Fundamental properties of late-type stars in eclipsing binaries

Citation: Morales, J.C.; Ribas, I.; Giménez, Á.; Baroch, D. Fundamental Properties of Late-Type Stars in Eclipsing Binaries. Galaxies 2022,10, 98. https://doi.org/ 10.3390/galaxies10050098 Academic Editors: Robert E. Wilson and Walter Van Hamme Received: 22 July 2022 Accepted: 14 September 2022 Published: 17 September 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). galaxies Article Fundamental Properties of Late-Type Stars in Eclipsing Binaries Juan Carlos Morales 1,2,* , Ignasi Ribas 1,2 , Álvaro Giménez 3and David Baroch 1,2 1Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB, C/ Can Magrans s/n, Bellaterra, E-08193 Barcelona, Spain; [email protected] (I.R.); bar[email protected] (D.B.) 2Institut d’Estudis Espacials de Catalunya (IEEC), C/ Gran Capità 2-4, E-08034 Barcelona, Spain 3Centro de Astrobiología (CSIC-INTA), Ctra. Ajalvir, Km 4, Torrejón de Ardoz, E-28850 Madrid, Spain; [email protected] *Correspondence: [email protected] Abstract: Evidence from the analysis of eclipsing binary systems revealed that late-type stars are larger and cooler than predicted by models, and that this is probably caused by stellar magnetic activity. In this work, we revisit this problem taking into account the advancements in the last decade. We provide and updated a list of 32 eclipsing binary or multiple systems, including at least one star with a mass . 0.7M  and with mass and radius measured to an accuracy better than 3%. The comparison with stellar structure and evolution theoretical models reveals an overall discrepancy of about 7% and − 4% for the radius and effective temperature, respectively, and that it may be larger than previously found below the full convection boundary. Furthermore, the hypothesis of stellar activity is reinforced by the comparison of different systems with similar components. Further eclipsing binaries with accurately determined masses and radii, and with estimated activity levels, as well as the implementation of magnetic activity in theoretical models will help to improve our knowledge of low-mass stars, which are prime targets for exoplanet surveys. Keywords: binaries: eclipsing; binaries: spectroscopic; stars: late-type; stars: fundamental parameters 1. Introduction Several astronomical surveys aim at late-type stars because they are excellent targets to look for Earth-like planets within the habitable zone. Due to their small mass, the radial velocity signature imprinted by planets around them are larger, and the habitable zone is closer, therefore their transit probability is also larger (see e.g., [ 1 – 3 ]). However, their structure is not fully understood yet due to the difficulty to accurately measure their fundamental properties, from which those of the exoplanet they host depend. Double-lined eclipsing binary systems (hereafter, DLEBs) were revealed to be a unique opportunity to derive the masses and radii of stars in a fundamental way, almost independent of any assumptions. This is achieved by analyzing both the photometric light curves showing the mutual eclipses of the components, which provide the relative size of the stars and the orbital inclination, and the radial velocities, which provide the minimum masses and absolute dimensions of the orbit. Such studies can yield the masses and radii of the component stars with a precision better than 3% that allows us to thoroughly test the predictions of stellar models (see e.g., [ 4 – 6 ]). Furthermore, due to their accuracy, the masses and radii of DLEBs are also used to calibrate empirical mass–radius relationships from which the properties of single stars are then computed (e.g., [7,8]). However, several studies of DLEBs composed of late-K and M dwarfs found discrepancies between the results of the analysis of observations and the theoretical structure and evolution models. The