit.>. Planetesimals, the building blocks of planets, form and evolve within the disc following the dynamics of either the gaseous or solid component, depending on their relative size and their coupling (or lack thereof) with the gas particles; on the other hand, the protostar is fed by the disc itself, through the accretion of material that loses angular momentum and drifts inwards. The ideal ground to explore the connection between protoplanetary discs and their host stars is provided by large surveys of entire star-forming regions, targeting the properties of both discs and protostars; the last decade has seen a significant observational effort in the direction of these population-level studies, also thanks to the advent of facilities like the Atacama Large Millimeter Array (ALMA) (see the PPVII reviews by ). 

Disc masses and accretion rates are arguably the most studied integrated disc properties. Accretion rates are inferred from the spectra of the central stars, which show an excess emission (especially prominent in the UV) when accretion is taking place; surveys performed across different star-forming regions <cit.> agree on the presence of a power-law correlation between the accretion rate and the stellar mass, Ṁ∝M_⋆^λ_acc, obs (following the notation of ). On the other hand, disc masses have traditionally been determined from observations of the sub-mm continuum emission of the solid component of discs; due to the large number of assumptions involved in converting sub-mm fluxes into total disc masses (see ), one of the current main goals of the protoplanetary disc community is the accurate determination of total disc masses - both from dynamical constraints <cit.> and direct measurements of the total gas content (e.g., ). Despite the systematic uncertainties involved in their determination, dust-based disc masses also seem to show a power-law correlation with the stellar mass, M_d∝M_⋆^λ_m, obs, across different star-forming regions <cit.>.

The existence of the disc mass-stellar mass and accretion rate-stellar mass correlations is now generally accepted; however, there is no consensus on the physical reason behind their establishment and their evolution with time. While the Ṁ - M_⋆ correlation appears to have a roughly constant slope[Throughout this work, we use 'slope' as a synonym of power-law index, referring to the correlations in the logarithmic plane.] of λ_acc, obs≈ 1.8 ± 0.2 (as first suggested by  and supported by many of the following works mentioned above), the M_d - M_⋆ correlations is claimed to be steepening with time <cit.>, from the lowest λ_m, obs ( t ∼ 1 Myr) = 1.7 ± 0.2 (Taurus) to the highest λ_m, obs (t ∼ 5 Myr) = 2.4 ± 0.4 (Upper Sco). Whether these correlations reflect the initial conditions of disc populations, or are rather a product of the secular evolution, is still under debate. Both possibilities have been discussed for the Ṁ - M_⋆ correlation: <cit.> have assumed it to hold as initial condition, favouring the correlation to be present in young populations, whereas <cit.> have derived it from a simple model of disc formation from a rotating collapsing core, which provided an explanation for evolved disc populations. At the same time, the claimed increase in the slope of M_d - M_⋆ does suggest an evolutionary trend; <cit.> have found that, assuming power-law correlations between both M_d and Ṁ and the stellar mass as initial conditions, secular evolution can indeed alter the slopes of the correlations themselves (see Section <ref> for details). However, their analysis was limited to the standard viscous evolution paradigm, whereas the driving mechanism of accretion is far from being constrained (see  for a review). 

The traditional viscous accretion model prescribes a macroscopic viscosity as the cause of redistribution of angular momentum within the disc <cit.>. In this scenario, while part of the material loses angular momentum and moves radially closer to the star, some other material gains the same amount of angular momentum and moves further away, effectively increasing the disc size. The viscous paradigm can explain many key features of disc evolution, but it cannot account for disc dispersal - as determined from the observational evidence of exponentially decreasing fraction of both disc-bearing <cit.> and accreting <cit.> sources in star-forming regions with time; furthermore, the low levels of turbulence detected in discs <cit.> appear incompatible with the observed evolution. While the discrepancy in the disc and accretion fraction can be mended considering mechanisms such as internal or external photoevaporation <cit.>, that effectively clear discs on timescales comparable with the observed decline, the tension between the expected and observed amount of turbulence does not appear to be solved yet. On the other hand, the MHD disc winds scenario offers a promising alternative. Pioneering work <cit.> supported by recent numerical simulations (e.g., ) demonstrated that MHD winds launched from the disc surface have the net effect of removing angular momentum as a consequence of the extraction of material; SMHD wind-driven accretion can even lead to disc dispersal <cit.>. Following disc evolution at population level in numerical simulations remains out of reach; however, three-dimensional core-collapse simulations have shown how non-ideal magnetohydrodynamics and ambipolar diffusion play a fundamental role in shaping the resulting population of early-type young stellar objects <cit.>. While some 3D studies of isolated disc formation have attempted to bridge the gap between Class 0/I and Class II stages <cit.>, the high numerical cost of the simulations for 3D population synthesis does not allow to follow the evolution of the discs up to very evolved stages where they can be considered isolated from the surrounding environment. MHD wind-driven disc populations can however be modelled in 1D using simple prescriptions as proposed by <cit.> or <cit.>. Detecting characteristic signatures of either of the two evolutionary prescriptions is a compelling issue <cit.>.

In the context of the evolution of the correlations between the disc properties and the stellar mass, while the purely viscous scenario has been extensively studied by <cit.>, the wind-driven paradigm remains unexplored; with this paper, we address this deficiency and investigate the impact of MHD wind-driven evolution on the M_d - M_⋆ and Ṁ - M_⋆ correlations, with a particular focus on their time evolution and the comparison with the purely viscous paradigm. We also extend the work of <cit.> by including internal photoevaporation to the viscous framework. We employ numerical simulations of populations of protoplanetary discs, performed with the population synthesis code , which we also introduce and release to the community. 

The paper is structured as follows: in Section <ref>, we present  and describe its main features, set up and solution algorithm; in Section <ref>, we discuss the time evolution of the M_d - M_⋆ and Ṁ - M_⋆ correlations in three evolutionary scenarios from the theoretical perspective; in Section <ref>, we show the impact of a spread in the initial conditions and dispersal mechanisms on the evolution of the slopes, and we present the numerical results obtained from realistic disc population synthesis; in Section <ref> we interpret the implications of our findings in the context of the observational determination of the slopes, and finally in Section <ref> we draw the conclusions of this work.







§ NUMERICAL METHODS: 


In this Section we present the 1D population synthesis code [ and the output analysis library  can be installed via the Python Package Index,  and . The full documentation and tutorials are available at https://alicesomigliana.github.io/diskpop-docs/index.html. If you use  in your work, please cite this paper (Somigliana et al. 2024).]. We describe the master equation for the secular evolution of discs (Section <ref>), the initial conditions to generate a synthetic population (Section <ref>), the solution algorithm (Section <ref>), and the user interface and output (Section <ref>). For a more detailed description, we refer to the code documentation; for a validation of the code, see Appendix <ref>.



 §.§ Master equation


The master equation of protoplanetary disc evolution,


    ∂Σ/∂ t = 3/r∂/∂ r[ 1/Ω r∂/∂ r( r^2 α_SSΣc_s^2 ) ] + 3/2r∂/∂ r[ α_DWΣc_s^2/Ω] 
    
                - 3 α_DWΣc_s^2/4 (λ-1)r^2 Ω - Σ̇_photo,


describes the time evolution of the gas surface density in the most general framework, where Σ is the gas surface density, Ω the Keplerian orbital frequency, α_SS the <cit.> α parameter, α_DW the MHD equivalent of α_SS <cit.>, c_s the sound speed, and λ the magnetic lever arm parameter, which quantifies the ratio of extracted to initial specific angular momentum. The four terms on the right hand side (RHS) refer to (i) the viscous torque, whose strength is parameterised by α_SS, (ii) the wind-driven accretion, which corresponds to an advection term, parameterised by α_DW, (iii) mass loss due to MHD disc winds, parameterised by λ and (iv) mass loss due to other physical phenomena (in our case, we consider internal and external photoevaporation). Depending on the values of the specific parameters, Equation (<ref>) can describe a purely viscous (α_DW = 0), purely MHD wind-driven (α_SS = 0) or hybrid (α_SS, α_DW≠ 0) evolution, with (Σ̇_photo≠ 0) or without (Σ̇_photo = 0) the influence of photoevaporation. In the following, we briefly describe the various evolutionary scenarios and the available analytical solutions.

Viscously evolving discs. In the case of purely viscous evolution, the MHD winds parameter α_DW is set to zero. If we also neglect the influence of photoevaporation, Equation (<ref>) reduces to the first term on the RHS and its solution depends on the functional form of the effective viscosity, parameterised as ν = α_SS c_s H (where H is the vertical height of the disc). A popular analytical solution for viscous discs is the <cit.> self-similar solution, which assumes viscosity to scale as a power-law of the radius (ν∝ R^γ).

MHD winds-driven evolution. There are two classes of analytical solutions to Equation (<ref>) in the MHD wind-driven scenario, associated with a specific prescription of α_DW <cit.>. We briefly describe their key features, and refer to the original paper for their derivation and an in-depth discussion.



    
  * The simplest class of solutions (so-called hybrid solutions), which highlight the main features of wind-driven accretion in comparison to the viscous model, assume a constant α_DW with time; these solutions depend on the value of ψ≡α_DW/α_SS, which quantifies the relative strength of the radial and vertical torque. 

    
  * Another class of solutions, which describe the unknown evolution of the magnetic field strength, assume a varying α_DW with time. To obtain these, <cit.> parameterised α_DW(t) ∝Σ_c (t)^-ω, with Σ_c = M_d(t)/2 πR_c^2 (t) (where R_c is a characteristic radius) and ω as a free parameter, and neglect the radial transport of angular momentum (α_SS = 0).
    


Photoevaporation. The generic Σ̇_photo term in Equation (<ref>) allows to account for photoevaporative processes, both internal and external. The exact form of Σ̇_photo depends on the specific model considered; therefore, the availability (or lack thereof) of analytical solutions needs to be considered case by case.


 allows to evolve populations of discs analytically. In particular, as of this release, it includes implementations of the <cit.> self-similar solution and all the analytical solution proposed by <cit.>. In the cases where Equation (<ref>) cannot be solved analytically, the code relies on the solution algorithm described in Section <ref>.





 §.§ Initial conditions and parameters


Every  simulation begins with the generation of a synthetic population of Young Stellar Objects (YSOs). Each YSO constitutes of a star and a disc, whose key initial parameters (stellar mass, disc mass, accretion rate, disc radius, evolutionary parameters α_SS, α_DW, λ, ω...) can be set by the user. In the following, we describe the standard case where we consider the stellar masses to be distributed according to an Initial Mass Function (IMF) and correlating with the disc mass and radius, and briefly mention the other possible choices; for a deeper discussion, we refer to the  documentation.

 assembles YSOs by determining their parameters as follows:


    
    
  * Stellar mass M_⋆: determined following the <cit.> IMF. Other possible choices are a constant mass for all the stars in the population, or a set of custom stellar masses.
    
    
  * Initial disc mass M_d, accretion rate Ṁ: determined from log-normal distributions of given width and mean value. In the standard case,  considers an initial power-law correlation between the initial M_d and Ṁ and the stellar mass (see Section <ref> for a detailed discussion), where the normalisation at 1 M_⊙, the slope and the scatter around the power-laws are free parameters. If the correlations with the stellar mass are neglected, the user sets the mean value and spread of the distributions. 

    
  * Accretion parameters (α_SS, α_DW, λ, ω): global properties of the whole population, given as input from the user. By setting the parameters controlling accretion,  determines the disc radius R_d and accretion timescale t_acc - which are instead disc-specific and linked to the disc mass and accretion rate.

    
  * Internal photoevaporation parameters (Ṁ_wind, L_X): the total photoevaporative mass-loss rate, Ṁ_wind, can either be set by the user or computed from the stellar X luminosity L_X as <cit.>
    
    
    Ṁ_wind = 6.25 × 10^-9×( M_⋆/M_⊙)^-0.068( L_X/10^30ergs^-1)^1.14 M_⊙yr^-1.


    The surface mass-loss profile Σ̇_photo (Equation B2 in ) is then scaled so that ∫ 2 π R Σ̇_photodR is equal to Ṁ_wind. Like for M_d and Ṁ, Ṁ_wind (or equivalently L_X) is extracted from a log-normal distribution whose mean is determined assuming power-law correlations with the stellar mass, while the normalisation at 1 M_⊙, the slope and the width of the distribution are free parameters.
    
    
  * External photoevaporation parameters (FUV): FUV flux experienced by each disc, in units of G_0[G_0 stands for the Habing unit <cit.>, the flux integral over the range of wavelengths [912 - 2400] A weighted by the average value in the solar neighbourhood (1.6× 10^-3 erg s^-1 cm^-2).]. This parameter can be set to any value accessible in the FRIEDv2 grid of mass loss rate <cit.>, spanning from 1 to 10^5 G_0.
    





 §.§ Solution algorithm


After generating the initial population of YSOs as described above,  proceeds to evolve it by integrating the master equation (<ref>). Our solution algorithm employs an operator splitting method: the original equation is separated into different parts over a time step, and the solution to each part is computed separately. Then, all the solutions are combined together to form a solution to the original equation. We split Equation (<ref>) into five different pieces, related to viscosity, wind-driven accretion onto the central star, wind-driven mass loss, internal and external photoevaporation respectively. Furthermore,  includes the possibility to trace the dust evolution in the disc, which is split in radial drift and dust diffusion. In the following, we describe the solution algorithm for each process.


 

    
  * Viscous accretion: the standard viscous solver is based on the freely available code by <cit.>. We assume a radial temperature profile T ∝ R^-1/2, which results in c_s∝ R^-1/4 and H/R ∝ R^1/4. Note that this implies ν∝ R (i.e., γ = 1), which will be the case from now on. We assume H/R = 1/30 at 1 AU and a mean molecular weight of 2.4. We refer to the original paper for details on the algorithm.
    
    
  * Wind-driven accretion: the second term in Equation (<ref>) is effectively an advection term. The general form of the advection equation for a quantity q with velocity v is ∂_t q(x, t) + v ∂_x q(x, t) = 0; in the case of wind-driven accretion, the advected quantity is R Σ, while the advection (inwards) velocity is given by v_DW = (3 α_DW H c_s)/2R. We solve the advection equation with an explicit upwind algorithm (used also for dust radial drift).

    
  * Wind-driven mass loss: the mass loss term (third in Equation <ref>) does not involve any partial derivative, and therefore is simply integrated in time multiplying by the time step.

    
  * Internal photoevaporation: effectively, internal photoevaporation (implemented through the model of ) is another mass loss term - therefore, as above, its contribution is computed with a simple multiplication by the time step. Once the accretion rate of the disc drops below the photoevaporative mass loss rate, a gap opens in the disc at the radius of influence of photoevaporation: in the model of <cit.>, the prescription changes depending on the radial location in the disc, with respect to the gap itself. Later, the gap continues to widen; when it eventually becomes larger than the disc, we stop the evolution and consider the disc as dispersed.

    
  * External photoevaporation: for a given stellar mass and FUV flux experienced by the disc, the mass loss rate arising from external photoevaporation is obtained, at each radial position, from a bi-linear interpolation of the FRIEDv2 grid <cit.> using the disc surface density at each radial cell. The outside-in depletion of material is implemented following the numerical approach of <cit.>: we define the truncation radius, R_t, as the position in the disc corresponding to the maximum photoevaporation rate (which is related to the optically thin/thick transition of the wind), and we remove material from each grid cell at R>R_t weighting on the total mass outside this radius. The mass loss attributed to the cell i can be written as:
    
    Ṁ_ext,i = Ṁ_totM_i/M(R>R_t),

    where M_i is the mass contained in the cell i, and Ṁ_tot is the total mass loss rate outside the truncation radius.

    
  * Dust evolution[As the dust evolution module was forked from Richard Booth's repository, users of  who wish to use dust in their work ought to cite <cit.> together with this paper.]: based on the two populations model by <cit.> and the implementation of <cit.>. We consider the dust grain distribution to be described by two representative sizes, a constant monomer size and a time-dependent larger size, which can grow up to the limit imposed by the fragmentation and radial drift barriers. We evolve the dust fraction of both sizes following <cit.>, and also include a diffusive term: the diffusion comes from the coupling with the turbulent gas, which has the effect of mixing the dust grains, counteracting gradients in concentration <cit.>. The dust-gas relative velocities are computed following <cit.> and include feedback on the gas component. We refer to <cit.> for details on the numerical implementation. Dust evolution is included in the release of , however the scientific results presented in this work are based on gas simulations only.
    


The separate pieces of Equation (<ref>) must be solved over the same time step to be joined in a coherent solution. We calculate the time step for each process imposing the Courant-Friedrichs-Lewy (CFL) condition. The CFL condition reads Δ t = C  min(Δ x / v) and ensures that, within one time step Δ t, the material moving at velocity v does not flow further than one grid spacing Δ x. The Courant number C must be positive and smaller than 1, with C = 1 corresponding to the maximum allowed timestep to keep the algorithm stable. In our implementation, we pick C = 0.5. We use zero gradients boundary conditions, setting the value of the first and last cell in our grid to that of the second and second to last. We solve the equation on a radial grid of 10^3 points with power-law spacing and exponent 1/2, extending from 3 × 10^-3 au to 10^4 au. From the physical point of view, this choice corresponds to assuming boundary layer accretion (see., e.g., ) - however the difference from magnetic truncation accretion is negligible beyond ∼ 10^-3 au.

After each process has been solved separately, all the pieces are put back together to compute the new surface density, from which the integrated disc quantities are then calculated. As each disc evolves independently of the others in the population, the solver can easily be run in parallel.




 §.§ User interface and output


The user interface of  is a .json parameters file which includes all the user-dependent parameters. Aside from the number of objects in the population and the evolutionary mechanism, the user can set the chosen IMF (either , single stellar mass, or custom input file), the distributions to draw the disc parameters from (single value, flat, normal, log-normal), as well as the normalisation, slope and spread of the correlations, the times at which snapshots are generated, and the initial dust-to-gas ratio. Furthermore, the user can determine a limit disc mass: this is to be intended as a threshold below which the disc would not be detectable anymore, and is therefore considered dispersed in the simulation as well. When a disc is dispersed, the corresponding YSO turns into a Class III object consisting of the central star only.

The output of  is a .hdf5 file containing the properties of both the disc and the star at all chosen time steps for each YSO in the population: this includes the stellar mass, luminosity, temperature, disc mass, accretion rate, accretion timescale, gas and dust surface density, disc radius, dust grain sizes. The output can be easily read and analysed with the dedicated library , released with the code. For a more in-depth description of the parameters, the user interface and the output, we refer to the  documentation.









§ THE TIME EVOLUTION OF THE CORRELATIONS BETWEEN DISC PROPERTIES AND STELLAR MASS UNDER DIFFERENT ACCRETION DRIVERS: ANALYTICAL CONSIDERATIONS







The existence of power-law correlations between the main integrated disc properties - namely the disc mass and stellar accretion rate - and the stellar mass is supported by various surveys across a number of different star-forming regions (e.g., on M_d-M_⋆: ; on Ṁ - M_⋆: ). However, whether the establishment and subsequent evolution of said correlations is a product of the secular evolution of discs, or rather an imprint of the initial conditions, remains unclear. <cit.> explored a combination of both possibilities, assuming the correlations to hold as initial conditions and investigating the impact of purely viscous evolution; we briefly recall their main theoretical results (Section <ref>) and extend their analysis to the hybrid (Section <ref>) and purely wind-driven (Section <ref>) models from the theoretical perspective. We note that the results presented in this work are based on gas simulations.

Following <cit.>, we assume power-law correlations between the disc properties and the stellar mass to hold as initial conditions. We focus on the disc mass M_d, the stellar accretion rate Ṁ and the disc radius R_d, and label the slopes of their correlations with the stellar mass λ_m, λ_acc, and ζ respectively. The initial correlations are set as follows:
    
    
    M_d (0) ∝M_⋆^λ_m, 0, 
    Ṁ (0) ∝M_⋆^λ_acc, 0, 
    
                R_d (0) ∝M_⋆^ζ_0.


To analyse the impact of secular evolution on this set of initial conditions, we analytically determine the evolved expressions for M_d (t), Ṁ (t) and R_d (t) in the three different scenarios. Table <ref> summarises the parameters introduced in this Section.





 §.§ Purely viscous model



The full calculations for the purely viscous case can be found in <cit.>. Here, we briefly remind the main assumptions and results, and we refer to the original paper for a detailed discussion.

