Paper Title: Atmospheric characterization of the super-Jupiter HIP 99770 b with KPIC.






0000-0003-0097-4414]Yapeng Zhang
51 Pegasi b fellow



0000-0002-6618-1137]Jerry W. Xuan





0000-0003-0774-6502]Jason J. Wang
Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA

0000-0002-5370-7494]Chih-Chun Hsu
Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA

0000-0003-2233-4821]Jean-Bapiste Ruffio


0000-0002-5375-4725]Heather A. Knutson


0000-0001-9164-7966]Julie Inglis





0000-0003-1728-8269]Yayaati Chachan
CITA National Fellow
Department of Physics and Trottier Space Institute, McGill University, 3600 rue University, H3A 2T8 Montreal QC, Canada
Trottier Institute for Research on Exoplanets (iREx), Université de Montréal, Canada

0000-0001-9708-8667]Katelyn Horstman









0000-0003-4737-5486]Benjamin Calvin







0000-0001-8953-1008]Jacques-Robert Delorme






0000-0002-1583-2040]Daniel Echeverri


0000-0002-1392-0768]Luke Finnerty



0000-0002-0176-8973]Michael P. Fitzgerald



0000-0001-5213-6207]Nemanja Jovanovic




0000-0002-4934-3042]Joshua Liberman




0000-0002-2019-4995]Ronald A. López



















0000-0003-1399-3593]Ben Sappey











0000-0001-5299-6899]J. Kent Wallace


0000-0002-4361-8885]Ji Wang


0000-0001-5173-2947]Clarissa R. Do Ó
Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA


§ ABSTRACT 

Young, self-luminous super-Jovian companions discovered by direct imaging provide a challenging test of planet formation and evolution theories. By spectroscopically characterizing the atmospheric compositions of these super-Jupiters, we can constrain their formation histories.
Here we present studies of the recently discovered , a 16  high-contrast companion on a 17 au orbit, 
using the fiber-fed high-resolution spectrograph KPIC (ℛ∼ 35,000) on the Keck II telescope. 
Our K-band observations led to detections of  and CO in the atmosphere of .
We carried out free retrieval analyses using  to measure its chemical abundances, including the metallicity
and C/O ratio,  projected rotation velocity (vsin i), and radial velocity (RV). 
We found that the companion's atmosphere has C/O =0.55_-0.04^+0.06 and [M/H] =0.26_-0.23^+0.24 (1σ confidence intervals), values 
consistent with those of the Sun and with a companion formation via gravitational instability or core accretion. 



The projected rotation velocity
vsin(i) < 7.8  is small relative to other directly imaged companions with similar masses and ages. 
This may imply a near pole-on orientation or effective magnetic braking by a circumplanetary disk.
In addition, we added the companion-to-primary relative RV measurement to the orbital fitting and 
obtained updated constraints on orbital parameters.
Detailed characterization of super-Jovian companions within 20 au like  is critical for understanding 
the formation histories of this population.








§ INTRODUCTION
 

Young, self-luminous super-Jovian companions discovered by direct imaging surveys have posed 
significant challenges to planet formation theories. 
These objects typically straddle the mass boundary between planets and brown dwarfs (∼13 ) and 
orbit their primary stars at large separations (>10 au). 
Their formation mechanisms remain under debate as they are not easily compatible with either
in-situ core accretion or gravitational instability <cit.>. 
The long timescale of core accretion in the outer protoplanetary disk makes it challenging to assemble  
massive cores to trigger runaway accretion before the gas disk dissipates <cit.>, while
disk instability and cloud fragmentation are expected to form more massive objects 
like brown dwarfs or stellar companions <cit.>.
For some wide-orbit companions at hundreds of au, 
formation at closer orbits followed by scattering events may be effective <cit.>.
Previous studies on orbital architectures of these super-Jupiters (<20 ) versus those of brown dwarfs (>20 )
found evidence for distinct distributions in their semimajor axes and orbital eccentricities <cit.>. 
This suggests that the two populations likely have distinct formation mechanisms, but it is unclear


whether there is a well-defined boundary separating them.


Atmospheric characterization provides an important avenue to distinguish between competing formation mechanisms for substellar companions.
Elemental abundances including the metallicty [M/H] and carbon-to-oxygen (C/O) ratios 
have been suggested as probes for planet formation <cit.>.
In general, formation via gravitational instability or cloud fragmentation is thought to result in compositions akin to stars,
while core accretion can result in a variety of compositions depending on the 
accretion mechanisms, formation location relative to disk icelines, migration histories 
<cit.>, among other factors. 
Although companions formed via gravitational collapse can also be enriched through late accretion, the impact on their compositions is not expected to be significant because of the high envelope masses <cit.>.

Super-Jupiters on wide orbits are excellent targets for directly probing planetary emission at high signal-to-noise, 
therefore allowing for robust constraints on their chemical compositions.



As measurements of atmospheric C/O and metallicity in directly imaged super-Jupiters have accumulated over the past few years,
we begin to build a sample large enough to test the hypotheses.
Spectral surveys of substellar companions with masses of 10-30  such as <cit.> suggest that they display broadly solar compositions, which are chemically compatible with
formation via gravitational instability or core accretion beyond CO icelines, where solids have approximately stellar C/O ratios.
Although the planetesimal accretion scenario is less likely for wide-orbit massive gas giants, pebble accretion, whose efficiency increases in the outer disk, is not precluded <cit.>.

At an individual system level, linking the atmospheric composition of a planet to its formation history 
is complicated by a chain of processes, including dust and ice/vapor chemistry evolution in disks, 
planet migration, late solid enrichment, and mixing of planet's core and envelope <cit.>. These convoluted processes make inferring a planet's formation history from its current atmosphere difficult.
The challenging nature of this task calls for combining different lines of evidence 
from atmospheric tracers such as refractory-to-volatile ratios  <cit.>
and isotopologue ratios  <cit.> as well as dynamical indicators 
such as spin and orbital architecture <cit.>.
High-resolution spectroscopy of super-Jovian companions provides information on these various aspects simultaneously, 
enabling a more comprehensive understanding of the current state of planetary system and its potential connection to  
formation pathways and evolutionary history.


Despite extensive studies of wide-orbit companions <cit.>, there are relatively few published spectral characterization of companions with orbital separation within 20 au, 
which represents a critical region of the parameter space for understanding 
population-level trends.
Most super-Jovian companions with compositional measurements are located beyond 50 au, 
with only a handful of exceptions such as β Pic b <cit.>, 
HR 8799 cde <cit.>, 

51 Eri b <cit.>, 
and AF Lep b <cit.>. 
Planets with cooler temperatures and/or smaller separations are more challenging 
to characterize because of the predominant stellar speckle contamination. 
Techniques that combine both the spatial and spectral resolving capability 
have proven to be powerful for enhancing the contrast limit of detection and characterization 
<cit.>.
New instruments coupling adaptive optics systems with high-resolution 
spectrographs, such as the Keck Planet Imager and Characterizer <cit.>, 
HiRISE on VLT <cit.>, and REACH on Subaru <cit.>, have recently enabled high resolution spectroscopy of faint companions on close-in orbits <cit.>. 


The recently discovered 16   is among only a handful of super-Jovian companions 
with an orbital separation within 20 au <cit.>. Its current location is expected to be within the
CO iceline of the protoplanetary disk around A-type host stars <cit.>.
In core-accretion scenarios, the C/O ratio likely varies as a function of companion's metallicity because of the partitioning of solid versus gas compositions.
Therefore, companions at smaller separations are
likely to display a larger variety of atmospheric compositions,
providing intriguing case studies for testing formation models by measuring both the C/O and metallicity.
In contrast, beyond the CO iceline, the companion's C/O ratio is expected to be solar for a wide range of metallicity because most volatiles are condensed out.
We note, however, if the companion has a solar metallicity, it is difficult to distinguish between core accretion and gravitational instability as both mechanisms can plausibly lead to this outcome, regardless of the planet's birth location.



In this paper, we present the atmospheric characterization of  
using KPIC observations.
When combined with its spin velocity and orbital properties, this allows us to
place constraints on its formation and evolution.
The paper is organized as follows. In Section <ref>, we introduce the target system
HIP 99770. The KPIC observations and data reduction are explained in Section <ref>.
Then, we carry out free retrieval analyses with the observations. 
The forward model and retrieval framework are described in Section <ref>.
We present the molecular detections and retrieval results in Section <ref>.
We discuss the measured chemical abundances, spin velocity, 
and their implications for planet formation in Section <ref>.
Finally, we summarize the findings in Section <ref>.




§ HIP 99770 SYSTEM
 

 is a recently discovered super-Jovian companion with a joint direct imaging and precision astrometry 
detection <cit.>. 
We summarize the properties of the  system in Table <ref>.
The 1.8  A-type primary star  has an effective temperature of ∼8000 K, an age of 40-400 Myr, 
and a distance of 40.74 pc 
<cit.>.
The primary star shows evidence of 
Hipparcos-Gaia astrometric acceleration induced by an orbiting companion, 
which was confirmed 
by direct imaging with the Subaru Coronagraphic Extreme Adaptive Optics Project 
(SCExAO/CHARIS) and Keck/NIRC2 at an angular separation of 0.44 <cit.>. 
The combination of the relative astrometry and
acceleration leads to a dynamical mass of 16.1^+5.4_-5.0  for the companion, 
or a companion-to-star mass ratio of ∼ 8×10^-3, at a semimajor axis of 
16.9^+3.4_-1.9 au and an orbital eccentricity of 0.25^+0.14_-0.16  
<cit.>. 
The companion's luminosity and dynamical mass are consistent with an age of 115-200 Myr. 

<cit.> carried out spectral characterization of  
using a CHARIS JHK broadband spectrum at 1.16-2.37 μm and Keck/NIRC2 L band photometry, 
which suggested a spectral type of L7-L9.5 near the L/T transition, 
an effective temperature of ∼ 1300-1600 K, and a surface gravity 
of log g =4-4.5. Evolutionary models <cit.> predict a radius of 1.1-1.2  while the 
spectral analysis tends to imply a smaller radius of up to 1.05  <cit.>.
A comparison to HR 8799 d's spectral shape shows that 's spectrum is 
less flat, indicating moderate cloudiness.


lcc
    
    Properties of the  system
    Property     Value     References
    3l 

    α_2000.0     20:14:32.032     1 

    δ_2000.0     +36:48:22.7     1 

    Distance (pc)     40.74 ± 0.15     1 

    v_sys ()       -20.52 ± 0.40     1 

    vsin i ()     81.1 ± 1.5     1 

    Age (Myr)     40 (Argus membership)      2 

         115-414 (astroseismology)     2 

    Mass (M_⊙)     1.85 ± 0.19     2 

    SpT     A5-A6     3 

    (K)      8000       3 

    K_s (mag)     4.42 ± 0.02     2 

    L_p (mag)     4.40 ± 0.05     2 

    
    [C/H]     0.18 ± 0.09     4 

    [O/H]     0.01 ± 0.09     4 

    
    
    3l 

    SpT     L7-L9.5     2

    (K)      1400^+200_-150      2

    K_s (mag)     15.66 ± 0.09     2

    L_p (mag)     14.52 ± 0.12     2

    a (au)     16.9^+3.4_-1.9     2 

    eccentricity     0.25^+0.14_-0.16     2 

    Mass ()     16.1^+5.4_-5.0     2 

    Radius ()     1.0- 1.1     2

    log g (cgs)     4.0-4.5      2

    
    vsin i ()     3.5_-1.9^+1.7     5

    [M/H]      0.26_-0.23^+0.24       5

    C/O      0.55_-0.04^+0.06      5 

    
    (1) <cit.>, 
    (2) <cit.>, (3) <cit.>, (4) <cit.>,
    (5) this work.




§ OBSERVATIONS AND DATA REDUCTION
 


We observed the 
system with KPIC <cit.> on UT 2023 June 16 and 21. 
KPIC uses a fiber injection unit (FIU) located downstream of the Keck II adaptive optics system 
to inject light into single-mode fibers, which are connected to the NIRSPEC high-resolution (ℛ∼ 35,000) spectrograph
<cit.> to disperse light onto the detector. 
Among the four illuminated fibers, one of them is aligned 
to the position of the companion to obtain its spectra. 
The use of single-mode fibers aids in the suppression of the stellar speckle and 
sky background, and ensure a stable Gaussian-like line spread function 
(LSF) during observations. 
A more detailed description of the instrument can be found in <cit.>.

ccccccc
    KPIC Observations of 
    Date     t_int (min)     Airmass     Throughput     S/N
    
    2023-06-16      90     1.1     3.4%     1.0 

        2023-06-21      60     1.1     3.3%     0.8 

    
    The S/N per wavelength channel for the companion is estimated from the residuals (see Fig. <ref>) that include both the background thermal noise and the photon noise due to stellar speckles.


Our KPIC observations of the  are summarized in Table <ref>.
We used an AB nodding scheme that alternately aligns the companion on one of two fibers to facilitate sky/background subtraction.  Eighteen frames of 300s exposure time  were recorded
on June 16 and 11 exposures of 600s on June 21. The total integration time on  was 90 minutes and 60 minutes, respectively.
We also switched to the primary star once every hour for modeling telluric absorption and instrument response. 
Each night we took short observations on an M giant star, HIP 95771 and HIP 81497, respectively, which display deep and narrow spectral lines, therefore convenient for the calibration of wavelength solution. 

We reduced the data using the public KPIC Data Reduction Pipeline[<https://github.com/kpicteam/kpic_pipeline>], including steps of 
nod-subtraction, bad pixel removal, spectral order tracing, optimal spectrum extraction, 
and wavelength calibration. We refer readers to <cit.> for
an in-depth description. 
The K-band observations span the wavelength range of 1.9-2.5 μm with a spectral 
resolution of ℛ∼35,000. We focused our analysis on the three reddest 
spectral orders from 2.29 to 2.49 μm, covering the main near-infrared absorbing molecules  
such as CO and  in the companion's atmosphere, as shown in Fig. <ref>. 

As a result of the companion's small separation (0.44) and high contrast 
(Δ K_mag=11.3), the extracted spectra are dominated by the 
diffracted starlight, which varies across wavelengths.
Therefore, the continuum of the companion's spectrum cannot be recovered. 
Taking into account the photon noise from the stellar speckles, we reach a signal-to-noise 
(S/N) of ∼1 per pixel for the companion's signal in 1.5-hour integration on the first night and S/N ∼0.8 on the second night.







§ RETRIEVAL ANALYSIS
 




 §.§ Spectral Model of 
 

The spectral model of the companion consists of three components, including the temperature profile, the chemistry model, and the cloud model. The models are set up as follows. 



  §.§.§ Temperature Model


We parametrize the temperature-pressure (T-P) profile using the gradients 
Δln T/ Δln P across 7 pressure knots 
set at 10^-5, 10^-3.5, 10^-2, 10^-1, 10^0, 10^1, 
and 10^2 bar,
and one absolute temperature value, T_0, at 100 bar.
This follows the approach in <cit.>.


We set broad prior ranges 
with a uniform distribution to ensure the flexibility of the temperature model (see Table <ref>). 
The full T-P profile is determined by a spline interpolation 
onto 60 layers evenly spaced in log pressure between 10^-5 and 100 bar.



  §.§.§ Chemistry Model


We explore two different chemistry models: one a free retrieval and the other a disequilibrium model.
The disequilibrium chemistry model first assumes chemical equilibrium to determine 
the chemical abundances at a given pressure, temperature, carbon-to-oxygen ratio (C/O) 
and metallicity ([M/H]), by interpolation of a precomputed table <cit.>. 
The model then parameterizes the disequilibrium chemistry of major species (CO, , and ) by enforcing constant 
volume mixing ratios (VMRs) at altitudes above a certain quenching pressure level (P_quench), 
which approximates the effect of rapid vertical mixing in disequilibrium chemistry <cit.>.
The free chemistry model allows the abundance of each chemical species to vary while assuming a vertically constant profile.

In the case of chemical quenching at a deep atmospheric region below the photosphere, and modest vertical sampling of the atmosphere, 
this constant VMR profile can be a reasonable approximation, which is therefore expected to lead to similar retrieval results as the disequilibrium chemistry model.




  §.§.§ Cloud Model


We adopt the condensate cloud model from <cit.> 
with  as the cloud opacity source – the expected dominant cloud 
species in L dwarfs <cit.>. 
The cloud is characterized by four parameters: the mass fraction 
of the cloud species at the cloud base X_0^MgSiO_3, the settling parameter f_sed 
(controlling the thickness of the cloud above the cloud base), 
the vertical eddy diffusion coefficient K_zz (effectively determining the particle size), 
and the width of the log-normal particle size distribution σ_g.
Following <cit.>, the location of the cloud base P_base is 
determined by intersecting the condensation curve of the cloud species with the T-P 
profile of the atmosphere.




 §.§ Fringing removal


Inspecting the observational residuals, i.e., the observed spectra minus the best-fit model as obtained through the 
retrieval analysis, we identified sinusoidal fringing features in the data. 
These fringing signals are believed to result from the entrance window of NIRSPEC and KPIC dichroics.
Although the fringing due to the NIRSPEC entrance window is static, 
the KPIC fringing signal shows temporal variation depending on the angle of incidence into the optics and 
the change of optical properties of the material with ambient temperature <cit.>. 
We found significant fringing in the periodogram of the residual data 
with a period of ∼4.5 Å originating from the KPIC dichroics, especially in the second epoch. 
To model the fringing, we used the function as described in
<cit.>:

    t = [1+ Fsin^2( 2π n d/λ)]^-1,

where n is the index of refraction of the material as a function of wavelength λ. F and d are two free parameters determining the amplitude and period of
the fringing signal.
We fit this function to the observational residuals using least-squares optimization to obtain the 
optimal fringing parameters in each spectral order and fiber.
Then, we applied this optimized fringing formula to the full spectral model during retrieval analysis. 
We note that the fringing signals do not bias the measurement of chemical composition as we obtained consistent results either with or without the treatment of fringing.




 §.§ Retrieval Framework



Given the model setup detailed in Section <ref>, we compute synthetic spectra 
of the companion using the radiative transfer code  <cit.>. 
The model accounts for the Rayleigh scattering of H_2 and He, the collision-induced absorption of H_2-H_2 and H_2-He, and
the scattering and absorption cross sections of crystalline, irregularly shaped  cloud particles. 
We use the line-by-line mode of  to calculate the emission spectra at high spectral resolution. 
To speed up the calculation, we downsample the original opacity tables (with λ/Δλ∼10^6) by a factor of 3. This downsampling factor has been tested to ensure that it does not bias the retrieval results <cit.>.
We include opacity from  <cit.>, ,  <cit.>,  <cit.>, 
 <cit.>, and  <cit.> 
in our model.

Subsequently, the synthetic high-resolution spectrum is radial-velocity (RV) shifted 
by the systemic and barycentric velocity, and rotationally broadened by vsin i using the method from
<cit.>.
The model spectrum at native resolution is then convolved with a Gaussian kernel and 
binned to the wavelength grid of the observed spectrum
to match the resolving power of the instrument (λ/Δλ∼35,000). 
The width of the LSF is measured from the point spread function of the stellar observations
in the cross-dispersion direction.
We apply a scaling factor ℛ_scaling to the width to account for 
potential inaccuracy in the measured values.

Then, the model spectrum is multiplied by the telluric transmission and instrument response, 
which is obtained by dividing the spectrum of the featureless A-type primary star by a 
PHOENIX stellar model <cit.>.
As the spectrum is contaminated by low-order stellar speckle noise, we discard the continuum information by 
high-pass filtering of both the observation and model using a 150-pixel (∼3 nm) wide Gaussian filter. We note that the retrieval results are robust to the choice of the kernel width.


To construct the full forward model and compare it directly to observations, we took the linear combination of the companion's component 𝐦_p with 
the observed stellar spectrum 𝐦_∗ following <cit.>. 
The full model is calculated as follows:

    𝐌 = 𝐦 𝐜 =
        [ 𝐦_∗ 𝐦_p ][ 𝐜_∗; 𝐜_p ],

where the coefficients 𝐜 of the two linear components 
can be analytically marginalized by solving the linear equation

    𝐝 = 𝐦 𝐜 + η,

with 𝐝 being the observed spectrum and η the random noise.
The least square solution for the linear equation can be found by solving

    𝐜^T 𝐦^T Σ_0^-1𝐦 = 𝐝^T Σ_0^-1𝐦,

where Σ_0 is the covariance matrix with the diagonal items populated by observation uncertainties.
Here, we do not account for the non-diagonal covariance matrix because of the low S/N of the observations. We test for the effects of correlated errors by modeling them with Gaussian Processes (GP) following <cit.>, and find no difference in the constraints on the free parameters (see Fig. <ref> in Appendix). This demonstrates that the uncorrelated noise dominates the uncertainty.




The log-likelihood of the model 𝐌 is formulated as:

    lnℒ =     -1/2[ Nln(2π)+ln(|Σ_0|)+Nln(s^2) 
        + 1/s^2(𝐝 - 𝐌)^T Σ_0^-1 (𝐝 - 𝐌) ],

where N is the number of data points, and s is an error bar inflation parameter, 
which accounts for the underestimated uncertainties of the observations. Following <cit.>, 
the optimal s for each model can be calculated by 
effectively scaling the reduced χ^2 to 1:

    s^2 = 1/N (𝐝 - 𝐌)^T Σ_0^-1 (𝐝 - 𝐌).

The error bar inflation parameter can essentially account for the model's systematic errors compared to the observations, which ensures that these uncertainties are propagated to the posteriors of the free parameters.
The log-likelihood is evaluated in each spectral order and fiber before being combined, 
allowing for different linear coefficients and error bar inflation factors optimized for each order.

In total, the retrieval with disequilibrium chemistry model has 22 free parameters and the free chemistry model has 23 free parameters, as summarized in Table <ref>.
For the Bayesian inference process, we use the nested sampling tool  <cit.>, 
which is a Python wrapper of the  method <cit.>.
The retrievals are performed in importance nested sampling mode with a constant efficiency of 5%. 
It uses 2000 live points to sample the parameter space and derives the posterior distribution of free
parameters. 


















cccc

    Priors and posteriors of the retrievals.
    Parameter     Prior     Disequilibrium     Free
    M_p ()     𝒩(16.1, 5.0)     17.2 ± 4.2     17.3 ± 4.2

    R_p (R_Jup)     𝒰(0.7, 1.3)     0.94_-0.16^+0.21     0.93_-0.15^+0.22

    vsin i     𝒰(0.1, 20)     3.5_-1.9^+1.7     3.4_-1.9^+1.7

    RV     𝒰(-30, 30)     17.9 ± 0.3     17.9 ± 0.3 

    ϵ_limb     𝒰(0, 1)    0.5 ± 0.3     0.5 ± 0.3 

    ℛ_scaling     𝒰(0.8, 1.2)    0.96_-0.10^+0.12      0.95_-0.10^+0.12 

    [M/H]     𝒰(-1.5, 1.5)     0.26_-0.23^+0.24     0.22 ± 0.23  

    C/O     𝒰(0.1, 1.5)     0.55_-0.04^+0.06     0.64 ± 0.05

    logX^H_2O     𝒰(-12, -1)     -     -2.47 ± 0.21 

    logX^CO     𝒰(-12, -1)     -     -2.03 ± 0.23 

    logX^CH_4     𝒰(-12, -1)     -     -5.2_-4.4^+0.6 

    logX^NH_3     𝒰(-12, -1)     -      -7.5_-2.9^+2.5

    logX^^13CO     𝒰(-12, -1)     -     -4.5_-4.2^+0.6 

    log(/)     𝒰(-12, -1)     -2.7_-5.6^+0.8     - 

    logP_quench (bar)     𝒰(-5, 2)      0.21_-0.83^+1.06     - 

    T_0 (K)     𝒰(1500, 5000)     3904 ± 785     4094 ± 724 

    (dln T /dln P)_0      𝒰(0.02, 0.05)     0.035 ± 0.01     0.035 ± 0.01 

    (dln T /dln P)_1      𝒰(0.03, 0.07)     0.05 ± 0.01     0.05 ± 0.01 

    (dln T /dln P)_2      𝒰(0, 0.15)     0.07 ± 0.05     0.08 ± 0.05 

    (dln T /dln P)_3      𝒰(0, 0.5)     0.22 ± 0.09      0.29 ± 0.12 

    (dln T /dln P)_4      𝒰(0., 0.5)     0.12 ± 0.05     0.13 ± 0.06 

    (dln T /dln P)_5      𝒰(0., 0.5)     0.22 ± 0.10     0.24 ± 0.09 

    (dln T /dln P)_6      𝒰(0, 0.5)     0.20 ± 0.15     0.17 ± 0.14 

    log(X_0^MgSiO_3)    𝒰(-12, -1)     -7.4± 3.0     -7.4± 3.0 

    f_sed     𝒰(0, 10)     6.0 ± 2.5     6.0 ± 2.5
 
    log(K_zz)     𝒰(5, 13)     6.0 ± 2.6     6.0 ± 2.6 

    σ_g     𝒰(1.05, 3)     2.0 ± 0.6     2.0 ± 0.6

    
    𝒰(a, b) represents a uniform distribution and 
    𝒩(a, b) represents a normal distribution. The last two columns show
    posteriors with 1σ uncertainties from the disequilibrium chemistry model and
    free chemistry model.






§ RETRIEVAL RESULTS
 

We jointly fit two epochs of KPIC observations by generating the model spectrum and calculating the likelihood for each epoch and summing up the two likelihood values.  
We show retrieval results for two model setups: disequilibrium and free chemistry.
The posterior distributions of the free parameters in these models are summarized in Table <ref>.
In general, these models result in consistent constraints on the atmospheric properties.
We also carry out analyses for the two epochs independently. The retrieved parameters are consistent within 1-2σ, as shown in Fig <ref>. The first epoch provides slightly tighter constraints because of its better S/N (see Table <ref>).



 §.§ Cross-correlation detection



We carried out a cross-correlation analysis to show the detection of molecules in .
The cross-correlation functions (CCF) were computed as 

    CCF(v) = 1/s^2𝐅(v) ^T  Σ_0^-1𝐑,

where 𝐑 is the observed spectra minus the best-fit disequilibrium chemistry model with the abundance of 
a specific molecule being set to zero; 𝐅 is the molecular template which was computed 
by differencing the best-fit model and the best-fit model without the contribution of that molecule;
v is the radial velocity shift between the model and data.
Both 𝐑 and 𝐅 were high-pass filtered using a median filter with a width of 100 pixels.
Then, the CCFs of individual orders were combined into a master CCF, as shown in Fig. <ref>.
The noise of the CCF was estimated by subtracting the model's auto-correlation function (ACF) and taking the standard deviation at |v|>150 .
We detected  and CO with an S/N of 14 and 23, respectively. 
We checked other molecules, including , , , H_2S, and CO_2, and found no significant detections (see Fig.<ref> in Appendix).












 §.§ C/O and Metallicity


The retrieved chemical abundances and metallicity [M/H] and C/O ratios using two models are 
shown in Fig. <ref>.
Using the disequilibrium chemistry model, we obtained a C/O=0.55_-0.04^+0.06 and 
[M/H]=0.26_-0.23^+0.24 (1σ confidence intervals).
The empirical level of systematic uncertainty of KPIC measurements is ∼ 20% for C/O ratio and 0.2 dex for metallicity,
as estimated from benchmark brown dwarf companions, HR 7672 B and HD 4747 B <cit.>. These are roughly consistent with our results on .
The free chemistry model resulted in a derived C/O of 0.64 ± 0.05 and metallicity of 0.22 ± 0.23, 
in line with the disequilibrium chemistry model.
In comparison, the free chemistry model retrieved 
consistent VMRs for the major molecules as shown in Fig. <ref>. 
The slightly different C/O ratio derived from the free chemistry 
model can be attributed to the cold trapping of oxygen in clouds <cit.>. 
Condensates such as  take a fraction of oxygen out of the gas phase.
The chemical model predicts a mass fraction of ∼ 2.4× 10^-3 for , which was not constrained with our analysis (see Section <ref>).
Taking this amount of clouds into account, we estimated that the actual C/O ratio in the free chemistry model would be ∼ 0.55, consistent with the result from the disequilibrium chemistry model. 

Regarding Bayesian evidence, the disequilibrium chemistry model slightly outperforms the free chemistry model
at 3σ significance <cit.>. The values are summarized in Table <ref>.
We found that the log likelihood of the best-fit disequilibrium model is the same as the free chemistry model. 
This suggests that both models fit the observations equally well, and the slightly lower Bayesian evidence of the free chemistry model is due to the increased number of free parameters. 
Additionally, we tested imposing a Gaussian prior of 1.1±0.1  on the radius of the companion following the evolutionary model <cit.> 
and found that the retrieval results remained entirely consistent.














 §.§ Disequilibrium Chemistry



The disequilibrium chemistry model retrieves a quench pressure of 
P_quench∼ 0.2 - 22.4 bar (1σ).
This quench pressure is needed in order to explain the observed under-abundance 
of  relative to CO in the atmosphere. 
In contrast, turning off chemical quenching results in a transition from CO-dominated
to a -dominated composition at pressures below 0.1 bar given the T-P profile 
(see Fig.<ref>). 
The KPIC observations do not support this equilibrium case, as the VMR of CO in our free chemistry retrieval is constrained to a higher 
value than the equilibrium prediction and we obtain a 3σ upper limit of <4×10^-5 on the VMR of .

We re-ran the retrieval assuming chemical equilibrium without quenching and compared the resulting fit to the nominal model with quenching. 
The equilibrium retrieval converged to a more isothermal T-P profile (i.e., hotter upper layers)
in order to avoid the transition to a -dominated atmosphere.
We calculated the Bayes factor for these two models and found that the nominal
disequilibrium chemistry model
is favored over that for equilibrium at a 2.4σ significance. 
Similar findings have been observed in several late L-type super-Jupiters 
such as HR 8799 cde <cit.> and brown dwarfs 
<cit.>. 
















 §.§ Clouds
 

The cloud-related parameters were not constrained in our retrieval analysis 
(see Table <ref> and Fig. <ref>). 
We calculated the Bayesian evidence for the clear versus cloudy models and found the clear model is slightly preferred at 2.7σ significance because of fewer free parameters.
We retrieved consistent chemical abundances from the cloudy and cloud-free models.
While clouds are expected in atmospheres of late L-type brown dwarfs <cit.>, 
the reason for our data's insensitivity to clouds may be twofold.
First, the limited S/N of the observations may not be sufficient to distinguish the small differences in line contrasts
caused by the cloud opacity. 
Second, the silicate cloud base in our model is located at ∼10 bar, below the photosphere of 
the atmosphere in K band, as shown in Fig. <ref>. 
Hence, the impact of cloud opacity is not significant at the 
pressure levels probed by the high-resolution K-band observations 
<cit.>.

















 §.§ Rotation and RV


The projected rotation velocity of  is measured to be vsin(i) = 3.5_-1.9^+1.7  in our retrievals. 
However, considering the spectral resolution of KPIC (ℛ∼ 35,000), 
the instrument broadening dominates over the rotational broadening of the target, 
meaning that the rotation is not resolved in our data. 
The likelihood of the best-fit retrieval model shows no difference whether including spin as a free parameter or not.
Therefore, we report a 3σ upper limit of vsin(i)<7.8  as derived from the posterior distribution. This is also consistent with the spectral resolution of KPIC. 
Adopting an isotropic distribution of inclination, we estimated its spin velocity to be <10 , which corresponds to a rotation velocity versus break-up velocity v/v_break∼0.06.
We further discuss this constraint in Section <ref>.


We measured a radial velocity of -17.9± 0.3 for HIP 99770 b. This allows us to compute the relative RV 
between the star and companion and add to the orbital constraints of the system. 
Since the A-type host star is almost featureless in K band, we cannot measure the stellar RV in our KPIC data.
The stellar RV is not fully consistent in the literature, likely because 
the fast rotation of the star <cit.> makes it challenging to measure the RV accurately. 
We adopted the RV measurement of  from Gaia DR3 of -20.5±0.4  <cit.>. This led to a relative 
RV of 2.6±0.5  between the companion and primary.
Adding this relative RV constraint, we carried 
out orbital fitting using the  code <cit.> following <cit.>
to refine the orbital properties of . 
We note that here we used the default 1/M_sec prior for the secondary mass, 
which is responsible for the lower M_sec (∼13 ) as was also mentioned in <cit.>.
We compared the posteriors of the orbital fitting with and without the relative RV measurement in Fig. <ref>.
Although the addition of relative RV did not significantly change the overall orbital constraints, 
it assisted in ruling out some solutions and improving the constraint on the argument of periastron ω, longitude of ascending node Ω, and orbital inclination i= 152_-11^+9 deg. 
Additional epochs of RV measurements in five-year baseline will 
further strengthen the constraining power
on other parameters, such as the secondary mass and orbital eccentricity.






§ DISCUSSION
 




 §.§ Rotation velocity
 

Putting our upper limit on rotation velocity into context, we note that the spin of  is at the low end
compared to literature measurements of other super-Jovian companions as shown in Fig. <ref>.
Previous studies suggest that younger companions generally have lower rotation rates, 
which are expected to 
increase with age as their radii contract following angular momentum conservation <cit.>. 
In contrast, field brown dwarfs display a larger scatter in spin <cit.>.
Although the age of  is older than 40 Myr, its rotation velocity (assuming isotropic inclination distribution) 
is comparable to that of very young companions (<10 Myr) such as GQ Lup b <cit.>, 
DH Tau b, and HIP 79098 b <cit.>, and is lower than those at older ages.
If the trend of increasing spin with increasing age holds, it may indicate a nearly pole-on orientation for . 
If we assume that its spin and orbit are aligned (i∼152), the spin velocity is <15  or v/v_break∼ 0.09, 
making it more comparable to other super-Jovian companions. 


