Paper Title: SPECtrophotometer for TRansmission spectroscopy of exoplanets (SPECTR).







0009-0005-5145-5165]Yeon-Ho Choi
Department of Astronomy and Space Science, University of Science and Technology, Korea, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea
Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea

0000-0003-1544-8556]Myeong-Gu Park
Department of Astronomy and Atmospheric Sciences, Kyungpook National University, 80 Daehak-ro, Buk-gu, Daegu 41566, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea

0000-0001-8969-0009]Jae-Rim Koo
Earth Environment Research Center, Kongju National University, 56 Gongjudaehak-ro, Gongju-si, Chungcheongnam-do 32588, Republic of Korea

0009-0008-2592-9437]Tae-Yang Bang
Department of Astronomy and Atmospheric Sciences, Kyungpook National University, 80 Daehak-ro, Buk-gu, Daegu 41566, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea


Division of Energy and Optical Technology Convergence, Cheongju University, 298 Daeseong-ro, Cheongwon-gu, Cheongju 28371, Republic of Korea


Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea




Department of Astronomy and Space Science, University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea
Korea Astronomy and Space Science Institute,
776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea


Byeong-Cheol Lee
bclee@kasi.re.kr


§ ABSTRACT 

The SPECtrophotometer for TRansmission spectroscopy of exoplanets (SPECTR) is a new low-resolution optical (3800 Å - 6850 Å) spectrophotometer installed at the Bohyunsan Optical Astronomy Observatory (BOAO) 1.8 m telescope. SPECTR is designed for observing the transmission spectra of transiting exoplanets. Unique features of SPECTR are its long slit length of 10 arcminutes which facilitates observing the target and the comparison star simultaneously, and its wide slit width to minimize slit losses. SPECTR will be used to survey exoplanets, such as those identified by the Transiting Exoplanet Survey Satellite (TESS), providing information about their radii across the wavelength range. In this paper, we present the design of SPECTR and the observational results of the partial transit of HD 189733 b and a full transit of Qatar8 b. Analyses show the SPECTR's capability on the white light curves with an accuracy of one ppt. The transmission spectrum of HD 189733 b shows general agreement with previous studies. 




































§ INTRODUCTION
 


The Kepler/K2 and the TESS space missions so far have confirmed about 3,800 exoplanet candidates, 67% of which are confirmed exoplanets[https://exoplanetarchive.ipac.caltech.edu/]. 
There are about 10,000 candidates yet to be confirmed. While these successful missions have discovered a substantial number of exoplanets, further follow-up observations, such as precise radial velocity measurements, are still necessary to determine their masses. Furthermore, there has been a growing importance placed on studying the atmospheric characteristics of exoplanets, such as atmospheric composition and the presence of clouds using transmission and thermal emission spectroscopy methods.

In particular, transmission spectroscopy, employed for transiting exoplanets, stands out as an exceptional tool for investigating the upper atmospheres of these planets. This method has been widely used with both space- and ground-based telescopes over a range of spectral resolutions. Space telescopes such as the Hubble Space Telescope <cit.> and the JWST <cit.>, along with ground-based telescopes such as the Very Large Telescope <cit.>, the Gemini Telescope <cit.>, and the Keck Telescope <cit.>, and programs such as the ACCESS program with the Magellan Telescope <cit.> and the LRG-BEASTS program with the William Herschel Telescope <cit.> have successfully studied the atmospheric characteristics of exoplanets. Until recently, such studies were limited to space- and ground-based large telescopes due to the faintness of the stars.

However, TESS has been particularly effective in discovering exoplanets orbiting bright stars. The atmospheres of these exoplanets can be studied using smaller telescopes like the BOAO 1.8 m telescope. This is because the 1.8 m telescope at BOAO, with its limited collecting area, is best suited for observing bright stars, which are also the primary targets of TESS due to its own small telescope size.

With these opportunities, we aimed to develop a spectrophotometer capable of studying the atmospheric characteristics of the TESS targets with the Bohyunsan Optical Astronomy Observatory (BOAO) 1.8 m telescope located in South Korea.

Korea Astronomy and Space Science Institute (KASI) and Kyungpook National University (KNU) initiated the development of the SPECtrophotometer for TRansmission spectroscopy of exoplanets (SPECTR). 
Considering the given situation and budget, we concluded that a long-slit spectrophotometer type is most suitable for studying the atmospheric characteristics of exoplanets at BOAO. While a long-slit spectrograph was operating at BOAO, it was not suitable for performing transmission spectroscopy due to its small field of view (FOV, 3.3× 3.3) and the absence of an imaging mode. The FOV determines the likelihood of finding a proper comparison star, necessary to perform precision differential spectrophotometry from the ground, due to the variability of Earth's atmosphere. Also, an imaging mode is required to confirm the position angle between the target and the comparison star to ensure that both stars are located within the same slit. 

In this paper, we describe the optical design of SPECTR and its first observational results.



§ SPECTR DESIGN AND SYSTEM


    

 §.§ Optical design

    SPECTR is designed to fit into the BOao Echelle Spectrograph (BOES) Cassegrain Interface Module <cit.>. 
    The CIM, attached to the Cassegrain focus of the BOAO 1.8 m telescope, includes a slit assembly, a slit-monitoring system, and a calibration lamp system. 
    The CIM is designed to facilitate high-resolution observation with BOES and medium-resolution observations with the Long Slit Spectrograph (LSS). BOES is a fiber-fed echelle spectrograph capable of providing precise radial velocity measurements <cit.>. BOES covers a spectral range from 3,500 Å to 10,100 Å. Its spectral resolution varies from 30,000 to 90,000, depending on the fiber diameter, which ranges from 80 microns to 300 microns.
    
    The original design of SPECTR had a straight optical path to enable the imaging mode utilizing the volume phase holographic grating (VPHG). 
    
    However, due to the limited space and compartments formed by walls inside the CIM as shown in Figure <ref>, it was changed to an L-shaped design using a flat mirror. Figure <ref> shows the optical layout of SPECTR. SPECTR consists of a slit assembly, a collimator assembly, a grism assembly, and an imaging assembly.

    The optical design of an L-shape can cause an imbalance at the center of mass in the CIM. To address this, we conducted simulations to test the amount of deflection due to changing gravity. We confirmed that the maximum deflection of the outer part of SPECTR at 1 g was 22 microns, which corresponded to a 2.6 Å shift in the spectral axis. Given that we use 400 Å bins for the spectroscopic light curves, this 2.6 Å shift is minor and will not severely affect the results.
    
    
    
    
    
    
    

 §.§ Slit assembly

    SPECTR has a wavelength coverage from 3,800 Å to 6,850 Å and a spectral resolution (R) ranging from 200 to 1,300 depending on the slit width. Details of the spectral resolutions for each slit are listed in Table <ref>.
    SPECTR contains eight slits of different widths, in addition to one pinhole for testing and one blank space for imaging to check the positions of the target and the comparison stars. Slit #1 is mainly used for taking the spectrum from the FeAr lamp for wavelength calibration and slits from #2 to #4 for general long-slit spectroscopy. Since transit observations require minimizing the light loss which may arise due to a finite slit width, we use slits from #5 to #8, much wider than the typical seeing at BOAO. 
    
    The slit plates are made of stainless steel (STS), measuring 60 mm × 22 mm, and they are coated with SiO_2 after metal polishing. The slits are manufactured by wire electric discharge machining and have the same physical length of 42 mm, which corresponds to 10 arcminutes in the FOV. 
    The blank space has the physical dimensions of 46 mm × 20 mm.
    
    
    
    
    In ground-based transmission spectroscopy, it is essential to observe a comparison star simultaneously with the target to correct for the effects of the changing of the Earth's atmosphere. A longer slit length will increase the probability of finding a suitable comparison star for the target. The LSS is already installed for low-resolution spectroscopy, but its FOV covers only one-third of that of SPECTR, limiting its capability to capture comparison stars simultaneously.
    In addition, the slit of SPECTR can be rotated to match the position angle of the two stars.
    To arrive at the optimal length of the slit within the optical and physical limitations of the telescope and the CIM, we estimated the number of suitable comparison stars available within a given FOV using the TESS Input Catalog <cit.>. During test observations, we found that a proper comparison star should have a difference in photometric magnitude of at most 1 magnitude in the V band and have colors or spectral types similar to the target. The TIC provides TESS magnitudes for about 471 million bright stars. Moreover, TESS provides light curves of stars, that can be used to check the photometric variability of potential comparison stars.
    Figure <ref> shows the number of stars with magnitudes within the interval of 1 magnitude for slit FOVs of 3, 5, and 10 arcminutes.
    
        
    
    
    
    

 §.§ Collimator assembly

    A collimator with an effective focal length of 280 mm was designed to change the f/8 beam from the 1.8 m telescope to a 35 mm diameter at the SPECTR grism. The collimator consists of two symmetric achromatic doublet lenses, which collimate the incident light towards the grism. The doublet lenses are fixed in place using two spacers, axial springs, radial springs, and retainers.
    

 §.§ Grism assembly

    The collimated beam is reflected by the folded mirror toward the grism and dispersed toward the imaging lens. The folded mirror has a physical dimension of 94 mm × 66 mm and is coated with enhanced aluminum for durability and reflectance with SPECTR's wavelength coverage.

    The grism assembly houses the grism. The grism consists of a VPHG and a wedge prism. The detailed parameters of the VPHG are given in Table <ref>. 
    
    It is manufactured by Wasatch Photonics, USA[https://wasatchphotonics.com/]. Two N-BK7 wedge prisms with an apex angle of 25.3 are glued to either side of the VPHG which together produces a straight optical path at 5261 Å. The grism is installed on the X-stage with a step motor and therefore can be moved the grsim away from the light path for the imaging mode.
       
    
    
    
    
    
    

 §.§ Imaging assembly

    We selected a Canon RF 85 mm f/1.2 lens as the imaging lens. The lens has excellent imaging capability despite its small form factor and large FOV. The imaging lens is backed by Andor iKon-L 936 CCD, which serves as the CCD for SPECTR. 
    This Peltier-cooled CCD has a pixel array of 2048 × 2048 and a pixel size of 13.5  × 13.5 . The designed FOV in imaging mode is 10× 4.6 with a plate scale of 0.62/pixel. The gain and readout noise are 2.3 e^-/ADU and 9 e^-, respectively. 
    
    

    
    

 §.§ Slit monitoring and lamp systems of CIM

    SPECTR installed on the CIM of the BOAO 1.8 m telescope is shown in Figure <ref>.
    We designed SPECTR in a cartridge form factor, which can be installed and removed easily, and has compatibility with existing systems. Such a design has three advantages. First, it allows us to utilize the observing time before and after the transit event. 
    Transit observations typically require only a fraction of a night. Hence, a simple procedure to exchange SPECTR with BOES allows efficient usage of the allocated time.
    
    Second, SPECTR can utilize the slit-monitoring and auto-guiding systems within the CIM. The FOVs of each system are 3 × 3 and 40 × 40, respectively. Precise guiding is crucial for transit observations lasting several hours to ensure that the whole point spread functions of both the target and the comparison star remain within the slit to avoid slit losses in the presence of changing seeing conditions.
    Although the slit-monitoring system has a limited FOV, it provides more precise guidance compared to the auto-guiding system. We utilize the auto-guiding system if no adequate star for guiding is available within this FOV.
    We checked the performance of each guiding system during test observations. The observations with the slit-monitoring system showed gradual positional drifts of approximately 1.8 over three hours, while the auto-guiding system showed drifts of approximately 3.6 shifts over four hours. 
    Third, it allows us to utilize the FeAr wavelength calibration lamp system in the CIM. Although SPECTR is designed to be integrated with the CIM, the existing calibration lamp system was not fully optimized for the SPECTR. When the calibration lamp is set to the optical axis, it only covers 50 which is 8% of the FOV of the SPECTR (red lines in Figure <ref>). We modified the lamp system so that the lamp along the spatial axis covers the whole FOV of SPECTR. Figure <ref> shows the resulting. The narrowest #1 slit width of 1.4 (R ∼ 1300) was selected to identify the wavelength solution and the rms value of the residuals after fitting was about 0.1 Å.
    
    
    





§ SYSTEM PERFORMANCE

We tested the performance of SPECTR by observing transiting close-in exoplanets. The first test target was the well-known exoplanet HD 189733 b <cit.>. 
The other was Qatar8 b, discovered by the Qatar Exoplanet Survey <cit.>. Our main purpose of the test was to validate the spectrophotometric stability and accuracy of SPECTR by comparing our results with existing spectroscopic light curves.


    

 §.§ Observations

    
    We observed with SPECTR one partial transit of HD 189733 b, starting at ingress, and one full transit of Qatar8 b. A summary of the observations is provided in Table <ref>. Observations were carried out with the #7 slit, which has a width of 15, to minimize slit losses.
    Although the spectral resolution R corresponding to this slit width is approximately 150, the effective spectral resolution is determined by the seeing when the seeing is smaller than the slit width. The mean values of seeing for each observation were about 1.5, resulting in a mean spectral resolution of approximately 1200.
    The auto-guiding system was used during the HD 189733 b observations, while the slit-monitoring system was used during the Qatar8 b observations. All observations were carried out with the target and a comparison star simultaneously within the slit. We collected calibration data such as bias, dark, wavelength calibration, and sky flat in both observations.
    
    The partial transit of HD 189733 b was observed on October 14, 2021. The airmass changed during the observations from 1.03 to 1.61. We selected the comparison star HD 345459 from <cit.>. This comparison star had been chosen for photometry and spectrophotometry in <cit.> and <cit.>. The target and the comparison stars have similar spectral types and magnitudes, K2V with V = 7.64 and K0V with V = 8.09, respectively <cit.>. A timeseries of 248 spectra of HD 189733 and the comparison star were obtained during in and out of transit with an exposure time of 15 seconds per frame. The cadence of the observations was 20 seconds, with five seconds of readout time, which was fixed throughout the entire test observation.

    One full transit of Qatar8 b was observed on April 8, 2022. The airmass changed during the observations from 1.22 to 1.81. Qatar8 is known to be a G0 V star <cit.>. Qatar8 is much fainter than HD 189733 and was chosen to test the spectrophotometric stability of SPECTR for a fainter system. We chose the comparison star TYC 4387-01409-1 (also known as TIC 103753216) from TIC v8 <cit.>. Although the spectral type of the comparison star is not known, we selected it based on its (B-V) color and checked its photometric variability from TESS light curves. The  () light curves of TYC 4387-01409-1 in TESS Sectors 40, 41, and 47 appear stable at the rms = 0.8 ppt across all sectors, as shown in Figure <ref>. 
    The target and the comparison star have similar magnitudes and colors, V = 11.71 with B-V = 0.534 and V = 11.65 with B-V = 0.573. A timeseries of 78 spectra of Qatar8 and the comparison star were obtained in and out of transit with each exposure time of 300 seconds without readout time.


    

    
    
 
    

 §.§ Data reduction
 

    Data reduction was performed with standard IRAF <cit.> procedures for bias, dark subtraction, illumination correction, aperture tracing, and wavelength calibration. From flat images of SPECTR, we found that uneven illumination does not significantly affect the results if the angular separation between the target and the comparison star is smaller than 7 arcminutes. For the HD 189733 b observations, the angular separation between the target and the comparison star was 8.72 arcminutes. Although we applied illumination correction to both targets, only the results for the HD 189733 b showed a significant difference.

    Aperture tracing and extraction were performed using the onedspec/apextract package. To confirm the aperture radii for both targets, we incrementally increased the radii from 3 to 15 pixels and extracted the aperture at each step. In this procedure, we fixed the background region as the pixel ranges from -30 to -20 and from 20 to 30 from the center to maintain consistency. Then we constructed white light curves of each aperture radius and compared the standard deviations of the out-of-transit light curves. The lowest scatters were found when we adopted aperture radii of 5.5 pixels for the HD 189733 and 11 pixels for the Qatar8 observations. These aperture radii were fixed for the subsequent analysis. The difference in aperture radii between the two observations seems to arise from different seeing conditions.

    Despite guiding, there were shifts along the spectral axis during the observations. We aligned the spectral axis of all spectra by comparing each spectrum with the reference spectrum using  function. Figure <ref> shows the pixel shifts over time in both spatial and spectral axes. Given that the integration time for Qatar8 b observations was approximately twice that of HD 189733 b observations, guiding with the slit-monitoring system appears significantly more effective. However, it is important to note that various confounding factors, such as atmospheric turbulence, the gravitational shifts depending on the position angle, and the position of the target in the sky, can impact guiding precision.
     
    Figure <ref> shows the extracted sample spectra of HD 189733 and Qatar8, along with those of the comparison stars. Qatar8 and its comparison star show quite similar spectral shapes, albeit with slightly different reported temperatures. The passbands of 400 Å width, used to construct seven spectroscopic light curves, are indicated by grayed vertical regions.

    


    
    










    

 §.§ White and spectroscopic light curves

    We integrated the flux across 4000 to 6800 Å to derive the white light curves for both targets, excluding wavelengths below 4000 Å  due to low SNRs.
    The white light curve of each target was obtained by dividing the integrated flux of the target by that of the comparison star. This removes the variations caused by the changing Earth's atmosphere as much as possible. Figure <ref> shows the resulting white light curves, illustrating the observed flux variations during the transit events of HD 189733 b and Qatar8 b, with model fits overlaying the data.
    
    We constructed the spectroscopic light curves by dividing the time-series spectra using uniform bins of 400 Å width.  
    Seven channels were extracted from 4,000 Å to 6,800 Å for each target and the light curve fits with the data from each channel are shown in Figure <ref>. 
    Spectroscopic light curves of both HD 189733 b and Qatar8 b show similar shapes to the white light curves of each target before detrending. Since the systematics appear consistent across wavelength bins, we detrended the white light curves and the spectroscopic light curves in a similar manner.


    

 §.§ Light curve fitting


    We fitted the transit light curves using a quadratic limb-darkening transit model <cit.> implemented in  <cit.>. 
    Given that only a single transit event was observed for each target, we adopted the orbital parameters such as the period, semi-major axis, and inclination from previous studies <cit.>.
    Mid-transit time, planet-to-star radius ratio R_ p/R_ *, limb-darkening coefficients, and baseline trend were modeled using .
    
    As shown in the middle panels of Figure <ref>, the relative fluxes show both downward and upward parabolic trends. 
    To account for these trends over time, we applied the Gaussian Process (GP) with a Matérn 3/2 kernel, implemented in  <cit.>.
    The transit parameters and baseline trends were determined using a Bayesian framework.
    The baseline trends were derived by dividing the fluxes by the transit models and subsequently fitting them with the GP.
    
    We set a normal prior for all parameters, except for R_ p/R_ *, which had a uniform prior range between 0.01 and 0.2.
    The mid-transit time prior was set using a normal distribution with a standard deviation of 0.01 days. For HD 189733 b, the center value was 1.96368 days, and for Qatar8 b, the center value was 178.125434 days, both referenced to BJD - 2,459,500 days, and the limb-darkening coefficients were calculated using the Python Limb Darkening Tool Kit <cit.> based on literature values for stellar parameters (, , and ) listed in Table <ref>. These values were used as means and standard deviations for the normal priors. 
    
    Optimal parameters were initially found using the differential evolution optimization method <cit.> implemented in . This was followed by MCMC sampling with  <cit.> to estimate the final parameters while fixing the baseline trends.
    The mid-transit times for each object were fixed for the spectroscopic light curves based on the white light curve estimates.
    The limb-darkening coefficients remained consistent with the  calculations after parameter estimation. 


    The best-fit models for the white light curves of HD 189733 b and Qatar8 b are shown in Figure <ref>. 
    The summary of the best-fit and fixed parameters is listed in Table <ref>. 
    The rms values of the residuals after the light curve model fitting are 0.969 ppt for HD 189733 b and 0.158 ppt for Qatar8 b. 

    We tabulate the spectral radius ratio from the spectroscopic light curve fits in Table <ref>. Our transmission spectra along with those of <cit.> and <cit.> are shown in Figure <ref>. The R_ p/R_ * values for HD 189733 b determined in this work show a general agreement with those reported in previous studies, although two points deviate significantly from the general trend. This discrepancy may have been caused by the effects of the absence of data before ingress. 
    For Qatar8 b, the R_ p/R_ * values show a relatively smooth trend across the wavelengths with small increases in the 5800 Å and 6200 Å bins. 
    By additional monitoring of these targets, we aim to confirm whether these anomalies are indeed real. The mean rms values of the spectroscopic light curve fit residuals are 1.060 ppt for HD 189733 b and 0.843 ppt for Qatar8 b. 


    To illustrate how sensitive our transmission spectra are to potential atmospheric features in these two planets, we calculated the atmospheric scale heights of our targets using the equation from <cit.>, under the assumption of hydrogen-dominated atmospheres with a mean molecular weight (μ_m=2.37). The atmospheric temperatures were taken to be equal to the equilibrium temperatures of the planets, and we used the known radii and masses of the planets. For HD 189733 b (R_ * = 0.776 R_ Sun from <cit.>), the calculated scale height is 192 km, which corresponds to 0.0004 R_ p/R_ *. For Qatar8 b (R_ * = 1.315 R_ Sun from <cit.>), the calculated scale height is 911 km, which corresponds to 0.0010 R_ p/R_ *. Figure <ref> shows the amplitudes that would correspond to ± 5 scale heights with respect to the median of the transmission spectrum for each planet. This illustrates that SPECTR is sensitive to that range of atmospheric heights
    
    
    
     



     



    
    


    







§ SUMMARY



We present the instrumental design and spectrophotometric performance of the SPECtrophotometer for TRansmission spectroscopy of exoplanets (SPECTR) to be used with the BOAO 1.8 m telescope.
With its 10 long-slit capability and spectrophotometrical stability, SPECTR will provide observational opportunities for 1 – 2 m class telescopes to contribute to the understanding of exoplanetary atmospheres, particularly those transiting bright stars. While many of these targets will be discovered by TESS, others will come from various surveys and instruments. 
The test observations showed that SPECTR can obtain precise spectroscopic light curves with a mean rms precision of 0.951 ppt. Although the analysis of the HD 189733 b and Qatar8 b can be further improved, our preliminary results highlight the potential of SPECTR to perform atmospheric characterization of transiting exoplanets.
As the importance of exoplanet atmospheric research is increasing along with the increasing number of bright exoplanet systems, we hope to make a meaningful contribution to the characterization of exoplanets at visible wavelengths with the SPECTR and the BOAO 1.8 m telescope.








§ ACKNOWLEDGMENTS


We would like to express our sincere gratitude to the referee for their extensive valuable and insightful comments, which have greatly improved the quality and clarity of this paper. This paper is based on observations collected with the BOAO 1.8 m telescope. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.
B.-C. Lee acknowledges partial support by the KASI (Korea Astronomy and Space Science Institute) grant 2024-1-832-03 and acknowledges support by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No-2021R1A2C1009501).
M.-G. Park was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No-2019R1I1A3A02062242 and No-2018R1A6A1A06024970) and KASI under the R&D program supervised by the Ministry of Science, ICT and Future Planning.
J.-R. Koo was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT; grant No-2022R1C1C2004102)


Software:  <cit.>,   <cit.>,  <cit.>,  <cit.>



aasjournal




 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 Trojan-like Colors of Low-Perihelion Kuiper Belt Objects.




0000-0003-4778-6170]Matthew Belyakov
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA
Matthew Belyakov
mattbel@caltech.edu
0000-0002-8255-0545]Michael E. Brown
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA

0009-0001-9229-4581]Alya al-Kibbi
Department of the Geophysical Sciences, University of Chicago, Chicago, IL 60637, USA

§ ABSTRACT 
An important testable prediction of dynamical instability models for the early evolution of the Solar System is that Jupiter Trojans share a source population with the Kuiper belt. Concrete evidence of this prediction remains elusive, as Kuiper belt objects (KBOs) and Jupiter Trojans appear to have different surface compositions. We address the long-standing question of Trojan origin by finding a dynamical sub-population in the Kuiper belt with Trojan-like colors. Combining existing photometric data with our own surveys on Keck I and Palomar P200, we find that the low-perihelion (q<30AU, a>30AU) component of the Kuiper belt has colors that bifurcate similarly to the Jupiter Trojans, unlike Centaurs (a<30AU) which have redder, Kuiper belt-like colors. To connect the Jupiter Trojans to the Kuiper belt, we test whether the distinct Trojan-like colors of low-perihelion KBOs result from surface processing, or are sourced from a specific population in the Kuiper belt. By simulating the evolution of the Canada-France Ecliptic Plane Survey synthetic population of KBOs for four billion years, we find that differences in heating timescales cannot result in a significant depletion of Very Red low-perihelion KBOs as compared to the Centaurs. We find that the neutrally-colored scattered disk objects (e>0.6 KBOs) contribute more to the low-perihelion KBO population rather than Centaurs, resulting in their different colors. 




