The X-Ray Emission Reveals the Coronal Activities of Semi-detached Binaries

X-ray emission is an important tracer of stellar magnetic activity. We carried out a systematic correlation analysis for the X-ray luminosity logLX , bolometric luminosity logLbol , and X-ray activity level log(LX /L bol) versus the binary parameters including orbital period P, Rossby number R O, effective temperature T eff, metallicity [Fe/H], the surface gravity logg , stellar mass M, and radius R, by assembling a large sample of semi-detached (EB-type) binaries with X-ray emission (EBXs). The fact that both logLX and logLbol change in accordance with logP indicates that X-ray emission originates from the convection zone, while logLX is proportional to the convection zone area. We found that EBXs with main-sequence components exhibit an upward and then a downward trend in both the logTeff – logLX and M– logLX relations, which is different from the monotonically decreasing trend shown by EBXs containing sub-giant and giant components. The magnetic activity level is negatively correlated with logTeff and stellar mass. Based on the magnetic dynamo model, the variations in the size and thickness of the surface convection zones can explain the observed relations. EBXs with main-sequence components have a similar R O– log(LX/Lbol) relationship to that of the binaries in the clusters as Praesepe and Hyade. We compared the X-ray radiation properties of EBXs with those of the X-ray-emitting contact binaries and found that EBXs have broader value ranges for logLX and log(LX /L bol).


INTRODUCTION
In general, an EB-type binary, also known as a β Lyrae-type binary, is a semi-detached close binary system with only one component filling its Roche lobe.Their typical spectral types range from late A to K (Zhang et al. 2019).The variable amplitudes of EB-type binaries are generally smaller than 1 magnitude, while their orbital periods span from 0.2 to several days (Zhang et al. 2019).The light curves exhibit fairly smooth and continuous eclipses.Furthermore, the luminosities at the two maxima are practically identical, whereas those at the minima differ considerably.In this work, our classification for the EB-type binary follows the All-Sky Automated Survey for Supernovae (ASAS-SN) survey 1 (Jayasinghe et al. 2018(Jayasinghe et al. , 2021)).EB systems include X-ray emitters (Szczygie l et al. 2008), which we refer to as EBXs in this work.By combining eclipsing binaries from the ASAS with the ROSAT All Sky Survey (RASS), Szczygie l et al. ( 2008) compiled a catalog with 266 X-ray-emitting EB binaries and expanded the coronal activity study from contact (EW-type) close binaries to semi-detached close binaries.Moreover, for a given optical color, the activity level of EB-type binaries is generally higher than that of contact binaries.
For late-type main-sequence (M-to F-types) single stars, previous studies (e.g., Stelzer et al. 2016;Wright et al. 2018;Pizzocaro et al. 2019;Wang et al. 2020;Magaudda et al. 2020Magaudda et al. , 2022) ) on the magnetic activity-rotation relations have revealed the relationship between X-ray emission and the stellar dynamo.The standard stellar dynamo located in the tachocline is powered by convection and rotation, connecting the solidly rotating radiative interior with the differentially rotating convective envelope (Parker 1955(Parker , 1993)).The X-ray emission can act as a proxy for the efficiency of the stellar dynamo (Magaudda et al. 2022) and serve as a manifestation of magnetic activity in the outermost atmospheric layer, namely the corona.The stellar dynamo magnetic activity produced by fast rotation and envelope convection, as well as the large-scale horizontal flow between the components, are considered to be the possible X-ray emission mechanism of EW-type close binaries (e.g., Stȩpień et al. 2001;Gondoin 2004;Chen et al. 2006;Liu et al. 2022).However, there is currently limited statistical research on the relationship between the X-ray emission properties of EBXs and the stellar dynamo model.Moreover, for late-type main-sequence stars and close binaries, the ratio of X-ray luminosity L X to bolometric luminosity L bol is used to represent the magnetic activity level of an individual system (Fleming et al. 1993;Güdel 2004;Chen et al. 2006;Liu et al. 2022).This ratio tends to reach a maximum level at 10 −3 , i.e., log(L X /L bol ) ∼ −3, which is called the saturation limit or saturation level (Vilhu 1984;Vilhu & Walter 1987;Fleming et al. 1993).Studying the X-ray radiation properties of EB-type binaries can advance our understanding of magnetic dynamo models of stellar activity.
In this study, we conduct the first systematic population study of EBXs for their coronal activities, primarily using samples selected from the ASAS-SN Variable Star catalog and the X-ray databases of XMM-Newton, RASS, and Chandra X-ray Observatory.The remainder of this study is structured as follows: Sections 2 and 3 describe the data selection and statistical analyses of EBX samples, respectively.Section 4 discusses the relationships among the period, stellar atmospheric parameters, and magnetic activity.Section 5 summarizes the main results.