general conclusion was that the observed radius of low-mass stars in eclipsing binaries is ∼ 5–10% larger than predicted by models, while effective temperatures are ∼ 10% cooler (see e.g., [ 9 – 12 ]). Interestingly, the luminosities are in agreement. Stellar magnetic activity was suggested as the probable cause of such discrepancies [ 10 , 13 , 14 ]. Galaxies 2022,10, 98. https://doi.org/10.3390/galaxies10050098 https://www.mdpi.com/journal/galaxies Galaxies 2022,10, 98 2 of 14 Known DLEBs with low-mass components are typically close systems with orbital periods of few days, with components tidally locked and rotating synchronously [ 15 ]. This fast rotation induces intense magnetic fields, which generate cold spots in the surface of the stars. Studies on theoretical models found that the presence of such spots significantly affects the structure of stars [ 16 – 20 ] and models including such effects have been recently published [ 21 ]. Moreover, some studies found that stellar spots can also cause some bias on the determination of the fundamental properties from light curve analyses [22,23]. In this paper, we compile a list of the up-to-date known late-type stars in eclipsing binary and multiple systems with well-determined masses and radii with uncertainties below the 3% level to revisit the comparison between the observations and updated lowmass stellar models. We discuss the present state of this long-standing issue and provide some ideas for future studies. 2. Sample of Low-Mass Systems Torres et al. [ 5 ] compiled a list of stars in DLEBs with reliable analyses providing the masses and radii of the components with uncertainties below the 3% limit. Only four systems with late-K and M dwarf components were included in this list, namely the well-known classical cases of GU Boo [ 13 ], YY Gem [ 10 ], CU Cnc [ 11 ], and CM Dra [ 24 ]. In the recent years, the large number of ground and space-based surveys to look for exoplanets yielded also several eclipsing binary systems as a byproduct. In Table 1, we list the properties of low-mass stars with accurately measured masses and radii. We have limited our sample to DLEBs with at least one component with a mass below ∼ 0.7 M  to focus our attention on the late-K and M dwarf domain, which is particularly interesting nowadays for exoplanet surveys. In order to perform a meaningful comparison with stellar structure models, we also restricted the sample to those systems with main sequence components and mass and radius uncertainties ≤ 3%. Only the reported uncertainties in the publications were considered, we did not apply any further selection based on the published analysis. This resulted in 28 DLEBs. The extremely precise and continuous monitoring of space telescopes also allows to measure the masses and radii of the components in multiple systems showing mutual eclipses. This has already been achieved either for triple stellar systems such as KOI126 [ 25 ], or binary systems with circumbinary planets such as Kepler-16 [ 26 ], Kepler-453 [ 27 ], and Kepler-47 [ 28 ]. The timing of the eclipses and transits between the different components becomes a very useful tool to accurately compute their masses without the necessity of spectroscopic orbits. On the other hand, the radius is determined from the eclipses and transit shapes as usual, with the additional constraint of the dynamical effects of the multiple systems. Those systems are also included in our sample. The 32 systems listed in Table 1contain a total of 52 late-type dwarfs with M.0.7M with fundamental properties determined to be a precision better than 3%. Most of the targets have short orbital periods; thus, their orbital and rotation periods are synchronized; thus, they are fast rotators and hence magnetically active. Indeed, studies about the tidal evolution of binary stars demonstrate that close binaries are synchronized