§ INTRODUCTION

The solar system's early large-scale dynamical instability described by the many iterations of the Nice Model has had great success in providing a single resolution to many phenomena in the solar system, including the excited eccentricities of the giant planets <cit.> and the structure of the Kuiper belt and asteroid belt <cit.> among many other properties of the solar system <cit.>. Since the initial publication of the Nice model in 2005, the nature of its dynamical instability model has evolved significantly. The first three landmark Nice model papers relied on Jupiter and Saturn acquiring excited eccentricities through slowly migrating through their mutual 1:2 resonance <cit.>. Resonance crossing was abandoned to due its inconsistency with the structure of inner solar system, as during the instability, Saturn's secular resonances would sweep out material needed to form the inner planets. The current model for the early instability involves the ejection of a fifth (or potentially more) giant planet by Jupiter, which suddenly shifts Jupiter's orbit inwards, avoiding the issues of granular migration and resonance crossing <cit.>.

One of the most robust predictions present in all versions of the Nice Model is that the Jupiter Trojan asteroids at 5.2 AU share a source population with the more distant Kuiper belt <cit.>. In the current iteration of the Nice Model, the giant planets undergo a series of dynamical instabilities, caused by the ejection of one or more planetary cores formed somewhere between Jupiter and Neptune <cit.>. At the time of these instabilities, small bodies that are in the Trojan region of any of the giant planets would be thrown onto now chaotic orbits <cit.>. In this picture, the current Jupiter Trojans could only have been captured during Jupiter's final large-scale instability. Jupiter Trojans are thus simply the objects that happened to be at Jupiter's L4/L5 points during its final “jump”, captured into librating orbits. Meanwhile, planetesimals originating in the primordial Kuiper belt disk would be scattering throughout the solar system, forming the dominant population of small bodies in the outer solar system. Consequently, the Nice model implies that progenitor objects for the Kuiper belt also became the Trojans we see today <cit.>. This prediction gives a clear observational test for the Nice model: if Trojans and Kuiper belt objects (KBOs) indeed have a shared origin, these two populations and should have common physical properties <cit.>. However, observational backing for a shared origin remains elusive, leaving a fundamental gap in the evidence for the Nice model's predictions regarding the evolution of the outer solar system.

One piece of evidence that supports the shared origin of KBOs and Jupiter Trojans is the similar size distribution of the excited component of the Kuiper belt and the Trojans <cit.>. A similar size distribution is, however, only a necessary but not sufficient condition to prove KBOs and Trojans have a common source. Complicating this prediction are the differences in the surface properties of KBOs and Trojans. Recent JWST spectroscopy has shown that KBOs have a rich surface chemistry, with water, carbon dioxide, methanol, and carbon monoxide ices frequently present <cit.>. By contrast, Jupiter Trojans are spectroscopically neutral in the 1-4 micron range, with only weak organic and hydration features at around 3.0 microns <cit.>. Resolving how two populations that ought to have the same origin have divergent spectral and photometric properties is necessary to prove the common origin of KBOs and Trojans – an essential component of the Nice Model of the formation of the solar system. 

In this paper we find that the low-perihelion Kuiper belt objects (objects with q < 30 AU, a > 30 AU) have Trojan-like visible colors, and are a likely link between the Jupiter Trojans and Kuiper belt. In <ref>, we discuss the colors of the various populations of the outer solar system, demonstrating how observations of the low-perihelion Kuiper belt place it apart from the rest of the Kuiper belt and Centaurs, and that the colors potentially bifurcate into those of the Jupiter Trojans. In <ref> we investigate the two possible reasons for the unique colors of the low-q Kuiper belt as opposed to the Centaurs – different heating histories, or different source populations. Both of these possibilities are examined through the lens of dynamical simulations, finding that the difference in source populations between Centaurs and low-q KBOs is the more likely reason for the more neutral colors of the low-q KBOs. 




§ COLORS OF OUTER SOLAR SYSTEM POPULATIONS



Objects in the outer solar system generally have featureless visible reflectance spectra ranging from neutral to red colors. To good approximation, these colors can be defined by a single parameter referred to as the spectral slope. The spectral slope is defined as the slope of the reflectivity at each observed wavelength, expressed in units of the percent change in the reflectivity over 100 nm, typically normalized at V band (550 nm). The slope can be calculated from photometric observations in different colors and filter systems, though it is primarily useful in the regime where the spectrum of objects tends to be linear, between 400-800 nm <cit.>. In order to obtain a full picture of the variety of color distributions in the many dynamical sub-populations that exist in the outer solar system, we combine four data sets. The first is from the Minor Bodies of the Outer Solar System (MBOSS) set of colors  <cit.>, the second is the Colors of the Outer Solar System Origins Survey (Col-OSSOS) <cit.>, the third is from the Dark Energy Survey (DES) <cit.>, and the fourth is our own small survey of the colors of low-perihelion Kuiper belt objects, the technical details of which we describe in <ref>. We restrict ourselves to objects which have colors measured well enough that the uncertainty in the spectral slope is smaller than ±7%, equivalent to g-i ± 0.2. We include only objects with absolute magnitude H>7 in order to remove larger objects whose colors are significantly different from smaller objects, hypothetically due to the geologic processing from accretional heating of larger objects <cit.>. Moreover, since we are comparing the colors of the Kuiper belt to the Trojans, we should be selecting only those KBOs which match the size range of the Jupiter Trojans, the largest of which are at H ∼ 7 (see figure 6 of ). In total, we include 12 low-q KBOs from OSSOS, 7 from MBOSS, 4 from DES, and 24 from our survey for a total of 47. For the Centaurs, 31 are taken from OSSOS, 38 from DES, none from DES, and one from our survey.
 


Kuiper belt objects with semi-major axis greater than Neptune's orbit have long been known to have bifurcated colors, split into a “Very Red” (KBO VR) population with slope greater than 20% and a “Red” (KBO R) population with slope less than 20%, or g-i = 1.15 <cit.>. The Jupiter Trojan asteroids are also bifurcated in color, but into a “Red” (JT R) population, with a mean spectral slope of around 10% or g-i = 0.87, and a “Less Red” (JT LR) population, with a mean spectral slope of around 5% or g-i = 0.74 <cit.>. In this classification scheme, the Kuiper belt's “Red” colors fully encompass the range of Jupiter Trojan colors. 

Centaurs are former KBOs recently injected into the giant planet region and on unstable orbits with lifetimes typically between 100 kyr to 10 Myrs <cit.>. For the purposes of this paper, we define Centaurs as having both perihelia and semi-major axes less than 30 AU. This population represents a transition from the cold temperatures of the KBOs to the warmer environments of the Jupiter Trojans. Though Centaurs have been substantially heated for many millions of years, no evidence exists that Centaurs are developing the surface colors of Jupiter Trojans – their colors are statistically indistinguishable from those of the parent KBO population (see <ref>).

Previous publications have suggested color changes in Centaurs with heating, but, in each case, as more data have appeared, the statistical evidence has gone away (see ). As an aside, KBO surfaces clearly do evolve if they are heated enough. Jupiter family comets which originate from the Kuiper belt have darker surfaces than those typically seen on KBOs and have spectra generally unlike most KBOs, presumably as a result of significant ice sublimation over many perihelion passages <cit.>. Comet-like activity is seen in some of the closest Centaurs, though the correlation with surface colors remains unclear <cit.>. JWST spectroscopy of the active comet 39P/Oterma, which is on a Centaur-like orbit, shows active CO_2 outgassing and a surface that has muted absorption features compared to KBOs observed with JWST <cit.>. Additionally, JWST spectroscopy by <cit.> has shown that the inactive Centaur Okyrhoe and Neptune co-orbital 2010 KR59 have spectra dissimilar to the larger KBO population, with broad 3.0 micron absorption features more similar to the Trojans shown in <cit.> rather than to the KBOs. The lack of color changes in the Centaurs further obfuscates the link between Trojans and the Kuiper belt. For example, hypotheses such as <cit.> suggested heating as the mechanism for the change from KBO R/VR colors to Jupiter Trojan LR/R colors, yet this transition is not observed in the colors of the Centaur population. 

A sub-population within the Kuiper belt that does, however, appear to have distinct colors is the low-perihelion Kuiper belt (q<30 AU, a>30AU). These objects are dynamically distinct from the Centaurs as they spend most of their orbits outside of Neptune, though both populations share similarly low perihelia. These low-q KBOs appear to have a distinct set of colors from the Red/Very Red KBO and Centaur-like colors to Jupiter Trojan-like Red colors inside of q<25 AU, as shown in <ref>. For perihelia between 25 and 30 au, there are both low-perihelion KBOs and resonant objects with large eccentricities that cross Neptune's orbit, so we exclude this region from our analysis.

We can statistically compare the difference in the two populations by computing the fraction of KBO “Red” versus “Very Red” objects in each population. Of the 70 Centaurs with q<25 AU in our sample, 28 are Very Red and 42 are Red, for a Red fraction of f_r=0.6 ± 0.06 (using the Wald interval for the binomial proportion confidence interval). Of the 47 low perihelion KBOs, 40 are Red, and only 7 are Very Red, for a Red fraction of f_r=0.85 ± 0.05. Drawing 10,000 samples from the Centaur proportion of Red and Very Red objects and asking whether the low-perihelion KBOs has a larger proportion of Red objects than the Centaurs gives a 99.94% confidence that the two populations have different proportions of each color for q<25 AU. Using even the very conservative one-sided Fisher exact test, the probability of the low perihelion KBOs having the fewer Red objects than the Centaurs is under p=0.003, or over a 99.7% probability that their colors are different. In <ref>, we iteratively repeat the Fisher test for each subset of the low-q population by successively adding objects from lowest to highest perihelion. We find a more than three sigma probability that the low-q KBOs have fewer Very Red objects than the Centaurs for any perihelion value above 17 AU. Moreover, our result for the Fisher exact test (or the more granular test in <ref>) gives a confidence of over 99% even when spectral slopes of 15-25% are excluded, indicating that the difference in color between Centaurs and low-q KBOs is robust and does not rely on a finely-tuned definition of the color bifurcation. 




 §.§ Testing the Trojan Bifurcation in Low-perihelion KBOs



We now test whether low-perihelion Kuiper belt objects have Trojan like colors by analyzing the distribution of their colors in the Trojan-like 0-15% spectral slope range (KBO Red, or Jupiter Trojan LR/R colors).The upper end of this range was set to be 15% as that is the maximum spectral slope of Trojans reported in <cit.>. First, we check whether the color distribution of Red low-q KBOs differs from that of similar spectral slope Centaurs. Selecting all Centaurs and low-q KBOs with spectral slope < 15% and comparing the slope distributions with a Kolmogorov-Smirnov test, we find that the probability of them being drawn from the same distribution is at p=0.058. Although this result is just under a two-sigma significance, the 94% probability of the two populations having a different distribution in the range of Jupiter Trojan colors invites us to test whether the low-q KBOs or Centaurs are bifurcated similarly to the Jupiter Trojans. Using a gaussian mixture model, we test how many gaussians are preferred to describe the distribution of Centaur and low-q KBO colors. For the low-q KBOs, we find that a bifurcated model has a significantly lower Akaike Information criterion than a single gaussian model, with Δ AIC = 14 <cit.>. For the Centaurs, the AIC is higher for any model with more than one Gaussian, indicating that a bifurcated model does not well-describe the slopes of the Centaurs. In <ref>, we show the result of mixture models for the spectral slopes of the Centaurs and low-q KBOs along with the Jupiter Trojan colors from the Sloan Digital Sky Survey <cit.>. We report both the AIC and BIC for the one, two, and three gaussian mixture models for all three populations. The two means of the mixture model for the slopes of the slope < 15% low-q KBOs are close to the two colors of the Jupiter Trojan population, implying that the “Red” low-q KBOs have a similar color distribution to the Jupiter Trojans. A factor that also suggests the similarity between low-q KBOs and Trojans is that the ratio of objects with Trojan Red colors to Trojan Less Red colors seems to be replicated in the low-q KBO population. However, a notable inconsistency between the Trojan and low-q KBO colors is that redder component of the two sub-populations in the low-q KBOs is slightly redder than the mean colors of the “Trojan Red” subpopulation, as seen in the right panel of <ref>. A similar effect was found in <cit.>, who examine the Hilda asteroid population and find that their colors are bifurcated similarly to the Trojans, but with the mean colors shifted towards slightly more neutral values. The Hildas lie between the asteroid belt but interior to Jupiter in the 3:2 mean motion resonance with Jupiter at 4 AU, and are thought to have a common origin story with the Trojans <cit.>. The successive shifts in the color bifurcation from the Trojan-like component of the low-q KBOs to the Jupiter Trojans to the Hildas indicates that spectral slopes could become slightly more neutral with decreasing heliocentric distances. Our evidence for the Trojan-like bifurcation among more neutrally-colored (Red, slope <15%) low-q KBOs suggests there exists a subset of objects in the Kuiper belt with Jupiter Trojan-like colors, giving an observational link between the two populations.



 §.§ Highly-inclined Objects

An interesting subset of objects in the Centaurs and the low-perihelion KBOs are those that have high or retrograde inclinations. The dynamical origin of i>50^∘ objects remains an open question <cit.>. Simulations of the Nice model fail to produce the high-inclination (i>50^∘) objects we observe in the Kuiper belt <cit.>, such as 2015 BP_159 or any of the numerous i>90^∘ Centaurs and low-perihelion KBOs <cit.>; simply put, the disk-like nature of a protoplanetary disk likely precludes the formation of any sizeable reservoir of highly-inclined objects. Moreover, starting from the currently observed Kuiper belt, there is no way to produce retrograde objects with q>5.2 AU with the four giant planets <cit.>. All of the 14 low-q KBOs and 3 Centaurs with i>50^∘ have remarkably similar colors, with spectral slopes between 5% and 20%. If we repeat our statistical comparison of the Centaurs and low-q KBOs without these inclined objects, the significance of the difference between Centaurs and low-q KBOs lessens from more than 3σ to over 2σ, largely due to the decreased sample size. Similarly, evidence for the bifurcation in <ref> weakens, with a less significant ΔAIC ∼ 7. However, these retrograde objects are a key feature of the low-q KBOs and the Centaurs – our current inability to assess the dynamical pathways of these i>60^∘ objects does not reduce the significance of the difference in colors of the Centaurs and low-q KBOs. Therefore, our dynamical analysis in the remainder of this paper excludes the high inclination (i>50^∘) objects from our sample, as these objects cannot be replicated in dynamical simulations of the Kuiper belt and the four gas giants <cit.>.




§ ORIGIN OF TROJAN-LIKE COLORS



We present two possible testable hypotheses for why the colors of the low-perihelion Kuiper belt are distinct from the Centaurs. The first hypothesis is that the average low-q KBO is heated longer than the average Centaur, causing their surfaces to undergo the chemical changes that produce Jupiter Trojan colors, resulting in the population-level color difference we observe. The second hypothesis is a purely dynamical one – the low-q KBOs and Centaurs have distinct source populations with distinct colors, which are then carried over to the two excited populations. To test these hypotheses, we need to obtain the distribution of possible Kuiper belt origins and dynamical lifetimes of our sample of Centaurs and low-q KBOs, and determine whether the populations have significantly different heating histories or source populations. 



Dynamical histories of unstable objects are impossible to directly ascertain as backwards integration gives radically different results from infinitesimally small changes to initial conditions. Even starting with thousands of clones for each object and integrating backwards does not correctly link observed Centaurs and low-q KBOs with the Kuiper belt. This method starts with a problematic assumption that these clones are all equally likely to appear from the scattering of Kuiper belt objects. Instead, we create a forward model, integrating a sample Kuiper belt for four billion years to see what differences exist in the dynamical behavior of the bodies that produce Centaurs and low-q KBOs. 

Our forward model begins with the Canada–France Ecliptic Plane Survey (CFEPS) L7 simulated Kuiper belt reference population <cit.>. We first integrate the reference population with the four giant planets using the  integrator <cit.>. We exclude objects that are initially low-q KBOs (and thus have an uncertain source) to avoid contaminating our sample of the excited population. We use a hybrid changeover radius of 6 Hill radii, a timestep of 150 days, and an ejection distance of 10000 AU. Every 10 million days (around 25,000 years), we record instantaneously whether each object fits our definition of a Centaur or low-q KBO and its initial orbital elements. Over four billion years, we obtain 150,000 snapshots of our populations of interest that contain 25.1 million synthetic low-q KBOs and 3.1 million Centaurs. With millions of simulated Centaurs and low-q KBOs each with a full dynamical history, we can begin to test whether a difference in heating or difference in source population is responsible for the distinct color distribution of the Centaurs and low-q KBOs. 

We cannot, however, compare all simulated low-q KBOs to all simulated Centaurs to answer our hypotheses. To explain the colors we observe in the specific sample from <ref>, we must select simulated low-q KBOs and Centaurs that are similarly distributed to the real low-q KBOs and Centaurs. For each low-q KBO and Centaur in the photometric data from <ref>, we compute statistics on its 1,000 nearest synthetic neighbors. Nearest neighbors are determined using the Euclidean distance in eccentricity/inclination/perihelion space, where we have normalized each parameter to be between zero and one. A schematic of this method is shown in <ref>.




 §.§ Heating Hypothesis

To test the heating hypothesis, we track how much insolation each synthetic object has experienced using a simple metric: time spent within 20 AU. Specifically, we track the time a given object is inside of 20 AU within its orbit, not the time an object has a perihelion of less than 20 AU. This specific threshold is an arbitrary one, and is not due to any sublimation line – repeating our analysis for other thresholds such as 10 or 15 AU produces similar results. By selecting the 1,000 synthetic objects closest to each real Centaur and low-q KBO in our sample, we can obtain a statistical distribution of the time spent inside of 20AU for each object we have colors for. The left panel of <ref> shows the median amount (with one sigma error bars) of time spent inside of r<20 AU as a function of current perihelion distance for the Centaurs or low-q KBOs used in our photometric analysis in <ref>. For Centaurs and low-q KBOs, the typical time spent within 20 AU is 1 to 10 million years, with the low-q KBOs past 15 AU spending more time within 20 AU than the average Centaur. 




Having found the distribution of time spent within 20 au for each real Centaur and low-q KBO using the synthetic population generated from the CFEPS simulations, we can evaluate whether the heating histories of the two populations produce the observed distinct colors. In order to determine whether distinct heating histories can reproduce the colors of the two populations, we test whether there exists a critical threshold for time spent within 20 au for the change from Very Red to Red colors that would create a significantly different proportion of Red to Very Red objects in the low-q KBOs as opposed to Centaurs. For each real low-q KBO and Centaur in the left panel of <ref>, we assign a base 50% chance of the object being Red. Starting from a threshold of 20 million years of heating to transition from Very Red to Red, we determine the probability of each object having transitioned to Red colors, given its distribution of time spent within 20 AU. In other words, for each real object, we determine its probability of being Red based on how many of its synthetic look-alikes have spent more time within 20 au within each given time threshold. We then simulate 10,000 draws for the colors of all the objects, thus producing 10,000 simulations of the colors of Centaur and low-q KBOs populations. We then get the median proportion of Red objects in each population as a function of heating threshold for transitioning between the two colors, which is shown in the right panel of <ref>. Regardless of the choice of the minimum amount of time it takes for an object to change color, heating of low-q KBOs and Centaurs results in the low-q KBOs having only slightly fewer Very Red objects than the Centaurs, not distinct enough to match the real proportions. In the 10,000 simulations performed at each heating threshold, at most 4% of the simulations yield a color difference between the two populations that is more than the observed one. All heating thresholds outside the 1-3 Myr range result in all of the simulations yielding a difference in the proportion of Red objects between the two populations that is less than the one observed. It is therefore unlikely that irradiation of surfaces results in the difference between the colors of low-q KBOs and Centaurs.



Heating and stability are a non-trivial result of dynamical interactions, but our modelling suggests that heating is insufficient as an explanation for the more neutral colors of low-q KBOs as compared to the Centaurs. Other, more physical, means of modelling heating produce the same result. Computing the solar flux accumulated over the lifetimes of these objects after they cross inside of 30 AU produces a nearly identical result to time spent within 20 AU. We use time spent within 20 AU as it is the most conservative metric for heating given the question we are testing – Centaurs are on average at lower perihelia than low-q KBOs, thus, if they spend a similar amount of time within 20 AU, low-q KBOs are heated less. We note that there are indeed spectral changes in the infrared observed on comets that are a result of surface processing, such as those that occur to the 1.5, 2.0, 3.0 micron water features, or the CO_2 absorption feature <cit.>. Rather, we find evidence that surface processing on timescales equal to or less than the dynamical lifetimes of these excited objects cannot reproduce the color difference in the visible between Centaurs and the low-q KBOs. Thus, the origin of Jupiter Trojan-like spectral slopes in the visible in the low-q KBO population has to be addressed by comparing the dynamical sources of the two populations. 




 §.§ Source Population Hypothesis



Since heating is unlikely to be responsible for the difference in color between Centaurs and low-q KBOs, we test whether the color distribution in the q>30 AU Kuiper Belt, which is the source population for the Centaurs and low-q KBOs, can account for the distinct colors of the two populations. We test whether there are differences in source location of the low-q KBOs and Centaurs from which they inherit their distinct colors. We show the distribution of source eccentricities and inclinations for Centaurs and low-q KBOs in <ref>. The source eccentricities and inclinations are determined by selecting synthetic low-q KBOs and Centaurs near the observed ones and tracing them back to their origin in CFEPS, recording the starting eccentricity and inclination. Despite objects commonly transitioning between these two dynamical classes, the two populations are unevenly sampling the same Kuiper belt source. Centaurs preferentially come from the low-e, low-i part of the Kuiper belt, while low-q KBOs are more likely than Centaurs to come from the high-eccentricity scattered disk or high-inclination populations such as the most excited of the hot classicals and Plutinos. The more excited populations in the Kuiper belt are known to have fewer VR objects, as shown in <ref>. For objects with eccentricity above 0.4, around 14% are VR, whereas 71% are VR for lower eccentricities. Similarly, among the objects with inclinations above 30^∘, only 8% are VR. Given that the Centaurs draw more frequently from the less excited population that has significantly more Very Red objects than the excited population, we can test whether these differences are enough to account for their different observed source colors.

Having found that low-q KBOs and Centaurs have differences in their source populations, we can test whether this difference is enough to reproduce the color difference of the two populations. First, we need to determine what is the probability of each synthetic object being Red or Very Red as a consequence of its Kuiper belt source in the CFEPS population. Since the CFEPS population is a model of the observed Kuiper belt, we can use observed colors from MBOSS, Col-OSSOS, and DES <cit.> to get a likelihood that a given CFEPS reference population object is R or VR. We exclude Neptune Trojans, H<7, and Slope error <7% objects from our Kuiper belt color sample. For each object in CFEPS, we determine its color using a K-nearest neighbors regression on the observed colors from the aforementioned sources. In other words, the color of each object in CFEPS is given by the distance-weighted average of nearby observed Kuiper belt colors. This distance is the Euclidean distance between the CFEPS object and any given object in the MBOSS/Col-OSSOS/DES sample, where inclination, eccentricity, and perihelion are normalized to be between 0 and 1, such that each dynamical parameter is equally weighted. For example, a synthetic low-q KBO that is sourced from the classical Kuiper belt is more likely to be Very Red than one coming from the Scattered disk.


Now that we have the likelihood of an object being R/VR for each synthetic object, we can determine the probability that our real objects are R/VR as described in the beginning of this section. We take the nearest 1000 simulated neighbors to each real low-q KBO and Centaur, and calculate a distance-weighted average of the R/VR probabilities of the synthetic objects. As a result, for each low-q KBO and Centaur in our photometric sample, we have the corresponding probability of any given object being R/VR based on its Kuiper belt source. 




Finally, to test our hypothesis, we compute the proportion of the low-q KBOs and Centaurs that we expect to be Red, similarly to the statistic obtained in <ref>, where 58± 6% of Centaurs and 78± 7% of low-q KBOs (now excluding the retrograde objects from the color sample) were found to be Red. Given that we have the probability of each real Centaur and low-q KBO being red, we can simulate 10,000 draws for the color of each object, thus producing 10,000 simulations of the colors of Centaur and low-q KBOs populations. In <ref>, we show the probability that each Centaur and low-q KBO used in our photometric analysis has Red/Very Red colors based on its Kuiper belt source. The shaded regions show the one sigma range for the population proportion of Red objects for the two populations, while the dashed lines show the true proportions from <ref>. We find that the proportion of Red low-q KBOs derived from propagating the colors of their source population is 0.73 ± 0.07 (where the observed value for the proportion of Red q<25 au low-q KBOs is 0.78 ± 0.07), while for Centaurs the Red proportion is 0.63 ± 0.06 (where the observed proportion is 0.58 ± 0.06). 

While the data proportions of Red objects for both Centaurs and low-q KBOs are within one sigma error bars of the simulated proportions, there is overlap between the simulated proportions for both sets of objects. This indicates that while we can match the Centaur proportion of Red objects or the low-q KBO proportion of Red objects, we do not necessarily match the difference between the two populations of 20%. Of our 10,000 simulations of the Centaur and low-q KBO populations, just over 17% have a difference in Centaur and low-q KBO Red proportion of 20% or more.  While this result indicates it is possible that source population differences explain color differences and that this hypothesis is a more likely one than the heating hypothesis, we cannot assert source population differences as the complete explanation without more evidence. One likely possibility for our model not explaining the full difference in colors of the Centaurs and low-q KBOs is that CFEPS does not include enough high eccentricity and high inclination objects. The scattered disk in the CFEPS model is known to be too small <cit.>, and more current models, such as those being developed from the OSSOS survey include notably more SDOs and higher-order resonant objects (Kavelaars, Personal communication, 2023). If we increase the number of both e>0.8 and i>40^∘ objects at the start of the CFEPS simulation by an order of magnitude (simply by duplicating them), we find that in over 35% of our simulations we match the observed difference between the two populations. 