As mentioned in Section <ref>, assuming a power-law scaling of viscosity with the disc radius (ν∝ R^γ) allows to solve the viscous evolution equation analytically, recovering the so-called self-similar solution <cit.>. In this case, the disc mass and accretion rate read


    M_d (t) = M_d, 0( 1 + t/t_ν)^1 - η,



    Ṁ (t) = (η - 1) M_d, 0/t_ν( 1 + t/t_ν)^- η,


where η = (5/2 - γ)/(2 - γ) and the viscous timescale t_ν = R_c^2/[3(2-γ)^2 ν(R = R_c)] at the characteristic radius R_c. Because Ṁ_0 ∝ M_d, 0/t_ν, 0, a power-law scaling of M_d, 0 and Ṁ_0 with the stellar mass implies the viscous timescale t_ν, 0 to scale as a power-law with the stellar mass as well, which we define as t_ν, 0∝M_⋆^μ_0; furthermore, this scaling corresponds to the difference between the scaling of the disc mass with the stellar mass and of the accretion rate with the stellar mass. Defining δ_0 = λ_m, 0 - λ_acc, 0, in this case μ_0 = δ_0[The definition of δ_0 might seem redundant at this stage, but it will become important in the following discussion.]; therefore, the scaling of t_ν, 0 with the stellar mass is determined by the relative values of λ_m, 0 and λ_acc, 0. The main results of <cit.> are that (i) viscous evolution maintains the power-law shape of the correlations between the stellar mass and the disc parameters, however (ii) the slope of said correlations may evolve with time, depending on the initial conditions. This is because in a purely viscous framework, the M_d-Ṁ correlation is bound to reach a linear correlation with slope unity <cit.>, which implies the two quantities to have the same dependence on the stellar mass, as M_d / Ṁ∝M_⋆^λ_m - λ_acc. Therefore, λ_m and λ_acc must eventually reach the same value, determined by the initial conditions as


    λ_m, evo = λ_acc, evo = 3 λ_m, 0 - λ_acc, 0/2.


Depending on the sign of δ_0 = λ_m, 0 - λ_acc, 0, the initial slopes can either


    
  * steepen, i.e. λ_m, evo > λ_m, 0, if δ_0 > 0 (implying also
    
    λ_acc, evo > λ_acc, 0);
    
  * flatten, i.e. λ_m, evo < λ_m, 0, if δ_0 < 0 (implying also 
    
    λ_acc, evo < λ_acc, 0);
    
  * remain constant, i.e. λ_m, evo = λ_m, 0, if δ_0 = 0 (implying 
    
    also λ_acc, evo = λ_acc, 0).


Because in the viscous case δ_0 = μ_0, where we note again that μ_0 is the slope of the correlation between the viscous timescale and the stellar mass (t_ν, 0∝M_⋆^μ_0), we can also interpret these scenarios from the viscous timescale perspective. If μ_0 > 0, meaning that the viscous timescale increases with the stellar mass, discs around less massive stars will have shorter viscous timescales, which leads to a faster evolution, compared to discs around more massive stars, which will in turn have longer viscous timescales. This uneven evolution across the stellar mass spectrum leads to a steepening of the linear correlation, as is visualised by <cit.> in Figure 1. The same reasoning, but with opposite/constant trend, applies to the other two scenarios.







 §.§ Hybrid model - omega = 0


In the hybrid viscous and MHD winds model, the general analytical solution by <cit.> gives 


    M_d (t) = M_0 ( 1 + t/(1 + ψ) t_acc, 0)^-(ψ + 2 ξ + 1)/2,



    Ṁ (t) = Ṁ_0 ( 1 + t/(1 + ψ) t_acc, 0)^-(ψ + 4 ξ + 3)/2,


where Ṁ_0 is defined as


    Ṁ_0 = ψ + 1 + 2 ξ/ψ + 1M_0/2 t_acc, 01/(1+f_M_0);




in this notation, ψ = α_DW/α_SS represents the relative strength of MHD winds and viscosity, 


    ξ = 1/4 (ψ + 1) [ √(1 + 4ψ/(λ-1)(ψ + 1)^2) - 1 ]


is the mass ejection index quantifying the local mass loss rate to the local accretion rate, and f_M, 0 = (R_c, 0/R_in)^ξ - 1 the dimensionless mass ejection-to-accretion ratio (with R_in initial disc radius). If we neglect the MHD-driven mass loss (ψ≪ 1 and ξ≪ 1, which correspond to f_M, 0≪ 1 as well), Equations (<ref>) and (<ref>) reduce to the viscous case; on the other hand, if mass loss is included, it depends on the radial extent of the disc through f_M, 0 + 1 - which has an impact on the initial accretion rate (Equation <ref>). Because the accretion timescale t_acc is a generalisation of t_ν in the MHD winds framework, the dependence of the two timescales on the stellar mass is exactly equivalent, and we will keep the same notation as above: t_acc∝M_⋆^μ. However, as mentioned above Ṁ_0 depends on the stellar mass not only through M_0 and t_acc, 0 as in the viscous case, but also through f_M,0 + 1. As f_M,0 + 1 ∝R_c, 0^ξ, and R_c, 0∝M_⋆^ζ_0, the additional dependence will have a slope of ζ_0 ξ. Therefore, in the MHD winds scenario we can link δ_0 with μ_0 as δ_0 = μ_0 + ζ_0 ξ. The practical meaning of this difference is that, while in the viscous scenario the difference in slope between the two correlations depends only on the scaling of the viscous timescale with the stellar mass, in the hybrid scenario it depends also on the scaling between the disc radius and the stellar mass. It is important to note that ξ is a small number, typically of the order of ∼ 0.1, therefore the difference between the viscous and hybrid case is not particularly prominent. For evolved populations, the disc mass and accretion rate read <cit.>


    M_d (t ≫ t_acc) ∼ M_0 (t/t_acc, 0)^-(ψ + 2 ξ + 1)/2,



    Ṁ (t ≫ t_acc) ∼Ṁ_0 (t/t_acc, 0)^-(ψ + 4 ξ + 3)/2;


this brings the evolved slopes λ_m, evo and λ_acc, evo to


    λ_m, evo = λ_m, 0 + 1/2μ_0 (ψ + 2ξ + 1), 
    λ_acc, evo = λ_acc, 0 + 1/2μ_0 (ψ + 4ξ + 3),





which reduces to Equation (<ref>) in the viscous case (ψ≪ 1, ξ≪ 1). Like viscosity, a hybrid secular evolution maintains the power-law shape of the correlation; moreover, Equation (<ref>) provides a theoretical prediction for the evolved slopes: they can steepen, flatten or remain the same as the initial conditions, depending on the involved parameters. As the terms in parentheses in Equation (<ref>) are sums of positive values, the sign of the evolved slopes depends on the sign of μ_0 like in the viscous case. However, there is a difference from the viscous case: as in the hybrid scenario μ_0 and δ_0 do not coincide anymore, a constraint on the value of μ_0 is translated into a constraint on δ_0 - ζ_0 ξ. In particular, the slopes will increase if δ_0 > ζ_0 ξ (corresponding to μ_0 > 0), whereas if δ_0 < ζ_0 ξ (corresponding to μ_0 < 0), the slopes will decrease; and finally, if δ_0 = ζ_0 ξ (corresponding to μ_0 = 0) the slopes will remain constant in time. Another difference from the viscous scenario is that λ_m, evo and λ_acc, evo are not expected to reach the same value anymore. The limit difference is given by δ_evo = λ_m, evo - λ_acc, evo: substituting the values from Equation (<ref>) one finds δ_evo = δ_0 - μ_0 (ξ + 1), which can be further reduced to δ_evo = ξ (2 β + 1/2) by using the definition of μ_0 as the slope of the t_acc, 0 - M_⋆ correlation. In this expression, β is the slope of the correlation between the disc aspect ratio and the stellar mass, which comes from definition of the accretion timescale; with a standard β = -1/2 (a reasonable approximation of the value derived by radiative transfer simulation, see e.g., ), we obtain δ_evo = -ξ/2. As ξ is positive by definition, in the hybrid scenario δ_evo is always negative, meaning that in an evolved population, the correlation between the accretion rate and the stellar mass will necessarily be steeper than that between the disc mass and the stellar mass. However, we stress once more that ξ is a small number and therefore the predicted difference is also small. Figure <ref> shows the evolution of the slopes from  simulations (with no spread in the initial conditions) for the hybrid model (coloured) compared with the viscous case (grey), which matches the theoretical expectations discussed above. The list of parameters used in the simulations is available in Table <ref>.

Summarising, both the hybrid and viscous secular evolution preserve the power-law shape of the correlations between the disc properties and the stellar mass. The main difference is that the hybrid model does not predict the slopes of the M_d - M_⋆ and Ṁ - M_⋆ correlations to reach the same limit value (unlike the viscous case). The predicted difference in the evolved slopes is given by δ_evo = -ξ/2 (of the order of 0.1). The current observational uncertainties on δ_evo range from 0.4 to 0.8 (, see Table <ref>), 4 to 8 times larger than the predicted difference, making it not observable at this stage.




 §.§ Pure wind - ω≠ 0


The solution to the pure wind model (i.e., time-dependent α_DW through α_DW(t) ∝Σ_c(t)^- ω) by <cit.> gives


    M_d (t) = M_0 ( 1 - ω/2 t_acc, 0 t )^1/ω,



    Ṁ (t) = M_0/2 t_acc, 0 (1+f_M, 0)( 1 - ω/2 t_acc, 0 t )^-1+1/ω;


in this case, the functional form of Equation (<ref>) and (<ref>) does not allow us to derive a simple analytical expression for M_d (t ≫ t_acc) and Ṁ (t ≫ t_acc). Therefore, to explore the evolution of the correlations in the pure wind case, we fully rely on  simulations. In order to account for the impact of secular evolution only, we input perfect correlations between the disc properties and the stellar mass - that is, we do not include any spread in the initial conditions. Figure <ref> shows the time evolution of M_d (left panel) and Ṁ (right panel) as a function of the stellar mass, from younger (darker) to older (lighter) populations. The input power-law correlation (light blue) corresponds to a line in the logarithmic plane; however, as early as ∼ 1 Myr (corresponding to ∼ 2<t_acc, 0> for this simulation), the input correlation starts to bend downwards at lower stellar masses. This behaviour reveals a significantly different trend from the viscous and hybrid model: in the pure wind scenario, the initial power-law shape of the correlations is not preserved by the secular evolution, but rather broken. In Figure <ref>, the faster evolution of discs around lower mass stars is the consequence of a positive μ_0, implying a positive correlation between the stellar mass and the accretion timescale; a negative correlation between M_⋆ and t_acc, 0 (i.e., a negative μ_0) would lead to faster evolution of discs around higher mass stars, causing the correlation to bend towards the other end of the stellar mass spectrum (see Figure <ref>). Another parameter that might impact the evolution of the correlation is the dispersal timescale, t_disp; in the wind-driven case, t_disp∝ t_acc, therefore t_disp does not introduce any further dependence on the stellar mass.





 §.§ Summary


In this Section we have discussed the time evolution of the correlations between the disc properties and the stellar mass from the theoretical point of view. We have performed analytical calculations for the viscous and hybrid scenario, finding that the power-law shape of the initial correlations is preserved during disc evolution. While the slopes are expected to evolve, increasing or decreasing depending on the initial conditions, the limit slopes for evolved populations differ in the two models. While viscosity requires the slope of the M_d - M_⋆ and Ṁ - M_⋆ correlations to reach the same value, in the hybrid scenario the two slopes always have different values; moreover, the evolved slope of Ṁ - M_⋆ is bound to be higher than that of M_d - M_⋆. Unfortunately however, the difference from the viscous model is a factor four too small to be observed within the current error bars. We have then confirmed our predictions running  simulations without any spread in the initial conditions (Figure <ref>). In the pure wind case instead, the functional form of the solution to Equation (<ref>) does not allow for an analytical determination of the evolved slopes; therefore, we entirely relied on  simulations. Our results show that, contrary to the viscous and hybrid case, wind-driven accretion does not maintain the power-law shape of the initial correlations: depending on the scaling of the accretion timescale with the stellar mass, the power-laws are broken because of a bending towards higher (t_acc, 0 decreasing with M_⋆) or lower (t_acc, 0 increasing with M_⋆) stellar masses.






§ POPULATION SYNTHESIS


In Section <ref> we have discussed analytical trends, and presented simulations with no spread to analyse the effect of secular disc evolution alone on the evolution of the slopes. In order to test our theoretical predictions against observational data, we need to account for both a spread in the initial conditions and disc dispersal mechanisms. In this Section, we discuss the impact of both factors on the M_d - M_⋆ and Ṁ - M_⋆ slopes and run realistic population synthesis simulations, to determine whether the model-dependent evolutionary features described in Section <ref> would be observable with the currently available data.



 §.§ Effects of a spread in the initial conditions




The introduction of an observationally-motivated spread in the initial conditions is crucial to produce realistic population synthesis model. In the purely viscous case, <cit.> have shown how the spread does not significantly impact the evolution of either the M_d - M_⋆ or Ṁ - M_⋆ correlations; the shape of the curves (grey in Figure <ref>) is unaffected, except for their starting point, and the statistical fluctuation - determined as the interval between the 25th and 75th percentile out of 100 realisations of numerical simulations with the same initial conditions - is of the order ∼ 0.1 for both slopes. This is a factor two less than the smallest observational uncertainty, and therefore does not produce a detectable difference in the predicted results.

Following <cit.>, we set σ_M_d(0) = 0.65 dex and σ_R(0) = 0.52 dex (determined from  and ) for the log-normal distributions of M_d(0) and R_d(0) in the hybrid scenario (with α_SS = α_DW = 10^-3 hence ψ =1, λ = 3, ω = 0). We find that, just like in the purely viscous case, a spread in the initial conditions only shifts the starting point of the curves (coloured in Figure <ref>) and does not have any significant effect on the shape of the evolution of the slopes (see Figure 5-6-7 in ). The statistical fluctuation for both slopes is again of the order of 0.1, and therefore below the observational error and not impacting our predictions.

On the other hand, wind-driven models with increasing α_DW in time and a spread in the initial conditions (Figure <ref>) behave quite differently from the theoretical expectation discussed in Section <ref>. As the spread introduces a stochastic component, the discs will have higher or lower masses and accretion rates with equal probability; the practical result for the initial correlations is that the bending towards lower stellar masses (approximately log_10(M_⋆/M_⊙) < 0.5 in Figure <ref>) is lost to the stochastic displacement of the discs in the M_d - M_⋆ and Ṁ - M_⋆ planes. Actually, the resulting distribution of discs both in the M_d - M_⋆ and Ṁ - M_⋆ log-log plane does simulate a linear correlation; this implies that, while the stellar mass and the disc properties should not exhibit a linear correlation already after a few t_acc, the presence of a spread mimics such correlation, making the wind-driven scenario indistinguishable from the viscous and hybrid ones. The one feature that remains observable, despite the presence of a spread, is the removal of discs around more or less massive stars - depending on the value of μ_0, as discussed in Section <ref>; in the simulation shown in Figure <ref> we have set μ_0 > 0, implying that discs around less massive stars evolve more rapidly and are therefore more readily dispersed, as is visualised by the lack of sources around lower stellar masses at evolved ages. 

Summarising, an initial power-law correlation between the disc properties and the stellar mass would keep its power-law shape under wind-driven evolution, like in the viscous or hybrid case; however, the interpretation of the observed correlations is different depending on the theoretical framework. While the viscous and hybrid models preserve an initially established correlation, making the characteristic evolution of the slopes λ_m and λ_acc a tracer of disc evolution itself, the apparent correlation observed in wind-driven populations is merely a signature of the initial spread, rather than the evolutionary mechanism at play.





 §.§ Accounting for disc dispersal: internal photoevaporation
 

Out of the three theoretical scenarios discussed so far, wind-driven evolution is the only one that manages to reproduce the disc and accretion fraction (as measured by  and  respectively) - and therefore, the only one whose predictions can reasonably be compared with observations. Traditionally, the problem of disc dispersal in viscous populations is addressed by including internal photoevaporation (see e.g., ): in this Section, we discuss the impact of internal photoevaporation on the previously described expectations for the evolution of the slopes in the purely viscous scenario. As the lack of analytical solutions to the general equation (<ref>) does not allow for analytical arguments, we base the following discussion on physical considerations.

Internal photoevaporation is a threshold process, that kicks in after the accretion rate drops below the photoevaporative mass-loss rate <cit.>. The moment where the effect of photoevaporation becomes non-negligible depends therefore on the initial accretion rate: assuming for simplicity a fixed photoevaporation rate for the whole population, as the accretion rate scales positively with the stellar mass we can expect discs around lower mass stars to show the effects of photoevaporation earlier. Moreover, given that the disc mass also scales positively with the stellar mass, said sources correspond to the less massive ones in the population. From these considerations, we can expect discs with lower initial mass to be the first ones to be affected by photoevaporation, causing a steepening of the M_d - M_⋆ and Ṁ - M_⋆ correlations. However, photoevaporation also disperses discs: removing sources from the population may alter the expected behaviour, hence the need to perform numerical simulations to understand the evolution of the correlations for a population of discs undergoing internal photoevaporation. Our simulations remove discs either because the photoevaporative gap becomes too large, or because the disc mass or accretion rate fall below a certain detectability threshold. As we mentioned above, the dispersal timescale t_disp might also play a role, if it has a different scaling with the stellar mass with respect to the accretion timescale (t_ν in the viscous case). Our numerical implementation of internal photoevaporation follows <cit.>, and we assume a mass-loss rate of 4 × 10^-10 M_⊙ yr^-1 - which allows us to reproduce the observed disc fraction for the set of parameters of our simulation - for all discs in the population; therefore, no further M_⋆ dependence is introduced, and the t_ν - M_⋆ scaling is the only one that matters. We stress that a stellar mass-dependent photoevaporative rate is expected <cit.>: we explore the influence of such dependence on disc observables in an upcoming work (Malanga et al. in prep.). We discuss the results of our  simulations in the following Section.





 §.§ What are the slopes tracing?






Figure <ref> shows the time evolution of the M_d - M_⋆ slope for 15 realisations with the same initial conditions for the viscous plus photoevaporative (central panel) and wind-driven (right panel) models, both reproducing the observed disc and accretion fractions, compared to the purely viscous scenario (left panel). The initial number of discs in the populations was determined to recover a certain number of objects at 5 Myr (increasing from top to bottom), and varies in the different simulations, as the decline (or lack thereof) of the disc fraction is model-dependent (see Figure 6 of ). 

The number of objects in the simulation displayed in the first row was set to obtain ∼ 30 discs at 5 Myr, corresponding to the currently available sample size in Upper Sco, the oldest observed star-forming region. In the left panel, we see how the evolution of the slope in the purely viscous model is not significantly affected by a spread in the initial conditions: the single realisations resemble each other remarkably well, the only difference being the starting point of the curve (as found by ). On the other hand, the photoevaporative and wind-driven models have a dissimilar behaviour: each realisation can deviate substantially from the others, as we can particularly notice by the location, amplitude and direction of the bumps. The key difference between these models and the purely viscous case is disc dispersal: the stochastic nature of the slope evolution suggests that it does not trace the underlying secular disc evolution, like in the viscous scenario, but rather carry the signatures of disc dispersal itself - making it impossible to use the evolution of the slopes as a proxy for accretion mechanisms. There are two main factor that play a role in this context:


    
  * Initial conditions and spread in the correlations. The exact evolution of the slopes will depend on the initial conditions, both of the disc mass-stellar mass correlations themselves and of the population-wide parameters. Furthermore, the removal of discs from the population would not impact the results of the fitting procedure only if there was perfect correlations between the disc parameters and the stellar mass; with a spread in the initial correlations, on the other hand, the results may differ depending on which discs in the population are dispersed;
    
    
  * Small number statistics. Depending on the initial number of objects, disc removal can lead to small samples - so small that it might lead to low number statistics issues. This is the case for the top row of Figure <ref>, where the number of objects at 10 Myr is of the order of 10 or lower.


In this work, we have used one specific set of parameters (summarised in Table <ref>), determined following <cit.> (viscous model) and <cit.> (hybrid and wind-driven), and we leave a deeper exploration of the parameters space to a future work. While the exact shape of the slope evolution, and therefore the accretion model signature, might depend on the initial conditions, the top panel of Figure <ref> shows that with the current sample sizes the noise dominates over the physical evolution. However, the currently available sample of the oldest star-forming region (Upper Sco) with both disc masses (derived from the millimetre flux, ) and accretion rate <cit.> estimates is highly incomplete; it is therefore worth investigating whether a higher level of completeness would help reducing the entity of the oscillations, allowing to disentangle between the different evolutionary models.

The central and bottom rows of Figure <ref> show how a larger sample would impact the oscillations of the slope evolution. The simulations in the middle row are performed imposing a double sample size at 5 Myr with respect to the current one (∼ 60 discs), while in the bottom row we assume to have the complete Upper Sco sample, totalling ∼ 140 discs (Carpenter et al. in prep.). We remind that we focus on Upper Sco as the oldest observed star-forming region, which makes it the most affected by disc dispersal.

As expected, statistical significance increases with a larger sample, leading to a decreased impact of the oscillations on the global slope evolution; with the complete sample, in particular, we can reduce the spread in the evolution by a factor of ∼ 2 compared to the current available data. This argument confirms the importance of larger sample sizes in discriminating between the viscous and wind-driven models, as already suggested by <cit.> in the context of the accretion rates distribution.

