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Elsa Ducrot

Publications and source records attributed to Elsa Ducrot.

At least 55 records · Page 3Linked to original sources

Thermal emission from the Earth-sized exoplanet TRAPPIST-1 b using JWST

The TRAPPIST-1 system is remarkable for its seven planets that are similar in size, mass, density, and stellar heating to the rocky planets Venus, Earth, and Mars in our own Solar System (Gillon et al. 2017). All TRAPPIST-1 planets have been observed with the transmission spectroscopy technique using the Hubble or Spitzer Space Telescopes, but no atmospheric features have been detected or strongly constrained (Ducrot et al. 2018; de Wit et al. 2018; Zhang et al. 2018; Garcia et al. 2022). TRAPPIST-1 b is the closest planet to the system's M dwarf star, and it receives 4 times as much irradiation as Earth receives from the Sun. This relatively large amount of stellar heating suggests that its thermal emission may be measurable. Here we present photometric secondary eclipse observations of the Earth-sized TRAPPIST-1 b exoplanet using the F1500W filter of the MIRI instrument on JWST. We detect the secondary eclipse in each of five separate observations with 8.7-$σ$ confidence when all data are combined. These measurements are most consistent with the re-radiation of the TRAPPIST-1 star's incident flux from only the dayside hemisphere of the planet. The most straightforward interpretation is that there is little or no planetary atmosphere redistributing radiation from the host star and also no detectable atmospheric absorption from carbon dioxide (CO$_2$) or other species.

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A reflective, metal-rich atmosphere for GJ 1214b from its JWST phase curve

There are no planets intermediate in size between Earth and Neptune in our Solar System, yet these objects are found around a substantial fraction of other stars. Population statistics show that close-in planets in this size range bifurcate into two classes based on their radii. It is hypothesized that the group with larger radii (referred to as "sub-Neptunes") is distinguished by having hydrogen-dominated atmospheres that are a few percent of the total mass of the planets. GJ 1214b is an archetype sub-Neptune that has been observed extensively using transmission spectroscopy to test this hypothesis. However, the measured spectra are featureless, and thus inconclusive, due to the presence of high-altitude aerosols in the planet's atmosphere. Here we report a spectroscopic thermal phase curve of GJ 1214b obtained with JWST in the mid-infrared. The dayside and nightside spectra (average brightness temperatures of 553 $\pm$ 9 and 437 $\pm$ 19 K, respectively) each show >3$σ$ evidence of absorption features, with H$_2$O as the most likely cause in both. The measured global thermal emission implies that GJ 1214b's Bond albedo is 0.51 $\pm$ 0.06. Comparison between the spectroscopic phase curve data and three-dimensional models of GJ 1214b reveal a planet with a high metallicity atmosphere blanketed by a thick and highly reflective layer of clouds or haze.

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A 1.55 R$_{\oplus}$ habitable-zone planet hosted by TOI-715, an M4 star near the ecliptic South Pole

A new generation of observatories is enabling detailed study of exoplanetary atmospheres and the diversity of alien climates, allowing us to seek evidence for extraterrestrial biological and geological processes. Now is therefore the time to identify the most unique planets to be characterised with these instruments. In this context, we report on the discovery and validation of TOI-715 b, a $R_{\rm b}=1.55\pm 0.06\rm R_{\oplus}$ planet orbiting its nearby ($42$ pc) M4 host (TOI-715/TIC 271971130) with a period $P_{\rm b} = 19.288004_{-0.000024}^{+0.000027}$ days. TOI-715 b was first identified by TESS and validated using ground-based photometry, high-resolution imaging and statistical validation. The planet's orbital period combined with the stellar effective temperature $T_{\rm eff}=3075\pm75~\rm K$ give this planet an instellation $S_{\rm b} = 0.67_{-0.20}^{+0.15}~\rm S_\oplus$, placing it within the most conservative definitions of the habitable zone for rocky planets. TOI-715 b's radius falls exactly between two measured locations of the M-dwarf radius valley; characterising its mass and composition will help understand the true nature of the radius valley for low-mass stars. We demonstrate TOI-715 b is amenable for characterisation using precise radial velocities and transmission spectroscopy. Additionally, we reveal a second candidate planet in the system, TIC 271971130.02, with a potential orbital period of $P_{02} = 25.60712_{-0.00036}^{+0.00031}$ days and a radius of $R_{02} = 1.066\pm0.092\,\rm R_{\oplus}$, just inside the outer boundary of the habitable zone, and near a 4:3 orbital period commensurability. Should this second planet be confirmed, it would represent the smallest habitable zone planet discovered by TESS to date.