Stepping back to the bigger question, the link between colors in the scattered disk, the low-perihelion component of the Kuiper belt, and the Jupiter Trojans provides observational support for the Nice model. According to the Nice model, the scattered disk is sourced from the inner part of the outer planetesimal disk (24-30AU) <cit.>, similarly to the Jupiter Trojans, which are the inner disk planetesimals that are scattered and then captured by Jupiter during its final instability <cit.>. The scattered disk indeed has more neutral colors relative to the rest of the Kuiper belt, similar to the Jupiter Trojans. Our observations of the low-perihelion component of the Kuiper belt find it to have the same preferentially “Red” colors. Then from dynamical simulations, we find that the low-q KBOs preferentially originate from the scattered disk, selecting mostly those objects which were part of the inner part of the planetessimal disk, from which Jupiter Trojans originate. Thus, the Trojan-like colors of the low-q KBOs suggest that the scattered disk and Jupiter Trojans have a common initial Nice model origin in the component of the planetesimal disk closest to the sun.



§ DISCUSSION



Combining our photometric observations on Keck and Palomar with all available data from MBOSS <cit.>, Col-OSSOS <cit.>, and DES <cit.>, we have isolated the low-perihelion Kuiper belt as a population with Jupiter Trojan-like colors in the outer solar system. The measured colors show that these KBOs with q<30 AU and a>30 AU have a proportion of Red to Very Red objects more than 3σ distinct from Centaurs, which have similar perihelion distances. Critically, we note that the low-perihelion KBOs display a similar bifurcation in colors to the Jupiter Trojans, with a slightly redder color, mimicking how the Hildas, interior to the Trojans, are also bifurcated, but with a slightly bluer set of colors. 

More importantly, we have found observational evidence for the Nice model, by linking the Jupiter Trojans and Scattered Disk Objects, which according to the Nice model, have a common source in the inner disk. The low-q KBOs share the more neutral colors of the Scattered Disk and Jupiter Trojans, while having a distinct source in the scattered disk, unlike the Centaurs, which are sourced from the more stable parts of the Kuiper belt. We also demonstrate that these colors cannot have arisen from heating of the low-perihelion population, as similar-perihelion KBO-colored Centaurs and Trojan-colored low-q KBOs have heating histories which do not result in the drastic difference in color shown in <ref>. Instead, we show that the difference in colors of Centaurs and low-q KBOs is plausibly explained by the two populations having different Kuiper belt progenitors. The low-q KBOs are more frequently sourced from the scattered disk and high-inclination population than the Centaurs. Our results indicate that the difference in source populations is preferable to heating as an explanation for the distinct proportions of Red objects in the low-q KBOs and Centaurs. Further improvements to both the simulated models of the Kuiper belt as well as our knowledge of its colors should help fully resolve the question of the origin of the difference in Centaur and low-q KBO colors.










§ OBSERVATIONAL DETAILS








In the course of our observing campaign, we obtained optical g/i photometry of 34 objects, 24 of which were q<25 AU KBOs, with the P200 telescope at Palomar Observatory and the Keck I 10 meter telescope at Maunakea Observatory. Observations taken at Palomar observatory used the WaSP instrument (Wafer-Scale Imager for Prime) and were collected in three days spread over 2018-20. Observations taken at Maunakea observatory used the LRIS instrument (Low Resolution Imaging Spectrometer) and were collected in two days, one in 2021 and one in 2022. Observations were done in series pairs of g,i or i,g observations from which colors were derived, with at least two but typically four colors obtained for each object, with two sets done an hour apart to confirm object on-sky motion. 

Reductions were performed using standard bias subtraction and flat fielding, followed by standard aperture photometry done using Pan-STARRS stars in the field of view of the images to calculate zero points. Apertures were selected to maximize the signal-to-noise, typically close to the FWHM seeing (typical values around 1.5” at Palomar and 1.0” at Keck). In our images, the average trailing of objects was sub-pixel for the Palomar observations and around one pixel for the Keck observations, with few objects trailing more than 2 pixels. Background photometric stars were characterized and selected using the  algorithm <cit.> as implemented in the  package <cit.>. Stars fainter than m_g = 21, with errors more than 0.03 mags, or with fewer than 5 observations per band were excluded from the zero-point calculations. Due to the differences between SDSS and Pan-STARRS filters, magnitudes were converted using the bandpass transformations given in <cit.>. Background subtraction was performed with  <cit.>. A notable point in the reduction is that we only use one of the four available detectors in the P200 WaSP images, as bias drift over the night causes each detector to scale in a visibly different way even after biasing, flat fielding, and background subtraction. We summarize the results of our observations in <ref> and <ref>. The errors provided for the observations are obtained by calculating the standard deviation from multiple measurements of the color of the object, rather than from poisson noise on the photometry, as the photometric error for a set of measurements in any one band is dwarfed by the variability of an object and observing conditions over the course of a night. We convert our g-i colors to spectral slopes normalized to V band at 544.8 nm, following <cit.>. Solar colors for g and i band are taken from <cit.>. 



aasjournal




 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: Spin-orbit coupling of the primary body in a binary asteroid system.



Spin-orbit coupling]Spin-orbit coupling of the primary body in a binary asteroid system










1,2]Hanlun Leileihl@nju.edu.cn


[1]School of astronomy and space science, Nanjing University, Najing, 210023, China

[2]Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University, Nanjing, 210023, China






Spin-orbit coupling is widespread in binary asteroid systems and it has been widely studied for the case of ellipsoidal secondary. Due to angular momentum exchange, dynamical coupling is stronger when the orbital and rotational angular momenta are closer in magnitudes. Thus, the spin-orbit coupling effects are significantly different for ellipsoidal secondaries and primaries. In the present work, a high-order Hamiltonian model in terms of eccentricity is formulated to study the effects of spin-orbit coupling for the case of ellipsoidal primary body in a binary asteroid system. Our results show that the spin-orbit coupling problem for the ellipsoidal primary holds two kinds of spin equilibrium, while there is only one for the ellipsoidal secondary. In particular, 1:1 and 2:3 spin-orbit resonances are further studied by taking both the classical pendulum approximation as well as adiabatic approximation (Wisdom's perturbative treatment). It shows that there is a critical value of total angular momentum, around which the pendulum approximation fails to work. Dynamical structures are totally different when the total angular momentum is on two sides of the critical value.











[
    [
    July 2024
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§ INTRODUCTION




Spin-orbit resonance happens if the rotational period of a considered object is commensurable with its orbital period <cit.>. In our Solar system, spin-orbit resonance is widespread and it plays an important role in sculpting dynamical structures of rotation. The well-known examples are the Mercury and our Moon. In particular, the Mercury is locked inside the 3:2 spin-orbit resonance <cit.> and our Moon is locked inside a synchronous (1:1) resonance with the Earth <cit.>. It is known that most regular satellites around giant planets in the Solar System are locked in synchronous resonances, with stable orbital and spin evolution. However, some natural satellites are found to have chaotic rotations, e.g. Hyperion <cit.>, Nyx and Hydra <cit.>. Additionally, spin-orbit resonance is a common phenomenon existing in binary asteroid systems <cit.>.

The classical model for describing spin-orbit resonances corresponds to a restricted case where the rotational angular momentum is negligible compared to the orbital angular momentum. Thus, it is assumed that the mutual orbits are decoupled from rotation, and such an approximation works well for the secondary bodies under Sun-planet and planet-satellite systems <cit.>. Under such a classical model, <cit.> investigated the onset of chaos through Chirikov's criterion <cit.> and they derived a critical value for asphericity parameter α_c, above which the large-scale chaos may occur. <cit.> studied the stability of periodic orbits under a nearly-integrable Hamiltonian model. <cit.> took advantage of Lie-series transformation to formulate a second-order Hamiltonian model for describing spin-orbit resonances. In particular, they derived an integral of motion for both resonant and non-resonant configurations <cit.>. According to pendulum approximations, <cit.> derived explicit expressions of resonant half-width in terms of eccentricity and asphericity parameter for spin-orbit resonances and they updated the Chirikov criterion given in <cit.>. Under a fourth-order Hamiltonian model, <cit.> provided a genealogy for spin-orbit resonances and they provided an analytical estimation for amplitudes of a set of spin-orbit-spin resonances. In addition, secondary spin-orbit resonances inside the synchronous resonance have been studied under the classical spin-orbit model. In this respect, <cit.> took a perturbative method to study the secondary 3:1 resonance of Enceladus. In particular, they discussed tidal dissipation inside secondary resonance and found that forced libration due to secondary resonance may be large and thus would imply large tidal heating. <cit.> formulated high-order normal Hamiltonian model by means of Lie-series transformation theory <cit.> and their analytical theory can fully reproduce the dynamics of the 3:1, 2:1 and 1:1 secondary resonances inside the synchronous spin-orbit primary resonance.

In binary asteroid systems, rotation and orbit are strongly coupled in evolution and it is necessary to introduce the spin-orbit coupling model. <cit.> adopted the technique of Poincaré sections to explore coupled motions of rotation and translation for binary asteroid systems. Under the planar ellipsoid-ellipsoid and ellipsoid-sphere configurations, <cit.> updated spin-orbit coupled Hamiltonian models and systematically studied spin-orbit, spin-spin and spin-orbit-spin resonances. They found that the resonance center may change with the mass ratio and the mutual distance between two asteroids. For the spin-orbit coupling problem, <cit.> formulated a fourth-order-and-degree Hamiltonian model, which determines a 2-DOF dynamical model depending on the total angular momentum. In particular, they provided a criterion for dynamical closeness of asteroid pairs based on the angular momentum ratio between rotation and translation.

Regarding the spin-orbit coupling problem, the dynamics are significantly different when the mass ratio of asteroid pairs and mutual distance are varied, as pointed out by <cit.>. It has been widely investigated for the spin-orbit coupling for the secondary in a binary asteroid system and it is shown that there is no qualitative difference compared to the classical restricted spin-orbit resonances <cit.>. However, it is totally different for the primary. Because of primary's strong coupling between rotations and orbits, the assumptions of invariant orbit made in the traditional model are no longer appropriate in the binary asteroid system. Regarding this topic, spin-orbit resonances and their dynamical stability are studied by taking advantage of the approach of periodic orbits in <cit.>. In theory, we know less about the primary's spin-orbit coupling. To this end, in this work we focus on the dynamics of spin-orbit resonances for the primary body in a binary asteroid system.

The remaining part of this work is organised as follows. In Sect. <ref>, two equivalent dynamical models for the spin-orbit coupling problem are formulated and numerical analysis are performed in Sect. <ref>. In Sect. <ref>, elliptic expansions of Hamiltonian function are made and Sect. <ref> discusses the pendulum approximations for spin-orbit resonances. Perturbative treatments are adopted in Sect. <ref> to formulate resonant Hamiltonian model and conclusions are summarised in Sect. <ref>.



§ DYNAMICAL MODEL



In this work, we concentrate on the spin-orbit coupling of the primary body under a binary asteroid system. The primary asteroid with mass m_p is of triaxial ellipsoid, the secondary asteroid with mass m_s is of spherical shape and they move around each other on perturbed Keplerian orbits under their mutual gravity field. The reduced mass of system is denoted by m = m_p m_s/m_p + m_s. Due to the spin-orbit coupling, angular momentum may exchange between orbit and rotation to conserve the total angular momentum, leading to the fact that their mutual orbit is changed <cit.>. It means that the semimajor axis, eccentricity and longitude of pericentre for describing orbits are varied in the spin-orbit coupling problem. This is distinctly different from the classical spin-orbit problem with an invariant orbit <cit.>.



In the secondary-centred reference frame, Fig. <ref> depicts the relative geometry of spin-orbit coupling problem considered in this work, where the true longitude θ = ϖ + f describes translation and the angle ϕ describes rotation. The mutual distance between the primary and secondary is denoted by r, and the physical semi-major axes of the primary are denoted by a_p, b_p and c_p, satisfying a_p ≥ b_p ≥ c_p. Under the homogeneous assumption, the primary's moments of inertia along principal axes are computed by

    {I_1,I_2,I_3}= 1/5m_p {b_p^2 + c_p^2,a_p^2 + c_p^2,a_p^2 + b_p^2}.

To simplify the spin-orbit coupling model, the following two assumptions are made <cit.>: (a) the primary asteroid spins about its principal axis of inertia, and (b) the principal axis of inertia is perpendicular to the mutual orbit plane. The mutual gravitational potential under the binary asteroid system truncated at the fourth order in a_p/r is expressed as <cit.>[It should be noted that there is a typo for the term associated with C_42 in the expression of potential function given in <cit.>.]

    U (r,θ,ϕ) =     -  G m_p m_s[ 1/r - a_p^2C_20/2r^3 + 3a_p^4C_40/8r^5 + ( 3a_p^2C_22/r^3 - 15a_p^4C_42/2r^5)cos( 2ϕ  - 2θ).
       + . 105a_p^4C_44/r^5cos( 4ϕ  - 4θ) ],

where G is the universal gravitational constant. The dynamical model up to the eighth order in a_p/r is discussed in Appendix. The spherical harmonics coefficients C_ij are defined by <cit.>

    C_20   = 1/5a_p^2( c_p^2 - a_p^2 + b_p^2/2),  C_22 = 1/20a_p^2( a_p^2 - b_p^2)
    C_40   = 15/7( C_20^2 + 2C_22^2), C_42 = 5/7C_20C_22, C_44 = 5/28C_22^2.

As a result, the Hamiltonian function, governing the orbital and rotational evolution of the primary body, can be written as the sum of rotational and transnational kinetic energy and potential energy,

    H =    T + U
    
    =   1/2 m (ṙ^2 + r^2θ̇^2 ) + 1/2I_3ϕ̇^2+U
    
    =   p_r^2/2m + p_ψ ^2/2 m r^2 + 1/2I_3(G_ tot + p_ψ)^2
       -  Gm_p m_s[ 1/r- a_p^2C_20/2r^3 + 3a_p^4C_40/8r^5 + 105a_p^4C_44/r^5cos 4ψ.
       .  + ( 3a_p^2C_22/r^3 - 15a_p^4C_42/2r^5)cos 2ψ]

where p_r = m ṙ is the conjugate moment of r, the angle ψ is ψ = ϕ - θ, its conjugate moment is p_ψ = - p_θ = - m r^2θ̇, and the total angular moment is a conserved quantity, given by

    G_ tot = p_θ + p_ϕ,

with p_θ = m r^2 θ̇ and p_ϕ = I_3 ϕ̇ as the orbital and rotational angular momentum.

The Hamiltonian (<ref>) determines a 2-DOF dynamical model, depending on the total angular momentum G_ tot. The Hamiltonian canonical relations <cit.> leads to the equations of motion, given by

    ṙ   = ∂ H/∂p_r,  ṗ_r =  - ∂ H/∂ r,
    ψ̇   = ∂ H/∂p_ψ,  ṗ_ψ =  - ∂ H/∂ψ.

In the Hamiltonian model, the state is denoted by X = (r,p_r,ψ,p_ψ)^T. Usually, it is convenient for us to describe the spin-orbit state by means of Z = ( a, e, f ,ϕ-ϖ ,ϕ̇)^T. It is not difficult to realise mutual transformations between X and Z.

For convenience of computation, we adopt the following units of length, mass and time:

    [L] = a_p, [M] = m_pm_s/m_p + m_s, [T] = √( G( m_p + m_s)/.-a_p^3).

Under the normalised system of units, the gravitational parameter μ =  G(m_p+m_s) is equal to unity and G m_p m_s = 1. In the entire work, we use LU to stand for unit of length and TU to stand for unit of time. It is noted that the system of normalised units adopted here is in agreement with that in <cit.> (see the statement before Eq. (14) of their work), but it is different from that shown in <cit.>.

Let us define the primary-to-secondary mass ratio as α_m = m_p/m_s > 1. As a consequence, the normalised mass of the primary and secondary are specified by

    m_s = 1 + 1/α _m∈( 1,2),  m_p = 1 + α _m∈( 2,∞),

and the total angular momentum can be expressed as

    G_ tot   = p_ϕ + p_θ
       = I_3ϕ̇+ m r^2θ̇
       = 1/5 m_p ( a_p^2 + b_p^2)ϕ̇+ m√(μ a (1-e^2))
       = 1/5( 1 + α _m)( 1 + b_p^2)ϕ̇+ √(a( 1 - e^2)).

In practice, we take the reference semimajor axis a_ ref and the reference eccentricity e_ ref to characterise the total angular momentum by

    G_ tot = 1/5( 1 + α _m)( 1 + b_p^2) ϕ̇_ ref  + √(a_ ref( 1 - e_ ref^2))

where the reference spin frequency is taken as ϕ̇_ ref=n_ ref for the synchronous (1:1) spin-orbit resonance, ϕ̇_ ref=1.5 n_ ref for the 2:3 resonance and ϕ̇_ ref= 0.5 n_ ref for the 2:1 resonance (i.e., at the nominal location of resonance centre). The reference mean motion is computed from n_ ref^2 a_ ref^3 = μ. Please refer to Table <ref> for physical parameters of those known doubly synchronous binary asteroid systems. It is observed that the mass ratio of the primary and secondary is not far from unity except the binary system (624) Hektor. It should be noted that a_ ref and e_ ref are just two parameters to characterise the total angular momentum. If we change a_ ref (fixing e_ ref), both the semi-major axis and the spin frequency of the primary are varied.

Unless otherwise stated, we take the following normalised parameters of system to perform all simulations in this work:

    a_p = 1.0,   b_p = 0.95,   c_p = 0.85,  α_m = 10.

Please refer to Table (<ref>) for the harmonics coefficients of the ellipsoidal primary. The moment of inertia along the shortest axis is equal to I_3=4.1855. The normalised mass of the primary is m_p=11 and the normalised mass of the secondary is m_s=1.1.







§ NUMERICAL EXPLORATIONS



The equations of motion represented by Eq. (<ref>) are numerically integrated over a certain period and the resulting trajectories are plotted in Fig. <ref>. It is observed that there is a strong coupling between translation and rotation. The argument σ = ϕ-λ (λ is the mean longitude) is librating around 0^∘, indicating that this example trajectory is located inside the spin-orbit synchronous resonance. From the evolution of eccentricity (see the top-right panel), we can see that the example trajectory is composed of long-period and short-period contributions. However, the short-period amplitudes of a, σ = ϕ-λ and ϕ̇ are much smaller than the corresponding long-period amplitudes. In addition, the evolution of a, e, σ = ϕ-λ and ϕ̇ shows that they have almost the same frequencies about the long-period components.



To see the global structure of spin-orbit coupling, we choose to produce Poincaré sections, defined by 

    mod (f,2π)=0.

To generate Poincaré sections, the total angular momentum and the Hamiltonian should be given. As for the 1:1 resonance, the total angular momentum is parameterised by equation (<ref>) in terms of a_ ref and e_ ref, and the Hamiltonian is also parameterised in terms of a_ ref and e_ ref in the following form,

    H = .  H|_a = a_ ref, e = e_ ref, f = 0, ϕ-ϖ = π/2, ϕ̇= n_ ref.

It means that a given set of parameters (a_ ref,e_ ref) is used to uniquely characterise the total angular momentum G_ tot and the Hamiltonian H.

Poincaré sections are produced and presented in the top two panels of Fig. <ref>, where the index shown in the colour bar stands for the magnitude of eccentricity e. The structures arising in Poincaré sections are completely different from the classical pendulum structures and also different from the ones associated with ellipsoidal secondary. It is observed that almost the entire phase space is filled with regular motions (chaotic layers are too thin to observe). In addition, numerical trajectories inside libration islands are shown in the bottom panels of Fig. <ref>.

In the case of a_ ref = 3.0 and e_ ref = 0.1 (see the top-left panel of Fig. <ref>), there are four islands of libration as well as four saddle points arising in the Poincaré section. In particular, two islands centred at ± 90^∘ hold higher values of ϕ̇ (lower values of eccentricity) than the other two centred at 0^∘ and 180^∘. An evident feature is that their dynamical separatrices are completely separable. Two trajectories marked by A and B inside libration islands are numerically integrated and the results show that both of them are locked inside the synchronous spin-orbit resonance (see the bottom-left panel of Fig. <ref>). In particular, the trajectory A is librating around σ = 0 and the trajectory B is librating around σ = 90^∘.

In the case of a_ ref = 4.0 and e_ ref = 0.1 (see the top-right panel of Fig. <ref>), there are also four islands of libration and four saddle points arising in the Poincaré section. However, the phase-space structures are distinctly different from the former case. In particular, the two islands centred at ± 90^∘ move up to ϕ̇/n_ ref∼ 1.5 and the other two centred at 0^∘ and 180^∘ move to ϕ̇/n_ ref∼ 1.0. Two trajectories marked by C and D inside libration islands are numerically integrated and the numerical results show that both of them are located inside the synchronous spin-orbit resonance (see the bottom-right panel of Fig. <ref>). In particular, the trajectory C is librating around σ = 0 and the trajectory D is librating around σ = 90^∘.



§ EXPANSION OF HAMILTONIAN FUNCTION



The elliptic expansion takes the form <cit.>,

    ( a/r)^lcos( mf) = ∑_s =  - ∞^∞X_s^ - l,m( e )cos( sM)

where X_c^a,b (e) is Hansen coefficient dependent on eccentricity <cit.> and its lowest order in eccentricity is equal to |b-c|. After performing elliptic expansion, the Hamiltonian (<ref>) can be written as


    H =     - 1/2a + p_ϕ ^2/2I_3
        - ∑_s =  - ∞^∞{[  - C_20/2a^3X_s^ - 3,0( e ) + 3C_40/8a^5X_s^ - 5,0( e )]cos( sM).
        + [ 3C_22/a^3X_s^ - 3,2( e )  - 15C_42/2a^5X_s^ - 5,2( e )]cos( 2ϕ  - 2ϖ  - sM)
       .  + 105C_44/a^5X_s^ - 5,4( e )cos( 4ϕ  - 4ϖ  - sM)}


where the normalised system of units has been used to simplify the expression (i.e., m = 1.0, a_p =1.0 and μ=1.0). For convenience, we introduce a set of canonical variables as follows:

    l    = M,   L = √(a),
    
    g    = ω ,   G = √(a( 1 - e^2)),
    
    h    = Ω,    H= Gcosi,
    ϕ   ,    p_ϕ,

and perform the following canonical transformation,

    ψ _1   = ϕ  - g - h = ϕ - ϖ,  Ψ _1 = p_ϕ,
    ψ _2   = l,  Ψ _2 = L,
    ψ _3   = g,  Ψ _3 = G + p_ϕ,
    ψ _4   = h,  Ψ _4 =H + p_ϕ.

In terms of canonical variables (ψ _1,2,3,4, Ψ _1,2,3,4), the Hamiltonian (<ref>) truncated up to the fourth order in eccentricity can be organised by

    H =     - 1/2Ψ _2^2 + Ψ _1^2/2I_3
        - ∑_n =  - 4^4 [  - C_20/2Ψ _2^6X_n^ - 3,0 + 3C_40/8Ψ _2^10X_n^ - 5,0]cos( nψ _2)
        - ∑_n =  - 2^6 [ 3C_22/Ψ _2^6X_n^ - 3,2 - 15C_42/2Ψ _2^10X_n^ - 5,2]cos( 2ψ _1 - nψ _2)
        - ∑_n = 0^8 105C_44/Ψ _2^10X_n^ - 5,4cos( 4ψ _1 - nψ _2)

We need to remember that, in equation (<ref>), Hansen coefficients X_c^a,b (e) are related to the eccentricity e, which can be expressed in terms of the canonical variables by e = √(1 - G^2/L^2)  = √(1 - ( G_ tot - Ψ _1)^2/Ψ _2^2).

It is observed that the angles ψ _3 and ψ _4 are cyclic variables under the dynamical model determined by the Hamiltonian (<ref>). Thus, their conjugate momenta Ψ _3≡ G_ tot (the total angular momentum) and Ψ _4 are motion integral. We can see that the dynamical model specified by the Hamiltonian (<ref>) is of 2 degrees of freedom, depending on the total angular momentum. When the total angular momentum is given (please recall that a_ ref and e_ ref are used to specify G_ tot), the first degree of freedom (ψ_1,Ψ_1) stands for rotation and the second degree of freedom (ψ_2,Ψ_2) stands for translation.

In particular, spin-orbit resonances may take place if the rotational period and orbital period of the primary are commensurable. In order to describe resonant dynamics, it is necessary to introduce the resonant argument. To this end, the following canonical transformation is introduced for the k_1:k_2 resonance,

    σ _1   = ψ _1 - k_2/k_1ψ _2,  Σ _1 = Ψ _1,
    σ _2   = ψ _2,  Σ _2 = Ψ _2 + k_2/k_1Ψ _1.