and circularized relatively fast within a few million years; thereby, systems with orbital periods below ∼ 20 days may be synchronized by ∼ 1 Ga (see e.g., [ 15 , 29 – 31 ]). Hints of stellar activity are reported for all of the systems, except for EBLM J0113+31 [ 32 ]. The most common signature is out-of-eclipse variability due to spots of a few percent in flux; however, other activity indicators such as chromospheric emission in the H α and CaH&K lines, enhanced X-ray emission, or flaring activity are reported for several systems (e.g., [ 10 , 12 , 24 ]). As mentioned, stellar activity may be driven by the fast rotation of the components, which is typically synchronized with the orbital motion for short period systems. For this reason, we included in Table 1the rotation velocity ( vsync ) of the components as computed from the radius of each star, and the orbital period and eccentricity of each system assuming periastron pseudo-synchronization [ 33 ]. Interestingly, large photometric surveys are also providing long-period systems that in principle should be less prone to be magnetically active. Galaxies 2022,10, 98 3 of 14 Actually, rotation rates slightly faster and significantly slower than expected from synchronization are reported for the long period systems Kepler-453 [ 27 ] and LSPM J1112+7626 [ 34 ], respectively, which may indicate that they have not yet reached spin-orbit synchronization. Table 1. List of DLEBs and multiple systems with a least one stellar component with a mass . 0.7M  and mass and radius uncertainties below 3%. Name Com. P M R Teff vsync [Fe/H] Ref. [d] [M] [R][K] [km s−1][dex] Eclipsing binary systems NGTS J052218.2-250710.4 A1.7477 0.17391 ±0.00126 0.2045 ±0.0048 2995 ±95 5.93 ±0.13 · · · [35] B 0.17418 ±0.00126 0.2168 ±0.00475 2997 ±84 6.29 ±0.14 CM Dra A1.2684 0.23102 ±0.00089 0.2534 ±0.0019 3130 ±70 10.22 ±0.08 −0.3 [5] B 0.2141 ±0.0008 0.2398 ±0.0018 3120 ±70 9.67 ±0.08 LP 661-13 A4.7044 0.30795 ±0.00084 0.3226 ±0.0033 · · · 3.47 ±0.04 −0.07 ±0.07 [36] B 0.194 ±0.00034 0.2174 ±0.0023 · · · 2.338 ±0.024 LSPMJ1112+7626 A41.0324 0.3951 ±0.0022 0.3815 ±0.003 3130 ±165 0.789 ±0.006 · · · [34] B 0.2749 ±0.0011 0.2999 ±0.0044 3015 ±166 0.620 ±0.009 NGTS 0002-29 A1.098 0.3978 ±0.0033 0.4037 ±0.0048 3372 ±40 18.62 ±0.22 0.04 ±0.04 [37] B 0.2245 ±0.0018 0.2759 ±0.0055 3231 ±34 12.72 ±0.24 CU Cnc A2.7715 0.4349 ±0.0012 0.4323 ±0.0055 3160 ±150 7.89 ±0.10 0.0 [5] B 0.3992 ±0.0009 0.3916 ±0.0094 3125 ±150 7.15 ±0.17 HAT-TR-318-007 A3.344 0.448 ±0.011 0.4548 ±0.0036 3190 ±100 7.07 ±0.06 0.298 ±0.08 [38] B 0.2721 ±0.0042 0.2913 ±0.0024 3100 ±100 4.53 ±0.04 MG1-2056316 A1.7228 0.469 ±0.002 0.441 ±0.002 3460 ±180 12.95 ±0.06 · · · [39] B 0.382 ±0.001 0.374 ±0.002 3320 ±180 10.98 ±0.06 MG1-646680 A1.6375 0.499 ±0.002 0.457 ±0.006 3730 ±20 14.12 ±0.18 · · · [39] B 0.443 ±0.002 0.427 ±0.006 3630 ±20 13.19 ±0.18 MG1-78457 A1.5862 0.527 ±0.002 0.505 ±0.008 3330 ±60 16.11 ±0.25 −1.55 ±0.05 [39] B 0.491 ±0.001 0.471 ±0.007 3270 ±60 15.02 ±0.22 NSVS01031772 A0.3681 0.53 ±0.014 0.559 ±0.014 3750 ±150 76.8 ±1.9 · · · [40] B 0.514 ±0.013 0.518 ±0.013 3600 ±150 71.2 ±1.8 MG1-116309 A0.8271 0.567 ±0.002 0.552 ±0.013 3920 ±80 33.76 ±0.76 −1.19 ±0.04 [39] B 0.532 ±0.002 0.532 ±0.008 3810 ±80 32.54 ±0.50 MG1-506664 A1.5484 0.584 ±0.002 0.56 ±0.005 3730 ±90 18.30 ±0.16 · · · [39] B 0.544 ±0.002 0.513 ±0.008 3610 ±90 16.76 ±0.26 YY Gem A&B 0.8143 0.5992 ±0.0047 0.6194 ±0.0057 3820 ±100 38.48 ±0.36 0.0 [5] GU Boo A0.4887 0.6101 ±0.0064 0.627 ±0.016 3920 ±130 64.9 ±1.7 · · · [5] B 0.5995 ±0.0064 0.624 ±0.016 3810 ±130 64.6 ±1.7 HIP 41431 A2.9300 0.625 ±0.010 0.588 ±0.012 4043 ±60 10.55 ±0.22 · · · [41] B 0.614 ±0.012 0.576 ±0.012 3986 ±60 10.33 ±0.22 KIC 9821078 A8.4294 0.67 ±0.01 0.662 ±0.001 · · · 4.233 ±0.007 · · · [42] B 0.52 ±0.01 0.478 ±0.001 · · · 3.056 ±0.007 BD-15 2429 A1.5285 0.7029 ±0.0045 0.694 ±0.011 4230 ±200 