The first low-mass eclipsing binary within the fully convective zone from TMTS

We present a comprehensive photometric and spectroscopic analysis of the short-period ($\sim$5.32 hours) and low-mass eclipsing binary TMTSJ0803 discovered by Tsinghua-Ma Huateng Telescope for Survey (TMTS). By fitting the light curves and radial velocity data with the Wilson--Devinney code, we find that the binary is composed of two late spotted active M dwarfs below the fully convective boundary. This is supported by the discovery of a significant Balmer emission lines in the LAMOST spectrum and prominent coronal X-ray emission. In comparison with the typical luminosity of rapidly rotating fully convective stars, the much brighter X-ray luminosity ($L_{X}/L_{\rm{bol}} = 0.0159 \pm 0.0059$) suggests the stellar magnetic activity of fully convective stars could be enhanced in such a close binary system. Given the metallicity of [M/H] = $-$ 0.35 dex as inferred from the LAMOST spectrum, we measure the masses and radii of both stars to be $M_{1} = 0.169 \pm 0.010~M_{\odot}$, $M_{2} = 0.162 \pm 0.016~M_{\odot}$, $R_{1} = 0.170 \pm 0.006~R_{\odot}$, and $R_{2} = 0.156 \pm 0.006~R_{\odot}$, respectively. Based on the luminosity ratio from the light curve modeling, the effective temperatures of two components are also estimated. In comparison with the stellar evolution models, the radii and effective temperatures of two components are all below the isochrones. The radius deflation might be mainly biased by a small radial velocity (RV) data or (and) a simple correction on RVs, while the discrepancy in effective temperature might be due to the enhanced magnetic activity in this binary.


INTRODUCTION
Low-mass stars are the most common stellar objects in our galaxy (Henry et al. 2006;Feiden & Chaboyer 2012) and understanding them is thus clearly an important endeavour.Henry et al. (2006) found that at least ∼ 70% of stars within 10 pc of the Sun are M dwarfs with  ≤ 0.6  ⊙ .Such low-mass stars are useful probes of their structures (e.g.Jurić et al. 2008), kinematics (e.g.Bochanski et al. 2007), and chemical evolution (e.g.Woolf & West 2012;Souto et al. 2022).The fundamental properties of M dwarfs have become an essential component to studies of the initial mass function (e.g.Li et al. 2023).M dwarfs are also attractive targets for the identification and characterization of exoplanets.Due to smaller sizes, it is easier to find Earth-size planets around M dwarfs compared to FGK stars (e.g.Nutzman & Charbonneau 2008;Charbonneau et al. 2009).
★ E-mail: liucheng@bjp.org.cn(CL) † E-mail: wang_xf@tsinghua.edu.cn(XW) In comparison with solar-type stars, stellar theory is not well understood for low-mass stars, especially in the regime of 0.08-0.3 ⊙ , because of complex and varied physics inside the stars and active magnetic phenomena on their surfaces (Mullan & MacDonald 2001).The measurements of fundamental properties (mass, radius and effective temperature) are crucial for calibrating stellar evolution models.Careful observations of detached double-lined eclipsing binaries can result in model-independent mass and radius estimates to a precision better than 1% in some cases (e.g.Morales et al. 2009;Kraus et al. 2011).The first precise determinations of the fundamental parameters, obtained by Torres & Ribas (2002) and Ribas (2003) for early and mid-M dwarfs in two eclipsing systems YY Gem and CU Cnc, respectively, indicated that their radii are inflated by up to 20% with respect to the model predictions.The discrepancies between the observed radii of CM Dra and that predicted by stellar models were also noticed (e.g.Morales et al. 2009;Terrien et al. 2012), however, CM Dra is only inflated by about 2% compared to the prediction based on an older stellar age and a near-solar metallicity (Feiden & Chaboyer 2014a).The M dwarfs with  ≳ 0.35  ⊙ with inflated radii is well established by observations (Kraus et al. 2011;Spada et al. 2013;Cruz et al. 2018).In the range  ≲ 0.35  ⊙ , however, the mass and radius of fewer stars can be accurately determined due to low brightness of cool and small objects (Zhou et al. 2014;Dittmann et al. 2017).
Recent studies suggested that most of the late M dwarfs in the fully convection regime follow the theoretical mass-radius relation (Hartman et al. 2018;von Boetticher et al. 2019;Maxted et al. 2022).Nevertheless, the effective temperatures measured for some low-mass cool stars were reported to be lower than those predicted by models (López- Morales & Ribas 2005;Morales et al. 2009;Zhou et al. 2015;Hartman et al. 2018).The study on white-dwarf and M dwarf binaries by Parsons et al. (2018) indicates that there is a 5% systematic bias towards larger radii for a sample of fully and partially convective low-mass stars, while the temperature measurements for the fully convection stars are in agreement with the theoretical predictions.The possible different trends of temperatures and radii between fully and partially convective stars may be caused by important transitions in inner structures of these stars (Chabrier & Baraffe 1997).This could be the explanation of a discontinuity in the effective temperature-radius relation discovered by Rabus et al. (2019) for M dwarfs.
It remains unclear whether the radius and temperature discrepancy is attributed to interior structure of low-mass stars or a systematic effect specific to the short-period binary systems.Stellar magnetic activity is a popular explanation of the inflation of radii of low-mass stars (Mullan & MacDonald 2001;Chabrier et al. 2007;Feiden & Chaboyer 2014b).The activity hypothesis is favored by observations of the short period binary systems (Morales et al. 2009;Kraus et al. 2011;Spada et al. 2013).The tidal interaction in such systems can give rise to a fast rotation of the companions, which is expected to generate strong magnetic fields and lead to larger stellar radii.The magnetic field might explain the radius inflation for M dwarfs with larger mass, however, its strength is unlikely stable in the fully convective interior of these stars (Feiden & Chaboyer 2014b).Other possible mechanisms include metallicity effects (Berger et al. 2006;von Boetticher et al. 2019) and magnetic star-spots on stars (e.g.Morales et al. 2010).A final solution to the discrepancies may be a combination of the above factors which likely all contribute to the interior structure of fully convective stars (Feiden & Chaboyer 2014a).
In order to probe different effects individually and in aggregate, more sample of low-mass eclipsing binary systems are needed.In this paper we report the study of a short-period eclipsing binary TMTS J08032285+3930509 (dubbed as TMTSJ0803) discovered during the TMTS survey (Lin et al. 2022), which consists of two detached M dwarfs in the fully convective mass range.As M dwarf system is very useful for testing theoretical models of low-mass stars, we utilize multicolor photometry from different telescopes and radial velocity (RV) measurements to constrain the masses and radii of both components of this binary system.
The paper is structured as follows.In Section 2, we describe the discovery, the follow-up photometric observations, and collection of the spectra.The light curve and period of the binary system is examined in Section 3. In Section 4, we estimate the properties of the binary via SED fitting and analysis of the LAMOST spectrum.In section 5, we calculate the absolute physical parameters of TMTJ0803 by a joint analysis of the light curves and RV data.In Section 6, we discuss the implications of these parameters in regards to the existing theoretical stellar models.We summarize in Section 7.