On the other hand, the slow spin may be indicative of effective magnetic braking by the circumplanetary disk (CPD)
<cit.>. 
Massive companions are expected to effectively ionize the CPD 
and interactions with magnetic fields then act to spin down the companions.
Therefore, the slow spin of  may 
indicate that it hosted an unusually long-lived CPD and/or that it formed early via gravitational instability, 
both of which allow for a longer time for 
the companion to spin down.
The large scatter in the measured spins of super-Jupiter companions may represent the consequences of 
mixed formation pathways and histories <cit.>.
Future rotation period measurements with light curves will help break this degeneracy with inclination.
More accurate measurements of rotation velocities using high-resolution, large spectral grasp instruments such as
VLT/CRIRES^+ and Keck/HISPEC will be essential for understanding the population-level trend.







 §.§ Implication for Formation


We retrieve an atmospheric [M/H] =0.26 ± 0.24 and C/O =0.55 ± 0.06 (1σ confidence intervals) for , consistent with the solar values. 
Previous studies reported stellar abundances of [C/H]∼0.18 ± 0.09 and [O/H]∼0.01 ± 0.09 
<cit.>. 
This corresponds to a C/O ratio of 0.8_-0.2^+0.3, roughly consistent with solar to super-solar value of 0.59 ± 0.08 <cit.>.
The uncertainties in the stellar composition result from the large rotational broadening of the primary and the deficit of oxygen lines in its spectrum. 
A more accurate stellar C/O constraint is needed in order to draw unambiguous conclusion on the formation of the companion.
For consistency, we simply assume a solar composition for the primary star  in the following discussion.

The companion's current semimajor axis of 17 au is well within the typical location of the CO iceline 
in protoplanetary disks around early-type stars <cit.>.
This makes it an interesting case for testing formation models.
Given our constraints on its atmospheric composition,
the companion is compatible with a formation via either core accretion or gravitational instability.
Based on the static picture of chemistry in smoothed protoplanetary disks <cit.>,
the core accretion scenario is expected to result in either: 1) a substellar C/O and metal-rich atmosphere 
because of the accretion of
oxygen-enhanced solids,  2) an elevated C/O combined with substellar metallicity 
due to gas-dominated accretion, or 3) a stellar C/O when the accreted material corresponds to an overall stellar metallicity for the atmosphere. 
In practice, protoplanetary disks are not static and have complicated substructures <cit.>. Dust drift and chemical evolution of gas and dust can alter the distribution of disk elemental abundances over time <cit.>. Planet formation models coupled with disk chemistry evolution and dust migration are needed to investigate these questions further. Qualitatively, disk observations support the simplified picture that the gas-phase oxygen is strongly depleted as more O-bearing species are locked in the ice phase beyond the snowline <cit.>. Therefore, planets formed under such conditions are expected to follow the scenarios outlined above.

In contrast, gravitational instability should lead to similar stellar and companion atmospheres
since it is thought to occur relatively early when the disk is massive and the reservoir of solids is pristine <cit.>. 
Although the companion may undergo late enrichment by solid accretion after the initial collapse, 
this would not significantly alter the composition due to the high envelope mass; for example,
to enhance the metallicity by one dex would require accreting ∼2  of solids 
<cit.>.
Similarly, we do not expect a large enrichment of metal in super-Jupiter atmospheres in core accretion models.
Therefore, the solar composition of  does not preclude either formation scenarios. 
If the companion was born ex-situ, it may have formed in the outer disk
followed by type II migration <cit.>. 
Such planet migration could potentially boost the accretion of planetesimals, leading to a slight metal enrichment, as we constrain for . 









The chemical composition of  also fits in the overall trend for super-Jupiter 
companions. 
<cit.> carried out retrieval analyses in a sample of 
10-30  companions using KPIC observations. They found the companions generally had
solar compositions, hinting at formation via gravitational collapse or plausible core 
accretion beyond CO iceline. In contrast, lower mass (<10 ) companions appear to display
a larger range in C/O and metallicity <cit.>. 
By comparing the C/O ratio in a sample of directly imaged companions with transiting exoplanets, <cit.> suggested that there are two distinct populations with a boundary at ∼5 , implying a transition of
formation pathways near a mass range of 5-10. 
However, more work is needed to determine whether there exists a clear mass boundary for distinct formation channels, as such compositional measurements get challenging towards lower-mass and closer-in companions.
The comparison between directly imaged super-Jupiters and transiting exoplanets with similar masses will be informative in distinguishing whether these populations are separated by different formation pathways or migration histories <cit.>.
It also calls for the joint understanding of trends in the orbital architectures of these systems <cit.>
to unravel their formation.






§ CONCLUSION


We carried out detailed spectral characterization of the recently discovered super-Jupiter 
companion  using the fiber-fed high resolution spectrograph KPIC (ℛ∼ 35,000). 
The K-band observations led to clear detections of  and CO in the atmosphere of .
We used atmospheric retrievals to constrain its atmospheric composition, including its C/O and metallicity, 
projected rotation velocity (vsin i), and radial velocity (RV). 


    
  * We found the companion's atmosphere has C/O =0.55 ± 0.06 and [M/H] =0.26± 0.24 (1σ confidence interval), which are 
    consistent with stellar values. 
    It joins the ensemble of super-Jovian companions showing broadly solar compositions.
    This is compatible with a formation 
    via either gravitational instability or core accretion. 
    
  * We found that the ratio of  to CO in the companion's atmosphere is lower than predicted by equilibrium chemistry models, and place a lower bound of 0.2 bars on the quench pressure.
    
  * Although the companion is expected to be cloudy, we found that our models do not require clouds. This may be because of the limited S/N, or or because the clouds are located below the photosphere for our K band observations.  
    The presence or absence of clouds could be better constrained by expanding the wavelength range of these observations, which would increase our sensitivity to the wavelength-dependent scattering signal from clouds.
    
  * We added the companion-to-primary relative RV measurement to the orbital fitting and obtained updated
    constraints on the companion's orbital i, ω, and Ω.
    
  * The projected rotation velocity
     vsin(i)<7.8  is small compared to other directly imaged companions with similar ages and masses. 
     This may indicate a nearly pole-on orientation or effective
    magnetic braking by a circumplanetary disk.


Located within 20 au and straddling the deuterium-burning mass boundary,  represents an intriguing
target to investigate the link between the present atmosphere and formation history.

The convoluted formation and evolution processes make it challenging to retrace the origin of planets
unambiguously.
Modeling studies on the formation of these well-characterized super-Jovian companions are crucial for advancing 
our knowledge. 
In the future, higher S/N spectroscopic observations will allow for tighter constraints 
on the metallicity 
by probing elements beyond C and O. The estimates of solid budgets will massively benefit the 
inference of formation history <cit.>.
Other formation tracers, such as carbon isotope ratios, will help pin down the formation
location relative to the CO iceline, therefore allowing us to lift the degeneracy 
between formation mechanisms <cit.>.
Detailed characterization of similar companions will play an essential role in
charting the parameter space to understand the formation of the super-Jovian population.




Y.Z. is thankful for support from the Heising-Simons Foundation 51 Pegasi b Fellowship (grant #2023-4298).
Y.C. acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC) through the CITA National Fellowship and the Trottier Space Institute through the TSI Fellowship.
Funding for KPIC has been provided by the California Institute of Technology, the Jet Propulsion Laboratory, the Heising-Simons Foundation (grants #2015-129, #2017-318, #2019-1312, #2023-4598), the Simons Foundation, and the NSF under grant AST-1611623.
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 Mauna Kea has always had within the indigenous Hawaiian community.  We are most fortunate to have the opportunity to conduct observations from this mountain. 
The computation was carried out on the Caltech High Performance Cluster.





Keck (KPIC)


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







§ RETRIEVAL RESULTS

We summarize the Bayesian evidence of various retrieval models mentioned in Section <ref>.
Fig. <ref> shows the posterior distributions of free parameters in the baseline retrieval with 
disequilibrium chemistry model.
In Table <ref>, we list the Bayesian evidence of various retrieval models and their comparisons to the baseline model.
The cross-correlation functions of molecular non-detections are shown in Fig.<ref>.
We carry out independent retrievals on two epochs of KIPC data and find broadly consistent posterior constraints as shown in Fig. <ref>.
We also explore the effect of correlated noise on the retrieval results by implementing Gaussian Processes. The corner plot of the posteriors is shown in Fig.<ref>. 





cccccc
    Summary of Bayesian evidence of various retrieval models
    Model     ln Z     Δln Z     Significance (σ) 
    
    Disequilibrium chemistry      -93748.5     -     -  

        Free chemistry      -93751.8     -3.3     3.1  

        Equilibrium chemistry      -93750.0     -1.5     2.4  

        Disequilibrium chemistry, no cloud      -93746.2     2.3     2.7  

    
    Δln Z represents the difference of Bayesian evidence ln Z of each alternative model compared to the nominal disequilibrium chemistry model.










aasjournal




 Paper Title: FCN4Flare: Fully Convolution Neural Networks for Flare Detection.



A-Li Luo
* lal@nao.cas.cn 
0009-0001-0257-3377]Ming-Hui JIa
CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories, Beijing 100101, China
University of Chinese Academy of Sciences, Beijing 100049, China
0000-0001-7865-2648]A-Li Luo^*
CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories, Beijing 100101, China
University of Chinese Academy of Sciences, Beijing 100049, China
Nanjing Institute of Astronomy and Optical Technology, University of Chinese Academy of Sciences, Nanjing 211135, China

University of Science and Technology Beijing,Beijing, 100083, People's Republic of China


§ ABSTRACT  
Stellar flares originate from the sudden reconnection of magnetic flux lines within the star's atmosphere, particularly in the chromosphere. Automated flare detection enables exploiting vast photometric datasets from missions like Kepler. Prior methods rely on outlier detection, facing challenges of complexity, detection accuracy, and scalability. This paper presents FCN4Flare, a deep learning approach using fully convolutional networks for precise point-to-point flare prediction regardless of light curve length. Key innovations include the NaN Mask to handle missing data, dilated convolutions to preserve local information, and the MaskDice loss to mitigate severe class imbalance. Experiments demonstrate significantly improved detection performance over previous models, with a 0.64 Dice coefficient on Kepler data. Applying FCN4Flare to Kepler and LAMOST, we compile a catalog of 30,285 high-confidence flares across 1426 stars. Flare energies are estimated and stellar/exoplanet properties analyzed, identifying pronounced activity for an M-dwarf hosting a habitable zone planet. This work overcomes limitations of prior flare detection methods via deep learning, enabling new scientific discoveries through analysis of photometric time-series data.






§ INTRODUCTION

Stellar flares may arise from the magnetic reconnection <cit.>. They offer profound revelations into the enigmatic magnetic dynamo processes transpiring deep within stellar cores, as well as the intricate mechanisms orchestrating stellar activity cycles <cit.>. By meticulously scrutinizing the distinctive properties of these flares, we gain invaluable insights into the intricate nature of stars themselves. Moreover, beyond their significance in comprehending stellar phenomena, flares emerge as pivotal agents molding the genesis and evolution of exoplanets <cit.>. The discernment of stellar flare occurrences thus assumes a critical role in our endeavors to assess the potential habitability of exoplanets <cit.>.

In the contemporary astronomical landscape, the advent of photometric space missions, exemplified by the likes of Kepler <cit.> and TESS <cit.>, has ushered in unprecedented opportunities for astronomers to delve into the study of stellar flares. These missions have facilitated the acquisition of vast troves of light curves, presenting us with an exhilarating prospect. However, the sheer magnitude of this data also presents us with a novel challenge: the development of efficient methods for automated flare detection.

To address this challenge, various methodologies, predominantly anchored in outlier detection, have been meticulously crafted <cit.>. These methods adroitly follow a systematic three-step procedure:

    
  (1) In the initial stage, a meticulous fitting of the quiescent flux or baseline is accomplished through the skillful deployment of a prewhitening procedure, complemented by an iteratively smooth filter. The combined effect of these techniques is to deftly eliminate extraneous noise and spurious data points, thus laying the foundation for a pristine analysis.
    
  (2) With the quiescent flux duly established, stringent criteria, encompassing amplitude, profile, and duration, are artfully applied to the candidate outliers, artfully winnowing out the most promising flare events. These selection criteria typically involve setting a threshold on the amplitude, mandating an impulsive surge followed by an exponential decay in the flare profile, and considering the temporal extent of the event.
    
  (3) The results are validated using mathematical tests or simulations to establish selection criteria and assess errors. 


To provide concrete instances, <cit.> adroitly employed an iterative light curve detrending approach to effectively segregate intrinsic variability from potential flares. Flare candidates were duly identified as instances where consecutive points surpassed 2.5 σ above the local mean flux for a sustained period. Subsequently, manual confirmation of flares ensued to ensure precision. In a similar vein, <cit.> meticulously employed thresholds governing the intensity increase and duration of the flares, stipulating a minimum of 3 consecutive points exhibiting an intensity above 3σ, accompanied by a rapid flux augmentation. Flares, once again, were judiciously validated through manual inspection. <cit.> adeptly resorted to iterative light curve detrending to model the quiescent brightness. Subsequently, flare candidates were discerningly extracted as outliers from the quiescent model, courtesy of a matched filter approach featuring an analytical flare template. <cit.>, meanwhile, adroitly detrended light curves via the proficient application of Fourier series fitting, mandating that flare candidates exceed a 3σ threshold for a minimum duration of 3 data points.

Undoubtedly, the method's utilization of traditional outlier detection comes with two primary limitations, impeding its broad application in large-scale astronomical data analysis. Firstly, the detection process's inherent complexity necessitates continuous iteration, leading to protracted computational times. Such extended processing periods can hinder timely analyses of extensive datasets, making it challenging to keep up with the ever-expanding volume of astronomical information <cit.>. Secondly, to effectively suppress the false positive sample rate, the methods rely on the development of highly detailed and stringent criteria. As observed in the study conducted by <cit.>, the earlier flare catalogs suffered from contamination by various false-positive signals and artifacts. Consequently, more rigorous criteria were introduced by <cit.> to discern flares on Kepler light curves. While stricter evaluation criteria enhance the accuracy of flare detection, they introduce further intricacy to an already involved detection process, exacerbating the challenges faced in analyzing large datasets.

The recent application of deep neural networks in detecting stellar flares has shown considerable promise. Notably, Stella <cit.> and Flatwrm2 <cit.> have revealed the effectiveness of using convolutional and recurrent networks to determine whether the center of a fixed-length segment of light curves indicates a flare event. By utilizing a sliding window method, these networks can accurately predict flare occurrences throughout a light curve.

The key advantage of employing deep neural networks lies in their inherent capacity for highly parallelizable computations. This attribute allows for expedient processing of substantial datasets, enabling the rapid and efficient detection of flares on a large scale <cit.>. The ability to analyze copious data points in parallel not only accelerates the identification process but also contributes to the overall scalability of the method. Consequently, these deep learning-based approaches hold great promise in tackling the challenges posed by the vast amounts of astronomical data acquired through modern photometric space missions like Kepler and TESS <cit.>.

However, the precision of these methods still need to be improved. Additionally, their use of a fixed sliding window to detect flares in an entire light curve is not efficient enough for large-scale datasets.

Drawing inspiration from the groundbreaking work of <cit.>, our endeavor is to accomplish point-to-point dense prediction, eschewing the sliding window approach. In this paper, we present a fully convolutional network designed for the purpose of detecting stellar flares—henceforth referred to as FCN4Flare. This approach holds the potential to revolutionize the domain of flare detection, propelling it towards unprecedented levels of accuracy and efficiency.

In the following article, Section <ref> will introduce the details of FCN4Flare, Section <ref> will discuss the model performance, Section <ref> will explore the practical applications of our approach, and Section <ref> is the conclusion.



§ FCN4FLARE





The crux of our task lies in predicting the probability that each flux value in a given light curve belongs to a flare event. To accomplish this, we employ neural networks denoted by f(X|Θ), where X={x_i ∈ℝ | i=1, 2, ⋯, T} represents the input light curve with a length of T. Our objective is to generate a probability vector P={ p_i ∈ [0, 1] | i=1, 2, ⋯, T }, where p_i signifies the likelihood that the flux value x_i is part of a flare event.

A notable feature of our approach is its flexibility regarding input length. Unlike recent methods that utilize equal-length segments of the light curve, generated by sliding a window point by point along the curve, our method can handle light curves of varying lengths. This adaptability allows us to analyze and predict flare probabilities effectively for light curves of different durations, which is particularly advantageous given the inherent variability in astronomical data.

By leveraging the power of neural networks and their capability to discern intricate patterns and relationships within data, our method aims to yield accurate and reliable predictions regarding the likelihood of flare events at individual flux values. With the ability to process light curves of diverse lengths, we anticipate that our approach will contribute significantly to the efficient analysis of large-scale astronomical datasets, unlocking new possibilities for the exploration and understanding of stellar phenomena.



 §.§ Architecture Overview


As depicted in Figure <ref>, FCN4Flare is composed of five essential components: NaN Mask, Mapping, Backbone, SegHead, and MaskDice. Each of these components plays a crucial role in the process of flare detection and optimization of the neural network:

    
  * NaN Mask This component serves to guide the model in learning where the flux values are NaN (Not-a-Number) and how to handle these NaN values during the flare detection process. Proper handling of NaN values is crucial for accurate predictions in light curves with missing data. See section <ref>.
    
  * Mapping The Mapping component utilizes a convolution layer with a 1 × 1 filter size to map the extracted features to a high-dimensional space. This step enhances the representation capabilities of the model. See Section <ref>.
    
  * Backbone The Backbone is the primary component responsible for feature extraction. It is formed by stacking ten Conv Blocks, each contributing to the extraction of relevant features from the input normalized flux sequences. See Section <ref>.
    
  * SegHead The SegHead is responsible for point-wise flare detection of the light curves, regardless of their variable lengths. This component is essential for generating predictions at each point in the light curve, enabling detailed and precise flare detection. See Section <ref>.
    
  * MaskDice MaskDice serves as the optimization objective function for the neural network. Its design is aimed at addressing the challenge of extremely imbalanced samples, which commonly occurs in astronomical datasets. See Section <ref>.


The input to FCN4Flare is a batch of entire normalized flux sequences of light curves. The normalization process involves dividing the flux values by the median flux, ensuring consistent and comparable representations.

The model's output consists of point-wise flare probabilities, which have the same shape as the input batch. This design allows FCN4Flare to achieve end-to-end training and point-to-point prediction. This advantage not only facilitates efficient model training but also enables accurate and granular predictions, making FCN4Flare a powerful tool for the comprehensive analysis of large-scale astronomical datasets and the identification of stellar flares with precision.



 §.§ NaN Mask


NaN values, which can cause abnormal calculation results and disrupt the normal training of neural networks are common in light curves. Previous studies <cit.> addressed this by simply removing NaN data points before feeding light curves into the model. Post-prediction, any flares distributed on both sides of the NaN timestemps were manually excluded. In contrast, our approach automates this process within the model itself.

We introduce the NaN Mask module to guide the model in learning to distinguish NaN values on its own. This module first replaces all NaN flux values with placeholder values (e.g., zeros). It then creates a concatenated matrix by combining the flux vector with an indicator vector, where `1' denotes NaN and `0' denotes non-NaN values. For instance, a light curve with a flux vector x=[0.1, NaN, 0.3, NaN, NaN] results in a matrix x̂=[[0.1, 0, 0.3, 0, 0], [0, 1, 0, 1, 1]]. This method allows the model to discern between valid data points and missing data, preventing NaN values from adversely affecting the training and prediction processes.



 §.§ SegHead and Mapping


The variable feature shape inherent in light curves poses a significant obstacle when attempting point-to-point prediction using neural networks. Previous approaches, such as those seen in <cit.> and <cit.>, were compelled to resort to a sliding window methodology with a fixed width due to the constraints imposed by fully connected (FC) layers. These FC layers demand input data with fixed shapes, rendering them incompatible with variable-length input curves.

In our quest to overcome this limitation, we introduce SegHead module, which enables us to generate variable-length outputs, thereby triumphing over the constraint of fixed input shapes.

Consider a flux sequence with an input shape of 1 × L, where L denotes the length of the sequence. Upon feature extraction, the resulting feature shape manifests as F × L^', where the dimension F represents the feature channels, and L^' denotes the spatial dimension. It is important to note that L^' may differ from the original length L due to the utilization of down-sampling layers, which are strategically designed to expand the receptive field of subsequent convolutional filters.

To achieve precise point-to-point prediction, it is crucial that the shape of the output matches that of the input. In other words, we need to convert the feature channels F to 1 and the spatial dimension L^' to L. To facilitate this process and for the sake of ease during prediction, we set L^' equal to L during the feature extraction phase (see section <ref> for further elucidation on this aspect). To achieve the conversion of feature dimensions, we resort to employing a convolutional layer, which adeptly modifies the feature channels without altering the spatial dimension. Specifically, we employ convolution filters of size 1 × 1 and set the output channels to 1. As a result, this convolutional layer adeptly accomplishes weighted summation of different channels for each data point, ultimately yielding probabilities through the application of the sigmoid function.

It is noteworthy that the use of a 1 × 1 convolutional layer is tantamount to employing a fully connected (FC) layer for each individual point's feature in mathematical terms. However, this convolutional approach imparts greater flexibility to both the input and output, facilitating the accommodation of variable-length input sequences. 

Additionally, in the context of neural networks, as the depth of features increases, the network gains the ability to capture more global and high-level patterns in the data. However, in the case of flare detection, it is crucial to focus on local and fine-grained information within the light curves. To address this challenge effectively, we introduce an additional convolutional layer with a small filter size before the 1 × 1 convolution layer as depicted in Figure <ref>. This strategic inclusion of a small filter size convolutional layer serves the purpose of capturing and preserving important local information within the data.

Similarly, the application of a 1 × 1 convolution layer (Mapping) before the Backbone serves to rapidly increase feature channels while preserving the spatial dimension, as depicted in Figure <ref>. This enables the model to effectively capture and leverage higher-level features without compromising the crucial spatial details necessary for accurate flare detection.




 §.§ Backbone


The Backbone, serving as the primary component for feature extraction, plays a pivotal role in the efficacy of the model. As mentioned in section <ref>, we need to keep feature spacial size the same during feature extracting. So unlike <cit.>, we use dilated convolution layers <cit.> instead of combination of traditional convolution and pooling layers.

Dilated convolution is a powerful convolutional operation widely employed in deep learning models for image processing tasks. In this technique, the filter is dilated by inserting zeros between its values, effectively creating a larger filter with gaps or holes between its elements. This dilation strategy augments the receptive field of the filter without necessitating pooling layers, thereby preserving the feature map size.

Inspired by the insightful work of <cit.>, our backbone architecture consists of ten residual convolutional blocks. Each of these blocks comprises two 1-D convolutional layers, with a dilation parameter, as illustrated in Figure <ref>.




 §.§ MaskDice


We have achieved point-to-point prediction on light curves of variable lengths using the aforementioned four components. However, this solution has given rise to a novel challenge, namely, an exceedingly pronounced sample imbalance. As is well-known, the number of flare points within a light curve is significantly smaller in comparison to the number of no-flare points. This profound disparity in the distribution of data may compel our neural networks to favor predictions of the no-flare label for all data points, as such an approach would conveniently yield high accuracy (for instance, in the case of a light curve where the flare point rate is merely 1%, a model that merely predicts the no-flare label would attain a seemingly impressive accuracy of 99%, although such a model would bear no meaningful significance).

It is important to emphasize that such an issue was not prevalent in previous methods <cit.>, as their input was confined to individual curve segments rather than the entire curve. Consequently, they were able to control the distribution of different segment types within the training set, effectively circumventing this predicament. However, in our particular scenario, we confront the daunting task of addressing the challenge of extremely imbalanced samples, necessitating a more sophisticated and nuanced approach.

We use the MaskDice component shown in Figure <ref> to solve the problem. It is a loss function based on Dice loss <cit.>.

The Dice loss, specifically tailored to address the significant imbalance between foreground and background pixels, is a widely employed loss function in the domain of image semantic segmentation. This loss is determined through the Dice coefficient, a metric employed to gauge the similarity between distinct sets. A higher Dice coefficient signifies a greater degree of similarity between the sets. The Dice coefficient is computed using the formula shown in Equation <ref>, where DiceCoef ∈ [0, 1] represents the Dice coefficient, and X and Y represent the respective sets involved.


    DiceCoef = 2|X ∩ Y|/|X| + |Y|


To ensure efficient computation and prevent potential division by zero, the loss is often calculated using the formulation shown in Equation <ref>, which is equivalent to Equation <ref>. In this context, N represents the length of a light curve, p_i and r_i correspond to the model's prediction and the ground truth for the i-th point in the light curve respectively, and ϵ is a small constant introduced to prevent the denominator of the Dice coefficient from becoming zero.


    DiceLoss = 1 - DiceCoef = 1 - 2|X ∩ Y|/|X| + |Y|



    DiceLoss = 1 - 1/N2 ∑^N_i=1 p_i · r_i + ϵ/∑^N_i=1 p_i + ∑^N_i=1 r_i + ϵ


Although Dice Loss has been effective in handling sample imbalance <cit.>, it may not perform optimally in the context of flare detection when dealing with a very small percentage of flare points (approximately 0.5% in our dataset). To better understand the issue, let's consider a prediction vector P={ p_i ∈ [0, 1] | i=1, 2, ⋯, T } and a label vector R = { r_i ∈{ 0, 1 } | i=1, 2, ⋯, T }, where r_i = 1 denotes that the timestamp is a flare point, and r_i = 0 denotes that it is not. In the ideal scenario, p_i → 0 (e.g., p_i = 1e-9) when r_i = 0, and it is similar when r_i = 1, ensuring that ∑_i=1^N p_i →∑_i=1^N r_i, and consequently, DiceLoss → 0.

However, during the initial stages of training, the values of p_i may not be ideal (e.g., p_i = 0.1 when r_i = 0). In this situation, ∑_i=1^N p_i might be much larger than ∑_i=1^N r_i due to the overwhelming proportion of non-flare points. Consequently, the DiceLoss might not decrease significantly, regardless of the nature of the label vector R for a given curve, making it challenging to optimize the neural network parameters effectively.

In essence, the main reason why Dice Loss does not perform well in this context is the exceedingly large proportion of non-flare timestamps. To address this issue, the paper presents a solution in the form of the Mask Dice Loss, which aims to tackle the sample imbalance problem more effectively and improve the performance of flare detection in light curves.

The Mask Dice Loss function is expressed in Equation <ref>, where u(x) represents a step function, as defined in Equation <ref>, and m is a hyperparameter. The objective of this function is to address the issue of sample imbalance in the context of flare detection. It achieves this by utilizing a simple step function to mask the timestamps for which the prediction probabilities are less than the threshold m. By doing so, the loss function penalizes the model for misclassifying non-flare timestamps, thereby mitigating the numerical advantages associated with the significant sample imbalance.



    Mask   Dice   Loss = 1 - 1/N2 ∑^N_i=1 u(p_i - m) p_i · r_i + ϵ/∑^N_i=1 u(p_i - m) p_i + ∑^N_i=1 r_i + ϵ




    u(x) = {[ 1, x > 0.5; 0, x ≤ 0.5 ].


The step function u(x, m) acts as a masking mechanism, identifying the timestamps for which the prediction probabilities (p_i) are less than the threshold m. These masked timestamps, for which the model can easily make correct predictions, have less impact on improving the overall performance compared to the probabilities near the classification threshold boundary. By selectively masking such timestamps, the model is encouraged to pay more attention to the flare samples, enabling the optimizer to guide the optimization of neural network parameters in the right direction from the initial stages of training, even in the presence of the significant sample imbalance problem. As a result, the Mask Dice Loss function helps improve the performance of the model for flare detection in light curves by mitigating the challenges posed by imbalanced data.




§ RESULTS AND DISCUSSION





 §.§ Data and Training

The data used in this study is sourced from the catalog presented by <cit.>, which identifies flares in Kepler's long-cadence light curves. The catalog records a total of 162,262 flare events across 3,420 distinct stars on 33214 quarters of light curves. The method of detecting flares used in <cit.> involves a rigorous and detailed approach to ensure the accuracy and credibility of the catalog. It is a multi-step process that includes the following key steps:

    
  (1) Each observation quarter is divided into blocks at points where there are discontinuities longer than 6 hours. Each block is independently adjusted by removing long-term instrumental trends using a third-order polynomial. A Lomb–Scargle periodogram determines the most significant frequency, and a spline filter fits the baseline. An iterative σ-clipping method removes outliers to clean the data further.
    
  (2) Flare candidates are identified under strict conditions:
    
        
  (i) At least three consecutive points must exceed a 3σ threshold without interruption within 3 hours, with the flare ending at the last point above 1σ.
        
  (ii) The decay phase of a flare must be longer than its rise phase.
        
  (iii) Flares must be observed in at least two different quarters if multiple quarters of data are available.
        
  (iv) Flares energy must be greater than 10^34 ergs and last least 2 hours if only single quarter is available.
    
    
  (3) A contamination check is performed to ensure the purity of the flare signal. This involves spatial analyses to rule out photometric contamination from nearby sources, correlation checks with other instruments to confirm the uniqueness of the events, and comparison with known artifacts to identify false positives. Critical cases undergo a manual review by experts to further validate the findings.


The stringent process can significantly reduce the false positive rate of flare detection. While label errors can not be completely cleared up in the catalog, it still provides a robust basis for effectively training and testing deep learning models. This issue is common in machine learning benchmarks, as highlighted by <cit.>, which shows that even widely used datasets like ImageNet comprises label errors at least 6% of the validation set.

The 33,214 light curves described in the catalog were divided into training, validation, and test sets using a 6:2:2 ratio. We train all the models including ours and previous methods on the training set from scratch. Model selection was based on performance in the validation set, and the chosen models were then evaluated on the test set. Comparative performance results are discussed in Section <ref>.



Our model is trained using two NVIDIA V100 GPUs with a batch size of 150, utilizing the AdamW optimizer at an initial learning rate of 2 × 10^-5. The training is set to run for a maximum of 50 epochs, incorporating an early stopping mechanism based on validation dice scores. The entire training process takes approximately one hour. Figure <ref> illustrates the loss and dice scores during training and evaluation.



 §.§ Detection Performance





To rigorously evaluate the efficacy of the FCN4Flare method for stellar flare detection, we employ a suite of statistical metrics that provide a multi-dimensional view of performance. Here are the metrics:


    
  * Recall measures the proportion of actual flare events that are correctly identified by the model. A higher recall rate indicates a lower likelihood of missing real flare events. It is calculated as:
    
    Recall = TP/TP + FN

    where TP and FN correspond true positives and false negatives, respectively.
    
    
  * Precision assesses the accuracy of the positive predictions by the model. A higher precision rate indicates that the flares detected are indeed true flares, thereby minimizing the time and resources spent on following up false leads. It is calculated as:
    
    Precision = TP/TP + FP

    where FP corresponds false positives.
    
    
  * F1 is the harmonic mean of recall and precision, providing a single measure to balance both metrics. It is calculated as:
    
    F1 = 2 ×Recall × Precision/Recall + Precision

    
    
  * AP is the average precision scores at each threshold. Each precision score is weighted by the increase in recall compared to the previous threshold. AP provides a comprehensive measure that captures the performance across all thresholds which we decide the probability as the flare or not. It is calculate as:
    
    AP = ∑ (R_n - R_n-1) × P_n

    where R_n and P_n represent precision and recall at n-th threshold, respectively.
    
    
  * AUC-ROC is the area under the curve plotted with true positive rate (TPR) against false positive rate (FPR) at various threshold settings. It tells us how likely the model is to correctly identify a positive case as being more likely than a negative one when picking randomly. It is calculated as:
    
    AUC-ROC = ∫_0^1 TPR(x)   dx

    where x corresponds to varying FPR from 0 to 1.
    
    
  * IoU also known as the Jaccard index, is the intersection of union. It provides a straightforward metric that quantifies how much the predicted set covers actual set while penalizing false positives and false negatives. It calculated by:
    
    IoU = |A ∩ B|/|A ∪ B|

    where A and B are the predicted and actual labels set.
    
    
  * Dice coefficient is similar with IoU, but it calculates the size of the overlap times two divided by the sum of the sizes of both sets. It is more sensitive to the size of the intersection relative to the total size of each individual set. It is calculated as:
    
    Dice = 2 × |A ∩ B|/|A| + |B|





Table <ref> presents a comparative analysis of FCN4Flare with established flare detection methods using selected performance metrics. FCN4Flare's superiority is evident as it outperforms other models in all metrics, particularly showing significant improvements in precision, AP, IoU, and Dice coefficient. These enhancements suggest greater specificity in flare detection, as demonstrated by fewer false positives. To further highlight the model's predictive accuracy, the confusion matrix is shown in Figure <ref>. This matrix clearly visualizes the reduction in false positives, providing solid evidence of FCN4Flare's enhanced detection capabilities.