23.20 ±0.35 · · · [43] B 0.6872 ±0.0049 0.699 ±0.014 4080 ±200 23.37 ±0.46 M55 V54 A9.2692 0.726 ±0.015 1.006 ±0.009 6246 ±71 7.24 ±0.06 −1.86 ±0.15 [44] B 0.555 ±0.008 0.528 ±0.005 5020 ±95 3.80 ±0.04 RXJ0239.1-1028 A2.0719 0.73 ±0.009 0.741 ±0.004 4645 ±20 18.09 ±0.10 · · · [40] B 0.693 ±0.006 0.703 ±0.002 4275 ±15 17.17 ±0.05 NGC2204-S892 A0.4518 0.733 ±0.005 0.719 ±0.014 4200 ±100 80.5 ±1.6 · · · [45] B 0.662 ±0.005 0.68 ±0.017 3940 ±110 76.2 ±1.9 UCAC3 127-192903 A2.293 0.8035 ±0.0086 1.147 ±0.01 6088 ±108 25.31 ±0.22 −1.18 ±0.02 [46] B 0.605 ±0.0044 0.611 ±0.0092 4812 ±125 13.48 ±0.21 KIC 6131659 A17.5278 0.922 ±0.007 0.88 ±0.0028 5789 ±50 2.540 ±0.008 −0.23 ±0.2 [47] B 0.685 ±0.005 0.6395 ±0.0061 4609 ±32 1.846 ±0.018 EPIC 247605441 A1.6534 0.934 ±0.017 1.058 ±0.023 5668 ±71 32.3 ±0.7 −0.26 ±0.26 [48] B 0.409 ±0.005 0.408 ±0.009 3590 ±100 12.48 ±0.28 ASAS J065134-2111.5 A8.2196 0.956 ±0.012 0.997 ±0.004 5500 ±100 6.277 ±0.024 0.09 ±0.13 [49] B 0.674 ±0.005 0.69 ±0.007 3970 ±110 4.34 ±0.04 IM Vir A1.3086 0.981 ±0.012 1.061 ±0.016 5570 ±100 41.01962 ±0.64 −0.3 [50] B 0.6644 ±0.0048 0.681 ±0.013 4250 ±130 26.33 ±0.51 V530 Ori A6.1108 1.0038 ±0.0066 0.98 ±0.013 5890 ±100 9.68 ±0.13 −0.12 ±0.08 [51] B 0.5955 ±0.0022 0.5873 ±0.0067 3880 ±120 5.80 ±0.06 EBLM J0113+31 A14.2768 1.029 ±0.025 1.417 ±0.014 3.787 ±0.003 10.00 ±0.10 −0.3 ±0.1 [32] B 0.197 ±0.003 0.215 ±0.002 3.528 ±0.005 1.517 ±0.014 Eclipsing triple systems KOI-126 A 33.9214 1.347 ±0.032 2.20254 ±0.0098 5875 ±100 · · · 0.15 ±0.08 [25]Ba 1.7671 0.2413 ±0.003 0.2543 ±0.0014 · · · 7.62 ±0.04 Bb 0.2127 ±0.0026 0.2318 ±0.0013 · · · 6.94 ±0.04 Circumbinary planets Kepler 16 A41.0792 0.6897 ±0.0035 0.6489 ±0.0013 4450 ±150 1.1166 ±0.0022 −0.3 ±0.2 [26] B 0.20255 ±0.00066 0.22623 ±0.00059 · · · 0.3893 ±0.0010 Kepler-453 A27.322 0.944 ±0.01 0.833 ±0.011 5527 ±100 1.715 ±0.022 0.09 ±0.1 [27] B 0.1951 ±0.002 0.215 ±0.0014 3226 ±100 0.4428 ±0.0029 Kepler-47 A7.4484 0.957 ±0.014 0.936 ±0.005 5636 ±100 6.738 ±0.037 −0.25 ±0.08 [28] B 0.342 ±0.003 0.338 ±0.002 3357 ±100 2.433 ±0.015 3. Models vs. Observations The standard theoretical models of stellar structure and evolution typically do not include the effect of intense magnetic fields, or the appearance of surface spots, which are associated to stellar activity. However, several studies analyzed their effect in the past years. For instance, Mullan and MacDonald [ 16 , 52 ] introduced the strength of the magnetic field in stellar structure models as an additional parameter changing the criterion of the onset of convection. Strong magnetic fields reduce the efficiency of convection and change the structure of the star, producing stars with a larger radius and cooler effective temperatures in line with DLEBs. In contrast, luminosities are also changed. Chabrier et al. [ 17 ] also studied the impact of magnetic activity in the Lyon stellar structure models [ 53 ]. They suggested that the inhibition of convection could be mod- Galaxies 2022,10, 98 4 of 14 eled changing the mixing length parameter ( α ) of the convection. Moreover, they also introduced a second parameter ( β ) that accounts for the effect of spots in the photosphere. They demonstrated that the presence of spots, which block the outgoing flux, does significantly change the structure of low-mass stars over all the mass range. On the other hand, the inhibition of convection is only important above the fully convective boundary. A subsequent comparison of these models with well-known DLEBs revealed that they would reproduce the radius of the components when a spot coverage of about 35% of the photosphere is assumed ( β= 0.17) after removing a possible ∼ 3% systematic bias on the radii determined from light curves due to the variability caused by stellar spots [ 22 ]. Magnetic activity effects were also inspected using the Dartmouth stellar models [ 18 ] by Feiden and Chaboyer [ 19 , 20 ]. They concluded that magnetic fields of few kG can explain the inflated radii of partially convective stars [ 19 ], but not for fully convective stars [ 20 ]. The conclusions of all these works confirm the hypothesis that stellar activity and magnetic fields play a key role in the stellar structure of low-mass stars, also from the theoretical point of view. More recently, a new set of stellar models, dubbed SPOTS [ 21 ], including the presence of stellar spots have been published. A parameter accounting for the filling factor of spots is implemented in these models [ 54 ]. The surface inhomogeneities block the outgoing flux of the stars so their structure is altered. They reproduce the results of the work by Chabrier et al. [ 17 ]. The filling factor ( f ) is computed assuming a phenomenological effective temperature difference between the spots and the photosphere [ 55 ]. This means that the actual fraction of photoshpere covered by spots may be different to f ; it depends on the real temperature contrast of the surface features with respect to the photosphere. Figure 1shows the mass–radius ( M−R ) and mass–effective temperature ( M−Teff ) relationships for low-mass stars from theoretical models compared with the measured values of the stars with M.0.7Min our sample of eclipsing and circumbinary systems in Table1. Here, we compare the observations only with models publicly available: the more recent version of the Lyon stellar structure models (http://perso.ens-lyon.fr/isabelle. baraffe/BHAC15dir/, accesed on 15 November 2021), with updated molecular lines and newly calibrated convection parameters (hereafter BHAC15, [ 56 ]), the Dartmouth stellar models (http://stellar.dartmouth.edu/models/, accesed on 15 November 2021) [ 57 ] for 1 and 5 Ga, and the 1 Ga SPOTS models (https://doi.org/10.5281/zenodo.3593339, accesed on 15 November 2021) with different values of the filling factor parameter [ 21 ]. Differences between these sets of stellar models not including the effect of photospheric spots are small except for the larger masses, where the modeling of the convective layer of the star plays a significant role on the stellar structure (see e.g., [ 17 ]). Taking as reference the 1Ga BHAC15 isochrone, the mean radii differences between observations and models are 6.6% (standard deviation, σ= 4.2%) for the 52 low-mass stars with available radii. In the case of the effective temperature, the mean difference with respect to the models is − 2.5% ( σ= 7.6%) for the 45 stars with measured temperatures; however, the determination of the absolute effective temperature from spectrophotometric data is less constrained and depends on the determination of the absolute temperature of one of the components and the temperature ratio derived from multi-band light curves of the eclipsing binary systems; therefore, its comparison is not as straightforward as for the radii. Actually, our computed mean Teff discrepancy is significantly influenced by the low-mass components of M55 V54 [ 44 ] and UCAC3 127-192903 [ 46 ], which are suggested to be members of the globular clusters M55 and M4, respectively. The proposed old age and poor metallicity of the systems, and possible biases in the estimation of the primary Teff from B−V colors may be responsible for the large difference with respect to theoretical stellar models (see [ 46 ] for further details). Removing these systems from the sample, the mean temperature difference is −3.9% (σ=4.4%). Galaxies 2022,10, 98 5 of 14 Figure 1. M−R ( top ) and M−Teff ( bottom ) relationships for the late-type stars listed in Table1. BHAC15, Darmouth, and SPOTS stellar models with different parameters are plotted as labeled. Inset plots show a zoom in into the lower mass domain. For the sample of stars in Table 1, the radii discrepancy between fully convective stars, those with a mass . 