EBXs in 4XMM-DR11
The EB-type binaries were first selected from the databases of ASAS-SN, which periodically scans the entire visible sky with a cadence of ∼2−3 days and a sensitivity limit of V ≲ 17 mag (Jayasinghe et al. 2018(Jayasinghe et al. , 2021)).Since 2018, the monitor of the sky has expanded to a depth of g ≲ 18.5 mag with ∼ 1 d cadence.ASAS-SN identifies new variable star candidates by applying a random forest classifier to the light curve characteristics (Jayasinghe et al. 2019).Until November 2023, the ASAS-SN Variable Star Database (AVSD)2 lists 25932 EB-type binaries classified from ∼ 680000 variable stars.By cross-matching variable stars with different external catalogs (Christy et al. 2023), such as Gaia EDR3 (Gaia Collaboration et al. 2021), 2MASS (Skrutskie et al. 2006) and ALLWISE (Wright et al. 2010), AVSD provides information about the Gaia DR3 IDs, 3 eclipsing periods, proper motion, photometry, and color/reddening for most sources.By combining the parallax information of Gaia DR3, we calculated the distance of each system and eliminated those with distance > 2 kpc or uncertainty of distance > 20% (parallax error/parallax > 20%).We selected objects within the 2 kpc distance to ensure accurate X-ray luminosity calculations, as Gaia DR3 provides most reliable distances up to 2 kpc (see Section 3.1 of Fouesneau et al. 2023).After these criteria were applied, 15039 EB-type binaries (ASAS-SN-EB) were selected as the primary catalog.
The 4XMM-DR11 catalog contains 895415 unique X-ray sources detected during 12210 pointed XMM-Newton EPIC observations (Webb et al. 2020).The high sensitivity and random nature of the observations of 4XMM-DR11 make it suitable for searching for the X-ray counterparts of EB-type binaries.Liu et al. (2022) verified the completeness of XMM-Newton and RASS samples for the study of eclipsing binaries.
We cross-matched the ASAS-SN-EB catalog with the 4XMM-DR11 full catalog with a matching radius of 15 ′′ .This process yielded 180 closest unique X-ray sources for the XMM-Newton detection.To further purify the sample, we visually examined the AVSD light curves with reference to the shape of the light curves given by ASAS-SN,4 and eliminated sources that did not exhibit EB-type light curve characteristics.A total of 150 subjects remained in the sample.The X-ray fluxes in the 0.2-12.0keV band in the 4XMM-DR11 catalog were calculated by assuming a power-law model with a photon index of 1.42 and the hydrogen column density (N H ) of 1.7×10 20 cm −2 .We derived the N H value for each object using the extinction A V obtained from Starhorse and the relation of N H (cm −2 ) = (2.21±0.09)×10 21A V (mag; Güver & Özel 2009).The average N H value is 1.41×10 21 cm −2 , which is higher than the above value adopted by 4XMM-DR11.Therefore, we re-calculated the flux values for each binary using its N H values with PIMMS.The flux error includes the uncertainty due to the choice of the spectral fitting model.