As we mentioned in Section <ref>, our simulations consider discs as dispersed if their masses or accretion rates fall below the imposed detectability threshold of 10^-12 M_⊙ yr^-1. We have also included a threshold in accretion rates in post-processing to account for non-accreting objects. From the observational point of view, this latter selection depends on both the instrumental sensitivity and the definition of disc itself: how Class II objects are defined, and in turn how Class III sources are removed from the observed samples, impacts the resulting slope. Summarising, with the current sample sizes, the evolution of the slopes is significantly more affected by disc dispersal than it is by secular evolution; therefore, at the state of the art, it cannot be used as a proxy to disentangle between the different evolutionary models. Increasing the sample size would allow to reduce the effect of low number statistics, potentially allowing to observe the different evolution of the slopes under the two evolutionary mechanisms; we further discuss this possibility in the following Section.










§ THE OBSERVATIONAL RELEVANCE OF THE SLOPES


Observed star-forming regions have both a spread in the initial conditions in addition to some disc dispersal mechanism (be it photoevaporation or MHD winds); as we discussed in Section <ref>, with the current sample sizes, the statistical significance of the observationally-determined slopes is undermined and their evolution traces disc dispersal, rather than the accretion mechanism. In this Section, we perform a statistical analysis of our simulated slopes and compare them with the currently available measurements; furthermore, we show the relevance of measuring the slopes despite these limitations and discuss the conditions under which they allow us to put constraints on disc evolution.




 §.§ Comparison of different evolutionary models
 

Figure <ref> shows the comparison of the evolution of the slopes between the viscous + photoevaporative (solid line) and wind-driven (dashed line) models for both the M_d-M_⋆ and Ṁ - M_⋆ correlations (top and bottom row, respectively), including the measured slopes in four star-forming regions from <cit.> as grey dots. As we mentioned above, both models lead to disc dispersal consistently with the observed disc and accretion fraction (shown in Figure 6 of ); the three columns show simulations performed with a different initial number of discs, increasing from left to right, to obtain a different sample size at 5 Myr according to the predicted decline of observed discs. Like in Figure <ref>, the number of objects at 5 Myr is ∼ 30, ∼ 60 and ∼ 140 from left to right, increasing from the currently available measurements in Upper Sco to the virtually complete sample. To estimate the effect of statistical fluctuations, given by the spread in the initial conditions, we ran 100 simulations for each set up: the solid and dashed lines represent the median evolution, while the intervals between the 25th and 75th percentile are visualised by the shaded areas. The growing shaded area, particularly visible with smaller sample sizes, is representative of the decreasing amount of sources on which the fit is performed: with the current sample size (left column), which leads to ∼ 30 discs at 5 Myr, we end up with a 1σ deviation from the median value of ∼ 0.5 - 0.6. Larger sample sizes significantly reduce the scatter, leading to σ∼ 0.4 with a double sample and σ∼ 0.2 for the complete sample, reducing the current one by a factor 3. As mentioned in Section <ref>, with the currently available number of objects the dominant role in the evolution of the slopes is played by disc dispersal, which makes it difficult to trace the imprint of the secular evolution. The expected Upper Sco complete sample (right column) allows for a better separation between the two models - particularly for the M_d-M_⋆ correlation: the expected slope in the two scenarios differs by ∼ 0.5, while the typical uncertainty of the currently measured slopes is between 0.2 and 0.3. Larger sample sizes would further decrease this uncertainty, allowing us to discriminate between the two models based on the slope evolution. 

The observed slopes (from ) are only included in the left column of Figure <ref> as they refer to the current sample size. The main source of uncertainty in the current measurements is given by Upper Sco, mainly due to the incomplete sample; moreover, it is worth pointing out that external photoevaporation is likely to play a significant role in this region (Anania et al. in prep.). This comparison is meant as a first glance of the parameters space of the observed slopes, and we anticipate a proper exploration of the initial conditions once the full sample will be available.

In the following Section, we discuss the other constraints that we can put on disc evolution, besides identifying the driving accretion mechanism.




 §.§ What are the slopes telling us, then?


Despite not allowing to conclusively discriminate between different evolutionary scenarios with the current sample sizes, the slopes of the M_d - M_⋆ and Ṁ - M_⋆ correlations can still help with constraining other properties from the theoretical considerations presented in Section <ref>, which we summarise in Table <ref>. If we assume an evolutionary model to begin with, and we can estimate (directly or indirectly) either μ_0 or δ_0, we can constrain the other parameter. When discussing the observational determination of what we have so far referred to as initial conditions, it is important to clarify the meaning of "initial".  deals with and evolves Class II, potentially Class III, objects; hence, the initial conditions we input refer to the beginning of the Class II phase, where the protostellar collapse is over and the disc is already formed. From the observational point of view, this means that we expect δ_0 and μ_0 to refer to young Class II objects - around, or younger than, approximately 1 Myr. Earlier phases like the Class 0 and I need a dedicated study, as the accretion of the protostellar envelope is expected to play a prominent role in those stages.

In the following, we discuss the constraints we can put in both directions and comment on their feasibility based on the currently available estimates of μ_0 and δ_0.




  §.§.§ Constraining delta0 from mu0


<cit.> claimed λ_m to be increasing with time. As shown by <cit.> and discussed in Section <ref>, increasing slopes imply that discs around less massive stars evolve faster than those around more massive stars; this can be interpreted in terms of increasing accretion timescale with stellar mass, which corresponds to μ_0 > 0 (with t_acc, 0∝M_⋆^μ_0). In this section we discuss the implications of the increasing slope scenario on the initial conditions, λ_m, 0 and λ_acc, 0.

The top panel of Table <ref> shows the relation between μ_0 and δ_0 in the different evolutionary models. As μ_0 = δ_0 - ζ_0 ξ, in the viscous case (corresponding to ξ = 0) we have μ_0 = δ_0 as mentioned in Section <ref>. This means that, to recover the suggested increasing slopes scenario, δ_0 necessarily needs to be positive - regardless of the value of any other parameters: this translates to the initial M_d - M_⋆ correlation being steeper than Ṁ - M_⋆. In the hybrid and wind-driven case, instead, the implication is less straightforward as a positive μ_0 can lead to opposite signs of δ_0: this is determined by the scaling of the disc radius with the stellar mass, which is suggested to be (weakly) positive from observational evidences (e.g., ). In principle, as δ_0 ∈ (ζ_0 ξ, + ∞), the sign of ζ_0 determines whether negative values of δ_0 are possible; however, as ξ is a small number (0.1 in this work), only a limited area of the parameters space would lead to a negative δ_0. Summarising, if we assume increasing slopes (μ_0 > 0) we can constrain the sign of δ_0 regardless of the evolutionary model assumed: in both cases δ_0 needs to be positive, which leads to an initially steeper M_d - M_⋆ than Ṁ - M_⋆ correlation.




  §.§.§ Constraining mu0 from deltaevo


Instead of assuming increasing slopes, we can start from the currently measured values of δ_evo and estimate δ_0 in the different evolutionary models. In the viscous case, because δ_evo = 0, we focus on the value of the single slopes instead: as λ_m, evo = λ_acc, evo = δ_0/2 + λ_m, 0, the measured final value of the slopes does not help in constraining δ_0 as it also depends on λ_m, 0. In the hybrid case, instead, we have δ_evo = ξ(ζ_0 - μ_0), meaning that if we can determine the sign of δ_evo we can constrain that of μ_0 as well. While in principle the sign of ζ_0 influences that of μ_0, as we mentioned above ζ_0 is likely a small number: therefore, effectively, δ_evo and μ_0 have opposite signs for the vast majority of the parameters space.

Assuming that the observed disc populations can be considered evolved enough for the above arguments to hold, we can estimate δ_evo from the most recent and homogeneous measurements available of λ_m and λ_acc <cit.>. The resulting values of δ_obs (which we label 'observed' as opposed to the theoretical expectation, 'evolved'), summarised in Table <ref>, are oscillating: out of the four regions L1668, Lupus, Chameleon I and Upper Sco, we find two positive and two negative median values. Moreover, in three cases out of four the uncertainties are so large that δ_obs would be compatible with both a positive and a negative value. The difficulty in assessing the sign of δ_obs from the current measurements of the slopes make constraining μ_0 from δ_evo not trivial. Larger sample sizes would give a better measurement of the slopes and reduce the uncertainty, leading to a more solid determination of the sign of δ_obs - which would possibly allow to constrain μ_0. 

Summarising, the (admittedly not robust) observational evidence pointing towards increasing accretion timescale with stellar mass allows us to constrain the initial correlations between the stellar mass and disc parameters; regardless of the evolutionary model considered, the initial slope of the M_d-M_⋆ correlation needs to be larger than that of Ṁ - M_⋆. The other way around, constraining the slope of the accretion timescale - stellar mass correlation from the difference between λ_m and λ_acc at the present time, requires sample sizes larger by at least a factor two.








§ CONCLUSIONS


In this paper, we have investigated the impact of disc evolution models on the correlations between the stellar mass and the disc properties - especially the disc mass and the accretion rate. We have explored the purely viscous, wind-driven, viscous and wind hybrid, and photoevaporative models. Assuming power-law correlations to hold as initial conditions, M_d(0) ∝M_⋆^λ_m, 0, Ṁ(0) ∝M_⋆^λ_acc, 0, we performed analytical calculations (where possible) and population synthesis simulations for both evolutionary scenarios, and compared them with the purely viscous case discussed in <cit.>. Our main results are the following:




    
  * The viscous and hybrid models change the slope of the initial correlations as function of the evolutionary time, but preserve their shape. In the wind-driven model, instead, the correlations deviate from the original power-law shape: this is visualised in the logarithmic plane as a bending of the linear correlation (see Figure <ref>). The bending direction is towards the less or more massive stars depending on the scaling of the accretion timescale with the stellar mass (positive and negative correlation respectively).
    
    
  * The characteristic behaviour of the slopes in the wind-driven model is concealed by the presence of a spread in the initial conditions, which introduces a scatter in the correlations and makes it no longer possible to detect the bending (Figure <ref>). This leads to a considerably similar evolution of the correlations in the different accretion models.

    
  * Performing our simulations with evolutionary models that match the disc dispersal timescales (intrinsic in the wind-driven model and including internal photoevaporation in the viscous case), we find that the evolution of the slopes is significantly impacted by the removal of discs from the population (Figure <ref>). Different realisations of the same simulation dramatically differ from one another, and show a stochastic behaviour with large variations (Figure <ref>). This has both a physical (presence of a spread in the initial conditions) and a statistical (low number of objects left after a few Myr of evolution) reason.

    
  * While a proper exploration of the parameters space, outside of the scope of this work, would be needed to assess the impact of the initial conditions, with the currently available sample sizes the noise dominates over the physical evolution.

    
  * Increasing the sample size can mitigate the effects of disc dispersal on the evolution of the slope by removing the stochastic effects. We find that, for our parameters choice, the complete sample of Upper Sco (∼ 140 sources) at 5 Myr would reduce the oscillations enough to make the slopes a proxy for the evolutionary model (Figure <ref>).

    
  * While the currently available sample sizes do not yet allow to distinguish between the different evolutionary models, we can use them to put some constraints on the initial conditions. We find that in all evolutionary scenarios, the observational claim of increasing slopes leads to an initially steeper correlation between the disc mass and the stellar mass than between the accretion rate and the stellar mass. The other possible way, measuring the current slopes and inferring the correlation between the accretion timescale and the stellar mass from them, provides weaker constraints because of the high uncertainties in the current measurements.

    
  * We have presented and released the 1D Python disc population synthesis code  and its output analysis library .
    


In this work, we have shown how large enough samples of protoplanetary discs can provide a way of distinguishing between the evolutionary models (with a standard set of parameters) through the observation of the time evolution of the correlations between the disc properties and the stellar mass. We have shown how the stochastic fluctuations seen with the currently available observations could be significantly reduced if we had access to the complete Upper Sco sample, consisting of approximately 140 sources at 5 Myr. We strongly support the observational effort in the direction of obtaining larger amounts of data for evolved star-forming regions, and encourage the exploration of the parameters space beyond the standard case.





We thank an anonymous referee for their useful comments that helped us improve the clarity of the manuscript. This work was partly supported by the Italian Ministero dell’Istruzione, Università e Ricerca through the grant Progetti Premiali 2012-iALMA (CUP C52I13000140001), by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Ref no. 325594231 FOR 2634/2 TE 1024/2-1, by the DFG Cluster of Excellence Origins (www.origins-cluster.de). This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska- Curie grant agreement No 823823 (DUSTBUSTERS) and from the European Research Council (ERC) via the ERC Synergy Grant ECOGAL (grant 855130) and ERC Starting Grant DiscEvol (grant 101039651). GR acknowledges support from Fondazione Cariplo, grant No. 2022-1217. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

















aa







§ PARAMETERS USED THROUGHOUT THE PAPER


Table <ref> shows the parameters used throughout the paper. The general simulation parameters refer to the initial correlations (distribution, slopes and spreads), while the following three tables are divided by evolutionary model and display the main parameters for each of them.






§ MHD MODEL WITH MU<0




As mentioned in Section <ref>, the breaking of the linear correlation between the disc properties and the stellar mass happens towards higher or lower stellar masses depending on the value of μ_0. Figure <ref> shows the a simulation μ_0 > 0, while in Figure <ref> we show the opposite case. As μ_0 is linked to δ_0 through μ_0 = δ_0 - ζ_0 ξ, if δ_0 < ζ_0 ξ the resulting μ_0 will be negative, leading to a specular bending of the correlation. Given that ζ_0 ξ is a small number (∼ 0.1 in our simulation), this generally requires a negative δ_0. The simulation in Figure <ref> has λ_m, 0 = 1.3 and λ_acc, 0 = 1.7, resulting in δ_0 = -0.4.







§ VALIDATION OF 
 

Figure <ref> shows the evolution of the gas surface density as a function of the disc radius in the cases of evolution driven by (i) viscosity, (ii) viscosity and internal photoevaporation, (iii) MHD winds, and (iv) viscosity and external photoevaporation for a single disc simulated with . The top left panel, corresponding to viscous evolution (i), shows the key feature of viscous spreading: while the global surface density decreases as a consequence of the accretion onto the central star, the radial extent of the disc increases. This is a consequence of the redistribution of angular momentum, that causes part of the disc material to move towards larger radii. The top right panel, where the disc evolves under the combined effect of viscosity and internal photoevaporation (ii), shows the characteristic two-timescales behaviour <cit.>: the evolution is effectively viscous in the earliest stages, as long as the accretion rate is larger than the photoevaporative mass-loss rate; then, photoevaporation opens a cavity within the disc, which gets divided into an inner and an outer disc. The inner disc is less extended and therefore has a shorter viscous timescale, which means it evolves much faster and is quickly completely accreted onto the protostar; the outer disc instead keeps on evolving on timescales comparable to the original one, making photoevaporation a two-timescales process. The bottom left panel shows a disc evolving due to MHD winds (iii): the absolute value of the surface density drops faster than in the viscous model, because of the increase of α_DW in time. Furthermore, as angular momentum is removed from the wind (together with material), the disc does not spread but rather shrinks in time, as expected from the theoretical prediction <cit.>. Finally, the bottom right panel shows the evolution of a disc undergoing external photoevaporation combined with viscosity (iv): the latter dominates at the earliest stages, producing the characteristic features like viscous spreading, while the effect of external photoevaporation is visible at later ages as a truncation of the disc that also halts its spreading. In this case, the disc truncation and the outside-in depletion of disc material is the consequence of the photo-dissociation of gas molecules due to the FUV radiation emitted by massive stars and experienced by the disc. The efficiency of this process depends primarily on the stellar mass and the FUV flux experienced: given a fixed FUV flux, a disc around a lower mass star will lose its material to external winds more easily compared to a disc around a higher mass star, because of the higher gravitational bond of the system. For the same reason, more extended and less massive discs are more prone to external truncation.



Figure <ref> shows the isochrones in the Ṁ - M_d plane at 0.1, 1 and 10 Myr for the three evolutionary models of viscosity, viscosity and photoevaporation, and MHD winds. Each dot represents a disc in the population, while the solid lines show the analytical prediction (when applicable): in the viscous case, the isochrones read <cit.>


    Ṁ = M_d/2(2 - γ)t[ 1 - ( M_d/M_0)^2(2-γ)],


while in the MHD wind-driven scenario <cit.>


    Ṁ = 1/ω (1 + f_M, 0)t M_d[ ( M_d/M_0)^- ω - 1 ].



The left panel shows the viscous model, where the discs tend more and more towards the theoretical isochrone during their evolution <cit.>; the central panel includes internal photoevaporation, which has the effect of bending the isochrones once the accretion rate becomes comparable to the photoevaporative mass-loss rate <cit.>; finally, the right panel shows an MHD wind-driven population, where the scatter in the Ṁ - M_d plane remains significant during the evolution - contrary to the viscous prediction <cit.>.











 Paper Title: The D/H ratio in Titan's acetylene from high spectral resolution IRTF/TEXES observations.


1
.001


D/H ratio in Titan's acetylene    


<B. Bézard et al.>  


mode = title]The D/H ratio in Titan's acetylene from high spectral resolution IRTF/TEXES observations  























1]B. Bézard[orcid=0000-0002-5433-5661]


[1]





bruno.bezard@obspm.fr





[1]organization=LESIA, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Cité, CNRS,
            addressline=Place Jules Janssen, 
            city=Meudon,
           citysep=, 
            postcode=92195, 
            country=France

2]C.A. Nixon

conor.a.nixon@nasa.gov



1]S. Vinatier
sandrine.vinatier@obspm.fr

1]E. Lellouch
emmanuel.lellouch@obspm.fr

3]T. Greathouse
thomas.greathouse@swri.org
[1]

3]R. Giles
rohini.giles@swri.org

2]N.A. Lombardo
nicholas.lombardo@yale.edu
[2]

4]A. Jolly
jolly@lisa.ipsl.fr


[2]organization=NASA Goddard Space Flight Center,
            city=Greenbelt,
 
            postcode=20771, 
            state=MD,
            country=United States
            
 [3]organization=Southwest Research Institute,
            city=San Antonio,
            citysep=, 
            state=TX,
            country=United States
            
[4]organization=LISA, CNRS UMR 7583, Université Paris-Est Créteil, Université Paris-Cité,  IPSL,
            city=Créteil,
            citysep=, 
            postcode=94010, 
            country=France             



[1]Corresponding author


[1]Visiting Astronomer at the Infrared Telescope Facility, which is operated by the University of Hawaii under contract 80HQTR19D0030 with the National Aeronautics and Space Administration
[2]Present address: Department of Earth and Planetary Sciences,Yale University, New Heaven, CT 06511, United States





§ ABSTRACT 
 We report observations of deuterated acetylene () at 19.3 μm (519 ) with the Texas Echelon Cross Echelle Spectrograph on the NASA Infrared Telescope Facility in July 2017. Six individual lines from the Q-branch of the ν_4 band were clearly detected with a S/N ratio up to 10. Spectral intervals around 8.0 μm (745 ) and 13.4 μm (1247 ) containing  acetylene () and methane () lines respectively, were observed during the same run to constrain the disk-averaged   abundance profile and temperature profile. Cassini observations with the Composite Infrared Spectrometer (CIRS) were used to improve the flux calibration and help to constrain the atmospheric model. The measured D/H ratio in acetylene, derived from the / abundance ratio, is (1.22^+0.27_-0.21) × 10^-4, consistent with that in methane obtained in previous studies. Possible sources of fractionation at different steps of the acetylene photochemistry are investigated.
  


















Titan  Planetary atmospheres  Isotopic abundances Atmospheric composition  Infrared spectroscopy


[
    [
    August 5, 2024
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§ INTRODUCTION


Titan's complex photochemistry produces a suite of hydrocarbons and nitrogen-bearing compounds, eventually leading to the formation of haze particles. This chemistry is initiated by the dissociation and ionization of dinitrogen (N_2) and methane (CH_4) by ultraviolet solar radiation, particles from Saturn's magnetosphere and galactic cosmic rays. The vertical and horizontal abundance profiles of many of these photochemical species along with their seasonal variations have been extensively characterized through Cassini observations <cit.>. These investigations are complemented by other space-based and ground-based millimeter and mid-infrared observations, which in particular allowed the detections of new photochemical compounds <cit.>. 

Observational data serve as constraints for photochemical models that aim at understanding the chemistry at work in Titan's atmosphere <cit.>. Besides chemical abundance profiles, isotopic ratios may provide valuable information on the chemical and physical processes involved in the production and loss of the species. For example, the large difference in the ^14N/^15N ratio observed in HCN <cit.> and in N_2 <cit.>, the main nitrogen reservoir, allows us to constrain the relative flux of fractionated N atoms from photolysis and non-fractionated N atoms from other processes, such as magnetospheric electrons or galactic cosmic rays <cit.>. 