 §.§ Ablation Study

We present an ablation study to systematically evaluate the contribution of individual components within the FCN4Flare architecture. By sequentially removing key components, we assess their impact overall performance of the model focusing on precision and IoU metrics.

The ablation study involves training variants of the FCN4Flare model, each with a specific component removed. Each modified version of the model is trained under identical conditions to the full FCN4Flare model to ensure comparability of the results. Performance metrics focusing on precision and IoU are recorded in Table <ref>. The table displays the metrics scores for each model alongside the percentage change from the full FCN4Flare model, which serves as the baseline. 



The ablation study provides clear evidence of the importance of NaN Mask and Mask Dice loss components in the FCN4Flare architecture. Each component significantly contributes to model's overall performance, and their removal leads to pronounced decreases in both precision and IoU.



§ APPLICATION



To advance our scientific investigation, we deploy FCN4Flare to detect flares on the Kepler-LAMOST dataset, which is generated by cross-matching the Kepler <cit.> DR25 long cadence catalog and LAMOST <cit.> DR9 low-resolution catalog with a tolerance of 3^''. This dataset encompasses a total of 1,282,142 light curves, representing 87,549 distinct stars.



It is imperative to acknowledge that the data partitioning approach we adopt, utilizing the complete Kepler flare catalog <cit.> for training and a subset of Kepler data for prediction, may be deemed unconventional in machine learning due to concerns of potential data leakage. Nevertheless, in the context of our study, this concern is mitigated. The primary objective of our research is the development of a novel flare detection methodology and the identification of candidates for flare events, rather than a direct comparison of various methods based on specific performance indicators.

In light of these considerations, we have included the outcomes of our model's predictions, which can be invaluable for further scientific research and the exploration of flare phenomena.



 §.§ Flare Catalog




Following the application of our model to the set of light curves, we obtained probabilities assigned to each time step, signifying the likelihood of it belonging to a flare event. To ensure a higher level of reliability and to minimize the risk of false positive events, we implemented specific screening rules on the prediction results. The rigorous screening process involved the following criteria:


    
  (1) A minimum of three consecutive points with probabilities above 0.5 was required to qualify as a flare event.
    
  (2) For these points, the duration from start to peak time must be shorter than from peak to end time, adhering to flares' typical “rapid rise and slow fall” pattern.
    
  (3) Flux levels before and after the event must be close to each other, indicating a clear deviation from background levels.
    
  (4) In the case of a target exhibiting multiple quarters, flares had to occur in at least two quarters to be considered valid.


These screening rules were systematically applied during a post-processing step and significantly contributed to enhancing the overall accuracy of our predictions. By carefully filtering the candidate flare events based on these criteria, we could mitigate false detections and ensure the reliability of our flare detection methodology.

In our investigation, we successfully identified a total of 30,285 distinct flare events occurring in 1426 stars. For each of these events, Table <ref> furnishes crucial details, including the start time, end time, and energy estimation.

To estimate the energy associated with each flare event, we adopted a well-established approach based on integrating normalized flux, as described in <cit.> and <cit.>. Initially, we normalized the PDC flux by dividing it by the median value. Next, we employed linear interpolation to determine the start and end times of the flare, using the stellar background flux during the flare as reference. These measures helped create a suitable baseline for the estimation process. Subsequently, the flare energy is computed by this equation:

    E_flare = 4 π R_*^2 σ_sb T^4 ∫ F_flare dt

where R_* and T represent the stellar radius and effective temperature, respectively, both of which were sourced from the Kepler Catalog. The symbol σ_sb denotes the Stefan-Boltzmann constant, while F_flare corresponds to the difference between the normalized flux and the interpolated background flux. The energy estimation is expressed in erg units.



Figure <ref> illustrates the energy estimation and the frequency distribution of flares, which intriguingly conform to a power-law relation, expressed as dN/dE ∝ E^α. We apply an ordinary least-square to fit the solid line, whose slope is the α index. Drawing from <cit.>, it is noteworthy that the exponent α was found to be approximately 2.

Except for the parameters of the flare events, the parameters of the 1426 flare stars are also listed in Table <ref>, in which the spectral type is taken from the LAMOST catalog, Teff, log g, [Fe/H] and radius are all taken from the Kepler catalog, and the rotation period is estimated by the Lomb–Scargle periodogram <cit.> algorithm, T_obs represents the observation time of Kepler, T_fl/T_obs represents the proportion of the flare duration, E_max represents the maximum detected flare energy, N_flare represents the number of detected flare events, and L_fl/L_bol represents the flare activity, which is calculated as the ratio of the total flare energy to the bolometric luminosity (i.e., ∑ E_flare / ∫ L_bol dt = L_flare / L_bol) <cit.>.



Subsequently, we computed the incidence of flare stars stratified by their spectral types, and the findings are thoughtfully presented in Table <ref>. To further encompass the intrinsic properties of stars, we deliberately differentiated between M-type dwarfs (dM) and M-type giants (gM). Intriguingly, our analysis revealed a pronouncedly elevated incidence of flare stars among dM, while the occurrence of flare events in gM stars was found to be nearly negligible. This intriguing observation seamlessly aligns with our current comprehension of the underlying flare mechanism inherent to M-type stars.



 §.§ Flare activity distribution




In the quest to elucidate flare activity across varied stellar spectral types, two quintessential diagrams have been proffered, delineated in Figure <ref>. The first panel presents a histogram detailing the flare activity distribution of different stellar types. Subsequently, the secondary illustration, rendered as a density plot, provides a normalized perspective, artfully portraying the probabilistic density of flare activity within each stellar type. This normalization ensures an equivalency in the integral area beneath each curve, thereby facilitating an unambiguous juxtaposition amongst the subclasses. It is noteworthy that the abscissa of both depictions adopts a logarithmic scale.

One striking observation emerges: the zenith of flare activity recedes progressively from M-type stars to F-type counterparts. Intriguingly, the flare activity linked with A-type stars surpasses that of both F and G spectral categories. Several postulations arise from this conundrum. It could be postulated that the cohort of A-type stars, wherein flares are detected, is minuscule, thereby deeming it statistically inconsequential, or perhaps an observational predilection is at play. Alternatively, the underpinnings governing the flare processes in Type A stars might diverge from other types, a phenomenon that remains shrouded in mystery.




 §.§ Chromospheric Dynamics and Stellar Eruptions




The intricate interplay between chromospheric dynamics and stellar flares has been a subject of sustained academic enquiry. Among the myriad indicators of such dynamism, the H_α equivalent width (EW) and the S Index emerge as particularly salient. Leveraging the LAMOST spectral data, we have discerned the H_α EW and the S Index for our stellar catalog, subsequently elucidating their correlation with stellar flare proclivities. These findings are delineated in Figure <ref>. Gleanings from the figure encompass:


    
  * Interplay of H_α and Flare: An overwhelming majority of the stellar types, with an especial emphasis on M-type stars, exhibit a pronounced trend wherein an uptick in the H_α EW directly corresponds to an escalation in flare activity. This observation is congruent with antecedent investigations underscoring the nexus between elevated chromospheric dynamism and augmented flare manifestations, particularly within this stellar classification.
    
  * Correlation between S Index and Flare: Analogous to the patterns discerned in H_α EW, the S Index also evinces a discernible positive association with flare activities. Notably, the symbiotic relationship for G-type and F-type stars is more palpable in the S Index than in the H_α EW.
    
  * Nuances between H_α and Max Flare Energies: The relationship between H_α EW and the max flare energy presents a nuanced scenario, challenging extant paradigms. It appears that elevated H_α EW metrics do not unerringly predicate the magnitude of flares, hinting at the presence of latent variables modulating flare energies. 
    
  * Interrelation of S Index and Peak Flare Energies: Mirroring the dynamics between S Index and stellar eruptions, the correlation between the S Index and the apogee of flare energies appears elusive, suggesting a potentially multifaceted interrelation between chromospheric dynamism and peak flare intensities.


It is imperative to underscore that the paucity of flare-star specimens in our compilation, especially within the Type A stars, could engender observational biases in discerning correlations.






 §.§ Habitable Zone Exoplanets

Planetary bodies represent the sole confirmed repositories for the phenomenon of life as we know it. The irradiative influence from a celestial body's central luminary plays a pivotal role in determining the habitability of any given planetary environment. Consequently, the scientific examination of stellar flares assumes paramount importance in the discrimination and selection of potentially life-bearing planets.



The NASA Kepler mission has facilitated the discovery of an extensive array of planetary candidates, many of which have been corroborated by subsequent observational analysis <cit.>. We have executed a cross-match between our flare star catalog and the cumulative table of Kepler Objects of Interest (KOI) <cit.>. The resulting data is delineated in Table <ref>. Within the table, `kepid' refers to the identification, `kepoi_name' designates a numerical value utilized to identify and monitor a KOI, `kepler_name' is the nomenclature assigned to the specific planet, and `subclass' denotes the spectral classification of the star as obtained from LAMOST. Additionally, `radius', `teff', `feh', and `logg' symbolize stellar parameters extracted from the Kepler catalog. The table is methodically organized in accordance with flare activity.

Numerous investigations have focused on the intricate subject of habitable zone exoplanets, contributing to a comprehensive understanding of these celestial bodies <cit.>. In this work, we execute a meticulous cross-reference between the previous table and the Catalog of Habitable Zone Exoplanets, as assembled by <cit.>. Remarkably, one stellar object situated in the inner habitable zone has emerged from this analysis, and it is distinctively highlighted in Table <ref>. This stellar entity, identified by the Kepler ID 5938970, manifests indications of pronounced magnetic activity.

In the present manuscript, we delineate an exhaustive investigation of this particular star, detailing the salient characteristics such as flare events, flare energy, flare activity indicators, and other pertinent parameters. We harbor the aspiration that our scholarly contribution will significantly augment and influence future research endeavors directed towards the understanding of the habitability potential of this specific stellar object.





§ CONCLUSION


In this paper, we have presented FCN4Flare, a novel deep learning approach for detecting stellar flares in light curves. By leveraging fully convolutional networks, our model achieves end-to-end training and precise point-to-point flare prediction regardless of input length. This overcomes limitations of prior methods relying on fixed-length input segments. 

Through components like NaN Mask, dilated convolutions, and MaskDice loss, FCN4Flare handles challenges unique to flare detection, including missing data, feature resolution, and class imbalance. Our experimental results demonstrate a significant improvement in flare detection performance compared to previous models, with a Dice coefficient of 0.64 versus 0.21 for the state-of-the-art.

Applying FCN4Flare to Kepler-LAMOST data, we have compiled a catalog of 30,285 flare events across 1426 stars. By estimating flare energies and cross-matching with exoplanets, our analysis provides new insights into stellar magnetic activity and planetary habitability. One noteworthy finding is the identification of pronounced flaring behavior on an M-dwarf hosting a planet within the optimistic habitable zone. 

Overall, this work makes multiple contributions - an innovative deep learning architecture for flare detection, a sizable catalog of high-quality flare predictions, and new perspectives on connections between stellar flares and exoplanetary environments. FCN4Flare demonstrates the power of deep learning to unlock new scientific discoveries in astronomical time-series data. 

Key directions for future work include enhancing the model, expanding the stellar sample analyzed, and combining our flare predictions with other astronomical data to answer open questions about stellar physics and planetary system evolution. By overcoming long-standing challenges in flare detection, this research opens new avenues for mining the wealth of photometric data from missions like TESS. The transformative potential of deep learning in astronomy is only beginning to be realized.




§ ACKNOWLEDGEMENTS

This work is supported by the National Science Foundation of China (Nos. U1931209, 12261141689) and the National Key R&D Program of China (No. 2019YFA0405502).  Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences.  
Funding for the project has been provided by the National Development and Reform Commission.  
LAMOST is operated and managed by the National Astronomical Observatories, the Chinese Academy of Sciences.


 



 Paper Title: The OATMEAL Survey. I. Low Stellar Obliquity in the Transiting Brown Dwarf System GPX-1.






















































Steven Giacalone
giacalone@astro.caltech.edu

0000-0002-8965-3969]Steven Giacalone
NSF Astronomy and Astrophysics Postdoctoral Fellow
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

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

0000-0002-9849-5886]J. J. Zanazzi
51 Pegasi b Postdoctoral Fellow
Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA

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

0000-0001-8189-0233]Courtney D. Dressing
Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA

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

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

0000-0001-6416-1274]Theron W. Carmichael
NSF Ascend Postdoctoral Fellow
Institute for Astronomy, University of Hawai`i, 2680 Woodlawn Drive, Honolulu, HI, 96822, USA

0000-0002-0701-4005]Noah Vowell
Center for Astrophysics  Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA
Center for Data Intensive and Time Domain Astronomy, Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824,
USA

0000-0002-3239-5989]Aurora Kesseli
IPAC, Mail Code 100-22, Caltech, 1200 E. California Boulevard, Pasadena, CA 91125, USA

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

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

0009-0008-9808-0411]Max Brodheim
W. M. Keck Observatory, Waimea, HI 96743, USA

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

0000-0003-3504-5316]Benjamin J. Fulton
NASA Exoplanet Science Institute/Caltech-IPAC, California Institute of Technology, Pasadena, CA
91125, 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, Kamuela, HI 96743, USA

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

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


Caltech Optical Observatories, California Institute of Technology, Pasadena, CA 91125, USA

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

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

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

0000-0003-0967-2893]Erik A. Petigura
Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, 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, CA 94720, USA


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

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
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI 96743, USA

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

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














§ ABSTRACT 

We introduce the OATMEAL survey, an effort to measure the obliquities of stars with transiting brown dwarf companions. We observed a transit of the close-in (P_ orb = 1.74 days) brown dwarf GPX-1 b using the Keck Planet Finder (KPF) spectrograph to measure the sky-projected angle between its orbital axis and the spin axis of its early F-type host star (λ). We measured λ = 6.88 ± 1.72 ^∘ (with additional unquantified systematic uncertainty), suggesting an orbit that is prograde and well aligned with the stellar equator. Hot Jupiters around early F stars are frequently found to have highly misaligned orbits, with polar and retrograde orbits being commonplace. It has been theorized that these misalignments stem from dynamical interactions, such as von Zeipel-Kozai-Lidov cycles, and are retained over long timescales due to weak tidal dissipation in stars with radiative envelopes. By comparing GPX-1 to similar systems under the frameworks of different tidal evolution theories, we argued that the rate of tidal dissipation is too slow to have re-aligned the system. This suggests that GPX-1 may have arrived at its close-in orbit via coplanar high-eccentricity migration or migration through an aligned protoplanetary disk. Our result for GPX-1 is the fifth measurement of the obliquity of a star with a transiting brown dwarf. By enlarging the number of such measurements and comparing them with hot Jupiter systems, we will more clearly discern the differences between the mechanisms that dictate the formation and evolution of both classes of objects.




























§ INTRODUCTION
 

Historically, giant planets and brown dwarfs have been delineated based on mass, with a dividing line at the deuterium-burning limit (13   M_ J) <cit.>. However, some argue that it is more sensible to classify these objects based on their formation mechanisms and subsequent orbital evolution. While different formation mechanisms likely correlate with mass, there is currently no theoretical prediction or empirical evidence that 13   M_ J has any special significance in the process. For instance, formation via core accretion <cit.>, which is thought to create most Jupiter-size planets, is possibly capable of forming objects 10–20 times more massive than Jupiter. Formation via gravitational instability <cit.>, which is believed to form very massive giant planets and low-mass stars efficiently, may be able to form objects as low-mass as 2   M_ J <cit.>. Post-formation evolution mechanisms, such as migration through the protoplanetary disk and high-eccentricity migration, are also believed to depend on companion mass <cit.>. We can infer these histories by statistically characterizing the occurrence rates and orbital properties of giant planets and brown dwarfs.

The first studies of giant planet and brown dwarf demographics came from radial velocity surveys of Sun-like stars. Early analyses of their occurrence rates found a notable dearth of “brown-dwarf-mass” (hereafter 13 - 80   M_ J, for the sake of convenience) objects compared to “planetary-mass” (hereafter < 13   M_ J) and “stellar-mass” (hereafter > 80   M_ J) objects at orbital separations less than ∼ 1 AU <cit.>. This dearth, referred to by some as the brown dwarf “desert,” suggests that brown dwarfs are either unable to form close to Sun-like stars as efficiently as giant planets, or that they are rarely transported inward from their birthplaces. <cit.> examined a sample of brown dwarfs detected via transit and radial velocity to explore the orbital properties of those in the desert, finding that those with masses below 42.5   M_ J have orbital eccentricities resembling giant planets and those with masses above about 42.5   M_ J have orbital eccentricities resembling stellar companions, potentially suggesting that this mass divides planet-like and star-like formations. <cit.> used a similar sample to inspect how the occurrences of giant planets and brown dwarfs depend on host star metallicity, finding evidence that objects with masses < 4   M_ J tend to orbit metal-rich stars and more massive objects tend to orbit stars with a wide range of metallicities, hinting that the lower-mass objects preferentially form via core accretion.

Direct imaging surveys have also provided insight into the differences between giant planet and brown dwarf formation. For instance, <cit.> showed that for orbital separations of 10-100 AU, the occurrence rate of brown dwarfs is lower than that of giant planets by a factor of ∼ 10, suggesting that the efficiencies of brown dwarf and giant planet formation mechanisms differ dramatically. <cit.> showed that wide-separation brown dwarfs tend to have orbits that are more eccentric and more misaligned with the equators of their host stars compared to wide-separation giant planets.[Although we note that the analysis of orbital eccentricities has been shown to be highly sensitive to the choice of priors <cit.>.] <cit.> showed through measurements of the stellar spin axes, inclinations of the orbit normals, and brown dwarf spin axes of multiple systems that both the spins and orbits of wide-separation brown dwarfs are often misaligned with the spins of their stars. In general, these observations support the notion that wide-separation brown dwarfs frequently form via fragmentation of the protoplanetary disk or molecular cloud core <cit.>. 

Analogous experiments can be carried out for transiting giant planets and brown dwarfs in close-in orbits. Because these close-in objects are thought to form farther from their stars than the locations in which they are observed today, their occurrence rates and orbits are likely linked more closely to their orbital evolutions than their formation <cit.>. Several calculations of the hot Jupiter occurrence rate around Sun-like stars have been conducted using data from the Kepler <cit.> and K2 <cit.> missions, revealing a frequency of ∼ 1 % <cit.>. More recently, TESS <cit.> has enabled investigations of hot Jupiter occurrence rate as a function of stellar mass. These studies have hinted that these planets may be less common around A-type and M-type stars than around G-type stars <cit.>, although the underlying cause of this trend is unknown. Similarly reliable occurrence rate calculations have yet to be performed for transiting brown dwarfs because relatively few have been found.



Stellar obliquity measurements that leverage the Rossiter-McLaughlin effect <cit.> and the Doppler shadow technique <cit.> have also provided a window into the dynamical histories of hot Jupiters. <cit.> first identified a trend in the obliquities of stars with hot Jupiters, noting that stars with effective temperatures above the Kraft break (T_ eff≳ 6250 K) tend to have spin axes that are misaligned with the orbital axes of their planets, whereas cooler stars generally exhibit spin-orbit alignment. This finding led to two key hypotheses: (1) the processes that transport hot Jupiters from their formation locations to their current close-in orbits often leave them in orbits that are misaligned with the equators of their host stars, and (2) the ability of a star to tidally realign with the orbit of a hot Jupiter depends on T_ eff. In relation to the former, several mechanisms have been proposed that are capable of tilting the orbit of a hot Jupiter, although the data seems most consistent with scattering or secular interactions with a massive outer companion followed by high-eccentricity migration <cit.>. In relation to the latter, it is believed that stars with radiative envelopes (T_ eff≳ 6,250 K) are far less efficient at damping tides excited by their close-in planets than cool stars, causing a drastic discrepancy in the tidal realignment timescale and creating the dichotomy in stellar obliquities observed today.

Stellar obliquity measurements for transiting brown dwarf systems are much less common than those for transiting planet systems, largely due to the rarity of transiting brown dwarfs. However, this limitation is being lifted thanks to TESS, which has significantly increased the number of known transiting brown dwarfs around relatively bright (V < 13) stars <cit.>. These discoveries provide the exciting opportunity to investigate the stellar obliquity distribution of stars with close-orbiting brown dwarfs and compare it to that of hot Jupiters. If close-in brown dwarfs tend to have orbits that are aligned with the equators of their host stars – especially when the stars are hot or the brown dwarfs orbit their stars at distances large enough for tides to be negligible – it would suggest that brown dwarfs spiral inwards via disk-driven migration <cit.> or co-planar high-eccentricity migration <cit.>. If these systems instead have a wide range of stellar obliquities, it would point to non-co-planar forms of secular excitation followed by high-eccentricity migration <cit.> or dynamical capture <cit.> as the primary migration mechanisms.

As of today, obliquities have been measured for only four of the roughly fifty stars known to have transiting brown dwarfs: CoRoT-3 <cit.>, KELT-1 <cit.>, WASP-30 <cit.>, and HATS-70 <cit.>. Of these four, three orbit stars hotter than the Kraft break (the exception being WASP-30), and they are all consistent with good alignment (i.e., all have sky-projected stellar obliquities ≲ 45^∘). These three results hint that transiting brown dwarf systems may generally have lower stellar obliquities than transiting hot Jupiter systems, but the sample size is too small to reliably probe the underlying obliquity distribution. Here, we introduce the OATMEAL (Orbital Architectures of Transiting Massive Exoplanets And Low-mass stars)  survey, a coordinated effort to increase the number of transiting brown dwarf systems with stellar obliquity measurements. We report the first result of this survey: the stellar obliquity of the early F-type star GPX-1, which hosts a 19.7 ± 1.6   M_ J brown dwarf with a 1.74 day orbital period <cit.>.

This paper is structured as follows. In Section <ref>, we describe our observations collected with the Keck Planet Finder (KPF) spectrograph and the related data reduction. In Section <ref>, we outline our analysis of the data and report the stellar obliquity. In Section <ref>, we examine the possibility that the system could have undergone spin-orbit realignment and discuss plausible formation and migration scenario for the brown dwarf. Lastly, in Section <ref>, we provide concluding remarks.



§ OBSERVATIONS
 

We observed a transit of GPX-1 b on 25 September 2023 UT using the KPF spectrograph on the 10-meter Keck-I telescope <cit.>. We took 29 500-second exposures between 10:28 and 15:10 UT, achieving signal-to-noise ratios (SNRs) between 80 and 90 within 500–600 nm after stacking the three slices of the science fiber. These observations spanned of the brown dwarf, which took place between 11:05 UT and 13:10 UT, and approximately 2 hours of post-transit baseline. Every 60 minutes, we acquired a calibration exposure using a Fabry–Pérot etalon in order to track and correct for intranight instrumental drift. The data was reduced using the publicly available KPF pipeline,[<https://github.com/Keck-DataReductionPipelines/KPF-Pipeline>] which produces wavelength-calibrated and barycentric-corrected spectra for each order on the CCD. 

We note that there is a gap in the data between 12:18 and 12:52 UT, which was due to a software malfunction related to the tip-tilt mirror used for guiding. However, as we show below, this gap in the data did not significantly impact our ability to measure the stellar obliquity of the system.



§ ANALYSIS AND RESULTS
 

The goal of our observations were to measure the sky-projected obliquity (λ) of the star GPX-1, or the sky-projected angle between its spin axis and the orbital axis of its transiting brown dwarf companion. For each spectrum, we calculated the stellar line profile using the least-squares deconvolution (LSD) method <cit.>. Calculation of the profile via LSD required a binary mask of stellar lines, which we obtained from version 3 of the Vienna Atomic Line Database (VALD3; ) for a star with properties matching those of GPX-1 reported in <cit.>. Next, we narrowed down this line list by inspecting the collected spectra for lines that are deep enough to see visually, are relatively free from blends arising from rotational broadening, and are uncontaminated by strong telluric features (e.g., the molecular oxygen A- and B-bands). These 184 lines provided the best characterization of the stellar line profile.





We examined the Doppler Shadow of GPX-1 using the method described in <cit.>. In short, we subtracted the average out-of-transit line profile from the in-transit line profile. Figure <ref> shows the line profile residuals as a function of time and stellar-centric velocity. The Doppler shadow of GPX-1 is the blue diagonal feature in the lower half of the diagram. The flux deficit in the line profile moves from the blueshifted to the redshifted limb of the host star. This is expected if the BD is on an aligned orbit around the host star. The star is assumed to have solid-body rotation, while GPX-1 is assumed to have a circular orbit <cit.> with elements fixed at the median values reported by <cit.>. The only exceptions are the impact parameter and sky-projected obliquity of the BD which were allowed to vary freely in our Doppler shadow model. In the model, we synthesized line profiles by pixelating the host star. Each pixel is assigned a line-of-sight rotational velocity, limb darkening, and macroturbulence, all of which are nuisance parameters that are marginalized in posterior sampling. Our posterior sampling is performed with nested sampling code using Dynesty <cit.> using the default sampling settings. We found λ = 6.88 ± 1.72 ^∘ and v sini_⋆ = 50.77^+2.27_-2.62 km s^-1, suggesting a prograde orbit that is well-aligned with the stellar equator. We note that the errorbar on λ = 6.88 ± 1.72 ^∘ is likely underestimated as it does not include any systematic errors associated with various above-mentioned assumptions, and due to fixing the Keplerian orbital elements. Lastly, we stress that because we do not know the inclination of the stellar spin axis, the true stellar obliquity may be higher. GPX-1 likely has a fully radiative exterior and a spotless surface, making it challenging to determine the rotation period (and therefore inclination) of the star. Indeed, we found no evidence of starspot modulation in the TESS data.



§ DISCUSSION
 

We display GPX-1 in the context of other transiting hot Jupiter and brown dwarf systems with λ measurements in the top panel of Figure <ref>. Like other stars with massive hot Jupiters and close-in brown dwarfs for which λ is known, GPX-1 is consistent with spin-orbit alignment. This measurement contrasts with those of hot-Jupiter hosts with T_ eff≥ 7000 K, for which the lowest measured λ is 21^∘_-4^+5 (HAT-P-69; ).

The observation of good alignment raises the question of whether GPX-1 b arrived at its close-in orbit on an already-aligned orbit or if the system was able to achieve spin-orbit realignment on a shorter timescale than systems with hot Jupiters. On one hand, GPX-1 b is more massive (m = 19.7 ± 1.6   M_ J; ) than most hot Jupiters, which should correspond to a faster tidal realignment. On the other hand, GPX-1 is hot enough to have a fully radiative outer envelope and is relatively young (Age = 0.27^+0.09_-0.15 Gyr; ), meaning that tidal damping in the star may not have been efficient enough to realign the system to its current orientation. Here, we consider whether or not enough time has transpired for the GPX-1 to have realigned tidally.





A common method of determining if a system has previously undergone significant tidal evolution is by assessing whether the system has reached tidal equilibrium, which is characterized by a circular orbit, spin-orbit alignment, and synchronous rotation <cit.>. In general, a tidal equilibrium state is achieved if the total angular momentum of the system L_ tot is larger than some critical value L_ crit. Following <cit.> and assuming i_⋆ = 90^∘, we calculated L_ tot/L_ crit = 1.032 ± 0.024 for GPX-1. This represents the minimum possible ratio, as lower values of i_⋆ correspond to faster stellar rotation speeds and therefore higher values of L_ tot. Assuming an isotropic distribution of possible stellar spin vectors, we found the possible system orientations divided roughly evenly between “Darwin stable” (at higher i_⋆; corresponding to a system where the orbit of the companion no longer evolves tidally) and “Darwin unstable” (at lower i_⋆; corresponding to a system where the orbit of the companion will eventually decay to within the Roche limit of the star). By comparison, we calculated minimum values of L_ tot/L_ crit = 1.198 ± 0.034 for CoRoT-3 <cit.>, L_ tot/L_ crit = 0.995 ± 0.011 for KELT-1 <cit.>, L_ tot/L_ crit = 1.51 ± 0.09 for WASP-30 <cit.>, and L_ tot/L_ crit = 0.951 ± 0.014 for HATS-70 <cit.>. Taken at face value, these numbers suggest that many close-in brown dwarfs have reached, or are close to reaching, a state of tidal equilibrium <cit.>. However, we also see that these ratios do not correlate with measured spin-orbit angles. For example, CoRoT-3 should be in tidal equilibrium but was measured to have a relatively high degree of misalignment with λ = 38^∘^+10_-22, indicating that tides have not yet fully realigned the spin of the star with the orbit of its companion.[Although we note that this measurement has relatively large error bars and that more precise observations could plausibly yield a much smaller λ.] Additionally, over half of hot stars with known misaligned (λ > 30^∘) hot Jupiters also have L_ tot/L_ crit > 1. This tidal equilibrium metric therefore does not appear to be a reliable predictor of spin-orbit realignment due to tidal damping in hot stars. 

Alternatively, we can estimate the timescale of spin-orbit realignment predicted by different tidal damping frameworks. Multiple theories have been proposed to explain the distribution of stellar obliquities for close-in exoplanet systems (we refer the reader to Section 4 of  for a summary); we explore a few of these theories here.

Firstly, we acknowledge that previous studies of spin-orbit realignment have often utilized classical equilibrium-tide theory <cit.>. However, as is discussed and demonstrated in <cit.>, equilibrium tides alone cannot reproduce the observed distribution of close-in companions around hot stars. One major issue with equilibrium-tide theory is that it predicts the obliquity realignment timescale to be comparable to the orbital decay timescale of the close-in companion. If this were the case, most hot Jupiters around cool stars that have tidally realigned would have also been engulfed, which is inconsistent with the data. In addition, realignment via equilibrium tides is typically thought to be dependent on interactions between the orbit of the close-in companion and the convective envelope of the star <cit.>, which hot stars like GPX-1 do not have. We therefore turn to other frameworks for estimating the realignment timescale.








<cit.> calculated the synchronization times of F and A binary stars, assuming internal gravity waves tidally excited at the base of the star's envelope dissipate by radiative diffusion. <cit.> used this synchronization time as a proxy for the obliquity realignment timescale for substellar companions:




    τ_ RA∝ Q_⋆(m/M_⋆)^-2(1 + m/M_⋆)^-5/6( a/R_⋆)^17/2.

Here, M_⋆ is the mass of the host star, R_⋆ is the radius of the host star, m is the mass of the close-in companion, a is the semi-major axis, and Q_⋆ is the stellar tidal quality factor. An honest calculation of Q_⋆ is complex (see e.g. ). Instead, we compare the relative value of τ_ RA for stars with T_ eff > 6250 K, assuming a common value of Q_⋆, in order to search for trends with λ and put GPX-1 into context with similar systems. The results of this calculation are shown in Figure <ref>.



According to this relative timescale, GPX-1 should realign quicker than most other hot Jupiters and brown dwarfs. The brown dwarf with a smaller τ_ RA than GPX-1 is KELT-1, which <cit.> argued has likely synchronized with the orbit of the close-in companion due to tidal interactions. This argument was made by comparing the approximate equatorial velocity of the star, calculated using v sini_⋆ and assuming i_⋆ = 90^∘, with the orbital velocity of the companion. <cit.> used the same reasoning to argue that HATS-70 is not synchronized with the orbit of its brown dwarf companion and therefore has not undergone significant tidal evolution. We can say the same for GPX-1 and its companion, which have an equatorial velocity of 50.77^+2.27_-2.62 km s^-1 (assuming i_⋆ = 90^∘) and an orbital velocity of 211 ± 2 km s^-1, respectively.[Of course, one could satisfy the synchronization condition with a smaller value of i_⋆, but a smaller i_⋆ would also result in a higher 3D stellar obliquity. There is therefore no situation in which both synchronization and spin-orbit alignment can be achieved.] Indeed, GPX-1 and HATS-70 have roughly equal values of τ_ RA. This suggests that either (1) the GPX-1 and HATS-70 systems have not yet undergone significant tidal evolution, or (2) tidal realignment occurs significantly faster than tidal synchronization in hot stars.


Another popular driver of tidal evolution is resonance locking, the process by which the orbit of the close-in companion couples with the gravity mode of the star <cit.>. Recently, <cit.> showed that resonance locking can explain the distribution of stellar obliquities as a function of stellar T_ eff. In short, this mechanism can explain the low obliquities of cool stars and the high obliquities of hot stars because g-mode frequencies increase substantially over the main sequence phase in stars with radiative cores, whereas g-mode frequencies remain relatively constant in stars with convective cores. For a system in resonance lock, this frequency evolution drives strong tidal evolution. Under this framework, the evolution of the 3D obliquity (ψ) goes like

    dψ/dt≃( J_ orb/J_⋆ + cosψ) 1/3 t_ evsinψ

where t_ ev is the timescale of g-mode frequency evolution, ,  for F and A stars, , and Ω_⋆ is the stellar spin frequency. Here, we estimate t_ ev for the star as a function of age by interpolating the data in Figure 6 of <cit.> and we assume ψ≈λ (i.e., i_⋆≈ 90^∘). The evolution of λ for four different initial misaligned orientations is shown in Figure <ref>. The model predicts that for an initial misaligned orbit, very little λ evolution would have occurred in the GPX-1 system thus far. In other words, realignment via resonance locking cannot explain the low obliquity of GPX-1 that we observe today.