Photometric data from TMTS, SNOVA, TESS and TNT
TMTS is a photometric survey with four 40-cm optical telescopes located at Xinglong Observatory in China.The survey operates in such an observation mode: uninterrupted observing of the LAMOST areas for the whole night with a cadence of about 1 min, resulting in discoveries of many interesting short-period variables and eclipsing binaries (see more details in Zhang et al. 2020;Lin et al. 2022Lin et al. , 2023Lin et al. , 2024;;Guo et al. 2024).A primary eclipse of TMTSJ0803 was observed on 2020 Jan. 15 in the first year TMTS survey (see Table 1).The system TMTSJ0803, alternative names are LP 208-19 and 2MASS J08032307+3930558, were first identified as an eclipsing binary by Palaversa et al. (2013).Based on the LAMOST spectrum and the continuous light curve observed on a whole night, we believe that TMTSJ0803 is a double M dwarfs binary system with a short period (∼ 5.32 hours).TMTSJ0803 is the one of the first 12 short period double M dwarfs binary candidates selected from the database of TMTS survey, based on their light curves, (B-R) colors, and photometric and/or spectroscopic temperatures.
Follow-up photometric observations of TMTSJ0803 have been taken by SNOVA and Tsinghua-NAOC 0.8-m telescope (TNT; Wang et al. 2008;Huang et al. 2012).The SNOVA is a 36 cm telescope located at Nanshan Observatory in China, and it is used to monitor TMTSJ0803 in white light (clear band) and standard I band for a total of 8 nights, as shown in Table 1.TMTSJ0803 was also monitored in standard R band on 5 nights with the TNT at the Xinglong Observatory in December 2022.To achieve better photometry, we adopted different exposure time, ranging from 60s to 180s, for different telescopes and under different seeing conditions.
The standard image processing, such as bias correction, flat correction and source extraction are performed with Ccdproc of Astropy (Craig et al. 2017) and SExtractor (Bertin & Arnouts 1996).Four comparison stars are used to calibrate the photometry of TMTSJ0803.
In addition, TMTSJ0803 has also been observed by TESS (Ricker et al. 2015) in 3 sectors (20, 47, and 60) with cadences from 30 minutes to 200 seconds (see Table 1).For the long cadence observations in the Kepler mission, it was demonstrated by Zola et al. (2017) that the shapes of light curves of short-period (< 1.5 days) eclipsing binaries are influenced by the smearing effect.As the binary has a very short period, long exposure time in TESS observations would cause a significant influence on the shape of light curve of the binary.Therefore, the light curve from sector 60, at a cadence of 200 s, is only used to measure the physical parameters of the binary in Section 5.

