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.