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The Effect of Stellar Contamination on Low-resolution Transmission Spectroscopy: Needs Identified by NASA's Exoplanet Exploration Program Study Analysis Group 21

Study Analysis Group 21 (SAG21) of NASA's Exoplanet Exploration Program Analysis Group (ExoPAG) was organized to study the effect of stellar contamination on space-based transmission spectroscopy, a method for studying exoplanetary atmospheres by measuring the wavelength-dependent radius of a planet as it transits its star. Transmission spectroscopy relies on a precise understanding of the spectrum of the star being occulted. However, stars are not homogeneous, constant light sources but have temporally evolving photospheres and chromospheres with inhomogeneities like spots, faculae, plages, granules, and flares. This SAG brought together an interdisciplinary team of more than 100 scientists, with observers and theorists from the heliophysics, stellar astrophysics, planetary science, and exoplanetary atmosphere research communities, to study the current research needs that can be addressed in this context to make the most of transit studies from current NASA facilities like HST and JWST. The analysis produced 14 findings, which fall into three Science Themes encompassing (1) how the Sun is used as our best laboratory to calibrate our understanding of stellar heterogeneities ("The Sun as the Stellar Benchmark"), (2) how stars other than the Sun extend our knowledge of heterogeneities ("Surface Heterogeneities of Other Stars") and (3) how to incorporate information gathered for the Sun and other stars into transit studies ("Mapping Stellar Knowledge to Transit Studies"). In this invited review, we largely reproduce the final report of SAG21 as a contribution to the peer-reviewed literature.

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A First Look at the JWST MIRI/LRS Phase Curve of WASP-43b

We observed a full-orbit phase curve of the hot Jupiter WASP-43b with MIRI/LRS as part of the Transiting Exoplanet Community Early Release Science Program. Here we report preliminary findings for the instrument performance from the team's MIRI Working Group. Overall we find that MIRI's performance for phase curve observations is excellent, with a few minor caveats. The key takeaways for Cycle 2 planning with MIRI/LRS are: (1) long-duration observations (> 24 hours) have now been successfully executed; (2) for phase curves, we recommend including a one-hour burn-in period prior to taking science data to mitigate the effects of the ramp systematic; and (3) we do not yet recommend partial phase curve observations. In addition, we also find that: the position of the spectrum on the detector is stable to within 0.03 pixels over the full 26.5-hour observation; the light curves typically show a systematic downward ramp that is strongest for the first 30 minutes, but continues to decay for hours; from 10.6-11.8 microns, the ramp effect has remarkably different behavior, possibly due to a different illumination history for the affected region of the detector; after trimming the integrations most affected by the initial ramps and correcting the remaining systematics with analytic models, we obtain residuals to the light-curve fits that are typically within 25% of the photon noise limit for 0.5-micron spectroscopic bins; non-linearity correction is not a significant source of additional noise for WASP-43, though it may be an issue for brighter targets; the gain value of 5.5 electrons/DN currently on CRDS and JDox is known to be incorrect, and the current best estimate for the gain is approximately 3.1 electrons/DN; new reference files for the JWST calibration pipeline reflecting these findings are under development at STScI.

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SPECULOOS Northern Observatory: searching for red worlds in the northern skies

SPECULOOS is a ground-based transit survey consisting of six identical 1-m robotic telescopes. The immediate goal of the project is to detect temperate terrestrial planets transiting nearby ultracool dwarfs (late M-dwarf stars and brown dwarfs), which could be amenable for atmospheric research with the next generation of telescopes. Here, we report the developments of the northern counterpart of the project - SPECULOOS Northern Observatory, and present its performance during the first three years of operations from mid-2019 to mid-2022. Currently, the observatory consists of one telescope, which is named Artemis. The Artemis telescope demonstrates remarkable photometric precision, allowing it to be ready to detect new transiting terrestrial exoplanets around ultracool dwarfs. Over the period of the first three years after the installation, we observed 96 objects from the SPECULOOS target list for 6000 hours with a typical photometric precision of $0.5\%$, and reaching a precision of $0.2\%$ for relatively bright non-variable targets with a typical exposure time of 25 sec. Our weather downtime (clouds, high wind speed, high humidity, precipitation and/or high concentration of dust particles in the air) over the period of three years was 30% of overall night time. Our actual downtime is 40% because of additional time loss associated with technical problems.