These interpretations are compatible with a primordially aligned system produced by inward migration via disk migration <cit.> or co-planar high-eccentricity migration <cit.>. We note that the latter preferentially occurs in the presence of a more massive outer companion responsible for the excitation of the orbital eccentricity of the inner companion. This would necessitate the existence of an additional undetected brown dwarf or star in the outer system, which may be detectable in upcoming Gaia data releases <cit.>. In either case, these migration mechanisms require the brown dwarf to have formed near the midplane of the protoplanetary disk, pointing to core accretion or disk fragmentation as the most likely formation mechanisms for GPX-1 b.



§ CONCLUSION
 

We reported a sky-projected obliquity of λ = 6.88 ± 1.72 ^∘ for the early F-type star GPX-1, which hosts a close-in transiting brown dwarf. This measurement suggests that the orbit of the brown dwarf is prograde and that the spin of the star and the orbit of the planet are well aligned. This orientation is consistent with other known transiting brown dwarf systems, but is unlike the many hot Jupiters orbiting hot stars that frequently have polar and retrograde orbits. We argued that, due to the relatively young age of the system (Age = 0.27^+0.09_-0.15 Gyr; ) and the inefficiency of tidal damping in hot stars, the brown dwarf is most likely primordially aligned. We encourage the measurement of stellar obliquity for more transiting brown dwarf systems, which would allow for a statistically robust characterization of the underlying obliquity distribution and provide novel clues for the origins of elusive close-in brown dwarfs.



We thank Luke Bouma for his useful feedback and suggestions.

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. 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.

We gratefully acknowledge the efforts and dedication of the Keck Observatory staff for support of KPF and remote observing.

SG is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-2303922.

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.


 

aasjournal



 Paper Title: Non-radial oscillations mimicking a brown dwarf orbiting the cluster giant NGC 4349 No. 127.

 


   

   

   

    Landessternwarte, Zentrum für Astronomie der Universität Heidelberg, Königstuhl 12, 69117    Heidelberg, Germany 

   dane.spaeth@lsw.uni-heidelberg.de
             

   



 
  
  
   Several evolved stars have been found to exhibit long-period radial velocity variations that cannot be explained by planetary or brown dwarf companions. Non-radial oscillations caused by oscillatory convective modes have been put forth as an alternative explanation, but no modeling attempt has yet been undertaken.
  
   We provide a model of a non-radial oscillation, aiming to explain the observed variations of the cluster giant NGC 4349 No. 127. The star was previously reported to host a brown dwarf companion, but whose existence was later refuted in the literature.
  
   We reanalyzed 58 archival HARPS spectra of the intermediate-mass giant NGC 4349 No. 127. We reduced the spectra using the SERVAL and RACCOON pipelines, acquiring additional activity indicators. We searched for periodicity in the indicators and correlations between the indicators and radial velocities. We further present a simulation code able to produce synthetic HARPS spectra, incorporating the effect of non-radial oscillations, and compare the simulated results to the observed variations. We discuss the possibility that non-radial oscillations cause the observed variations. 
  
    We find a positive correlation between chromatic index and radial velocity, along with closed-loop Lissajous-like correlations between radial velocity and each of the spectral line shape indicators (full width at half maximum, and contrast of the cross-correlation function and differential line width). Simulations of a low-amplitude, retrograde, dipole (l=1, m=1), non-radial oscillation can reproduce the observed behavior and explain the observables. Photometric variations below the detection threshold of the available ASAS-3 photometry are predicted. The oscillation and stellar parameters are largely in agreement with the prediction of oscillatory convective modes.
  
   The periodic variations of the radial velocities and activity indicators, along with the respective phase shifts, measured for the intermediate-mass cluster giant NGC 4349 No. 127, can be explained by a non-radial oscillation.

   

   Non-radial oscillations mimicking a brown dwarf orbiting the cluster giant NGC 4349 No. 127
    Dane SpaethFellow of the International Max Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD)
          Sabine Reffert
          Emily L. Hunt
          Adrian Kaminski
          Andreas Quirrenbach
    Received 28 March 2024 / 
   Accepted 09 July 2024
==========================================================================================================================================================================================================================================================






§ INTRODUCTION

With the advent of large-scale transit surveys such as Kepler <cit.>, K2 <cit.>, and TESS <cit.>, the number of known exoplanets has increased drastically to over 5600 confirmed planets to date.[<https://exoplanetarchive.ipac.caltech.edu/> (June 2024)] Despite this immense wealth of newly discovered planets, less than 200 of these have been found orbiting evolved stars.[<https://www.lsw.uni-heidelberg.de/users/sreffert/giantplanets/giantplanets.php>] Due to intrinsic biases of the transit surveys, the majority of planets orbiting giant stars have been found using the radial velocity (RV) method. 

Targeting giant stars in RV surveys offers important additions to surveys targeting main-sequence stars. Specifically, it allows the detection of planets around the evolved counterparts of stars more massive than about 1.5 M_  <cit.>, which are increasingly inaccessible to RV measurements during their main-sequence phase, due to high effective temperatures and rotation rates <cit.>. Furthermore, analyzing the planet population in later evolutionary stages allows  studies of the effect of stellar evolution on planetary systems. Therefore, detecting a statistically meaningful sample of planets around giant stars is crucial in order to capture a full picture of the general planet population. For simplicity, we also refer to low-mass brown dwarfs (M ≲ 30 M_Jup) as planets within this work.   

However, detecting planets orbiting giants using the RV method has its own challenges. First, most giant stars are known to undergo short-term, stochastically driven p-mode oscillations, called solar-like oscillations. For giants, these typically occur on timescales of hours to days and lead to RV jitter on the order of ∼ 10 m s^-1 to ∼ 20 m s^-1, even for giant stars considered to be relatively stable <cit.>. This intrinsic noise can often be overcome with sufficient statistics, but limits the detectable orbital companions to those with rather high masses (typically Jovian or higher). 

Less understood, and therefore more challenging, is the recent discovery of several giant stars showing long-term, periodic RV variations, some of which were already attributed to planets or brown dwarfs, yet were later shown to be incompatible with orbital companions. These include γ Dra <cit.>, Aldebaran <cit.>, ϵ Cyg <cit.>, 42 Dra <cit.>, Sanders 364[SIMBAD identifier: BD+12 1917] <cit.>, 41 Lyn, 14 And (both ), and the cluster giants NGC 2423 No. 3, NGC 2345 No. 50, NGC 3532 No. 670, and NGC 4349 No. 127[SIMBAD identifiers: NGC 2423 3, NGC 2345 50, HD 96789, and NGC 4349 127] <cit.>. <cit.> show that doubtful planet detections around giant stars mostly accumulate at periods between 300 d and 800 d and around stars with radii R > 21 R_, pointing at a common origin of these RV variations.

<cit.> present the planet occurrence rate around evolved stars as a function of the orbital period, which follows a broken power law relation peaking at P ∼ 720 d, very close to the periods of the identified false positive detections. If a significant number of as-yet-unidentified false positives contaminate the comparably small sample of planets around giants, significant implications on the planet occurrence rate and its interpretation are to be expected <cit.>.

Moreover, as the phenomenon is still poorly understood and can mimic planets quite convincingly in RV data, it is often impracticable to unambiguously confirm new planets around giant stars (see, e.g., ). This is due to the fact that it is not yet clear which observables are the most decisive diagnostics to separate intrinsic signals from orbital companions for giants. This problem is made even worse by the varying availability, in data sets from different spectrographs, of spectral diagnostics targeting such intrinsic variations. These are commonly referred to as activity indicators, and we adopt this term even though we do not (mainly) target magnetic activity in this work. 

So far, several intrinsic origins have been discussed. Radial oscillations can usually be ruled out due to the long-period nature of the suspected signals, which appear at periods much longer than the fundamental radial mode <cit.>. In the binary system ϵ Cyg, the heartbeat phenomenon was considered <cit.>; however, it  is  not applicable to single stars. Magnetic surface structures, such as cool spots, can in most cases equally be ruled out, as they should manifest themselves in much larger photometric variations than were reported for the above stars <cit.>. <cit.> (and references therein) propose magnetic plages or other magnetic structures that locally reduce convection and could cause RV variations without associated photometric variations. However,  the magnetic fields on the surface of most giant stars are still  not well understood  (see, e.g., , for notable exceptions). 

Non-radial oscillations, on the other hand, can cause large amplitude RV variations at long periods <cit.>. <cit.> link the aforementioned phenomenon of false positive planet detections to oscillatory convective modes presented by <cit.>. These modes were proposed as the origin for the  sequence D of the long secondary periods (LSPs) observed for variable bright giant stars <cit.>. These are dipole (l=1) g^--modes that become oscillatory in the non-adiabatic conditions present in the envelopes of luminous (log L/L_≳ 3) giant stars <cit.>. <cit.> build on this argument, showing in their Fig. 8 that both γ Dra and Aldebaran fall into a region in the period-luminosity diagram in which an extrapolation of the models by <cit.> could potentially explain the observed variations. 
Nevertheless, no satisfying attempt to apply models of non-radial oscillations to the observables of a known false positive evolved planet host has been published to date.

In this work we reexamine the RV variations of the cluster giant NGC 4349 No. 127, previously thought to host a brown dwarf companion in a P=677.8 d orbit originally published by <cit.>. However, <cit.> show that, while the RV signal is stable in HARPS (High Accuracy Radial velocity Planet Searcher) spectra, variations of the full width at half maximum (FWHM) of the cross-correlation function (CCF) and the Hα index at the orbital period are present. The authors therefore refute the companion's existence, favoring rotational modulation of magnetic activity as the most likely alternative, while not ruling out non-radial oscillations. <cit.> affirm these findings by presenting 11 additional RV measurements.

Here, we present a reanalyis of the HARPS data set adding measurements of the chromatic index (CRX) and differential line width (dLW) variations by using the SERVAL <cit.> pipeline, as well as the contrast of the CCF by using the RACCOON <cit.> reduction software. We reveal a linear correlation between the RV, on the one hand, and the CRX and the Hα indicator, on the other hand, respectively. We further show that the dLW, the FWHM of the CCF, and the contrast of the CCF each correlate with the RV in a “closed-loop” relation. We additionally present simulations of the observational effects of non-radial oscillations that can closely reproduce the periods and amplitudes of the activity indicator variations, as well as the phase differences with the RV.

This paper is structured as follows. In Sect. <ref> we give an overview of the stellar parameters, observations, and data reduction. In Sect. <ref> we present the  simulation suite developed to simulate the effect of non-radial oscillations on HARPS spectra.  We present newly detected correlations between the activity indicators and the RVs in Sect. <ref>, and show that they are consistent with a retrograde, dipole, non-radial oscillation model.
In Sect. <ref> we present general observational properties of different oscillation modes. We discuss our non-radial oscillation model in the context of oscillatory convective modes in Sect. <ref>, before summarizing our findings in Sect. <ref>.



§ OBSERVATIONS AND STELLAR PARAMETERS




 §.§ Stellar parameters

NGC 4349 No. 127 is the most evolved star of the open cluster NGC 4349 <cit.>, located at a distance d = 1788.9±2.9 pc <cit.>, with an estimated age of around 300 Myr <cit.>. The stellar parameters vary considerably between different studies and are summarized in Table <ref>. 

In Fig. <ref>, we plot a color-magnitude diagram based on Gaia DR3 photometry <cit.>, highlighting the position of NGC 4349 No. 127 as the orange star marker. The membership list and isochrone fit were taken from <cit.>. The star's exact evolutionary state is ambiguous between the first or second ascent on the red giant branch (RGB), as was discussed in the literature <cit.>. One reason for this ambiguity is the differential extinction 0.57^+0.30_-0.31 mag in the V band present in the cluster <cit.>, which is also evident from the spread at the turn-off of the main sequence in Fig. <ref>. This complicates the determination of a specific extinction value for NGC 4349 No. 127, adding uncertainty to its intrinsic photometry. The difficulty in determining an accurate extinction value could also (partially) explain the variability of the stellar parameters summarized in Table <ref>.

We also tested the stellar parameters using Bayesian inference based on Gaia DR3 parallaxes and photometry using [<https://github.com/StephanStock/SPOG>] <cit.>. We use extinction values taken from the Starhorse catalog <cit.>. The metallicity was taken from <cit.>. The tool yields a probability of 99.1% for the star being on the horizontal branch (HB) or second ascent on the RGB. The stellar parameters derived by  are listed in Table <ref>.

While the results for T_eff are roughly consistent with other determinations in the literature,  estimates a significantly lower mass, radius, and luminosity. The age determination of the star τ = 930^+300_-330 Myr also significantly deviates from the estimated age for the whole cluster from other studies. We assume that stellar parameters derived from spectroscopy or ones that take the whole cluster into account should generally be regarded as more accurate.  is also quite sensitive to the choice of extinction values, which are difficult to determine. However, as masses of giant stars derived from evolutionary models are suspected to be overestimated (see, e.g., ), we note this somewhat lower mass estimate, as well as the lower mass estimates by <cit.> and <cit.>, and the general variability of the parameters in Table <ref>.

The star has furthermore been found to show enhanced Li abundance compared to other giant stars in the cluster and compared to field stars <cit.>, which was proposed to be caused by planet engulfment. However, <cit.> notes that the determined Li abundance is still below the traditional limit for Li-rich giant stars logϵ(Li) ≥ 1.50. 

Given the precision of the results and favoring parameters derived by spectroscopy, we base our analysis and discussion on the stellar parameters derived by <cit.> and <cit.>. For the simulations presented in Sect. <ref>, we chose (T_eff=4500 K, log g = 2.0, [Fe/H]=0.0) as the closest grid point of the PHOENIX spectral library. We refrain from interpolating the base spectrum of the simulations to the exact parameters due to the differences shown in the literature (Table <ref>) and to avoid interpolation errors. Using the relation by <cit.>, we determine the value of the macroturbulence to be ζ=4822±39 m s^-1 at this effective temperature. 








 §.§ Data


We downloaded 58 publicly available, extracted HARPS <cit.> spectra from the ESO science archive[<http://archive.eso.org/scienceportal/home>] for NGC 4349 No. 127. The spectra were taken between 2005 and 2022 and have signal-to-noise ratios (S/N) per extracted pixel between 25 and 62 (with one outlier at 14.7) at the peak of the blaze function of (physical) order 102, which is centered around 6000 Å. The data were taken as part of a long-term RV survey of intermediate-mass giant stars in open clusters and were originally published by <cit.> and <cit.>.

We reduced the spectra using the SERVAL pipeline <cit.>, which derives radial velocities via a least-squares optimization of the RV shift relative to a high signal-to-noise stellar template obtained by coadding the available observations. SERVAL was shown to produce slightly more precise RV results than the default HARPS data reduction software (DRS) <cit.>. The SERVAL pipeline furthermore computes additional activity indicators, namely the CRX and the dLW <cit.>, which have proven to be effective indicators to detect activity for main-sequence dwarfs, especially in context of the CARMENES (Calar Alto high-Resolution search for M dwarfs with Exoearths with Near-infrared and optical Echelle Spectrographs) survey <cit.>. It moreover calculates several line indices, of which the Hα index will be used within this work.  

We furthermore utilized the RACCOON reduction software <cit.> to obtain radial velocities using a CCF with a weighted binary mask. The mask was created from the SERVAL template via the routines provided by the RACCOON software. It furthermore provides three activity indicators targeting the contrast and FWHM of the CCF and calculates the Bisector Inverse Slope (BIS) <cit.>. We prefer the RACCOON pipeline over the default HARPS DRS, since the latter is not publicly available and can therefore not be used to reduce the simulated data presented in Sect. <ref>. The results of the RACCOON pipeline are generally in good agreement with the HARPS DRS results, but yield somewhat higher RV uncertainties compared to HARPS DRS and SERVAL (see Table <ref>). 

We separated the spectra taken before (46 spectra) and after (12 spectra) 2 June 2015,  to account for the HARPS fiber change.[<https://www.eso.org/sci/facilities/lasilla/instruments/harps/news.html>] We did not separate the spectra further, accounting for the HARPS warm-up on 23 March 2020, as only three spectra were taken after this date. We treated the two subsets as separate RV time series, allowing for a relative offset. 

We restricted the analysis of the activity indicators to the 46 spectra acquired before the HARPS fiber exchange, as we noted significant offsets of the activity indicators by analyzing the activity time series of quiet stars using the HARPS RVBANK <cit.> (see also Appendix A of ). As these offsets are (to our knowledge) poorly understood and constrained, we decided to focus merely on the spectra acquired prior to the fiber change. Both the SERVAL and RACCOON RVs and activity indicators are listed in Table <ref>. 



§ SIMULATIONS


To study whether non-radial oscillations could explain the variations of the radial velocities and the activity indicators, we simulate the observational effect of these on HARPS spectra. The simulations are part of the simulation suite .[<https://github.com/DaneSpaeth/pyoscillot>]



 §.§ Description of non-radial oscillations

We start out with a model of the stellar photosphere, using spherical coordinates (θ, ϕ), with θ being the polar angle (or colatitude), measured from the oscillation pole and ranging from 0 to π, and ϕ being the azimuthal angle, ranging from 0 to 2π. Each of the grid points has a local effective temperature T_eff and a local oscillation velocity vector v⃗_⃗o⃗s⃗c⃗ defined by the three components <cit.>


    v_r    = ∂ξ_r/∂t = i 2 πνa/C Y_l^m(θ, ϕ)exp(i2πν t), 
    
        v_θ   = ∂ξ_θ/∂t = i 2 πνKa/C∂ Y_l^m(θ, ϕ)/∂θexp(i2πν t),
    
        v_ϕ   = ∂ξ_ϕ/∂t = i 2 πνKa/Csinθ∂ Y_l^m(θ, ϕ)/∂ϕ)exp(i2πν t),

which are oriented in the directions of the local unit vectors of r, θ, and ϕ, respectively. We use the real part of the velocity components to calculate the final local velocities. Here, ξ describes the local displacement vector, ν is the oscillation frequency, a is the displacement amplitude in the radial direction, K is the ratio between the radial and the horizontal displacement, and Y_l^m are the spherical harmonics defined as 

    Y_l^m(θ, ϕ) = √(2l+1/4π(l-m)!/(l+m)!)P_l^m(cosθ)exp(imϕ).

Here we introduce the quantum numbers l and m. l is the number of line of nodes on the stellar surface, while m is the azimuthal order quantifying how many of these line of nodes run through the oscillation pole. It ranges from -l ≤ m ≤ l.
P_l^m(cosθ) are the Legendre Polynomials defined by

    P_l^m(cosθ) = (-1)^m/2^ll! (1-cos^2θ)^m/2d^l+m/dcos^l+mθ (cos^2θ-1)^l.

Thus, the velocity amplitude in the radial direction is given by v_osc = 2πνa/C, and consequently K · v_osc is the velocity amplitude in the θ and ϕ directions. We introduce the normalization factor C=max(Re(i Y_l^m(θ, ϕ))) for m ≠ 0, and C=max(Re(Y_l^m(θ, ϕ))) for m=0, such that v_osc can be regarded as the amplitude of the radial component at the position of maximum variation on the stellar surface, giving it a straight-forward interpretation. Both v_osc and K are input parameters to be defined by the user. In the Cowling approximation, neglecting perturbations of the gravitational potential, K can be expressed as <cit.>

    K = GM_⋆/(2πν)^2R_⋆^3.



Associated with the (physical) radial displacement are variations of the local effective temperature of a scale δ T_eff and at a phase ψ_T. Assuming non-adiabatic conditions, we can express the local effective temperature as <cit.> 

    T_eff, local = T_eff + δ T_effRe(Y_l^m(θ, ϕ)/max(Re(Y_l^m(θ, ϕ)))exp(i(2πν t+ψ_T))),

defining δ T_eff as the amplitude of the temperature variation at the position of maximum variation on the stellar surface. The scale of the temperature variation is not influenced by the choice of the oscillation velocity v_osc, such that a reasonable value for δ T_eff has to be chosen by the user.
  

As the velocity components of interest are generally low, we neglect geometrical distortions of the stellar sphere and keep the stellar radius at unity at all times. 
Thus, the simulations can be visualized as mapping the oscillation's velocity components onto the unit sphere, neglecting the displacements of the individual positions. We further do not include surface-normal variations which were found to play a minor role on the variation of line widths by <cit.>. However, we note that <cit.> advocates to include both surface-normal and surface-area variations to accurately reproduce photometric continuum variations. 



 §.§ Calculation of the synthetic spectrum

After choosing an inclination angle, defined as the angle between the polar axis of the oscillation and the line of sight, the velocity and temperature fields are projected onto a flat, uniformly spaced 2D grid of size N_grid× N_grid, using triangulation and linear interpolation. The directional velocity components are further projected onto the line-of-sight vector, and can thereafter be summed up to yield the local combined oscillation velocity for each element on the grid. 

We further add a projected rotational velocity for each position characterized by the (unprojected) rotational 
velocity v_rot. We fix the axis of rotation to coincide with the polar axis of the simulated star and thus with the symmetry axis of the oscillation. The summation of the projected, combined oscillation velocity and the projected rotational velocity yield the final, local velocities along the line of sight v_i for each position i. This local velocity will subsequently be used to Doppler shift the local spectra as detailed below. 

The projected positions on the grid can be regarded as the center points of small surface areas with the same projected area. For cells at the limb of the star, a geometric weight w_i is calculated as these are only partially covering the stellar disk. 

For each grid cell, a synthetic stellar spectrum is calculated by performing a cubic spline interpolation with respect to the local effective temperature T_eff,i within a grid of synthetic spectra taken from the PHOENIX spectral library[<https://phoenix.astro.physik.uni-goettingen.de/>] <cit.>. For computational speed, the cubic spline interpolation is performed in 0.1 K steps.  For simplicity, we do not interpolate the spectra with respect to the surface gravity log g and the metallicity [Fe/H]. Both are fixed to the closest value for NGC 4349 No. 127, that is log g=2.0 and [Fe/H]=0.0. 

Ideally, one would like to use the specific intensity spectra available from the PHOENIX library, which were calculated for a model atmosphere under different observation angles μ = cosγ, with γ being the angle between the line of sight and the surface normal. However, these are only available with a sampling rate of 1.0 Å, too low to simulate high-resolution Echelle spectra. We are therefore forced to use the high-resolution spectra of the PHOENIX spectral library. As these are already disk-integrated spectra, the final simulated spectrum can only be regarded as a relatively crude approximation. 



  §.§.§ Implementation of the limb-darkening correction

Since we are interested in the observational effects of non-radial oscillations, which can be dominated by their horizontal components and thus strongest toward the limb of the stellar disk, we implement a wavelength-dependent limb-darkening correction. <cit.> fit the intensity profile of the solar disk as a function of μ and wavelength λ using data provided by <cit.> and <cit.> and provide the relation 

    I(μ, λ) = 1 - u(1-μ^α(λ)).

<cit.> use u=1 and determine α∼ -0.023 + 0.292 λ^-1 for λ in units of μ m and 416 nm≲λ≲ 1099 nm. Since the wavelength range of HARPS extends down to 378 nm at the blue end, we extrapolate the model, despite the discontinuity at λ∼ 390 nm discussed by <cit.>.

As the underlying PHOENIX spectra are already disk-integrated and therefore include the effect of limb darkening, we first divide each local spectrum by the wavelength-dependent intensity profile for the mean angle μ. Next, the limb-darkening effect is added back into the models by multiplying with the respective intensity profile for each cell according to its local angle μ_i. By this approach, we add an implicit weight based on the intensity according to the angle μ_i during the spectrum combination process detailed below. Ideally, one would like to use μ and wavelength-dependent models for a star more similar to NGC 4349 No. 127, but such models are not available in the literature.

We further include the effect of macroturbulent broadening by convolving the local spectra with a wavelength-dependent Gaussian kernel, specified by a user-defined macroturbulent velocity ζ. Microturbulent broadening has already been included during the synthesis of the base PHOENIX spectrum, albeit at a fixed value for each grid point within the PHOENIX library <cit.>.



  §.§.§ Implementation of the convective blueshift correction

We also include an estimation for the effect of convective blueshift on the spectral line bisectors, following the approach by <cit.>. As the PHOENIX spectral library is derived from 1D atmospheric models, they do not properly reproduce line bisector shapes <cit.>. The intrinsic bisectors of the PHOENIX spectra therefore have to be removed prior to adding in a more plausible bisector shape stemming from convective processes.

First, the PHOENIX spectra are normalized using RASSINE <cit.>. Using the normalized spectra we first calculate the small wavelength shifts necessary to remove the intrinsic PHOENIX bisector by measuring the bisectors of five FeI lines (FeI 5250.2084 Å, FeI 5250.6453 Å, FeI 5434.5232 Å, FeI 6173.3344 Å, FeI 6301.5008 Å) and calculating the average. Next, the wavelength shifts from one of the giant stars presented by <cit.> are calculated using the bisector-removed, normalized spectra. Both the wavelength shifts are applied to each grid point in the unnormalized PHOENIX spectra. These are then used during the spectrum combination process. By testing all available bisectors, we found that the observed CCF bisectors are well reproduced using the bisector model of the giant star β Boo. While this procedure creates CCF bisectors quite similar to the observed ones, it has little effect on the observables presented in Sect. <ref>.



  §.§.§ Combination of the local spectra

Finally, after including each of the effects on the local spectra (local temperature variation, limb darkening, convective blueshift), the individual spectra have to be Doppler-shifted before being summed to yield the disk-integrated spectrum.

To yield accurate Doppler shifts, we first oversample the wavelength grid at equidistant 0.001 Å steps and perform a cubic spline interpolation for the flux values at the oversampled grid points. After the spectra are Doppler-shifted according to the summed local velocities of oscillation and rotation v_i, the shifted flux values are then linearly interpolated back onto the original wavelength grid, which is common for all surface grid points. Finally, the spectra are summed, accounting for the reduced geometrical weights w_i for cells at the edge of the star, and divided by the sum of all geometrical weights to maintain a reasonable normalization of the spectra. The final, disk-integrated spectrum can be expressed as 

    I_integrated(λ) = 1/∑_i w_i∑_i I_i(T_eff,i, v_i, μ_i, λ) · w_i .


Applying the above steps, we calculate synthetic stellar spectra including the effect of non-radial oscillations. The code is also able to study other effects such as star spots, which is however beyond the scope of this publication.



  §.§.§ Conversion to simulated HARPS spectra


The combined spectrum is next smoothed with a Gaussian kernel to bring the spectrum to the required resolution for each spectrograph. For HARPS, R = 115 000 <cit.> is used (CARMENES VIS and NIR channels are also available). The smoothed spectra are then rebinned onto an extracted wavelength grid of a real HARPS observation of NGC 4349 No. 127. Conceptually, converting the combined spectrum from an energy flux to a photon flux would more closely mimic the acquisition process of a real CCD. However, tests revealed that the differences are negligible and we thus decided to omit the conversion for simplicity. Next, a measured blaze function is applied to each spectral order and the necessary FITS header keywords are altered to mimic a real observation. Finally, the spectrum is saved in the FITS format and the RVs can be reduced from the simulated, extracted spectra. The simulation is performed at user-defined epochs. 

The final spectra are then reduced using the RACCOON and SERVAL pipelines – identical to the processing applied to real observations – yielding the simulated RVs and activity indicators. For the RACCOON pipeline, we use the weighted mask created from the real observations. For the SERVAL reduction, a new template is created for the simulation. 

For the simulation results presented below, we performed the simulation with a grid of size 150 × 150. This number was chosen for computational speed, but tests at higher resolutions revealed negligible differences.



§ RESULTS




 §.§ Confirmation of the intrinsic nature of the RV variations

From the 46 spectra acquired prior to the HARPS fiber change in 2015, two (BJD = 2454323.471811, BJD = 2454349.472032) were found to be strong outliers in the dLW time series. These have dLW = 96.6 and dLW = 96.4, respectively, while the dLW time series for the other spectra varies between -21 and 24. These two spectra were discarded from all time series. We note that the two data points do not have particularly low S/N and are not conspicuous in the RVs or any of the other activity indicators. It remains unclear, why the two spectra are strong outliers in only one indicator.

Another strong outlier (BJD = 2458849.842858) in the RV time series was removed from the 12 spectra acquired after 2015. Again, it remains uncertain why the spectrum deviates strongly in the RVs. It has sufficient S/N, albeit slightly below average.

The remaining 44 spectra taken before the fiber change and 11 spectra taken after the fiber change are plotted in Fig. <ref> as blue and orange data points, respectively. We treated the two data sets independently and fitted for an offset.
We plot a fitted sinusoid with period P=674.0±0.1 d in black to illustrate that the RV signal is consistent with a periodic variation stable over at least fifteen years. This RV variation by itself could easily be attributed to an orbiting companion, which was however shown not to exist by <cit.>, based on significant periodicity of the FWHM of the CCF close to the proposed orbital period. <cit.> further report periodicity of the Hα index close to the orbital period but with a false alarm probability (FAP) less significant than 1.0%, which is further reduced in significance when including the spectra acquired after 2015 <cit.>. Moreover, the authors find a weak but significant correlation between Hα and RV.



With the additional activity indicators available through the SERVAL and RACCOON reductions, we can strengthen these findings. Figure <ref> shows a generalized Lomb-Scargle (GLS) periodogram <cit.> of the 44 spectra acquired prior to the HARPS fiber change and reveals that the RV periodicity in the pre 2015 data set (red dashed line) is accompanied by further significant peaks of the activity indicators close to the orbital period. Besides the FWHM, also the contrast of the CCF and the dLW of the SERVAL reduction show strong peaks more significant than the FAP=0.1% level, which was determined via bootstrapping with 10 000 reshuffles. The peak in the contrast GLS is slightly offset at P=645.4 d. 

Both the Hα- and the CRX-periodograms furthermore have peaks at the orbital period, or slightly offset in case of the CRX (P = 645.4 d). While both peaks are formally less significant than the FAP=5% level, the fact that they appear very close to the RV period and are the largest peak in their respective periodograms is certainly not a coincidence and indicates a signal at the RV period in Hα and CRX.
Only the BIS periodogram is inconspicuous. Taken together, the additional spectral diagnostics strengthen the findings by <cit.>, rule out a physical companion, and thus confirm the intrinsic origin of the RV variations.





 §.§ Correlations between the activity indicators and the RVs






To find an alternative explanation for the radial velocity variations, we searched for correlations between the activity indicators and the RVs. 
Figure <ref> shows the CRX, dLW, contrast, and FWHM of the CCF plotted against the RV for the 44 HARPS spectra taken prior to the 2015 fiber exchange. For all indicators, we plot the absolute deviation from the mean value, which was determined by fitting sinusoids and determining the offset, to allow an easier comparison with the simulated variations.  

All data points in Fig. <ref> are color-coded with the RV period P=674.0 d present in the full data set (including the spectra taken after 2015). It is evident from the top left panel that the CRX is positively correlated with the RV with a Pearson's r value r=0.54 with p-value p(r)=0.015%. We fitted a linear relation using orthogonal distance regression and perform an F-test <cit.> to validate the relation's significance against a constant model, resulting in a p-value p(F-test)=0.002%. Following <cit.> and <cit.>, we regard the correlation as significant. As <cit.>, we further note a significant positive correlation of the Hα index with the RV with r=0.48 and p(F-test)=0.2% (shown in Fig. <ref>).

Furthermore, we observe closed-loop correlations with the RV for the dLW, contrast, and FWHM of the CCF. That is, the activity indicators correlate elliptically with the RVs with each data point's position on the ellipse given by the phase of the variation, indicated by the color-coding. Such ellipses are effectively Lissajous curves resulting from sinusoidal variations at the same period but with different amplitudes and phase shifts close to π/2.

The color-coding further reveals the direction of correlation, which is anti-clockwise for FWHM and dLW but clockwise for the contrast, which is to be expected as FWHM and contrast should be inversely dependent on each other. We note that the dLW measures a similar line variation as the FWHM and the contrast and should therefore not be regarded as an entirely independent indicator <cit.>. However, as the results stem from different reduction pipelines (RACCOON vs SERVAL) with different approaches to derive the RVs, it is reassuring that we find a similar behavior for both reductions.  



 §.§ Simulation of a dipole, retrograde (l=1, m=1) oscillation mode


In order to explain these peculiar correlations, we simulated a set of models using the  simulation suite detailed in Sect. <ref>. 
Motivated by oscillatory convective modes as published by <cit.>, we tested different configurations for dipole (l=1) modes using the stellar parameters (T_eff=4500 K, log g = 2.0, [Fe/H]=0.0,ζ=4822 m s^-1) for the base spectrum as discussed in Sect. <ref>. 

We include the chromatic effect of limb darkening and the effect of convective blueshift. For the latter, we calculated the CCF bisectors for all measured bisectors of giant stars from <cit.> and compared them to the observed CCF bisectors for NGC 4349 No. 127. The best match was found for the bright giant star β Boo. We note, however, that the inclusion of the convective blueshift plays a minor role in modeling the relative variations of the activity indicators and mostly provides an offset for the absolute value of the BIS. 


Using the RV period and the stellar properties, the ratio between the horizontal and radial oscillation components can be estimated as K=1856±404. Therefore, the oscillation is dominated by the horizontal components, such that the product K · v_osc is the decisive input parameter and should be interpreted in conjunction (since changes in v_osc can be balanced out by changing K). This behavior was confirmed through test simulations. K was fixed to K=1856 when searching for the best fitting parameters. 