EBXs in RASS
We applied the same data screening procedures as described in Section 2.1 to select 96 EBXs from the Second RASS source catalog (2RXS; Boller et al. 2016), except using a 20 ′′ matching radius.The X-ray flux in the 0.1−2.4keV band was converted from the count rate using the energy conversion factor calculated from the hardness ratios provided by 2RXS (Huensch et al. 1996).In this catalog, the N H value for each source based on Dickey & Lockman (1990) was applied.Using P IM M S5 , we transformed the unabsorbed X-ray fluxes from the 0.1−2.4keV band to 0.2−12 keV, assuming a photon index of 2.0.This assumption was made because the distribution of photon indices in the power law model fitting for 2RXS objects peaks at 2.0 (Boller et al. 2016).The flux error values incorporate the uncertainties from the spectral model fitting.

EBXs in Chandra
We utilized the Chandra Source Catalog 2.0 Quick Search6 (Evans et al. 2019(Evans et al. , 2020) ) to look for X-ray counterparts, resulting in 24 sources with a matching radius of 1 ′′ , after visual screening of the ASAS-SN light curves.Their X-ray fluxes and uncertainties in the 0.5-7.0keV band were derived under the power law model with a fixed photon index 2.0 and the Galactic N H in the direction of each source, obtained from the Colden tool7 .Using the P IM M S, we converted the unabsorbed X-ray flux in the 0.5 − 7.0 keV band into that in the 0.2 − 12.0 keV band.The flux error was also calculated, including the uncertainties caused by the assumed underlying spectral model.

The full sample size and X-ray source matching background
When combining the samples from different X-ray missions, we adopted the average flux for duplicate sources in the three catalogs.The total number of sources is 255, which constitutes our Full Sample.We adopted Gaia DR3 distances to calculate the X-ray luminosity, along with corresponding uncertainties for each object.The ASAS-SN names, common names, Gaia DR3 IDs, J2000 coordinates (R.A. & Dec.), orbital periods (P ), distances, and X-ray luminosity (log L X ) with the lower and upper errors are presented in Table 1.
We estimated the expected "background" random match to the X-ray sources using the package astropy.coordinates.Firstly, we add a 1 • offset to each object in a random direction.Then, we employ the same cross-matching method as described in Sections 2.1, 2.2 and 2.3.The random "background" rate of X-ray matching is 1.18% (3/255), which is negligible for our analysis.

Effective Temperature, Gravity and Metallicity
To further obtain the stellar atmospheric parameters and bolometric luminosity for EBXs, we cross-matched our Full Sample with the Large Sky Area Multi-Object Fiber Spectroscopic Telescope Data Release 9 catalog (LAMOST DR9; Zhao et al. 2012;48 counterparts), and the Gaia DR3 catalog (Gaia Collaboration et al. 2023).LAMOST DR9 provides one set of the atmospheric parameters T eff , log g, and [Fe/H], while the Gaia DR3 Astrophysical Parameters Supplement Catalog8 in Gaia Collaboration et al. (2023) provided two sets of T eff , log g and [M/H],9 each from GSP-Phot Aeneas, for the MARCS (named Gaia3M ; 190 counterparts) and PHOENIX (named Gaia3P ; 138 counterparts) libraries, respectively, using BP/RP spectra.The parameter values from the LAMOST DR9, Gaia3M , and Gaia3P catalogs are generally consistent with each other.We finally chose the T eff , [M/H] and log g values from Gaia3M because it provides the largest number of counterparts to our Full Sample, while choosing the single optimal catalog can avoid the heterogeneity by combining multiple catalogs.The adopted stellar parameter values are listed in columns 9, 12, and 15 of Table 1.
We calculated the bolometric luminosity for the 190 counterparts in the Gaia3M catalog, following the method provided by the Gaia data release documentation10 as follows: where M G and G are the absolute and apparent G−band magnitudes, while M sun is the solar bolometric magnitude of 4.74 mag, BC is the bolometric correction, and D is the distance.A G is the StarHorse extinction in the G-band  1.