The angle σ_1 stands for the resonant argument. In particular, it is σ_1 = (ϕ-ϖ) -M for the synchronous spin-orbit resonance and, it is σ_1 = (ϕ-ϖ) - 3M/2 for the 2:3 spin-orbit resonance and it is σ_1 = (ϕ-ϖ) - M/2 for the 2:1 spin-orbit resonance. As a result, the Hamiltonian (<ref>) becomes


    H =     - 1/2( Σ _2 - k_2/k_1Σ _1)^2 + Σ _1^2/2I_3  - ∑_n =  - 4^4 [  - C_20X_n^ - 3,0/2( Σ _2 - k_2/k_1Σ _1)^6.
       .  + 3C_40X_n^ - 5,0/8( Σ _2 - k_2/k_1Σ _1)^10]cos( nσ _2) - ∑_n =  - 2^6 [ 3C_22X_n^ - 3,2/( Σ _2 - k_2/k_1Σ _1)^6.
       .  - 15C_42X_n^ - 5,2/2( Σ _2 - k_2/k_1Σ _1)^10]cos[ 2σ _1 + ( 2k_2/k_1 - n)σ _2]
        - ∑_n = 0^8 105C_44X_n^ - 5,4/( Σ _2 - k_2/k_1Σ _1)^10cos[ 4σ _1 + ( 4k_2/k_1 - n)σ _2]


For convenience, we denote the Hamiltonian (<ref>) by the following compact form,

    H= ∑_k_1,k_2 C_k_1,k_2( G_ tot;Σ _1,Σ _2)cos( k_1σ _1 + k_2σ _2)

which specifies a 2-DOF Hamiltonian model, depending on the total angular momentum G_ tot.

Based on the Hamiltonian formulation given by Eq. (<ref>), in the following two sections we will take advantage of pendulum approximation as well as adiabatic approximation to analytically study spin-orbit resonances.



§ PENDULUM APPROXIMATIONS



In this section, we make some discussions about pendulum approximations to the spin-orbit coupling problem <cit.>.

For the synchronous (1:1) spin-orbit resonance (i.e., k_1 =k_2 =1), performing average for Hamiltonian (<ref>) over one period of σ_2 leads to the resonant Hamiltonian,

    H_1:1 =     - 1/2( Σ _2 - Σ _1)^2 + Σ _1^2/2I_3 + C_20/2( Σ _2 - Σ _1)^6 - 3C_40/8( Σ _2 - Σ _1)^10
        - [ 3C_22/( Σ _2 - Σ _1)^6 - 15C_42/2( Σ _2 - Σ _1)^10]cos 2σ _1
        - 105C_44/( Σ _2 - Σ _1)^10cos 4σ _1

which is truncated at the first order in eccentricity. The resonant Hamiltonian (<ref>) is coincident with that provided in <cit.> (see equation 23 in their study). Ignoring the influence of C_20 and C_40 upon the nominal resonance, we can obtain the nominal location of 1:1 resonance at Σ_1, res = I_3 √(μ/a_ ref^3). By expanding resonant Hamiltonian (<ref>) around the nominal resonance in Taylor series and removing high-order terms <cit.>, the approximate resonant Hamiltonian takes the form,

    Δ H_1:1 =   1/2( 1/I_3 - 3/( Σ _2 - Σ _1)_ res^4)ΔΣ _1^2
       - [ 3C_22/( Σ _2 - Σ _1)_ res^6 - 15C_42/2( Σ _2 - Σ _1)_ res^10]cos 2σ _1
       - 105C_44/( Σ _2 - Σ _1)_ res^10cos 4σ _1

where the constant terms have been removed, the term (Σ_2 - Σ_1)_ res is evaluated at the nominal location of resonance centre and it is equal to (Σ_2 - Σ_1)_ res = √(a_ ref) for the synchronous resonance. According to <cit.>, the fourth-order harmonic term associated with C_44 has no contribution to the width of the synchronous resonance. Thus, it can be removed from the Hamiltonian. As a consequence, we can reach the pendulum approximation for the resonant Hamiltonian (<ref>), given by

    Δ H_1:1 = 1/2( 1/I_3 - 3/a_ ref^2)ΔΣ _1^2 - [ 3C_22/a_ ref^3 - 15C_42/2a_ ref^5]cos 2σ _1.

Similarly, the pendulum approximation of resonant Hamiltonian for the 2:3 resonance (k_1=2 and k_2=3) can be written as

    Δ H_2:3 = 1/2( 1/I_3 - 27/4a_ ref^2)ΔΣ _1^2 - [ 21C_22e_ ref/2a_ ref^3 - 135C_42e_ ref/4a_ ref^5]cos 2σ _1,

and the pendulum approximation for the 2:1 resonance (k_1=2 and k_2=1) can be expressed as

    Δ H_2:1 = 1/2( 1/I_3 - 3/4a_ ref^2)ΔΣ _1^2 + [ 3C_22e_ ref/2a_ ref^3 + 15C_42e_ ref/4a_ ref^5]cos 2σ _1.

From the pendulum approximations, the location of libration centre is dependent on the sign of S_1:1 for synchronous (1:1) resonance, of S_2:3 for the 2:3 resonance and of S_2:1 for the 2:1 resonance,

    S_1:1 = 1/I_3 - 3/a_ ref^2,  
    S_2:3 = 1/I_3 - 27/4a_ ref^2,  
    S_2:1 = 1/I_3 - 3/4a_ ref^2.

In particular, the libration centre is located at

    2σ _1,c = {[ 0,   if S_1:1 > 0; π,   if S_1:1 < 0 ].

for the synchronous (1:1) resonance, it is located at

    2σ _1,c = {[ 0,   if S_2:3 > 0; π,   if S_2:3 < 0 ].

for the 2:3 resonance and it is located at

    2σ _1,c = {[ π,   if S_2:1 > 0; 0,   if S_2:1 < 0 ].

for the 2:1 resonance.

When I_3 is given, the sign of S_1:1 (or S_2:3 and S_2:1) is determined by the value of a_ ref. It is observed that the libration centre may be changed if the reference semimajor axis is different. This is consistent with the conclusion made by <cit.>.

In particular, for the synchronous resonance, the critical reference semimajor axis a_c,1 is

    S_1:1 = 0 ⇒a_c,1 = √(3/5( 1 + α _m)( a_p^2 + b_p^2))

for the 2:3 resonance, the critical semimajor axis a_c,2 is 

    S_2:3 = 0 ⇒a_c,2 = 3/2a_c,1,

and, for the 2:1 resonance, the critical semimajor axis a_c,3 is

    S_2:1 = 0 ⇒a_c,3 = 1/2a_c,1.

As a result, the libration centre σ_1,c is dependent on the reference semimajor axis,

    2σ _1,c = {[  0,   if a_ ref > a_c; π,   if a_ ref  < a_c ].

for the synchronous (1:1) resonance and the 2:3 resonance, and it is 

    2σ _1,c = {[  π,   if a_ ref > a_c; 0,   if a_ ref  < a_c ].

for the 2:1 resonance. It takes a_c = a_c,1 for the synchronous resonance, a_c = a_c,2 for the 2:3 resonance and a_c = a_c,3 for the 2:1 resonance. Please refer to Table <ref> for the critical value of reference semimajor axis a_c,1 for those known doubly synchronous binary asteroid systems. It is observed that it satisfies a_ ref < a_c,1 for (624) Hektor meaning that its resonance centre is at 2σ _1,c = π, and it satisfies a_ ref > a_c,1 for other binary asteroid systems, meaning that their resonance centres are at 2σ _1,c = 0. This is in agreement with the statement made by <cit.>.

If the equatorial elongation is fixed at a_p/b_p = 1/0.95 (the elongation adopted in the entire work), the critical semimajor axis is an increasing function of the primary-to-secondary mass ratio α_m, as shown in the top-left panel of Fig. <ref>. In particular, when α_m = 10, the critical semimajor axes are a_c,1=3.53, a_c,2=5.31 and a_c,3=1.76 in normalised unit of length. As the critical semimajor axis a_c,3=1.76 is close to contact, it shows that the exchange of libration centre is physically impossible for the 2:1 resonance in the case of α_m = 10. Thus, in the following simulations, we will not present simulation results for the 2:1 resonance.



To see the influence of reference semimajor axis (with fixed reference eccentricity at e_ ref=0.1) upon dynamical structures, we take the synchronous (1:1) resonance as an example and adopt three representative reference semimajor axes at a_ ref = 3.0 (below the critical value), a_ ref=3.5 and 3.6 (near the critical value), and a_ ref=4.0 and 5.0 (above the critical value) to produce Poincaré sections. With a given set of (a_ ref, e_ ref), the total angular momentum and the Hamiltonian are determined. Please refer to Fig. <ref> for Poincaré sections. The cases of a_ ref = 3 and a_ ref=4 show that there are four libration islands and four saddle points (but their structures are different). Dynamical structures are distinctly different for the cases of a_ ref=3.5 and 3.6 which are near the critical semimajor axis. Please refer to the middle-row panels of Fig. <ref>, which show that there are no libration islands in the phase space. In the case of a_ ref=5, there are two islands of libration and two saddle points in the Poincaré section.

From the pendulum approximations, we can further derive the expression of resonant half-width. The resonant half-width in terms of Δϕ̇ is given by

    ( Δϕ̇/n)_1:1 = √(12C_22a_ ref^2 - 30C_42/I_3( a_ ref^2 - 3I_3)),

for the synchronous resonance, it is

    ( Δϕ̇/n)_2:3 = √((42C_22a_ ref^2 - 135C_42)e_ ref/I_3( a_ ref^2 - 27/4I_3)),

for the 2:3 spin-orbit resonance and it is

    ( Δϕ̇/n)_2:1 = √(( 6C_22a_ ref^2 + 15C_42)e_ ref/I_3( a_ ref^2 - 3/4I_3))

for the 2:1 spin-orbit resonance. Resonant half-width of the synchronous resonance is independent on the eccentricity while resonant width of the 2:3 (and 2:1) resonance is an increasing function of the eccentricity. Comparing to Table 1 of <cit.>, we can see that the expressions of resonant half-width are different[The author of <cit.> confirmed that there exist typos in their expressions (private communications).].

Under the pendulum approximations (<ref>-<ref>), dynamical separatrix corresponds to the level curve of Hamiltonian passing through saddle points and it plays an important role in dividing the entire phase space into regions of libration and circulation. As a result, the dynamical separatrix is expressed by

    1/2( 1/I_3 - 3/a_ ref^2)ΔΣ _1^2 - [ 3C_22/a_ ref^3 - 15C_42/2a_ ref^5]( cos 2σ _1± 1) = 0

for the synchronous (1:1) resonance, it is expressed by

    1/2( 1/I_3 - 27/4a_ ref^2)ΔΣ _1^2 - [ 21C_22/2a_ ref^3 - 135C_42/4a_ ref^5]e_ ref( cos 2σ _1± 1) = 0

for the 2:3 resonance and it is expressed by

    1/2( 1/I_3 - 3/4a_ ref^2)ΔΣ _1^2 + [ 3C_22/2a_ ref^3 + 15C_42/4a_ ref^5]e_ ref( cos 2σ _1∓ 1) = 0

for the 2:1 resonance. In the expressions of separatrix, the upper sign is for the case of a_ ref>a_c and the lower sign is for the case of a_ ref<a_c.

Dynamical separatrices of synchronous resonance with reference semimajor axes at a_ ref=3.0, a_ ref=3.5, a_ ref=3.6 a_ ref=4.0 and a_ ref=5.0 are compared with the associated Poincaré sections, as shown in Fig. <ref>. It should be noted that, at the section defined by M=0, it holds σ_1 = ϕ-ϖ. We can see that, under the pendulum approximations, the libration centre is located at 2(ϕ-ϖ) = π for the cases of a_ ref < 3.53 and it is located at 2 (ϕ-ϖ) = 0 for the cases of a_ ref>3.53. This is in consistent with our previous discussion.

In the case of a_ ref=3.0, the dynamical separatrix under the pendulum approximation can qualitatively describe the high-ϕ̇ libration islands but it cannot describe those low-ϕ̇ islands of libration. For the cases of a_ ref=3.5 and 3.6 (near the critical value), there are no islands of libration in the Poincaré section. Evidently, the dynamical separatrix under the pendulum approximation cannot describe the associated structures arising in the Poincaré section. In the case of a_ ref=4.0, there are four islands of libration. The separatrix under the pendulum approximation cannot describe the phase-space structures. In the case of a_ ref=5.0, the pendulum approximation can well describe the numerical structures. 

To conclude, we can see that the pendulum approximations can only describe numerical structures for those cases with reference semimajor axes far away from the critical value. For those cases with reference semimajor axes which are not far from the critical value, pendulum approximations fail to describe the numerical structures. The failure of pendulum approximations near the critical case is due to the strong coupling between rotation and translation under the considered problem. Thus, we may ask: can we formulate a general resonant Hamiltonian which is applicable for describing spin-orbit coupling in the entire range of reference semimajor axes? The purpose of Section <ref> is to address this problem.



§ ADIABATIC APPROXIMATION



In this section, we intend to formulate resonant Hamiltonian to describe resonant structures under the spin-orbit coupling problem by taking advantage of perturbative treatments developed by <cit.>.



 §.§ Resonant Hamiltonian model



When the primary body is locked inside a certain k_1:k_2 spin-orbit resonance, the associated argument σ_1 becomes a long-period variable and the other angle σ_2 is a short-period variable. With a given total angular momentum, (σ_1,Σ_1) corresponds to the slow DOF and (σ_2,Σ_2) corresponds to the fast DOF. As a result, the Hamiltonian (<ref>) determines a typical separable dynamical model <cit.>. During one period of the fast DOF, the variables associated with the slow DOF have negligible changes. According to the concept of perturbative treatment proposed by <cit.>, the variables associated with the slow DOF can be treated as parameters when considering the evolution of the fast DOF over its timescale. Such an approximation is usually known as adiabatic approximation, which has been widely applied in varieties of problems <cit.>. 

In practice, the variables (σ_1,Σ_1) are considered as parameters when considering the evolution of the fast DOF (σ_2,Σ_2). Under the assumption, the subsystem associated with the fast DOF becomes integrable and the solutions are level curves of the Hamiltonian in the (σ_2,Σ_2) space <cit.>. As a result, the action-angle variables can be introduced as follows <cit.>:

    σ _2^* = 2π/Tt,  Σ _2^* = 1/2π∫_0^2πΣ _2 dσ _2

where T is the period and the Arnold action Σ_2 stands for the signed area bounded by the solution curve during one period of the fast DOF. Performing the canonical transformation given by equation (<ref>), we can write the Hamiltonian (<ref>) as an integrable Hamiltonian form,

    H( G_ tot;σ _1,Σ _1,σ _2,Σ _2) ⇒ H( G_ tot;σ _1,Σ _1,Σ _2^*).

The angle σ_2^* is absent from the Hamiltonian H( G_ tot;σ _1,Σ _1,Σ _2^*), resulting in that the Arnold action Σ_2^* becomes a motion integral. For convenience, we denote the motion integral as the adiabatic invariant J = Σ_2^*.

During the long-term evolution, there are two conserved quantities including J and H besides the total angular momentum. Thus, the original 2-DOF Hamiltonian model is integrable. Naturally, global structures can be explored by analysing phase portraits. In practice, phase portraits are produced by plotting level curves of adiabatic invariant J with given Hamiltonian H and total angular momentum G_ tot <cit.>. 



 §.§ Analytical and numerical structures



According to the discussions made in Section <ref>, H and G_ tot can be characterised by reference semimajor axis a_ ref and reference eccentricity e_ ref. Please refer to equation (<ref>) for the total angular momentum and refer to equation (<ref>) for the Hamiltonian H in the case of synchronous resonance. For the 2:3 resonance, the Hamiltonian is characterised by

    H = .  H|_a = a_ ref, e = e_ ref, f = 0, ϕ-ϖ = π/2, ϕ̇= 1.5 n_ ref.

A pair of (a_ ref,e_ ref) determines the set of conserved quantities (G_ tot, H). It should be noted that the 2:1 resonance is not considered in practical simulations because, in the case of α_m = 10, the critical semimajor axis a_c,3 = 1.76 is smaller than the allowed minimum distance.

For the synchronous (1:1) resonance, phase portraits and Poincaré sections are produced and compared in Fig. <ref> for the cases of (a_ ref=3,e_ ref=0.1), (a_ ref=3.5,e_ ref=0.1), (a_ ref=4,e_ ref=0.1) and (a_ ref=5,e_ ref=0.1). Analytical and numerical results for the 2:3 resonance are shown in Fig. <ref> where the cases of (a_ ref=4.5,e_ ref=0.2), (a_ ref=5,e_ ref=0.2) and (a_ ref=5.5,e_ ref=0.2) are considered. In both figures, left-column panels represent phase portraits and the right-column panels stand for Poincaré sections. The colour index shown in phase portraits stands for the magnitude of the adiabatic invariant J and the colour index shown in Poincaré sections correspond to the orbital eccentricity. The dynamical separatrices corresponding to the level curves passing through saddle points are marked by red dots and they divide the entire phase space into regions of libration and circulation.

From Figs. <ref> and <ref>, we can observe an excellent correspondence between analytical structures arising in phase portraits and numerical structures arising in Poincaré sections. It validates the resonant Hamiltonian model for describing spin-orbit resonances for the primary body under strong coupling environments.



For the synchronous (1:1) resonance (see Fig. <ref>), dynamical structures are different when the reference semimajor axis is changed. In the case of a_ ref = 3, the high-ϕ̇ islands of libration are bounded by the separatrix stemming from those high-ϕ̇ saddle points and those low-ϕ̇ islands of libration are bounded by the separatrix stemming from those low-ϕ̇ saddle points. In the case of a_ ref = 3.5, there are no islands of libration and saddle points in both the phase portraits and Poincaré sections. In the case of a_ ref = 4, those high-ϕ̇ islands of libration are bounded by the separatrix stemming from those low-ϕ̇ saddle points and vice versa. In the case of a_ ref = 5, the islands of libration in the high-ϕ̇ branch disappear and only those islands centred at the nominal resonance remain.



For the 2:3 resonance (see Fig. <ref>), the dynamical structures are changed at different reference semimajor axes. In the case of a_ ref=4.5, the high-ϕ̇ branch and the low-ϕ̇ branch are distinctly separable. The high-ϕ̇ islands are bounded by the separatrix stemming from high-ϕ̇ saddle points and vice versa. It is different for the case of a_ ref=5. The high-ϕ̇ islands are bounded by the separatrix stemming from low-ϕ̇ saddle points and those low-ϕ̇ islands are bounded by the separatrix stemming from high-ϕ̇ saddle points. In the case of a_ ref=5.5, the islands of libration centred at ϕ-ϖ = ±π/2 disappear. However, the islands centred at ϕ-ϖ = 0 become very small and additional islands centred at ϕ-ϖ = 0 bifurcate in the high-ϕ̇ space.



In fact, it is still possible to discuss phase-space structures based on the resonant Hamiltonian by Eq. (<ref>) for the synchronous 1:1 spin-orbit resonance. In Fig. <ref>, we produce dynamical structures from the resonant Hamiltonian and from adiabatic approximation and compare analytical structures to the corresponding Poincaré sections. As desired, dynamical structures produced from adiabatic approximation have better agreement with numerical structures. 



 §.§ Distribution of spin equilibrium



By analysing phase portraits produced from adiabatic approximation, we can further determine the location of libration centres and saddle points, as shown in Fig. <ref>. Two cases of reference eccentricities at e_ ref = 0.1 and e_ ref = 0.2 are considered for the synchronous (1:1) resonance and 2:3 resonances. The resonant centres are shown in blue dots and saddle points are marked in red dots. The vertical dashed lines stand for the cases considered in Figs. <ref> and <ref>. According to the discussion made in previous section, the critical semimajor axis is a_c,1 = 3.53 for the synchronous (1:1) resonance and a_c,2 = 5.31 for the 2:3 resonance. 

For the synchronous resonance (see the top panels of Fig. <ref>), the characteristic curves show that there is a gap of a_ ref in the vicinity of a_c,1. In the space with a_ ref < a_c,1, there are two branches of libration centres: one is close to the nominal synchronous resonance with argument at 2σ_1 = π and the other branch with argument at 2σ_1 = 0 occupies a lower-ϕ̇ space. In the space with a_ ref > a_c,1, there are also two branches of libration centres: one is close to the nominal synchronous resonance with argument at 2σ_1 = 0 and the other branch with argument at 2σ_1 = π occupies a higher-ϕ̇ space. When the reference semimajor axis a_ ref is higher (or lower) than a certain value, the high-ϕ̇ (or low-ϕ̇) branch will disappear. 

For the 2:3 resonance (see the bottom panels of Fig. <ref>), there is no gap for the characteristic curves in the case of e_ ref=0.1 and there is a gap in the vicinity of a_c,2 for the case of e_ ref=0.2. Similar to the case of synchronous resonance, the high-ϕ̇ (or low-ϕ̇) branch may disappear if the reference semimajor axis a_ ref is higher (or smaller) than a certain value.





§ CONCLUSIONS



In this work, spin-orbit resonances of the primary body are studied in a binary asteroid system, composed of a triaxial ellipsoidal primary and a spherical secondary. The Hamiltonian is expanded up to a high order in eccentricity based on elliptic expansions. The resulting Hamiltonian determines a 2-DOF dynamical model, depending on the total angular momentum. We introduce two parameters including the reference semimajor axis a_ ref and the reference eccentricity e_ ref to characteristic the total angular momentum G_ tot and the Hamiltonian H.

Numerical simulations show that (a) the dynamical structures of spin-orbit resonances are distinctly different from the classical structures under the spin-orbit problem for the secondary body and (b) the main structures may significantly change at different reference semimajor axes.

According to pendulum approximations, there are three critical semimajor axes (denoted by a_c,1 for the synchronous resonance, a_c,2 for the 2:3 resonance and a_c,3 for the 2:1 resonance), below and above which the dynamical structures of spin-orbit resonance are totally different. However, it is observed that pendulum approximations of Hamiltonian fail to describe dynamical structures of spin-orbit resonances when the semimajor axis is not far from the critical value.

To address this problem, adiabatic approximation is used to formulate a `general' resonant Hamiltonian for describing spin-orbit coupling. In particular, an adiabatic invariant denoted by J is introduced to describe the resonant dynamics. The conserved quantities including the adiabatic invariant J and the Hamiltonian H make the original 2-DOF dynamical model be integrable. As a result, the global structures in phase space can be explored by analysing phase portraits. Results show that there is an excellent agreement between analytical structures arising in phase portraits and numerical structures arising in Poincaré sections. By analysing phase portraits, it is possible to produce characteristic curves of resonant centre and saddle points, which can be used to predict the variations of dynamical structures when the reference semimajor axis is changed.

Regarding the spin-orbit coupling problem, <cit.> arrived at an intriguing conclusion that the centre of spin-orbit resonance may change when the mass ratio and mutual distance are varied to alternate the sign of the index S (S=S_1 for the synchronous resonance, S=S_2 for the 3:2 resonance and S=S_3 for the 1:2 resonance). In <cit.> the author pointed out the change of libration centre is physically impossible through an alternation of the orbital semimajor axis (please see point (4) in their introduction part). Through our analysis, we observe that, for the secondary body in a binary asteroid system, it is indeed physically impossible to change the libration centre through an alternation of semimajor axis. This is because, for the secondary, the sign of S_1,2,3 remains unchanged at arbitrarily allowed semimajor axes. However, for the primary body, it becomes possible to alternate the resonance centre by changing the semimajor axis when the primary-to-secondary mass ratio is greater than unity.











Acknowledgements
The author wishes to thank Prof. Xiyun Hou for helpful dicussions on numerical structures arising in Poincaré sections, Dr. Jafari-Nadoushan for confirming the expression of mutual gravitational potential for spin-orbit coupling problem and Dr. Shunjing Zhao for his help in deriving high-order spherical harmonic coefficients showin in Appendix. This work is financially supported by the National Natural Science Foundation of China (Nos. 12073011 and 12233003) and the National Key R&D Program of China (No. 2019YFA0706601).




§ DECLARATIONS

The authors declare that they have no Conflict of interest.





§ HIGH-ORDER DYNAMICAL MODEL


In the spin-orbit coupling problem, the primary is of ellipsoid. Truncating the mutual potential at the 8th order in (a_p/r), we can write the Hamiltonian governing the evolution of rotation and translation in the following form:

    H =   1/2mp_r^2 -  Gm_pm_s/r + ( 1/2mr^2 + 1/2I_3)p_ψ ^2 + G^2_ tot/2I_3 + G_ totp_ψ/I_3
       -  Gm_pm_s[  - a_p^2C_20/2r^3 + 3a_p^4C_40/8r^5 - 5a_p^6C_60/16r^7 + 35a_p^8C_80/128r^9.
       + ( 3a_p^2C_22/r^3 - 15a_p^4C_42/2r^5 + 105a_p^6C_62/8r^7 - 315a_p^8C_82/16r^9)cos 2ψ
       + ( 105a_p^4C_44/r^5 - 945a_p^6C_64/2r^7 + 10395a_p^8C_84/8r^9)cos 4ψ
       + ( 10295a_p^6C_66/r^7 - 135135a_p^8C_86/2r^9)cos 6ψ
       .  + 2027025a_p^8C_88/r^9cos 8ψ]

where the second- and fourth-order spherical harmonic coefficients are given in Sect. <ref>. Here we only provide the sixth- and eighth-order coefficients as functions of C_20 and C_22 in the following form <cit.>:

    C_60 =   125/7( 1/3C_20^2 + 2C_22^2)C_20
    C_62 =   25/21( C_20^2 + C_22^2)C_22
    C_64 =   25/252C_20C_22^2,  C_66 = 25/1512C_22^3

and 

    C_80   = 625/11[ 1/3C_20^4 + 2C_22^2( 2C_20^2 + C_22^2)]
    C_82   = 625/77( 1/3C_20^2 + C_22^2)C_20C_22
    C_84   = 125/462( 1/2C_20^2 + 1/3C_22^2)C_22^2
    C_86   = 125/16632C_20C_22^3,  C_88 = 125/133056C_22^4

Please refer to Table <ref> for the spherical harmonic coefficients for the ellipsoid-shaped primary with semimajor axes a_p = 1.0, b_p=0.95 and c_p = 0.85. 



