The EBLM Project XII. An eccentric, long-period eclipsing binary with a companion near the hydrogen-burning limit

In the hunt for Earth-like exoplanets it is crucial to have reliable host star parameters, as they have a direct impact on the accuracy and precision of the inferred parameters for any discovered exoplanet. For stars with masses between 0.35 and 0.5 ${\rm M_{\odot}}$ an unexplained radius inflation is observed relative to typical stellar models. However, for fully convective objects with a mass below 0.35 ${\rm M_{\odot}}$ it is not known whether this radius inflation is present as there are fewer objects with accurate measurements in this regime. Low-mass eclipsing binaries present a unique opportunity to determine empirical masses and radii for these low-mass stars. Here we report on such a star, EBLM J2114-39\,B. We have used HARPS and FEROS radial-velocities and \textit{TESS} photometry to perform a joint fit of the data, and produce one of the most precise estimates of a very low mass star's parameters. Using a precise and accurate radius for the primary star using {\it Gaia} DR3 data, we determine J2114-39 to be a $M_1 = 0.998 \pm 0.052$~${\rm M_{\odot}}$ primary star hosting a fully convective secondary with mass $M_2~=~0.0986~\pm 0.0038~\,\mathrm{M_{\odot}}$, which lies in a poorly populated region of parameter space. With a radius $R_2 =~0.1275~\pm0.0020~\,\mathrm{R_{\odot}}$, similar to TRAPPIST-1, we see no significant evidence of radius inflation in this system when compared to stellar evolution models. We speculate that stellar models in the regime where radius inflation is observed might be affected by how convective overshooting is treated.


INTRODUCTION
The key to estimating any exoplanet's physical properties is to first determine its host star's parameters accurately.Typically, host star parameters are determined by finding the closest fit between the stellar observables and stellar models (such as Baraffe et al. 2015;Dotter et al. 2008;Fernandes et al. 2019).Thus, any biases or missing physics present in stellar models will lead to biased stellar and planetary parameter estimates.One issue of particular concern is the M-dwarf radius inflation problem where M-dwarfs with masses in the range 0.3 to 0.5 M ⊙ appear to have radii a few percent larger than predicted by typical stellar models (Torres & Ribas 2002;López-Morales 2007;Feiden & Chaboyer 2012).Additionally, M-dwarf effective temperatures appear to fall below predictions (Parsons et al. 2018).It has been claimed that these factors compensate for each other such that the luminosity of M-dwarfs is accurately predicted by stellar models (Torres 2007), but recent results have cast doubt on this claim (Swanye et al. 2023).
Although M-dwarf stars are the most common type of star in our galaxy (Kroupa 2001;Chabrier 2003;Henry et al. 2006), they are not as well-studied as solar-type stars.The Detached Eclipsing Binary Catalogue (DEBCat) 1 archive reports only one set of double-lined eclipsing binaries with either component below 0.2 M ⊙ (Southworth 2015;Casewell et al. 2018), meaning that empirical mass-radius relations, determined from detached, double-lined, eclipsing binaries typically do not include fully convective M-dwarfs (≤ 0.35 M ⊙ ; Torres et al. 2010;Moya et al. 2018).One of the main objectives of the EBLM project (Eclipsing Binaries -Low Mass; Triaud et al. 2013) is to measure accurate masses and radii for a sufficiently large sample of very low-mass M-dwarfs such that an empirical mass-radiusmetallicity relation can be calibrated for the lower main-sequence.
Due to their low masses and low brightnesses, M-dwarf stars can be difficult to study.However, as an eclipsing secondary star they can be investigated using the same methods routinely implemented for exoplanets.For this reason, the EBLM project (Triaud et al. 2013) can address the low number of studied M-dwarfs and begin to fill the poorly populated region of parameter space in the mass-radius diagram for stars with masses < 0.35 M ⊙ .Populating this space is of particular importance for the study of exoplanets as low-mass stars approaching the hydrogen burning limit are ideal candidates for the detection of temperate Earth-sized planets (Nutzman & Charbonneau 2008;Rodler & López-Morales 2014;Gillon et al. 2017;Triaud 2021;Delrez et al. 2022).
Results from the EBLM survey have in the past shown that fully convective M-dwarf radii are correlated with their primary host star's metallicity, with very little evidence for a radius inflation problem (von Boetticher et al. 2019).Radius inflation for high-mass M-dwarfs has been explained as the result of stellar magnetic activity whereby close-in binaries are synchronised, increasing the dynamo effect (e.g.Feiden & Chaboyer 2013).If true, one expects a reduction of inflation with increasing orbital period, but results from the EBLM project show no relation between radius and orbital separation (von Boetticher et al. 2019;Swanye et al. 2023).
Here, we present a new system from the EBLM sample, EBLM J2114-39 (hereafter J2114-39).In the past, we published radial-velocities (RVs) modelled together with eclipses from groundbased telescopes (e.g.Triaud et al. 2013;von Boetticher et al. 2019).More recently, we combined radial-velocities with photometry from two satellites CHEOPS and TESS (Swayne et al. 2021;Sebastian et al. 2023;Swanye et al. 2023).From now on, we will publish the rest of the EBLM systems combining HARPS and/or SOPHIE radial-velocities (as well as any available supplementary data), with photometry from TESS.This is the first such paper in the series.In Section 2 we describe the observations collected.Section 3 details how we estimate parameters for the primary star, and the modelling of all data is in Section 4. Section 5 explains how we ensure we extract the most accurate parameters for the secondary star.We discuss the results and conclude in Section 6.