Mass and Radius
We developed the single-and binary-star spectral models with machine learning (Liu et al. 2024, in preparation) to facilitate the spectral fitting for the LAMOST DR9 data.We utilized the spectra from LAMOST DR9 and the stellar parameters obtained from the Apache Point Observatory Galactic Evolution Experiment (APOGEE) DR16 as our training dataset.To create the single-star spectral model, we employed the neural network version of the Stellar Label Machine (SLAM) (Zhang et al. 2020a,b).Subsequently, the binary-star model was constructed by combining two single-star models while considering their respective radial velocities.In our model, we also used the MIST model trained from stellar evolutionary tracks (Dotter 2016;Choi et al. 2016) to convert mass, age, and metallicity to effective temperature, gravity, and radius.Our single-and binary-star spectral models have been applied in the spectral fitting in the search for compact objects (Zhao et al. 2023).
We directly performed binary-star model fitting on the LAMOST DR9 spectra of 40 EBXs with the signal-to-noise ratio S/N > 30 (out of a total of 48 sources as mentioned in Section 2.5).As an example, in Figure 1, we present the spectral fitting results for five sources, indicating that the observational spectra can be well-fitted by our model.Eventually, the fitting yielded parameters of each component, i.e., the mass M and radius R, are presented in Table 2.We use subscripts 1 and 2 to denote the more massive primary and less massive secondary stars, respectively.
We did not apply the temperature, metallicity, and surface gravity derived simultaneously from the binary-star model fitting to Sections 2.5, which reduces the potential impact on the statistical results due to the differences in parameter derivation methods.Furthermore, the analysis of the relationship between mass and radius and the properties of X-ray radiation can serve as an independent validation for the analysis of other parameters (e.g., period and effective temperature), as they are obtained through mutually independent methods.

Rossby numbers
The Rossby number R O , first defined by Noyes et al. (1984), is a dimensionless quantity used to describe the stellar rotation-activity relation, specifically in the context of stellar activity and the dynamo processes.It is defined as the ratio of the rotation period (P ) of a star to its convective turnover time τ t , i.e., R O =P /τ t , which represents the characteristic timescale for convective motions in the star's interior.For the calculation of τ t , we used the empirical color-log(τ ) relation developed by Wright et al. (2018), which is applicable in the range 1.1 < V − Ks < 7.0, where V and Ks are the average magnitudes collected from ASAS-SN and 2MASS surveys, respectively.The A V and A Ks (= 0.596×A V ; Wang & Chen 2019) from Starhorse (Anders et al. 2022) was used to apply the extinction correction for the observed average V -band and Ks-band magnitudes, respectively.The values of color V − Ks and τ t are listed in columns 27 and 28 of Table 1, while the Rossby number R O values are listed in the last column.
The 190 objects in our Full Sample with stellar parameters are used in the subsequent analyses.We plot these objects on the log T eff − log g diagram in Figure 2, where each solid line shows the theoretical isochrone for stars with the same age and different masses.These isochrones, in the range of 10 8.8 to 10 10.0 years with intervals of 0.2 dex, are derived from stellar evolutionary tracks computed with PARSEC (version 1.2S, Bressan et al. 2012), using solar metallicity.The sample is divided into two sub-samples based on the values of log g: Sample 1 with log g > 4.0 and Sample 2 with log g < 4.0.Sample 1 contains objects where surface gravity generally decreases with increasing temperature, indicating that the radii of these stars increase with increasing temperature.In contrast, Sample 2 have objects whose surface gravity increases with increasing temperature, suggesting that their radii decrease with increasing temperature.We can infer that Sample 1 mainly consists of main-sequence stars, while Sample 2 is mainly composed of sub-giants and giants, as well as a portion of stars about to depart from the main sequence.The investigations in log L X and log(L X /L bol ) in the following sections also show that they have different X-ray emission properties.It is worth noting that the determination of spectral parameters (T eff , log g, and [Fe/H]) did not consider the influence of binaries.El-Badry et al. (2018) pointed out that the temperature difference when fitting binaries with a single-star model in LAMOST is around 100 K, and the difference in log g is about 0.1 dex, both of which are much smaller than the overall parameter distribution range (4300 K< T eff <7900 K; 2.90< log g <4.60).We do not expect a significant impact of these uncertainties on the analysis presented in this work.We investigate the correlation between the orbital period and the X-ray emission for our EBXs.We performed linear regression for the correlation analysis between log P versus log L X , log L bol and log(L X /L bol ) using the Markov Chain Monte Carlo (MCMC) fitting procedure (see Figure 3).The results are as follows:
The statistical distributions of log L X , log L bol and log(L X /L bol ) are shown in the right panels of Figure 3.They generally follow the normal distributions with the best-fit parameters of µ = 30.56± 0.02, 34.22±0.05and -3.43±0.04,and σ = 0.48 ± 0.02, 0.44±0.06and 0.48±0.04,respectively.The log L X (erg/s) range from 29.3 to 32.1, while the log(L X /L bol ) ranges from -5.2 to -2.3.
It is evident that the surface gravity is negatively correlated with the period (see the color map in Figure 3), which is not surprising given that lower surface gravity stars tend to have larger radii and thus longer orbital periods of the binary system (also see Section 3.3.4).