Dissociation of methane by solar ultraviolet radiation is the primary source of the hydrocarbons produced on Titan. Methane is also lost by reactions with various photochemical products and, most importantly, by reaction with the C_2H radical below 600 km. These processes are potentially a source of hydrogen fractionation in the photochemical products as they preferentially break the C-H bond over the stronger C-D bond <cit.>. Thus measuring the D/H ratio in various hydrocarbons can shed light on different photochemical pathways at work on Titan.

Besides methane, the only hydrocarbon in which the D/H ratio has been measured is acetylene (). Fitting simultaneously the 14.7-μm (678 ) ν_5 band of  and the nearby 13.7-μm (729 ) ν_5 band of  observed in spectral averages of nadir Cassini/CIRS spectra, <cit.> inferred a D/H ratio in  of (2.09±0.45) × 10^-4. This ratio appears significantly larger than that in methane, which is in the range (1.1–1.6) × 10^-4  <cit.>. However, from the analysis of CIRS limb spectra, <cit.> derived a lower value of the D/H ratio in , (1.63±0.27) × 10^-4, closer to the value in . It can be noted that the  emission feature is weak and mixed with  emission features at the CIRS resolution (0.5 ). <cit.> also detected a weak emission from the ν_4 band of  centered at 19.3 μm (519 ) in a large average of CIRS spectra (mixing disk and limb data). However, they did not analyze it due to the low S/N ratio and the presence of spurious instrumental features.

To overcome these difficulties and improve the precision in the D/H in acetylene, we used the Texas Echelon Cross Echelle Spectrograph (TEXES) on the NASA Infrared Telescope Facility (IRTF) to observe the ν_4 band of   at an unprecedented resolving power λ / Δλ of ∼65,000. We also observed spectral ranges around 8.0 and 13.4 μm to constrain the temperature profile and the  abundance profile in our radiative transfer analysis, in order to improve the accuracy of the /  ratio determination.




§ TEXES OBSERVATIONS


Observations of Titan were conducted with TEXES <cit.> mounted at the NASA/IRTF on July 8 and 14, 2017 UT under programmes 2017A045 (PI: Bézard) and 2017A109 (PI: Nixon) respectively. We targeted three spectral intervals containing lines from  (518.4–520.0 ),  (742.9–746.7 ) and  (1244.3-1250.7 ). TEXES was used in the high-resolution cross-dispersed mode, achieving a resolving power between 65,000 and 95,000. Details of the observations are given in Table <ref>. Titan's angular size was 0.78 arcsec, less than the slit width (2.0 or 1.4 arcsec), so that our measurements pertain to disk-averaged conditions. The sub-Earth and sub-Sun latitudes were both 27N. Observations occurred around Titan's summer solstice.

 

The  observations were performed by nodding Titan 5 arcsec along the 2.0 × 15 arcsec slit in order to remove the sky emission. For the  and  settings, the nodding amplitude was 3 arsec along the 1.4 × 9 and 1.4 × 7 arcsec slits respectively. The individual spectra were flat fielded, calibrated and co-added using the TEXES pipeline software package with the procedure described in <cit.>. To remove the residual telluric absorption, we divided the Titan spectra by spectra of the asteroid 10 Hygiea recorded shortly after. 

In a first step, we also relied on the 10 Hygiea spectra to obtain a flux calibration of our Titan observations. To do so, we used the asteroid Standard Thermal Model (STM) <cit.> with the Earth distance, Sun distance and phase angle corresponding to our observations. We used the asteroid radius determined by <cit.> (216.5 km) and the beaming parameter determined by <cit.> (0.81). The Bond albedo and emissivity were taken as 0.07 and 0.98 respectively, as in <cit.>. The STM spectrum, calculated for zero solar phase angle, was corrected for phase by 0.01 magnitude per degree (10 Hygiea's phase angle was 3.5 for the July 8 observations and 5.8 for the July 14 observations). Titan's flux (in Jy) is then given by:

    F_T= S_T/S_aF_a  f,

where S_T and S_a are respectively the Titan and asteroid measured fluxes (in TEXES units), F_a is the asteroid flux (in Jy) and f a possible corrective factor (see Section <ref>), for now set to 1.

Figure <ref> shows the TEXES spectra in the three settings. In the  setting, we clearly detect several lines from the Q-branch of the  ν_4 band with a signal-to-noise (S/N) ratio up to 10. These lines appear on a continuum mostly due to N_2– and N_2–H_2 collision-induced absorption and to haze emission. The  setting exhibits  lines of different intensities along with a H^12CN line, a weak H^13CN line and two weak emission features from C_3H_8. The  setting shows many strong and weak methane lines. The Titan-Earth Doppler shift of 15 km s^-1 allowed us to observe the full low-frequency half of the Titan  strong lines. In all plots, regions of low telluric transmission have been removed.






§ CASSINI/CIRS OBSERVATIONS


Because the ,  and  lines were not recorded simultaneously in the TEXES spectra, the precision of the D/H ratio we are able to achieve is sensitive to any error in the relative flux calibration of the three settings. We expect that our flux calibration, based on the ratio between Titan and 10 Hygiea observations, is no more than 10-15 % accurate. Error sources come from possible flux variations of asteroid 10 Hygiea with rotational phase and possible spectral variations of its emissivity between the three TEXES settings. Also, variations in the atmospheric seeing and pointing errors can induce flux calibration errors.

To get around this difficulty, we employed spectra recorded by the Cassini Composite Infrared Spectrometer <cit.> to calibrate our TEXES data. CIRS was a Fourier transform spectrometer, composed of three focal planes covering the spectral ranges 10–600  (FP1), 580–1100  (FP3) and 1050–1500  (FP4) with a spectral resolution adjustable between 0.5 and 15 . FP1 had a single detector with a circular field of view (FOV) of 3.9-mrad diameter while FP3 and FP4 each included a linear array of ten detectors providing an instantaneous field of view (IFOV) of 0.27 mrad (square). The CIRS absolute radiometric calibration is expected to be precise at  the 1% level <cit.> but great care must be taken in selecting the CIRS spectra for comparison with our disk-averaged TEXES observations.

Regarding the calibration of the  TEXES setting, we selected FP1 CIRS spectra recorded during a TEA (Titan Exploration at Apoapse) observational sequence designed to place Titan fully within the FP1 pixel <cit.>. Three TEA sequences are useable (CIRS_182TI_TEA001_PRIME, 
CIRS_202TI_TEA001_PRIME and CIRS_219TI_TEA001_PRIME). We chose to average the last two sequences in which the sub-spacecraft latitude is 0 and 51N, the first one being 41S, far from the 27N of our TEXES observations. Details of the CIRS observations are given in Table <ref>. The resulting spectrum is shown in Fig. <ref> along with a second-order polynomial fit over the range 460–560  (a degree 2 for the polynomial fit is the one that minimizes the reduced χ ^2). This fit indicates a continuum intensity of 0.396 ± 0.002 mW m^-2 sr^-1 / cm^-1 at 519  that we need to convert into a flux in Jy at the Titan-Earth distance of the TEXES observations (9.124 AU). 
However, this is not straightforward because the CIRS FP1 FOV is not homogeneous, being greatest at the center, reaching half power at 1.2 mrad from center and zero at 3.2 mrad <cit.>. To do so, we used the FOV sensitivity function determined by <cit.> and a synthetic spectrum of Titan's emission calculated as a function of distance from disk center. This spectrum was generated from the radiative transfer and atmospheric models described in Section <ref>, with the haze opacity adjusted to reproduce the CIRS radiance at 519 . Doing so, we expect a continuum flux of 23.0 Jy in the TEXES observations around this wavenumber. This is about 6% lower than obtained by using the flux from the STM of 10 Hygiea, and we have then to set the corrective factor f in Eq. <ref> to 0.94.





The CIRS FP3 and FP4 detectors have a much lower IFOV than FP1's FOV and there are no CIRS observing sequences in which Titan is fully included in their linear arrays. We then turned to the COMPMAP (Composition Map) sequences in which the FP3 and FP4 arrays were positioned to span Titan's disk in one to five positions <cit.>. We selected the one noted in Table <ref> as it is closest in terms of time and in sub-solar and sub-instrument latitudes to the TEXES observations. We binned the observations into concentric rings of equal area, centered on Titan's disk and covering altogether 0 to 3050 km in radius (i.e up to 475 km above the surface). We then summed these bins and converted the result into a flux (Jy) at a Titan-Earth distance of 9.124 AU, as relevant for the July 8 TEXES observations. The FP3 and FP4 final spectra are shown in Fig.<ref>. We degraded the TEXES observations at the resolution of the CIRS spectra (0.53 ) with the help of a synthetic spectrum (see Section <ref>) to fill in the troughs in the TEXES data. From a least-squares fit approach, we then determined the corrective factor f in Eq. <ref> to be 1.09 for the  setting and 0.86 for the  setting, taking into account the Titan-Earth distances of the observations (Table <ref>).  



In the radiative transfer analysis of the TEXES observations, we used this CIRS-based flux calibration to which we assign an uncertainty of ±5%, arising from the construction of the CIRS disk averages and possible variations of the integrated disk flux. We note that the corrective factors we derived vary from 0.86 to 1.09, which seems reasonable given the uncertainty in the 10 Hygiea STM and possible pointing errors.



§ RADIATIVE TRANSFER ANALYSIS


Spectral modeling of the observations was performed with a line-by-line radiative transfer code coupled with an iterative inversion scheme <cit.> as described in <cit.>. Sources of molecular opacity are the collision-induced opacity (CIA) from N_2,  and H_2 pairs, ro-vibrational bands of , CH_3D, , , HCN, C_2H_6 and C_3H_8, and absorption from aerosol particles. We used the CIA absorption coefficients from <cit.>, <cit.> and  <cit.>. Line positions, energy levels and intensities of  and  come from GEISA2020 <cit.>, the  parameters being based on the analysis of <cit.>. For the N_2 pressure-broadened line widths and temperature exponents, we used the m-dependent values determined by <cit.>. Line parameters for the other molecules are the same as in <cit.>. As regards the aerosol opacity, we used the spectral dependence recommended by <cit.> (their Fig. 4), which is based on <cit.> beyond 600 . The integrated optical depth at the reference wavenumber of 1090  is 0.029.

The atmospheric model includes vertical profiles of gas abundances, aerosol absorption and temperature. The  profile is fixed to that derived by <cit.> from Huygens in situ measurements. For the other molecular species and the aerosol opacity, we adopted the profiles retrieved by <cit.> from Cassini/CIRS limb and nadir measurements recorded on September 11, 2017 at 21N. Note that the corresponding  profile is only used as an a priori (first guess) profile in the retrievals of the  profile from TEXES measurements. This is also true for the HCN profile. All other absorber abundance profiles are left unchanged in the retrieval code.

The temperature profile that serves as an a priori profile in the inversion process is the 21N profile (September 11, 2017) from <cit.> down to the 10-mbar level. In the troposphere, the profile is constrained to reproduce two selections of FP1 spectra (15- resolution) from 55 to 545 , recorded in 2016–2017 between 0 and 40N and from a distance of (7.5–9.5) × 10^4 km from Titan. The first one is characterized by a “low” mean emission angle of 13.1 and the second one a “high” mean emission angle of 54.2.
 
Spectra are calculated using an atmospheric grid of 70 layers equally spaced in log p from the surface up to p = 0.15 μbar (∼600 km). For Cassini/CIRS modeling, monochromatic spectra are convolved with a Hamming function of resolution 0.528  (distance of first zero). For TEXES modeling, the convolution function is a Gaussian with the full width at half-maximum given in Table <ref>. Radiance spectra are calculated for five lines-of-sight intercepting the surface from disk center to the solid radius and for six lines-of-sight above the surface at altitudes up to 475 km. These two regions are divided into respectively five and six circular rings of equal area. The distance of the lines of sight to disk center is taken as the average of the radii of the concentric circles delimiting each annulus. The intensities are then summed with weights equal to the circular ring areas to produce an integrated flux spectrum. 

The inversion algorithm minimizes a weighted sum of the residuals between synthetic and observed spectrum (i.e. the χ^2) and between solution and a priori profile (temperature or mole fraction) <cit.>.  A correlation length (L) of one pressure scale height is further used for filtering the solution profile. The disk-averaged temperature profile in the stratosphere was retrieved from the methane emission spectrum in the 1200–1370  range for the CIRS measurements and 1244.7–1250.5  for the TEXES measurements, assuming that the stratospheric methane mole fraction is constant with altitude and over the disk <cit.>. In fact, by simultaneously analyzing FP1 and FP4 Cassini/CIRS spectra at different latitudes, <cit.> showed that the stratospheric methane abundance varies within a range of 1 to 1.5% as a function of latitude. In particular, the ∼1% mole fraction derived at low latitudes agrees with the reanalysis of the Huygens Descent Imager/Spectral Radiometer (DISR) data by <cit.>. The  abundance profile they derived is lower than the Huygens/GCMS profile <cit.> above ∼39 km and reaches ∼1% above 110 km. Consequently, we also ran an alternative case in which we used the <cit.> methane profile in place of the <cit.> profile (see Section <ref>).

The disk-averaged acetylene profile was retrieved from the 679–780  interval for the CIRS measurements and 742.8–746.6  for the TEXES measurements. The  /  ratio (assumed to be constant with altitude) was inferred from the TEXES measurements combining the intervals 518.38–518.73 and 519.08–519.17 . The aerosol optical depth at the reference wavenumber was simultaneously retrieved.

The procedure we followed in this analysis is the following:





  * retrieve the temperature profile from the disk-averaged CIRS FP4 selection using the <cit.> 21N profile as a first guess ⟶ T_CIRS(p)

  * retrieve the temperature profile from the TEXES measurements ( setting and continuum in the  setting) taking T_CIRS(p) as the a priori profile ⟶ T_TEXES(p)

  * retrieve the  profile from the disk-averaged CIRS FP3 selection with the <cit.>  21N profile for a priori profile and T_CIRS(p) for the temperature profile ⟶ q_CIRS(p)

  * retrieve the  profile from the TEXES measurements ( setting) using q_CIRS(p) for the a priori profile and T_TEXES(p) for the temperature profile ⟶ q_TEXES(p)

  * determine the  /  ratio from the TEXES measurements ( setting) using T_TEXES(p) for the temperature profile and q_TEXES(p) for the  profile
  

We also performed some tests in which we used the Huygens/HASI temperature profile <cit.> recorded in situ at 10S as the a priori profile in Step a or directly in Step b. As regards Step e, we tested an additional case in which we use the CIRS-derived temperature (T_CIRS(p)) and acetylene (q_CIRS(p)) profiles to derive the / ratio (see next section).





§ RESULTS

The temperature profile retrieved from the inversion of the CIRS FP4 average (T_CIRS(p)) is shown in Fig. <ref> and the corresponding model fit to the data is shown in Fig. <ref>b. The observations are sensitive to the 0.03–11 mbar range (95–350 km) (Fig. <ref>). In the 0.1-mbar region, this profile is warmer than the a priori profile, which refers to latitudes around 21N, while the situation is reversed in the 1–10 mbar range. This is expected as the CIRS spectral average incorporates latitudes that are on the average warmer than mid-northern latitudes around 0.1 mbar and cooler in the 1–10 mbar pressure range, as can be seen in Fig. 2 of <cit.>.

In a second step, this CIRS-inverted profile was used as an a priori profile to retrieve a temperature profile T_TEXES(p) from the TEXES observations around 1248 . This profile is shown in Fig. <ref> and the corresponding fit of the data in Fig. <ref>a. The higher spectral resolution of the TEXES data allows us to somewhat tighten the constraints in the lower stratosphere down to about 15 mbar (85 km) (Fig. <ref>). In the inversion procedure, we also used data in small spectral intervals in the continuum around 518.5–518.7  to constrain the (near-)surface temperature. The CIRS– and TEXES–retrieved temperature profiles are very similar and differ by at most 1.5 K in the 2–mbar region. 





A  profile q_CIRS(p) was first obtained from the CIRS FP3 spectral average using the 21N profile of <cit.> as the a priori in the inversion process (Fig. <ref>). The fit to the data is shown in Fig. <ref>a. The region probed ranges from 0.12 mbar (275 km) to 18 mbar (80 km) (Fig. <ref>). The TEXES data around 745  were then used to retrieve the final  profile (q_TEXES(p)) shown in blue in Fig. <ref> using the CIRS-retrieved profile as the a priori. The fit of the data and the residuals are shown in Fig. <ref>b. As for temperature, the high resolution of the TEXES observations allows us to slightly improve the constraints on the disk-averaged  profile, extending the region probed to ∼0.10–20 mbar (75–285 km). The TEXES-derived  mole fraction is some 35% smaller than the CIRS-derived one in the 0.2–0.5 mbar range and about 20% larger from 4 to 15 mbar.



The formal 1–SD noise errors on the TEXES-retrieved profiles are 0.2 K in the 0.03–15 mbar range for the temperature and 2% for the  mole fraction in the 0.10–20 mbar. We evaluated the model errors from the variations induced in the retrieved profiles by a change in the a priori temperature and  abundance profiles used in Steps a) and c). To do so, we considered a priori profiles that differ from the <cit.> 21N profiles above the 20-mbar level by ±10 K (temperature) and factors of 0.5 or 2 (). The so-derived error bars are shown in Fig. <ref> at a few pressure levels.

Finally, we used the TEXES  setting around 519  to retrieve the  /  ratio using the TEXES-derived temperature profile T_TEXES(p) and  profile q_TEXES(p). In the inversion process, the  /  ratio is assumed to be constant with altitude and we thus solved for only one parameter (technically this is done by fixing a very large value for the correlation length L, 100 times the pressure scale height). We derived a / ratio that corresponds to a D/H ratio of (1.14±0.04) × 10^-4, where the error bars account only for the noise equivalent flux in the TEXES  observations and correspond to a χ^2 variation of 1.  The model fit to the observations is shown in Fig. <ref>. Our measurement is most sensitive to the region between 0.7 and 16 mbar (Full Width at Half Maximum of the contribution function), with the contribution function peaking at 6 mbar (115 km) (Fig. <ref>).

We investigated different sources of error. We made a first test in which we used the CIRS-derived atmospheric model (T_CIRS(p) and q_CIRS(p)) to model the  TEXES lines. We then derived a D/H ratio of (1.09±0.04) × 10^-4, i.e. 5% smaller than when the temperature and acetylene profiles are derived from the  and  TEXES settings. We also tested the sensitivity of the derived D/H ratio to assumptions about the a priori profiles used in the retrieval process. First we used the Huygens/HASI profile <cit.> as the a priori profile for the temperature retrievals from Cassini/CIRS data (Step a) or directly for the temperature retrievals from TEXES data (Step b). In either case, the so-derived D/H ratio is within 1% of our nominal value. Secondly, we considered the propagation of the model errors mentioned above into the retrieved D/H ratio. A variation of the a priori temperature model by ±10 K induces a change by ±3% in the derived D/H ratio while a variation of the a priori acetylene model by a factor of two causes negligible variations of the retrieved D/H ratio (<1%).

As discussed in Section <ref>, we also ran a case in which the temperature was retrieved using the <cit.> methane profile, which has a stratospheric  abundance of 1% instead of the 1.48% mole fraction from <cit.>. In this case, the derived temperature profile is warmer by at most 2.6 K at 1.3 mbar and by 1.2 K at 0.25 and 10 mbar. The derived  mole fraction is consequently lower by up to -32% around 1 mbar while the D/H ratio reaches 1.31 × 10^-4, the  lines being less sensitive to temperature change than the optically thick  lines. Because the methane mole fraction varies over Titan's disk between 1 and 1.5% <cit.>, we retained for the nominal value of the D/H ratio the (geometrical) mean of the values retrieved for the <cit.> and for the <cit.>  profiles, these values providing the conservative upper and lower limits: D/H (in ) = (1.22^+0.09_-0.08) × 10^-4.

We also considered a 5% flux calibration error in each of the three TEXES settings and finally took into account the ±4% uncertainty of the  band intensity reported by <cit.>. The errors induced on the D/H ratio are listed in Table <ref>. Combining quadratically these different error sources, we obtain a relative error of +24% / -18%, leading to:


D/H (in ) = (1.22^+0.27_-0.21) × 10^-4



We note that an important source of error is that due to the flux calibration of the  setting. A 5% variation of the flux scale causes a change of the retrieved  abundance profile of 24% at 1mbar, 11% at 5 mbar and 6% at 10 mbar. The induced change on the  /  ratio is then +17% / -12%. The second most significant source of error is the uncertainty on the disk-averaged stratospheric  used in the temperature retrievals.