0.35M  , and partially convective ones is not as different as in previous studies. They are 5.8% (15 stars) and 7.0% (37 stars), respectively. It is clear from the figure that this is due to some of the new very-low-mass systems showing larger discrepancies than the long-ago well-known systems such as CMDra, for instance. This stresses the need to still increase the number of well-characterized DLEBs with late-M type components. As expected, the discrepancies between the observations and models are reduced when the effect of photospheric spots is taken into account as in the case of the SPOTS theoretical models with f> 0. Stars are predicted to be larger and cooler, so that the mean radii discrepancies are reduced to 4.1% and 0.5% for models with filling factor 0.17 and Galaxies 2022,10, 98 6 of 14 0.51, respectively. This is illustrated in Figure 2, where the relative difference between the measured radius of DLEB components and those predicted by the 1 Ga BHAC15 model ( ∆R/Rmodel ) is plotted. About 2/3 of the stars with M. 0.7M  in our sample could be explained assuming filling factors between 0 and 0.51; however, there are still few systems that show larger discrepancies, which may point to a larger effect of magnetic activity on their component stars. This makes crucial the implementation of such effects in stellar theoretical models as suggested in several works [17,19,22]. Figure 2. Relative radius difference with respect to the BHAC15 1Ga stellar model as a function of the mass of the star. Red lines correspond to SPOTS models as labeled in Figure 1. The horizontal blue dot-dashed line is shown as a reference of agreement with the BHAC15 1Ga model. 4. Discussion Different issues are still making it difficult to obtain a general picture of the problem between the observations of DLEBs and theoretical stellar structure models. For instance, the fundamental properties of stars depend on their metallicity and age. Both these properties are not as fundamentally constrained as the masses and radii from light and radial velocity curves. Typically, the age and metallicity are estimated from the membership of the system to moving groups or stellar clusters, or in the case of metallicity from spectroscopic analysis. However, few of the low-mass stars with accurate masses and radii listed in Table 1have reported values. This is the case of the classical systems CMDra, whose age is estimated from the cooling sequence of a white dwarf companion [ 24 ], YYGem and CUCnc, in which cases their age is assumed from the membership to the Castor moving group [ 10 , 11 ]; however, assuming the corresponding ages and metallicities, the radii discrepancies with respect to the Lyon stellar models are reduced by less than 1% in the case of CMDra and they are even larger for YYGem and CU Cnc. This clearly demonstrates that stellar activity is playing a key role in the structure of such stars. The case of NGTS0002-29 is also remarkable. The system is a member of the Blanco1 open cluster [ 37 ] with and estimated age of 90 − 150Ma. Comparing with the BHAC15 models at this age range, radius discrepancies are reduced to ∼ 5.5% and ∼ 3.2% for components A and B, respectively. Still, there is a significant difference, although in this case, it cannot yet be excluded that the system is still in the last stages of the pre-main sequence phase. On the other hand, the ages reported from isochrone fitting for the systems M55V54 [ 44 ] and UCA3127-192903 [ 46 ] Galaxies 2022,10, 98 7 of 14 show inconsistencies between the components of each binary. This points towards and additional effect causing the inflation of their radii. The main conclusion is that it is crucial to know the age and the composition of the DLEB and multiple systems in order to thoroughly test the stellar structure models. Efforts are on the way to characterize binary stars in clusters for that purpose. For