Effective Temperature
In Figure 4, panel (1) shows the distribution of effective temperatures of EBXs in Gaussian fitting with µ = 3.76±0.04(∼ 5700 K) and σ = 0.06 ± 0.04.The log T eff versus log L X and log(L X /L bol ) relationships are plotted with small circles in panels (2) and (3), respectively.Based on the classification in Section 3.1, we separately plotted Sample 1 and Sample 2 in panels ( 4) and ( 6).It is evident that there is a "turning" point at log T eff ∼ 3.73 in the log L X -log T eff relation for Sample 1, while for Sample 2, log L X is anti-correlated with log T eff .The thin arrows in panels ( 4) and ( 6) indicate the directions of decreasing log g.As defined in Section 3.1, objects in Sample 2 have smaller log g values than those in Sample 1.It is worth noting that Sample 2 objects generally have higher X-ray luminosity than Sample 1 objects.We used the segmented linear function to fit this distribution, and the marginalized posterior probability distributions are shown in Figure 10 in the Appendix.The fitting result is listed as follows, where k 1 and k 2 are the slopes of two fitting lines, respectively; log T eff,break is the breakpoint, and b is the intercept of the first part of the segmented linear fitting.The marginalized posterior probability distribution in Figure 10 shows the break point log T eff,break = 3.73 (T eff ∼ 5400K).The log L X -log T eff anti-correlation of Sample 2 is best represented by the following equation, log The Kendall's τ test (Kendall 1990) shows that τ is -0.42 with confidence 1 − P τ > 99.99% (> 5σ).
In contrast to the unusual behavior of the log T eff − log L X relation, both Sample 1 and Sample 2 exhibit highconfidence (> 5σ) anti-correlations between the X-ray activity level (log(L X /L bol )) and effective temperature.The difference lies in that when we consider along the increasing log T eff (and thus decreasing log(L X /L bol )), the surface gravity log g decreases in Sample 1 but increases in Sample 2. The results of the linear fitting and Kendall's τ test are listed in Table 3.

Metallicity and Surface Gravity
The statistical distributions of the metallicity and surface gravity values of our sample are both modeled with Gaussian profiles, resulting in the best-fit parameters of (µ, σ) = (−0.28± 0.03, 0.28 ± 0.07) for [Fe/H], and (µ, σ) =  (4.11± 0.01, 0.20 ± 0.02) for log g (see panels 1 & 4 in Figure 5).The tail towards smaller log g indicates the giant and sub-giant star population.Linear regressions with MCMC and Kendall's τ test are performed on the [Fe/H] − log L X , [Fe/H] − log(L X /L bol ), and log g − log L X relations.The best-fit parameters are listed in Table 4.The metallicity [Fe/H] is marginally correlated with log(L X /L bol ) at < 3σ significance.The surface gravity log g has strong anti-correlation with log L X at a confidence level of > 5σ.For the log g − log(L X /L bol ) relation, the segmented linear fitting with MCMC was employed and listed in Equation 7. The marginalized posterior probability distributions of this fitting are shown in Figure 11, which indicates that the log g − log(L X /L bol ) relationship has a breakpoint at log g break = 4.03 ± 0.02.This value is consistent with the dividing value log g = 4.0 for Sample 1 and Sample 2 within 2σ, reinforcing the differences in the X-ray activity level of the two subsamples.log (2.43 ± 0.07) × log g − (13.50 ± 0.10), (log g > 4.03). (7)