Spectroscopic data from LAMOST
An optical spectrum of TMTSJ0803 was observed on 26th December 2014 by LAMOST (Cui et al. 2012) in low resolution mode ( ∼ 1800; wavelength range 370 -900 nm; Luo et al. 2015).This LAM-OST spectrum has a bad quality in the blue arm due to the faintness of two these late M dwarfs at the blue end.While the average signalto-noise (SNR) ratio of the spectrum is ∼ 66 in the red end.The H  emission can be clearly seen in Figure 3, where this emission feature observed at an orbital phase of about 0.48 was identified.The presence of prominent H  emission line indicates that TMTSJ0803 could be very active (see more discussions in Sect.6.1).

Spectroscopic data from APOGEE
TMTSJ0803 was also observed by the APOGEE (Apache Point Observatory Galactic Evolution Experiment) project in high-resolution mode, involving the APOGEE-N (north) spectrograph (Wilson et al. 2019) which cover a spectral range of 1.51-1.7 m with an average resolution of R ∼ 22,500 (Wilson et al. 2010(Wilson et al. , 2019)).A total of seven infrared spectra can be extracted from the APOGEE DR16 database (Ahumada et al. 2020;Jönsson et al. 2020).Among them, four spectra were observed in 2016, while other three spectra were taken in 2012, 2013 and 2017, respectively, as listed in Table 1.The multiple-epoch spectra enable the measurements of RVs for individual components of TMTSJ0803 (see Table 2).
As the APOGEE DR16 pipeline constructed cross-correlation function (CCF) for each infrared spectrum, thus an automated code called apogeesb22 can be used to measure RV for each component by deconvolving the custom CCF (Kounkel et al. 2021).To identify the primary and secondary components, the apogeesb2 analyzes each CCF by utilizing the autonomous Gaussian deconvolution Python routine, GaussPy (Lindner et al. 2015).In this work, the log parameter is set to 3.1 to 3.8, depending on different spectrum rather than the recommended value 1.5 given in Kounkel et al. (2021).This is due to the relatively low quality of CCFs resulting from the low SNR of the APOGEE spectra.Green dots show velocities of the primary relative to those of the secondary.Black dash line shows the best fit to the data; the slope of this line relates to the mass ratio.Red line is the line of equality between  prim and  sec ; the intersection of these two lines corresponds to the barycentric velocity of the system.
The exposure time of each spectrum is usually longer than an hour except for the earliest spectrum observed in 2012.As mentioned in Section 2.1, the phase smearing effect caused by long exposure time can not be ignored.To evaluate the smearing effect on RV, we simply assume an identical sine curve for both primary and secondary stars because they have almost the same stellar mass as discussed in Section 6. Comparing the mean RV within the total exposure time with the one at the middle time, we then obtain the corrected RV value as listed in Table 2.
Although there is an insufficient number of epochs to construct a full orbital velocity variation, current measurements allow us to determine the mass ratio (  =  2 / 1 ) and the central velocity () for this binary by constructing a Wilson plot (Wilson 1941).We find that the mass ratio and the central velocity are  = 0.961 ± 0.039 and  = −1.0 ± 2.0 km s −1 , respectively.In Figure 1, the RV pairs from four epochs are used to measure  and .The first RV pair from epoch BJD 2456264.0093 is firstly removed because it is an outlier in the phased RV curves.We find that the absolute RVs (> 57 km s −1 ) at the first epoch are much higher than the expected values (∼ 7 km s −1 ) at phase 0.51.The contradiction might be caused by a wrong measurement of RVs at the first epoch.Two spectra observed at epoch BJD 2456381.7143 and 2457690.9798are also abandoned, because the correction for phase smearing has a very large effect (ΔRV/RV 1,2 * > 100%) on their RVs and makes the RVs from these two spectra rather unreliable.Therefore, in Table 2 the first three RV pairs are ignored when the RV curves are used to constrain the semi-major axis () in Section 5.