Replacing the Hamiltonian (<ref>) in Eq. (<ref>) leads to the equations of motion for the spin-orbit coupling problem. Then, we integrate the equations of motion with the same initial conditions as the ones adopted by Fig. <ref>. Numerical trajectories propagated under the dynamical models truncated at different orders in a_p/r are compared in Fig. <ref>. It shows that the deviation between 4th-order and 8th-order models is of 1.0 × 10^-4, meaning that it is accurate enough to truncate the Hamiltonian at the fourth order in a_p/r.


































 Paper Title: Candidate Distant Trans-Neptunian Objects Detected by the New Horizons Subaru TNO Survey.





0000-0001-6680-6558]Wesley C. Fraser
National Research Council of Canada, Herzberg Astronomy and Astrophysics Research Centre, 5071 W. Saanich Rd. Victoria, BC, V9E 2E7, Canada
Department of Physics and Astronomy, University of Victoria, Elliott Building, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada

0000-0003-0333-6055]Simon B. Porter
Southwest Research Institute, 1301 Walnut St., Suite 400, Boulder, CO 80302, USA

0000-0002-9179-8323]Lowell Peltier
National Research Council of Canada, Herzberg Astronomy and Astrophysics Research Centre, 5071 W. Saanich Rd. Victoria, BC, V9E 2E7, Canada
Department of Physics and Astronomy, University of Victoria, Elliott Building, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada
Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada

0000-0001-7032-5255]JJ Kavelaars
National Research Council of Canada, Herzberg Astronomy and Astrophysics Research Centre, 5071 W. Saanich Rd. Victoria, BC, V9E 2E7, Canada
Department of Physics and Astronomy, University of Victoria, Elliott Building, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada

0000-0002-3323-9304]Anne J. Verbiscer
Department of Astronomy, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904-4325, USA
Southwest Research Institute, 1301 Walnut St., Suite 400, Boulder, CO 80302, USA

0000-0003-0854-745X]Marc W. Buie
Southwest Research Institute, 1301 Walnut St., Suite 400, Boulder, CO 80302, USA

0000-0001-5018-7537]S. Alan Stern
Southwest Research Institute, 1301 Walnut St., Suite 400, Boulder, CO 80302, USA


Southwest Research Institute, 1301 Walnut St., Suite 400, Boulder, CO 80302, USA

0000-0001-8821-5927]Susan D. Benecchi
Planetary Science Institute, 1700 East Fort Lowell, Suite 106, Tucson, AZ 85719, USA

0000-0003-4143-4246]Tsuyoshi Terai
Subaru Telescope, National Astronomical Observatory of Japan 650 North Aòhoku Place, Hilo, HI 96720, USA

0000-0002-0549-9002]Takashi Ito
Center for Computational Astrophysics, National Astronomical Observatory of Japan, Osawa 2-21-1, Mitaka, Tokyo, 181-8588, Japan

0000-0002-3286-911X]Fumi Yoshida
University of Occupational and Environmental Health, 1-1 Iseigaoka, Yahata, Kitakyushu 807-8555, Japan
Planetary Exploration Research Center, Chiba Institute of Technology, 2-17-1 Tsudanuma, Narashino, Chiba 275-0016, Japan

0000-0001-6942-2736]David W. Gerdes
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA

0000-0003-4827-5049]Kevin J. Napier
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

0000-0001-7737-6784]Hsing Wen Lin
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

0000-0001-8221-8406]Stephen D. J. Gwyn
National Research Council of Canada, Herzberg Astronomy and Astrophysics Research Centre, 5071 W. Saanich Rd. Victoria, BC, V9E 2E7, Canada


0000-0002-7895-4344]Hayden Smotherman
DIRAC Institute, Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195, USA

0000-0003-2239-7988]Sebastien Fabbro
National Research Council of Canada, Herzberg Astronomy and Astrophysics Research Centre, 5071 W. Saanich Rd. Victoria, BC, V9E 2E7, Canada
Department of Physics and Astronomy, University of Victoria, Elliott Building, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada


0000-0003-3045-8445]Kelsi N. Singer
Southwest Research Institute, 1301 Walnut St., Suite 400, Boulder, CO 80302, USA

0000-0003-0193-9151]Amanda M. Alexander
Southwest Research Institute, 1301 Walnut St., Suite 400, Boulder, CO 80302, USA

0000-0003-1260-9502]Ko Arimatsu
National Astronomical Observatory of Japan, Tokyo, Japan

0000-0002-8236-7396]Maria E. Banks
NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA

0000-0002-7277-3980]Veronica J. Bray
Lunar and Planetary Laboratory, University of Arizona, 1541 University Blvd, Tucson, AZ 85721, USA

0000-0002-8262-0320]Mohamed Ramy El-Maarry
Department of Earth Sciences, and Planetary Science Research Group, Khalifa University, Abu Dhabi, UAE

 0000-0002-2530-0427]Chelsea L. Ferrell
Independent


National Astronomical Observatory of Japan, Tokyo, Japan

0000-0002-7851-3186]Florian Glass
Independent

0000-0003-0437-3296 ]Timothy R. Holt
Centre for Astrophysics, University of Southern Queensland,  West St, Darling Heights QLD 4350, Australia


Planetary Exploration Research Center, Chiba Institute of Technology, 2-17-1 Tsudanuma, Narashino, Chiba 275-0016, Japan

0000-0001-7918-118X]Ryo Ishimaru
Planetary Exploration Research Center, Chiba Institute of Technology, 2-17-1 Tsudanuma, Narashino, Chiba 275-0016, Japan

0000-0001-6255-8526]Perianne E. Johnson
Institute for Geophysics, Jackson School of Geosciences, University of Texas at Austin,10100 Burnet Road (R2200), Austin, Texas, USA

0000-0003-3234-7247]Tod R. Lauer
NOIRLab, P.O. Box 26732, Tucson, Arizona USA, 85726

0000-0002-6477-1360]Rodrigo Leiva
Instituto de Astrofísica de Andalucía, Consejo Superior de Investigaciones Científicas, Glorieta de la Astronomía s/n, 18008 Granada, Spain

0000-0003-0926-2448]Patryk S. Lykawka
Kindai University, Shinkamikosaka 228-3, Higashiosaka, Osaka, 577-0813, Japan

0000-0002-0362-0403]Raphael Marschall
CNRS, Observatoire de la Cõte d'Azur, Laboratoire J.-L. Lagrange, CS 34229, 06304 Nice Cedex 4, France

0000-0003-0930-6674]Jorge I. Núñez
Johns Hopkins University Applied Physics Laboratory 11100 Johns Hopkins Road, Laurel, MD 20723-6099, USA

0000-0002-9365-7989]Marc Postman
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA

0000-0003-2768-0694]Eric Quirico
Université Grenoble Alpes, CNRS, Institut de Planetologie et Astrophysique de Grenoble (IPAG), UMR 5274, Grenoble, F-38041, France

0000-0003-2805-4994]Alyssa R. Rhoden
Southwest Research Institute, 1301 Walnut St., Suite 400, Boulder, CO 80302, USA

0000-0001-8994-032X]Anna M. Simpson
Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Department of Physics University of Michigan Ann Arbor, MI 48109, USA

0000-0003-1316-5667]Paul Schenk
Lunar and Planetary Institute (USRA), Houston, TX 77058, USA

0000-0001-8671-5901]Michael F. Skrutskie
Department of Astronomy, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904-4325, USA

0000-0002-5358-392X]Andrew J. Steffl
Southwest Research Institute, 1301 Walnut St., Suite 400, Boulder, CO 80302, USA

0000-0002-1988-223X]Henry Throop
Independent





Wesley C. Fraser
wesley.fraser@nrc-cnrc.gc.ca


§ ABSTRACT 
We report the detection of 239 trans-Neptunian Objects discovered through the on-going  survey for distant minor bodies being performed with the Hyper Suprime-Cam mosaic imager on the Subaru Telescope. These objects were discovered in images acquired with either the r2 or the recently commissioned EB-gri filter using shift and stack routines. Due to the extremely high stellar density of the search region down stream of the spacecraft, new machine learning techniques had to be developed to manage the extremely high false positive rate of bogus candidates produced from the shift and stack routines. We report  discoveries as faint as r2∼26.5.  We highlight an overabundance of objects found at heliocentric distances R≳70 au compared to expectations from modelling of the known outer Solar System. If confirmed, these objects betray the presence of a heretofore unrecognized abundance of distant objects that can help explain a number of other observations that otherwise remain at odds with the known Kuiper Belt, including detections of serendipitous stellar occultations, and recent results from the Student Dust Counter on-board the  spacecraft. 







§ INTRODUCTION 


NASA's  spacecraft launched in January 2006 and carried out a successful fly-through of the Pluto system in July 2015 <cit.> and continues to be fully operational. 
In 2004, two years before launch, the  team began a ground-based observational search to identify a Kuiper Belt Object (KBO) that the spacecraft could fly by at close range after the Pluto encounter.  Due to Pluto's location, the trajectory of  sent it toward the apparent position of Sagittarius, near the galactic center. Any possible encounter target would be located along or near the trajectory of  beyond Pluto. In 2009, any potentially encounterable KBOs had sky locations that projected to 0 degrees Galactic latitude and 12 degrees longitude.  The dense stellar-background created by the alignment of the projected sky location of the spacecraft trajectory with the galactic centre makes searches very difficult owing to contamination of the KBOs as they inevitably traverse background stars.  Likewise, most trans-Neptunian Objects (TNOs) are small (most have diameters <200 km), distant (>30 au) and faint (m_R<25.0 mag R-band), and therefore require large telescopes to discover them. Searches therefore used the largest telescopes available paired with instruments having large fields of view. 
    
The high stellar background densities thwarted the initial observation campaigns conducted in 2004-2005 from Subaru using its Suprime-Cam imager <cit.>. 
By 2011 the target field had moved to a slightly less dense background, and the apparent locations of TNOs downstream of the spacecraft occupied a smaller area on the sky <cit.>. The team undertook a new search, employing a combination of facilities, including Subaru and its Suprime-Cam imager <cit.>, the Canada-France-Hawaii Telescope MegaPrime imager <cit.>, and Megacam <cit.> and IMACS <cit.> on the Magellan telescopes. 
In 2014, the Hubble Space Telescope (HST) conducted a highly targeted search, resulting in the discovery of (486958) Arrokoth (2014 MU69), and enabling the first flyby of a cold classical Kuiper Belt Object in January 2019 <cit.>. The results of these early searches are reported in <cit.>. 
    

After the successful Arrokoth flyby,  remains healthy and continues on a trajectory through the trans-Neptunian region, maintaining the capability for the spacecraft to observe additional TNOs, with current heliocentric distance of R∼59 au. 
Beginning in 2018, a renewed search began in earnest to identify targets, not only for close encounter flyby but also for observation by the spacecraft at distances, resolutions and phase angles not possible from Earth-based observatories. The first survey was with the Dark Energy Camera on the Blanco Telescope <cit.>. 
The 8.2-m Subaru Telescope with its 1.77 deg^2 Hyper Suprime-Cam (HSC) <cit.> field of view is an excellent facility for carrying out such a search and so, in 2020 
we began using that camera. At first we observed with the facility r2 filter, and in fall 2022, commissioned an extra wide “EB-gri" facility filter, which allows the search to detect objects ∼0.5 magnitudes fainter than with the r2 filter. Depending on the night, our survey reached 26^th-27^th magnitude, even at the high stellar densities of the near-galactic plane search fields. 
    
The following sections describe the survey strategy, the Subaru-HSC instrument as applied to this program,  the data reductions,  and the novel search techniques developed using machine learning tools to optimize object discovery. We finish by discussing the detections found beyond the nominal “edge” of the Kuiper Belt (R∼50 au) thus far and the implications they have for the  mission as well as for Kuiper Belt science in general. 
 


§ THE CAMERA AND OBSERVING STRATEGY  

In this section, we describe the observing program, including a description of the camera we heavily utilized for the search (<ref>), and the observing strategy used to provide both discovery and tracking observations (<ref>). 



 §.§ Subaru-HSC and the EB-gri filter



The Subaru-HSC is a wide-field camera whose focal plane is populated with a mosaic of 116 CCDs, 103 of which are available for science.

The camera has a pixel size of 15 μm and a pixel scale of 0.168. The ∼ 1.5^∘ field of view camera is designed to conduct large-scale surveys of the sky with significant depth and area to high photometric and astrometric precision.

The delivered image quality is excellent across the field of view, with a median seeing in the i-band of about 0.6” <cit.>.


For technical details of HSC, see
<cit.> for system design,
<cit.> for camera dewar design,
<cit.> for filter system,
and <cit.> for  on-site quality-assurance system.[
the dedicated Subaru-HSC web-page is available at <https://www.subarutelescope.org/Observing/Instruments/HSC/>.]



As part of this project, our group commissioned a new broad-band EB-gri filter that roughly spans the g, r2, and i2 filters provided by Subaru (see Figure <ref>). This is similar to the “w-band” (GRI.MP9605) filter in use on CFHT-Megacam.[<https://www.cfht.hawaii.edu/Instruments/Imaging/MegaPrime/specsinformation.html>] From photon noise considerations, we anticipated a ∼0.5 mag increase in equivalent r survey depth compared to use of the r2 filter. We provide estimates of r- and g-band color terms in the appendix. 





 §.§ The Observing Strategy 





The primary objective of these observing campaigns is to discover and track TNOs whose space locations will be accessible by   enabling a close fly-by.   
TNOs that were too far away to allow an encounter could be observed at larger distances, some at resolutions that exceed what might be achievable from Earth-localized observers. Those observations are useful for distant high phase angle and satellite searches.
We utilize TNO orbital and size distribution models to determine the optimal sky search location based on computed encounter trajectories between model objects and the spacecraft trajectory. 
The current sky locations of model objects with which  could have a close encounter, allowing for small course corrections (Δ-V ∼ 100s m/s), determined the optimal search area.
The TNOs discovered in these searches were not encounterable, but many were close enough to be observable using the  LORRI camera <cit.>.

The orbit model substantially influences the outcome of these searches. 
For planning the initial searches for Arrokoth <cit.>, the bulk orbital properties of the known classical KBOs were used; these searches did not result in the discovery of an encounter target but did discover numerous TNOs that could be observed remotely.
The HST-based search in 2014 <cit.> transitioned to using debiased orbit distribution models from the Canada-France Ecliptic Plane Survey <cit.>. 
The discovery of Arrokoth, a few arc minutes from the predicted optimal search location, confirmed the veracity of the model-based search field.

In 2020, after  successfully encountered Arrokoth in 2019 <cit.>, we began a new campaign using the Subaru-HSC to search for new targets observable with the  LORRI.
The exact telescope pointings were determined using a debiased population model derived from the orbital and size distributions determined by the Outer Solar System Origins Survey <cit.>.
The transition to the OSSOS++ model was triggered by the evolving understanding of the shape of the size-distribution of TNOs, which were strongly informed by the cratering records on Pluto and Arrokoth <cit.> and the OSSOS measured population distributions in distant Neptune mean-motion resonances <cit.> and includes the latest measurements of the absolute magnitude distributions from OSSOS <cit.>. 
The wide latitude and faint magnitude distributions of potential targets, caused by the broader inclination distribution of these distant objects, necessitated a wide field search for faint TNOs, making Subaru-HSC an excellent search platform for the task. 
With the sample of observable model objects determined, we computed ephemerides and determined the optimal sky locations for each night of allocated observing. 
These optimal sky locations shifted each night to ensure maximal recovery of the sample of observable TNOs, as predicted by our model inputs. See Figure <ref> for examples of the observing layouts used in 2020 through 2023. Each night followed essentially the same observing strategy: acquiring as many consecutive 90 s exposures as possible within the allotted time. The sequence was broken only for refocusing exposures or loss of tracking.
The post-Arrokoth searches provided a sample of 239 TNOs (submitted to the Minor Planet Center) whose observed orbital distributions can be used to test the completeness of the model used in planning those surveys. 




§ REDUCTIONS AND SEARCH TECHNIQUES


This section describes the pipeline steps required to go from raw pixels to detected TNOs. 
These steps include image preprocessing and calibrations (<ref>), the shift and stack detection techniques (<ref>), culling of bad candidate sources using machine learning techniques and human confirmation of good sources (<ref>), and finally intra-night source linking to verify detections (<ref>). 



 §.§ Image Preprocessing  


Raw imagery was preprocessed and calibrated using the LSST pipeline v19[As the results were satisfactory for our needs and to maintain consistency, we chose to stick with LSST pipeline v19 throughout the project rather than update to newer releases from year to year.] <cit.>. For each night, an average bias frame was constructed from bias frames acquired during the twilight of that night, using the  task. An average flat field for the night was generated using the archival dome flats that were temporally closest to the night in question (usually acquired in the previous dark run), using the 
 task. Using the average flats and biases constructed for each night, the  package was used to debias, flat-field, and perform photometric and astrometric calibrations. Calibrations were performed against the merged PS1-GAIA catalogue,[<https://community.lsst.org/t/pan-starrs-reference-catalog-in-lsst-format/1572>] resulting in ∼0.02 mag RMS photometric calibration quality and ∼0.04 RMS astrometric calibration quality. Similar astrometric quality was found with the EB-gri filter, though with a few tenths of a magnitude worse absolute photometric calibration. Roughly ∼2% of cases were catastrophically compromised through causes such as guider loss or mid-exposure focus changes, though with the low occurrence rate; this held no significant bearing on our survey depth.

To enable the characterization of the detection efficiency as a function of brightness and orbital parameters, a significant number of artificial moving bodies were implanted in the data. Objects were randomized across a range of semi-major axes, eccentricities, and inclinations, and with randomized absolute magnitudes to result in brightnesses 20≤ m ≤ 28 in the observed filter. Between 20 and 40 sources were generated per chip. Point spread function (PSF) characterization was done with the TRailed Imagery in Python (TRIPPy) package <cit.>. All non-saturated sources were examined for low ellipticity and no neighboring sources within 8 FWHM, and those with <0.5 mag difference between their Kron <cit.> and circular aperture magnitudes were used to generate the PSFs. The PSF lookup tables were then cleaned of any residual signal by replacing pixels that had variations more than 2σ away from the mean and outside 2 FWHM radius, with a random value drawn from the distribution of residual-free pixels. This later step was necessitated by the highly crowded fields but resulted in PSFs of sufficient fidelity for our needs. 

Source injection was also done with TRIPPy. Each source was injected with trailing matching its simulated orbital trajectory. Poissonian noise was added.

The LSST pipeline was used to produce difference images of the implanted imagery. The , , and  routines were used to generate the subtraction template, itself generated from the 30% of images from a night with the best image quality (IQ). The frames used in the template generation should span the full temporal range of the nightly sequence. This generally resulted in excellent removal of bright stationary sources without catastrophic removal of the flux from moving sources. The subtraction was performed with . The full template and subtraction routine was done on a nightly basis, with each CCD treated individually. The latter choice was made due to some bugs discovered during the image subtraction process that manifests only when multiple CCDs were considered together during the template generation and subtraction phases. 

For the 2020 and some of the 2021 of the implanted and subtracted imagery, pixels with large subtraction residuals (variance >25× the variance of source-free regions) were masked in order to reduce the number of spurious detections and reduce the visual inspection load.  Starting part way through 2021, however, this masking step was removed as it was found that the ResNet (see Section <ref>) we made use of during the vetting step was able to remove the vast majority of bogus sources resulting from these poorly subtracted regions. 



 §.§ Shift and Stack  

Subtracted data were searched for moving point sources using a shift-and-stack technique, whereby the nightly frame sequences are shifted at linear rates of motion to counteract the movement of the bodies of interest and then digitally stacked. Near a motion vector that matches the apparent movement of a source, said source appears as a near-point source in the stack, allowing simple detection and achieving depths comparable to the limit of the sidereal stack. 

The first search effort of data observed in 2020 consisted of classical CPU stacking methods <cit.>, and vetting of many thousands of candidate sources with a large team of volunteer vetters.[for whom we are forever grateful] This enabled a quick search but with poor performance and extreme human and computational cost. It was quickly recognized that such a method was untenable, and so we turned to more modern stacking and vetting techniques. The techniques implemented evolved from run to run to adjust for lessons learned as we went.

The Kernel-Based Moving Object Detection <cit.> software package was used to perform the shift-and-stack search for the 2021 and later data. This software uses graphics processing units to perform the computational heavy lifting, reducing the stack time from days per chip (even on a supercomputer) to hours. For near opposition fields, a grid of 35 rates uniformly spanning 0.5<v<4.0/hr – corresponding to heliocentric distances 30≲ R ≲ 300 au – and 15 angles uniformly spanning -20 ≤ a ≤ 20^∘ from the ecliptic to be sensitive to objects on prograde orbits. The grid spacing was chosen to ensure that an object's flux was spread over a region no wider than 1.4, or roughly twice the typical seeing over Maunakea. It is possible that the use of a finer grid would result in a fainter search depth, but comes at the cost of increased false positive rate and compute time, and so this spacing was chosen as a practical balance between these considerations.

The KBMOD likelihood threshold was set to 5, an ambitious level at which the majority of candidates are noise sources. This threshold resulted in 3,000-5,000 candidate sources per CCD.

For each candidate source, mean and median stacks were produced at the rate and angle that maximized the likelihood as reported by KBMOD. The stacking routine ignored all pixels flagged by the LSST pipeline as , , , , , which are pixels marked as on an edge, containing no real data, saturated, interpolated over, or rejected by the pipeline for other reasons. These stacks acted as the main products for training and execution of the machine learning tools, which we discuss in the next section.

For the October 2022 data, which were observed a few days before or after the stationary point at R∼45 au, the grid of rates and angles was adjusted to 0.5<v<3.0/hr and -60 ≤ a ≤ 60^∘ to be sensitive to the objects of interest. The sign of the rates grid was adjusted accordingly for observations that occurred after the stationary point. 



 §.§ Vetting Through Machine Learning


After the initial quick search of the 2020 data using a team of humans, we implemented machine-learning techniques to provide an initial cull of bad sources from the candidate list, thereby saving enormous amounts of human cost in the final vetting.

The process of vetting is essentially one of binary classification, where a source is labeled as good and kept, or bad and rejected. As such, vetting of shift-stacks represents a task ideally suited for convolutional neural networks. As the search commenced, many different neural networks were explored. Our experimenting included basic convolutional neural networks (CNNs) of varying complexity and depth. 
The 2020 data (nights 03071, 03072, 03093, 03147, 03148; see Table <ref>) were searched using
basic CNNs. An ensemble of three identical CNNs that was trained (we discuss the training routine below) using all implanted artificial sources was first used, but with sub-par performance of the network within ∼1 magnitude of the nominal detection limit of r_2∼27 mag. As such, a second “deep” ensemble of 3 CNN networks was trained using only implants with r_2>25.5. This deep network resulted in significantly increased performance at the faint end of the implants. 

We further explored other networks, including ensembles of the above two networks, and networks trained with implanted sources divided in 1.5 mag-wide bins. Only minimal performance gains were found, however, not warranting a re-search of the data. 

A remarkable performance boost was found when we implemented Residual Networks <cit.>. Our adopted ResNet (see Fig <ref>), which was trained on all detected implants regardless of brightness, met or outperformed all previous networks as measured by depth and peak detection efficiency, and came with significantly reduced numbers of trainable parameters compared to all other networks we tried. This is important for overfitting considerations, which we discuss further below.



We adopted an ensemble of networks <cit.> of identical structure and trained three master networks independently. Each master network itself is an ensemble of separate singular networks which we call branches. Each branch of a master is trained on the same data, but initialized separately. The final ensemble prediction probability is an average of prediction probabilities from each master network, each itself outputting an ensemble of probabilities from its singular networks.

To train each network (all CNNs and ResNets) we utilized supervised learning. Determining a successful training procedure was an extremely difficult task. The successful procedure we found was as follows. A candidate KBMOD source detection was labelled good if it was within 3 pixels of a known implanted source location with rate of motion within 0.2/hr of the implant's rate of motion. All other sources were labelled as bad. The full candidate sample was vastly dominated by approximately 200:1 junk candidates to good detections, which is much too unbalanced for training. As such, a random subset consisting of all good detections, and a subset of the junk detections of size 1.6× the good detections was generated. Any more unbalancing of the sample towards junk detections would result in failure to train. We utilized a sample weighted loss, with sample weight of 0.7 for junk candidates and 1 for good sources, as these values resulted in excellent network outputs. Training proceeded through usual means: the ADAM[ADAM is not an acronym; it is a name given to the optimization routine.] minimizer was used for 200 epochs <cit.>, with a batch size of 256 (any smaller resulted in failed training), and a categorical entropy loss function given by log L = 1/N∑_i=1^N (1-y_i) log(1-p_i) - y_i log(p_i) where p_i is the probability that sample i is good, and y_i is the ground truth for sample i, equalling 1 if the sample is good, 0 if bad. 