OBSERVATIONS
For our analysis of J2114-39 (other designations: TIC 159730525, 2MASS J21143061-3918506, Gaia DR3 6583420566747746176) we used photometric data collected by TESS (Transiting Exoplanet Survey Satellite; Ricker et al. 2014), alongside spectra from the 1 https://www.astro.keele.ac.uk/jkt/debcat/HARPS (Mayor et al. 2003) and FEROS (Kaufer et al. 1999) instruments.This is a single lined binary (SB1) with an eclipsing M-dwarf on a  = 44.92471± 0.00025 day period orbiting a G-type primary.The J2114-39 system is located at  = 21 ℎ 14 ′ 30.61 ′′ and  = −39 • 18 ′ 50.6 ′′ on the sky with  mag = 11.1.J2114-39 was not originally included in the EBLM catalogue (and is thus missing from Triaud et al. 2017) as it was identified as a likely long-period transiting gas giant, but FEROS observations quickly revealed the companion to be too massive and therefore non-planetary.The star was then added to our observing programme to be monitored by HARPS to seek circumbinary exoplanets in the context of the BE-BOP survey (Binaries Escorted By Orbiting Planets; Martin et al. 2019;Standing et al. 2023).
The observations of J2114-39 with TESS were accessed using the Mikulski Archive for Space Telescopes (MAST) web service2 and downloaded using the lightkurve software package (Lightkurve Collaboration et al. 2018).The target was observed in Sector 1 where one eclipse is clearly present and Sector 28 which only contains outof-eclipse data.The Sector 28 data is not included as it added no additional information to our fit.The phase-shifted lightcurve can be seen in Fig. 1.Due to the long orbital period compared to the ∼27 day TESS sector length, the secondary eclipse falls outside of the sectors observational windows.This analysis uses the 30-min cadence data products from the TESS-SPOC authors (Caldwell et al. 2020) which was processed with the Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) algorithm (Stumpe et al. 2012;Smith et al. 2012) to remove systematic trends caused by instrumental noise.
We have six recorded radial-velocity measurements of J2114-39 with FEROS, mounted on the 2.2m telescope at La Silla Observatory between 2019 Sep 12 and Nov 2019 Nov 28, and 16 radial-velocity measurements with the HARPS spectrograph, mounted on the ESO 3.6m telescope also at La Silla Observatory between 2020 Nov 18 and 2022 Nov 14.The FEROS observations were performed in the context of the Warm gIaNts with tEss collaboration (WINE, Hobson et al. 2021;Trifonov et al. 2023;Brahm et al. 2023).The radial velocities of the primary star measured from these spectra using cross-correlation against a numerical mask based on the solar spectrum are given in Tables B1 and B2 for the HARPS and FEROS observations respectively, and the phased radial-velocity curve can be seen in Fig. 2.
These six FEROS measurements were acquired with the simultaneous wavelength calibration technique where the second fibre is illuminated by the ThAr lamp to trace instrumental radial velocity drifts during the science exposure.We adopted an exposure time of 900 s, which produced spectra with a typical signal-to-noise ratio of 100 per resolution element.FEROS data were processed with the ceres (Brahm et al. 2017) pipeline, which generates optimally extracted, wavelength calibrated, and drift corrected spectra from the raw science images.Then we computed the radial velocities with the cross-correlation technique by using a G2 binary mask as template, where a gaussian is fitted to the cross-correlation peak.
The HARPS data were reduced by the standard (now public) data reduction software (DRS; Lovis & Pepe 2007).After reduction, the spectra were cross-correlated with a G2 mask.The resulting crosscorrelation function (CCF) is fitted with a Gaussian with its mean as the radial-velocity.