PERIOD STUDY
To derive the orbital ephemeris of TMTSJ0803, we calculate the time of minimum light by fitting its light curve by a Gaussian function.A total of 389 minimum light epochs, 192 primary and 187 secondary ones, are obtained and listed in Table 3.With the least-square method, a linear ephemeris is derived by fitting the minimum light time as: Min.I = 2459579.765702(±0.000009) Figure 2 shows the linear fitting and the corresponding observed minus computed O − C residuals as a function of the epoch number E. It should be noted that the minimum-light time has a large error (more than 0.0007) and those values derived from eclipses without enough observation data are abandoned.

Broad-band photometric characterization
A total of 19 broad-band photometric datapoints are available for TMTSJ0803, including GALEX_NUV (Bianchi et al. 2011) (Schlegel et al. 1998;Schlafly & Finkbeiner 2011) and adoption of extinction law from Fitzpatrick (1999), the best-fit parameters estimated for the system are:  eff = 2924 ± 11 K, log g = 5.62 ± 0.11 dex, [M/H] = -0.10 ± 0.07 dex,  = 43.57± 0.08 pc, and A V = 0.07 ± 0.01 mag.These stellar parameters, except for metallicity, are consistent with the results from a single-star model presented in the next subsection.Here, the error on  eff may be underestimated by the methodology.To estimate the possible systematic errors, we calculate the differences between the recommended values from literatures (see Table 3 in Vines & Jenkins 2022) and  eff from the SED-fitting for 13 benchmark M dwarfs (< 3600 K) from Vines & Jenkins (2022), with no systematic offset being found (i.e.,∼ 1 ± 180 K).

Spectroscopic characterization by comparison with synthetic spectra
Using the full-spectrum fitting method developed by Kovalev et al. (2022), we analyze the LAMOST spectrum in binary and singlestar spectral models (see details in Section 2.3 in Kovalev et al. 2022).For double M-type binary, the synthetic spectra from BT-Settl (AGSS2009) models (Allard et al. 2011(Allard et al. , 2012) ) rather than the NLTE MPIA models are used in the analysis.Since there is no reliable absolute flux calibration for the LAMOST spectra, the observed spectrum is flux-renormalized by a fourth-order polynomial recommended from the loss function (Zhang et al. 2021).The synthetic spectrum of binary system is generated as a sum of two Dopplershifted, normalized, and scaled single-star model spectra which are function of atmospheric parameters and stellar size.Comparing the synthetic binary spectrum with the observed one yields estimations of optimal  eff , log g, [M/H], RV of each component, mass ratio  and one set of four coefficients of polynomials.On the other hand, the observed spectrum is also analysed with a single-star model, which is identical to a binary model when the parameters of both components are equal.
Owing to the quality of the spectrum in the blue end, only the spectrum of the red end ranging from 6800 Å to 8500 Å is fit by models, as shown in Figure 3.As the mass ratio  is measured from the RVs data discussed in Section 2.3, we fix  = 0.96 in order to find solution with minimal  2 in the spectral fitting program.Finally, we obtain the best atmospheric parameters in the single-star model:  eff = 2930 K, log g = 5.33 cgs, and [M/H] = − 0.35 dex.For the binary model, we find  eff = 3000 K, log g = 5.34 cgs, and [M/H] = − 0.44 dex for the primary star, and  eff = 2882 K, log g = 5.27 cgs, and [M/H] = − 0.44 dex for the secondary star.The minimal  2 of these two fittings are shown in Figure 3.As the  2 from the binary model is larger then that of the single-star model, therefore, the atmospheric parameters from the single-star model would better constrain the binary properties by fitting to the light curves.Furthermore, by considering uncertainties, our estimated  eff is consistent with the effective temperature of 3003±156 K and 2974±194 K measured from the LAMOST and APOGEE spectra, respectively.The effective temperatures of two components in the binary model are consistent with the results (see Table 4) derived from the light curve modeling (see Section 5).
Errors on  eff , log g and [M/H] are provided by the full-spectrum fitting method for both single-star and binary models.However, they are underestimated and nominal, such as 1 -10 K in  eff , and 0.01 dex in log g and [M/H], because the systematic errors are not included.Since the typical errors have been evaluated by Kovalev et al. (2022) using simulated datasets (see more details in their Sect.2.3.3),we thus adopt the standard deviation of the test sample as our errors estimation.In this case, for the single-star model and primary component, the typical errors on  eff , log g and [M/H] is less than 150 K, 0.1 cgs, and 0.1 dex, respectively.For the secondary component, the typical errors on  eff , log g and [M/H] are less than 350 K, 0.2 cgs, and 0.2 dex, respectively.