Experiments with this training demonstrated small variations in performance from training run to training run even with the same training set, and so from a given training set, an ensemble of three branch networks was trained, but each with a different random initialization resulting in slightly different classification probabilites from each branch. These three commonly trained branch networks make up one master network (see Figure <ref>). 

To ensure the ability to reject the broad range of junk sources that appear in the data set, multiple unique randomized training sets were generated as described above, with the same set of good sources, but a newly randomized set of junk sources. For each unique training set, a master network was trained. After training ∼10 master networks on different randomized training sets, the best performing three were accepted for production. 

The training data consisted of the KBMOD outputs from all chips of a field, from nights 03072 and 03093 (see Table <ref>). After generation of the random set, but before training, shift and rotation augmentations were implemented, whereby a given source was duplicated, but with 3 separate cardinal rotations, and as well, shifted by 1 pixel left, right, up, and down producing a training sample of size 419,605 samples. This augmentation resulted in a factor of ∼2 reduction in mislabels of junk sources. 

To ensure a satisfactory search depth, it was required to select a unique acceptance threshold for each night separately. This value (usually between 0.05 and 0.6) was manually tuned to select the point where a reduction in threshold did not result in an appreciable increase in depth, but did result in a significant increase in number of sources requiring human vetting. For a few nights, the performance of the network was less than satisfactory. These circumstances seemed to result from highly variable seeing conditions, or poor subtractions, making the stacks quantitatively different than those used for the training. One solution would be to train a network with a more diverse range of observational conditions. We reserve this exploration for future work.

The performance of the network ensemble was admirable. The exact performance varied from field to field, but typically the use of the ensemble ResNet resulted in a ∼ 500 - 1000× decrease in bogus sources in the candidate source list, with a loss of ∼0.2 mag in total depth compared to the raw KBMOD outputs. Without such a decrease in rate of bogus sources, the final human vetting task would be practically impossible.

The sources labelled as good were passed to a human for final visual vetting. For each source, the mean and median stacks were displayed side-by-side and labelled with the KBMOD likelihood and flux estimates. The junk:good purity at this stage was usually better than 3:1, and a field could be vetted by a trained operator in less than a day. 

The list of human vetted sources was typically of order ∼ 200 sources long. Above likelihood ∼7 all sources labelled good by a human were obviously real. The final human-vetted list contained some bright false positives caused primarily by subtraction residuals of bright stars. The list however, was  dominated with low flux sources with likelihoods <7, many of which were not real. These sources were predominately driven by the extremely high stellar density and consequent high density of subtraction residuals. In other words, nightly false positive rates were high because of our near–galactic-plane observing. Fortunately, a final false positive rejection occurred during the linking process, which we describe below. 

One major concern we had with the use of machine learning was overfitting. Overfitting can come in multiple flavours. One particularly damaging form is one in which the network is able to identify injected sources better than real sources, which leads to an incorrect measure of detection efficiency. Detection of overfitting in neural network training is notoriously difficult and is an ongoing field of study. We are confident that overfitting is not an issue from two findings in our search. Namely, real objects are discovered across different nights of the search with a frequency that appears consistent with expectations from the nightly detection efficiencies. As well, we found that the searches of the majority of nights not used in the training perform equally as well as those used during the training by measures such as peak detection efficiency, limiting magnitude, number of real detections, and number of junk detections requiring human rejection. That is to say, with few exceptions, the ResNet seemed to perform equally well across many nights of data, not just those for which it was trained. This implies that no sensitivity to peculiarities of those nights – which may include processing or PSF oddities, for example – were not what the machine latched onto, but rather, properties of the HSC data in general. We also point out that during training, the number of observations (unique pixels) in the training set was more than 50× larger than the number of trainable parameters, giving confidence that overfitting was extremely unlikely. It seems that if  overfitting did occur, its influence on the results must be minor. The similarity in performance of the network between nights used for training and those other nights which served as a fully independent validation set supports our conclusion that overfitting is not significant. The final detection efficiencies after vetting shift-stack sources with a neural network, and then human vetting are shown in Table <ref>.

We highlight that the above technique is a more advanced version of the neural network implemented to vet KBMOD sources presented by <cit.>. They utilized the ResNet50 <cit.> to cull sources from some DECam shift-stack data, but used less sophisticated training techniques and Gaussian PSFs for implants. The simpler model we present here outperforms this prior attempt and avoids overfitting risks incurred with the huge increase in trainable parameters in the ResNet50 model.



llllccccccc

HSC Observing Epochs.

        
        Observation    
        Pipeline    
        R.A.    
        Dec.    
        Number    
        Sequence    
        
        A     
        c    
        M_o    
        σ    
        Network

        Time (MJD)    
        Night    
        (^∘)    
        (^∘)    
        Exposures    
        Duration (hrs)    
        
             
            
            
            
        Type




58995.4     03068     287.37585     -20.22724      81     2.7     -     -     -     -     - 
 
58997.5     03070     288.74057     -20.57457      76     2.7     -     -     -     -     - 
 
58998.5     03071     287.33743     -20.12725     123     4.1     0.6757      0.0070      26.6652      0.2205     c 
 
58999.5     03072     288.71466     -20.37960     127     4.3     0.7570      0.0056      26.4101      0.2486     c 
 
59000.5     03073     287.31192     -20.12834     126     4.2     -     -     -     -     - 
 
59001.5     03074     288.68802     -20.38238      58     4.1     -     -     -     -     - 
 
59019.5     03092     287.14477     -20.04175     129     4.4     -     -     -     -     - 
 
59020.5     03093     288.66147     -20.05024     117     4.0     0.8295      0.0054      26.3803      0.2016     c 
 
59021.5     03094     287.11257     -20.05892     113     3.8     -     -     -     -     - 
 
59022.5     03095     288.62929     -20.05384     101     3.4     0.8568      0.0010      26.3944      0.1838     f 
 
59024.5     03097     287.06207     -20.06483     129     4.3     0.8192      0.0006      26.2047      0.1815     f 
 
59025.5     03098     288.57787     -20.06015     119     4.0     -     -     -     -     - 
 
59073.3     03145     286.25079     -20.16488     116     3.9     -     -     -     -     - 
 
59074.3     03146     287.67551     -20.12296     128     4.3     -     -     -     -     - 
 
59075.3     03147     286.22292     -20.16881     127     4.3     0.7642      0.0085      26.3015      0.1758     f 
 
59076.3     03148     287.74292     -20.16470     121     4.3     0.7841      0.0091      26.5365      0.1989     f 
 
59374.5     03447     288.54961     -21.09995     115     4.2     0.8214     0.0     26.6310     0.4176     r 
 
59382.5     03455     288.44956     -21.10000      94     3.4     0.8302     0.0     26.5853     0.2644     r 

59400.4     03473     288.00157     -20.28053     117     4.1     0.8826     0.0039     26.9686     0.2573     r 

59462.2     03535     288.35953     -21.35001     119     4.0     -     -     -     -     -  

59463.2     03536     286.58291     -21.34999     116     3.9     -     -     -     -     -  

59732.5     03805     288.50000     -21.66665     111     3.9     0.8343     0.0     26.6119     0.2941     r 

59733.5     03806     288.49996     -21.66668     119     4.1     0.8275     0.0     27.0161     0.3851     r 

59759.5     03832     288.25939     -21.42730     116     3.8     0.8902     0.0044     26.7224     0.2565     r 

59760.5     03833     288.25943     -21.42732     114     3.9     0.8887     0.0055     26.3220     0.3435     r 

59872.2     03945     288.42499     -21.40842      56     2.0     -     -     -     -     - 

59873.2     03946     288.42503     -21.40844      65     2.2     0.8090     0.0078     26.32     0.2532     r 

59875.2     03948     288.37854     -21.41377      64     2.1     -     -     -     -     - 

60196.3     04269     288.25001     -21.44998     125     4.4     0.8208     0.0000     25.78     0.7214    r 

60197.3     04270     288.25002     -21.44998     136     4.5     -     -     -     -     - 
 

List of observing sequences gathered and the survey discovery efficiency. Nights without reported discovery efficiencies were not searched, but used only for follow-up. Reported detection efficiencies were fit using the functional form η(m) = A - c(m-20)^2/1+e^m-M_o/σ <cit.>.  A is the peak efficiency, at m=20. This function was required over simpler representations such as the tanh representation <cit.> to accurately represent the detection efficiency which has a multimodal behaviour that exhibits a gradual decrease in efficiency with increasing magnitude (quantified by parameter c) followed by much more precipitous decline nearer the limiting magnitude (quantified by parameter M_o, with a width quantified by w).  The network type column describes which type of network CNN (c), flux sorted CNN (f), or ResNet (r) was used for initial KBMOD source rejection. Nights marked with `-' were not searched directly, but used only for follow-up and linking. For all sequences, exposure times were 90 s. Field 03833 encountered an issue whereby the efficiency dropped to ∼0% brighwtward of r∼22 and so the efficiency parameters quoted apply to only fainter values. These observations were acquired in Subaru programs S20A-OT04, S20B-TE019-G, S20B-081, S20B-TE204-K, S20B-OT04, S21A-OT04, S21A-TE216-K, S21B-OT04, S22A-TE049-K, S22B-TE071-K, and S23B-OT55.








 §.§ Astrometry and Linking 



Turning our initial detections into confirmed TNOs with measurable orbits required a process to expand the single-night detections into multi-night arcs.
This arc-expansion process was necessarily less automatic than the initial detection process, owing both to the idiosyncratic nature of the data and the smaller number of potential TNOs to be processed.
The vetted results of the machine learning process were summarised as a list of detections with a nominal center-time position and velocity in right ascension and declination.
For most detections, if the shift-and-stack step reported multiple `best' velocities, each was initially treated as a separate detection before the later detection-merger steps.


The first step in the linking process was to filter out already-known objects from the detection list.
While this was attempted for the 2020 data, only three previously-known TNOs were visible in the entire 2-field 2020 dataset (2001 BL_41, 2011 HF_103, and 2011 JX_31).
The filtering process turned up significantly more known objects in the 2021 and 2022 detections, all of them being discovered in previous datasets from this program.
This filtering process ensured that known TNOs were not re-linked and saved considerable time in later years. 
Once the linking process was complete, this same filtering code was used to determine which fraction of reported detections corresponded to TNOs with determined orbits; see Table <ref>.

The next step was to vet the detections visually using the unimplanted star-subtracted HSC data.
These star-subtracted data were produced from the same  output as was used for the detection step, but fed through an independent pipeline to produce star-subtracted images.
This independent star subtraction pipeline was previously developed for recovery of  targets in HSC data obtained in 2016 and 2017 <cit.>.
The images were coregistered to a common sidereal frame and the best 90%  were stacked to produce a nominal “star stack”.
The vignetted areas on the edge CCDs and the areas marked as saturated by the LSST pipeline are masked out in the coregistration step.
The effective PSF was extracted for each image as well as for the star-stacked image.
The PSFs were created using a list of 100 stars from Gaia DR3 <cit.>, extracting a window of 51×51 pixels centered on each star, dividing by the nominal flux of the star based on its Gaia g' magnitude, and then median-combining all the stars to filter out outlier pixels.
For each image, a transformation was calculated from the star stack PSF to the image PSF using the method from <cit.>, the transformation was applied to the star stack, and the transformed star stack was subtracted from each coregistered image.
This is the same process in principle as applied for the machine learning detections, but using custom Python-based software originally developed for the manual search process before the machine learning was developed.
The dissimilar redundancy of this process would likely not make a difference for fields with much fewer stars, but the extremely high star density of the  search areas required extra steps to ensure that nominal TNO detections were not just residuals from poor star subtractions.

For each filtered detection, 41×11 pixel shift-stacked windows were extracted using the nominal position and velocity information from the detection list.
Each of these shift-stacked windows was then visually vetted, through a process that required the user to either mouse-click on the TNO or press a key to discard the detection.
Typically, 10-30% of detections were discarded in this step, with the number being much higher for nights with poor seeing which resulted in much less effective star subtraction.
The manual “click on the TNO” process was required for earlier iterations of the machine learning process, which tended to only be accurate to within a few pixels, and then retained for resilience as the machine learning process subsequently improved.
The pipeline then used the manual clicked positions as initial conditions to fit the position and flux of the TNO in the shift stack, using the empirical star-based PSF as a model.
After an initial best-fit using downhill simplex as an optimiser, the pixel position and flux of the TNO were fit as a discretely sampled probability distribution function (PDF) using the emcee implementation of a Markov Chain Monte Carlo (MCMC) maximum likelihood technique.
Both the best-fit and MCMC likelihood used a χ^2 estimate based on the sum of the square of the difference between the data and a model-shifted PSF, tuned to the noise of the image, and calibrated on known good detections.
This discretely-sampled pixel and flux cloud was transformed into position and magnitude clouds for the first, middle, and last observations used in the stack.
While this was technically transforming four parameters (mean position RA/Dec and RA/Dec rate) into six, we found it to be best way to capture the motion of the object, since just using two positions resulted in effectively overestimating the motion uncertainty.
This was particularly true for the observations away from opposition, as the motion of the KBOs close to quadrature is very non-linear.
Using three points (instead of two) also effectively encodes the correlation between the points as they were all drawn from the same stacked image. These three clouds were then turned into mean positions and covariance matrices, and the means and covariances saved as astrometry files.

The next step was to fit an initial orbit to this astrometry.
The orbit fitting code used is a derivative of the one described in <cit.>, and provides orbit solutions and MCMC clouds that have full planetary perturbations.
To bootstrap the orbit solutions, rough circular orbits were first fit to the data.
These were then immediately used as initial conditions for a second pass that allowed the eccentricity to vary up to 0.1,
followed by a full MCMC solution that allowed eccentricity to vary up to 0.9.
The orbit fitter and MCMC used Cartesian state vectors internally and were thus fully capable of handling parabolic and hyperbolic orbits.
Using Cartesian state vectors was particularly useful for the MCMC, as they represent a fully linearally independent basis, allowing for accurate calculation of their covariance.
However, the observation fields were being shifted at rates optimized for bound, stable objects at similar distances to  and thus any long-period cometary or interstellar objects would likely have been unrecovered anyways. 
In addition, most of the detections with very low rates of motion were likely star residuals rather than real-but-distant objects, so imposing an effective rate restriction of e≤0.9 filtered out many spurious detections.
Once the initial orbit clouds for the detections were produced, they were then used to create predictions of the positions of the TNO for the images in both the discovery night and in the other observations.
These predictions were produced by propagating the MCMC orbit cloud to the night, determining the RA/Dec of each element of the cloud, and then taking the mean and covariance of the RA/Dec cloud.

At this point, the next iteration of the process begins, a process that is repeated until the orbits are all as well determined as they can be with the dataset.
The predicted positions are first used to find which HSC CCD (if any) the TNO is projected to be on for all the other data being considered, initially just observations close to discovery, then later the full dataset.
The covariance matrices for all the observations on which a TNO is on a CCD are then used to calculate a 3σ uncertainty ellipse for the TNO on each night.
If the minor axis of the 3σ uncertainty ellipse was greater than 1000 pixels, then that night was skipped as being too uncertain.
This provided an effective filter for short arcs and initial detections that were projected too far, but still allowed for objects that had a few-month arc (and therefore a well-determined orbit plane) to be projected to data in the previous years, with uncertainty ellipses that might be thousands of pixels long, but only tens of pixels wide.
These ellipses were used to produce World Coordinate System (WCS) projections sized to cover the 3σ uncertainty and that were rotated to be aligned to the error ellipse, with the horizontal image axis aligned to the major uncertainty axis.
This WCS was then re-centered to the mean predicted locations of the TNO in each image and stacked, to produce a non-linear shift-stack covering the possible locations where the TNO may reside.
Rotating the images to the uncertainty axis saved considerable disk space and CPU time, and meant that most of the shift stack was searchable pixels.
These images were then processed with the same manual inspection process as the initial detections.
The vast majority of detections were just as visible in their discovery night as before (if not moreso), though a small fraction were discarded, as the eccentricity restriction in their initial orbit caused them to no longer appear as a viable detection.
On nights other than the discovery night, the manual process allowed the user to click any source that appeared as the object, or discard that night for that iteration.
This second click would provide a link, and extend the orbit beyond just the discovery night.
In practice, only very bright and obvious TNOs were linked in the first iteration, with more links being attempted as the iterations were repeated and confidence was gained in the detections.
The manual clicks were then used to produce MCMC-fit astrometry just as before, and the astrometry fit to initial orbits with e<0.9.
The residuals of all of these initial orbits for an iteration were then visually inspected. 
If a newly added point had significant tension with the existing points for an orbit, it was discarded and the orbit refit.
Orbit clouds and predictions were then generated for all the objects and the process repeated.

Subsequent iterations would allow the observation arcs of the object to be gradually increased, a night or two of data at a time.
Likewise, if an initial link is attempted, and the object does not appear on a third night where it should be visible, that link can be discarded as bad and the object returned to a single night detection.
The detection process is best run on several nights of nearby data simultaneously, as this allows duplication of detection and crosslinking between nights.
The initial duplicate detection checks if any of the not-discovery detections corresponds to known detections for other nights.
Since the machine learning process discussed above can report a detection twice with two different rates, this can mean that up to four detections on two nights can be combined into a single TNO for future iterations.
The next extension is crosslinking, which is only performed on objects that still appear single after the initial manual linking stage, and attempts to fit an orbit for every combination of two single-night detections from different nights.
This is very CPU-intensive, but typically results in a few extra links that were missed by the manual linking process.
Finally, there is a process to merge single night detections with multiple rates, to see if the effective average between their rates is more likely to produce a link than the separate rates by themselves.
This is again not very efficient, but sometimes found additional links that had previously been missed because the original rates reported by KBMOD were incorrect.
After all of these merger or crosslinking steps, the prediction/manual linking/orbit fitting iteration was performed again.
This process was repeated until all possible links had been investigated and the orbits were as good as possible with the data near the discovery.

Once the linking process was finished, the objects with at least two good nights were shifted to a second (but very similar) pipeline that enabled a search of all relevant HSC data back to 2014, as well as include in their orbits any precoveries/recoveries discovered in other datasets, such as the DEEP Survery <cit.>.
This second process was repeated to extend the orbital arcs of the TNO discoveries as much as possible, as well as to find any duplicates with known objects that had not previously been uncovered.
Finally, the resulting astrometry of the second pipeline was exported to MPC submission format and submitted to the MPC.



§ THE DETECTIONS 

We present 239 multi-night detections from our search to date in Tables <ref> and  <ref> and Figure <ref>. Many of these objects have only a few linked individual detections and of relatively short arcs, and so the fidelity of the derived orbital information is low. However, even for the short-arc detections, the distance at discovery is sufficiently constrained for this discussion.




The orbits are mean barycentric solutions for a cloud of all possible solutions, and since very short arcs have more possible solutions at high inclinations and/or eccentricities, their mean solutions can show exaggerated eccentricity or inclination. True eccentricities and inclinations tend to be biased to lower values than predicted from short arcs, a behavior that is seen by other orbit fitting techniques <cit.>. Forward modelling of simulated ephemerides reveal the tendency for fits to report inflated inclinations for arcs that are a few days in length (see Appendix <ref>). However, many of the objects at this point have at least a one-month observational arc, which helps exclude the most inclined solutions. In addition, we excluded solutions with an eccentricity greater than 0.9 for any objects with an arc shorter than two months, again to reduce the number of physically unlikely orbits in the solution cloud. These two constraints together help to produce mean orbits that are not greatly overly inflated by unlikely cases,
and have been proven out in our ability to recover numerous short-arc discoveries in data taken in the following or preceding opposition.


The distance and inclination distributions of our discoveries follow expectations from the known Trans-Neptunian Objects (TNOs) <cit.>, including the abundance of low-inclination objects between the 3:2 and 2:1 resonances, and the so-called Kuiper Cliff – the well known rapid decrease in detections outside R∼55 au <cit.>. Interestingly our detections include an unexpectedly high number of objects at larger heliocentric distances. These objects are the focus of our discussion.

Eleven of our multi-night detections have best-fit heliocentric distances R>70 au. Of these distant objects, 6 have linkages spanning three or more epochs with three spanning four epochs. The remaining five objects have linkages spanning only two epochs. The high recovery rate lends to the veracity of the detections. Admittedly, our analysis did not include provisions for characterizing the rate of false positive detections and the rate of false linkages that might result. We emphasize however, that three of the five objects with only two epochs of detection have brightnesses r<26 which are easy detections in our data suggesting that their lack of more frequent recoveries is merely a reflection of the high stellar density in the field. Indeed, three objects linked across three epochs have brightnesses r>26.5 demonstrating that such faint objects can be detected and linked. Furthermore, if the distant detections were indeed driven by false positives, one would expect a pile-up of faint detections near the detection threshold of our survey (typically r∼26.5 mags), but instead, we see that six of the eleven distant objects have brightnesses <26.0. 
In general, the rates of motion used for detection by KBMOD are consistent with the rates from final orbits, particularly for the faintest detections, which would only be detectable at nearly the correct rates.


The short arcs (77 days or less) of the distant objects preclude high-accuracy orbit determination (see Table <ref>). The distance determinations, however, are accurate enough to reveal a valley in the density of detections between R∼ 55 au and R∼70 au. To the best of our knowledge, an increase in the number density of objects outside R∼70 au is a feature of the radial distribution that has not been reported by other surveys to similar depths <cit.>. Given the limited accuracy of the measured orbital elements, simulation of the detailed intrinsic structure implied by our detections would not be useful. Use of the Outer Solar System Origins Survey (OSSOS) Simulator <cit.> however, can still prove informative. A simulation of the pointing history used in this survey and the most up-to-date model of the Outer Solar System planetesimal populations <cit.>, which includes a distant belt that is well matched to the OSSOS detections <cit.> finds that for every 1 detection beyond ∼70 au we expect between 72 and 83 detections[This range is dominated by the uncertainty in the distant population model.] inside that distance. Thus, the 11 distant detections in our survey would imply between ∼790 and ∼900 detections inside 70 au, which is 3.5-4× larger than we observe. In other words, the 11 distant objects seem to betray the presence of a larger than expected population of bodies beyond R∼70 au. This inference assumes that all of those distant objects are real detections. 
We may have a non-zero false detection rate, though we point out that we have a near 100% success rate in the follow-up of targets that have been selected for imaging with the HST (22 targets, 1 failed recovery; see Table <ref>), and the  spacecraft itself (7 targets). The only two cases of failed follow-up are: a non-detection of a source in LORRI observations; and a non-detection of a source with detection distance ∼50 au that was followed up with Gemini imagery. We suspect the latter was due to the short Gemini imaging sequence used which prevented the motion of the source from being detectable. This follow-up track record suggests that the false positive rate cannot be too high. In the extreme circumstance that of the 11 distant sources, the only real objects are those with at least month-long arcs and with at least three epochs of detection, we still detect 1.5-1.8× more distant objects than implied by the OSSOS++ model and simulations using our measured detection characteristics. 
Excluding all of our distant detections to bring our survey into compatibility with the OSSOS model would imply an exceptionally high false positive rate. Evidence of a much lower false positive rate comes from the 6 detections with distances 55<r<70 which have rates of motion only marginally faster than objects with R>70 au. Four of those objects have arcs that are months long, with three having arcs years long. Rather, we conclude that many if not most of our distant detections are real. Confirmation of their presence in future observations is highly desirable.























lcccccccc
        
        MPC    
        Arc Length    
        # Epochs    
        mag (r2)    
        R (au)    
        q (au)    
        a (au)    
        i (^∘)    
        e 

            
         (days)    
            
            
            
            
            
            
         


2017 OW166     2242     8     24.52± 0.18     43.78± 0.00     29.36± 0.00     39.50± 0.00     2.33± 0.00     0.26±0.00 

2017 OX166     1121     8     26.21± 0.26     41.42± 0.00     40.84± 0.26     44.21± 0.01     2.43± 0.00     0.08±0.01 

2017 OY166     1417     11     24.75± 0.24     44.20± 0.00     40.94± 0.00     44.50± 0.00     2.22± 0.00     0.08±0.00 

2020 KA54     77     6     26.32± 0.21     41.34± 0.10     40.20± 0.90     47.33± 0.59     11.08± 0.14     0.15±0.01 

2020 KA55     77     6     26.45± 0.19     33.85± 0.05     33.79± 0.08     36.82± 0.46     2.95± 0.01     0.08±0.01 

2022 LU16     28     2     26.12± 0.52     71.48± 3.65     19.78±18.61     84.48±54.48     28.29±19.05     0.82±0.16 

2022 LT16     28     3     25.79± 1.29     72.81± 2.64     29.30±17.25     105.16±76.29     7.04± 2.01     0.68±0.24 

2022 LR15     28     2     26.32± 0.35     74.28± 3.90     21.73±18.89     86.64±52.83     41.00±24.45     0.80±0.15 

2020 KX11     77     5     26.09± 0.56     74.81± 0.29     4.11± 0.35     110.07±42.70     55.52± 4.45     0.96±0.01 

2022 LY15     27     2     25.64± 1.00     76.37± 3.80     23.13±19.94     116.38±108.28     31.94±19.89     0.80±0.17 

2022 LQ16     29     4     26.43± 0.48     76.81± 2.04     44.22±16.60     147.52±123.79     7.89± 1.47     0.60±0.25 

2020 PD95     3     2     25.39± 1.72     80.08± 4.74     22.11±20.41     98.18±66.82     35.50±24.97     0.83±0.14 

2022 LX15     28     3     25.97± 0.52     82.15± 1.56     9.13± 2.09     83.13±53.49     58.07±16.12     0.86±0.08 

2022 LS16     29     2     25.59± 0.94     87.65± 4.20     27.74±21.95     122.65±97.52     44.19±23.82     0.79±0.16 

2020 MJ53     4     2     26.25± 0.44     87.72± 4.06     23.91±24.35     114.66±89.28     14.91± 9.30     0.84±0.15 

2020 MK53     4     2     25.99± 0.49     162.03± 5.35     46.54±45.03     212.81±161.42     31.44±20.77     0.83±0.15 


Barycentric orbital elements of the discovered objects. Short example of the list of TNOs discovered in our 2020, 2021, and 2022 Subaru-HSC searches.
These discoveries have been submitted to the Minor Planet Center. 
See Section <ref> for details on the orbit fitting. The full list is presented in the Appendix. The point clouds generated as part of the orbit fitting are available from the authors upon request. For convenience we have included the 11 objects with R>70 au. 