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TESS discovery of a sub-Neptune orbiting a mid-M dwarf TOI-2136

We present the discovery of TOI-2136b, a sub-Neptune planet transiting every 7.85 days a nearby M4.5V-type star, identified through photometric measurements from the TESS mission. The host star is located $33$ pc away with a radius of $R_{\ast} = 0.34\pm0.02\ R_{\odot}$, a mass of $0.34\pm0.02\ M_{\odot}$ and an effective temperature of $\rm 3342\pm100\ K$. We estimate its stellar rotation period to be $75\pm5$ days based on archival long-term photometry. We confirm and characterize the planet based on a series of ground-based multi-wavelength photometry, high-angular-resolution imaging observations, and precise radial velocities from CFHT/SPIRou. Our joint analysis reveals that the planet has a radius of $2.19\pm0.17\ R_{\oplus}$, and a mass measurement of $6.4\pm2.4\ M_{\oplus}$. The mass and radius of TOI2136b is consistent with a broad range of compositions, from water-ice to gas-dominated worlds. TOI-2136b falls close to the radius valley for low-mass stars predicted by the thermally driven atmospheric mass loss models, making it an interesting target for future studies of its interior structure and atmospheric properties.

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Complex Modulation of Rapidly Rotating Young M Dwarfs: Adding Pieces to the Puzzle

New sets of young M dwarfs with complex, sharp-peaked, and strictly periodic photometric modulations have recently been discovered with Kepler/K2 (scallop shells) and TESS (complex rotators). All are part of star-forming associations, are distinct from other variable stars, and likely belong to a unified class. Suggested hypotheses include star spots, accreting dust disks, co-rotating clouds of material, magnetically constrained material, spots and misaligned disks, and pulsations. Here, we provide a comprehensive overview and add new observational constraints with TESS and SPECULOOS Southern Observatory (SSO) photometry. We scrutinize all hypotheses from three new angles: (1) we investigate each scenario's occurrence rates via young star catalogs; (2) we study the features' longevity using over one year of combined data; and (3) we probe the expected color dependency with multi-color photometry. In this process, we also revisit the stellar parameters accounting for activity effects, study stellar flares as activity indicators over year-long time scales, and develop toy models to simulate typical morphologies. We rule out most hypotheses, and only (i) co-rotating material clouds and (ii) spots and misaligned disks remain feasible - with caveats. For (i), co-rotating dust might not be stable enough, while co-rotating gas alone likely cannot cause percentage-scale features; and (ii) would require misaligned disks around most young M dwarfs. We thus suggest a unified hypothesis, a superposition of large-amplitude spot modulations and sharp transits of co-rotating gas clouds. While the complex rotators' mystery remains, these new observations add valuable pieces to the puzzle going forward.

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prose: A Python framework for modular astronomical images processing

To reduce and analyze astronomical images, astronomers can rely on a wide range of libraries providing low-level implementations of legacy algorithms. However, combining these routines into robust and functional pipelines requires a major effort which often ends up in instrument-specific and poorly maintainable tools, yielding products that suffer from a low-level of reproducibility and portability. In this context, we present prose, a Python framework to build modular and maintainable image processing pipelines. Built for astronomy, it is instrument-agnostic and allows the construction of pipelines using a wide range of building blocks, pre-implemented or user-defined. With this architecture, our package provides basic tools to deal with common tasks such as automatic reduction and photometric extraction. To demonstrate its potential, we use its default photometric pipeline to process 26 TESS candidates follow-up observations and compare their products to the ones obtained with AstroImageJ, the reference software for such endeavors. We show that prose produces light curves with lower white and red noise while requiring less user interactions and offering richer functionalities for reporting.