Modeling light curves and radial velocity shifts
To determine the properties of the binary, we use the 2015-version Wilson-Devinney (WD) code (Wilson & Devinney 1971;Wilson 1979Wilson , 1990Wilson , 2012;;Wilson & Van Hamme 2014) to fit the light curves and the RVs simultaneously.In the subsequent analysis, the primary and the secondary stars are indicated with the subscripts 1 and 2, respectively.The small asymmetry (O'Connell effect (O'Connell 1951)) seen in the light curves, especially in the I band light curve shown in Figure 4, could be related to cool spots and/or hot spots on the surface of the stars.Since TMTSJ0803 is a detached system and the Balmer emission lines emerge in the LAMOST spectrum, we thus believe that the asymmetries in the light curves can be attributed to cool spots.Therefore, the binary model with spots (called Model B later) is applied to model the light curves.For comparison, another binary model without spot (called Model A) is also used to fit the light curves.
In the modelling, the effective temperature of the primary ( eff,1 =  m ) is fixed to 2930 K according to the single-star spectral fitting result.According to the early studies by Lucy (1967) and Ruciński (1969), the gravity-darkening exponents  1 =  2 = 0.32 and the bolometric albedo  1 =  2 = 0.5 are adopted in the fitting, respectively.The bolometric and bandpass logarithmic limb-darkening coefficients are interpolated from the tables given in van Hamme (1993), Claret & Bloemen (2011) and Claret (2017).As the mass ratio (q) and the center-of-mass velocity () were determined from the RV data, we therefore fix them in the binary modeling process.Moreover, the limited RV data can not provide enough constraint on the eccentricity (e), we thus simply assume it to be zero.During the fitting, the adjustable parameters are the semi-major axis of the binary (a), the inclination (i), the effective temperature of the secondary ( eff,2 ), the surface potential (Ω 1 and Ω 2 ), the phase shift, the dimensionless luminosity of the primary ( 1 ), the spot parameters, longitude (), spot angular radius (r), and temperature factor ( s / * ).Since the spot area and the latitude are highly correlated with temperature and radius of the star (Zhang et al. 2014), respectively, the latitude of the spot is assumed to be at 90 • ( ≃ 1.571 in radian).
It is well known that errors on the parameters provided by the WD code are underestimated.To overcome the drawback, firstly the observational errors4 are used as weights in our modeling processes to estimate the random errors.To estimate the systematic errors caused by the choices of input physics in the WD model, secondly numerous solutions are generated by choosing different gravity darkening exponents, limb darkening law and coefficients, albedo values, reflection effect, and whether or not to include spots.Here the systematic errors for i,  eff,2 , Ω 1 , Ω 2 , a, and four equivalent radius () for each star are estimated and given in Table 4 (see the second error).We find that the systematic errors is at least twice times larger than the random errors, except for the errors of semi-major axis.Both the random and systematic errors are propagated to the uncertainties of the final absolute parameters, such as mass and radius.Although there is a small effect on some fitted parameter values caused by the spot, two WD models give us the same absolute parameters (see Table 4).This suggests that the free parameters are well constrained by the multiple light curves in this work.

Results
The best-fit light cures and RV curves (red solid lines) from the binary model and their corresponding  −  residuals are shown in Figure 4 and 5, respectively.The results of the best-fit models are listed in Table 4, where the converge solutions without spot are listed in Model A and the solutions with one spot on the secondary are listed in Model B. The smaller value of ( − ) 2  in Model B gives us a better solution.This is consistent with the conclusion in Sect.6 that the binary system is active from the phenomenon of the Balmer emission lines, X-ray radiation and the flare events.Here, a third body around the binary system is tested, while the contribution of the third light to the total system is estimated to be almost zero in all bands.
Taking the semi-major axis of the binary (a) and the mass ratio (q) into the Kepler's third law ( 1 +  2 = 0.0134 3 / 2 ) and the equivalent radius ( = /), we calculate the mass and radius of primary and secondary star, respectively.The units of  1,2 , , and  in the Kepler's equation should be  ⊙ ,  ⊙ , and days, respectively.Although the uncertainties of the final absolute parameters are carefully estimated in the previous section, it is likely underestimated due to the limitation of RV data.From Table 4, we find that the ratio of the temperature and luminosity between the secondary and the primary is determined to be ∼ 0.96 and 1, respectively.As the low-resolution spectrum observed by LAMOST covers a phase range of 0.14 to 0.84, the light contribution from the secondary star to the observed spectrum can not be neglected.According to the study by Kjurkchieva et al. (2018), we then calculate the final temperatures: where Δ =  eff,1 −  eff,2 and  =  2 / 1 are calculated based on the final solutions from Model B. We find that the final  eff,1 = 2971 K and  eff,2 = 2869 K are consistent with the temperatures given by the binary spectral fitting method.Both the two empirical relations between effective temperature and spectral type, given by Bessell (1991) and Rajpurohit et al. (2013), suggest similar spectral types -2.830±0.006-2.830±0.006a The systematic errors that are propagated to the uncertainties of the final absolute parameters are estimated for some parameters, such as, i,  eff,2 , a and so on.b The primary temperature and spot latitude are fixed when modeling the light curves and spot on the star surface.