§ DISCUSSION 


Our use of Subaru Telescope, which was needed to search the wide area where candidates might be located, resulted in a sample of 239 TNOs being discovered as of December 2023, including those TNOs that were suitable for observations from . 

The observing strategy and search technique we employed enabled precise determination of the detection and tracking efficiency of the sample resulting in a well-characterized sample of TNO orbits. Comparison of our sample with outputs from the OSSOS Survey Simulator reveals a ∼3.5-4× larger-than-expected sample of objects at distances R≳70 au.






As noted in <cit.>, there is a growing body of evidence hinting at the presence of a yet-to-be-discovered mass of distant planetesimals. This evidence includes the dust impact rate measurements of the Student Dust Counter on-board  which are not decreasing at the rate expected given the known TNOs <cit.>, the serendipitous occultation detections <cit.> which seem too improbable given the known extent of the dynamically excited Kuiper Belt, and the report of a few distant TNOs detected in archival HST imagery <cit.> all of which which appear to be incompatible with the observed luminosity function reported from other ground-based surveys <cit.>.  As Shannon et al. point out, models that include a more massive distant population with either or some combination of decreasing maximum size distribution with increasing semi-major axis, or increasing surface density of material, can reconcile the occultation detections with the Pluto-Charon cratering record <cit.>. Indeed, at least in principle, all of these results may be reconciled if there is a population of planetesimals beyond the main Kuiper Belt that has yet to be recognized.

Some rough constraints on the possible distant population come from past TNO surveys. The deepest published wide area surveys with good distance measurements for their sources have a brightness limit of r∼25.5 mag <cit.> and do not report the detection of a substantial distant population that could explain the abundance of objects at distances ∼70 au that we report here. If the distant population was in a belt form like that of the classical TNOs, it must be the case that the majority of objects must be fainter than r=25.5, or H∼7. These are significantly smaller than the largest cold classical objects, which have H∼5.0 <cit.>, and is consistent with the suggestion by Shannon that the size distribution of an outer population is likely to consist of smaller bodies than found in the main Kuiper Belt. With the use of the OSSOS Survey Simulator <cit.>, we have found that the 11 detections with distance R>70 au that we report are consistent with a belt of material in a 5 au wide ring centered at 70 au with a surface density of 6.2 objects per square au with H_r_2<8.66. This estimate assumes the objects have a steep size distribution like that of the hot classical TNOs <cit.> of the form N(<r_2)∝10^α r_2 with α=1.1 for objects with H_r_2<7.5 and α=0.3 for objects with H_r_2>7.5 and are on circular orbits with inclinations distributed as a sin-normal with width of 12^∘ <cit.>. This density increases to 20.5 objects per square au if the ring is instead centered at 80 au. These density estimates suggest a mass that is comparable with the mass of the known excited TNO population but these estimates are highly uncertain, and so provide only a sense of the scale of the required population, as the orbital and size distributions of such a population remain mostly unconstrained. The orbital elements of our distant detections imply at least moderate inclinations for those bodies, and so we do not expect the underlying population to be dynamically cold. This is consistent with the moderate ecliptic latitudes of the detections of serendipitous stellar occultations reported by <cit.>. 


While the presence of an undiscovered belt is compatible with the distribution of distant objects we report, it is incompatible with the results of the DECam Ecliptic Exploration Project <cit.>. For an example, we show a simulated distribution of detections for both DEEP and this survey, for a belt or annulus of distant bodies in Figure <ref>. DEEP reports a similar limiting magnitude and survey area and utilizes very similar search techniques as this survey does, but detected only 1 object with distance R>60 au <cit.>.  The density of detections of objects at R>60 au in DEEP is roughly an order of magnitude lower than we report here, where the density of objects should be the same for both surveys if the population of distant bodies is in a longitudinally symmetric distribution such as in the annulus model. This result implies  either that DEEP are unexpectedly insensitive to slow-moving bodies, or, more likely, that if there is a large population of distant bodies that their apparent sky density is low at the DEEP survey pointing locations. 


The differences in the observed distributions of distant objects between DEEP and what we report here may be caused by longitudinal variations in the radial distribution of objects in distant mean-motion resonances. To demonstrate how this might be the case, we present a toy model realization of the 5:2 resonant population in the OSSOS++ model. This population uses a uniform semi-major axis distribution between 54.75≤ a≤ 55.75 au, a Gaussian distribution in eccentricity with mean of 0.43 and standard deviation of 0.03 limited to 0.2 ≤ e ≤ 0.5, a sin-normal inclination distribution with width of 20^∘ limited to i≤ 45^∘, and a triangularly distributed resonant amplitude with peak at 75^∘ that goes to zero at 0^∘ and 180^∘. We adopted the size distribution for the hot classical population reported by <cit.>.
A realization of the aforementioned 5:2 model was generated with the OSSOS Survey Simulator <cit.>, drawing objects randomly through the Solar System until 1,000 total objects were drawn between the  Survey, and the DEEP pointing. The former used the exact pointing history that we report in Table 1, and the latter was a single epoch pointing at 23:32:00, -04:00:00 with an area of 7.9 square degrees. Those drawn objects with magnitude r<27 were recorded, and are presented in Figure <ref>. 
This simple realization broadly reflects the notable properties of the real detections we report here, including an absence of objects near R∼65 au with a smattering of detections at further distances. 
Such a longitude dependent distribution is also consistent with the lack of detections in the DEEP sample beyond R∼70 au. 

We emphasize that our 5:2 population is a toy model that uses a single resonance with a relatively poorly constrained eccentricity distribution <cit.> to demonstrate the influence of pointing longitude on the observed sample. The true Solar System includes many other resonant populations.  Confirmation that the distribution we have observed is driven by a heretofore previously unrecognized massive resonant population awaits significant future modeling efforts. Though we point out that a simulation of the DEEP and  pointings shown in Figure <ref> that utilizes the full OSSOS++ model  – which includes an up-to-date estimate of of all distant resonances for which at least one object was detected <cit.> – presents a similar lack of distant detections compared to more proximate TNO populations. 

Whatever the intrinsic dynamical structure distant (R≳70 au) objects occupy – disk, annulus, resonant population, etc. – the discovery rate in the survey we present here implies more mass in that distant population than previously recognized. A larger than expected population of distant TNOs can help explain the unexpected and consistently high Student Dust Counter impact rates, and the serendipitous occultations detected by <cit.> and <cit.>, both of which appear somewhat at odds with the small known mass of TNOs at the required distances. With an increased mass, these observations may not be at odds after all. 

The work we have presented here is not an in-depth analysis of the broad and varied range of potential forms a distant population could take. Any detailed analysis will be hampered by the limited fidelity of the orbits we report which are simply insufficient to provide strong constraint on the orbital distributions the distant objects may take.  Rather, we have limited our discussion mainly to our methods of discovery and stick to the main point – there appears to be more distant bodies than previously recognized.

Constraints on the orbital structure of the distant population remain weak, primarily as a result of the insufficient depth limits of surveys which have provided the strongest orbital constraints. A survey with depth limits similar to what we report here, but over a wider field and at different ecliptic longitudes is necessary to confirm the presence of this distant population, and to reveal its intrinsic orbital structure. The wait for such a survey should not be too long however, as combined ongoing surveys like the DEEP survey and the Classical and Large-A Solar System Survey <cit.>, the pencil beam survey with the James Webb Space Telescope <cit.>, and the efforts of this team which include on-going observations and the application of additional search techniques to the data we present here <cit.>, should achieve sufficient depth, coverage, and spread in pointings to detect large numbers of these distant bodies, and test the findings we report here.





















§ SUMMARY  

We report the discovery of 239 new TNOs discovered with the Subaru Telescope and the Hyper SuprimeCam as part of the  distant search.  Beyond the Kuiper Cliff, these objects show an unexpectedly high fraction of distant R≳70 au hinting at a previously unrecognized large population of distant bodies. The fidelity of the orbits for these distant objects is too low to provide meaningful constraints on the intrinsic orbital structure of the distant population, which is compatible with a homogeneous distribution (disk or annulus) or a large population of resonant bodies. The inferred mass however, could be as high as the known excited Kuiper Belt. Confirmation of this signal is required, and will likely come from ongoing surveys, including this one. 

aastex631
aasjournal


 parallel <cit.>, numpy <cit.>, matplotlib <cit.>, TRailed Imaging in Python, <cit.>, astropy <cit.>, SExtractor <cit.>




§ ACKNOWLEDGEMENTS



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

This research is based on data collected at the 8.2m Subaru Telescope (<https://subarutelescope.org/en/>) using its prime focus camera (the Hyper Suprime-Cam), operated by the National Astronomical Observatory of Japan. The Subaru Telescope provided the main source of data for this work. We would like to thank the director and the TAC for working with us to accommodate our complex scheduling requirements.  

This research is based on observations from the programs GS-2019B-FT-102, GN-2020A-DD-103, and GS-2020A-DD-103 obtained at the international Gemini Observatory, a program of NSF’s NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).


This research used the Canadian Astronomy Data Centre facilities operated by the National Research Council of Canada with the support of the Canadian Space Agency. This work also made use of the Gemini Observatory Archive, NASA’s Astrophysics Data System Bibliographic Services, and the JPL HORIZONS web interface (<https://ssd.jpl.nasa.gov/horizons.cgi>).

 scientists Stern, Benecchi, Buie, Fraser, Lauer, Kavelaars, Peltier, Porter, and Verbiscer thank NASA's New Horizons Mission (NASW-02008) for funding.

This work made use of the Pan-STARRS1 Surveys (PS1) and the PS1 public science archive, which have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.

This paper makes use of LSST Science Pipelines software developed by the Vera C. Rubin Observatory. We thank the Rubin Observatory for making their code available as free software at <https://pipelines.lsst.io>.

We also thank the former and current directors of the Subaru Telescope, Michitoshi
Yoshida and Satoshi
Miyazaki, for opening the Subaru Telescope to collaborate with the New
Horizons mission. And we thank the TAC and Subaru staff for working with us to
accommodate our complex scheduling requirements.

Finally, W. C. Fraser would like to thank B. C. Pontus and D. Ragozzine for their insightful comments.






§ FULL DETECTION LIST




lcccccccccccccc
The fitted barycentric orbital elements of the TNO discoveries reported in this paper. Objects are sorted in order of increase heliocentric distance. 

MPC     Arc      Epochs     mag     σ_mag     R      σ_R     q      σ_q     a     σ_a     inc     σ_inc     e     σ_e 

     (day)     #             (au)     (au)     (au)     (au)      (au)     (au)     (deg)     (deg)         

2017 OW166     2242     8      24.52      0.18       43.78     0.00       29.36      0.00       39.50      0.00        2.33      0.00       0.26     0.00  

2017 OX166     1121     8      26.21      0.26       41.42     0.00       40.84      0.26       44.21      0.01        2.43      0.00       0.08     0.01  

2017 OY166     1417     11      24.75      0.24       44.20     0.00       40.94      0.00       44.50      0.00        2.22      0.00       0.08     0.00  

2020 KA54     77     6      26.32      0.21       41.34     0.10       40.20      0.90       47.33      0.59       11.08      0.14       0.15     0.01  

2020 KA55     77     6      26.45      0.19       33.85     0.05       33.79      0.08       36.82      0.46        2.95      0.01       0.08     0.01  

2020 KA56     79     8      24.81      0.21       38.37     0.04       32.11      0.32       39.69      0.67        4.57      0.03       0.19     0.02  

2020 KB54     76     8      25.53      0.27       44.47     0.07       41.42      0.63       44.04      0.75        2.73      0.01       0.06     0.02  

2020 KB55     77     6      26.06      0.18       45.40     0.10       44.80      0.58       47.74      0.83       18.61      0.26       0.07     0.02  

2020 KB56     79     7      25.93      0.25       43.39     0.11       42.83      0.28       143.42      6.11        7.80      0.09       0.70     0.01  

2020 KC53     78     9      25.84      0.25       44.42     0.09       42.25      1.52       43.96      0.63        2.39      0.01       0.04     0.03  

2020 KC54     79     7      24.73      0.29       42.82     0.05       41.08      1.19       42.54      0.34        2.59      0.00       0.03     0.02  

2020 KC55     77     6      26.08      0.25       44.17     0.09       38.64      2.04       43.91      0.38        2.70      0.00       0.12     0.04  

2020 KC56     79     5      25.17      0.25       38.22     0.12       36.80      1.33       47.91      1.31        9.86      0.15       0.23     0.04  

2020 KD53     467     8      25.37      0.19       35.67     0.00       34.86      0.02       96.98      0.17       15.29      0.00       0.64     0.00  

2020 KD54*     423     8      26.28      0.25       42.14     0.00       37.35      0.03       66.82      0.09       17.85      0.00       0.44     0.00  

2020 KD55     76     6      25.95      0.24       32.53     0.06       31.74      0.42       36.48      0.68       15.00      0.15       0.13     0.03  

2020 KD56     27     4      25.60      0.22       37.92     0.28       32.71      3.83       113.54     81.73       17.65      0.65       0.64     0.20  

2020 KE53     406     9      26.37      0.21       39.17     0.00       34.48      0.20       39.33      0.05        5.06      0.00       0.12     0.01  

2020 KE54     80     6      23.03      0.21       33.41     0.01       31.03      0.11       39.59      0.21       21.41      0.05       0.22     0.01  

2020 KE55     77     4      26.43      0.21       42.93     0.19       39.35      3.01       47.99      2.64        2.98      0.04       0.18     0.09  

2020 KE56     75     5      24.85      0.21       38.86     0.05       33.08      0.32       39.01      0.59        6.52      0.04       0.15     0.02  

2020 KF53     5     3      26.06      0.17       49.90     3.30       16.73     12.19       61.10     36.58        1.35      0.10       0.72     0.23  

2020 KF54     80     7      24.96      0.20       40.17     0.04       38.42      0.60       43.79      0.26        2.91      0.00       0.12     0.01  

2020 KF55     75     5      25.99      0.33       40.93     0.10       36.36      1.09       46.18      1.96       12.82      0.18       0.21     0.06  

2020 KF56     78     5      25.97      0.23       41.68     0.10       40.86      0.70       42.84      0.89       10.16      0.14       0.05     0.03  

2020 KG53     5     3      25.61      0.25       49.54     2.89       17.70     11.33       60.64     34.83        2.22      0.66       0.70     0.23  

2020 KG54*     462     8      25.66      0.22       32.42     0.00       32.19      0.00       78.33      0.02       22.47      0.00       0.59     0.00  

2020 KG55     24     4      26.14      0.20       38.27     0.35       28.96      5.39       79.23     55.58       15.49      0.75       0.53     0.27  

2020 KG56     76     9      26.13      0.21       35.22     0.05       31.88      0.40       47.44      1.38        2.29      0.00       0.33     0.03  

2020 KH42*     400     8      22.36      2.64       49.53     0.00       37.32      0.01       77.55      0.05       25.88      0.00       0.52     0.00  

2020 KH54     80     7      25.18      0.26       44.10     0.07       38.19      0.69       44.46      0.96        3.11      0.01       0.14     0.03  

2020 KH55     27     4      25.73      0.21       44.06     0.37       28.24      6.52       82.60     67.31        3.77      0.12       0.53     0.29  

2020 KH56*     426     9      25.33      0.34       35.53     0.00       32.63      0.14       55.68      0.36        3.06      0.00       0.41     0.01  

2020 KJ54     77     5      25.42      0.17       43.63     0.08       40.85      1.65       42.81      0.54        2.45      0.00       0.05     0.03  

2020 KJ55     27     4      25.89      0.24       44.86     0.39       30.00      6.37       68.07     33.60        2.06      0.00       0.50     0.28  

2020 KJ56*     371     9      22.60      0.30       42.80     0.00       42.79      0.01       44.09      0.01        2.73      0.00       0.03     0.00  

2020 KK54*     155     7      25.92      0.20       49.53     0.00       37.25      0.26       43.97      0.15        5.99      0.00       0.15     0.01  

2020 KK55*     369     7      26.43      0.24       34.22     0.00       33.75      0.09       76.21      0.35        3.83      0.00       0.56     0.00  

2020 KL53*     388     8      24.14      0.37       44.59     0.00       42.46      0.04       43.59      0.01        2.41      0.00       0.03     0.00  

2020 KL54     77     6      25.89      0.23       43.67     0.11       38.67      2.60       43.77      0.36        7.09      0.10       0.12     0.05  

2020 KL55     75     5      26.07      0.40       46.56     0.17       23.63      1.53       176.05     63.48        2.51      0.03       0.86     0.06  

2020 KM53     388     6      25.14      0.28       44.06     0.00       41.93      0.64       44.53      0.05        1.96      0.00       0.06     0.02  

2020 KM54     76     6      26.09      0.24       41.73     0.10       39.17      1.47       45.89      0.58        6.33      0.08       0.15     0.02  

2020 KM55     77     6      25.70      0.19       44.41     0.09       40.90      1.81       45.88      0.46        2.67      0.01       0.11     0.03  

2020 KN53     78     10      24.76      0.19       39.63     0.04       38.73      0.40       42.96      0.25        9.25      0.04       0.10     0.01  

2020 KN54     78     7      24.27      0.21       20.19     0.01       11.29      0.05       16.83      0.01        9.53      0.02       0.33     0.00  

2020 KN55     77     6      24.22      0.17       40.56     0.03       37.16      0.57       41.13      0.13       26.54      0.11       0.10     0.01  

2020 KO11*     154     8      25.24      0.25       49.42     0.00       38.54      0.11       44.75      0.05        6.75      0.00       0.14     0.00  

2020 KO53     77     6      25.32      0.14       45.93     0.09       42.64      1.16       48.28      1.20       12.26      0.14       0.12     0.05  

2020 KO54     29     5      26.15      0.44       29.30     0.25       23.59      3.98       54.51     23.15        6.51      0.26       0.53     0.23  

2020 KO55     77     6      25.86      0.19       44.94     0.09       37.26      1.59       47.74      0.45        5.47      0.05       0.22     0.03  

2020 KP11*     1202     27      23.74      2.57       51.45     0.00       38.10      0.00       45.47      0.00       22.20      0.00       0.16     0.00  

2020 KP53     77     6      26.03      0.22       45.87     0.11       38.66      0.76       46.11      1.53        4.37      0.05       0.16     0.04  

2020 KP54     77     7      26.17      0.24       41.11     0.09       38.63      1.08       43.02      1.12        2.93      0.02       0.10     0.05  

2020 KP55*     366     8      25.04      0.27       32.82     0.00       30.31      0.12       47.87      0.18        3.49      0.00       0.37     0.00  

2020 KQ11*     156     8      23.20      0.35       48.23     0.00       39.56      0.05       44.55      0.01        3.32      0.00       0.11     0.00  

2020 KQ54     77     7      26.36      0.30       42.47     0.11       41.85      0.61       44.45      0.88        2.89      0.03       0.06     0.02  

2020 KQ55     77     6      26.09      0.21       43.86     0.09       42.39      1.24       45.59      0.67        9.11      0.11       0.07     0.02  

2020 KR11*     1771     15      22.19      2.73       50.09     0.00       42.67      0.00       46.38      0.00        5.89      0.00       0.08     0.00  

2020 KR54     77     7      25.84      0.22       41.06     0.08       40.79      0.27       44.07      0.70        2.04      0.00       0.08     0.01  

2020 KR55     80     7      25.95      0.17       38.89     0.06       38.77      0.11       42.60      0.46       18.30      0.14       0.09     0.01  

2020 KS11*     792     9      26.42      0.30       57.86     0.00       34.92      0.02       99.86      0.19        2.67      0.00       0.65     0.00  

2020 KS53*     370     10      24.74      0.25       45.82     0.00       43.49      0.06       44.69      0.02        5.67      0.00       0.03     0.00  

2020 KS54     79     7      25.92      0.33       40.77     0.09       38.14      1.67       42.01      0.39       18.50      0.23       0.09     0.03  

2020 KS55     80     7      25.74      0.31       42.14     0.06       37.20      0.53       59.75      1.96       24.94      0.18       0.38     0.03  

2020 KT11*     1391     14      24.40      2.61       51.30     0.00       33.74      0.00       42.90      0.00       27.65      0.00       0.21     0.00  

2020 KT53     75     5      25.22      0.15       45.61     0.09       40.01      0.69       43.83      0.90        4.96      0.05       0.09     0.02  

2020 KT54     77     7      26.51      0.24       40.18     0.07       36.56      0.77       40.74      0.78       18.30      0.18       0.10     0.03  

2020 KT55     29     6      24.98      0.36       43.56     0.29       31.26      6.14       62.30     25.04        5.55      0.21       0.46     0.28  

2020 KU11*     162     6      26.57      0.41       50.71     0.04       36.40      0.54       44.31      0.48       15.48      0.07       0.18     0.02  

2020 KU53     77     5      25.25      0.19       47.84     0.11       39.87      1.78       44.19      0.64        2.93      0.01       0.10     0.03  

2020 KU54     75     5      26.39      0.15       38.37     0.10       38.17      0.21       45.02      0.97        6.67      0.08       0.15     0.02  

2020 KU55     76     6      25.83      0.20       44.39     0.09       43.05      0.97       44.65      0.81        5.49      0.06       0.04     0.02  

2020 KV11*     794     9      25.29      0.31       65.94     0.00       35.96      0.00       95.87      0.04        4.64      0.00       0.62     0.00  

2020 KV53     80     7      25.56      0.53       44.87     0.07       40.94      1.87       45.15      0.36        4.37      0.03       0.09     0.04  

2020 KV54     26     5      26.11      0.35       43.37     0.30       29.95      6.55       81.14     58.00        4.80      0.15       0.52     0.29  

2020 KV55     78     9      24.72      0.19       41.78     0.04       38.36      0.38       41.58      0.41       17.43      0.10       0.08     0.02  

2020 KW11*     160     6      25.75      0.30       48.71     0.00       39.93      0.15       45.20      0.04        4.80      0.00       0.12     0.00  

2020 KW53     77     7      25.96      0.21       46.52     0.10       41.38      0.95       44.35      0.77        5.15      0.06       0.07     0.02  

2020 KW54     75     5      26.38      0.20       43.29     0.14       42.85      0.52       182.55     33.88        6.20      0.08       0.76     0.05  

2020 KW55     24     4      25.10      0.21       40.33     0.33       28.74      5.86       65.87     32.94        2.39      0.04       0.50     0.29  

2020 KX11     77     5      26.09      0.56       74.81     0.29        4.11      0.35       110.07     42.70       55.52      4.45       0.96     0.01  

2020 KX53     30     5      25.99      0.55       46.54     0.30       29.69      5.93       66.21     31.36        4.09      0.11       0.50     0.28  

2020 KX54     75     6      26.44      0.13       44.09     0.10       41.49      0.99       43.92      0.97        2.89      0.02       0.05     0.03  

2020 KX55*     422     11      24.21      0.21       35.55     0.00       35.47      0.00       67.01      0.00        2.73      0.00       0.47     0.00  

2020 KY11     3     2      25.35      0.76       55.34     3.91       13.28     12.90       67.72     47.70       25.37     18.44       0.84     0.14  

2020 KY53*     427     9      25.07      0.28       30.06     0.00       23.26      0.01       48.77      0.03        4.72      0.00       0.52     0.00  

2020 KY54     27     4      26.52      0.22       43.17     0.38       30.87      6.53       75.91     42.57        4.12      0.13       0.51     0.28  

2020 KY55     79     7      25.72      0.35       39.20     0.06       38.60      0.44       46.41      0.92        3.46      0.02       0.17     0.03  