Magnetic activity and Stellar Mass
Mass, a fundamental stellar parameter, dictates a star's temperature across various evolutionary stages, offering insights into the relationship between temperature and magnetic activity.It should be noted that each binary system corresponds to one X-ray counterpart.We carried out the analysis for the masses of primary component M 1 versus the binary X-ray luminosity log L X and the magnetic activity level log(L X /L bol ) in Figure 6, which is classified based on Sample 1 and Sample 2. Overall, in panel (1), there is an increase followed by a decrease in X-ray luminosity in the direction of increasing mass.
In panel (3), the 'peak-like' relationship between M 1 and log L X in Sample 1 can be well described by the segmented linear model specified in Equation 8 with a peak at M 1,break = 1.04 +0.03 −0.04 M ⊙ (corresponding to a temperature of ∼ 5900 +80 −120 K in the main sequence; Cox 2000).The marginalized posterior probability distribution is shown in the left panel of Figure 12.For sources in the main-sequence stage with M 1 < 1.04 M ⊙ , their X-ray luminosity increases with the primary star's mass, whereas for sources with M 1 > 1.04 M ⊙ , this trend is the opposite.In panel ( 5), for the sub-giants and giants sources in Sample 2, their fitted line presented in Equation 9implies only a decrease in their X-ray luminosity with increasing mass.The Kendall's τ test (Kendall 1990) shows that the τ is -0.46 with confidence 1 − P τ = 98.85% (> 2σ).Moreover, the objects from Sample 2 have higher X-ray luminosity compared to those from Sample 1 in the mass range of ∼0.8 to 1.6 M ⊙ , as indicated by the former consistently being located above the latter at a certain mass.This indicates that the X-ray radiation luminosity of the sub-giants and giants is likely higher than that of the main-sequence stars with similar mass. log log L X = (−0.49± 0.05) × M 1 + (31.34 ± 0.06).( 9) Figure 6 panel (2) shows a trend of monotonous decrease with primary stars' masses, indicating that as the mass increases, the level of X-ray activity weakens.In Figure 6 panels (4) and ( 6), all the mass-magnetic activity level relationships follow similar negative correlations with high significance (> 2σ) as listed in Table 5, which means that the lower-mass EBX holds higher levels of magnetic activity compared to a higher-mass one.
The same analytical processes are also applied to the secondary component in Figure 7, revealing that the statistical results of the secondary star's mass versus the binary X-ray luminosity and magnetic activity level are nearly similar to those of the primary star.The fitting result of M 2 − log L X for Sample 1 with the peak located at M 2,break = 0.64 ± 0.02 M ⊙ (corresponding to ∼ 4310 ± 60 K; Cox 2000) is listed in Equation 10, while the marginalized posterior probability distribution is shown in the right panel of Figure 12.The fitting result of Sample 2 is presented in Equation 11 with τ = −0.27and confidence 1 − P τ > 83.47% (> 1σ).In The study of radius can directly link the magnetic activity properties with the geometric structure of EBXs.In Figure 8, we investigated the correlation between the radii of primary components R 1 , secondary components R 2 and binary systems' equivalent radii of EBXs with the binary X-ray luminosity log L X  3) and ( 4) are for the Sample 1, while panels ( 5) and ( 6) are for the Sample 2. The dashed lines in all panels represent 95% uncertainty ranges of the MCMC fitting.4) are for the Sample 1, while panels ( 5) and ( 6) are for the Sample 2. The dashed lines in all panels represent 95% uncertainty ranges of the MCMC fitting.
and activity level log(L X /L bol ).Sample 1 and Sample 2 are distinguished by triangles and circles, respectively.The fitting results for R − log L X are listed in Table 7.Compared to R 2 (with confidence < 1σ), the X-ray luminosity shows a high-confidence (∼ 2σ) positive correlation with R 1 and R 1+2 .All of the above positive correlations establish that X-ray luminosity is proportional to the radii of EBXs, meaning that X-ray luminosity is proportional to the surface area of EBXs.
For the magnetic activity level, we find that all the R −log(L X /L bol ) can be described by the segmented linear fitting (using the same fitting procedure as Equation 3.3.1)with breaks at R 1,break = 1.42 ± 0.05 R ⊙ , R 2,break = 1.29 +0.03 −0.02 R ⊙ and R 1+2,break = 2.27 +0.05 −0.03 R ⊙ .The fitting results are listed in Equations 12, 13 and 14 with the marginalized posterior probability distributions in Figure 13.These results indicate that the magnetic activity level of EBXs first decreases and then increases with the growth of the radius.
Along the direction of increasing radius and decreasing surface gravity, one can find that the distributions of R−log L X versus log g −log L X and R − log(L X /L bol ) versus log g −log(L X /L bol ) are consistent.These two consistencies establish the correspondence between the magnetic activity of EBXs at two levels: geometric structure (R) and atmospheric parameters (log g), and mutually validate each other.
Moreover, as shown in panels (2), ( 4) and ( 6), the clear differences in magnetic activity levels between Sample 1 and Sample 2 are evident in all R − log(L X /L bol ) relationships.The sources in Sample 1 (triangle points) mostly follow a negative correlation distribution, while those in Sample 2 (circle points) follow a positive correlation distribution.This once again confirms the necessity and correctness of our sample classification (as described in Section 3.1) from the perspective of radius.From the color map of surface gravity overlaid on the radius, it's evident that there is a negative correlation between surface gravity and the radii of the component stars of EBXs.All of the above indicates that the components with lower surface gravity tend to have larger radii, confirming the statement made in Sections 3.1 and 4 (the first paragraph): lower surface gravity sources tend to have larger radii, consequently affecting the eclipsing orbital radii and periods.