Activity analysis
Chromospheric activity and star-spot activity could be triggered by magnetic fields and maintained by a magnetic dynamo.The spectral lines (H  , H  , H  , H  and Ca II H&K) are useful diagnostic indicators of chromospheric activity for late-type stars.The chromospheric activity of M stars shows emissions above continuum or core emissions in Balmer lines.
In our analysis, we first created a subtracted spectrum (the observed spectrum minus the synthetic spectrum from the binary model in Sect.4.2) based on the observed LAMOST spectrum.When calculating the equivalent width of the H  line (EW), we integrate the emission profile using the following formula where   and   represent the fluxes of spectral line and the contin-uum.Comparing the criteria (0.75 Å) for determining the chromospheric activity of M-type stars (West et al. 2011), the EW of H  (2.63 Å) confirms the eclipsing binary is active.
The ratio of the H  luminosity to the bolometric luminosity of the star,  H  / bol , enables a better mass-independent comparison between activity levels in M dwarfs than EW alone (Newton et al. 2017).If we simply consider TMTSJ0803 as a single star,  H  / bol could be easily calculated:  H  / bol = EW ×  by adopting  factor from Douglas et al. (2014).Our result ( H  / bol ∼ 0.65 ± 0.13 × 10 −4 ) is consistent with the values for most of rapidly rotating and full convective stars (Douglas et al. 2014;Newton et al. 2017).
Like the spectral lines, coronal X-ray emission is also a useful diagnostic of stellar activity.A close relation between the surface magnetic flux and X-ray radiance indicates that X-ray emission is a reliable proxy of magnetic activity (Wright & Drake 2016).Crossmatching with the updated ROSAT point-source catalogue (2RXS Boller et al. 2016) within 2 arcsecond, we find star ROSAT 2RXS J080322.0+393050 has a X-ray spectrum in the 0.1-2.4keV energy band.With the SPIDERS program (Dwelly et al. 2017;Comparat et al. 2020), the observed instrumental ROSAT count rates can be converted into physical flux at 6.42965 ± 2.40437 ×10 −13 mW/m 2 .Given a parallax distance of 43.584 ± 0.10 pc (Bailer-Jones et al. 2021;Gaia Collaboration et al. 2021), the flux can be used to calculate the X-ray luminosity which is   = 3.8175 ± 1.4276 × 10 −5  ⊙ .Combining with the Bolometric luminosity from the results of SED fitting in Section 4.1, we finally obtain   / bol = 0.0159 ± 0.0059.Comparing with the fractional X-ray luminosity (  / bol < 0.005) of rapidly rotating fully convective stars, we find that the luminosity of TMTSJ0803 is 3 times brighter than that of the strongest X-ray stars in the catalog of Wright et al. (2011) and much higher than the mean saturation level of 10 −3 .This suggests that stellar magnetic activity of fully convective stars could be significantly enhanced in a very close-by binary system such as TMTSJ0803.It is highly possible that the two components of this binary system have synchronized their rotation periods with the orbital period (i.e., ∼ 5.32 hours) via tidal interaction.This is consistent with the discoveries that tidal locking leads to larger magnetic fields due to faster rotation rate (Spada et al. 2013;Gehan et al. 2022).
Flares are sudden and violent events that release magnetic energy and hot plasma from the stellar atmosphere.It is an indicator of the inherent activity of M dwarf stars.A clear flare event emerged around phase 0.23 on BJD 2459957.4416 was observed by TESS in sector 60.The flare duration was found to be ∼50 min.Two more flares located around phase 0.69 and 0.83 were observed by TESS in sector 20 and 47, respectively.Those flares confirm that the system has stellar activity.