STELLAR PROPERTIES
We used the HARPS spectra to determine the stellar atmospheric parameters.The 16 available spectra for J2114-39 were shifted into the laboratory frame and normalised to a continuum level of 1.0.The spectra were then co-added to achieve a combined signal-tonoise ratio,   = 257.The following atmospheric parameters for the primary star were derived using iSpec (Blanco-Cuaresma et al. 2014;Blanco-Cuaresma 2019): the effective temperature,  eff , surface gravity, log  1 , metallicity, [Fe/H], and microturbulent velocity,  mic .We applied the curve-of-growth equivalent widths method with both the WIDTH (Sbordone et al. 2004) and MOOG (Sneden et al. 2012) radiative transfer codes separately.Solar parameters were input, using those outlined by Blanco-Cuaresma (2019).The SPECTRUM (Gray & Corbally 1994) line list was used in addition to the ATLAS (Kurucz 2005) set of model atmospheres.To reduce systematic error, the final stellar atmospheric parameters displayed in Table 2 are weighted averages of the results from both the WIDTH and MOOG code.The primary's mass and radius are derived by interpolation of MIST isochrones (Dotter 2016;Choi et al. 2016) based on the atmospheric parameters T eff , [Fe/H], together with the parallax and infrared colours as input parameters and using a nested sampling approach implemented in the isochrones package (Morton 2015).The stated uncertainty is the average of the errors calculated from the 16 and 84 percentiles of the resulting distribution.All values are reported in Table 2.

GLOBAL MODELLING
To analyse the data from J2114-39 we use allesfitter (Günther & Daylan 2021, 2019) to perform a simultaneous Markov Chain Monte Carlo (MCMC) modelling of the primary and secondary stars.allesfitter amalgamates many useful python packages frequently used in modelling stellar or planetary systems.At the heart of allesfitter is ellc (Maxted 2016) which generates lightcurves and celerite (Foreman-Mackey et al. 2017) for any modelling with Gaussian processes (GPs).To obtain most likely parameters, two types of samplers are used; a nested sampler (dynesty; Speagle 2020) and an affine-invariant MCMC sampler (emcee; Foreman-Mackey et al. 2013;Goodman & Weare 2010).Here, as in other EBLM papers we use MCMC, since it is less computationally intensive and we have no requirement for model comparison.The nested sampling approach gives similar results to those from the MCMC.All results presented in the tables and in Section 6 are from a MCMC fit.
Orbital eccentricity, , in allesfitter is reparameterised with respect to the argument of periastron, , as  c =