Magnetic activity and Rossby number
Describing the relationship between R O and log(L X /L bol ) provides a more direct way to study coronal activity and rotation in low-mass stars (Núñez et al. 2022).For fitting the R O -log(L X /L bol ) relation, a widely used model is a constant region connected to a power-law model (e.g., Wright et al. 2018;Núñez et al. 2022), which is shown as follows where R O,sat is the Rossby number at which X-ray saturation occurs, (L X /L bol ) sat is a constant indicating the saturated X-ray activity level at R O ≤ R O,sat ; β is the index of the power law model for the unsaturated part of the X-ray activity, while C is a constant.Núñez et al. (2022) pointed out that there is no difference in the coronal parameters (R O versus log(L X /L bol )) between single and binary stars.In Figure 9, we only use the R O -log(L X /L bol ) distribution (black open circles) for binary stars as a background.As shown in Figure 9, the data of Sample 1 and Sample 2 are presented with red and blue circles, respectively.We attempted to fit these parts separately using the above model and also tried a combined fit.However, our data shows little sign of the power-law component.This leads us to focus on only the constant part represented by (L X /L bol ) sat .The fitting results log(L X /L bol ) sat = −3.14 ± 0.10 and −3.66 ± 0.10 for Sample 1 and Sample 2 are shown in Figure 9 as red and blue lines, respectively.An overall fit to the data of both parts yields log(L X /L bol ) sat = −3.40 ± 0.10.For EBXs overall, as the Rossby number increases from 0.01 to 0.5, the range of log(L X /L bol ) gradually widens from [−3.5, −2.5] to [−5.0, −2.5]; there is no obvious turning point R O,sat in the entire R O range (see Figure 9).Núñez et al. (2022) point out that for the single-star sample, R O,sat appears around 0.19, while for the dwarf binary, it appears around 0.15.One reason that EBXs do not show R O,sat at a similar location may be that the sample has few sources at those locations to form an effective model-fitting constraint.