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NGTS clusters survey -- III: A low-mass eclipsing binary in the Blanco 1 open cluster spanning the fully convective boundary

We present the discovery and characterisation of an eclipsing binary identified by the Next Generation Transit Survey in the $\sim$115 Myr old Blanco 1 open cluster. NGTS J0002-29 comprises three M dwarfs: a short-period binary and a companion in a wider orbit. This system is the first well-characterised, low-mass eclipsing binary in Blanco 1. With a low mass ratio, a tertiary companion and binary components that straddle the fully convective boundary, it is an important benchmark system, and one of only two well-characterised, low-mass eclipsing binaries at this age. We simultaneously model light curves from NGTS, TESS, SPECULOOS and SAAO, radial velocities from VLT/UVES and Keck/HIRES, and the system's spectral energy distribution. We find that the binary components travel on circular orbits around their common centre of mass in $P_{\rm orb} = 1.09800524 \pm 0.00000038$ days, and have masses $M_{\rm pri}=0.3978\pm 0.0033$ M$_{\odot}$ and $M_{\rm sec}=0.2245\pm 0.0018$ M$_{\odot}$, radii $R_{\rm pri}=0.4037\pm 0.0048$ R$_{\odot}$ and $R_{\rm sec}=0.2759\pm 0.0055$ R$_{\odot}$, and effective temperatures $T_{\rm pri}=3372\,^{+44}_{-37}$ K and $T_{\rm sec}=3231\,^{+38}_{-31}$ K. We compare these properties to the predictions of seven stellar evolution models, which typically imply an inflated primary. The system joins a list of 19 well-characterised, low-mass, sub-Gyr, stellar-mass eclipsing binaries, which constitute some of the strongest observational tests of stellar evolution theory at low masses and young ages.

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A transit timing variation observed for the long-period extremely low density exoplanet HIP 41378f

HIP 41378 f is a temperate $9.2\pm0.1 R_{\oplus}$ planet with period of 542.08 days and an extremely low density of $0.09\pm0.02$ g cm$^{-3}$. It transits the bright star HIP 41378 (V=8.93), making it an exciting target for atmospheric characterization including transmission spectroscopy. HIP 41378 was monitored photometrically between the dates of 2019 November 19 and November 28. We detected a transit of HIP 41378 f with NGTS, just the third transit ever detected for this planet, which confirms the orbital period. This is also the first ground-based detection of a transit of HIP 41378 f. Additional ground-based photometry was also obtained and used to constrain the time of the transit. The transit was measured to occur 1.50 hours earlier than predicted. We use an analytic transit timing variation (TTV) model to show the observed TTV can be explained by interactions between HIP 41378 e and HIP 41378 f. Using our TTV model, we predict the epochs of future transits of HIP 41378 f, with derived transit centres of T$_{C,4} = 2459355.087^{+0.031}_{-0.022}$ (May 2021) and T$_{C,5} = 2459897.078^{+0.114}_{-0.060}$ (Nov 2022).

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Refining the transit timing and photometric analysis of TRAPPIST-1: Masses, radii, densities, dynamics, and ephemerides

We have collected transit times for the TRAPPIST-1 system with the Spitzer Space Telescope over four years. We add to these ground-based, HST and K2 transit time measurements, and revisit an N-body dynamical analysis of the seven-planet system using our complete set of times from which we refine the mass ratios of the planets to the star. We next carry out a photodynamical analysis of the Spitzer light curves to derive the density of the host star and the planet densities. We find that all seven planets' densities may be described with a single rocky mass-radius relation which is depleted in iron relative to Earth, with Fe 21 wt% versus 32 wt% for Earth, and otherwise Earth-like in composition. Alternatively, the planets may have an Earth-like composition, but enhanced in light elements, such as a surface water layer or a core-free structure with oxidized iron in the mantle. We measure planet masses to a precision of 3-5%, equivalent to a radial-velocity (RV) precision of 2.5 cm/sec, or two orders of magnitude more precise than current RV capabilities. We find the eccentricities of the planets are very small; the orbits are extremely coplanar; and the system is stable on 10 Myr timescales. We find evidence of infrequent timing outliers which we cannot explain with an eighth planet; we instead account for the outliers using a robust likelihood function. We forecast JWST timing observations, and speculate on possible implications of the planet densities for the formation, migration and evolution of the planet system.