§ DISCUSSION


The value we retrieved for the D/H ratio in acetylene (1.22^+0.27_-0.21 × 10^-4) is significantly smaller than that derived by <cit.> from different selections of Cassini/CIRS nadir spectra over Titan's disk (2.09±0.45 × 10^-4). It is also marginally smaller than the value derived by the same authors from Cassini/CIRS limb data at four different latitudes, averaging to 1.63±0.27 × 10^-4, although their error bars clearly overlap with ours. These CIRS determinations are made difficult by the weakness of the  emission in the 678- ν _5 band at the instrument's resolution of 0.5 . In addition, the part of the analysis limited to surface-intercepting measurements suffers from the lack of a precise determination of the  vertical profile from the data, which has a strong influence on the retrieved  /  ratio. On the other hand, the precision of our measurement is mostly limited by the fact that we do not observe the  and  lines simultaneously. We have then to rely on a relative calibration of the two instrumental settings for which we used Cassini/CIRS spectral averages as representative as possible of the observing conditions of TEXES (period, sub-solar and sub-instrument latitudes).

The D/H ratio in acetylene we obtained is consistent, within error bars, with the D/H ratio measured in methane from Cassini/CIRS measurements by <cit.> (1.32^+0.15_-0.11 × 10^-4), <cit.> (1.17^+0.23_-0.28 × 10^-4), <cit.> (1.58±0.16 × 10^-4) and <cit.> (1.59±0.27 × 10^-4) and from the ground-based measurements by
<cit.> (1.25±0.25 × 10^-4) and <cit.> (1.13±0.25 × 10^-4), indicating altogether a value in the range (1.1-1.6) × 10^-4. 

This result suggests no significant fractionation of deuterium in acetylene compared with methane, the main atmospheric reservoir of hydrogen. On the other hand, as first investigated by <cit.> and then <cit.>, a source of fractionation in the methane photochemistry lies in the higher energy of the C–D bond compared with the C–H bond. These authors considered that methane is most efficiently destroyed by the abstraction reactions: 

C_2H + CH_4  →  + CH_3 (k_1)

C_2H + CH_3D  →  + CH_2D (k_2a)

  → C_2HD + CH_3 (k_2b)

From consideration of chemical kinetics of some deuterated species, <cit.> argued that q, equal to (k_2a + k_2b)/k_1, lies in the range 0.80–0.88. Because the ratio of the reaction coefficients q is lower than 1,  destruction is favored over CH_3D destruction, opening up the possibility of a deuterium fractionation in the photochemical products. We define q' as the ratio of the intrinsic probability of breaking a C-D bond to that of breaking a C-H bond, or equivalently by the ratio 3 k_2b/k_2a. If we further assume that k_2a is equal to 0.75 k_1, given that Reaction 2a does not break the C–D bond, then q' =3 k_2b/k_2a =  4 k_2b/k_1 is in the range 0.2–0.52.

According to the photochemical model of <cit.>, photolysis of ethylene () is the main source of acetylene in the bulk of the atmosphere. This production peaks around 300 km. Ethylene is mostly produced in the upper atmosphere (∼800 km) from the reaction of the CH radical with methane and flows down to the stratosphere where it is mostly lost by photolysis. In Appendix <ref>, we use a simplified 0-D photochemical model to identify possible fractionation processes in the production of  linked to the expected higher binding energy of the C–D bond. We found that fractionation of deuterium can occur at two stages: in the formation of the CH radical from the ^3CH_2 radical and from the photolysis of . Both of them tend to enhance the D/H ratio in  and thus in . If the q' ratios associated to these two reactions were similar to that for C_2H + CH_4 mentioned above (0.2–0.52), the D/H in  would be as high as 1.4–1.9 times that in  (Eq. <ref>), which is not what we observe. On the other hand, if fractionation were limited to the CH radical formation stage (i.e., assuming no fractionation through the  photolysis), the fractionation factor would be lower (1.08–1.17) and compatible with the D/H measurement errors.This implies that either the kinetic isotope effect (KIE) in the photodissociation of  is not as strong as in the C_2H + CH_4 reaction considered by <cit.>, or that other photochemical processes that we did not consider in Appendix <ref> mitigate this KIE.



§ CONCLUSIONS


We report here the first observation of several lines from the Q-branch of the ν_4 band of   on Titan  at 19.3 μm. Combining observations of ,  and  lines with the same instrument (IRTF/TEXES) and using Cassini measurements for the flux calibration, we derived a value for the D/H ratio in acetylene of (1.22^+0.27_-0.21) × 10^-4. This value is consistent within error bars with the D/H ratio measured in methane. Our main source of uncertainty arises from the relative flux calibration of the three different settings we used to record the ,  and  lines. Improving on the flux calibration, e.g. by observing these emissions simultaneously, would allow us to reduce significantly the error bars on the D/H ratio in acetylene. Recent observations by the MIRI medium-resolution spectrometer (MRS) aboard the James Webb Space Telescope (JWST) conducted in July 2023 (Titan project GTO 1251) may help in this regard once issues in fringe removal and flux calibration have been resolved.

While our result indicates no significant deuterium fractionation in acetylene relative to methane, at least in the region probed (0.7-16 mbar; 80-190 km), we have identified two possible processes in the photochemical production of  that should in theory enhance the D/H ratio in acetylene. These are the formation of the CH radical from the ^3CH_2 radical and the photodissociation of . This potential enhancement arises from the fact that the C–D bond is slightly stronger than the C–H bond. Laboratory measurements of the KIEs for these two reactions, most importantly for the  photolysis, would be very valuable. It would be also useful that complete photochemical models incorporate deuterium species, even with approximative KIEs, to investigate the deuterium fractionation in various hydrocarbons. From an observational standpoint, signatures from other deuterated hydrocarbons (e.g. C_2H_6, ) could also be sought in Titan's spectrum to better constrain its atmospheric photochemistry.



§ DATA AVAILABILITY


The raw TEXES data used in this study can be obtained from the NASA/IRTF archive at <https://irsa.ipac.caltech.edu/applications/irtf/>. The product IDs for the three spectral settings (,  and ) are:


  Titan C_2HD: TX17A0708.1038-1045

  10 Hygiea C_2HD: TX17A0708.1035-1036

  Titan C_2H_2: TX17A0714.6004-6017

  10 Hygiea C_2H_2: TX17A0714.6018-6019

  Titan CH_4: TX17A0708.1030-1033

  10 Hygiea CH_4: TX17A0708.1017-1028

The raw data were calibrated using the 2017 version of the TEXES pipeline software (current version is available at <https://github.com/TEXESArch/pipecode>). The reduced data presented in this study are archived <cit.> as ascii files.



§ ACKNOWLEDGMENTS


BB, SV and EL acknowledge support from the Programme National de Planétologie (PNP) of CNRS-INSU co-funded by CNES. TK was a Visiting Astronomer at the Infrared Telescope Facility, which is operated by the University of Hawaii under contract 80HQTR19D0030 with the National Aeronautics and Space Administration.






















§ FRACTIONATION IN THE PRODUCTION OF 


Potential sources of isotopic fractionation in the photochemistry of methane include photochemical reactions (molecular, ion-molecule and photodissociation), atmospheric escape, condensation/evaporation and molecular diffusion. We investigate here how the photochemical production of  from methane photochemistry could lead to a deuterium fractionation due to the greater strength of the C–D bond compared to the C–H bond. To do so, we only considered the major chemical steps leading to the formation and loss of  (through 0-D calculations), based on the photochemical model of <cit.>.  We further assumed that a reaction of a species A (radical or molecule) with a deuterated hydrocarbon B has the same reaction rate k as with the main isotopologue if it does not break the C–D bond and a rate q' k, with 0≤ q' ≤1, if the C–D bond is broken. The same is assumed for a photodissociation reaction.

According to 's () model,  is mostly formed by photodissociation of  in the upper stratosphere (∼300 km):

 + hν  →  + 2H (J_1)

  →  + H_2, (J_2)

and accordingly for  by:

 + hν  →  + 2H (0.5 J_1)

  →  + H_2 (0.5 J_2)

Note that in addition,  can be dissociated as:

 + hν  →  + H + D (0.5 q'_d J_1)

  →  + HD (0.5 q'_d J_2)


 and  are both lost by vertical transport to the condensation region in the lower stratosphere (∼70 km), with a loss rate proportional to the species number density in this formation region: K_1 [] and K_1 []. We neglect here the small vapor pressure isotope effect in the relative condensation loss of the two isotopologues. Equaling the production and loss rates for both isotopologues yields:
[] = J_1 [] / K_1 and [] = 0.5 J_1 [] / K_1, so that 

    [D/H]_C_2H_2 = 0.5 [ C_2HD]/[ C_2H_2] = 0.25 [C_2H_3D]/[C_2H_4] = [D/H]_C_2H_4^1,

where the latter term is the D/H ratio in ethylene in the formation region of acetylene (upper stratosphere).

Ethylene is transported from its formation region in the thermosphere where it has a D/H ratio that we denote [D/H]_C_2H_4^0. The  used to produce  is therefore supplied by a source having this D/H ratio and lost by photolysis with a  to  loss ratio equal to  0.5 (1+q'_d) J [] / (J [] ) = 0.5 (1+q'_d) [] /  [], with J = J_1 + J_2. Therefore,  in steady state and neglecting other sinks for ethylene, we obtain:

    [D/H]_C_2H_4^1 = 2/(1+q'_d) [D/H]_C_2H_4^0

Photodissociation of ethylene thus appears as a potential source of enhancement of the D/H ratio in this species and consequently in acetylene, provided that q'_d <1.

Following <cit.>, we assume that the bulk of ethylene is formed in the thermosphere, around 800 km, through the reaction:

CH +  →  + H,(k_1)

while  is accordingly formed by:

CH +  →  + H, and(0.75 k_1)

CD +  →  + H(k_1)

The CH radical is mostly formed from the suite of reactions:

 + hν  → ^1CH_2 + 2H(J_3)

^1CH_2 → ^3CH_2

^3CH_2 + H → CH + H_2(k_2),

where ^3CH_2  is the methylene radical in the ground electronic state X^3B_1 and ^1CH_2 the methylene radical in the excited state a^1A_1.
The CHD radical is formed through:

 + hν → ^1CHD + 2H(0.5 J_3)

and lost through:

^3CHD + H → CD + H_2(0.5 k_2)

 → CH + HD(0.5 q'_f k_2)

The former reaction also forms CD, which is lost through the CD +  reaction mentioned above.

Balancing production and loss rates for each radical and assuming [CH_3D] << [CH_4] yields:[^3CH_2] = J_3 [] / (k_2 [H]),[CH] = J_3 / k_1,[^3CHD] = J_3 [] / (k_2 (1 + q'_f) [H]), and[CD] = (0.5 / (1 + q'_f)) J_3 [] / (k_1 []).

Ethylene is mostly lost by downward transport, a process that is not isotope-selective and proportional to the number density. Denoting the loss rates for the two isotopologues as K_0 [] and K_0 [] and again balancing production and loss rates, we obtain:[] = J_3 [] / K_0,[] = J_3 [] (0.75 + 0.5 / (1 + q'_f)) / K_0

The D/H ratio in ethylene in its production region is then given by:

    [D/H]_C_2H_4^0 = (0.75 + 0.5/(1+q'_f)) [D/H]_CH_4


A fractionation of deuterium may then occur in the production of ethylene, linked to the reaction of the methylene radical (CH_2) with the H atom. Combining Eqs. <ref>–<ref>, the D/H ratio in acetylene in this simplified model is equal to: 

    [D/H]_C_2H_2 = 2   (0.75 + 0.5/(1+q'_f)) / (1 + q'_d)  [D/H]_CH_4


If the q' factors associated to  photodissociation (q'_d) and to reaction of CH_2 with H (q'_f) were in the range of that for the C_2H + CH_4 reaction <cit.>, the D/H ratio in acetylene would be enhanced by a factor in the range 1.42–1.94 compared to the ratio in methane. This is not consistent with our measurement. On the other hand, if fractionation were only effective for the CH_2 + H reaction and not for the  photodissociation (q'_d = 1), the fractionation factor would amount to 1.08–1.17, which is compatible with our result, given the errors associated with D/H ratio measurements in acetylene and methane. The kinetic isotope effect (KIE) in  photodissociation appears to be potentially the most important factor in the deuterium fractionation in acetylene. Our result of D/H in   being so similar to that in  suggests that the D/H fractionation in  via the photolysis of  should be weaker than in the catalytic destruction of methane through the C_2H + CH_4 reaction.







cas-model2-names












 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: Clouds on partial atmospheres of lava planets and where to find them.





0000-0001-6171-200X]T. Giang Nguyen
Department of Physics,
McGill University,  3600 rue University, Montréal, QC H3A 2T8, Canada

0000-0001-6129-5699]Nicolas B. Cowan
Department of Physics,
McGill University,  3600 rue University, Montréal, QC H3A 2T8, Canada
Department of Earth & Planetary Sciences,
3450 rue University, Montréal, QC H3A 0E8, Canada

0000-0003-4987-6591]Lisa Dang
Trottier Institute for Research on Exoplanets and Department of Physics, Université de Montréal, 1375 Ave.Thérèse-Lavoie-Roux Montréal, QC H2V 0B3, Canada

§ ABSTRACT 

With dayside temperatures hot enough to sustain a magma ocean and a silicate atmosphere, lava planets are the best targets to study the atmosphere of a rocky world. In the absence of nightside heating, the entire atmosphere collapses near the day-night terminator, so condensation seems inevitable, but the impact of clouds on radiative transfer, dynamics, and observables has not yet been studied in the non-global atmospheric regime. Therefore, we simulate cloud formation and determine which lava planets should be most affected by clouds. We find that despite the scattering of visible light by clouds, heat advection compensates for the cooling effect of clouds in the atmosphere. On the other hand, surface temperatures are significantly affected and can drop 100-200 K under a cloudy sky. We find that among our targets, HD213885b and HD20329b are most affected by cloud formation: there is a discernable difference between having clouds and not having them, but the precision required to make such an inference is at the limit of current instruments.







§ INTRODUCTION
 



 §.§ Background


Hundreds of Earth-sized planets have been discovered <cit.> but inferring their surface and atmospheric properties remains a significant challenge. Our best options for in-depth characterization of rocky planets are currently ultra-short period planets. Phase curves obtained with the Spitzer Space Telescope have enabled tentative inferences of atmospheres on highly irradiated super-Earths K2-141b <cit.> and 55 Cnc-e <cit.>.

A more powerful instrument is needed for definitive atmospheric detection and characterization of thin silicate atmospheres. Indeed, the recently launched James Webb Space Telescope (JWST) has observed ultra-short period planets K2-141b <cit.> and 55 Cnc-e <cit.>, and will observe TOI-561b <cit.> and K2-22b <cit.>. Such observations can provide unique clues about terrestrial planet evolution.

Ultra-short period rocky planets with dayside temperatures >1800K are theorized to harbour a permanent dayside magma oceans and are often referred to a lava planets. Since the silicate melt is directly exposed to the atmosphere, atmospheric dynamics are intrinsically dependent on interior dynamics <cit.>. Magma oceans are a common result of planetary formation <cit.> and play a critical role in sequestering volatiles <cit.>, so lava planets provide a unique window into the early solar system and Earth <cit.>. For a complete overview of silicate outgassing and the corresponding chemistry and radiative-convective dynamics, see <cit.>.

Tidally locked planets risk having their atmosphere condense out entirely on the permanent nightside <cit.>. However, tidally-locked terrestrial planets may sustain a global atmosphere if the initial outgassed atmosphere is thick enough <cit.>. Therefore, general circulation models (GCMs) are adequate for simulating a thick-atmosphere regime for lava planets such as 55 Cnc-e <cit.>. A general source of atmospheric circulation on tidally locked planets can be found in <cit.>. Despite such robust work on lava planet atmospheres, the non-global atmospheric regime is less studied and cannot be simulated with a GCM. Two approaches are therefore used to simulate a thin lava planet atmosphere: either focusing on the dayside radiative transfer or the hydrodynamics.

Radiative transfer simulations of lava planets show a silicate atmosphere with a strong vertical temperature inversion (e.g., ). These studies often find the atmosphere hotter than the surface as heat cannot be advected to cooler regions. On the other hand, hydrodynamical models of non-global atmospheres often simplify, if not outright ignore radiative transfer (e.g., ). However, this method can trace the horizontal heat and mass transport due to day-night winds. Both approaches ultimately lack self-consistent cloud formation which affects both the fluid dynamics and radiative transfer.



 §.§ Objective


In anticipation of JWST observations of lava planets, we simulate cloud formation and quantify their effect on thin-atmospheric dynamics and observations. Although there are many published works specific to exoplanet clouds <cit.>, it is impossible to use a sophisticated cloud model within an existing non-global atmosphere modelling framework. We therefore use a simple parameterization of cloud formation consistent with the turbulent boundary layer model of <cit.> which is the only published work that have coupled radiative transfer with hydrodynamical processes for non-global atmospheres.



We use this model to simulate the atmospheres of five lava planets: K2-141b <cit.>, TOI-431b <cit.>, HD20329b <cit.>, HD213885b <cit.>, and 55-cnc e (, but we note that its recently-published eclipse spectrum suggests a volatile-full planet: ). The selected candidates have some of the best signal-to-noise ratios for rocky exoplanets and they will likely be observed extensively. Therefore, we compare these lava planets to predict where cloud formation might be expected.



§ METHODS
 



 §.§ The hydrodynamical model


Thin and tenuous atmospheres are common on moons and dwarf planets of the outer solar system. Huge temperature fluctuations on these bodies induce periods and locations of atmospheric collapse and massive surface outgassing, e.g., on Io <cit.> or Pluto <cit.>. Due to the stark changes in fluid regime, it is difficult to implement GCMs and hydrodynamics are best approximated using boundary layer analysis  <cit.>. Because the atmospheres of lava planets are expected to behave similarly, the same calculations are useful, if not necessary.

The expectation of synchronous rotation due to tidal locking makes lava planets easier to deal with than solar system objects. Having a permanent dayside and nightside allows for steady-state flow. Due to the complexities in handling non-linear terms around evaporation and condensation, we model a pure atmosphere.
We are also forced to neglect atmospheric escape which should happen for lighter volatiles <cit.> but is justified if we track heavy molecules like SiO. Finally, we neglect Coriolis force as to preserve the axial symmetry of tidally locked objects. Generally, for planets with rotational period of multiple days, the km/s wind speeds are strong enough to make Coriolis force insignificant <cit.>. However, our targets have a period of ≤ 1 day and so we acknowledge that Coriolis force may play a large role in dynamics but implementing it is beyond our current capability.

By applying the approximations described above, we can construct a framework that describes the flow along the planetary boundary layer. The formulation of the base evaporation-driven flow is akin to the shallow-water equations of <cit.>. The equations essentially calculate pressure, winds, and temperature of the atmosphere at half the scale height. Detailed formulation can be found in the Appendix.



 §.§ Radiative transfer


Our implementation of radiative transfer largely follows <cit.> and a detailed formulation can be found in the appendix. <cit.> split up the radiative transfer into three bands (UV, visible, IR) with an averaged absorption cross-section within each band. To improve on this scheme, we now calculate radiative absorption and emission line by line in the spectrum, doing away with averaging spectral bands. We validate our radiative transfer scheme by directly comparing our wind-free silicate atmosphere with <cit.>. We find that our atmospheric temperatures match well with theirs (within 5%) whereas the old scheme of <cit.> can be off by up to 25%, especially for hotter and brighter stars.

We adopt the absorption cross-section of SiO from in <cit.>. As for the stellar spectra, we found proxy analogues in the MUSCLES treasury surveys for K2-141, TOI-431 and 55-Cnc: their closest analogues are HD85512, HD40307, and HD97658, respectively <cit.>. For our targets of HD213885 and HD20329, their properties are closest to the Sun and so we use the solar spectrum; the spectra are shown on the bottom panel of Fig. <ref>. Absolute magnitude of each analogue's spectrum is scaled to the target's effective stellar temperature and orbital distance.






 §.§ Cloud implementation


Clouds are abundant on exoplanets but their modelling and detection have largely been confined to hot Jupiters <cit.> and sub-Neptunes <cit.>. Some of these clouds likely consist of silicate material like SiO and MgSiO_3 <cit.>. Although lava planets have exotic atmospheric dynamics, their cloud composition is not, at least by the standards of exoplanets. Cloud particles on lava planets may be analogous to dust found on catastrophically evaporating rocky worlds <cit.>.

Rapid mass exchange between the surface and the atmosphere of a lava planet may lead to meteorological processes driven by the recycling of rocky material between the surface and atmosphere <cit.>. Volatile components like Na and K may form clouds due to interactions with stellar winds but these light material are subjected to atmospheric loss <cit.> or freeze out in nightside alkali glaciers <cit.>.