Initial tests at different inclination angles revealed that retrograde (m=1) modes are the only dipole modes capable of reproducing the observed correlations for NGC 4349 No. 127, as the other allowed modes would either lead to much smaller amplitudes in the indicators (m=0) or to inverse phase relations (m=-1). We examine this behavior more closely in Sect. <ref> and focus on l=1, m=1 modes for the current discussion. We examine the amplitudes for inclination i=90 (equator-on), which maximizes the amplitude for the |m| = 1 modes. We also restrict the models to only include one isolated oscillation mode. 

We further find that the scale of the temperature variations δ T_eff directly influences the amplitudes of all activity indicators and the RV. Nevertheless, as the CRX captures the wavelength dependence of the RV, it is the most sensitive indicator to the δ T_eff parameter. At the same time, the variations of the CRX are strongly influenced by the chromatic limb-darkening law. At maximum RV, the visible and projected θ and ϕ components of the l=1, m=1 oscillation are entirely positive. As the RVs are dominated by the horizontal components, the strongest contribution stems from the cells at the limb of the star due to the projection onto the line of sight. The limb darkening suppresses their contribution, such that the overall RV is reduced compared to a model without limb-darkening correction. However, as the limb-darkening law is chromatic, this reduction is stronger at shorter wavelengths, leading to a larger RV at longer wavelength and hence a positive CRX at positive RV. This accounts for the majority of the positive CRX-RV correlation.

Only small temperature variations are therefore consistent with the data, as an increase in δ T_eff at phase shift ψ_T=0 increases the CRX-RV slope. However, at such small scales the scatter within the data complicates an exact determination. We therefore estimated a value δ T_eff=2.5 K predicted for an oscillatory convective mode as presented in Fig. 6 of <cit.>. We determined this value based on the luminosity of the star and the radial amplitude that minimized the residuals between the simulations and the real data for the other observables. We note that the models presented by <cit.> at the luminosity of NGC 4349 No. 127 are only available for lower stellar masses. A higher temperature variation is therefore possible. 

For small temperature variations, the simulations become rather insensitive to small deviations of the phase shift between the temperature variation and the radial displacement, given the scatter in the real data. This phase shift is predicted to be small, ψ_T ≲ 0.1 ·π <cit.>. We therefore fixed this value to ψ_T=0, such that the temperature is largest when the radial displacement is at its maximum.  


With these initial considerations, we ran a large number of models varying v_rot and v_osc, finally selecting the model (by eye) that minimizes the residuals between the simulations and the real data for all observables. 
We simulated 40 synthetic spectra spread over the time span of the real observations (prior to the fiber change) and reduced these using SERVAL and RACCOON. We find that the amplitudes of the activity indicators and the RVs and their phase relations are reproduced using v_rot=1700 m s^-1 and v_osc=0.30 m s^-1. We summarize the best values for all simulation parameters in Table <ref>.





 §.§ Comparison between the data and the oscillation model

Equipped with the non-radial oscillation model, we compare the observed correlations with the simulated ones in Fig. <ref>. The aim is to test whether such an oscillation is able to explain the observed behavior.


 The oscillation model predicts a positive correlation between CRX and RV (top left panel). We fitted the correlation with a linear relation and determined the slope m_sim=0.129±0.001 Np^-1 (black line). The overplotted, colored lines encode the phase information on the linear relation. The simulated data points deviate only marginally from the linear relation and were left out for visual clarity. The slope is consistent with the actual slope measured for the real data m_real=0.101±0.021 Np^-1 with a deviation of 1.3 σ.

For the dLW, contrast, and FWHM, closed-loop correlations very similar to the observed correlations are predicted. We plot fitted ellipses to the simulated indicator-RV correlations that closely follow the synthetic data points (which were therefore left out for visual clarity) in Fig. <ref>, applying the same color-coding. An offset for the RV zero points was fitted and removed between the real and synthetic data sets. We stress again that we show the absolute deviation from the respective mean of each time series. The absolute values between the FWHM and contrast of the CCF of the real and synthetic observations are slightly offset as not all influences can be modeled adequately. 

We also had to downscale the simulated dLW variations by a multiplicative factor of 0.098 to match the real dLW variations. As the dLW is a differential quantity sensitive to the second derivative of the SERVAL templates, which are separate for the real and simulated data sets, such a multiplicative factor is expected when comparing separate dLW time series <cit.>. We  find that the scaling factor is on the order of unity when using the same template for both reductions, which however reduces the precision of individual RV and indicator determinations. 
The multiplicative factor is presumably caused by the (slightly) different widths of the real and simulated spectral lines (see also Sect. <ref>). It is furthermore linked to deviations from the assumption of Gaussian spectral lines due to the oscillation, which are fundamental to the definition of the dLW <cit.>. We note that, due to this scaling factor, the amplitude of the dLW holds little quantitative information. The phase relation with the RV, however, holds valuable information to validate the simulations.       

It can be seen from Fig. <ref> that all variations of the activity indicators, the relative phases, and the directions of correlations are well reproduced. We tested whether radial, p-mode (solar-like) oscillations can explain the residual scatter present in the activity indicators. From the residual scatter in the RVs, we calculated associated temperature variations of δ T_eff∼ 0.9 K using the scaling relations by <cit.>. Fine-tuning v_osc to reproduce the residual scatter in the RVs, we find that the activity indicators are in principle sensitive to the p-mode oscillations, but the amplitudes are too small to explain the observed residuals. We assume that instrumental effects and other intrinsic, stellar noise sources, such as granulation, dominate the residual scatter. 

The CRX is sensitive to the wavelength dependence of the RVs and thus to temperature variations on the stellar surface, as well as the limb-darkening coefficients. We find the interplay of limb darkening and the oscillations to be the dominating influence. As the limb-darkening contrast decreases toward redder wavelength and the resulting RV is dominated by the horizontal components at the limb of the star, the resulting RV amplitude is larger in the redder part of the spectrum and hence the CRX is positively correlated with the RV. Its slope is commonly referred to as chromaticity <cit.>. 
  
In comparison, the temperature variations have only minor influence on the observed CRX-RV slope. If we simulate an (unphysical) oscillation model setting δ T_eff=0 K, the resulting slope of the CRX-RV correlation is reduced slightly to m_δ T=0K=0.1213±0.0003 Np^-1, even closer to the real correlation. A limb-darkening model adapted to the stellar parameters of NGC 4349 No. 127 could therefore be beneficial, but is beyond the scope of this work. 
 
The variations of the line shape indicators (dLW, FWHM, contrast) are mainly influenced by the interplay between the oscillation at the edge of the stellar disk (for which the dominating horizontal components are most directly oriented along the line of sight) and the rotational broadening. This can be easily understood as the oscillation velocities at the limb directly affect the flanks of the broadened spectral lines, thus introducing variations of the line width and depth, or introducing asymmetries. 

Figure <ref> shows the RV and indicator time series data acquired pre 2015. We fitted sinusoids to the simulated time series and color-code these in the same way as the real data. The simulations are able to reproduce the amplitudes and phases of the RVs and all activity indicators, although large scatter is present in some of the real indicators. We note that the simulations also predict small variations of the BIS at the same period as the RVs that are consistent with the observed data, thus adding another argument in favor of the oscillation model. Large errors and high scatter in the real data likely obscure these variations, so that they are not detected in the GLS periodogram search in Fig. <ref>.

No meaningful Hα variations can be predicted by the models since the simulations focus solely on the stellar photosphere, while the Hα variations are mostly caused by chromospheric processes <cit.>. Whether such chromospheric (and likely magnetic) processes could be linked to the oscillations is unknown. We note that similar connections between stellar oscillations and magnetic fields have been proposed (see, e.g., ).



 §.§ Cross-correlation profiles



In Fig. <ref> (top), we plot the real, observed CCF profiles from the RACCOON reduction for NGC 4349 No. 127 (black) as well as the modeled profiles (red). While the CCF shapes are reasonably well reproduced, slight differences in the absolute values of the CCF width (FWHM) and its contrast remain. We therefore plot the differences from the respective means in Fig. <ref> and Fig. <ref>. 

Offsets between the simulated and the real CCF profiles are to be expected as not all influences can be modeled accurately. These include the real instrumental profile of the HARPS spectrograph, which we model as a Gaussian, and the intrinsic line shapes inherent to the PHOENIX models. These are given with fixed microturbulent broadening values for each model. For the base PHOENIX model employed in the simulation, microturbulence ξ = 1.49 km s^-1 was used, somewhat lower than expected for our star (see Table <ref>). Due to these inherent differences, the line shapes cannot be reproduced accurately. 

We also find that the rotational velocity v_rot plays an important role in reproducing the amplitudes of the line shape indicators (dLW, FWHM, contrast), with higher rotation velocities generally producing larger variations. At the same time, v_rot influences the absolute width of the CCF profile. We find a smaller rotation velocity (v_rot = 1.7 km s^-1) to be generally in good agreement with the observed variations for NGC 4349 No. 127. We note that more recent determinations of the star's rotation velocity are larger (see Table <ref>), but find good agreement with the rotation velocity determined by <cit.>. However, due to the inherent differences of the PHOENIX spectra to the real spectrum of the star, we caution against using this value as a determination of the rotation velocity of NGC 4349 No. 127. 

The bottom panel of Fig. <ref>  portrays a zoomed-in image of the calculated bisector profiles. We observe that the general bisector shape is well reproduced by the convective blueshift model based on β Boo <cit.>. However, we find the bisector shape to have negligible effect on the observables presented in Fig. <ref> and Fig. <ref>, apart from providing an offset for the absolute value of the BIS. 






 §.§ Photometric variations

For many types of intrinsic variations manifested in RV data, such as stellar spots or radial oscillations, photometric variability is expected and can provide independent constraints on their properties (see, e.g., ).
One V-band photometric data set, already discussed by <cit.>, is available for the star from the All Sky Automated Survey (ASAS) at Las Campanas Observatory (Chile) <cit.>. We use only grade A and B results as suggested by <cit.>. 

In Fig. <ref> (top), we plot the ASAS-3 data (dots) against time. We overplot binned data points (black rectangles) with bin size 67.4 d (10% of the RV period) to add some visual clarity. There is significant scatter at the ∼ 0.1 mag level, albeit without obvious periodicity. We overplot the predicted V-band variations from the simulated dipole oscillation caused by the temperature fluctuations. These were derived by integrating the product of the simulated spectra and the Bessel V-band filter curve[<http://spiff.rit.edu/classes/phys440/lectures/filters/bess-v.pass>] and rescaling to the median magnitude present in the ASAS-3 data set. The simulations predict a sinusoidal variation at the RV period with an amplitude of 0.003 mag. This lies well below the average ASAS-3 error for NGC 4349 No. 127 of 0.051 mag and below the scatter present in the ASAS-3 data set. It would therefore be plausible that the photometric variation stemming from the oscillation would not be detected by the ASAS-3 photometry. We thus conclude that the available photometry does not argue against the oscillation hypothesis. We discuss the available photometry in more detail and comment on the findings by <cit.> in Appendix <ref>.

ASAS-3 is reported to achieve a differential accuracy of 0.01 mag for ideal, bright targets <cit.>, which is still larger by a factor of 3 than the photometric amplitude predicted by the simulation. The star was further observed in the TESS full-frame images in sectors 11, 37, 38, 64, and 65. However, the short duration of each individual sector, the long gaps between the available sectors, and instrumental offsets between them prevent us from analyzing the TESS photometry on timescales similar to the RV period. Future Gaia data releases might provide the necessary photometric precision and timescales to detect such variability.[<https://www.cosmos.esa.int/web/gaia/science-performance>] However, as the star is relatively bright, systematic effects might hinder this analysis.

The calculation of the photometric variability is based solely on the temperature variations on the stellar surface, neglecting surface-area and surface-normal variations. The latter are small at the velocity amplitudes considered but would violate the assumption of spherical symmetry of the star, complicating the calculation of the velocity fields. Given the small amplitudes, we find this simplification justified. 

We note, however, that <cit.> argues to include surface-area and surface-normal variations when predicting photometric variations of non-radial oscillations. However, the oscillation amplitudes considered are much larger than in the case of NGC 4349 No. 127 and their K values range between 0 and 1. The differences between simulations considering these geometrical surface variations and those that do not (see their Fig. 6) seem to become less pronounced toward larger K (as in our case). The surface-normal variations mainly act to avoid an unphysical photometric minimum at K ∼ 0.85 with the differences being on the order of a factor 2 to 3 otherwise. It can therefore be assumed that the overall photometric variability would not be affected enough to be easily detectable by the ASAS-3 photometry. 

With the presented results, we show that a retrograde, dipole (l=1, m=1) oscillation mode is fully consistent with the radial velocity variations and the variations of all studied activity indicators, including the amplitudes and phase relations, as well as the available photometry. We therefore conclude that non-radial oscillations are indeed present in the star and cause the observed periodic patterns. 




§ OTHER OSCILLATION MODES


Having established that a retrograde l=1 mode is able to reproduce the observables, we further aim to qualitatively address the question whether, alternatively, other oscillation modes are also capable to reproduce the same behavior.


 §.§ l=1 modes





Motivated by the models for dipole oscillatory convective modes presented by <cit.>, we first study the characteristics of l=1 modes, for which the azimuthal orders m=(-1, 0, 1) are possible. We chose the same parameters as presented above, aiming to understand the qualitative differences for the different modes. We simulated the different oscillation modes at 20 epochs each. 

We first note that different modes have different inclination angles at which the RV amplitudes are maximized or minimized. In case of the l=1 modes, the RV amplitudes are maximized when viewed equator-on (i=90) for |m| = 1, while the m=0 mode has maximum RV amplitude when viewed pole-on (i=0) <cit.>. We therefore ran the simulations at different inclination angles and present the intermediate inclination angle i=45, for which the amplitudes for the |m| = 1 and m=0 modes are comparable.

Figure <ref> shows the results for radial velocity (top left) and the spectral diagnostics for the l=1, m=-1 mode (red), l=1, m=0 mode (black), and l=1, m=1 mode (blue) plotted against time. We fitted sinusoids to the variations (except for FWHM, dLW, and contrast in the case of m=0) to highlight the phase relations. The differences between the exact simulated data points and the sinusoids are negligible. 

We first note that all three modes are capable of producing RV variations with similar amplitudes and would thus, in principle, be able to explain the RV variations for NGC 4349 No. 127. While the |m|=1 modes are in phase, the m=0 mode is phase-shifted. However, we note that an artificial phase shift can be added (and was used to match the retrograde mode with the real observations) to match the m=0 mode to the real RVs. 

What cannot be altered, though, are the phase relations between the RV and the spectral diagnostics. First, we observe that for the line shape diagnostics (FWHM, dLW, contrast) the m=1 and m=-1 modes are roughly in antiphase with each other. The m=0 mode shows much smaller line shape variations that are not perfectly sinusoidal (and therefore presented here by connecting the simulated data points) and, most notably, vary at roughly half the RV period. The behavior can be explained as the azimuthal (ϕ) component of the axisymmetric m=0 mode is in all cases 0. Thus, the interplay between the ϕ oscillation component and the rotation vector, which causes the majority of the line shape variations as it influences the flanks of the rotationally broadened line profiles, is not present in the m=0 mode. The now solely dominating θ component of the m=0 oscillation is symmetric with respect to the rotation axis and thus leads to much smaller line shape variations.

We also show the phase relations and their directions of correlations in Fig. <ref> (left column). The dLW, contrast, and FWHM are plotted in the bottom three panels, respectively. We fitted ellipses to the correlations and present the direction of correlation when appropriate. Figure <ref> reveals that both |m| = 1 modes create closed-loop correlations, while the m=0 mode leads to an “arc-like” correlation, clearly not consistent with the real data for NGC 4349 No. 127. 

For the two |m|=1 modes, we note that the time dependence of the correlation is reversed (indicated by the arrows), a consequence of the phase shifts of the line shape indicators in Fig. <ref>. While the retrograde m=1 mode shows an anti-clockwise correlation for the dLW and FWHM (and therefore clockwise for the contrast, as for NGC 4349 No. 127), the prograde m=-1 mode correlates with a reversed time dependence and thus cannot explain the real variations. 

It is furthermore evident from Fig. <ref> (bottom left panel), that the CRX is in all cases roughly in phase with the RV with only slightly different amplitudes. These lead to similar positive correlations between CRX and RV in Fig. <ref> (top left panel). The similarity of the CRX variations stems from the fact that the wavelength-dependent limb darkening in interplay with the horizontal components at the limb of the star is dominating the CRX amplitude. The CRX variation is therefore, for all l=1 modes considered, in phase with the RV variation. This relation holds true as long as the temperature variation δ T_eff is small. The CRX-RV correlation thus seems to be a sensitive indicator for the l-mode. 

The slight differences in the CRX amplitudes (and therefore the slopes of the correlation) stem from the interplay between the small temperature variation δ T_eff=2.5 K and the rotational broadening of the spectral lines. That is, the temperature variation by itself has a small effect on the RVs and the CRX, as it slightly increases or decreases the flux from either side of the rotationally broadened line profile, introducing line asymmetries that are measured as RV shifts. As the contrast (flux ratio) between the slightly hotter and cooler halves of the star is less pronounced at longer wavelengths, this effect is wavelength dependent and thus captured by the CRX. For the m=-1 and m=1 modes, for instance, the temperature variation is phase-shifted by 180. Thus, the effect of temperature alone enhances the RV and CRX amplitudes in the m=1 case, while it decreases both amplitudes in the m=-1 case. The CRX-RV correlation for the m=0 mode coincides with the effect given by the limb darkening alone.


Figure <ref> finally presents the BIS variations (bottom right panel). Sinusoidal variations at the period of the RV are predicted for all three m-modes, albeit with different amplitudes and phase relations. However, as the variations for NGC 4349 No. 127 cannot be detected given the noise of the data, no strong conclusion can be drawn from the BIS indicator in this case. 
All of the above findings are valid at all inclination angles that produce non-zero RV variations for the individual modes. 

The fact that m=0 modes lead only to small line shape variations is somewhat discouraging when considering the possibility of applying the same analysis to other evolved stars, suggested to be false positive planet hosts. If an m=0 mode is equally likely, and considering the typical scatter in the real time series of activity indicators, it seems very challenging to detect the line shape variations predicted for this mode. Only the CRX (and BIS) variation and its correlation with the RV are predicted to be similar as for NGC 4349 No. 127 and could provide the most direct hint for non-radial oscillations in the case of an m=0 mode. However, if stemming from non-radial oscillations, the amplitude of the CRX variations also depend directly on the amplitude of the oscillation velocity v_osc. Therefore, identifying oscillations based on the CRX at much smaller RV amplitudes could be challenging with the typical data sets. 

In principle, all m-modes can be expected to be excited, albeit possibly at different amplitudes and periods in the presence of rotation. For NGC 4349 No. 127, considering only dipole (l=1) modes, it seems most straight-forward to explain the data with a single l=1, m=1 mode. In reality though, a combination of all three m-modes could be the most plausible scenario. Such a combination, however, could also be difficult to detect. 

If all three m-modes are excited to the same amplitude, and neglecting at first the change in oscillation periods due to rotation, cancellation effects between the m=1 and m=-1 modes remove the largest part of the variations present in FWHM, contrast, and dLW (as these are in antiphase for the m=1 and m=-1 modes), while resulting in a large RV variation. Only the variations in CRX would likely remain above a detectable threshold, a behavior that we confirmed with test simulations. The resulting RV curves of a combination of all m-modes can be well approximated as the simple sum of the individual RV curves for each mode shown in Fig. <ref>. Therefore, a combination of all three modes at equal and constant amplitudes leads to a phase shift of the RV curve.

In the presence of rotation, the frequencies ν_l,m for different azimuthal quantum numbers m are split (see, e.g., ). This frequency split can be approximated as 

    ν_l,m - ν_l,0 = m/P_rot(1-1/l(l+1))
 <cit.>.
For oscillation periods comparable to the rotation period, this can lead to large period changes. 

 A superposition of sinusoidal signals at different periods and amplitudes can lead to RV curves potentially resembling multi-planetary signals. Moreover, if such signals are insufficiently sampled in context of exoplanet surveys and interpreted as single-planet signals, a change in period, amplitude, or phase of the RV curve might be deduced. This could potentially offer an explanation for the amplitude changes and phase shifts detected in the cases of γ Dra <cit.> and Aldebaran <cit.>, although more thorough modeling would be necessary. We also note that a period modulation has recently been reported for some long-period variables showing long secondary periods <cit.>. Of course, in the context of oscillations, it is also plausible to assume that the oscillation periods themselves are not constant.



 §.§ Higher-order l-modes

While <cit.> discuss that dipole oscillatory convective modes have properties that resemble the period-luminosity relations of sequence D variable giant stars, higher-order l-modes could also cause false positive planet detections. To understand qualitatively how different quantum numbers l affect the simulation results, we simulated (with the same settings) all possible modes up to l=4. 

Motivated by the finding that the l=1, m=1 is the only dipole mode consistent with the data for NGC 4349 No. 127, we plot the correlations between the indicators and the RV for the m=l modes in the central column of Fig. <ref>. We also show the variations plotted against time in Fig. <ref>. As in the previous section, we present the results at the inclination angle i=45, but note that the results are qualitatively consistent across all inclination angles that produce significant amplitudes in the RVs. We fitted linear correlations for the CRX plotted against RV (top) and elliptical relations for the line shape indicators. All relations follow the simulated data points closely, such that the latter were omitted. Only slight deviations from the linear CRX correlations are present. However, as these are much smaller than the scatter in the real data, we still approximated these with linear relations for visual clarity.

We first note that different RV amplitudes are predicted. While naively one would expect the RV amplitude to decrease for higher-order l-modes due to increasing cancellation effects, the behavior is not as straight-forward for non-radial oscillations dominated by the horizontal components. This behavior can be understood as the partial derivatives in Equ. <ref> and <ref> introduce a factor m that increases the respective amplitudes as well as (slightly) different normalization factors that were introduced in Equ. <ref>. The physical horizontal velocity components therefore differ for different modes with the same input velocity v_osc. As v_osc is a-priori not well constrained, differences in the RV amplitudes can be overcome when attempting to match a mode to the real data. 

Furthermore, sinusoidal variations at the period of the RV are predicted for all indicators. The phase differences between the indicators and the RVs again provide insights to identify the modes. For the line shape indicators FWHM and dLW, the phase shift is positive for the l=1, m=1 mode, while it is negative for the higher-degree m=l modes. The behavior of the contrast is reversed. This leads to an inversion of the temporal direction of the elliptical correlations in Fig. <ref>. This already disqualifies the higher-degree m=l modes from being consistent with NGC 4349 No. 127. 

However, we find the opposite behavior for the m=-l modes, plotted in the right column of Fig. <ref> and Fig. <ref>. For l≥2, modes with m=-l produce sinusoidal variations with an anti-clockwise correlation for dLW and FWHM with the RV. The amplitudes of these line shape variations generally depend on the oscillation amplitude v_osc and the rotation velocity v_rot and could therefore (to some extent) be adapted in an attempt to match NGC 4349 No. 127. From this finding alone, these higher-order modes would therefore be able to explain the observables for NGC 4349 No. 127. 

Again, we find the CRX to be the most decisive indicator to identify the mode. As presented in the top panels of Fig. <ref>, the CRX correlates linearly with the RV for all modes. However, the slope of the correlation depends critically on the mode. While the dipole l=1 (red) mode has a positive slope very similar to the real data, the l=2 (orange) mode has a decreased but still positive slope. Increasing the quantum number l decreases the slope further, making it progressively negative for l=3 (black) and l=4 (blue). This behavior is nearly identical for both m=l and m=-l modes, and is roughly consistent for all m-modes for the same order l. The same effect is also evident in the phase shifts between the different CRX time series and the RVs in Fig. <ref>. 

The behavior can be understood due to the geometry of the projected and combined oscillation velocities in interplay with the wavelength-dependent limb darkening. The modes are in all cases dominated by the horizontal components (due to the high K factor). For the low-order m=l=1 mode, maximum RV was reached when the projected ϕ component of the oscillation was fully positive. The wavelength-dependent limb darkening then decreases the overall maximum RV to a larger extent in the blue vs. the red part of the spectrum. This leads to a chromatic RV, captured by a maximum positive CRX at time of maximum positive RV, and therefore the observed positive chromaticity. 

For modes with l=|m|=3 and l=|m|=4 the dominating ϕ component facing the observer is never fully positive. Maximum RV is then reached when the central part of the disk is receding from the observer, while at the same time thin strips at the limb of the star have negative RV values. As these thin strips of cells are most affected by the limb darkening, its wavelength-dependent reduction of RV now has the inverse effect. That is, it diminishes the negative parts and thus increases the overall RV. As this effect is more prominent in the blue vs the red, the phase of maximum CRX is now phase-shifted by half a period with respect to the phase of maximum RV. As a consequence, a negative CRX-RV correlation is observed. 

The l=|m|=2 mode is  intermediate. While the ϕ component can still entirely be positive, the θ component already presents a similar effect as detailed for the l>2 modes. As the θ component is generally smaller in summed projected velocities, this acts only to decrease the CRX-RV slope. Of course, the exact CRX-RV slopes depend on the choice of limb-darkening parameters, but the overall trend can be expected to be insensitive to the exact set of parameters. 

To further study this behavior, we simulated even higher l-modes, observing that the slopes start to alternate between uneven (positive slope) and even (negative slope) l-modes, as can be expected from geometrical considerations. However, due to the ever increasing cancellation effects, simulated RV and indicator amplitudes quickly drop and would thus require much larger values of v_osc to be detectable.  

Finally, we note that these considerations are only valid for small temperature variations (here δ T_eff = 2.5 K), for which the limb darkening dominates the chromatic behavior. Increased temperature variations change the CRX-RV slope and thus make mode identification based on CRX measurements ambiguous. 

As the overall behavior of the l=2, m=-2 mode resembles the retrograde, dipole mode discussed in Sect.  <ref>, we tested whether we could reproduce the data for NGC 4349 No. 127 with this mode. While the CRX-RV correlation can be reproduced with a slightly increased temperature variation δ T_eff∼ 4 K, the amplitudes of the line shape indicators, especially the FWHM and contrast of the CCF, were somewhat smaller than presented for the l=1, m=1 simulation for all sets of parameters considered. However, given the scatter of the real data, they provide a fit nearly as good as presented in Sect. <ref>. Due to the increased rotation velocity (v_rot = 4.5 km/s), necessary to increase the amplitude of the FWHM variations, and the altered oscillation geometry, the amplitude of the BIS variation is increased by a factor ∼9.6 compared to Fig. <ref>, yielding a sinusoidal variation slightly larger but still comparable to the scatter in the BIS data. It is unclear whether this signal would have been picked up in the HARPS data set. We therefore cannot exclude the possibility of an l=2, m=-2 mode as the cause of the variations.

In general, we find that most modes with m ≠ 0 for l≤4, produce similar sinusoidal variations of the RVs and indicators which lead to elliptical correlation plots with reversed directions for the two opposite m-modes. One exception are |m|=1 for l≥3 at specific inclination angles that resemble the arc-like correlations for m=0. However, examining this behavior more closely is beyond the scope of this work. The CRX-RV slopes are always very similar for all m-modes for one quantum number l. The m = 0 modes always show a similar behavior to the l=1, m=0 mode presented in the previous section and can thus always be refuted for NGC 4349 No. 127.
  




§ DISCUSSION


The aim of this study was to test whether non-radial oscillations are capable of explaining the variations of the RVs and the activity indicators of the cluster giant NGC 4349 No. 127. With the data and model presented in Sect. <ref>, we can confirm that the variations of the RVs and indicators are indeed of intrinsic origin and are consistent with a model of a retrograde, dipole oscillation mode. Here, we put these findings into the context of previously reported false positive exoplanets and discuss whether the non-radial oscillations could be oscillatory convective modes. We further discuss that magnetic activity, as suggested by <cit.>, is incompatible with the observations. 



 §.§ Oscillatory convective modes




In Fig. <ref>, we adapt Fig. 8 of <cit.>, showing the location of NGC 4349 No. 127 along with the locations of other giant stars (orange star markers) with established intrinsic RV variations (some of which were formerly thought to host planets or brown dwarfs) in a period-luminosity plot. We do not include ϵ Cyg, as the RV variations of the star are likely caused by the heartbeat phenomenon <cit.>. Luminosities and periods were taken from the respective publications that refute the planets. For Sanders 364, the luminosity was taken from the Gaia DR3 determination of astrophysical parameters <cit.>. We note that <cit.> find RV periodicity at several periods (224 d, 281 d, 333 d, 530 d, 777 d) longer than the published planet period at 121 d by <cit.>, such that the position of the star could move significantly parallel to the x-axis of Fig. <ref>.

It is evident that NGC 4349 No. 127 lies in close vicinity to the refuted planet hosts γ Dra <cit.> and Aldebaran <cit.>. We also show the locations of variable giant stars in the Large Magellanic Cloud from the OGLE survey <cit.> in green and stars which show long secondary periods in black (see  for details). We further plot the models taken from <cit.> for a mixing length parameter of α=1.2. 

Overplotted in red are known planets from the Lick survey (see a recent overview in ), along with their Hipparcos identifiers, and planet candidates identified in the same survey in blue. The latter group has (semi-)periodic RV signatures, but either high RV jitter or otherwise complicated RV curves, which are challenging to model with Keplerian orbits. The two systems HIP 53261 and HIP 39117 were classified as planet candidates by <cit.> and are also listed with their Hipparcos identifiers. 

There appears to be a perceptible distinction between the groups of confirmed planets on the one hand and planet candidates and false positives on the other. The latter mostly accumulate close to the region with periods between 500 d and 1000 d and luminosities 2.3 ≲log_10(L/L_) ≲ 3.2. We find two refuted planetary systems, 42 Dra and NGC 2423 No. 3, to be quite close to established planet hosts. We also find a group of three false positive planet hosts (Sanders 364, 41 Lyn, 14 And) to be located at much lower luminosities and much shorter periods.  As <cit.>, we note that increased luminosity is also linked to increased levels of RV jitter, which makes the confirmation of exoplanets more challenging. 

However, increased RV jitter alone cannot explain the individual cases of most refuted planet systems, which show large-amplitude RV variations, such as the star discussed in this work. As <cit.> and <cit.> argue, if one were to extend the low-mass models of oscillatory convective modes toward longer periods, they would presumably cross the region covered by Aldebaran, NGC 4349 No. 127, γ Dra, and NGC 3532 No. 670. Another cluster giant, NGC 2345 No. 50, appears to be close to the high-luminosity end of the Saio models. 42 Dra, NGC 2423 No. 3, Sanders 364, 41 Lyn, and 14 And seem to have  luminosities that are too low to be in good agreement with the Saio models. As the last three also exhibit shorter periods, one can suspect that another mechanism could be present in these stars, such as the rotational modulation of stellar spots as suggested by <cit.>.

We also note that many of the stars with longer-period RV variations, including NGC 4349 No. 127 (M=3.01±0.24 M_), are significantly more massive than predicted by the Saio models. These include γ Dra (M=2.14±0.16 M_; ), NGC 3532 No. 670 (M=3.05±0.23 M_), NGC 2423 No. 3 (M=2.03±0.14 M_), and NGC 2345 No. 50 (M=5.84±0.61 M_)<cit.>. All of these would need to have higher luminosities to be consistent with the Saio models. Only Aldebaran (M=0.91^+0.04_
-0.02 M_; ) and 42 Dra (M=0.98±0.05 M_; ) have lower masses. Therefore, only Aldebaran is in good agreement with the low-mass models of <cit.>. 

For NGC 4349 No. 127, as detailed in Sect. <ref>, a lower mass M=2.20^+0.38_-0.20 M_ is also possible, which is however also linked to a lower luminosity log_10(L/L_) ∼ 2.3. We also note that the models could be altered significantly by a more sophisticated treatment of convection <cit.>.

Overall, we find that the properties (mode, amplitude of temperature variation) of the presented oscillation model seem to be consistent with the predictions for oscillatory convective modes. We nevertheless are cautious to conclude that the Saio models provide a good match given the stellar properties and period of the variation. More detailed modeling of their properties would be very valuable. 

As discussed by several authors (e.g., ), a suspicious accumulation of false positive planet detections in the vicinity of NGC 4349 No. 127 is evident. As the non-radial oscillation model convincingly explains the variations of the star, a generalization to other stars with similar parameters seems plausible. 

On the other hand, an alternative explanation for the long secondary periods of stars on sequence D is binarity. In this scenario, a brown dwarf or low-mass stellar companion enveloped by a comet-like dust cloud causes the photometric variations by obscuring the stellar disk <cit.>, a proposal supported by the detection of secondary eclipses in mid-infrared photometry for a subsample of variable stars showing long secondary periods <cit.>. 

While a brown dwarf is indeed consistent with the RVs, this scenario nevertheless seems implausible for NGC 4349 No. 127, as it would not be clear how the line shape and CRX variations could be explained. If we, for the sake of the argument, assume that the brown dwarf exists, is engulfed in a dusty cloud, and is transiting the star, then transit durations are on the order of T_transit∼ 22 d, even if one were to assume a radius of 10 R_ for the dust cloud. In principle, the Rossiter-McLaughlin effect can explain the line shape variations during the transit itself. But since we observe a near sinusoidal variation, for instance, for the FWHM, the variations out of transit would remain unaccounted for. It would be even more problematic in the cases of Aldebaran or γ Dra, which show RV amplitude changes or phase shifts that are hard to explain with an orbital companion. This scenario can thus not explain the observables. 