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$π$ Earth: a 3.14-day Earth-sized Planet from $\textit{K2}$'s Kitchen Served Warm by the SPECULOOS Team

We report on the discovery of a transiting Earth-sized (0.95$R_\oplus$) planet around an M3.5 dwarf star at 57$\,$pc, K2-315b. The planet has a period of $\sim$3.14 days, i.e. ${\sim}π$, with an instellation of 7.45$\,$S$_{\oplus}$. The detection was made using publicly available data from $\textit{K2}$'s Campaign 15. We observed three additional transits with SPECULOOS Southern and Northern Observatories, and a stellar spectrum from Keck/HIRES, which allowed us to validate the planetary nature of the signal. The confirmed planet is well suited for comparative terrestrial exoplanetology. While exoplanets transiting ultracool dwarfs present the best opportunity for atmospheric studies of terrestrial exoplanets with the $\textit{James Webb Space Telescope}$, those orbiting mid-M dwarfs within 100$\,$pc such as K2-315b will become increasingly accessible with the next generation of observatories.

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TRAPPIST-1: Global Results of the Spitzer Exploration Science Program {\it Red Worlds}

With more than 1000 hours of observation from Feb 2016 to Oct 2019, the Spitzer Exploration Program Red Worlds (ID: 13067, 13175 and 14223) exclusively targeted TRAPPIST-1, a nearby (12pc) ultracool dwarf star orbited by seven transiting Earth-sized planets, all well-suited for a detailed atmospheric characterization with the upcoming JWST. In this paper, we present the global results of the project. We analyzed 88 new transits and combined them with 100 previously analyzed transits, for a total of 188 transits observed at 3.6 or 4.5 $μ$m. We also analyzed 29 occultations (secondary eclipses) of planet b and eight occultations of planet c observed at 4.5 $μ$m to constrain the brightness temperatures of their daysides. We identify several orphan transit-like structures in our Spitzer photometry, but all of them are of low significance. We do not confirm any new transiting planets. We estimate for TRAPPIST-1 transit depth measurements mean noise floors of $\sim$35 and 25 ppm in channels 1 and 2 of Spitzer/IRAC, respectively. most of this noise floor is of instrumental origins and due to the large inter-pixel inhomogeneity of IRAC InSb arrays, and that the much better interpixel homogeneity of JWST instruments should result in noise floors as low as 10ppm, which is low enough to enable the atmospheric characterization of the planets by transit transmission spectroscopy. We construct updated broadband transmission spectra for all seven planets which show consistent transit depths between the two Spitzer channels. We identify and model five distinct high energy flares in the whole dataset, and discuss our results in the context of habitability. Finally, we fail to detect occultation signals of planets b and c at 4.5 $μ$m, and can only set 3$σ$ upper limits on their dayside brightness temperatures (611K for b 586K for c).

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An Eclipsing Substellar Binary in a Young Triple System discovered by SPECULOOS

Mass, radius, and age are three of the most fundamental parameters for celestial objects, enabling studies of the evolution and internal physics of stars, brown dwarfs, and planets. Brown dwarfs are hydrogen-rich objects that are unable to sustain core fusion reactions but are supported from collapse by electron degeneracy pressure. As they age, brown dwarfs cool, reducing their radius and luminosity. Young exoplanets follow a similar behaviour. Brown dwarf evolutionary models are relied upon to infer the masses, radii and ages of these objects. Similar models are used to infer the mass and radius of directly imaged exoplanets. Unfortunately, only sparse empirical mass, radius and age measurements are currently available, and the models remain mostly unvalidated. Double-line eclipsing binaries provide the most direct route for the absolute determination of the masses and radii of stars. Here, we report the SPECULOOS discovery of 2M1510A, a nearby, eclipsing, double-line brown dwarf binary, with a widely-separated tertiary brown dwarf companion. We also find that the system is a member of the $45\pm5$ Myr-old moving group, Argus. The system's age matches those of currently known directly-imaged exoplanets. 2M1510A provides an opportunity to benchmark evolutionary models of brown dwarfs and young planets. We find that widely-used evolutionary models do reproduce the mass, radius and age of the binary components remarkably well, but overestimate the luminosity by up to 0.65 magnitudes, which could result in underestimated photometric masses for directly-imaged exoplanets and young field brown dwarfs by 20 to 35%.