Comparisons with other M dwarf systems and stellar evolution models
Besides TMTSJ0803, there are 41 other M dwarf stars in eclipsing binaries with masses between 0.1 and 0.4  ⊙ and with masses and radii measured to have an accuracy better than 5%.Parameters of these binary systems are collected in Table 5. Figure 6, 7, and 8 show the mass-radius, mass- eff , and  eff -radius relations for these objects, respectively.Inspecting the mass-radius and mass- eff plots reveals that the isochrones from the Dartmouth theoretical stellar evolution models show the best match with the observations.Although almost all the observed radii are above the age = 1 Gyr isochrone, they are consistent the old and metal-rich isochrone ([Fe/H] = +0.5 dex, age = 10 Gyr) predicted by the Dartmouth models.This suggests that most of the field stars may be old and/or metal-rich, while stars from open clusters (NGTS J0002-29A, B and PTFEB132.707+19.810A,B) are close to the young isochrone (age = 0.1 Gyr) in the massradius plot.Unlike the masses and radii are more consistent with the models, the observed effective temperatures are systematically lower than the models, especially the BHAC98 and BHAC15 models.
Figure 6 shows the locations of TMTSJ0803 in Radius−Mass relation as inferred from Models A and B, together with those of known sample of double M dwarf binaries (see also Table 4).Figure 7 shows distribution of TMTSJ0803 and the other sample in the  eff -Mass relation.The outstanding feature is that the observed radii of two components of TMTSJ0803 are below the isochrones, while their effective temperatures are not outliers in the M dwarfs sample.Comparison with the stellar evolution models, smaller radii might be due to limited  data or (and) a simple correction for RVs based only on the exposure times (see Section 2.3).We find from Table 2 that the difference between the original RV and corrected RV could be larger than 30 km s −1 .According to the third Kepler law, the semimajor axis is in proportion to the maximum RV of the binary when we fix the period.In this case, the uncorrected RV may decrease the radius by about 20% at least.Moreover, there is no RV data in the third quarter of the phase (see Figure 5).This might also cause bias in the final stellar radius.Therefore, it is not surprising that the radii of two components are slightly below the isochrones.
We find that different methodologies give different metallicities by fitting to the spectrum of TMTSJ0803.According to the parameters given by LAMOST DR7, the metallicity of this binary system is [M/H] = −0.62 dex with a large uncertainty.Comparing with the single-star model in Section 4.2, the binary model gives a [M/H] = −0.44 dex.From the combined infrared spectrum, we find that the best stellar atmospheric parameters have been provided by the pipeline ASPCAP in APOGEE DR14 (García Pérez et al. 2016;Abolfathi et al. 2018) and the system metallicity is [M/H] = −0.94dex.Those suggest that the metallicity of TMTSJ0803 might be more poor than [M/H] = −0.35dex used in this work.In this case, it would not be surprised that the estimated radii and temperatures are all below the isochrones.
Unlike a sharp transition identified by Rabus et al. (2019) for single M dwarfs, the M dwarfs in eclipsing binaries display a roughly linear feature between between stellar radius and  eff , as seen in Figure 8.The above discrepancy can be explained by several reasons.Firstly, estimating the effective temperatures of a binary system is more challenging in comparison with single star, and the temperatures might suffer a larger bias and uncertainty, such as the system CU Cnc, LSPM J1112+7626, and MG1-2056316 listed in Table 5. Secondarily, the data do not show a 'discontinuity' due to the lack of low-mass M dwarfs (< 0.2  ⊙ ) with measurements of  eff .Finally, there is a possibility that M dwarfs in different environments might have different characteristics.For instance, the stellar activity of individual components could be enhanced in close-binary as discussed in Section 6.1.The strong magnetic fields could inhibit convections in the atmosphere of stars (MacDonald & Mullan 2012).