It is, of course, possible that the variability on sequence D, mostly detected via photometry, is simply not related to the intrinsic RV variations of the discussed giants. Moreover, it is possible that even within the small group of these false positive planet hosts, different intrinsic mechanism are at work. For NGC 4349 No. 127, non-radial oscillations provide the most convincing explanation for all observables to date. Whether these are linked to oscillatory convective modes or not remains uncertain. 



 §.§ Magnetic activity

<cit.> propose the rotational modulation of magnetic spots as the origin of the RV variations. However, the available photometry and the newly presented activity indicators lead us to refute the spot hypothesis. 

If magnetic surface spots cause the RV periodicity, significant photometric variations are to be expected. Following <cit.>, we can roughly estimate the required spot filling factor to produce RV variations with amplitude A_RV=235.6 m/s (see Fig. <ref>) as f=7.3%, using vsin i=4.81 km/s <cit.>. <cit.> used T_eff=5800 K and T_spot=4600 K for the effective temperatures of the stellar photosphere and the star spot, respectively. Downscaling the spot temperature to yield the same flux ratio as considered by <cit.>, but at effective temperature T_eff=4417 K <cit.>, yields a temperature difference Δ T_spot=914 K between photosphere and spot. Assuming black body radiation, we calculate the flux in a Bessel V-band filter for a photosphere with a star spot of filling factor f to be reduced by 5.7% compared to the flux of a photosphere with no spot present. This corresponds to a magnitude difference of Δ V ∼ 0.06 mag. 

We searched for the largest amplitude consistent with the ASAS-3 V-band photometry by fitting sinusoids with fixed periods in a box of size 100 d around the best RV period P=674.0 d. The largest amplitude present in the data is A_V=0.006±0.003 mag, an order of magnitude lower than predicted by our rough estimate. We further show in Appendix <ref> that the apparent periodicity at roughly half the RV period is caused by the window function of the ASAS-3 observations. We therefore conclude that there are no photometric variations at periods related to the RV period in the ASAS-3 data set that are consistent with the prediction.

Another strong argument against the RV variations being caused by star spots is the positive correlation between CRX and RV in Fig. <ref> (top left panel). If spots were to cause the RV variation, one would expect the CRX to be anticorrelated with the RV (negative chromaticity), as the contrast (flux ratio) between the cool spot and the stellar photosphere decreases toward longer wavelengths <cit.>. The same relation holds for hot active regions <cit.>. Thus, the positive chromaticity observed for NGC 4349 No. 127 is challenging to explain with a spot model, while it is a natural consequence of the limb darkening in case of dipole non-radial oscillations.

We note that <cit.> recently discussed other magnetic structures that locally reduce convection and lead to RV variations without photometric variations. However, it is not clear if such structures could explain any of the correlations present for NGC 4349 No. 127. We therefore find the oscillation hypothesis more convincing.



§ SUMMARY


We presented a reanalysis of archival HARPS spectra of the cluster giant NGC 4349 No. 127, which was originally thought to host a brown dwarf orbital companion <cit.>. <cit.>, however, refute the brown dwarf's existence based on variations of the FWHM of the CCF and the Hα index. This makes NGC 4349 No. 127 another example of a group of giant stars with RV periodicities that mimic orbital companions, but are of unknown intrinsic origin. 

We first presented additional activity indicators available through the use of the SERVAL and RACCOON reduction pipelines, revealing that, along with the FWHM,   the contrast of the CCF, the dLW, and the CRX also have significant periodicity at the proposed orbital period. We therefore affirm the findings of <cit.> and unequivocally refute the orbital companion. Furthermore, we find a significant, positive correlation between the CRX and the RVs, as well as closed-loop correlations between the RVs and dLW, FWHM, and contrast, respectively, a result of sinusoidal variations with phase shifts close to π/2.

We presented the simulation software , capable of simulating synthetic HARPS spectra, along with their respective RVs and activity indicators, based on a model of non-radial oscillations in the photosphere of a star. Such non-radial oscillations were previously proposed to cause the intrinsic RV variations that mimic orbital companions. 

We showed that an isolated, low-amplitude, retrograde, dipole (l=1, m=1) oscillation reproduces the periodicities and amplitudes of the RVs and all activity indicators, along with their respective phase relations and thus the Lissajous figures of their correlations. The oscillation model also predicts low-amplitude photometric variations at the orbital period, which are well below the detection threshold of the available ASAS-3 data, and thus cannot be ruled out.

We also showed that other dipole (l=1) modes, while producing similar RV variations, are inconsistent with the activity indicators measured for NGC 4349 No. 127 and present the expected variations for all modes. We showed that the CRX is the only indicator yielding variations for all azimuthal orders m allowed for dipole (l=1) modes, while the line shape diagnostics are mostly inconspicuous for the l=1, m=0 mode. In the presence of scatter in the CRX data, such modes and combinations of different l=1 modes could therefore be difficult to detect. 

Moreover, we explored higher-order modes, showing that modes with m=l and m=-l show similar behavior to the l=1, m=1 mode.  The l=2, m=-2 mode cannot entirely be ruled out as the  cause of the observed variations. In general, the CRX provides the most direct approach to differentiate between modes, but it is also affected by the scale of the temperature variations employed, and  thus cannot be used to rule out modes unambiguously. 

We finally showed that the stellar parameters of NGC 4349 No. 127 are broadly consistent with the models of oscillatory convective modes <cit.> put forth to cause non-radial oscillations of bright giant stars. We showed that reliable planet detections, on the one hand, and more uncertain planet systems and false positives, on the other, are separated in a period-luminosity plot. This suggests that the same intrinsic phenomenon might be present in the latter group. We note, however, that the masses of most giant stars with intrinsic RV signatures are higher than predicted by the models of <cit.>.
We further give arguments that the correlations observed for NGC 4349 No. 127 cannot be explained by magnetic spots. 

We conclude that the RV variations of NGC 4349 No. 127 are likely caused by a retrograde, dipole (l=1, m=1) non-radial oscillation. We also provide, for the first time,  a testable model capable of explaining the intrinsic RV variations discussed in the literature. 

The case of NGC 4349 No. 127 shows the importance of acquiring a set of reliable and precise activity indicators targeting the shape of spectral lines and the wavelength dependence of the RV. However, as m=0 modes do not show large variations of the shape of spectral lines, and would only be detected by the CRX, which often suffers from significant scatter, long-term RV monitoring remains crucial in order  to detect any changes in the amplitude, period, or phase of the RV variation. 

Applying the same analysis to other evolved stars with similar luminosities and RV periods could offer valuable insights to constrain their (potential) intrinsic nature. This analysis may thus contribute to verifying the reliability of the planet population orbiting evolved stars. 


We thank the anonymous referee for their helpful comments that improved the quality of this work. We further thank M. Zechmeister for fruitful discussions regarding the SERVAL reduction results. 
      This work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the Priority Program SPP 1992 “The Diversity of Exoplanets” (RE 2694/7-1). Based on data obtained from the ESO Science Archive Facility with DOI: <https://doi.eso.org/10.18727/archive/33>. This research has made use of the VizieR catalog access tool and the SIMBAD database, operated at CDS, Strasbourg, France. This work has made use of data from the European Space Agency (ESA) mission
Gaia (<https://www.cosmos.esa.int/gaia>), processed by the Gaia
Data Processing and Analysis Consortium (DPAC,
<https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research made use of Astropy,[<http://www.astropy.org>] a community-developed core Python package for Astronomy <cit.>. We further acknowledge use of the Python packages NumPy <cit.>, SciPy <cit.>, Matplotlib <cit.>, pandas <cit.>, PyAstronomy[<https://github.com/sczesla/PyAstronomy>] <cit.>, lsq-ellipse <cit.>, the RV fitting tool Exo-Striker <cit.>, and the color map “vikO” <cit.>. 


aa 





§ LINEAR CORRELATION BETWEEN H ALPHA AND RV





§ DISCUSSION OF THE ASAS-3 PHOTOMETRY





In Fig. <ref> (top), we plot the available ASAS-3 data against time, color-coding the individual data points with respect to the RV period. For enhanced visibility, we further show binned data points as black rectangles. The photometric variation predicted by the l=1, m=1 oscillation model discussed in Sect. <ref> is overplotted as the colored line and is much smaller than the scatter present in the ASAS-3 data. It is therefore reasonable to assume that the photometric variations would not have been detected in the ASAS-3 data set. 

In Fig. <ref> (top), we plot the GLS periodogram of the ASAS-3 data, labeling the four most significant peaks in units of days. <cit.> discuss the same data set and argue that the peak at 338 d is close to the first harmonic (P/2) of the RV period or could be linked to the rotation period of the star. However, we find that the most significant peak at 2592 d provides a much more convincing fit to the ASAS-3 data set, which we plot in black in the top panel of Fig. <ref>. The sinusoidal fit has an amplitude of 0.05 mag and a period of 2571 d. The discrepancy to the 2592 d period found in the GLS periodogram stems from the sampling of the periodogram. We show a phase-folded plot of the ASAS-3 data in the bottom panel of Fig. <ref>.

In the middle panel of Fig. <ref>, we show the GLS periodogram of the residuals after removing the fit with period 2571 d. The bottom panel further shows the periodogram of the window function computed as a Lomb-Scargle (LS) periodogram, as suggested by <cit.>. The formerly second strongest peak at 430 d coincides with the alias of the 2592 d signal and the strong yearly period present in the window function. It is therefore much reduced in significance by removing the long-period signal. 

The interesting 338 d period also changes significantly by removing the fit with period 2571 d. It is reduced in significance and shifted to 361 d, coinciding quite closely with the yearly period. The two neighboring peaks at 248 d and 532 d are furthermore close to the alias period of the yearly period and the 950 d period present in the window function. We therefore argue that the 338 d signal discussed by <cit.> is likely not real and instead an artifact of the window function, as well as its neighboring peaks. 

The origin of the fourth peak at 1160 d present in the original GLS periodogram is less certain. While it is close to the first harmonic of the long-period signal, it is not entirely suppressed by removing the 2571 d fit. It is, however, shifted closer to the 950 d peak present in the window function and could therefore potentially be explained as another windowing artifact. 

For the 2592 d peak, there is no obvious peak present in the window function. It is also unlikely to be caused by the typical rise of the periodogram power of the window function at periods longer than the observing window. We therefore argue that the 2592 d signal is the only signal that cannot be explained by the window function. Whether this signal is of instrumental or stellar origin is uncertain. If it is real, it seems unlikely to be related to the non-radial oscillation discussed in Sect. <ref>. 

It could, in principle, be related to the rotation period of the star. However, given the radius of the star, this would lead to a rotation velocity of v_rot∼ 0.75 km s^-1, considerably lower than most estimates in the literature (see Table <ref>), but consistent with the estimate by <cit.>. We note that the presented simulation of a l=1, m=1 oscillation mode also requires a somewhat lower rotation velocity to reproduce the variations of the line shape indicators, but caution to use this value to determine the actual rotation velocity of the star. 



§ OSCILLATION MODES WITH M=L




§ OSCILLATION MODES WITH M=-L




§ HARPS DATA SET







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

 


   

   

   Valerin et al.
   

   Affiliations are listed after the acknowledgements
             

   



 
  
  
   
  
   We investigate the spectroscopic characteristics of Intermediate Luminosity Red Transients (ILRTs), a class of elusive objects with peak luminosity between that of classical novae and standard supernovae. Our goal is to provide a stepping stone in the path to unveil the physical origin of such events, thanks to the analysis of the datasets collected.
  
   We present the extensive optical and near-infrared (NIR) spectroscopic monitoring of four ILRTs, namely NGC 300 2008OT-1, AT 2019abn, AT 2019ahd and AT 2019udc. First we focus on the evolution of the most prominent spectral features observed in the low resolution spectra, then we discuss more in detail the high resolution spectrum collected for NGC 300 2008OT-1 with the Very Large Telescope equipped with UVES. Finally we analyse late time spectra of NGC 300 2008OT-1 and AT 2019ahd through comparisons with both synthetic and observed spectra.
  
   Balmer and Ca lines dominate the optical spectra, revealing the presence of slowly moving circumstellar medium (CSM) around the objects. The line luminosity of Hα, Hβ and Ca II NIR triplet presents a double peaked evolution with time, possibly indicative of interaction between fast ejecta and the slow CSM. The high resolution spectrum of NGC 300 2008OT-1 reveals a complex circumstellar environment, with the transient being surrounded by a slow (∼30 km s^-1) progenitor wind. At late epochs, optical spectra of NGC 300 2008OT-1 and AT 2019ahd show broad (∼2500 km s^-1) emission features at ∼6170Å and ∼7000Å which are unprecedented for ILRTs. We find that these lines originate most likely from the blending of several narrow lines, possibly of iron-peak elements.
  
   

   

   A study in scarlet - II. Spectroscopic properties of a sample of Intermediate Luminosity Red Transients
    G. Valerin1E–mail: giorgio.valerin@inaf.it,
          A. Pastorello1,
          E. Mason2,
          A. Reguitti1,3,
          S. Benetti1,
          
          Y.-Z. Cai4,5,6,             
          T.-W. Chen7,         
          D. Eappachen8,9, 
          N. Elias-Rosa1,10,
M. Fraser11, 
A. Gangopadhyay12,13,  
E. Y. Hsiao14,           
D. A. Howell15,16,
C. Inserra17,             
L. Izzo18,19,
J. Jencson20,         
E. Kankare21,
R. Kotak21, 
P. Lundqvist12,22,
P. A. Mazzali23,24,         
K. Misra25, 
G. Pignata26,
S. J. Prentice27,           
D. J. Sand28,        
S. J. Smartt29,30,
M. D. Stritzinger31, 
L. Tartaglia32,
S. Valenti33,
J. P. Anderson34,35,
J. E. Andrews28,
R. C. Amaro28,              
C. Barbarino12,
S. Brennan12,
F. Bufano36,          
E. Callis12,
E. Cappellaro1,
R. Dastidar37,35,
M. Della Valle18,38,               
A. Fiore1,39,40,          
M. D. Fulton30,
L. Galbany10,41,
M. Gromadzki42,
T. Heikkilä21,            
D. Hiramatsu14,15,43,44,             
E. Karamehmetoglu12,31,    
H. Kuncarayakti21,45,
G. Leloudas46,  
M. Limongi47,48,49
M. Lundquist28,                
C. McCully15,
T. E. Müller-Bravo10,41, 
M. Nicholl30,
P. Ochner1,50, 
E. Padilla Gonzalez15,16,    
E. Paraskeva51,
C. Pellegrino52,     
A. Rau53,
D. E. Reichart54,
T. M. Reynolds20,55,56,      
R. Roy57,                                    
I. Salmaso1,    
M. Shahbandeh58,59,
M. Singh60,
J. Sollerman12,
M. Turatto1,               
L. Tomasella1,                                 
S. Wyatt28,
D. R. Young30
          
    Received —-; accepted —-
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================





§ INTRODUCTION


The transient sky is brimming with a wide variety of phenomena, and spectroscopic observations are a crucial tool to navigate through such a complex landscape. Indeed, remarkable efforts have been made in order to promptly classify newly discovered transients, culminating in the formation of dedicated spectroscopic surveys such as the “Public European Southern Observatory Spectroscopic Survey of Transient Objects" (PESSTO) consortium <cit.>.
The swift collection of spectra is rewarded with key insights on the ongoing physical processes which could not be otherwise probed. In the past years there has been a growing number of studies on objects displaying narrow and intense emission lines coupled with a slow spectral evolution: these are the hallmarks of optically thick circumstellar medium (CSM) surrounding the transient (e.g. ). Such a scenario has been investigated primarily in Supernovae IIn (SNe IIn): terminal stellar explosions within an opaque CSM formed during the years prior to the event <cit.>. 

Critically, SNe IIn are not the only events showing evidence of optically thick CSM. Violent eruptions of Luminous Blue Variables (LBVs) mimic the characteristics of a genuine SN, despite being non-terminal events: for this reason they earned the designation of “SN impostors"  (e.g. ). 

The picture was further complicated by the discovery of other classes of transients that display clear signs of CSM in their spectra. Luminous Red Novae (LRNe) have been argued to arise from mergers of non-degenerate stars after a common envelope phase where the final product of the merger event is expected to survive <cit.>. Finally, perhaps the most poorly understood among the objects discussed so far, there are the subjects of the present study: the Intermediate Luminosity Red Transients (ILRTs), whose nature is still debated. Together, LBVs, LRNe and ILRTs make up a wider class of objects with luminosity intermediate between Classical Novae and standard SNe, called “Gap Transients" (see  and  for a review).
While their physical origin may be remarkably different, all these transients have one thing in common: the presence of optically thick CSM surrounding the system.
The resulting similarity among the spectra of the Gap Transients and SNe IIn makes a proper classification of a newly discovered transient quite difficult: it can be challenging to unmask an impostor <cit.>.




Photometric follow-up can provide some important clues for this type of investigation. The light curve shape can be a useful indicator for classification: LBVs alternate periods of quiescence with erratic and powerful outbursts <cit.>, LRNe often show distinctive double peaked light curves, while ILRTs are characterized by SN-like light curves <cit.>. 
It is still essential to complement the photometric follow-up with a spectroscopic study to characterize each class, despite the partial degeneracy in the properties of these transients.
From a spectroscopic point of view, ILRTs can be summarily described as a black body continuum with narrow emission lines superimposed. Strong Balmer emission lines dominate their spectra together with several Ca features: Ca II H&K in absorption, strong Ca II NIR triplet in emission as well as the [Ca II] forbidden doublet. Opposed to LRNe, ILRTs show no sign of molecular bands in the late phases of their evolution <cit.>.



A key difference between ILRTs and SNe IIn is that both the peak observed temperature (T_ILRT∼7500 K) and peak luminosity (L_ILRT ≲ 5x10^41 erg s^-1) of ILRTs are significantly lower compared to SNe IIn, which instead tend to be hot (T_IIn∼10000 K, , but up to ∼15000 K, e.g. ) and even brighter than average SNe II (e.g. ). The photospheric temperature and luminosity observed for SNe IIn can be sustained even for years (e.g. ), thanks to the interaction between fast ejecta and slow moving CSM, a mechanism that can efficiently convert kinetic energy into radiation <cit.>. When the interaction region between ejecta and CSM is visible, specific spectral features can be identified, most notably a blue shoulder on the Hα line <cit.>. ILRTs, however, do not show clear signs of strong dynamical interaction between ejecta and CSM in their spectra: their low temperature and luminosity could imply a rather weak ejecta-CSM interaction, or at least that such interaction is hidden underneath layers of unshocked CSM.




As previously mentioned, the physical mechanism behind ILRTs is still debated. Since the photosphere lies within an optically thick gas shell, it is not possible to probe the material underneath the CSM, which could reveal key information on the nature of the transient itself. This uncertainty has led to a variety of interpretations regarding their true nature.
Some researchers have advocated outbursts of post-main sequence stars <cit.>, while others have suggested that ILRTs are the result of jets originating from interacting binary stars <cit.>. Finally, ILRTs were also associated with Electron Capture Supernovae (ECSNe, e.g. ) originating from Super AGB stars <cit.>.
ECSNe are expected to have some distinctive features that separates them from classic Fe core collapse events. As was discussed for SN 2018zd, nebular spectra of ECSN are expected to show Ni features stronger than Fe lines, due to the different nucleosynthesis yields in the two scenarios <cit.>. Indeed, <cit.> provide synthetic nebular spectra of a 9 M_⊙ Fe core collapse SN, and approximate the behaviour of an ECSN by enhancing the Ni content in their Fe core collapse model. The Ni enriched spectra display particularly strong [Ni II] emission lines.
Until now, such an analysis has not been performed for ILRTs due to the lack of high quality nebular spectra, as well as the lack of detected features beside the ones originating in the CSM.

In this paper we present and analyse spectra of four ILRTs, following the photometric analysis of the same targets performed in “A study in scarlet I. Photometric properties of a sample of Intermediate Luminosity Red Transients" (Valerin et al. 2024, hereafter "Paper I"). The work is structured in the following manner: in Section 2 the data reduction techniques are described. In Section 3  we present the optical and near infrared (NIR) data collected. In Section 4 the line luminosity and velocity are measured and compared within the sample. Section 5 is dedicated to the study of the high resolution spectroscopy obtained for NGC 300 2008OT-1, while in Section 6 we discuss the broad features appearing in the late time spectra of NGC 300 2008OT-1 and AT 2019ahd. Finally, in Section 7 we summarize our work and draw the conclusions.




§ DATA REDUCTION

The analysis presented in the following sections is based on the spectroscopic observations of four different ILRTs over the course of several months. Our follow-up campaigns are focused on the optical and Near-Infrared (NIR) domains, employing several telescopes equipped with different instruments. The data reduction procedure is based on the Image Reduction and Analysis Facility (, ), a software specifically designed for image processing developed within the astronomical community.
Every image undergoes the same preliminary reduction procedure: trimming, bias and flat-field correction. NIR spectra go through an additional sky subtraction step when combining the different nodding positions.
A one dimensional spectrum of the target is extracted from the two dimensional image, while simultaneously performing a background subtraction in order to avoid contamination from the sky emission lines and the host galaxy signal.
The one dimensional spectrum is then calibrated in wavelength on standard spectral lamps, typically showing He, Ne, Ar, Hg and Kr lines.


These procedures are repeated on the spectrum of a standard star (belonging to the list of ESO spectrophotometric standards[https://www.eso.org/sci/observing/tools/standards/spectra/stanlis.html]), which is routinely obtained during every observation run. Thanks to the comparison between the observed and tabulated flux values of a standard star, it is possible to accurately flux calibrate the spectrum of the target. Standard star observations are also useful to remove the telluric absorption features which affect ground-based observations. An additional, wavelength independent flux calibration is performed, ensuring that the fluxing of the spectrum is compatible with the values measured from coeval photometric observations. Finally, each spectrum is corrected for redshift and reddening, in particular assuming a reddening law R_V = 3.1 both for Galactic and internal extinction <cit.>. The distance, reddening and redshift adopted for each transient are summarized in Table <ref>: for a discussion on the choice of the values used, see Paper I.









Many facilities and instruments were involved in the follow-up campaigns of our targets. Indubitably, the backbone of our spectroscopic monitoring was obtained through the Nordic Optical Telescope (NOT) within the “NOT Unbiased Transient Survey 2" collaboration (NUTS2, ) and the instruments available within the “Global Supernova Project" (GSP, ). Other telescopes used during this project include the Gran Telescopio Canaria equipped with OSIRIS, the William Herschel Telescope equipped with ACAM and LIRIS, the South African Large Telescope equipped with RSS, and the New Technology Telescope plus EFOSC2, thanks to the ePESSTO+ consortium <cit.>. Additionally, multiple spectra of NGC 300 OT were taken with the 6.5 m Magellan Telescopes, and the 4.0-m R-C Spectrograph at the Cerro Tololo Inter-American Observatory, and with the Very Large Telescope, most notably a high resolution spectrum with UVES <cit.> and a late spectrum with X-Shooter <cit.>. We also integrated our dataset with some publicly available spectra to have a more complete sampling of each event ( for AT 2019ahd and  for AT 2019abn).
The detailed logs listing all the spectroscopic observations for each target are reported in Tables  <ref>, <ref>, <ref> and <ref>.





















§ OVERVIEW OF THE DATA




 §.§ Optical Spectra
 
 Spectra obtained for NGC 300 OT are displayed in Figures <ref> and <ref>, while Figures <ref> and <ref> show the spectral sequence of AT 2019abn. Figures <ref> and <ref> present the spectral evolution of AT 2019ahd and AT 2019udc, respectively. All spectra were corrected for redshift and total extinction along the line of sight: values adopted for these corrections are reported in Table <ref>. The most prominent features noticeable in these objects are the emission lines of the Balmer series. Their narrow profiles (≲1000 km s^-1) indicate that the spectral lines are forming in a region where the material is moving slowly, compatible with an opaque CSM. We do not detect clear P-Cygni profiles in the emission lines, which would be caused by absorbing gas approaching the observer along the line of sight. The absence of the P-Cygni features may imply that the photosphere of the transient is located at the outer edge of the CSM, that the transient is intrinsically asymmetric, or that the covering fraction of the absorbing material is low.
The Ca II H & K absorption lines (λλ 3934,3968) are found in all objects, especially at early phases. The photons absorbed through the Ca II H & K resonance lines excite the electrons of the Ca II atoms from the ground state 3s^2 3p^6 4s ^2S_1/2 to 3s^2 3p^6 4p ^2P^o_1/2,3/2. From this energy level the electrons can radiatively decay to the level 3s^2 3p^6 3d ^2D_3/2,5/2, giving rise to a triplet of lines (λλλ 8498,8542,8662). Indeed this Ca II NIR triplet is detectable at most epochs in ILRTs, and it becomes dominant at later times, as displayed in Figures <ref> and <ref>. The compatible shape of the Ca II NIR and Ca II H& K lines in velocity space corroborates this interpretation (e.g. see Figure <ref>).

















 Finally, the transition that closes this “loop" envisions the de-excitation of the electrons from the level 3s^2 3p^6 3d ^2D_3/2,5/2 back to the ground state 3s^2 3p^6 4s ^2S_1/2. Such transition, which produces the [Ca II] doublet (λλ7291,7323), is forbidden.
 Even if the metastable energy level 3s^2 3p^6 3d ^2D_3/2,5/2 is populated, the formation of [Ca II] requires a large amount of gas at moderate excitation and most of all at low density (e.g. ). In fact, in a dense environment, collisions with other atoms or ions de-excitate the electrons back to the ground state before the radiative decay process can take place, consequently suppressing the formation of these lines (e.g. ). Therefore, this spectral feature is only found in regions where the gas density is below the critical density N_crit ∼ 10^7 cm^-3 (for temperatures typically found in ILRTs, adopting the collision strength reported by ).
 Based on these premises, the ubiquitous presence of the [Ca II] doublet is one of the most characteristic features in ILRTs spectra. While not unique to ILRTs, the persistent presence of the [Ca II] forbidden doublet in all the objects analysed so far has been proposed as a key feature to distinguish ILRTs from other transients that present similar spectra <cit.>. 

 
 
 

 Another noteworthy feature in the low resolution optical spectra of our sample of ILRTs is the presence of O I lines. In the spectra of all the considered objects the O I triplet λλλ 7771.9,7774.2,7775.4 is found in absorption. This triplet is detected at multiple epochs in NGC 300 OT (Figure <ref>), AT 2019abn (Figures <ref> and <ref>) and AT 2019ahd (Figure <ref>), while in the case of AT 2019udc it is confidently detected only in the GTC spectrum obtained on 2019-11-22, 8 days after maximum. The O I triplet in absorption is a sign that the metastable level 2s^22p^3 (^4S^o)3s^5S^o_2 is populated. We do not detect any emission lines from transitions that may feed this energy level: first of all the λλλ 7771.9,7774.2,7775.4 triplet is never clearly detected in emission at any epoch, but we also do not detect transitions from higher energy levels (e.g. O I λ3947). This suggests that the electrons populated the metastable energy level s^22p^3 (^4S^o)3s^5S^o_2 by recombination after an initial excitation. With time, the O I absorption triplet becomes less pronounced, until its detection becomes uncertain in the latest phases. The absence of a clear emission component of the O I triplet could be due to efficient absorption by electrons in the metastable energy level.
 Few months after maximum luminosity there is also the appearance of the O I λ8446 emission line, which becomes progressively more prominent with time. In AT 2019abn this feature appears around +155 d, in AT 2019ahd around +93 d and in NGC 300 OT around +123 d (but this emission line is detected already in the high S/N spectrum at +47 d discussed in Section <ref>). This feature is not found in the spectra of AT 2019udc, but its spectroscopic monitoring had to be stopped at +67 d, therefore this emission line could have developed after the end of our observational campaign. The appearance of the O I λ8446 line is particularly interesting, since it cannot be the result of the de-excitation of O I atoms, which would result in the coeval formation of the λλλ 7771.9,7774.2,7775.4 triplet in emission. Instead, as discussed above, the O I triplet has only been detected in absorption.
 We conclude that the O I λ8446 emission line found in ILRTs is the consequence of UV pumping. 
 This process is also known as Bowen fluorescence, and it is caused by the resonance between the Lyβ λ1025.72 Å and the O I λ1025.76 line, which excites the electrons from the ground level 2p^4 ^3P_2 to 2p^33d ^3D^0_3 <cit.>. The electrons subsequently decay to the levels 2p^33p ^3P_2 (emitting λ11287 Å), 3p^33s ^3S^0_1 (emitting λ8446 
 Å) and finally back to the ground state (through the UV triplet λλλ1302.2,1304.9,1306.0). Indeed, in the NIR spectra of NGC 300 OT presented in the following section we detect the O I λ11287 Å emission line, corroborating the fact that in these ILRTs we are observing the fluorescence process described above, rather than a different pumping mechanism (e.g. see the different processes described by ). Since the excitation of the O I electrons to the 2p^33d ^3D^0_3 energy level requires Lyβ photons, the appearance of prominent O I λ8446 Å at relatively late phases (and not at early ones) needs to be addressed. In the context of faster ejecta expanding underneath a slow moving CSM, the Lyβ could be shifted off resonance due to Doppler effect: in fact, to have a perfect wavelength match, the Lyβ photons should be redshifted by 11.7 km s^-1 with respect to the oxygen atoms. 
 In the early phases, when the resonant lines are opaque, the Lyβ escapes the ejecta from a compact pseudophotosphere which is moving faster than the CSM above, greatly reducing the coupling between the Lyβ and the O I λ1025.76 line. As time progresses and the ejecta expand, the opacity of the lines drops and CSM is finally reached by the redshifted Lyβ photons that can trigger the Bowen fluorescence, hence explaining the delayed appearance of the O I λ8446 line. 

 
 
 

 
 
 
 It is worth noting that all spectra of ILRTs in our sample become bluer during the rise to maximum: such behaviour is observed in other interacting transients (e.g. SN 2023ixf, ) and predicted by models of interacting SNe <cit.>, and therefore it could be a result of increasing interaction between the ejected material and the CSM. This interpretation is also supported by the increasing flux of the emission lines during the rise to maximum luminosity in all observed ILRTs (see Section <ref>). Finally, the late time spectra (≳100 days) collected for NGC 300 OT, AT 2019abn and AT 2019ahd do not show clear blueshift nor change in the shape of the emission lines. These are telltale signs of dust along the line of sight, and dust formation was inferred for these objects in Paper I from the NIR excess in their SED. A possible explanation for the lack of these spectroscopic signatures may be that the dust is not located along the line of sight: in this scenario, ILRTs would be intrinsically asymmetric events.

In Figure <ref> we display spectra of six ILRTs taken in the first weeks after maximum luminosity. Beside the four targets in our sample, SN 2008S <cit.> and AT 2017be <cit.> are also shown. SN 2008S is considered to be a prototype of the ILRT class thanks to its prominent Ca lines, with a peak luminosity well in line with our sample of objects. On the other hand, AT 2017be lies in the lower end of the luminosity distribution of ILRTs (see Figure 5 of Paper I).
On the right panels of Figure <ref> we zoom in on the most characterising features of this class, in particular the [Ca II], Ca H&K and Hα lines. The strength of each feature varies among the sample: for instance, AT 2019udc is characterised by a strong Ca H&K absorption, while the [Ca II] doublet is most prominent in NGC 300 OT and SN 2008S. Albeit present, the Ca II NIR triplet becomes more prominent in all ILRTs at later phases.










 §.§ Near Infrared Spectra




Beside the optical spectra, we gathered a total of five NIR spectra of our targets. NIR spectra of AT 2019abn were collected during the early rise, immediately after discovery. The single spectrum available for AT 2019udc was taken in a more advanced phase, three weeks after peak luminosity. Finally, the two spectra of NGC 300 OT monitor the object at late phases, hundreds of days after maximum light. As displayed in Figure <ref>, AT 2019abn and AT 2019udc appear to be quite similar: a nearly featureless continuum, without strong emission or absorption lines superimposed. 
Conversely, in the spectra collected for NGC 300 OT it is possible to identify several emission features. Most notably, these include hydrogen lines of the Paschen (Pa β,γ,δ,ϵ) and Brackett series (Br γ), He I (λ10830), O I (λ11287) and Fe I (λλλ15831,16679,16693). As mentioned in the previous section, the detection of O I λ11287 Å has been key to confirm that the optical line O I λ8446 Å is produced by the Bowen fluorescence.






§ LINE PARAMETERS




 §.§ Line luminosity
 





In this section we estimate the parameters of the main lines observed in our spectra. In order to avoid an additional source of systematic uncertainty, we do not assume a specific line profile (e.g. Gaussian or Lorentzian): instead we estimate the luminosity contribution of a given feature by integrating the flux over the continuum in the considered spectral region. This procedure is repeated multiple times in order to estimate the error associated with the measurement. The measured line flux is then converted into the luminosity of the line by accounting for the distance of the transient (the spectra were already corrected for extinction during the calibration procedure).
The luminosity evolution of Hα, Hβ, Ca II NIR triplet and [Ca II] doublet is shown in Figure <ref>. In each panel we also represent the continuum level in the proximity of each spectral feature as a solid line. For clarity, the continuum level is scaled to match the line luminosity at the first measured epoch. The continuum was estimated in the range 4725-4825 Å for Hβ, 6700-6850 Å for Hα and [Ca II] and 8200-8350 Å for Ca II NIR triplet. In the same Figure, the bottom panels report the equivalent widths (EW) for Na ID and Ca II H&K in absorption. We remark that the spectral features examined can present multiple components in high resolution (see in particular Section <ref>), therefore the values reported here track the behaviour of the sum of all components of each line. An additional complication is given by the fact that in the case of optically thick lines we are not sampling the whole emitting volume.