√
cos  and  s = √  sin  (as in Triaud et al. 2011).For limb darkening, we apply the quadratic law with  eff , log  1 and [Fe/H] as stellar properties, and adopt the output from PyLDTk (Parviainen & Aigrain 2015) (using the PHOENIX stellar atmosphere library; Husser et al. 2013).Limb darkening coefficients are reparameterised to  1 and  2 following Kipping (2013) for the fitting process.All priors are described in Table 1.
From a visual inspection it is clear that there is a level of variability in the TESS photometry, likely caused by stellar activity.We first tried to apply polynomial and spline functions, but none returned a good fit.To account for intrinsic variability of the star and instrumental noise we fit a GP to the out-of-eclipse photometry assuming a Matérn 3/2 kernel which detrends short and long term fluctuations.We fit for two hyper-parameters; the amplitude scale  and the length scale , as required for this choice of kernel.For the radial-velocity data we add a jitter term (as part of ln  RV ) in quadrature with the instrumental white noise error to account for any stellar variability effects, as well as normalised scaling parameters (included in ln  phot ) for the photometry.Other parameters within the model include; the ratio of radii ( 2 / 1 ), inverse scaled semi-major axis (( 1 +  2 )/), cosine of the orbital inclination (cos ), eclipse epoch ( 0 ), orbital period (), radial-velocity semi-amplitude for the primary star ( 1 ), and constant baseline offsets for the radial-velocity instruments (Δ RV ).
Prior to executing the MCMC sampling, we first perform a visual inspection of the photometric and spectroscopic data to ascertain suitable initial values and priors for the walkers such that the initial fit produces an acceptable and sensible result.When initial values are far from the solutions, walkers take significantly longer to converge, increasing the computational time needlessly for entirely comparable results.For similar reasons, we perform a short MCMC on the radialvelocity data only, before carrying out any joint sampling with the photometry.This allows for more informed estimates of the orbital period and eccentricity of the system.Once satisfied with the fit, the prior space is widened to ensure biases were not introduced to the fit by limiting its explorable parameter space.
For the final analysis the MCMC has 60 walkers with 30,000 steps, including 8000 burn-in steps which we discard.The final solutions, considered to be most probable, are determined by the median value of each fitted parameter's posterior distribution where the quoted upper and lower estimates of uncertainty represent the 16/84 percentiles.We consider the chains to have converged to a solution as they were at least 100 times the autocorrelation length for each parameter as well as the chains visually converging in trial plots.
All fitted parameters included in the fit are presented in Table 1 alongside further details on the type of priors used and the selected bounds for the sampling.Physical parameters, derived from the fitted parameters are found in Table 2.

PARAMETER DERIVATION
To extract accurate parameters on the secondary star, we need accurate parameters on the primary.Thanks to Gaia DR3 (Gaia Collaboration et al. 2016) the stellar radius ( 1 ) is better estimated than the mass ( 1 ) that traditionally relies on applying stellar models.However, with our lightcurve modelling we can constrain  1 using the stellar density.From Kepler's law, stellar density ( 1 ) is defined as where  is the gravitational constant, and  is the semi-major axis (Triaud et al. 2013).
In planetary cases the second term ( 2 / 3 1 ) is typically considered negligible.However, for binary stars, it must be included, since its contribution is no longer insignificant.For eclipsing binaries we are able to determine the primary and secondary masses independent of any assumption of  1 .
We start by rearranging Eqn.1 for  2 and define the total mass term as (2) Then using the observables fit via the method described in Section 4 from each step in the MCMC we calculate the mass function,   Table 1.Priors used in the fit and resulting output for the fitted parameters.Uniform priors are marked as U(lower limit, upper limit) and normal priors marked as N(mean, standard deviation).All times are given in BJD UTC .The quoted upper limits correspond to the 99.7 percentile of the posterior distribution.Transformed limb darkening ;  2;TESS N (0.520898, 0.5) 0.47 +0.27   −0.20 (Hilditch 2001),

Parameter
As the mass function can also be expressed as we can substitute in our expression from the stellar density equation to calculate a secondary mass Finally with a value for  2 we can now use Eqn.1 to calculate the primary mass as From the eclipse signal analysis, / 1 is a modelled variable along with all other observables in Eqn 1 and 3 which allows for the stellar masses to be calculated.To ensure the error in the the primary radius,  1 (the only assumed value from outside the modelling), is reflected in the final mass uncertainty we assign a normal Gaussian distribution and draw random samples to be worked through in the same manner as all other variables.