CONCLUSIONS
We present a photometric and spectroscopic analysis of the shortperiod (∼5.32 hours) eclipsing binary TMTSJ0803 detected by the TMTS.The analysis reveals that it is a late M dwarf binary whose components are below the fully convective boundary.Comparing with a normal eclipsing binary, the binary model (Model B) with one spot on the secondary provides the best fit to the light curves and RV data.Under the assumption of the Kepler's third law, we measure the masses and radii of both stars to be  1 = 0.169 ± 0.010  ⊙ ,  2 = 0.162 ± 0.016  ⊙ ,  1 = 0.170 ± 0.006  ⊙ , and  2 = 0.156 ± 0.006  ⊙ , respectively.Based on the luminosity ratio from the light curve modeling, the effective temperatures of two components of binary system are determined as  eff,1 = 2971 K and  eff,2 = 2869 K, respectively.These are consistent with the results derived by fitting to the LAMOST spectrum with binary model.
The significant Balmer emission lines seen in the LAMOST spectrum of TMTSJ0803 suggest that this eclipsing binary is very active.Furthermore, we find that TMTSJ0803 has coronal X-ray emission and the fractional X-ray luminosity is   / bol = 0.0159 ± 0.0059, which is much brighter than that of the typical rapidly rotating fully convective stars.This indicates that the stellar magnetic activity of the fully convective stars could be enhanced in close-by binary environment.
We find that both the radii and temperatures of the two components of TMTSJ0803 are below the isochrones.In comparison with the stellar evolution models, the radius deflation might be mainly biased by the limited RV data or (and) a simple correction for RVs.The effective temperature suppression might be due to enhanced magnetic activity in the binary.To better understand the origin of discrepancy in effective temperature and radius for this system, the higher-precision measurements of RV and photomteric data are required.Combined with high-cadence photometric data, the LAMOST medium resolution survey gives us more opportunities to explore the nature of low-mass eclipsing binaries like TMTSJ0803 from TMTS.   4. Black open circles are used for systems without a measured metallicity.The red dash and dot-dash lines show theoretical mass-radius relations from the Dartmouth (Dotter et al. 2008) models with different ages and metallicities.The blue and green lines represent the solar metallicity isochrones of BHAC15 (Baraffe et al. 2015), while the blue and green dash lines represent the sub-solar metallicity ([Fe/H] = -0.5 dex) isochrones of BHAC98 (Baraffe et al. 1998) age 1 and 10 Gyr.(Zhou et al. 2014;Jennings et al. 2023) * Normally, the listed metallicity is the value determined spectroscopically for the primary.a NGTS J0002-29 and PTFEB132.707+19.810 are the member of the Blanco 1 and Praesepe open cluster, respectively, and the adopted metallicity is the value for the cluster.b The metallicity of the LP 661-13 eclipsing binary system was not determined spectroscopically, but was estimated using the absolute  s magnitude and the MEarth- S broadband color following Dittmann et al. (2016).c Assuming that both components have the same metallicity, the listed value is the binary system metallicities.d KOI-126B and KOI-126C are components of a triply eclipsing hierarchical triple system.The listed metallicity is the value determined spectroscopically for the primary.

DATA AVAILABILITY
The minimum times of TMTSJ0803 are available in its online supplementary material.The TESS data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute (STScI) (https://mast.stsci.edu).Funding for the TESS mission is provided by the NASA Explorer Program directorate.

Figure 1 .
Figure1.The Wilson plot.Green dots show velocities of the primary relative to those of the secondary.Black dash line shows the best fit to the data; the slope of this line relates to the mass ratio.Red line is the line of equality between  prim and  sec ; the intersection of these two lines corresponds to the barycentric velocity of the system.

Figure 2 .
Figure 2. The corresponding O − C residuals after the linear fitting.

Figure 3 .Figure 4 .
Figure 3.Comparison of the single-star (offset + 1.0) and binary model fits for TMTSJ0803.The difference of observed spectrum and single-star model spectrum is shown as gray line.

Figure 5 .
Figure 5. Upper panel: Differential radial velocity curves of the primary (blue squares) and secondary stars (green squares) of TMTSJ0803.The red lines represent corresponding fitting curves.Lower panel: residuals of the best-fit curves relative to the radial velocities.

Figure 6 .
Figure 6.Mass-radius diagram for M dwarfs with 0.09  ⊙ <  < 0.40  ⊙ .The color scale of the points indicates different metallicity.Large filled blue and red triangles show the components of TMTSJ0803 in Model A and B, respectively.Smaller circles show other M dwarfs with parameters given in Table4.Black open circles are used for systems without a measured metallicity.The red dash and dot-dash lines show theoretical mass-radius relations from the Dartmouth(Dotter et al. 2008) models with different ages and metallicities.The blue and green lines represent the solar metallicity isochrones of BHAC15(Baraffe et al. 2015), while the blue and green dash lines represent the sub-solar metallicity ([Fe/H] = -0.5 dex) isochrones of BHAC98(Baraffe et al. 1998) age 1 and 10 Gyr.

Figure 7 .
Figure 7. Similar to Figure 6, but we show the mass- eff diagram.The observed temperatures are systematically below the theoretical models, with even the 10 Gyr, [Fe/H] = +0.5 dex model being above about the half of the observations.

Figure 8 .
Figure 8. Similar to Figure 6, but here we show the  eff -radius diagram.The dash blue lines represent the discontinuity of the  eff -radius relation at stellar mass 0.23 M ⊙ (Rabus et al. 2019).It looks like that there is no discontinuous behaviour for M dwarfs from binaries.
e The listed [Fe/H] is the [M/H] value determined spectroscopically for the primary.State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

Table 2 .
Radial velocities from the APOGEE spectra.RV 1 stands for RV of the primary star, while RV 2 stands for that of the secondary star.Both RVs are corrected based on the exposure times. *

Table 4 .
Light curve solution and physical parameters of the binary TMTSJ0803.

Table 5 .
The known M dwarfs in eclipsing binary systems with masses between 0.1 and 0.4  ⊙ , and with better determinations of masses and radii (i.e., < 5% error).