instance in a series of papers, Torres et al. analyzed several DLEBs pertaining to the Ruprecht147 open cluster [ 58 – 61 ]. None of the systems fulfill the mass or accuracy threshold to be in Table 1. However, it is worth mentioning that, interestingly, the large mass components are generally well fitted by an isochrone at the expected age of the cluster, while the lower mass components still show radius discrepancies that can be attributed to the effect of stellar activity (see Torres et al. in this volume for more details [62]). Throughout the paper we have also mentioned that there is broad evidence that stellar magnetic activity plays a crucial role in the stellar structure and evolution. As stated in Section 3 , several authors have taken its effects on stellar structure into account [16,17,19,21,62] by implementing the impact of photospheric spots, of strong magnetic fields, or both in the theoretical models; however, this adds a new free parameter to the models that need to be tested against observations. However, it is still uncertain how the stellar activity can be quantified and how it correlates with the radii inflation. DLEBs with active components typically show photometric variability caused by stellar spots with changing amplitude due to the evolution of spots. Frequently, they also show flaring activity on the light curves and also X-ray emission. Actually, López-Morales et al. [ 14 ] obtained a linear relation between the X-ray to bolometric luminosity ratio ( LX/LBol ) and the radius inflation of DLEBs ( ∆R/Rmodel ) that could be used as a correction to the observed radius values. Taking advantage of the second ROSAT all-sky survey [ 63 ] and the recent parallax determinations from Gaia DR3 [ 64 , 65 ] we revisited such calibrations. Table2lists the binary systems with X-ray data. Bolometric luminosities were computed from the radius and effective temperature of each star. ROSAT data were converted to X-ray luminosities following the prescriptions in Schmitt et al. [ 66 ] and the distance reported in the Gaia archive. We estimated the luminosity corresponding to each star by weighting by v2 sync [ 14 ]. This is equivalent to assuming a weighting according to R2 , i.e., that the X-ray luminosity of each component depends on the surface of the star, which can be used as a proxy for the surface of the chromosphere. Table 2. List of DLEBs with X-ray data and parallax determination from Gaia. Name vX HR com. LX/LBol [mas] [cts−1] [×10−4] CM Dra 67.288 ±0.034 0.210 ±0.017 −0.344 ±0.061 A 8.9 ±1.2 B 9.0 ±1.2 NGTS 0002-29 13.80 ±0.26 · · · · · · A 14.8 ±1.6 B 17.6 ±2.1 CU Cnc 60.060 ±0.036 0.732 ±0.050 −0.084 ±0.054 A 16.3 ±4.1 B 17.1 ±4.6 NSVS01031772 16.572 ±0.018 0.076 ±0.013 0.016 ±0.17 A 7.0 ±2.2 B 8.2 ±2.5 YY Gem 66.310 ±0.023 3.716 ±0.091 −0.152 ±0.020 A&B 13.5 ±1.7 GU Boo 6.187 ±0.011 0.039 ±0.012 0.16 ±0.32 A 17.3 ±7.0 B 19.4 ±8.1 BD-15 2429 23.58 ±0.14 0.495 ±0.042 −0.249 ±0.068 A 7.0 ±1.7 B 8.0 ±2.3 RXJ0239.1-1028 8.144 ±0.016 0.036 ±0.014 1.00 ±0.59 A 5.2 ±2.4 B 7.3 ±3.3 EPIC 247605441 9.99 ±0.30 0.129 ±0.019 −0.06 ±0.10 A 2.7 ±0.5 B 16.7 ±3.8 ASAS J065134-2111.5 10.819 ±0.021 0.049 ±0.012 −0.35 ±0.21 A 0.69 ±0.22 B 2.56 ±0.86 IM Vir 11.136 ±0.017 0.261 ±0.031 −0.232 ±0.089 A 3.38 ±0.56 B 10.0 ±2.0 V530 Ori 9.763 ±0.018 0.0252 ±0.0092 −0.41 ±0.26 A 0.36 ±0.17 B 1.90 ±0.94 1LX/LBol is estimated assuming log LX=29.29ergs−1[37]. Galaxies 2022,10, 98 8 of 14 The comparison between ∆R/Rmodel and LX/LBol is illustrated in Figure 3. ∆R/Rmodel is again computed taking the 1Ga BHAC15 model as reference. The correlation factor of these dataset, ρ = 0.12 (p-value = 0.64), points towards uncorrelation. Actually, the slope of the best fit (blue solid line) is not significantly different from zero. From this result, we conclude that a linear relation is not supported by our sample of late-type dwarfs in DLEBs. We stress