RESULTS AND CONCLUSION
After the global modelling and parameter derivation, we determine a mass function   = 0.0008127 ± 0.0000028 M ⊙ , indicating a low-mass companion.Then we calculate the secondary's surface gravity log  2 = 5.222 ± 0.0135, which is found in absolute terms (Southworth et al. 2007) from parameters fit with the MCMC.This value alone confirms the secondary is a dense star, near the bottom of the main-sequence.Assuming parameters for the primary star (as in Section 3;  1 = 1.283 ± 0.012 R ⊙ and Section 5:  1 = 0.998 ± 0.052 M ⊙ ), we find the secondary stellar companion to be a late M-dwarf with a mass  2 = 0.0993 ± 0.0033 M ⊙ and a radius  2 = 0.1250 ±0.0016 R ⊙ and thus a mass ratio  = 0.0995±0.0019with the primary.If we recalculate the surface gravity of the primary star using the obtained mass and radius, we find that the value is consistent within 1 sigma with the spectroscopic surface gravity, regardless of the choice of primary mass value.From the fit, we also find the system has a clearly detected orbital eccentricity  = 0.56733 ± 0.00027, which is not unusual for a binary with this period (e.g.Triaud et al. 2017).
Without following the parameter derivation procedure described in Section 5, we would have calculated the mass of the secondary star through equation 4 where we would have assumed the denominator to be  2 1 as commonly done when  2 ≪  1 .By using the primary mass value calculated from isochrones, we would then have found the mass of the secondary companion to be  2 = 0.0999 ± 0.0045 M ⊙ instead of  2 = 0.0993 ± 0.0033 M ⊙ .Although these results are consistent within error margins (as can be seen in Fig. 3) this rep-resents a ∼ 1.0% systematic bias towards higher masses (and would therefore bias the overall population towards less inflated objects).Appendix A details why the fractional uncertainty on mass is smaller for the secondary star than for the primary.
EBLM J2114-39b has a mass approaching the hydrogen burning limit, which is a key region for empirical calibrations of the massradius relation.As illustrated in the mass-radius plot (Fig. 3), our target is consistent within 1  with the Baraffe et al. (2015) models continuing a trend observed in von Boetticher et al. ( 2019) and Swanye et al. (2023), where fully convective secondary stars do not appear to be systematically inflated whereas higher mass M-dwarfs are.Several mechanisms have been proposed to explain radius inflation, however these explanations do not predict why fully convective stars would be different and not show inflation.Here we speculate that this might have to do with convective overshooting, which is known to affect modelled stellar parameters (e.g.Baraffe et al. 2023).For a fully convective star such as EBLM J2114-39 B, convective overshooting is essentially irrelevant since there is no penetration of convective materials into a radiative interior.However slightly more massive stars, with a small radiative core, might be very sensitive to how convective overshooting is modelled, since any plume of convective material into the small radiative interior would likely affect a greater fraction of the core, and might produce large differences in stellar evolution.
Another aspect that makes J2114-39B stand out in this context is its long orbital period.Typically other objects of a similar mass are found in binary systems with periods < 10 days.As such EBLM J2114-39B is likely not much influenced by its primary.RB acknowledges support from FONDECYT project 11200751 and additional support from ANID -Millennium Science Initiative -ICN12_009.
A    The lower panel highlights differences between the objects and the 5 Gyr isochrone.Black and grey data points from compendium (and cited papers) in Triaud et al. (2020), with additional points from Morales et al. (2009).

APPENDIX A: ERROR CALCULATION FOR STELLAR MASSES
The fractional error on the secondary star is smaller than the fractional error on the primary mass.This may be surprising, but is actually a direct result of the fit to the RVs being very good and the different powers in the mass ratio of Eqn 4. From Eqn 5, we can write that Using the rules of error propagation, we can show that The error in  can be calculated from Eqn 6:

Figure 1 .Figure 2 .
Figure 1.Top panel: Raw TESS data in purple with 20 random samples of the GP from the MCMC modelling the additional noise in blue.Middle panel: The phase-shifted eclipse of EBLM J2114-39, data shown in purple after the signal modelled by the GP has been removed.Twenty random samples from the MCMC run are displayed in teal colour.Bottom panel: The residuals to the best fit model.

Figure 3 .
Figure 3. Mass-radius diagram with logarithmic mass axis, showing published single-lined systems (SB1) in grey, double-lined binaries (SB2) in black and EBLM J2114-39 results overlapping in turquoise and coral colour.The 1 Gyr and 5 Gyr isochrones are shown in purple and teal respectively.The lower panel highlights differences between the objects and the 5 Gyr isochrone.Black and grey data points from compendium (and cited papers) inTriaud et al. (2020), with additional points fromMorales et al. (2009).
This research is supported from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement n • 803193/BEBOP), and by a Leverhulme Trust Research Project Grant (n • RPG-2018-418).

Table 2 .
Parameters of EBLM J2114-39. (* ) from Sec. 3, and ( * * ) from GAIA DR3.Primary mass values given from the MIST isochrones  1,iso , as well as from the method described in Section 5.The quoted upper limits correspond to the 99.7 percentile of the posterior distribution.

Table B1 .
HARPS observations of J2114-39.All times are given in BJD UTC .

Table B2 .
FEROS observations of J2114-39.All times are given in BJD UTC .