Because we are focusing on lava planets with Earth-like bulk composition (excluding 55 Cnc-e), we can reasonably assume that volatiles such as water or CO_2 are absent from the atmosphere <cit.>. Silicate condensates remain the dominant constituent but if the dew point is too close to the surface, the silicate vapour may not condense midair but simply deposit directly on the surface of the magma ocean <cit.>. We do not know how much of the SiO vapour will condense into cloud particles nor do we know the specific properties of these particles. Since clouds might have significant impact on the atmosphere of a lava planet, ignoring them altogether is not a viable option. We therefore opt to parameterize cloud effects into our hydrodynamical model in the same spirit as how radiative transfer was implemented, i.e., as simply as possible.

As cloud formation is inherently tied to the condensation rate C, we define a cloud optical depth proportional to C: τ_c = f_c  C. The dimensionless f_c parameterizes the fraction of condensible material that actually turns into cloud particles as well as the particle properties that affects optical depth such as particle size and density. Although much nuance is lost in this formulation, current observations only allow us to probe the temperature of the atmosphere and surface. Therefore, our main aim is to estimate how much clouds affect the overall thermodynamics and see if clouds can produce a detectable signal.

By creating and testing a range of f_c, we are essentially inferring how the planet behaves under varying strength of cloud coverage. This helps to identify important aspects such as minimum detectable cloud formation, which planets are most affected by clouds, and direct inference of cloud properties.

We calculate the resultant opacity (ϵ_c) following Eq. <ref>. Here we make another simplifying cloud assumption: cloud particles scatter visible light isotropically so that half the incident visible stellar light (F_*,vis) is reflected upwards when the cloud opacity is 1. This assumes that the clouds are transparent in the IR and that most of the UV is absorbed by the upper levels of the atmosphere. Note that ignoring cloud absorption in the IR neglects their greenhouse effect, removing significant complexities to the system of equations. This yields F_c, the stellar flux that is reflected because of clouds:


    F_c = 1/2 ϵ_c F_*,vis.


Because SiO is nearly transparent at the visible wavelengths (Fig. <ref>), the atmospheric energy balance is unaffected by F_c. However, the surface energy balance is significantly affected as visible light is absorbed entirely by the surface. This stems from treating the surface as a blackbody but is justified in the expected low geometric albedo of magma surfaces <cit.>: the surface energy is calculated from the bulk bolometric flux minus stellar radiation absorbed by the atmosphere plus the emissions from the atmosphere minus visible stellar flux that is reflected by clouds minus latent heat when evaporating. The surface energy balance then becomes:


    σ T_s = F_* - F_stel + F_RC - F_c - Q_lat,


where σ is the Stefan-Boltzmann constant. The surface temperature, T_s depends on the overlying atmospheric properties and vice versa (i.e. Eq. <ref>, Eq. <ref>), coupling the surface and atmosphere. F_* is the bulk stellar flux, F_stel is the stellar flux that has been absorbed by the atmosphere, F_RC is the thermal emission from the atmosphere, F_c is the stellar flux that was reflected by clouds, and Q_lat is the latent heat of vaporization.



 §.§ Simulated observations


This subsection will describe how state variables P, V and T, computed as in the Appendix, are used to simulate observations. Because a lava planet's atmosphere is so thin and does not extend much past the terminator, transmission spectroscopy of the gravitationaly bound atmosphere is challenging <cit.>. We therefore focus on planetary emission. We first calculate the top-of-atmosphere outgoing radiative flux F_TOA:


    F_TOA(λ) = ϵ_λ B(λ,T) + (1-ϵ_λ) B(λ,T_s),


where the first term is atmospheric emission and the second surface emission. We take into account the adiabatic profile where T(P) is calculated at the pressure where optical depth is 2/3. For any location θ, the flux can be calculated and so we convert temperature maps into flux maps.

We convert F_TOA(θ) into a map of co-latitude (α) and longitude (ϕ) assuming polar symmetry about the sub-stellar point. We then integrate the flux across the visible hemisphere following <cit.>:


    F_e(λ,ζ) = ∫_0^π∫_ζ-π/2^ζ+π/2 F(α,ϕ,λ) sin^2(α) cos(ϕ-ζ) dϕ dα,


where F_e is the emission signal, ζ is the position of the planet in its orbit (ζ=0 at superior conjunction and ζ=π at inferior conjunction). Keeping λ constant and varying ζ between 0 and 2 π yields a phase curve at wavelength λ. Keeping ζ at 0 while varying λ yields the emission spectrum at eclipse.



§ RESULTS
 



 §.§ Output from model


All the equations from the previous section together form a self-consistent model that couples the atmosphere and surface along with rudimentary cloud formation. To simulate our specific targets, we apply the planetary parameters listed in Table <ref>.

ccccccc[t!]
System parameters used for the hydrodynamical model. Although these values do not show up in the equations mentioned in the previous section with the exception of g, the orbital properties are needed to calculate the incident stellar flux pattern, F_*(θ), following <cit.>.
0pt

Planet Name     Radius (R_E)     Stellar Radius (R_*)     Distance (a/R_*)     Stellar Temp. (K)     g (m/s^2)     Source

K2-141b     1.54     0.681     2.30     4373     21.8     <cit.> 

TOI-431b     1.28     0.731     3.32     4850     18.4     <cit.> 

HD20329b     1.72     1.13     3.42     5596     24.6     <cit.> 

HD213885b     1.74     1.10     3.93     5978     28.5     <cit.> 

55 cnc e     1.87     0.964     3.52     5317     22.3     <cit.> 



We show the initial results without any cloud formation in Fig. <ref>. We see that atmospheric pressure is directly related to surface temperature: the hotter the surface, the thicker the atmosphere. However, despite having cooler surface temperature, HD213885b and HD20329b have much hotter atmospheres; they also have stronger winds.



We plot the contributions of all radiative terms in Fig. <ref>. With the exception of K2-141b, most of the atmospheric absorption is in the UV owing the absorption features of SiO in that range. Most of the radiative cooling occurs in the IR although HD213885b and HD20329b are hot enough to emit significant UV radiation.

Despite the enormous magnitude of the radiative fluxes, incoming and outgoing radiation tend to reach equilibrium: the net radiative balance is 1–2 order of magnitude smaller than each individual radiative terms.





 §.§ Results with clouds


To review, we parameterized cloud optical effects using the parameter f_c in the Methods section. Note that our simple implementation of clouds neglect certain feedbacks such as the greenhouse effect of clouds. With the assumption that cloud formation must be proportional to the condensation rate C, optical depth and opacity must subsequently also depend on C. Therefore, f_c acts as a substitute to estimate how much of the condensible material becomes cloud, as well as cloud properties such as droplet size; we explore these properties in the Discussion section.

We restrict f_c between 10^2–10^4; any lower and cloud effects are negligible across all planets while anything higher will make opacity uniformly 1. Because the SiO vapour absorption is weak in the visible, clouds do not directly affect the radiative budget of the atmosphere but do so via atmosphere–surface interactions. The biggest impact, therefore, is on the surface energy budget as expected from our formulation (see Eq. <ref>). We show the condensation rate, optical depth, opacity, and surface temperature of cloudy simulations in Fig. <ref>.





 §.§ Simulated observations


Given a global map for surface and atmospheric temperature, we calculate the outgoing radiative flux at the top of the atmosphere using Eq. <ref>. We fix the time at eclipse and integrate the flux over the visible hemisphere using Eq. <ref> at different wavelength λ to yield Fig. <ref>. 





§ DISCUSSION
 



 §.§ On radiative energy balance


We see that the stellar spectrum predictably dictates how hot the atmosphere gets. Despite the surface of K2-141b receiving more bolometric stellar flux than HD20329b and HD213885b, its atmosphere absorbs less stellar flux as the star K2-141 emits mostly in the visible and IR. Meanwhile, HD213885b and HD20329b absorbs more stellar UV and visible radiation, thus providing more heat to their atmopsheres.

The coarser radiative scheme of <cit.> tended to overestimate UV radiative balance while underestimating the IR. With a finer spectral resolution, we see that UV emission never dominates and that SiO spectral features in the IR are strong enough to emit flux even when the pressure is low.



 §.§ Vertical temperature profile


One-dimensional radiative-convective models predict that while the lower silicate atmosphere of lava planets does follow an adiabatic profile, there is a sharp temperature inversion at the top of the atmosphere due to shallow UV penetration in the atmosphere <cit.>. Hydrodynamical studies, on the other hand, predict that advection of heat weakens this inversion <cit.>.

Although we adopted an adiabatic vertical temperature profile, we can guess what happens when the vertical temperature lapse rate is inverted. This can be done by increasing the β value (this parameter is calculated from vertical pressure integration in Eq. <ref>). <cit.> showed that arbitrarily increasing β leads to stronger horizontal flow but an overall cooler atmosphere. Within the calculations, raising β increases the horizontal pressure gradient force, thus increasing wind velocity, V. However, the energy balance does not change so any gains in the kinetic term, V^2/2, translate to a loss in the internal heat term, Cp T (see Eq. <ref>). The physical reasoning is that because the atmospheric inversion inhibits mixing, horizontal winds are significantly increased, but since energy must be conserved, the air's internal heat energy must be lower.

When balancing the incoming radiation in the atmosphere, we find that stellar absorption is comparable to surface blackbody absorption for K2-141b, TOI-431b, and 55 Cnce; this suggests a rather isothermal T-P profile as there is as much heat from the top as there is at the bottom. However, for HD213885b and HD20329b, there is significantly more stellar absorption than surface absorption. With significantly more heating from the top, most of which is UV flux, we would expect a strong temperature inversion as predicted by <cit.>. While we have kept a consistent vertical temperature profile in this study to isolate the effects of clouds, we note that future simulations of lava planets should aspire to use self-consistent vertical profiles given the spectrum of the host star.



 §.§ Latent and chemical heat


As the radiative terms tend to cancel out, the net energy budget of the atmosphere is controlled by dynamics terms such as latent heat and sensible heat. We show in Fig. <ref> that the latent heat is negligible, as predicted by <cit.>. Even when coupling the latent heat with the atmosphere where radiative transfer is hampered by the absorption cross-section, latent heating changes the atmosphere by less than 5%. However, K2-141b has a relatively larger condensation rate and a relatively lower radiative absorptivity due to its cooler star; temperatures can increase up to 10% when latent heating is accounted for. Sensible heating is still the main driver of the flow and it is composed of heat flux from two directions: horizontally via flow-induced heat advection and vertically via temperature mixing between the surface and atmosphere.



Although latent heat is relatively small, the vaporization of the surface and subsequent condensation is not a simple phase change. The magma ocean is principally composed of SiO_2 (assuming a Bulk Silicate Earth composition; ), while the atmosphere contains mostly SiO, hence evaporation and condensation involve the dissociation and recombination of oxygen atoms.  In the context of hot gaseous planets, <cit.> showed that the energy of dissociation and recombination of hydrogen molecules can be significant. We estimate the energy of the reaction SiO_2 ⇒ SiO + (1/2)O_2 based on the difference in enthalpy of formation between SiO and SiO_2 (enthalpy of O_2 is 0). With this difference being 811 kJ/mol <cit.>, the energy per unit mass of SiO is 2×10^7 J/kg, an order of magnitude magnitude greater than the latent heat of vaporization (10^6 J/kg, ). Unlike the latent heat, this chemical energy may dominate the energy budget, but we leave this problem to future studies, since our current model can only accommodate a single gas species.



 §.§ The effects of clouds on lava planet climate


The formation of clouds has a weaker effect on the atmosphere than on the surface, due to the fact that the atmosphere is optically thin at visible wavelengths even in the absence of clouds. Visible light reflected by clouds would have mostly passed through the atmosphere in any case; it is the surface that loses that energy and thus can cool by 200 K. 

Smaller f_c correspond to larger cloud particles, which are radiatively less important. Assuming spherical droplets, we can decompose f_c into cloud particle properties via:


    f_c  = 3/8 t/ρ d p_e.


The density, ρ, of the condensed SiO_2 is 2650 kg/m^3,  p_e is the fraction of condensible material that forms cloud droplets instead of condensing directly on the surface of the magma ocean—this depends on the altitude at which clouds form. The particle diameter, d, is typically sub-micron to 10s of microns for silicate clouds in hot Jupiter atmospheres <cit.> while dust grain analysis of catastrophically evaporating rocky world yield 0.1–1 micron <cit.>. Finally, t is the timescale over which the cloud droplet stays in the atmosphere. Two timescales could be at play: the time it takes for the cloud droplet to fall to the surface or the time it takes for the cloud droplet to be advected from the point of nucleation to where the atmosphere collapses. 

We estimate the terminal velocity of free-falling cloud particles falling through 2500 K gas under an acceleration g = 25 m/s^2 using Stokes' law: terminal velocity, V = (2/9) (ρ_c-ρ_a)g r^2 / η, where ρ_c is the particle density and ρ_a is the air density. With micron-size particles, the terminal velocity is a very slow 2×10^-4 m/s. We surmise that the horizontal advection timescale is likely the limiting factor. It takes ∼ 5000 s to transport a parcel of gas 45^∘ of angular distance for a planetary radius of 1.5 R_E and a wind speed of 1.5 km/s. Therefore we get f_c∼10^4 given d=10^-6 m, and p_e=0.1. 

The effect of clouds on the atmospheric energy budget is minimal because we have assumed clouds are transparent in the IR, but the surface is influenced significantly. Because we have treated the magma ocean as a still surface, it does not have any mechanism to advect heat (cf. ). Therefore, the surface is entirely dependent on instellation and losing up to half of all visible light due to clouds results in significant temperature decreases that might be detectable due via astronomical observations.



In the simulated eclipse spectra shown in Fig. <ref>. We see that the greater the cloud opacity, the lower the overall emission. Clouds have the biggest impact on the dynamics of HD213885b and HD20329b due to the high incident stellar radiation at visible wavelengths. However, this difference in overall signal is too small for current instruments to precisely characterize. Ultimately, K2-141b remains the best target lava planet for atmospheric detection because of signal-to-noise but is the worst target to differentiate between cloudy and clear sky conditions. 





§ CONCLUSIONS
 

Clouds are a fundamental part of a planet's meteorology. Lava planets are most likely to have, at minimum, non-global atmospheres. In this regime, clouds should appear as the atmosphere starts to condense out. Therefore, we simulated the atmosphere of lava planets K2-141b, TOI-431b, HD20329b, HD213885b, and 55-cnc e to see which of these planets are best for cloud detection and characterization.

With or without clouds, our calculations show that atmospheric temperature is more dependent on the stellar spectrum than bolometric flux. HD213885b and HD20329b, whose stars emit more UV light, have a hotter atmosphere than K2-141b despite receiving less instellation.

We implement clouds by scaling the optical depth to the condensation rate using the f_c parameter to roughly approximate weak, medium, to strong cloud effects. We find that atmospheric properties are not significantly affected by visible light scattering of clouds: the existing heat advection from the evaporating region through sensible heat as well as latent is enough to offset clouds. The surface is more affected by clouds and the effect is predictably stronger for greater optical instellation. As such, the overall effect of clouds is best seen on HD213885b and HD20329b but is still challenging to distinguish.

Implementing an adiabatic profile up to the top of atmosphere create deep absorption features which render the spectroscopic signature of clouds to be indistinguishable; adopting the silicate T-P profiles of <cit.> and <cit.> should yield more accurate observables which may reveal an opportunity to detect clouds but the results of such 1D models must be reconciled with atmospheric dynamics.



§ ACKNOWLEDGEMENTS


This research was partially funded by the Canadian Space Agency’s James Webb Space Telescope observer’s program and the Heising-Simons Foundation. NBC acknowledges support from an NSERC Discovery Grant, a Tier 2 Canada Research Chair, and an Arthur B. McDonald Fellowship. The authors also thank the Trottier Space Institute and l’Institut de recherche sur les exoplanètes for their financial support and dynamic intellectual environment.


aasjournal



§ APPENDIX




 §.§ Base hydrodynamical formulation


The framework for the hydrodynamical calculation is
a system akin to the shallow-water equations describing the conservation of mass (Eq. <ref>), momentum (Eq. <ref>), and energy (Eq. <ref>) following <cit.>:


    ∂_x (ρ h V) = m  E,


    ∂_x (ρ h V^2) = -∂_x ∫_z P dz + τ,


    ∂_x (ρ h V (V^2/2 + C_p T)) = Q_sens + Q_lat + Q_RC,


where x and z denote the horizontal and vertical direction coordinates. P, V, and T are state variables pressure, velocity, and temperature. Variables ρ and h are air density and column thickness. Parameters m and C_p are the molecular mass and heat capacity respectively. Eq. <ref> describes how the change of material moving along the flow is controlled by the mass exchange between the surface and the atmosphere via evaporation, E. Eq. <ref> describes how the momentum of the air parcel is controlled by the horizontal pressure gradient force and surface drag, τ. Eq. <ref> describes how the kinetic energy and internal heat energy of the parcel is controlled by the energy fluxes which includes sensible (Q_sens), latent (Q_lat), and radiative heating terms (Q_RC) and is expanded upon later in the text.

The mass flux, E [molecules/s/m^2], is the term that describes what is added and what is removed from the atmosphere. This is controlled by evaporation:


    E = P - P_v(T_s)/m √(2 π R T_s),


where P_v is the saturated vapour pressure taken from <cit.> and T_s is the surface temperature. R is the gas constant. To further calculate the fluxes, such as surface drag τ and components of Q, we define a mean flow velocity V_e = m E/ρ and an eddy velocity V_d = V_*^2/V. The frictional velocity V_* is calculated iteratively from the implicit relation with V:




    V = 2.5 V_* log(9.0  V_*  H ρ/2η),


where H is the scale height, ρ is the density, and η is the temperature dependent viscosity [η = 1.8×10^-5(T/291)^3/2(411)/(T+120) following Sutherland's ideal gas formula <cit.>]. With these velocities, we further define advection coefficients that describe how heat and momentum is dynamically transferred between the surface and atmosphere, w_a and w_s:



    w_a    = 2V_d^2/V_e+2V_d,        (V_e > 0)
      
       w_s    = V_e^2+2 V_d V_e + 2 V_d^2/V_e + 2V_d,





    w_a    = V_e^2-2 V_d V_e + 2 V_d^2/-V_e + 2V_d,        (V_e ≤ 0)
     
    w_s    = 2 V_d^2/-V_e + 2 V_d.



The above equations yield the dimensionless advection factors when there is net evaporation (V_e > 0) and deposition (V_e ≤ 0). These parameters are then used to calculate the frictional drag, τ and sensible heat, Q_sens:


    τ = -ρ V  w_a,



    Q_sens = ρ(w_s C_p T_s + w_a(V^2/2 + C_p T)).


In Eq. <ref>, the first term is the heat contribution from the surface and the second term is the atmospheric advection. The latent heat term Q_lat, is simply:


    Q_lat = L  m  C,


where L is the latent heat of vaporization and C is the condensation rate. This term is defined such that latent heat is added to the atmosphere when condensing but is taken away from the surface when evaporating; this term will subsequently show up in the surface energy balance equation. The last remaining flux term is the pressure gradient force where we must integrate the vertical pressure with respect to height from a specified altitude to infinity:


    ∫_z=z_0^∞ P dz = ∫_0^P_0Φ/g dP = C_p/g∫_0^P_0 (T_0 - T) dP.


Eq. <ref> switches from the local z altitude coordinates to P levels by substituting the geopotential variable Φ. By expresing Φ as work done by moving the air parcel vertically under the gravitational acceleration g, we can relate it to enthalpy which yields a function dependent on the pressure-temperature profile. For an adiabatic profile, completing the vertical integration yields:


    C_p/g∫_0^P_0 (T_0 - T)  dP = C_p  P_0  T_0  β/g,


where β = R/(R + C_p). Eq. <ref> essentially provides an analytical solution to the pressure integral of a specific profile but one can generate another temperature profile and solve Eq. <ref> numerically.

With the exception of Q_RC, equations <ref>, <ref>, <ref>, and <ref> show how all of the fluxes are calculated in equations <ref>-<ref>. The next step is to convert the local coordinates to global coordinates in terms of θ, the angular distance from the substellar point. Steps detailing the full conversion can be found in <cit.> but the net result is shown:


    1/r  sin(θ)d/dθ(V_*  P_*  sin(θ)/g) = m  E,



    1/r  sin(θ)d/dθ((V_*^2 + β C_p  T_*)P_*  sin(θ)/g) = β C_p  T_*  P_*/g  r  tan(θ) + τ,



    1/r  sin(θ)d/dθ((V_*^2/2 +  C_p  T_*)V_*  P_*  sin(θ)/g)  = Q_sens + Q_lat + Q_RC,


The asterisk subscript on the state variables denotes its value at the of the turbulent boundary layer. Because surface drag induces a logarithmic vertical wind profile, the wind speed at half the scale height is closest to the weighted mean flow of the air column. Therefore, the turbulent boundary layer is taken to be at half the scale height to harmonize our treatment of mass, momentum, and energy advection.