here that this may be caused by the fact that the low-mass stars in our sample show saturated levels of X-ray emission. Indeed, the X-ray to bolometric luminosity ratio of the stars illustrated in this figure is in the range − 3.6 <log(LX/LBol)<− 2.7, well in the saturated regime. The updated X-ray data and distances are also responsible for some of the differences with the previous work [ 40 ]. For instance, the distance to the CUCnc system provided by Gaia is about 20% farther than previously reported [ 11 ], therefore its X-ray emission is stronger. Figure 3. Relative radius difference with respect to the BHAC15 stellar models at 1Ga as a function of the X-ray to bolometric luminosity ratio. The solid-blue line depicts the best linear fit to the data, whose slope is not significantly different from zero. The red dot-dashed line illustrates the slope reported in López-Morales et al. [ 14 ] fitted to our data for comparison. The horizontal blue dot-dashed line is shown as a reference of agreement with the BHAC15 1Ga model. Certainly, there is a well known correlation between X-ray emission (and activity level) and the rotation of stars [ 67 ] suggesting that the emission of late-type stars is saturated for rotation periods . 10days. This may be the case for most of the systems in Table 1, whose orbital periods are below few days and the rotation of their components may be synchronized to the orbital motion as mentioned in the previous section. DLEBs with long period orbits, whose components may not be rotating synchronously, may be less magnetically active and more consistent with models. This was statistically studied by Coughlin et al. [ 68 ] using eclipsing binaries detected in the Kepler mission field-of-view. From the analysis of only their light curves, they concluded that the mean radius discrepancy between the DLEBs measurements and theoretical models diminishes for stars in systems with longer orbital periods. For comparison, the panels in Figure 4illustrate the ∆R/Rmodel with respect to the orbital period of the system and vsync . It is evident from these plots that the DLEBs with vsync above ∼ 20km s −1 show larger discrepancies on average, although they are a few. On the contrary, the radii of DLEBs components with vsync . 4kms −1 and orbital periods above ∼ 8 days are closer to model predictions. This reinforces the hypothesis of stellar activity, caused by fast stellar rotation, as playing a crucial role in the structure of stars. Galaxies 2022,10, 98 9 of 14 Figure 4. Relative radius difference with respect to the BHAC15 stellar models at 1Ga as a function of the orbital period of the system ( top ) and the pseudo-synchronization rotation velocity at periastron ( bottom ). The horizontal blue dot-dashed line is shown as a reference of agreement with the BHAC15 1Ga model. The DLEBs with long orbital periods discovered in the past decade deserve some attention. For instance, Kepler-16 [ 26 ] and LSPM J1112+7626 [ 34 ], both have an orbital period of ∼ 41 days, the longest in our sample. Although this long orbital period, ∆R/Rmodel is about 3.25% and 3.35% for Kepler-16 A and B components, respectively, and 3.7% and 10.1% for LSPM J1112+7626 A and B, respectively. However, their light curves still show 1% and 2% out-of-eclipse photometric variability, respectively. The period of such variability is consistent with spin-orbit pseudo-synchronization for Kepler-16, while it is about twice larger for LSPM J1112+7626, which points towards slowly rotating components; however, the photometric variability suggests a moderate activity level. On the contrary, Kepler-453, with a shorter orbital period of ∼ 27.3 days, shows a lower level of stellar activity and a low-mass component fairly in good agreement with a 1 Ga theoretical model. The only caveat is a slightly different radii ratio than predicted by models [ 27 ]. This is