Considering the emission features, we note that the luminosity evolution of Hα and Ca II NIR triplet is not monotonic: after an early peak, the luminosity declines sharply over the course of a few weeks. A minimum is then reached, and subsequently the lines start to strengthen again, forming a sort of broad, long lasting second peak. This trend is even more clear when compared to the evolution of the spectral continuum at the same phases, which is monotonically declining after maximum brightness. The same evolution can be detected also in Hβ, although less pronounced. For NGC 300 OT, only the second broad peak is visible, because the object was behind the Sun during the first phases of its evolution. This line rebrightening has already been noticed in other ILRTs and it was tentatively attributed to CSM interaction <cit.>. In this scenario, there is a collision between rapidly expanding ejecta and the slower CSM lying above. The energy released from this interaction could lead to the re-ionization of the gas above, and its subsequent recombination. The rebrightening of the hydrogen emission lines being coeval with the Ca II NIR strengthening (see Figure <ref>) suggests that these features originate in a similar environment and are produced by material undergoing similar physical processes. 
At the same time, we note that during this rebrightening of the lines there are no relevant changes in the ionisation state nor in the profile of the lines in the low resolution spectra, which are typical telltale signs of ongoing interaction. This means that either the CSM along the line of sight is optically thick enough to hide the interaction region, or that the secondary peak in line luminosity is caused by a mechanism other than CSM interaction. An alternative scenario to CSM interaction could be presence of material that is reached by the photons produced by the transient only after some time after its onset (in the order of one or two months). This material would become excited as it is overflown by photons, producing Balmer and Ca II NIR emission lines as it de-excites. Based on the onset time of the second peak (30 days for AT 2019udc, up to 55 days for AT 2019abn) and the speed of light, in this scenario the material providing the line luminosity for the secondary peak would be located at least at 8 x 10^16 cm (1.4 x 10^17 cm) from the transient. In any case, the secondary luminosity peak observed for the emission lines favours the scenario of a complex environment surrounding ILRTs, which could have been formed by a stellar wind or eruptive episodes of the progenitor star.

 Whatever the mechanism driving its evolution, the marked rebrightening shown by the Ca II NIR and Hα lines is a characteristic commonly found in ILRTs.
 In contrast, the luminosity of the [Ca II] forbidden doublet displays an overall monotonic decline. We point out that the [Ca II] evolution appears to mimic the shape of the continuum luminosity in all four objects. This behaviour is consistent with the fact that the [Ca II] doublet originates from low density gas, farther from the dense CSM. Its evolution is therefore dictated by the incoming radiation from the transient, in particular by the radiation absorbed by the Ca H& K lines. As previously described, the emission of the [Ca II] doublet can only happen if the medium is below the critical density. In this configuration, the hypothetical interaction between ejecta and CSM can cause the collisional excitation of Ca and H atoms in the dense CSM, which is over the critical density for the production of the [Ca II] doublet, therefore resulting in a rebrightening only for Balmer and Ca II NIR triplet lines. At the same time the slow material producing [Ca II] lines is too distant to be affected by the collision, and keeps following the evolution of the spectral continuum.












The Na ID EW depends on the properties of the CSM, the interstellar gas along the line of sight, possible ejecta and even the viewing angle in case of non-spherically symmetric ejecta: for these reasons, its behaviour presents a large variance within the sample. AT 2019udc shows a strong Na ID feature in the first two observed epochs (EW∼5 Å), which become rapidly weaker during in the following days: a similar behaviour was observed for the ILRTs AT 2010dn and AT 2012jc <cit.>. On the contrary, AT 2019abn initially displays a weak Na ID absorption feature (EW∼1 Å), which becomes more significant in the following weeks (up to EW∼3 Å). AT 2019ahd and NGC 300 OT present a similar behaviour, although scaled down compared to AT 2019abn. The increase and following decrease in EW of the Na ID over time can be qualitatively reproduced by considering the column density of Na I laying in front of a shock travelling through a thick CSM <cit.>: in this simple scenario the recombination of Na II is responsible for the initial increase in Na ID EW, while the shock travelling outward and sweeping the gas eventually causes its decline.
Finally, the EW evolution of Ca II H&K appears to be quite similar in AT 2019ahd and AT 2019udc: an initial rapid decline followed by a somewhat broad "peak". NGC 300 OT, missing the first phases of evolution, displays a slow decline in EW that is resembling the evolution of AT 2019udc in particular. In stark contrast with the behaviour of the other three ILRTs, AT 2019abn displays Ca II H&K absorption features that become significantly more pronounced after the first month of evolution. In principle, high resolution and multi-epoch observations of Na ID and Ca II H&K components in ILRTs would allow us to discriminate if the evolution of these absorption features is driven by a change in the ionization state of the CSM or by the presence of clumps of matter, as prescribed by <cit.>. However, Na ID and Ca II H&K absorption features often saturate in ILRTs, even despite the broadening introduced by the velocity field: this inevitably hinders the physical interpretation of these measurements.







AT 2019abn, the brightest and longest-lasting ILRT observed to date, also displays the brightest Hα and [Ca II] features among the sample, reaching 1.6 × 10^39 erg s^-1 and 10^38 erg s^-1 at peak luminosity, respectively. Conversely, NGC 300 OT is characterised by the least luminous Balmer lines within the sample. However, beside the simple luminosity of these emission features, it is useful to investigate their evolution relative to other lines. 
For this purpose, in Figure <ref> we track the change in value of the Balmer decrement (Hα / Hβ) over time for our sample of ILRTs.
In their first spectra, both AT 2019ahd and AT 2019udc display large values for the Balmer decrement (>4 and >6, respectively), which quickly decrease down to 2.1 and 2.4 in the following two weeks. AT 2019abn and NGC 300 OT do not show this early decline, but this can be caused by the absence of very early spectra, especially for NGC 300 OT. 
With time the Balmer decrement of the whole sample grows significantly, reaching values of 7 and 23 for AT 2019abn and NGC 300 OT, respectively. 
The Balmer decrement evolution just described matches the one observed in SN 2008S. Based on the measured values of Balmer decrement and temperature at early phases, <cit.> constrain the electron density in the CSM to be 10^10 cm^-3 < N_e < 10^12 cm^-3, according to the work performed by <cit.>.
Such high electron densities are not unheard of in the context of SNe IIn: for example SN 2006gy presented an electron density in the CSM above 6×10^10 cm^-3 ().

However, these considerations are based on the assumption of case B recombination, meaning that the gas is optically thick for lines of the Lyman series and optically thin for lines of the Balmer series. Under this conditions, it has been shown that the Balmer decrement is close to ∼3 for a wide range of temperature and density values of astrophysical interest <cit.>. As reported in Figure <ref>, ILRTs display a ratio Hα / Hβ which varies with time, reaching values well above the expected one of ∼3.
There are several possible explanations for this departure from case B recombination, such as the collisional excitation of H atoms, which would boost the Hα luminosity, and dust formation, which would extinguish the Hβ more than the Hα <cit.>. Another mechanism that can cause an increase in the Balmer decrement is an increased opacity of the lines of the Balmer series, as explained by <cit.> (Chapter 4.5). 















 §.§ Width of the spectral features


Measuring the full width at half maximum (FWHM) of the emission lines of ILRTs can be challenging: narrow features such as the [Ca II] doublet are typically unresolved in our spectra. As previously done for the line flux, we do not assume a specific fitting function, but we manually measure the FWHM of the lines. The resulting FWHM is corrected for the resolution of the instrument (typical values of R∼500) and then converted into a FWHM velocity, which can be taken as an order of magnitude estimate of the speed at which gas is moving in the emitting region. This value, however, should not be treated as a precise velocity measurement: the broadening of the line profiles can be caused by the combination of Doppler shift due to bulk motion of the emitting material as well as by the electron scattering of the photons within a hot medium <cit.>. Electron scattering is more marked at higher electron temperatures as well as higher optical depths, resulting in the appearance of extended wings <cit.>.
In Figure <ref> we display the FWHM velocities inferred from the Hα line for our sample of ILRTs in the first 150 days of evolution. Analysing the Hα line has the significant advantage that this feature is bright at all epochs, therefore allowing for an accurate FWHM measurement. Furthermore, especially at early times, the Hα line is resolved in most spectra, and it is not blended with other features (as happens for the Ca II NIR triplet), making it the ideal feature for comparing the evolution of different objects with our dataset. 


During the earliest phases, when the luminosity is still rising, the FWHM velocity of Hα quickly increases in AT 2019abn and AT 2019ahd and AT 2019udc. AT 2019udc, on the other hand, displays the broadest FWHM in the first spectrum; it is impossible to confirm if this is the case also for NGC 300 OT, since we are missing the pre-maximum part of its evolution. The ILRTs in our sample reach a peak FWHM velocity between ∼700 and 1100 km s^-1, with AT 2019abn as a clear outlier and the other three objects clustering around v_FWHM∼800 km s^-1 at peak. 
After the maximum FWHM is reached, ILRTs display a rapid decline over the course of 3-5 weeks. Afterwards, NGC 300 OT settles on an almost constant v_FWHM∼550 km s^-1 for over 100 days, while the other three transients are characterised by a secondary peak in v_FWHM after a local minimum, increasing  by ∼200 km s^-1 in about 5 weeks. During this phase, AT 2019ahd displays the broadest lines, with a v_FWHM up to 750 km s^-1. Eventually the Hα narrows once again, down to 320 km s^-1 at 135 days for AT 2019abn (and unresolved thereafter), 470 km s^-1 at 302 days for AT 2019ahd and 170 km s^-1 at 536 days for NGC 300 OT.
The behaviour of the FWHM velocity just described shows remarkable differences from the photometric features of the transients. For example, NGC 300 OT is two magnitudes fainter than AT 2019abn (see Paper I), yet its FWHM velocity is not only compatible, but oftentimes larger than the one observed for AT 2019abn.
It is important to stress that these FWHM velocities are linked to the properties of the CSM where the line is produced: as the line formation region moves within the CSM, different values of gas speed, electron temperature and optical depth mold the profile of the lines, which is reflected in the evolution of the FWHM velocity shown in Figure <ref>. However, as happens in SN IIn, the low FWHM velocities measured do not exclude the presence of fast ejecta lying below the CSM (e.g. ).














§ HIGH RESOLUTION SPECTROSCOPY OF NGC 300 OT
 

NGC 300 OT is the closest ILRT observed (see  and  for a review of known ILRTs). Thanks to the proximity of its host galaxy (1.92±0.14 Mpc), it was possible to obtain a high resolution optical spectrum of the transient on 2008-07-02 with the Very Large Telescope (VLT) equipped with UVES, yielding a resolution between 0.12 and 0.36 Å. In Figure <ref> the identification of the main spectral lines on this high resolution spectrum is shown. The measurements pertaining to the most prominent features are reported in Table <ref>, in particular their observed central wavelength, their FWHM and their estimated flux.
Apart from the H, Ca, O and Na lines discussed in the previous sections, easily recognizable also in the low resolution spectra, in this high resolution and high S/N we find a number of metal lines, in particular Fe I (λλλλ 5535,8688,8757,8824), Fe II (multiplets 40,42,46,49,73,74 and forbidden line λ 7155), Mg I (λ8806) and Sc II lines (Moore multiplet 29). Additionally, we identify [O I] (λλ 6300,6364), faint Ca II absorption doublet (λλ8912,8927) and He I (λλλ 5876,6678,7065). All these features are narrow (FWHM ≲ 10 Å) but resolved, therefore originating within the opaque CSM or above it. 
Thanks to the high resolution spectrum, it is possible to accurately compare the FWHM velocities associated with different emission features. The broadest feature observed at this epoch is the Hα emission line, with a FWHM velocity of 520 km s^-1. The Ca II NIR line λ 8498 is marginally narrower, with a FWHM velocity of 440 km s^-1: the fact that Hα and Ca II NIR have compatible shapes in velocity space corroborates the scenario outlined in the previous section, where both these features are generated in neighbouring environments. The [Ca II] forbidden doublet, instead, is characterised by a FWHM of 80 km s^-1, pinpointing its origin in a separated region, likely characterized by lower gas velocity, electron temperature and optical depth compared to the region where permitted lines are formed. The [O I] forbidden lines are comparable in FWHM and shape with the [Ca II] doublet.
In general, the range of FWHM velocity displayed by the lines (Table <ref>) suggests a stratification in the emission lines of different ions, and even within the emission lines of the same ions. 









Spectral features that appear simple in low resolution may actually hide a more complex structure, which can only be unveiled through high resolution observations. The shape of such lines carries relevant information on the geometry of the emitting region. In Figure <ref> we show the profiles of the lines of the Balmer series, of some prominent Fe II lines, of the Ca features (NIR λ8498, [Ca II] λ 7291 and Ca II K absorption) and the Na ID absorption doublet. The Hα line in the UVES spectrum shows a composite profile: superimposed to the emission line are two narrow absorption components. The most prominent absorption component has a FWHM of 30 km s^-1 and is only 15 km s^-1 redward of the rest wavelength of Hα. The second absorption component is even narrower, with a FWHM close to the spectral resolution, and it is situated 140 km s^-1 blueward of the emission peak. In the top left panel of  Figure <ref>, the most prominent and redward of the two absorption features can be seen in other coeval lines of the Balmer series, although with lower S/N. The second, more shallow and blueward absorption does not have a counterpart in the other H lines, and therefore it could result from a transition of a different element. The Hγ line presents a unique shape, which deviates from the trend of the other Balmer lines, showing in particular a marked absorption in the blue wing which also results in a narrower FWHM. The same behaviour for the Hγ line has been observed in the yellow supergiant IRC+10420, where it has been explained as the result of absorption components found in stellar atmosphere models (Figure 6 in ). 

In the top right panel of Figure <ref> it is possible to study the different profiles of the Fe II lines. Despite originating from the same atomic species, some lines display a dominant emission component, with only traces of absorption at around the rest wavelength (e.g. λ6517), while other lines display a more complicated profile, with marked absorption features and a redshifted emission component, compatible with a P-Cygni profile (e.g. λλ5018,5169). The Fe II lines that display a stronger absorption, such as the multiplet 42, are characterized by a larger oscillation strength, which favours self absorption. Additionally, the spatial stratification of the emission lines may further enhance this effect.






















The analysis of the Ca II lines (bottom left panel of Figure <ref>) offers valuable insight on the geometry of the line forming region.
The Ca II NIR line λ8498 displays, as said, a prominent absorption component with a FWHM of 30 km s^-1 situated just 10 km s^-1 blueward of the rest wavelength position of this line. 
On the other hand, the [Ca II] doublet presents a very different line profile: the line is markedly asymmetric, with the blue wing almost completely missing. <cit.> tentatively attribute this behaviour to self absorption, in a scenario where the [Ca II] doublet originates in an low density, inflowing gas. However, <cit.> state that self absorption is quite rare for a forbidden transition such as [Ca II] (although not impossible, as shown for example by ): the peculiar line profile is postulated to be formed through electron scattering in an expanding envelope, which gives rise to the extended red wing observed.
We expand on this interpretation, based on the similarity between the shape of the absorption feature superimposed to the Ca II (λ8498) emission line and the [Ca II] doublet: a steep blue edge and an extended red wing (lower left panel of Figure <ref>). We deduce that both features are produced in the same region, in which the gas density must be low to allow the production of the [Ca II] doublet. 
The velocity field of the gas in this region molds the shape of both spectral features. We speculate that the observed asymmetry reflects the dynamics of a stellar wind which reached the terminal velocity. The blueward sharp cutoff would indicate the terminal wind velocity.
By extension, all the observed narrow absorptions superimposed to the emission lines (Balmer lines, Ca II NIR triplet, He I, Fe II and O I) can be traced back to this low density, pre-existing wind produced by the progenitor star. With the redshift we adopted for NGC 300 OT, the blue cutoff for the [Ca II] is located at ∼-30 km s^-1, which provides an estimate of the terminal wind velocity in the scenario just described. Such value is well in line with the wind velocities observed for AGB stars (e.g. ).
On the other hand, the red wing of the [Ca II] extends over 200 km s^-1 redward: this could be the result of photons scattered into the line of sight by interaction with free electrons. Indeed, a similar line profile has been obtained by simulating the scattering of an emission line within a stellar wind (Figure 10 in  and Figure 4 in ). 

In the Na ID absorption profile (bottom right of Figure <ref>) it is possible to differentiate multiple components: a narrow, barely resolved (v_FWHM<10 km s^-1) component associated with the Milky Way gas (which is unchanging with time, according to ), a second narrow component associated with the host galaxy, NGC 300, and a broader absorption (FWHM velocity ∼200 km s^-1 for each component of the doublet) originating from the gas expanding around the transient (for a more detailed discussion of Na ID in ILRTs, see ).
Lastly, the Ca II H line presents a FWHM velocity of 420 km, ruling out an interstellar origin in favour of production within the more turbulent CSM surrounding the transient. The other line of the doublet, Ca II K, is contaminated by the Hϵ emission line, resulting in a misleading asymmetric profile with a short blue wing and an extended red wing.






















§ BROAD FEATURES IN LATE TIME SPECTRA


Late time spectra of ILRTs display some intriguing features. Both in NGC 300 OT and AT 2019ahd, two broad features start appearing at 6168 Å and 7002 Å a few months after peak luminosity (the position of the peak was estimated through a Gaussian fit). Notably, so far such broad lines were never detected in any spectrum of ILRTs. In Figure <ref> we show the late time spectra of NGC 300 OT and AT 2019ahd superimposed, in order to highlight the similarities between the two, in particular regarding these broad features. The red wing of the bluer feature (at 6168 Å) partially overlaps with the narrow [O I] λ6300 line, while another emission feature can be detected at λ6247. On the redder feature (at 7002 Å) there is a minor contamination from He I λ7065. It is challenging to exclude the presence of absorption components, similar to the ones detected in the H and Ca II NIR lines, given the rather low S/N. Both broad features present a FWHM velocity of ∼2500 km s^-1.
The lower panels of Figure <ref> show their evolution over time in NGC 300 OT, from their first detection at +123 d up to +243 d. Their development is remarkably slow, as the features appear barely different after 120 days. As for AT 2019ahd, we only detect the broad emission components in a single spectrum, therefore it is not possible to study their evolution in time.
Given the rather symmetrical distribution of these broad features around the Hα line, in principle they could be explained through Hα emission from two fast jets of matter moving at ∼20000 km s^-1 (based on the displacement of their peak with respect to the Hα rest wavelength), one approaching and one receding along the line of sight. However, since the features appear to be almost identical in both AT 2019ahd and NGC 300 OT, it is extremely unlikely to have observed two jets oriented exactly in the same direction relative to the Earth, and therefore we disfavour this scenario.









In the context of a terminal, explosive event, it is tempting to associate these features with the products of nucleosynthesis emerging from the slow moving CSM that was obscuring our line of sight during the photospheric phase. However, these broad features are challenging to identify, as they do not match the common nebular lines found in SNe (e.g. ).
Few atomic species can account for both the emission lines: according to the tabulated values reported by <cit.>, a candidate could be Ni I λλ 6177,7002, which would corroborate the scenario in which explosive nucleosynthesis took place during the ILRT events.
In order to test this hypothesis, in Figure <ref> we compare the late time spectra of NGC 300 OT and AT 2019ahd with the models of nebular spectra presented by <cit.>. The synthetic spectra have been scaled down for clarity. These models aim to reproduce the late time spectra of SNe arising from the core collapse of a 9 M_⊙ star (at zero age main sequence). The first synthetic nebular spectrum considered is relative to 300 days after explosion and has a solar abundance of elements. A high Ni content is expected to be an indicator of an ECSN event <cit.>: for this reason we consider a second synthetic spectrum (again provided by ), at the same evolutionary stage, in which the ratio Ni/Fe is 30 times the solar value. In the lower panels of Figure <ref> we display a zoomed in view of the broad lines found in NGC 300 OT, together with the same spectral region of the synthetic nebular spectra. In the region of the feature peaking at ∼7002 Å, there is a noticeable contribution in flux that is present in the Ni rich spectrum. 
However, the broad emission component at 6168 Å is not well reproduced by either synthetic spectrum, and an enhanced abundance of Ni does not seem to significantly influence this spectral region. Furthermore, a notable shortcoming of this interpretation is the lack of prominent [Ni II] lines (λλλ6667,7378,7412, marked with dashed lines in the upper panel of Figure <ref>) in our ILRTs spectra, which would instead be expected in the nebular spectra of a Ni-rich explosion <cit.>. Having considered these elements, we deem unlikely that the broad features observed originate from the nebular emission of fast, Ni-rich ejecta.








We now perform comparisons with empirical data, starting with the well studied SN 2005cs, a low luminosity SN IIP <cit.>. In the upper panel of Figure <ref> we plot the nebular spectrum of SN 2005cs, taken 280 days after explosion, together with the late time spectra of NGC 300 OT and AT 2019ahd. We observe a promising match for the broad feature at 6168 Å: interestingly, during the photospheric phase of SN 2005cs there are two absorption components compatible with P-Cygni profiles in that region (bottom left panel of Figure <ref>). The broad emission components could therefore originate from the blend of multiple Fe II (and possibly Fe I) lines, which form a P-Cygni profile during the photospheric phase of SN 2005cs (+64 d), but then turn to pure emission when the continuum fades (+280 d). Subsequently, almost one year after explosion (+336 d), the broad emission component appears weakened. On the other hand, SN 2005cs does not present strong emission around the region around 7002 Å, as shown in the bottom right panel of Figure <ref>, although there could arguably be a feeble P-Cygni profile in this region during the photospheric phase.

As a final comparison, we present the red nova V1309 Scorpii (hereby V1309 Sco, ), which is interpreted as the result of a merger of a contact binary system <cit.>. The spectrum taken 48 days after the start of the outburst shows remarkable similarities with the late time spectra of our ILRTs (Figure <ref>). Thanks to the high resolution provided by the VLT equipped with UVES, it is possible to distinguish multiple narrow features in the spectrum of V1309 Sco which corroborate the scenario in which the broad emission components originate from the blend of multiple narrow lines. Fe I and II lines are likely candidates, although pinpointing the exact transitions involved is quite challenging. As shown in the bottom panels of Figure <ref>, the insurgence of this forest of emission lines is sudden, since there is no hint of emission over the continuum in the regions around 6168 Å and 7002 Å in the spectrum of V1309 Sco taken 26 days after the outburst. 
Based on this comparison, we favour the interpretation that the broad emission lines observed in late time spectra of ILRTs are blends of multiple narrow (possibly Fe) lines, as observed for V1309 Sco.




This newly discovered resemblance between the spectra of V1309 Sco, AT 2019ahd and NGC 300 OT shows that similar chemistry and physical conditions of the emitting regions can be found in red novae and ILRTs. However, while there are some objects showing hybrid properties (e.g. AT 2022fnm ), this does not mean that ILRTs and LRNe can be easily blended into one class.
First of all, the timescales over which V1309 Sco and the ILRTs develop these broad emission features are quite different. This could be linked to the different scale of the phenomena: the peak optical luminosity of V1309 Sco is about ∼150 times lower than the peak luminosity of NGC 300 OT and AT 2019ahd, and it is characterised by a faster decline rate, dimming by 3 magnitudes in 50 days. 
Furthermore, late time spectra of red novae such as V1309 Sco are characterised by a red continuum eroded by the absorption of countless molecular bands, most notably TiO, VO, CO and H_2O (e.g. ).
 On the contrary, ILRTs never display the absorption features typical of molecular bands. This difference is made evident by comparing the spectrum of NGC 300 OT taken 514 days after maximum with X-Shooter with the spectrum of a red nova that already developed molecular band absorptions. Even at 514 days after maximum luminosity, the surviving continuum of NGC 300 OT does not show signs of molecular bands, which are instead prominent in the case of V1309 Sco, 244 days after the outburst (see Figure <ref> in Appendix).



§ ON THE NATURE OF ILRTS


The physical mechanism driving ILRTs has been a puzzle since the identification of this class of objects, 15 years ago. We quickly outline the main observed features of these transients before reviewing some scenarios that could explain their origin:


  
  ∙ ILRTs present a single peaked, monotonically declining light curve. The total radiated energy spans from few 10^47 to few 10^48 erg (Paper I).
  

  
  ∙ Whenever detected, their progenitors were heavily obscured in the optical and NIR domain, but bright sources in the mid-infrared. These objects were associated with dust-enshrouded, extreme AGB stars in the range of 8-12 M_⊙ <cit.>.
  

  
  ∙ When monitored for several years, ILRTs decline well below the progenitor level before disappearing, even in the MIR domain (, Paper I). This favours the interpretation of ILRTs as terminal events.
  

  
  ∙ Their spectra evolve slowly and are characterized by low excitation, narrow lines, with no signs of broad features even 17 months after the discovery.


Eruption of an intermediate-mass star – A first scenario proposed for the formation of ILRTs was the non-terminal eruption of a ∼9-15 M_⊙ star, which reached its Eddington limit and became dynamically unstable after shedding a considerable amount of mass <cit.>. The similarity between NGC 300 OT and the yellow hypergiant IRC+10420 () corroborated the idea of a stellar outburst producing an optically thick expanding envelope. However, this scenario struggles to explain the disappearance of the star observed years later in the mid infrared domain, since in the original view the progenitor would survive the eruption.

Stellar mergers – Thanks to the careful analysis of V1309 Sco (e.g. ) the class of transients known as red novae was soundly associated with non-degenerate stellar mergers. 
Depending on the masses of the progenitor stars, the peak magnitude of red novae can span from being significantly fainter than a classical nova (M_r∼–4.5 mag, e.g. OGLE-2002-BLG-360, ) up to competing with genuine core collapse SNe (M_r∼–15.5 mag, e.g. AT 2017jfs, ). On the other hand, ILRTs known to date peak consistently in the range –15 mag ≲ M_r≲–11 mag (Paper I, ).
Thus, there is some overlap in the energy radiated by ILRTs and the brightest red novae: this degeneracy is worsened by the similar spectral features shown by ILRTs and red novae during their photospheric phase. This can give rise to ambiguous objects, for which a clear classification is uncertain (e.g. M85 OT2006-1, ). At the same time, there are some peculiarities in these two classes of transients that help with their identification: several (although not all) red novae present a double-peaked light curve, typically weaker or absent [Ca II] emission lines, and most importantly red novae always display molecular absorption bands in their late time spectra <cit.>, while ILRTs never show such bands. Furthermore, the progenitor systems of different red novae were observed in the optical domain (e.g. ), excluding the presence of large dust quantities along the line of sight, which is instead a key feature of ILRTs progenitors. In principle, these differences could be tentatively explained with a different viewing angle on the same type of progenitor system (such as binary stars surrounded by a dust torus): it was indeed suggested that the same physical mechanism could power both ILRTs and red novae (e.g. ). However, the expected result of a stellar merger struggles to be reconciled with the severe mid infrared dimming observed for ILRTs over the course of a few years. Prolonged monitoring is therefore crucial to confirm the complete disappearance of the object, since there are known cases of transients settling on a lower luminosity compared to their progenitor (e.g. ).

Failed supernovae – A hypothetical link was suggested between ILRTs and fallback supernovae, events in which a large mass fraction of the exploding star in unable to escape the gravitational potential of the compact object formed during the core collapse (e.g. ). <cit.> found that the low energy radiated and the low photospheric temperature reached by ILRTs is compatible with radiation hydrodynamical simulations of matter ejected by a failed supernova which is interacting with pre-existing CSM. This interpretation has the merit of linking together a terminal event with the low luminosity and the narrow lines spectra of these transients. However, as the authors remarked, their simulations involve Wolf-Rayet and blue supergiants, which are not compatible with the progenitors of ILRTs discovered so far. Additionally, the monitoring of NGC 300 OT and SN 2008S spanning several years has not shown signs of fallback accretion on the black hole, which would be expected in this scenario. 


Electron capture supernovae – Multiple works have proposed the association between ILRTs and ECSNe <cit.>. The 8-12 M_⊙ extreme AGB progenitors identified by <cit.> are the ideal candidates for giving rise to an ECSN explosion, and the intrinsic weakness of the event expected from the theory <cit.> is encouraging when considering these underluminous objects. ILRTs, despite showing some degree of individuality, appear to be a relatively homogeneous family of objects, especially compared with the profound variability of the observable parameters within the red novae class. This can be interpreted as transients arising from similar progenitors (8-12 M_⊙ extreme AGB stars) whose differences are caused by the interaction with different environments.
The declining of the transient below the progenitor level is definitely expected in this scenario, given the terminal nature of ECSNe. In this context, the late time spectra characterized by narrow emission lines which were shown in this work deserve additional commentary. Type IIn SNe have shown that large quantities of CSM can bottle up the ejecta of violent stellar explosions for several years (at least 4 years in the case of SN 2015da; ). Actually, among SNe IIn it is rare to witness the emergence of the ejecta from the CSM, as observed for example for SN 1995N years after its explosion <cit.>. In analogy, the absence of ejecta signatures in our latest ILRTs spectra does not exclude their presence underneath the CSM. The ECSN origin of these transients therefore remains a valid interpretation of the observed data, also corroborated by indirect indicators such as event rates (, ) and dust chemical composition ().



§ SUMMARY AND CONCLUSIONS


We analysed a rich collection of optical and NIR spectra of four ILRTs. Their slow evolution coupled with the presence of narrow (few 10^2 km s^-1) emission lines is indicative of the presence of optically thick CSM surrounding the transients. Balmer lines as well as Ca features (Ca H&K absorption doublet, [Ca II] doublet and Ca II NIR triplet) dominate the optical spectra and are recognizable at most epochs.
The O I λ 8446 line appearing at late time is a consequence of the Bowen fluorescence, and could be the result of coupling between the O I λ1025.76 line in the CSM and the Lyβ line emitted by underlying ejecta.
The luminosity evolution of the H emission lines and Ca II NIR triplet display a characteristic double peaked behaviour, an opposite trend compared with the neighbouring continuum flux: this is possibly linked to the interaction between fast ejecta and the slow moving CSM. 

The high resolution spectrum of NGC 300 OT obtained with VLT+UVES shows the presence of a complex environment, possibly multiple shells of gas, causing narrow absorption features to appear on the most prominent emission lines. The characteristic shape of the [Ca II] lines is likely a result of electron scattering in an expanding medium, and allows us to postulate the presence of an extended, slow (∼30 km s^-1) wind surrounding the progenitor star.
Late time spectra of NGC 300 OT and AT 2019ahd display broad features never observed in any ILRT so far. We find a striking similarity with the behaviour of the red nova V1309 Sco, and we tentatively explain these broad emission components with a blend of several narrow lines, possibly Fe transitions. 
In Paper I we discussed how the low energy released by ILRTs coupled with their terminal nature (at the very least for NGC 300 OT) make them appealing ECSN candidates. The spectroscopic analysis performed in this work provides a characterisation of the CSM surrounding these transients: while we do not find a smoking gun evidence for the explosion scenario, our findings are overall consistent with a weak ECSN event within a thick CSM.