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The L 98-59 System: Three Transiting, Terrestrial-Sized Planets Orbiting a Nearby M-dwarf

We report the Transiting Exoplanet Survey Satellite (TESS) discovery of three terrestrial-sized planets transiting L 98-59 (TOI-175, TIC 307210830) -- a bright M dwarf at a distance of 10.6 pc. Using the Gaia-measured distance and broad-band photometry we find that the host star is an M3 dwarf. Combined with the TESS transits from three sectors, the corresponding stellar parameters yield planet radii ranging from 0.8REarth to 1.6REarth. All three planets have short orbital periods, ranging from 2.25 to 7.45 days with the outer pair just wide of a 2:1 period resonance. Diagnostic tests produced by the TESS Data Validation Report and the vetting package DAVE rule out common false positive sources. These analyses, along with dedicated follow-up and the multiplicity of the system, lend confidence that the observed signals are caused by planets transiting L 98-59 and are not associated with other sources in the field. The L 98-59 system is interesting for a number of reasons: the host star is bright (V = 11.7 mag, K = 7.1 mag) and the planets are prime targets for further follow-up observations including precision radial-velocity mass measurements and future transit spectroscopy with the James Webb Space Telescope; the near resonant configuration makes the system a laboratory to study planetary system dynamical evolution; and three planets of relatively similar size in the same system present an opportunity to study terrestrial planets where other variables (age, metallicity, etc.) can be held constant. L 98-59 will be observed in 4 more TESS sectors, which will provide a wealth of information on the three currently known planets and have the potential to reveal additional planets in the system.

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The EBLM Project V. Physical properties of ten fully convective, very-low-mass stars

Measurements of the physical properties of stars at the lower end of the main sequence are scarce. In this context we report masses, radii and surface gravities of ten very-low-mass stars in eclipsing binary systems, with orbital periods of the order of several days. The objects probe the stellar mass-radius relation in the fully convective regime, $M_\star \lesssim 0.35$ M$_\odot$, down to the hydrogen burning mass-limit, $M_{\mathrm{HB}} \sim 0.07$ M$_\odot$. The stars were detected by the WASP survey for transiting extra-solar planets, as low-mass, eclipsing companions orbiting more massive, F- and G-type host stars. We use eclipse observations of the host stars (TRAPPIST, Leonhard Euler, SPECULOOS telescopes), and radial velocities of the host stars (CORALIE spectrograph), to determine physical properties of the low-mass companions. Companion surface gravities are derived from the eclipse and orbital parameters of each system. Spectroscopic measurements of the host star effective temperature and metallicity are used to infer the host star mass and age from stellar evolution models. Masses and radii of the low-mass companions are then derived from the eclipse and orbital parameters of each system. The objects are compared to stellar evolution models for low-mass stars, to test for an effect of the stellar metallicity and orbital period on the radius of low-mass stars in close binary systems. Measurements are in good agreement with stellar models; an inflation of the radii of low-mass stars with respect to model predictions is limited to 1.6 $\pm$ 1.2% in the fully convective regime. The sample of ten objects indicates a scaling of the radius of low-mass stars with the host star metallicity. No correlation between stellar radii and orbital periods of the binary systems is determined. A combined analysis with comparable objects from the literature is consistent with this result.

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Near-resonance in a system of sub-Neptunes from TESS

We report the Transiting Exoplanet Survey Satellite ($TESS$) detection of a multi-planet system orbiting the $V=10.9$ K0 dwarf TOI 125. We find evidence for up to five planets, with varying confidence. Three high signal-to-noise transit signals correspond to sub-Neptune-sized planets ($2.76$, $2.79$, and $2.94\ R_{\oplus}$), and we statistically validate the planetary nature of the two inner planets ($P_b = 4.65$ days, $P_c = 9.15$ days). With only two transits observed, we report the outer object ($P_{.03} = 19.98$ days) as a high signal-to-noise ratio planet candidate. We also detect a candidate transiting super-Earth ($1.4\ R_{\oplus}$) with an orbital period of only $12.7$ hours and a candidate Neptune-sized planet ($4.2\ R_{\oplus}$) with a period of $13.28$ days, both at low signal-to-noise. This system is amenable to mass determination via radial velocities and transit timing variations, and provides an opportunity to study planets of similar size while controlling for age and environment. The ratio of orbital periods between TOI 125 b and c ($P_c/P_b = 1.97$) is slightly smaller than an exact 2:1 commensurability and is atypical of multiple planet systems from $Kepler$, which show a preference for period ratios just $wide$ of first-order period ratios. A dynamical analysis refines the allowed parameter space through stability arguments and suggests that, despite the nearly commensurate periods, the system is unlikely to be in resonance.

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