2020 KZ53     77     6      25.73      0.30       37.43     0.08       37.38      0.06       59.76      1.50        4.09      0.03       0.37     0.02  

2020 KZ54     27     5      26.16      0.18       48.60     0.42       28.30      6.30       80.48     57.97       10.56      0.58       0.54     0.28  

2020 KZ55     77     8      24.27      0.19       37.16     0.03       36.82      0.18       40.72      0.28       17.13      0.07       0.10     0.01  

2020 LA21     25     4      25.99      0.25       42.82     0.38       29.97      6.38       68.54     33.79        3.56      0.09       0.50     0.28  

2020 LB21     73     3      26.12      0.37       45.35     0.18       16.42      1.28       46.27      3.19        6.89      0.20       0.64     0.05  

2020 MA53     56     5      24.01      0.21       28.90     0.08       28.59      0.14       40.48      1.02        2.11      0.01       0.29     0.02  

2020 MB53     54     4      25.33      0.18       36.50     0.45       36.10      0.52       48.39      5.96       16.71      1.07       0.25     0.10  

2020 MC53     6     3      25.02      0.15       49.66     2.73       15.49     12.62       60.08     36.98        8.54      3.33       0.76     0.21  

2020 MD53     54     4      25.78      0.29       40.12     0.64       37.10      2.57       46.22      6.68        4.76      0.35       0.21     0.10  

2020 ME53     54     4      25.84      0.18       44.07     0.73       34.38      6.24       42.26      5.50        3.59      0.23       0.20     0.10  

2020 MF53     6     3      25.75      0.16       59.62     3.00       20.65     13.92       78.98     54.62       41.13     19.11       0.73     0.18  

2020 MG53     51     3      25.76      0.16       45.26     0.46       29.06      6.45       66.87     32.45        5.03      0.29       0.51     0.28  

2020 MH53     76     8      25.58      0.23       41.72     0.08       40.65      0.85       44.60      0.57        2.45      0.01       0.09     0.01  

2020 MJ53     4     2      26.25      0.44       87.72     4.06       23.91     24.35       114.66     89.28       14.91      9.30       0.84     0.15  

2020 MK53     4     2      25.99      0.49       162.03     5.35       46.54     45.03       212.81     161.42       31.44     20.77       0.83     0.15  

2020 ML53     4     2      26.40      0.21       48.89     3.04       11.82     12.76       54.78     33.74        7.52      4.57       0.86     0.15  

2020 MM53     54     4      26.47      0.18       42.77     1.08       26.48      8.77       38.26      5.41        2.11      0.15       0.33     0.16  

2020 MN53     4     2      26.44      0.37       41.94     2.59        9.67      6.51       49.16     30.42       41.54     20.82       0.79     0.15  

2020 MO53     4     2      26.27      0.19       54.61     3.29       14.53     15.04       74.21     61.77        3.26      0.81       0.85     0.16  

2020 MP53     52     3      26.40      0.48       53.23     2.16        9.73      7.01       39.47      9.31       25.56     11.55       0.74     0.19  

2020 MQ53     56     5      25.98      0.39       56.75     0.59       55.69      1.17       979.04     660.60       73.94      2.99       0.93     0.03  

2020 MY52     54     4      25.53      0.17       39.18     0.75       34.91      3.27       42.21      6.04       22.80      2.49       0.19     0.09  

2020 MZ52     52     3      26.00      0.26       48.67     1.90       23.58     12.11       40.68      7.85       16.57      4.66       0.47     0.22  

2020 PC95     3     2      24.94      2.19       40.53     3.16        5.64      5.42       43.99     27.60       21.20     16.35       0.87     0.12  

2020 PD95     3     2      25.39      1.72       80.08     4.74       22.11     20.41       98.18     66.82       35.50     24.97       0.83     0.14  

2020 PE95     3     2      25.68      0.99       46.09     3.60       11.48     11.69       53.48     34.57       14.53      8.80       0.85     0.15  

2020 PF95     3     2      25.83      0.36       54.65     3.98       15.25     12.58       66.88     44.20       41.92     25.64       0.81     0.14  

2020 PG95     3     2      26.55      0.18       49.24     3.70       10.30     11.27       53.47     31.85        7.89      5.66       0.87     0.14  

2021 LA44     501     7      23.34      0.43       38.40     0.00       31.58      0.13       39.44      0.08        4.79      0.00       0.20     0.00  

2021 LA45     89     3      24.11      0.28       43.22     0.21       40.02      1.78       42.39      1.54        1.64      0.01       0.06     0.03  

2021 LB44     880     9      25.14      0.50       43.58     0.00       37.68      0.04       42.24      0.01        2.72      0.00       0.11     0.00  

2021 LB45     499     5      25.45      0.56       36.60     0.00       36.38      0.11       47.75      0.11        9.04      0.00       0.24     0.00  

2021 LC44     3042     9      25.54      0.58       49.39     0.00       40.61      0.01       46.47      0.00        2.43      0.00       0.13     0.00  

2021 LD44     878     9      24.11      0.44       39.96     0.00       27.48      0.01       39.40      0.01        1.80      0.00       0.30     0.00  

2021 LE44     501     7      25.00      0.55       44.97     0.00       40.70      0.57       44.29      0.10        2.86      0.00       0.08     0.01  

2021 LF44     89     4      25.50      0.29       38.03     0.07       37.11      0.77       41.84      0.78       17.78      0.17       0.11     0.03  

2021 LG44     89     4      25.93      0.24       44.06     0.11       40.86      2.37       43.87      0.50        0.96      0.01       0.07     0.05  

2021 LH44     501     7      25.42      0.50       42.32     0.00       42.19      0.12       44.16      0.02        2.06      0.00       0.04     0.00  

2021 LJ44     501     6      25.01      0.54       44.50     0.00       41.80      0.58       44.25      0.06        3.99      0.00       0.06     0.01  

2021 LK44     89     4      24.54      0.25       43.41     0.05       37.01      0.89       87.31      3.43       16.17      0.10       0.58     0.03  

2021 LL44     89     4      25.45      0.32       42.33     0.08       40.92      1.11       42.86      0.61        1.80      0.01       0.05     0.03  

2021 LM44     501     7      24.95      0.61       47.56     0.00       40.08      0.18       43.96      0.03        1.35      0.00       0.09     0.00  

2021 LN44     501     5      24.51      0.41       32.69     0.00       31.38      0.08       39.33      0.05        5.19      0.00       0.20     0.00  

2021 LO44     468     7      24.94      0.28       35.27     0.00       29.47      0.11       39.26      0.09       18.57      0.00       0.25     0.00  

2021 LP44     468     7      24.48      0.33       35.43     0.00       31.79      0.10       55.42      0.29        4.54      0.00       0.43     0.00  

2021 LQ44     501     6      23.60      0.40       51.86     0.00       43.29      0.07       47.62      0.02       20.89      0.01       0.09     0.00  

2021 LR44     501     6      24.55      0.64       38.69     0.00       32.62      0.23       39.50      0.11        7.35      0.00       0.17     0.01  

2021 LS44     501     7      24.91      0.50       43.83     0.00       42.75      0.01       43.30      0.00        2.18      0.00       0.01     0.00  

2021 LT44     89     3      25.77      0.41       41.62     0.48       39.19      1.82       47.58      5.35        4.40      0.27       0.18     0.11  

2021 LU43     880     6      22.42      0.40       42.60     0.00       42.00      0.02       42.84      0.00        3.13      0.00       0.02     0.00  

2021 LU44     89     3      26.22      0.45       36.53     0.10        4.47      0.18       30.63      1.99       44.54      1.23       0.85     0.02  

2021 LV43     501     7      25.95      0.40       38.65     0.00       33.01      0.35       47.52      0.46        1.68      0.00       0.31     0.01  

2021 LV44     501     5      24.40      0.47       32.80     0.00       32.25      0.10       39.57      0.05        6.85      0.00       0.18     0.00  

2021 LW43     499     6      25.80      0.50       43.83     0.00       43.75      0.04       62.37      0.04       24.05      0.00       0.30     0.00  

2021 LW44     501     6      24.35      0.61       45.23     0.01       41.12      0.74       44.03      0.10        2.43      0.00       0.07     0.02  

2021 LX43     501     7      25.47      0.59       43.69     0.00       42.55      0.50       43.46      0.02        1.99      0.00       0.02     0.01  

2021 LX44     501     6      22.72      0.52       36.56     0.00       34.01      0.20       39.62      0.08        3.18      0.00       0.14     0.01  

2021 LY43     499     6      24.72      0.44       40.96     0.00       39.98      0.25       46.05      0.08        5.27      0.00       0.13     0.01  

2021 LY44     501     6      25.20      0.60       44.93     0.01       35.80      0.77       58.83      1.63        4.13      0.00       0.39     0.03  

2021 LZ43     501     7      23.65      0.49       38.67     0.00       38.56      0.04       43.60      0.02        1.49      0.00       0.12     0.00  

2021 LZ44     501     5      24.59      0.51       42.10     0.00       41.92      0.18       43.83      0.01        1.65      0.00       0.04     0.00  

2021 MA23     491     5      24.44      0.47       41.91     0.00       41.78      0.10       45.62      0.03        2.02      0.00       0.08     0.00  

2021 MW22     468     4      23.50      0.21       30.70     0.00       30.00      0.04       39.54      0.03        7.76      0.00       0.24     0.00  

2021 MX22     493     5      23.65      0.58       30.71     0.00       30.65      0.01       39.52      0.01        6.47      0.00       0.22     0.00  

2021 MY22     491     5      24.37      0.60       30.16     0.00       29.96      0.03       39.51      0.03        3.00      0.00       0.24     0.00  

2021 MZ22     80     3      24.79      0.30       41.27     0.19       38.82      2.26       45.20      0.82        3.67      0.10       0.15     0.06  

2021 NS71     63     2      24.98      0.32       48.91     2.33       13.44      8.72       51.90     26.21       10.05      3.77       0.73     0.20  

2022 LA14     29     4      26.29      0.34       46.62     0.83       28.53      7.54       72.77     40.39        1.60      0.19       0.54     0.27  

2022 LA15     465     6      25.63      0.47       39.19     0.01       38.70      0.36       42.00      0.09       19.64      0.01       0.08     0.01  

2022 LA16     28     3      25.66      0.31       39.55     0.58       29.20      5.93       83.00     58.56        1.24      0.03       0.54     0.27  

2022 LA17     464     5      25.53      0.60       41.80     0.04       39.62      2.01       46.22      0.59        2.16      0.00       0.14     0.05  

2022 LB14     465     6      24.17      0.51       43.21     0.01       39.57      1.72       43.05      0.32        2.26      0.00       0.08     0.05  

2022 LB15     465     6      24.95      0.50       32.01     0.01       31.87      0.12       39.80      0.14        5.48      0.00       0.20     0.01  

2022 LB16     29     4      25.47      0.17       32.90     0.42       25.45      4.75       62.21     31.69        6.21      0.41       0.53     0.26  

2022 LB17     2     2      26.20      0.22       46.55     3.60        9.87     10.83       51.69     32.30        7.86      4.97       0.87     0.14  

2022 LC14     465     6      25.53      0.56       37.36     0.02       33.92      2.11       45.01      1.83        1.76      0.00       0.25     0.08  

2022 LC15     465     5      26.19      0.53       47.63     0.08       36.53      4.89       49.09      3.66        1.77      0.01       0.26     0.17  

2022 LC16     29     4      24.84      0.25       32.71     0.37       25.74      4.29       72.49     52.08        8.12      0.48       0.54     0.26  

2022 LD14     298     5      24.54      0.20       44.79     0.04       38.32      4.32       47.63      2.26        2.29      0.01       0.20     0.14  

2022 LD15     29     3      26.04      0.21       43.69     0.72       29.49      6.84       68.02     32.28        2.76      0.21       0.52     0.27  

2022 LD16     464     4      25.30      0.54       41.96     0.04       37.39      3.74       47.69      2.24        2.27      0.01       0.22     0.12  

2022 LE14     28     3      26.20      0.13       46.19     0.84       31.06      7.44       91.70     72.60        1.10      0.13       0.54     0.27  

2022 LE15     29     3      26.16      0.17       35.20     0.49       29.78      3.84       101.60     72.16       20.58      1.35       0.62     0.21  

2022 LE16     465     6      24.48      0.41       32.99     0.00       30.40      0.30       39.46      0.26       10.87      0.00       0.23     0.01  

2022 LE17     464     5      25.37      0.21       43.72     0.06       33.13      5.94       53.19     12.48        1.24      0.01       0.37     0.26  

2022 LF14     465     5      25.21      0.78       37.15     0.01       36.86      0.25       44.82      0.08       13.10      0.01       0.18     0.01  

2022 LF15     29     3      26.55      0.44       50.10     1.20        4.91      2.90       52.89     40.57       13.14      4.05       0.89     0.09  

2022 LF16     29     4      23.96      0.16       51.22     0.47       29.46      5.97       71.60     36.63       26.63      1.77       0.53     0.26  

2022 LG14     465     5      24.61      0.46       35.34     0.00       23.65      0.03       12792.44     11980.15       11.66      0.00       1.00     0.00  

2022 LG15     465     6      25.15      0.63       43.01     0.03       40.41      2.14       45.05      0.45        1.53      0.00       0.10     0.06  

2022 LG16     29     4      25.37      0.20       40.15     0.51       30.96      5.70       74.42     36.74       28.48      1.96       0.53     0.26  

2022 LH14     29     3      25.71      0.17       44.13     0.68       29.42      6.81       72.37     40.78        5.93      0.55       0.52     0.27  

2022 LH15     465     6      22.77      0.58       43.55     0.01       42.66      0.81       45.59      0.12        1.24      0.00       0.06     0.02  

2022 LH16     29     4      23.55      0.22       41.53     0.34       30.16      6.41       86.36     65.14        0.48      0.00       0.53     0.28  

2022 LJ14     465     5      26.23      0.49       40.30     0.05       35.12      3.51       45.20      2.43        1.97      0.01       0.23     0.13  

2022 LJ15     465     6      24.80      0.56       44.87     0.02       41.65      2.14       45.14      0.31        3.26      0.00       0.08     0.06  

2022 LJ16     29     4      25.21      0.22       31.82     0.39       24.11      4.60       54.83     25.81       12.01      0.80       0.51     0.27  

2022 LK14     465     5      24.10      0.62       48.95     0.01       47.06      0.26       148.87      3.23       32.57      0.01       0.68     0.01  

2022 LK15     465     5      24.43      0.53       42.22     0.02       40.94      1.19       46.34      0.25        3.05      0.00       0.12     0.03  

2022 LK16     29     4      22.28      0.15       37.10     0.30       30.90      4.54       66.31     25.68       31.16      1.33       0.50     0.23  

2022 LL14     465     6      23.51      0.51       48.68     0.01       37.32      0.37       43.33      0.11        9.26      0.01       0.14     0.01  

2022 LL16     464     5      25.20      0.51       34.54     0.03       29.93      3.78       55.55     12.46        0.63      0.00       0.45     0.19  

2022 LM14     735     7      24.39      0.55       46.98     0.00       45.12      0.12       46.76      0.01        2.55      0.00       0.04     0.00  

2022 LM15     465     6      24.83      0.63       42.44     0.02       40.34      1.73       45.98      0.40        3.77      0.00       0.12     0.05  

2022 LM16     465     6      25.97      0.56       50.60     0.06       37.43      2.61       46.61      1.20        3.58      0.01       0.20     0.08  

2022 LN14     465     5      24.61      0.41       41.06     0.02       38.30      2.44       45.23      0.76        1.89      0.00       0.15     0.07  

2022 LN15     29     4      24.56      0.16       34.84     0.37       25.12      5.01       53.70     23.55        5.48      0.34       0.49     0.28  

2022 LN16     465     5      25.30      0.58       33.74     0.00       33.40      0.31       39.54      0.19        6.87      0.01       0.16     0.01  

2022 LO14     464     4      23.67      0.43       40.81     0.05       36.10      3.80       46.09      2.27        0.76      0.00       0.22     0.12  

2022 LO15     29     4      24.89      0.20       36.86     0.39       30.00      4.64       101.84     95.40        5.61      0.32       0.58     0.24  

2022 LO16     465     6      25.60      0.60       35.43     0.01       33.71      0.85       64.77      3.22        3.96      0.01       0.48     0.04  

2022 LO17     735     7      24.53      0.63       46.25     0.00       41.90      0.00       44.08      0.00        2.34      0.00       0.05     0.00  

2022 LP14     29     4      26.12      0.33       42.13     0.64       26.91      6.28       72.71     50.22        0.68      0.02       0.53     0.28  

2022 LP15     465     6      24.88      0.44       32.52     0.01       32.36      0.17       36.44      0.03        4.22      0.00       0.11     0.01  

2022 LP16     28     3      23.32      0.15       42.09     0.35       29.58      6.22       68.49     35.00        2.75      0.12       0.50     0.28  

2022 LP17     28     3      25.87      0.28       46.01     1.18       26.11      8.49       77.29     54.65        2.06      0.18       0.57     0.26  

2022 LQ14     465     5      24.81      0.51       38.35     0.00       34.34      0.78       58.27      1.68        6.95      0.00       0.41     0.03  

2022 LQ15     29     4      26.48      0.33       42.89     0.67       30.99      6.71       92.88     74.45        2.70      0.23       0.54     0.27  

2022 LQ16     29     4      26.43      0.48       76.81     2.04       44.22     16.60       147.52     123.79        7.89      1.47       0.60     0.25  

2022 LR14     465     5      25.24      0.60       45.74     0.03       40.06      1.85       44.30      0.35        4.97      0.01       0.10     0.05  

2022 LR15     28     2      26.32      0.35       74.28     3.90       21.73     18.89       86.64     52.83       41.00     24.45       0.80     0.15  

2022 LR16     465     6      25.60      0.53       43.34     0.02       38.84      2.20       42.95      0.42        4.65      0.01       0.10     0.06  

2022 LS14     465     5      24.85      0.36       48.23     0.05       38.45      3.36       46.86      1.33        0.82      0.00       0.18     0.10  

2022 LS15     29     4      26.36      0.29       47.37     0.79       23.65      6.27       64.99     37.35        0.70      0.00       0.57     0.24  

2022 LS16     29     2      25.59      0.94       87.65     4.20       27.74     21.95       122.65     97.52       44.19     23.82       0.79     0.16  

2022 LT14     28     2      26.66      0.19       64.82     2.88       36.33     10.22       109.47     77.79       58.01     15.72       0.59     0.24  

2022 LT15     735     5      23.80      0.41       42.51     0.00       42.31      0.05       43.41      0.00        5.26      0.00       0.03     0.00  

2022 LT16     28     3      25.79      1.29       72.81     2.64       29.30     17.25       105.16     76.29        7.04      2.01       0.68     0.24  

2022 LU14     28     3      24.86      0.17       35.40     0.44       29.36      4.29       82.34     42.67        0.73      0.00       0.59     0.22  

2022 LU15     28     3      25.96      0.40       38.33     0.58       30.05      5.36       92.75     68.80        6.19      0.51       0.57     0.25  

2022 LU16     28     2      26.12      0.52       71.48     3.65       19.78     18.61       84.48     54.48       28.29     19.05       0.82     0.16  

2022 LV13     29     4      24.95      0.32       42.01     0.48       28.20      6.24       69.11     40.46        1.16      0.08       0.51     0.28  

2022 LV14     28     3      25.50      0.24       45.64     0.71       29.31      6.92       78.83     52.67        0.89      0.05       0.54     0.28  

2022 LV15     465     6      25.85      0.58       44.45     0.04       38.17      3.33       44.87      1.13        3.03      0.01       0.15     0.10  

2022 LV16     465     5      24.51      0.54       44.88     0.05       34.32      5.66       50.96      8.13        0.97      0.01       0.34     0.24  

2022 LW13     465     6      24.80      0.44       31.95     0.00       31.92      0.03       39.43      0.01       11.54      0.00       0.19     0.00  

2022 LW14     465     5      25.64      0.71       49.57     0.07       36.06      3.34       46.62      1.89        1.09      0.00       0.23     0.11  

2022 LW16     465     5      25.84      0.63       42.17     0.05       36.06      3.94       45.02      1.85        1.52      0.00       0.20     0.13  

2022 LX13     29     4      24.32      0.19       45.08     0.45       29.80      6.34       89.05     80.13        1.95      0.12       0.52     0.29  

2022 LX14     29     4      25.11      0.27       44.77     0.51       28.99      6.52       66.82     32.06        1.53      0.10       0.51     0.28  

2022 LX15     28     3      25.97      0.52       82.15     1.56        9.13      2.09       83.13     53.49       58.07     16.12       0.86     0.08  

2022 LX16     464     5      23.79      0.62       46.83     0.02       43.58      1.94       47.17      0.32        4.48      0.01       0.08     0.05  

2022 LY13     465     6      25.21      0.63       34.96     0.00       33.93      0.36       39.63      0.17       14.41      0.01       0.14     0.01  

2022 LY14     465     5      26.17      0.66       39.89     0.04       37.54      2.16       45.87      0.86        1.88      0.00       0.18     0.06  

2022 LY15     27     2      25.64      1.00       76.37     3.80       23.13     19.94       116.38     108.28       31.94     19.89       0.80     0.17  

2022 LY16     464     5      26.05      0.46       46.27     0.07       38.26      3.86       46.14      1.42        1.93      0.01       0.17     0.12  

2022 LZ13     29     4      25.72      0.27       45.15     0.65       27.77      6.62       65.13     31.70        1.07      0.08       0.52     0.27  

2022 LZ14     29     4      25.32      0.25       30.09     0.38       24.44      3.84       72.57     55.73       11.28      0.70       0.56     0.24  

2022 LZ15     465     5      24.71      0.54       40.32     0.02       38.46      1.42       48.05      0.70        3.53      0.01       0.20     0.04  

2022 LZ16     463     4      25.05      0.55       41.46     0.02       39.05      2.14       44.64      0.52        3.39      0.01       0.13     0.06  
 
List of TNOs discovered in our 2020, 2021, and 2022 Subaru-HSC searches.
These discoveries have been submitted to the Minor Planet Center. 
See Section <ref> for details on the orbit fitting. Arc and Epochs denote the span of each arc, and number of different nights on which the object was detected and linked. Mag and σ_mag are an estimate of the source brightness and its uncertainty. a, inc, and e are the fitted orbit semi-major axis, inclination, and eccentricity, and σ_a, σ_inc and σ_e uncertainties in each, respectively. Objects marked with a * were successfully recovered with follow-up observations with HST. Attempted recoveries of 2020 KY11 and 2020 KB56 with HST were unsuccessful. 





§ COLOUR TRANSFORMS FOR THE EB-GRI FILTER 




In Figure <ref> we present estimates of the color transformations between the (g-i) and the wide filter (gri-g) for the SDSS and PS filter systems (for clarity we forgo including “EB” in equations). These estimates were made with with pysynphot package. For each model stellar spectrum included in the pysynphot data files, the (g-i) and (gri-g) colors were estimated using the transmission provided by the manufacturer of the EB-gri filter (see Figure <ref>) and the appropriate filter transmission curves for the other filters. 3rd degree polynomials were fit to the outputs, resulting in the color terms:


    (gri-g)_sdss   = -0.047 -0.475(g-i)_sdss -0.0713(g-i)_sdss^2
    
    (gri-g)_ps   = -0.027 -0.464(g-i)_ps -0.0587(g-i)_ps^2.

For reference, the solar color (g-i)_sdss=0.55 translates to (gri-g)_sdss=0.36.


The above estimate does not include the transmission curve of the rest of the Subaru-HSC system,[<https://www.naoj.org/Projects/HSC/data/HSCWFCTx.dat>] which shows a ∼8% low-to-high variation across the EB-gri bandpass. This variation will somewhat affect the above result; we adopt a generous 0.05 mag uncertainty in the estimated EB-gri magnitude when converting tabulated (g-i) colors for photometric calibration purposes. 



§ SIMULATED ORBIT FITTING 



To assess the accuracy of our orbit-fitting scheme, we performed a set of orbit fits to simulated objects from the 5:2 resonance model discussed in Section <ref>. To do so, we selected those objects that fell within the survey footprint (see Table <ref>), and for each object simulated ephemerides for each date of observation. This large amount of synthetic data was then subsampled to mimic the detections we report and, in particular, the cases with short data arcs (see Table <ref>). After a random synthetic object was selected, its ephemerides were reduced to a random subset that had less than four nights of data spanning no more than 40 days. There were sets of nights that fulfilled these conditions in 2021, 2022, and 2023, although not in 2020. The synthetic positions for the object were then turned into astrometry files for input into the orbit fitter, with randomized astrometry with an assumed uniform 150 milliarcsecond uncertainty (typical for low SNR detections in this data). The short data arcs were then fit, initially with a rough circular orbit to provide an initial condition, and then with the full MCMC estimator (see Section <ref>) until it reached convergence. The results of the recovered orbits from fitting the synthetic data are shown in Figure <ref>. The varied qualities of the simulated fitted orbits exhibit the same behaviour of our real detections, including the numerous short-arc links with matching large uncertainties. Importantly, the orbit fits to short-arc ephemerides tend to be somewhat biased to higher inclinations, in that the fitted values are higher than the true values beyond what the uncertainties reported by the fits would allow.