The equations are solved as a set of ordinary differential equation with respect to θ. We use the “shooting" method where we guess the boundary conditions at the substellar point. Because we are dealing with non-linear mass, momentum, and energy fluxes, a steady-state solution is particularly sensitive to the initial conditions. The only definitive condition is V(0)=0; for T(0) and P(0), we iteratively have to solve for the conditions that is most numerically stable as described in <cit.>. Once a stable solution is established, the flow inevitably reaches supersonic speed and we linearly interpolate the transition from subsonic to supersonic flow. 

As the atmosphere gradually condenses out, there comes a point when the atmosphere transitions into an exosphere where molecules behave less like a continuous fluid and more like independent particles in ballistic motion. As the flow regime changes, our calculations fail. The result is a spatially bounded atmosphere spanning more or less a hemisphere with steady state evaporation and condensation.



 §.§ radiative transfer formulation


To calculate the radiative contributions to the atmosphere and surface, we first define the vertical optical depth, τ_d as a function of pressure:


    τ_d = x_a P_*/mg,


where x_a is the wavelength-dependent absorption cross-section (see Fig. <ref>). The absorptivity/emissivity ϵ of the overlying atmospheric column is related to the optical depth:


    ϵ = 1 - e^-τ_d.


The emissivity ϵ helps to calculate the individual radiative contribution from stellar radiation absortion (F_stel), surface blackbody radiation (F_surf), and radiative cooling (F_RC). The total heat contribution from radiative transfer, Q_RT becomes:



    Q_RT  =     F_stel + F_surf - 2F_RC - F_c
    
        =    ∫_λϵ(λ) F_*(θ,λ) dλ + ∫_λϵ(λ) B(T_s,λ) dλ - 2∫_λϵ(λ) B(T,λ) dλ - ϵ







where F_* is the stellar flux and B is the Planck blackbody function which depends on both wavelength and temperature. Since lava planets are so close to their star, parallel ray approximations no longer hold and F_* is not easily solvable; we use the formulation of <cit.> to account for finite light source effect. Note that the radiative cooling term is doubled to signify that the atmosphere radiates both down into the ground and outwards into space.

Although Eq. <ref> is identical to what we had in <cit.> barring cloud reflection, we have significantly improved the numerical integration of the stellar flux and blackbody radiation. Previously, we averaged the emissivity across the entire ultraviolet and infrared wavelength bands, while neglecting wavelengths in the optical completely. The problem is that the absorption cross-section of SiO varies enormously with changing temperature and pressure. We therefore calculate absorption and emission line by line with varying pressure and temperature ahead of the simulation. Stellar absorption depends mostly on pressure and a function of F_*(P) is extracted for easy implementation. Radiative cooling is much more tricky as it depends on both pressure and temperature, therefore a lookup table of flux is generated (see Fig. <ref> to quickly determine the radiative transfer terms.















 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: 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: KPF Confirms a Polar Orbit for KELT-18 b.












0000-0003-3856-3143]Ryan A. Rubenzahl
NSF Graduate Research Fellow


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

0000-0003-1312-9391]Samuel Halverson
Jet Propulsion Lab, Pasadena, CA 91125, USA

0000-0001-8638-0320]Andrew W. Howard


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

0000-0003-3504-5316]Benjamin Fulton


0000-0003-0012-9093]Aida Behmard
Kalbfleisch Fellow
American Museum of Natural History, 200 Central Park West, Manhattan, NY 10024, USA

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

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

0000-0003-3133-6837]Abby P. Shaum


0000-0002-0531-1073]Howard Isaacson
501 Campbell Hall, University of California at Berkeley, Berkeley, CA 94720, USA

 

0009-0008-9808-0411]Max Brodheim


0009-0000-3624-1330]William Deich


0000-0002-7648-9119]Grant M. Hill


0000-0002-6153-3076]Bradford Holden


0000-0003-2451-5482]Russ R. Laher


0009-0004-0592-1850]Kyle Lanclos


0009-0008-4293-0341]Joel N. Payne


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


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




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

0000-0002-6092-8295]Josh Walawender


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

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

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

0000-0002-4265-047X]Edward Wishnow














§ ABSTRACT 

We present the first spectroscopic transit results from the newly commissioned Keck Planet Finder on the Keck-I telescope at W. M. Keck Observatory. We observed a transit of KELT-18 b, an inflated ultra-hot Jupiter orbiting a hot star (= 6670 K) with a binary stellar companion. By modeling the perturbation to the measured cross correlation functions using the Reloaded Rossiter-McLaughlin technique, we derived a sky projected obliquity of λ = (ψ = 93.8_-1.8^+1.6^∘ for isotropic ). The data are consistent with an extreme stellar differential rotation (α = 0.9), though a more likely explanation is moderate center-to-limb variations of the emergent stellar spectrum. We see additional evidence for the latter from line widths increasing towards the limb. Using loose constraints on the stellar rotation period from observed variability in the available TESS photometry, we were able to constrain the stellar inclination and thus the true 3D stellar obliquity to ψ =. KELT-18 b could have obtained its polar orbit through high-eccentricity migration initiated by Kozai-Lidov oscillations induced by the binary stellar companion KELT-18 B, as the two likely have a large mutual inclination as evidenced by Gaia astrometry. KELT-18 b adds another data point to the growing population of close-in polar planets, particularly around hot stars.





§ INTRODUCTION
 

KELT-18 b is an ultra-hot Jupiter discovered by the KELT transit survey <cit.>. The 1.57 planet orbits its F5 type (6670 K) host star every 2.87 days. Hot stars (≳ 6250 K) with hot Jupiters (HJs) have been observed to have a broad range of obliquities, where the obliquity is defined as the angle between the host star's rotation axis and the planet's orbital plane. Conversely, HJs orbiting cooler stars (≲ 6250 K) tend to be aligned with their host star's rotation axis <cit.>. The transition temperature is near the Kraft Break <cit.>, suggesting realignment mechanisms that are effective for cooler stars–which have convective envelopes and strong magnetic fields–but are ineffective for hotter stars–which have radiative envelopes and weak magnetic fields <cit.>. 

While it is still an unsolved problem, the origins of HJs are likely a combination of multiple formation channels (see  for a review), namely in-situ formation, disk migration, and high eccentricity migration (HEM). HEM likely plays a significant role in shaping the overall HJ population <cit.> but must be triggered by an additional body in the system. This could be another planet, in the case of planet-planet scattering <cit.> or von-Zeipel-Kozai-Lidov[See <cit.> for a historical monograph.] oscillations <cit.> induced by an outer planetary <cit.> or stellar companion <cit.>. In the HEM scenario, the HJ originally formed beyond the water ice-line (∼2 AU) where giant planet formation is efficient <cit.>. The orbital eccentricity was increased through interactions with a perturbing companion until the planet's periastron distance became small enough for tides to dissipate energy and transfer orbital angular momentum to the star, causing the orbit to shrink and circularize. Either this HEM process, or perhaps a primordial misalignment of the protoplanetary disk <cit.>, leaves the HJ on an orbit that may be tilted by a large angle relative to the stellar equatorial plane. Only the HJs around stars cooler than the Kraft Break were then able to realign their host star's rotation axis.


The spin-orbit angle is usually measured as projected on the plane of the sky (λ), but for systems in which the inclination of the host star's rotation axis () can be inferred, the true 3D obliquity can be derived (ψ). Recently, <cit.> noted that hot Jupiters around hot stars do not span the full range of ψ, but instead show a preference for near polar orbits (80^∘–120^∘). However, there are still too few systems to be sure that the obliquity distribution has a peak near 90^∘  <cit.>. If there is a “polar peak” it would have important theoretical implications on plausible HJ formation mechanisms, which predict different obliquity distributions <cit.>. For small planets with massive outer companions, secular resonance crossing in the disk dispersal stage may produce polar orbits <cit.>. For giant planets, an initially inclined orbit inherited from a torqued protoplanetary disk in the presence of a binary companion can give the necessary starting point for subsequent ZKL-driven migration to create a polar HJ <cit.>.

The most commonly employed method for measuring the projected obliquity of a star with a transiting planet is to obtain high-resolution spectra throughout a transit and model the Rossiter-McLaughlin effect <cit.>,
which results from
the planet's obscuration of part of the rotating stellar photosphere.
This effect is often
modeled as an anomalous radial velocity (RV) signal, but measuring precise RVs for hot stars can often be challenging due to their fast rotation rates. The projected equatorial rotation velocity, , broadens (and blends) spectral lines, diminishing the Doppler information content <cit.>. As a result, stars with ≳ 10  are usually not amenable to anomalous-RV modeling. However, these fast rotating stars lend themselves to more detailed and direct methods of measuring the stellar obliquity. 
The Reloaded RM (RRM) method, developed by <cit.>, directly models the distortion of the line profile by the transiting planet. By subtracting an out-of-transit reference CCF (representing the star alone) from each in-transit CCF (corresponding to the stellar line profile integrated over the full disk, minus the integrated line profile from within the patch of the star beneath the planet's shadow), the resulting signal represents the “local” CCF, i.e., the spectrum originating from the portion of the star obscured by the transiting exoplanet (CCF_loc). The RRM method is also sensitive to stellar differential rotation, should the planet be highly misaligned so that it transits a wide range of stellar latitudes <cit.>.

In this paper we report our derivation of the obliquity of the host star in the KELT-18 system based on a time series of spectra taken during a transit of KELT-18 b with the Keck Planet Finder (KPF). By modeling the spectra according to the RRM method, we found the orbit of KELT-18 b to be nearly perpendicular to the star's equatorial plane. In Section <ref> we derive stellar properties, reexamine the rotation period with TESS photometry, and identify the nearby star KELT-18 B as a bound companion. We describe the Keck Planet Finder and our observations in Section <ref>, the RRM modeling procedure in Section <ref>, and dynamical implications for the KELT-18 system in Section <ref>.





§ KELT-18 SYSTEM




lrrr
Parameters of the KELT-18 System


  Parameter     
  Value     
  Unit    
  Source 

KELT-18 

              6670± 120                 K             M17 

M_∗             1.524^+0.069_-0.068     M_⊙     M17 

R_∗             1.908^+0.042_-0.035     R_⊙     M17 

u_1                0.337_-0.010^+0.011                   M17 

u_2                0.3229_-0.0059^+0.0066                M17 

RV_sys      -11.7 ± 0.1                       This work 
   
KELT-18 b 

             2.87169867 ± 0.00000085     days        I22 

t_c               2458714.17773 ± 0.00011     JD          I22 

b                 0.10^+0.10_-0.07                      M17 
 
             88.86^+0.79_-1.20         degrees     M17 

R_p/R_⋆       0.08462 ± 0.00091                       M17 

a / R_⋆       5.138^+0.038_-0.078                   M17 

e                 0                                         M17 

M_p               1.18 ± 0.11                 M_J       M17 

KELT-18 B 

              3900                        K             M17 

M_∗             0.575_-0.026^+0.025     M_⊙     B22 

sep                  1082                          AU            B22 

Δμ_RA     0.21±0.05        mas yr^-1     B22 

Δμ_Decl.     0.41±0.05     mas yr^-1     B22 

Δparallax     0.07±0.04             mas               B22 

ΔG            -5.4                    mag               B22 

ln(ℒ_1/ℒ_2)     4.83    —               B22 

(M17) <cit.>; (I22) <cit.>; (B22) <cit.>.


KELT-18 is a rapidly rotating F4 V star with one known transiting exoplanet, discovered by <cit.> (hereafter M17), and a stellar neighbor. M17 derived robust stellar properties using high resolution spectra, SED fitting, photometry, and evolutionary modeling in a global fit with their transit model. We adopted their best-fit stellar and transit parameters for our analyses herein, with two distinctions noted below. Table <ref> lists the full set of adopted parameters.

For the transit midpoint and orbital period of KELT-18 b, we adopted the improved ephemeris of <cit.> which implies an uncertainty of only 20 sec in the predicted transit midpoint on the night of our spectroscopic observations (Section <ref>).

For the projected stellar rotation velocity , M17 noted that their value of 12.3 ± 0.3  obtained from a TRES <cit.> spectrum is likely an overestimate as the method they used conflates macroturbulence and rotation. M17 also measured a value of 10 ± 1  using a HIRES <cit.> spectrum and the  <cit.> framework, but did not adopt this value because the best-fit  was outside the range 4800–6500 K over which the code had been calibrated. Since then, the  <cit.> tool was developed to derive stellar properties for a wider range of effective temperatures (3000–7000 K) by interpolating a grid of library spectra obtained with Keck/HIRES. We ran the same HIRES spectrum obtained by M17 through  to obtain new estimates of , Fe/H, and R_⋆. The resulting = 6330 ± 110 K is cooler than the 6670± 120 K value of M17 and is within the  regime. We therefore ran  on the same HIRES spectrum and found = 6530 ± 100 K and = 10.4 ± 1.0 . Thus, the true value of  is likely in the 9–12  range. All this together informs our adoption of an informed prior on  of 10.4 ± 1  for our spectroscopic transit analysis in Section <ref>.

















 §.§ Stellar rotation period


A significant peak at 0.707 days was observed in the Lomb-Scargle periodogram of the KELT photometry, which M17 interpreted as the rotation period of KELT-18. Given the measured stellar radius, this implied an equatorial rotational velocity of = 134 . While large, it is not atypical for stars of KELT-18's  and log g to have rotation speeds on the order of 100 . Combining this with their measured  of 12.3 ,  noted that the star must have an inclination of ∼5^∘. In other words, we are observing KELT-18 nearly pole-on. Since the planet's orbit is viewed at high inclination, the implication was that the planet's orbit is nearly polar.



Since the rotation signal in the KELT photometry appeared small compared to the measurement noise, we downloaded the available TESS photometry for KELT-18 to search for variability. KELT-18 was observed by TESS as TIC 293687315 (TOI-1300) in sectors 15, 16, 22, 23, 48, 50, 75, and 77. We downloaded the 1800 s cadence data for sectors 48 and 50, 120 s data for sector 75 and 77, and the 600 s data for the other sectors. We selected data processed by the TESS Science Processing Operations Center pipeline <cit.>, removed flagged values, stitched the six sectors together, and binned to a common 1 hour cadence using  <cit.>. The resulting light curve is shown in Figure <ref> along with its Lomb-Scargle periodogram.



The TESS periodogram does not contain any significant peaks below ∼3 days. There is a clustering of peaks around ∼5 days with a maximum power at 4.76 d, and variability on this timescale is visible by eye in the TESS photometry. Stars with KELT-18's  tend to have rotation rates < 8 d at >3σ <cit.>, so a ∼5 day rotation period would be reasonable. However, further inspection at the per-sector level (see Appendix <ref>) reveals that this ∼5 day periodicity is only dominant in sectors 15, 23, and 48. Other sectors show muted variability or strong peaks at other periods. Interactions between multiple spot groups, and/or the presence of differential rotation, can yield multiple periodogram peaks which may further wander in time <cit.>. Altogether, a precise and robust value for the stellar rotation period is not well-determined by the TESS photometry, though it does bring the previous value of 0.707 days into question.  

A rotation period of 4.76 days would imply an equatorial rotation speed of ∼ 20 . Given our spectroscopic measurement of  is only 10.4± 1 , it remains likely that KELT-18 is viewed at low inclination. Conservatively, we adopt a uniform < 8 d prior on the rotation period based on the 3σ empirical boundary defined by  and the TESS photometry showing variability on the order of several days. Using the methodology of <cit.>, this loose constraint translates into a stellar inclination of = 31^+35_-15^∘.




 §.§ KELT-18 B: Neighbor or Companion?


M17 noted a stellar neighbor at 3”.43 separation from KELT-18 B. The neighbor is fainter at K = 12.9 ± 0.2. Under the assumption that the neighbor is at the same distance as KELT-18, M17 obtained _B ∼ 3900 K. The available astrometry were not precise enough to identify the neighbor as comoving based on its proper motion, though the relatively small sky density of stars in the field around KELT-18 (at high galactic latitude) makes a chance alignment at such a small angular separation unlikely (>3σ).

KELT-18 and KELT-18 B appear in the catalog of stellar companions to TESS Objects of Interest of <cit.> (B22). B22 applied the methodology of <cit.> to the Gaia DR3 astrometry <cit.> to determine the likelihood the two stars are comoving. The method propagates the astrometric uncertainties from Gaia into a likelihood ratio comparing the comoving hypothesis (ℒ_1) to the null (not comoving) hypothesis (ℒ_2). B22 added a jitter term to account for any unknown systematic effects to improve the reliability of this hypothesis testing. B22 computed ln(ℒ_1/ℒ_2) =  4.83 for KELT-18 and KELT-18 B, giving strong evidence for the comoving hypothesis. They computed a stellar mass for KELT-18 B of 0.575_-0.026^+0.025 using  <cit.> and the Gaia magnitudes, in agreement with the _B estimate from M17. Given the consistent parallaxes (3.16±0.01 mas for the primary and 3.23±0.04 mas for the secondary), B22 computed a binary separation from the Gaia astrometry of 1082 AU. The relative velocity vector in the sky plane is thus 0.695 ± 0.075 . For reference, if both stars were orbiting in the sky-plane on circular orbits, their relative velocity would be 1.8 . So, unless the two stars have a significant relative radial velocity (which Gaia did not measure), they are likely bound.




§ OBSERVATIONS




We observed a transit of KELT-18 b on UT May 22, 2023 with the Keck Planet Finder <cit.>. KPF is a newly commissioned, optical (445–870 nm), high-resolution (R ∼ 98,000), fiber-fed, ultra-stabilized radial velocity system on the Keck I telescope at W. M. Keck Observatory. Our observations began 25 min before transit ingress and continued until 2 hours after transit egress, only being interrupted by hourly calibration exposures (described below) and a ∼20 min window near transit egress during which issues with the tip/tilt system prevented precise fiber positioning on the stellar PSF.

We chose a fixed 600 sec exposure time to balance averaging over p-mode oscillations (14 min from the scaling relations of ) with temporal resolution of the transit, while reaching a spectral signal-to-noise ratio (S/N) of at least 100 (typical values were 130 ± 11 in the green channel and 140 ± 13 in the red). The KPF “SKY” fiber collected background sky contamination from a position offset several arcsec from KELT-18. We simultaneously acquired broadband Fabry-Pérot etalon spectra in the “CAL” fiber to track instrumental drift, and periodically (once per ∼hour) took a single internal frame with etalon light in the “SKY”, “CAL”, and science fibers as an additional sanity check on drift. We observed a stable linear drift as traced by the simultaneous etalon spectra of -0.92±0.01  per hour in the green channel and -0.36±0.02  per hour in the red channel. This was well-matched by the hourly all-etalon RVs across each fiber. As a result, we drift-corrected our stellar spectra by Doppler-shifting the derived CCFs by estimated drift using our linear fit to the simultaneous etalon RVs.


We independently extracted 1D stellar spectra from each of the three science “slices” using the public KPF data reduction pipeline (DRP)[https://github.com/Keck-DataReductionPipelines/KPF-Pipeline/https://github.com/Keck-DataReductionPipelines/KPF-Pipeline/]. Wavelength calibration was performed for each spectral order using a state-of-the-art laser frequency comb (for ≳ 490 nm) and a ThAr lamp (for ≲ 490 nm) using calibration frames taken during the standard KPF calibration sequences performed that day. We used the F9 ESPRESSO mask <cit.> to derive cross-correlation functions <cit.> for each spectral order. This dataset was obtained before a significant charge transfer inefficiency (CTI) in one of the green CCD amplifiers was diagnosed using solar data <cit.>. Because of this, the green CCD readout utilized the original four-amplifier scheme and was thus affected by significant CTI. We masked the flux corresponding to the quadrant of the raw 2D image read by the affected amplifier when deriving CCFs. This affects half of the bluest 20 orders (roughly 445–530 nm). Fortunately, about 78% of the affected wavelengths also appear in the “good” amplifier of the subsequent order, so much of the spectral information is still contained in the final 1D spectrum. The CCFs from each slice were combined in a weighted sum, taking the weights to be proportional to the total flux in each slice from a representative high S/N spectrum. We repeated the same process across all orders, and then again across the green and red CCDs to obtain the final CCF for each observation. We also calculated the unweighted summed CCF to derive photon-noise uncertainties, which we scaled by the relative total flux in the weighted vs. unweighted CCF to yield appropriate uncertainties in each CCF.

We independently verified the systemic velocity reported by M17. We measured this by fitting the CCF of each of the three science traces, for each out-of-transit spectrum. The result was -11.7 ± 0.1 , in agreement with -11.6 ± 0.1  from M17. We found a slightly smaller value of -11.3 ± 0.1  by comparing the HIRES spectra used in Section <ref> to a telluric model <cit.>. Gaia <cit.> reports a smaller but less certain -10.83 ± 0.46  for KELT-18. We adopt the -11.7 ± 0.1  value from our KPF spectra for our analysis.





§ OBLIQUITY OF KELT-18 B




 §.§ Reloaded Rossiter-McLaughlin Modeling

To measure the obliquity of KELT-18 b, we applied the Reloaded Rossiter-McLaughlin technique <cit.> to our KPF spectra. We breifly summarize the process here.