We thank Steve Shore and Valentina D'Orazi for the useful and lively discussions.
We thank the staff of the various observatories where data were obtained for their assistance.
Based on observations made with the Nordic Optical Telescope, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and the University of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.
Observations from the Nordic Optical Telescope were obtained through the NUTS2 collaboration which are supported in part by the Instrument Centre for Danish Astrophysics (IDA) and the Finnish Centre for Astronomy with ESO (FINCA) via Academy of Finland grant nr 306531. The data presented here were obtained in part with ALFOSC, which is provided by the Instituto de Astrofisica de Andalucia (IAA).
Time-domain research by the University of Arizona team, and D.J.S. is supported by NSF grants AST-1821987, 1813466, 1908972, 2108032, and 2308181, and by the Heising-Simons Foundation under grant #2020-1864.
H.K. was funded by the Research Council of Finland projects 324504, 328898, and 353019.
This work makes use of data from the Las Cumbres Observatory network. The LCO team is supported by NSF grants AST–1911225 and AST–1911151, and NASA SWIFT grant 80NSSC19K1639. 
Data were also obtained at the Liverpool Telescope, which is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos with financial support from the UK Science and Technology Facilities Council. 
Based on observations made with the Gran Telescopio Canarias (GTC), installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the island of La Palma.
Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile, as part of ePESSTO+ (the advanced Public ESO Spectroscopic Survey for Transient Objects Survey).
ePESSTO+ observations were obtained under ESO program IDs 1103.D-0328 and 106.216C (PI: Inserra).
LCO data have been obtained via OPTICON. The OPTICON project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant no. 730890.
Time domain research by the University of Arizona team and D.J.S.  is supported by NSF grants AST-1821987, 1813466, 1908972, & 2108032, and by the Heising-Simons Foundation under grant #20201864. 
G.V., N.E.R. and I.S. acknowledge financial support from PRIN-INAF 2022 “Shedding light on the nature of gap transients: from the observations to the models".
A.R. acknowledges financial support from the GRAWITA Large Program Grant (PI P. D’Avanzo) and the PRIN-INAF 2022 "Shedding light on the nature of gap transients: from the observations to the models.
Y.-Z. Cai is supported by the National Natural Science Foundation of China (NSFC, Grant No. 12303054), the Yunnan Fundamental Research Projects (Grant No. 202401AU070063) and the International Centre of Supernovae, Yunnan Key Laboratory (No. 202302AN360001).
L.G. acknowledges financial support from the Spanish Ministerio de 
Ciencia e Innovación (MCIN), the Agencia Estatal de Investigación 
(AEI) 10.13039/501100011033, and the European Social Fund (ESF).
This work was funded by ANID, Millennium Science Initiative, ICN12_009
M.N. is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 948381). Part of the funding for GROND (both hardware as well as personnel) was generously granted from the Leibniz-Prize to Prof. G. Hasinger (DFG grant HA 1850/28-1).
H.K. was funded by the Academy of Finland projects 324504 and 328898. 
R.K. acknowledges support from the Research Council of Finland (340613).



















aa




§ AFFILIATIONS


^1INAF – Osservatorio Astronomico di Padova, Vicolo 
             dell’Osservatorio 5, I-35122 Padova, Italy

^2INAF – OATS, Via G.B. Tiepolo 11, 34143 Trieste, Italy

^3INAF – Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807, Merate (LC), Italy 

^4Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, P.R. China 

^5International Centre of Supernovae, Yunnan Key Laboratory, Kunming 650216, P.R. China 

^6Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, P.R. China 

^7Graduate Institute of Astronomy, National Central University, 300 Jhongda Road, 32001 Jhongli, Taiwan 

^8SRON, Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA, Leiden, The Netherlands 

^9Department of Astrophysics/IMAPP, Radboud University Nĳmegen, P.O. Box 9010, 6500 GL, Nĳmegen, The Netherlands 

^10Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, E-08193 Barcelona, Spain 

^11School of Physics, O’Brien Centre for Science North, University College Dublin, Belfield, Dublin 4, Ireland 

^12The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden 

^13Hiroshima Astrophysical Science Center, Hiroshima University, Japan 

^14Department of Physics, Florida State University, 77 Chieftan Way, Tallahassee, FL 32306, USA 

^15Las Cumbres Observatory, 6740 Cortona Dr. Suite 102, Goleta, CA, 93117, USA

^16Department of Physics, University of California, Santa Barbara, CA, 93106, USA

^17Cardiff Hub for Astrophysics Research and Technology, School of Physics & Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK 

^18INAF, Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, I-80131 Napoli, Italy 

^19DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark 

^20Caltech/IPAC, Mailcode 100-22, Pasadena, CA 91125, USA 

^21Tuorla Observatory, Department of Physics and Astronomy, University of Turku, 20014, Turku, Finland 

^22Department of Astronomy, AlbaNova University Center, Stockholm University, SE-10691 Stockholm, Sweden 

^23Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK 

^24Max-Planck-Institut für Astrophysik, Karl-Schwarzschild Str. 1, D-85748 Garching, Germany 

^25Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital 263001 India 

^26Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile

^27School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland 

^28Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA

^29Department of Physics, University of Oxford, Keble Road, Oxford, OX1 3RH 

^30Astrophysics Research Centre, School of Mathematics and Physics, Queens University Belfast, Belfast BT7 1NN, UK 

^31Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark 

^32INAF – Osservatorio Astronomico d’Abruzzo, via M. Maggini snc, Teramo, I-64100, Italy 

^33Department of Physics, University of California, Davis, CA 95616, USA 

^34 European Southern Observatory, Alonso de Córdova 3107, Casilla 19, Santiago, Chile 

^35Millennium Institute of Astrophysics, Nuncio Monsenor Sótero Sanz 100, Providencia, 8320000 Santiago, Chile 

^36INAF-Osservatorio Astrofisico di Catania, Via Santa Sofia 78, I-95123 Catania, Italy 

^37Instituto de Astrofísica, Universidad Andres Bello, Fernandez Concha 700, Las Condes, Santiago RM, Chile 

^38ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy 

^39Institut für Theoretische Physik, Goethe Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany 

^40INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Via Sommarive 14, I-38123 Trento, Italy 

^41Institut d’Estudis Espacials de Catalunya (IEEC), E-08034 Barcelona, Spain. 

^42Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland 

^43Center for Astrophysics, Harvard & Smithsonian, Cambridge, Massachusetts, MA 02138, US 

^44The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, USA 

^45Finnish Centre for Astronomy with ESO (FINCA), University of Turku, Väisäläntie 20, 21500 Piikkiö, Finland 

^46DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, 2800 

^47Istituto Nazionale di Astrofisica - Osservatorio Astronomico di Roma, Via Frascati 33, I-00040, Monteporzio Catone, Italy 

^48Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 

^49INFN. Sezione di Perugia, via A. Pascoli s/n, I-06125 Perugia, Italy 

^50Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli studi di Padova Vicolo dell’Osservatorio 3, I-35122 Padova, Italy 

^51IAASARS, National Observatory of Athens, Metaxa & Vas. Pavlou St., 15236, Penteli, Athens, Greece 

^52Department of Astronomy, University of Virginia, Charlottesville, VA 22904, USA 

^53Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstraße 1, 85748, Garching, Germany 

^54Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA 

^55Cosmic Dawn Center (DAWN), Denmark 

^56Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 København N, Denmark 

^57Manipal Centre for Natural Sciences, Manipal Academy of Higher Education, Manipal - 576104, Karnataka, India 

^58Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA 

^59Space Telescope Science Institute: Baltimore, MD, US 

^60Indian Institute Of Astrophysics, 100 Feet Rd, Santhosapuram, 2nd Block, Koramangala, Bengaluru, Karnataka 560034, India 









§ APPENDIX: ADDITIONAL TABLES AND FIGURES
 




































 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
============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================





§ 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: Neutron star kicks plus rockets as a mechanism for forming wide low-eccentricity neutron star binaries.


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0000-0002-8032-8174]Ryosuke Hirai (平井遼介)
Astrophysical Big Bang Laboratory (ABBL), Cluster for Pioneering Research, RIKEN, Wako, Saitama 351-0198, Japan
School of Physics and Astronomy, Monash University, Clayton, VIC3800, Australia
OzGrav: The ARC Centre of Excellence for Gravitational Wave Discovery, Clayton, VIC3800, Australia

0000-0002-8338-9677]Philipp Podsiadlowski
University of Oxford, St Edmund Hall, Oxford OX1 4AR, UK

0000-0002-3684-1325]Alexander Heger
School of Physics and Astronomy, Monash University, Clayton, VIC3800, Australia
Argelander-Institut für Astronomie, University of Bonn, Auf dem Hügel 71, 53121 Bonn,	Germany

0000-0002-7205-6367]Hiroki Nagakura (長倉洋樹)
Division of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan



§ ABSTRACT 
Recent neutron star surface observations corroborate a long-standing theory that neutron stars may be accelerated over extended periods after their birth. We analyze how these prolonged rocket-like accelerations, combined with rapid birth kicks, impact binary orbits. We find that even a small contribution of rocket kicks combined with instantaneous natal kicks can allow binaries to reach period–eccentricity combinations unattainable in standard binary evolution models. We propose these kick + rocket combinations as a new channel to form wide low-eccentricity neutron star binaries such as Gaia NS1, as well as inducing stellar mergers months to years after a supernova to cause peculiar high-energy transients. 




§ INTRODUCTION


Neutron stars (NSs)ns, the compact remnants of massive stars, are known to be born with high spatial velocities of a few×100 km s^-1 <cit.>. In the current paradigm, this velocity is given at birth due to asymmetries in the sn explosion, driven by hydrodynamical instabilities <cit.> and asymmetric neutrino emission <cit.>. Since this velocity is related to the core-collapse sn explosion mechanism itself, the acceleration occurs on dynamical timescales of τ_kick≲10 s and the resulting velocity is often called a “natal kick”.

On the other hand, some kick mechanisms have been proposed in the past that involve post-natal processes. A famous example is the “electromagnetic rocket” model, where the new-born ns is propelled along the spin axis by asymmetric spin-down radiation from off-centred or higher-order magnetic fields <cit.>. 
Recently, the NICER X-ray telescope revealed the non-antipodal orientation of hot spots on a neutron star surface, strongly supporting the existence of such higher-order magnetic fields <cit.> and therefore it is very likely that the electromagnetic rocket mechanism contributes to the ns kick to some degree. The total magnitude of the velocity achieved through this mechanism scales as

    v_roc≈260 km s^-1ϵ̅/0.1(R/10 km)^2(P_spin,ini/1 ms)^-2,

where ϵ̅ expresses the degree of asymmetry of the outgoing Poynting flux, R and P_spin,ini are the ns radius and birth spin period respectively <cit.>. In the most optimal magnetic field configuration, this can reach up to v_roc∼1,400 km s^-1[It should be noted that the strength of the magnetic field only influences the acceleration timescale in this model. The final velocity is determined by the reservoir of rotational energy and asymmetry of the radiation, hence the field strength is irrelevant.]. Another post-natal model is the so-called “neutrino rocket” mechanism, where neutrinos are asymmetrically emitted via neutrino cyclotron radiation from neutrons circulating superfluid vortices in the ns interiors <cit.>. Under favourable assumptions, this mechanism can accelerate the ns to velocities reaching v_roc≲1,000 . Although some difficulties have been pointed out for these “rocket” mechanisms, such as the requirement of a high birth spin (P_spin∼1 ms) that is in contradiction with observationally inferred values <cit.>, these models have not been firmly ruled out <cit.>. In particular, there are no arguments against the rocket mechanisms making minor contributions to the total kick.


A distinct property of the rocket mechanisms is its long acceleration timescale. Since it is driven by spin-down, the acceleration timescale can be τ_roc∼0.1–100 yr (≫τ_kick). Whereas this is still negligible compared to the lifetime of pulsars (∼10^7-8 yr), meaning it will be difficult to detect ongoing acceleration, it is longer than typical orbital periods of binary systems containing ns. Traditionally, ns kicks in the context of binary evolution have been treated as an impulsive process, in which the post-sn orbit can be analytically derived. The resulting orbital properties depend on the magnitude and direction of the kick, but in most cases it causes the orbit to become wider and eccentric or disrupt it completely. However, when the kick cannot be treated impulsively but instead the acceleration timescale is comparable to, or longer than, the orbital period, these impulsive solutions no longer hold. Instead, we need to consider the opposite extreme where the ns acceleration is a small perturbation compared to the orbital acceleration.

The two-body problem with a constant acceleration on one or both of the bodies is sometimes called the “accelerated Kepler problem” <cit.>, which is analogous to the Stark effect in quantum mechanics. Due to the secular nature, the orbit can be altered in ways that are qualitatively different from instantaneous kicks. The impact on ns binaries, however, has not been well quantified and the existing studies only deal with special cases where the rocket mechanisms act on initially circular orbits <cit.>. 

In this paper, we explore the implications of rocket-like acceleration in tandem with classical impulsive kicks, in particular, how it influences our understanding of the formation channels for various ns binaries. 
One of the main consequences is that it provides a channel to form the wide low-eccentricity ns binary Gaia NS1 <cit.> and other similar systems <cit.> that cannot be understood in standard binary evolution models. This may provide direct testable evidence for the rocket mechanisms in ns binaries. 

This paper is structured as follows. In Section <ref>, we introduce the secular accelerated Kepler problem and derive the analytic solution. The combined effect of rapid ns kicks and the rocket effect are demonstrated in Section <ref>, and its astrophysical implications are discussed in Section <ref>. We summarize our conclusions in Section <ref>.



§ THE ACCELERATED KEPLER PROBLEM


In this section we introduce the accelerated Kepler problem (or classical Stark problem), which is a two-body problem with a fixed acceleration on one of the bodies. The equation of motion of our two-body problem is

    m_1r̈_1   =F_12+m_1a_roc,
    
        m_2r̈_2   =F_21,

where m_i are the masses, r_i are the position vectors where subscripts i=1,2 denote the primary and secondary, and a_roc is the rocket acceleration acting on the new-born ns. F_12=-F_21 is the gravitational force between the two stars. Dots denote time derivatives. To solve for the orbit, we separate the equations into the centre of mass and relative motion by defining

    r_com≡m_1r_1+m_2r_2/m_1+m_2,
       r≡r_1-r_2.

We rewrite the gravitational force as

    F_g≡F_12=-F_21=-Gm_1m_2/r^2r̂=-k/r^2r̂,

where we have defined k≡ Gm_1m_2 and the unit vector r̂=r/|r|. The equation of motion for the centre of mass and relative position can hence be expressed as

    r̈_com=m_1a_roc/m_1+m_2,
       r̈=F_g/μ+a_roc,

where μ≡ m_1m_2/(m_1+m_2) is the reduced mass. The same expression for Eq. (<ref>) will hold even for cases where both stars have separate accelerations, by replacing a_roc with the difference of the two accelerations.





 §.§ Deriving the analytical solution

The problem has been known to be integrable for more than two centuries, but it was only in the work by <cit.> that the general closed-form solution to this problem was shown using elliptical coordinates. Under the secular approximation (|a_roc|≪|F_g/μ|), the semi-major axis (or orbital period) is preserved whilst the orbit is torqued by the rocket acceleration. The general behaviour is that the orbital plane librates around the direction of the rocket and the eccentricity oscillates up and down, while the binary as a whole is also accelerated. In this section, we derive an alternate analytical solution to the secular accelerated Kepler problem in vector form. Our much simpler form allows for an intuitive understanding and makes it suitable for statistical investigation. 

Our solution can be summarized as follows. The energy and angular momentum of the orbit are defined as

    E   ≡1/2μṙ^2-k/r=-k/2a,
    L   ≡μr×ṙ,

where a is the semi-major axis.
The energy gives an average orbital velocity

    v̅_orb=√(-2E/μ)=√(k/μ a).

Let us define a dimensionless angular momentum vector l≡L/L_0 where L_0≡ k/v̅_orb is the angular momentum of a circular orbit (e=0) with the same energy. The magnitude of this vector is |l|=√(1-e^2).
By taking the time derivative of the normalized angular momentum and using Eq. (<ref>), the torque on the orbit by the rocket acceleration can be expressed as

    l̇=μ/L_0r×a_roc.

Notice that l̇·a_roc=0, so the angular momentum in the direction of the rocket is strictly constant over time, whereas the transverse components are torqued by the rocket acceleration. Making the secular approximation, this becomes

    <l̇>   =μ/L_0<r>×a_roc,
       =-3/2v̅_orbe×a_roc,

where the angle brackets represent quantities averaged over an orbit 

    <x>≡1/P_orb∫_0^P_orbx(t)dt,

and the eccentricity vector (or Laplace-Runge-Lenz vector) is defined as

    e≡1/kṙ×L-r̂.

We have also used that <r>=-3ae/2 and k/(μ a)=v̅_orb^2.

Similarly, if we take the time derivative of the eccentricity vector, we obtain

    ė=1/k(a_roc×L+ṙ×L̇),

and the orbit-averaged value is

    <ė>   =1/v̅_orb(a_roc×l+<ṙ×l̇>),
       =-3/2v̅_orbl×a_roc.

Note here that <ė>·a_roc=0, so the component of the eccentricity vector parallel to the rocket is constant under the secular approximation.

We now define two support vectors

    h_±≡l±e,

There are several properties to these vectors. First, |h_±|=1, which is trivial from |e|=e, e·l=0 and |l|=√(1-e^2). Secondly, <ḣ_̇±̇>·a_roc=0, because l̇·a_roc=<ė>·a_roc=0, meaning that the component of the h_+ and h_- vectors parallel to the rocket are constant over time. Using Eqs. (<ref>) & (<ref>), the secular evolution of these vectors can be expressed as

    <ḣ_̇±̇>=∓3/2v̅_orbh_±×a_roc.

This implies that as the ns is accelerated by the rocket mechanism, both h_+ and h_- precess about the direction of the thrust a_roc in opposite circular motions. Let us take the direction of the rocket a_roc as the z axis. If the cumulative rocket velocity up to some time t is

    Δ v_roc(t)≡∫_0^t a_rocdt,

 the post-rocket state can be described by 

    h_±'   =R_z(±3Δ v_roc/2v̅_orb)h_±,

where R_z(θ) is a rotation matrix about the z-axis for an angle θ. We can see that the two vectors will circulate around the z-axis in opposite directions. The post-rocket orbit can then be completely described by

    l'   =1/2(h_+'+h_-'),
    e'   =1/2(h_+'-h_-').

The post-rocket eccentricity is simply e'=|e'|.

Although we have assumed that a_roc is constant, our above solution still holds with variable acceleration magnitudes as long as it does not change significantly over an orbital period. In the rocket mechanism, the magnitude should slowly decline over time as the ns spins down. Eventually, a_roc should become negligible and the eccentricity oscillation would cease. The asymptotic value of the accumulated velocity Δ v_roc determines the final eccentricity.

There are some complications if the rocket direction (or the spin axis) precesses. Our solution would still likely hold if the precession timescale is much shorter than the orbital period. On average, the rocket acceleration should be pointed along the axis of precession with a slightly lower magnitude, and the orbital evolution can be computed using this averaged acceleration. When the precession timescale is comparable to or longer than the orbital period, the above solution no longer holds. It has been shown in previous studies that in such cases the precessing rocket acceleration can lead to resonant eccentricity excitations <cit.>. For even longer precession timescales, the rocket acceleration ceases before the spin direction changes appreciably, so again our above solution may be applicable.




 §.§ Numerical validation





We verify that our analytic solution describes the true solution well by comparing against direct numerical integrations of the accelerated Kepler problem.
Figure <ref> compares an example of the evolution of individual components of l and e based on Eqs. (<ref>)–(<ref>) (dashed curves) against direct numerical integration of Eqs. (<ref>)–(<ref>) (solid curves) with a low acceleration magnitude (|a_roc|=|F_g/μ|/50). We start the integration from a binary with an eccentricity e=0.3 and the rocket direction is tilted from l by 60^∘ in the x direction. We can see that the xy components of the vectors l and e both oscillate sinusoidally with respect to the accumulated rocket velocity Δ v_roc. We show the trajectory of the vectors l and e projected on the xy-plane in Figure <ref>. Both vectors follow elliptic trajectories around the z-axis with the same semimajor and semiminor axes. When converted to the h_± vectors, we can see they follow circular trajectories moving in opposite directions. As a result of these circulations, the eccentricity oscillates, although not as a sine curve. The analytic solution is in perfect agreement with the numerical result.



The amplitude of the eccentricity oscillation depends on the initial eccentricity and rocket angle. For example, when the initial orbit is circular and the rocket is aligned with the orbital angular momentum (h_+=h_-=â_roc), there is no orbital evolution. On the other hand, if the initial orbit is circular and the rocket is perpendicular to the orbital angular momentum, the eccentricity can reach unity e=1.




To assess when the secular approximation breaks down, we perform numerical integrations with various acceleration timescales τ_acc≡v̅_orb/|a_roc|. The results are shown in Figure <ref>. As we demonstrated above, when the acceleration timescale is taken to be sufficiently longer than the orbital period τ_acc/P_orb≳10, the evolution of the eccentricity closely follows our secular solution (dashed curve). At τ_acc/P_orb≲1, the secular approximation completely breaks down and hence the binary is dissociated within a single orbit. For the intermediate acceleration timescales where it is larger than but comparable to the orbital period (2≲τ_acc/P_orb≲10), the numerical results slightly deviate from the analytic solution while maintaining the periodic oscillation behaviour. We conduct similar tests for other orbital configurations and find similar results; the secular solution describes the numerical results quite well as long as τ_acc/P_orb≳2.





§ COMBINED EFFECT OF MASS-LOSS, NATAL KICK AND ROCKET


When sn occur in binaries, the orbit will be altered by a combination of various effects. First, there is the immediate recoil due to the mass loss by sn ejecta <cit.>, which causes the orbit to widen and change its eccentricity. Another effect is the instantaneous kick on the ns associated with core collapse. The rocket mechanisms should then act on binaries that have already been perturbed by these rapid effects. The pre-rocket perturbations are crucial, as the rocket does not change the orbit unless the acceleration is tilted from the orbital angular momentum and/or there is initially non-zero eccentricity. Therefore, it is important to consider the full combined effect of the various forms of ns acceleration.


Figure <ref> illustrates an example of how an initially circular orbit is altered by different combinations of rapid and rocket-like kicks. Initial system parameters are set to m_1=3 , m_2=1 , P_orb,0=1 d, which has an average orbital velocity v̅_orb,0∼338 [Note that this is not the v̅_orb relevant for the post-rocket orbit calculation]. The sn progenitor is assumed to have lost its hydrogen envelope through binary interactions and has become a helium star <cit.>. We assume the primary star instantaneously loses m_ej=1.5 in the sn. After mass loss, the Blaauw effect causes the binary to jump both in period and eccentricity from the open circle to the filled circle. We further apply either an instantaneous kick with Δ v_kick=0.5 v̅_orb,0 (blue dots), a rocket-like kick with Δ v_roc=0.5 v̅_orb,0 (cyan dots), or a mixture of both where we apply a Δ v_roc=0.1 v̅_orb,0 rocket kick after the Δ v_kick=0.5 v̅_orb,0 rapid kick (red dots). For each model, we sample 10^4 systems assuming an isotropic probability distribution for the kick/rocket direction. For the mixture model, we independently sample both the kick and rocket directions from isotropic distributions. For the kick-only model (blue dots), the distribution spreads out diagonally with a strong correlation between the post-sn period and eccentricity. In the rocket-only model (cyan dots), the eccentricity is altered whereas the orbital period is unchanged. It is clear that the instantaneous and rocket-like kicks impact the orbit in qualitatively different ways.

In the kick+rocket mixture model (red dots), the period distribution is purely determined by the rapid component (see histograms on the top panel). The rocket effect then spreads the distribution out in the eccentricity direction. This enables the binary to cover a much wider area in period–eccentricity space from a given progenitor system, despite the rocket magnitude being rather small (Δ v_roc=0.1 v̅_orb,0∼34 ). As the rocket effect depends on the average orbital velocity (Eq. (<ref>)), its effects are most pronounced in the longer period regions where it can allow the binary to reach low eccentricities, down to e=0.







§ IMPLICATIONS


There are many implications of this wide phase space coverage by the kick+rocket mixture model, but in this paper we highlight a few select interesting applications.



 §.§ Wide low-eccentricity binaries


An important distinct feature of the rocket effect is that the periastron distance can widen after the sn (see in Figure <ref> how dots are distributed to the right of the black curve). Note that it is impossible to widen the periastron distance beyond the pre-sn separation with only rapid kicks. This effect is enhanced when the rocket acts on orbits already strongly perturbed by the rapid kick because the systems that are kicked to wider orbits are more susceptible to the rocket effect and can be circularized more easily. Therefore, the kick+rocket combination can in principle widen the periastron distance by arbitrary amounts as long as the secular approximation holds in the rocket phase. 

Recent observations by the Gaia satellite have discovered a number of wide (P_orb∼200–1000 d) ns binaries with ∼1 stellar companions <cit.> (see black crosses in Figure <ref>)[Gaia is only sensitive to long-period systems. The majority of systems in Figure <ref> would be the progenitors of low- or intermediate-mass X-ray binaries with much shorter orbital periods.]. Most notable is Gaia NS1, which has an orbital period of P_orb∼731 d and the lowest eccentricity e∼0.122 <cit.>. These systems are considered to be progenitors for symbiotic X-ray binaries, which also tend to be on wide and low-eccentricity orbits, e.g. GX 1+4 (P_orb=1160.8±12.4 d, e=0.101±0.022) <cit.>, 4U 1700+24 (P_orb=404±3 d, e=0.26±0.15) <cit.> and IGR J16194-2810 (P_orb=192.73 d with a red giant companion, implying a rather large periastron distance) <cit.>. It is challenging to devise formation channels for such systems in the traditional binary evolution framework, as the orbital periods are too short to avoid binary mass transfer from the sn progenitor that will shrink their orbits down to much tighter periastron distances. Even if the post-mass-transfer separations can be kept wide enough, the rapid kick needs to be Δ v_kick≲30 km s^-1 in order to keep the eccentricity low enough, whereas typical observed kick velocities are Δ v_kick≳100 km s^-1 <cit.>. Dynamical assembly channels in dense cluster environments have also been ruled out due to their low eccentricity <cit.>.

With rocket mechanisms, these systems can be formed more naturally (see Figure <ref>). The system can experience mass transfer or common envelope phases to strip the envelope of the primary and dramatically shrink the orbit. At this point the ejecta mass is small and the orbit is tight enough so that the sn mass loss and natal kick tend to not disrupt the binary but instead send it on a wide and eccentric orbit. Once the orbit has widened, the average orbital velocity is now slow enough that even small rocket-like kicks (Δ v_roc≲30 km s^-1) can induce large changes to the eccentricity. Although calculating detailed formation rates of low-eccentricity ns binaries is out of the scope of this letter, we note that classical binary evolution scenarios will need to be stretched to extremes whereas our scenario does not require any non-standard assumptions other than the rocket mechanism. Based on simple Monte Carlo simulations of kicks+rockets, we expect ∼0.05–0.1 % of ns + 1 binaries to end up with wide (P_orb≥100 d) and low-eccentricity (e≤0.3) orbits if their pre-sn periods are 1 d≤ P_orb≤30 d. See Appendix <ref> for details of the Monte Carlo calculations.








 §.§ Induced mergers

Whereas some systems lower their eccentricities due to the rocket effect, some systems will increase their eccentricities. If the rocket acceleration is directed at an appropriate angle, the eccentricity can reach very close to e=1, inevitably causing a tidal disruption of the companion or merger between the two components. These interactions may explain the origin of some superluminous sn or tzo if the companion is a non-degenerate star <cit.>.

In cases where the companion is also a compact object, it may induce a binary merger event that emits gravitational waves and/or electromagnetic signatures like short-duration grb, kilonovae or other peculiar transients. The resulting transient will be separated from the sn that gave birth to the ns by a span of weeks to years, as the merger is induced by a secular effect. If, for example, a binary ns merger occurs with a sufficiently short delay time, the merger will occur with unique conditions (e.g. relatively hot ns, dense circumstellar material due to sn ejecta), possibly giving rise to peculiar grb or late-time bumps in the sn light curve. Additionally if the delay time is long enough so that the grb occurs after the peak of the associated sn light curve, it may cause the accompanying sn to be undetected like in some known long grb <cit.>.





 §.§ Rockets in triples

There are possibly even more interesting effects in the context of triple star evolution. For example if the sn progenitor was in the inner binary of a hierarchical triple, the mass loss and natal kick not only alters the orbital period of the binary but also the outer orbit. In particular, the change in the outer orbit strongly depends on the orbital phase at which the explosion occurs, as it determines the direction of the mass-loss recoil. The subsequent rocket effect will cause the inner orbit to librate and change its eccentricity. As long as the ns acceleration timescale is longer than the outer orbital period, the same libration will occur on the outer orbit too, as the inner binary as a whole will also be accelerated. It is not trivial which libration will be more significant. Both the inner and outer orbits have different initial configurations and angle with respect to the rocket. Also, the degree of libration and eccentricity change depends on Δ v_roc/v̅_orb, which is also different between the inner and outer orbits. The multitude of scales involved can lead to a wide variety of possible outcomes.

Once the kick and rocket effects have sufficiently tilted the inner and outer orbits, the mutual inclination may exceed the critical inclination required for secular evolution like Kozai-Lidov oscillations <cit.> and higher-order effects <cit.>. Or it could take the triple all the way to an unstable configuration <cit.>, which should eventually terminate through stellar mergers or ejection of one of the stars <cit.>. Therefore, even if the rocket effect itself cannot modify the orbits enough to cause interesting phenomena, it can still be used as an efficient mechanism to hand over the system to where triple effects will finish off the job. Alternatively, the Kozai-Lidov oscillations may start to operate before the rocket effect has ceased. It may cause further interesting modulations and/or resonances if the oscillation timescales of the triple and rocket effects are comparable. We leave exploration of such phenomena for future work.






§ CONCLUSION


Motivated by recent observations of non-axisymmetric magnetic fields on ns, we explored the implications of combinations of rapid and rocket-like kicks in binary systems. We derive an analytic solution for the secular accelerated Kepler problem in vector form. This simple-formed solution is useful for other applications such as planetary science and artificial satellite maneuvering <cit.>.

We demonstrate how the combined effect of mass loss, rapid natal kicks, and rocket-like kicks impact the post-sn binary orbits. We find that even with only small contributions from the rocket component, the resulting binary orbit can be qualitatively different from what is achievable with purely rapid kicks. This can transform our understanding of the formation of many astronomical objects.

The wide low-eccentricity ns binaries such as Gaia NS1 and GX 1+4 are challenging to form in classical binary evolution models. Our proposed new scenario based on rapid kick+rocket mixtures can more robustly form these systems without requiring drastic changes to how we understand binary interactions. We also expect that there could be many types of peculiar transients from mergers induced by the rocket effect between ns and their companions. By studying these observables in detail, we may in turn gain insight into the physics that drive the rocket mechanism, such as the ns birth spins, configuration of magnetic fields in new-born ns or efficiency of cyclotron neutrino emission from superfluid vortices.


There are potentially numerous other applications where the rocket effect may be relevant, for example the formation of X-ray binaries, gravitational-wave sources, and triple star evolution. We plan to explore many of these other avenues in future studies.



RH thanks Ilya Mandel, Shoichi Yamada, Yongzhong Qian and Andrew Melatos for useful discussions during the early stages of this project. HN is supported by Grant-in-Aid for Scientific Research (23K03468) and also by the NINS International Research Exchange Support Program.
AH is supported by the Australian Research Council through grants DP240101786 and DP240103174.








gnuplot <cit.>
          






§ MONTE CARLO SIMULATION OF FORMATION RATES


Given that the magnitude of rocket-like kicks are still poorly constrained both observationally and theoretically, it is tricky to estimate formation rates of the various objects formed in our kick+rocket scenario. Here, we conduct a simple Monte Carlo simulation to provide ballpark estimates of the formation probabilities of Gaia NS1-like systems from given pre-sn binary configurations. We define any system with P_orb≥100 d as “wide” and e≤0.3 as “low-eccentricity” binaries. 

We first start from a system with a 3 sn progenitor and a 1 companion star on a circular (e=0) orbit. We assume that 1.5 is lost in the sn and the primary now becomes a 1.5 ns. The progenitor is assumed to have undergone some form of binary interaction to lose a large fraction of its envelope so that the ejecta mass is within the typical range of observed stripped-envelope sn ejecta masses <cit.>. The Blaauw mechanism causes the binary to become wider and more eccentric after this. We then randomly draw the rapid kick velocity Δ v_kick from a Maxwellian distribution with an average velocity σ_kick=265 <cit.> and a random direction from a spherically isotropic distribution. After applying this rapid kick to the ns, we check whether the binary is still bound and has an orbital period satisfying the wide condition (P_orb≥100 d). If the period is long enough we then draw a random rocket velocity Δ v_roc from a Maxwellian distribution with an average velocity σ_roc and a random direction from a spherically isotropic distribution. We then compute the post-rocket orbit and check if the resulting eccentricity satisfies the low-eccentricity condition (e≤0.3).


Several models are computed with varying rocket velocity distributions (σ_roc=0, 30, 100 km s^-1) and initial orbital periods P_orb,0∈[0.1,1000] d. For each model we draw 10^6 samples to obtain the probability of forming wide and low-eccentricity systems. 

Firstly, we see that the fraction of systems that stay bound after the sn decreases with increasing initial period (dashed). Among the surviving systems, only a fraction of them achieve long orbital periods satisfying the wide condition when the pre-sn orbital period is not wide. Out of the total number of samples, the fraction of wide systems ranges 0.1–1 % (dotted), peaking at around P_orb,0∼4 d. 
When there is no subsequent rocket phase (solid black), it is basically impossible to form wide low-eccentricity systems unless the initial period is already long (P_orb,0≳50 d). On the other hand, with a rocket phase there is finite probability to form such binaries even if the pre-sn period is significantly shorter than the threshold (solid red and pink). Among the wide binaries, the fraction that end up having low eccentricities only weakly depends on the initial period, ranging ∼7–10 %.




Our main conclusion is that as long as the binary can achieve long orbital periods after rapid kicks, there is almost a fixed probability (∼7–10 %) of forming low-eccentricity binaries through the rocket mechanism as long as the rocket magnitude is comparable to the post-kick orbital velocity. This is in contrast to pure-kick models where it is impossible to create wide low-eccentricity binaries unless the pre-sn orbit is already wide. Orbital periods around P_orb,0∼100 d is still too short to avoid any binary interactions between the massive sn progenitor and the low mass companion. Typically, such interactions would tighten the orbit down to much shorter orbital periods. Therefore, if we integrate our distribution in Figure <ref> weighted by the distribution of post-binary-interaction periods, pure-kick models have a near-zero chance of forming wide low-eccentricity systems, especially Gaia NS1. Given that dynamical assembly in clustered environments have been proven to be difficult to create Gaia NS1-like systems <cit.>, triple evolution or our kick+rocket scenario may be the only plausible explanations for the formation of Gaia NS1-like binaries.


In this simple analysis we have ignored various possible effects that may influence the probabilities. For example, we have assumed that the kick, rocket and orbit are independent whereas there may be correlations between the direction of these components. There may be correlations between the pre-sn orbital periods and the rocket magnitudes due to tidal spin-up of the sn progenitor. Different forms of kick velocity distributions may impact the overall survival and wide-orbit fractions. The thresholds we use for “wide” and “low-eccentricity” are arbitrary choices. We leave detailed exploration of formation rates of Gaia NS1-like binaries to future work.



natexlab#1#1




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