First, we transformed into the stellar rest frame by Doppler shifting the CCFs by our measured systemic velocity and by the expected Keplerian RV induced by KELT-18 b's orbital motion (using the planet's mass M_p from M17). The aligned CCFs were then normalized to a continuum value of 1. We then created a stellar template, CCF_out, by averaging the out-of-transit normalized CCFs. CCF_out describes the unperturbed average stellar line profile of KELT-18. To isolate the shadow of KELT-18 b, we subtracted each in-transit observation (CCF_in) from the template to obtain the local line profile within the planet's shadow, CCF_loc =CCF_out -CCF_in. Since we normalized the CCFs, we multiplied each CCF_loc by the calculated flux at that time according to a white-light synthetic transit light curve model, integrated over the exposure time of each observation. This placed each CCF_loc on the appropriate flux scale.

The resulting CCF_loc time series is shown in Figure <ref>. Each CCF_loc is fit with a Gaussian profile using  from  <cit.>, where the continuum, amplitude (i.e., depth), width, and centroid are free parameters. The centroid corresponds to the flux-weighted integrated stellar velocity profile within the shadow of the transiting planet; i.e., the local RV. The local RV is modelled by Eq. 9 in <cit.>,

    RV_loc = ∫ I(x, y)  v_stel(x, y) dA/∫ I(x, y) dA,

where I is the limb-darkened intensity, the integral is over the patch of star within the planet shadow, and v_stel is the stellar velocity field

    v_stel = x_⊥ (1 - α y_⊥'^2),

where α is the relative differential rotation rate (the difference in rotation rates at the poles compared to the equator, divided by the equatorial rotation rate). For the Sun, α = 0.27. We used the same coordinate definitions for (x_⊥, y_⊥') as in <cit.>, but we adapted the implementation of the integral for improved resolution. Instead of defining a Cartesian grid of points (x_k, y_k) centered on the planet's shadow spanning -R_p/R_∗ to +R_p/R_∗ and only keeping the points in the grid which satisfied x^2 + y^2 < (R_p/R_∗)^2, we generated a “grid” of N points (x_k, y_k) according to the sunflower pattern,

    r_k = √(k - 1/2/N-1/2),  θ_k = k(2πϕ), 
    
        x_k = r_kcosθ_k,  y_k = r_ksinθ_k,

where ϕ = (1+√(5))/2 is the golden ratio. The result is a set of points (x_k, y_k) uniformly spaced over a circle of radius unity. The points can then be scaled to R_p/R_∗ and centered at the position of the planet to quickly obtain a densely packed grid of points for which each point represents the same projected area dA of the stellar photosphere. The improved resolution of this grid at the shadow and disk limbs helped to reduce artifacts during ingress/egress, and greatly boosted performance when simulating full line profiles for stars with surface inhomogeneities 

<cit.>
.



To fit the local RVs, we modified the  package <cit.> to accept a new function that computes Eq. <ref> for each observation. The  framework automatically enabled us to perform maximum a-posteriori (MAP) fitting, MCMC sampling using  <cit.>, and model comparison with the BIC and AIC. We tested several different models: solid body (SB) rotation vs. differential rotation (DR), and with/without center-to-limb variations (CLVs); see <cit.> for more details. The former is a matter of fixing α to zero (SB) or letting it float (DR), while the latter requires adding an additional term to Eq. <ref> of the form

    v_conv = ∑_i=0^n c_i ^i.

This polynomial in the intensity-weighted center-to-limb position  is a good model for the velocity field introduced by granulation, which is azimuthally symmetric around the disk and varies with center-to-limb position as the line-of-sight intersects the tops of granules at disk-center and the sides of granules at the limb <cit.>. Since CCF_loc has the out-of-transit template subtracted, the net convective blueshift integrated across the full stellar disk has also been removed from the data. Consequently, c_0 must be constrained according to Eq. 13 in <cit.>. The additional model parameters are thus c_1 for a linear (n=1) CLV and (c_1, c_2) for a quadratic (n=2) CLV.




 §.§ MCMC Sampling


We tested a suite of models within the RRM framework corresponding to each combination of SB or DR, and no CLV, linear CLV (CLV_lin), and quadratic CLV (CLV_quad). The free parameters in all models were the sky-projected obliquity λ, the projected rotational velocity , the sine of the stellar inclination , and the impact parameter b. Models with DR include the degree of differential rotation α, and models with CLVs include either c_1 (linear) or c_1 and c_2 (quadratic). We allowed for anti-solar differential rotation by permitting α to be negative, with a uniform prior over (-1, 1).

To improve the sampling efficiency, we make the change of coordinates for our fitting basis into a polar coordinate system with λ as the azimuthal angle and √() as the radial dimension. The parameters for the fit are thus √()cosλ, √()sinλ, sin, b, α, and the CLV coefficients.

Because of the suspected polar orientation of the transit chord, we placed an informed Gaussian prior on  of 10.4 ± 1  based on our analysis of the spectroscopic  (Section <ref>). We also found that this prior, in conjunction with a prior on  based on previous transit fits (Table <ref>), was necessary to discourage the sampler from wandering to solutions of extremely low  (∼1 ) with (unrealistic) grazing transits. The final distributions for λ were unaffected by the exact boundaries chosen for these priors. We note that <cit.> found values of  (82.90^+0.62_-0.54^∘) and a/R_∗ (4.36^+0.11_-0.09) that were significantly discrepant with those measured by M17. We tried setting a prior on  to this value and found the MCMC to both not converge and produce a bimodal  posterior around 2 , which is highly inconsistent with the observed width of lines in the KPF spectra and the HIRES, TRES, and APF spectra of M17.

We first found the MAP solution for each model using  <cit.>. This best-fit solution was used as the initial location (plus a small Gaussian perturbation) for a MCMC exploration of the posterior. We ran  <cit.> as implemented in  <cit.> with 8 ensembles of 32 walkers each for a maximum of 10,000 steps, or until the Gelman–Rubin statistic <cit.> was < 1.001 across the ensembles <cit.>. In all cases, the G-R condition for convergence was satisfied and the sampler was terminated with a typical total number of posterior samples around 100,000. A second and final MAP fit was then performed using the median parameter values from the MCMC samples. The MAP fit for each model is plotted over the RV_loc time series in Figure <ref>, and Table <ref> lists our derived best-estimates for each parameter.



 §.§ Center-to-Limb Variations




Fig. <ref> plots the depth, equivalent width (EW), and full-width-at-half-maximum (FWHM) of the CCF_loc as a function of . A strong trend in the FWHM (Pearson correlation coefficient R=-0.87, p-value of 1.5×10^-7) and the EW (R=-0.94, p-value 1.6×10^-10) can be seen from center to limb, with the local line profiles narrowest at disk center and widening towards disk limb. The RV_loc time series (Figure <ref>) likewise has significant curvature with a maximum (minimum blueshift) at disk center (mid-transit) and minimum (maximum blueshift) at disk limb (ingress/egress). The CCF_loc contrast is only marginally correlated with CLV position during the first half of the transit (R=0.66, p-value 0.04) and is uncorrelated during the second half of the transit (R=0.03, p-value 0.94).

Only CLVs can simultaneously explain the local RV curvature and the change in line width. While differential rotation alone can reproduce the observed curvature in the local RV time series, it cannot explain the large (factor of two) increase in FWHM from disk center to limb. <cit.> showed using spectral line synthesis in 3D radiative magnetohydrodynamical (MHD) simulations that surface-layer granulation causes the width of spectral lines to increase towards disk limb, an effect that is relatively muted for GKM stars but is significant for F-type stars. The F3 V star in their simulation showed increases to the FWHM of iron lines by a factor of 1.5–2 from disk center to μ = 0.2. This effect is caused by the horizontal flows, which dominate the line-of-sight velocity at disk limb, having roughly three times as high a velocity dispersion compared to the vertical velocity <cit.>. They also found line cores to have a net convective blueshift about 500  less (i.e., larger RV) at disk limb than at disk center. We see the opposite in the modelled CLV vs. μ (Fig. <ref>) for the SB scenario, whereas in the DR scenario we find negligible convective velocities. 









llllllll

Best-fit Parameters and Model Comparison 


Parameter      Prior     
SB     SB+CLV_lin     SB+CLV_quad      DR     DR+CLV_lin     DR+CLV_quad

Model Parameters 

λ (^∘)     —     -93.4_-0.6^+0.6     -94.6_-0.7^+0.6     -94.8_-0.7^+0.7     -95.2_-1.2^+1.2     -95.0_-1.2^+1.0     -94.9_-1.3^+1.0 

 ()     Gaussian(10.4, 1)     10.2_-1.0^+1.0     10.2_-1.0^+1.0     10.2_-1.0^+1.0     10.3_-1.0^+1.0     10.2_-1.0^+1.0     10.2_-1.0^+1.0 

     Uniform(0, 1)     0.03_-0.63^+0.6     0.06_-0.64^+0.59     0.0_-0.61^+0.6     0.2_-0.05^+0.06     0.29_-0.12^+0.15     0.22_-0.81^+0.38 

b     Uniform(-1, 1)     0.035_-0.005^+0.006     0.069_-0.008^+0.01     0.07_-0.009^+0.009     0.127_-0.016^+0.018     0.104_-0.021^+0.023     0.075_-0.015^+0.023 

α     Uniform(-1, 1)      —     —     —    0.98_-0.03^+0.01     0.8_-0.27^+0.14     0.15_-0.45^+0.42 

c_0 ()     Uniform(-10, 10)      —    2.1_-0.3^+0.2     -3.7_-1.6^+1.7      —    0.9_-0.4^+0.6     -3.7_-1.8^+1.8 

c_1 ()     Uniform(-10, 10)      —     —    4.0_-1.1^+1.1      —     —    3.9_-1.4^+1.4 

ΔBIC     —     74.0      9.2      0.0      1.5      3.4      3.0  

ΔAIC     —     73.0      8.5      0.0      0.7      3.4      4.3  

Derived 

ψ (^∘)     —     89.7_-2.0^+2.1      89.5_-2.7^+2.8      89.2_-2.8^+3.0      89.6_-0.3^+0.4      90.2_-0.6^+0.9      90.1_-3.5^+2.1 

Values and their uncertainties represent 68% credible intervals as defined by the 16th, 50th, and 84th percentiles of the sampled posterior. Models with SB rotation do not constrain , whereas models with DR prefer large α and small  (Fig. <ref>). Posteriors in ψ are derived self-consistently using only the posterior samples in . That is, they do not incorporate any information about the rotation period. The column with the preferred model is highlighted in bold.

^∗Uniform in cos.






 §.§ Model Comparison
 

We computed the Bayesian information criterion (BIC) and Akaike information criterion (AIC) for each model. Their relative values to the minimum are listed in Table <ref>. The only model that is confidently ruled out is SB rotation with no CLV, at ΔBIC=74 and ΔAIC=73. The slightly preferred model is SB rotation with CLVs quadratic in . The models with DR all have ΔBIC and ΔAIC <5, and thus are statistically similar descriptions of the model.

The model with DR alone requires an extremely high values of α, in the 0.9–1 range. While it is not impossible for the stellar poles to rotate at just 10% the rate of the equator, it seems extremely unlikely for a star to have a rotational shear of this magnitude. Slowly rotating (≲ 50 ) F-type stars do commonly show signs of differential rotation (α≥ 0.1), whereas rapid rotators do not <cit.>. Our analysis of the TESS photometry in Section <ref> more likely places KELT-18 in the former category of slow rotators. Interestingly, the DR model with linear CLVs finds a smaller α=0.8_-0.27^+0.14, and the flexibility of the quadratic CLV model results in an unconstrained α, as the two effects are degenerate. In contrast, the SB models rely on strong CLVs to generate the measured curvature (Fig. <ref>). It is therefore ambiguous from the RV_loc time series alone whether the curvature is coming from a significant differential rotation, CLVs, or a mixture of the two. The difference between these two cases is greatest at low ⟨μ⟩, i.e. near the disk limb. This is where CLV effects are strongest whereas DR is only affected by the subplanet stellar latitude. However, the data near disk limb (i.e. near ingress/egress) are the lowest S/N observations, weakening their utility as a discriminatory lever-arm.

If the star is differentially rotating, the varying line-of-sight rotational velocities as a function of stellar latitude break the  degeneracy, allowing an independent constraint on the stellar inclination. In the DR-only model, the resulting stellar inclination is = 11.8_-3.2^+3.3. This value is in agreement with the 5–30^∘ range expected from the estimated rotation period and known R_∗ (Section <ref>). Consequently, the true 3D obliquity can be determined via the equation

    cosψ = coscos + sincosλ.

The corresponding values of ψ for each of fitted models are listed in Table <ref>. For the SB models, the posterior in  is unconstrained (uniform in 0 to 1) so this represents the maximally uncertain value of ψ. For instance, if we adopt λ = from the best-fitting SB+CLV_quad model, and take  to be isotropic (uniform in cos), then ψ = 93.8_-1.8^+1.6^∘. If instead we take  to be isotropic within 5–30^∘ as suggested from photometric analyses, we get ψ =. Our estimated obliquity measurements are consistent within 1σ across all models, thus we adopt the SB+CLV_quad model as our preferred fit as it is both the best-fitting model by the ΔBIC and ΔAIC tests, is more physically justified than the α∼ 0.9 models, and utilizes photometric constraints on the stellar inclination rather than the values from the RRM posteriors.





§ ORBITAL DYNAMICS


KELT-18 b joins the population of close-in planets orbiting hot stars in polar orbits <cit.>. There are no other known transiting planets in the system <cit.>, though there is a stellar companion, KELT-18 B (Section <ref>). It is likely that KELT-18 B influenced the orbital evolution of KELT-18 b. In this section we examine possible mechanisms.




Hierarchical triple systems such as KELT-18, KELT-18 b, and KELT-18 B will experience von-Zeipel-Kozai-Lidov (ZKL) oscillations if the mutual inclination between the two orbiting bodies is larger than 39.2^∘ <cit.>. ZKL oscillations are usually suppressed for planets on HJ-like orbits because of the general relativistic precession of the argument of periapsis. However ZKL oscillations may have taken place early in the system's history if KELT-18 b formed at a further orbital distance. <cit.> showed that such a scenario can lead to HEM, producing the HJ we see today in a close-in, highly misaligned orbit.

The Gaia astrometry enables a constraint on the inclination of KELT-18 B's orbit. The angle between the vector connecting the astrometric positions of the two stars (r⃗) and the difference in velocity vectors (v⃗), called γ <cit.>, encodes information about the companion's orbital inclination (though is degenerate with eccentricity). If γ∼ 0^∘ or 180^∘, then the companion's orbit is viewed edge-on. Otherwise, the companion's orbit may be viewed face-on or at some intermediate inclination. Since KELT-18 b transits, we know it has ∼ 90^∘, so γ can test for mutual (mis)alignment. We calculated γ = 95.2 ± 6.4^∘, which is most consistent with a low orbital inclination for a circular KELT-18 B and thus a large mutual inclination (Fig. <ref>).



For ZKL to excite KELT-18 b's orbit, it must have formed far away from its host star. By equating the timescales for GR precession and ZKL oscillations, one can solve for the minimum orbital separation at which ZKL oscillations are not quenched by GR <cit.>. We computed this value for the KELT-18 system and obtained ≳ 6.1±0.26 AU. In other words, if KELT-18 b was born beyond 6.16±0.26 AU, for instance via traditional core accretion beyond the ice-line, then it could have plausibly migrated to its current orbit via ZKL-induced HEM. B22 calculated typical minimum formation distances of 0.5–10 AU across the broader population of HJs for binary star induced ZKL HEM migration, which conspicuously aligns with the peak in cold Jupiter occurrence around 1–10 AU <cit.>. Future studies of KELT-18 b's atmosphere via transmission spectroscopy (Householder, A., Dai, F., et al. in prep) will seek additional evidence for KELT-18 b's birth conditions by measuring its inventory of refractory and volatile elements, the fingerprints of the original planetary building blocks <cit.>.

It may also be the case that KELT-18 b formed in a protoplanetary that was primordially misaligned, as can be the case when an outer stellar companion is involved <cit.>. Alternatively, the stellar companion may have torqued the outer regions of the protoplanetary disk into a misalignment, producing a broken protoplanetary disk which itself can play the role of an outer perturber in exciting large stellar obliquities <cit.>. <cit.> showed that such an initial configuration, in which the star has a disk-induced nonzero obliquity relative to the proto-HJ before ZKL oscillations induced by the stellar companion initiate HEM, the final obliquity distribution of the HJ is broadly retrograde with a peak near polar orbits. This is in contrast with the classical picture of ZKL starting with initially aligned planetary orbits, which produce HJs with a bimodal obliquity distribution near 40^∘ and 140^∘ <cit.>. Thus it may be that the orbit of KELT-18 b was already misaligned with the star's rotation before undergoing HEM into its present day HJ orbit, the result of which is an orbit with ψ∼ 90^∘ rather than 40^∘ or 140^∘.




§ CONCLUSION



We have presented the first science results from KPF on the Keck-I telescope: a transit of the inflated ultra-hot Jupiter KELT-18 b. We found the orbit to be nearly perpendicular to the stellar equatorial plane: ψ =. This result is robust to model choice and is largely constrained by the tight posterior on λ = and the relatively low value of . Taken in context with the binary stellar companion, which we find to be on a likely bound orbit that could be orthogonal to KELT-18 b's orbit, a history of ZKL-induced HEM is plausible if KELT-18 b formed beyond about 6 AU from its host star. Our main observational takeaways are as follows:

    
  * We searched the available TESS photometry for clues as to the rotation period of the host star and found evidence for modulation around ∼5 days. We did not see variability at 0.707 days as previously reported by M17 using KELT photometry. Both values are consistent with the tendency for F-type stars to have < 8 day rotation periods, and when combined with the measured  imply a near pole-on viewing geometry (≲ 30^∘).

    
  * The stellar neighbor KELT-18 B is highly likely to be a bound companion, based on Gaia DR3 astrometry. Its orbit is also likely orthogonal to KELT-18 b's orbit, based on the angle between the on-sky position vector and proper motion vectors.

    
  * We observed evidence of CLVs, as traced by the FWHM of the local line profile beneath the planet's shadow. The FWHM increased towards the disk limb by nearly a factor of two, in agreement with previous 3D MHD simulations of velocity flows in the near-surface layers of F-type stellar atmospheres. 
    
    
  * We modelled the centroid of the local line profile using the RRM technique and found that either strong differential rotation (α=0.9) or CLVs are needed to explain the curvature in the local RV time series. However, all of the models produce consistent, well-constrained posteriors for the sky-projected obliquity. Ambiguity between DR and CLVs is a common challenge of the RRM technique <cit.> and is complicated by the uncertainty in the stellar inclination (and thus the stellar latitudes occulted). A firm detection of a stellar rotation period from additional photometry would enable a better constraint of the degree of DR needed to explain the data. As it stands, the polar transiting geometry requires a low (near pole-on) stellar inclination in order to generate the observed curvature in the local RV timeseries, with maximum blueshift occulted at ingress/egress and near-zero velocity occulted at mid-transit. An edge on stellar inclination would produce the opposite effect, since the lowest velocity latitudes at the poles would instead be occulted at ingress/egress, while the maximum velocity latitude (the equator) would be occulted at mid-transit. If KELT-18 does have an edge-on stellar inclination, then DR would be inconsistent with the data and CLVs would be strongly favored.

    
  * The 3D orbital geometry of the KELT-18 system is explainable by a history of ZKL-induced migration, providing support for the HEM formation pathway for HJs. Future work will further test this by inventorying the elemental abundances in KELT-18's atmosphere, connecting the planet to the disk in which it formed.







§ ACKNOWLEDGEMENTS


We are grateful to Heather Cegla and Michael Palumbo for illuminating discussions on center-to-limb variations. Some of the data presented herein were obtained at Keck Observatory, which is a private 501(c)3 non-profit organization 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. Keck Observatory occupies the summit of Maunakea, a place of significant ecological, cultural, and spiritual importance within the indigenous Hawaiian community. We understand and embrace our accountability to Maunakea and the indigenous Hawaiian community, and commit to our role in long-term mutual stewardship. We are most fortunate to have the opportunity to conduct observations from Maunakea. 

R.A.R. acknowledges support from the National Science Foundation through the Graduate Research Fellowship Program (DGE 1745301). A.W.H. acknowledges funding support from NASA award 80NSSC24K0161 and the JPL President's and Director's Research and Develop Fund. 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). All the TESS data used in this paper can be found in MAST: [10.17909/t9-nmc8-f686]http://dx.doi.org/t9-nmc8-f686 <cit.>. 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.


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§ SECTOR-BY-SECTOR ROTATION PERIOD


Here we present, in Figure <ref>, the sector-by-sector periodogram of the TESS photometry.