DISCUSSION
Based on the distribution of log T eff − log g, we divided the sample of EBXs into two sub-samples, Sample 1 (log g > 4.0) and Sample 2 (log g < 4.0).The former (mainly main-sequence stars) shows a positive correlation trend between temperature and log g, while the latter exhibits a negative correlation trend (mainly sub-giants and giants, as well as a portion of stars about to depart from the main sequence).The log g − log(L X /L bol ) and R − log(L X /L bol ) relationships shown in Figure 5 panel (6) (log g break = 4.03 ± 0.02 within the 2σ range of log g = 4.0) and Figure 8 panels ( 2), ( 4) and ( 6) confirm the existence of differences in the X-ray magnetic levels between these two samples, validating the necessity of this classification for studying the X-ray radiation properties of EBXs.Figures 3 and 8 show that surface gravity maintains a clear negative correlation with both period and radius, respectively.In other words, the period, radius, and surface gravity change almost synchronously, meaning that sources with longer periods typically have larger radii and lower surface gravity.

Relation of X-ray Emission with Period
For EBXs, the component stars are generally considered to be tidally locked.Their rotational periods are the same as the systemic orbital period (Mazeh 2008).Despite the changes in the stellar evolutionary process of EBXs owing to the filling of the Roche lobes, this would not result in their complete loss of single-star-like temperature and luminosity properties (Yakut & Eggleton 2005).The unbiased distribution for various parameters of close binaries collected in Zhang et al. (2019) suggests that the period of EBs is proportional to the radius.Our EBXs sample also suggests that an increase in the orbital period indicates an increase in the EBXs' radius and surface area.Because the bolometric luminosity is proportional to the star's surface area, it is positively correlated with the period, as shown in Figure 3, panel (2).
For the X-ray emission of EBXs, the linear correlation of the X-ray luminosity with the period has an almost equal slope to that of the bolometric luminosity, as shown in Figure 3, which makes the X-ray activity level log(L X /L bol ) weakly negatively correlated with log P .This suggests that the X-ray emission of the EBXs also originates from the stellar surface and that the X-ray luminosity is proportional to the surface area, which can be verified by the positive relationship between radius and log L X in Section 3.3.4.In this scenario, the X-ray emission of EBXs is produced by the overall surface convection zone of the star via the magnetic dynamo mechanism rather than concentrated in certain dense regions, although we cannot exclude the presence of dense active regions (e.g., spots), which could enhance the X-ray emission to some extent.Additionally, the relationships listed in Equation 3 provide an empirical method to quickly determine the X-ray luminosity of an EBX based on the period, and can also be used to compare this empirical prediction with actual observed X-ray luminosity; the relationship also provides an observational constraint for the construction of a model of the magnetic activity radiation for the EBXs.

Relation of X-ray Emission with Surface Gravity and Radius
From the perspective of atmospheric parameters, along the direction of decreasing log g, log g − log L X shows an overall increasing trend.The magnetic activity levels of main-sequence components in Sample 1 (log g > 4.0 in Figure 5) exhibit a consistent decrease with decreasing log g, while for sub-giants and giants in Sample 2 (log g < 4.0 in Figure 5), this trend is the opposite.From the perspective of the geometric structure of binary systems, with increasing radius, R − log L X also generally exhibits a positive correlation trend.For the magnetic activity level, in Sample 1 (triangle points in Figure 8), the magnetic activity levels of main-sequence stars show a decrease with increasing radius, while for sub-giants and giants in Sample 2 (circle points in Figure 8), this trend is the opposite.It is evident that log g and R maintain a high degree of consistency with the magnetic activity properties of EBXs.These two parameters (log g and R) obtained independently connect atmospheric parameters with the geometric structure of EBXs, confirming that the magnetic activity of EBXs likely originates from the convection zone on the stellar surface.

Local Structure in log
The distribution of color-log L X has been used in studies of stellar magnetic activity, such as those published by Güdel (2004) and Núñez et al. (2022).In contrast, as shown in panels (4) of Figure 4, we directly describe the relationship between the effective temperature log T eff and the X-ray luminosity log L X .
As implied by the best-fit line in Figure 4 panel (4) for Sample 1 objects, the X-ray luminosity log L X generally increases with temperature until reaching log T eff ∼ 3.73 (T eff ∼ 5400 K).Afterward, the X-ray luminosity begins to decrease with increasing temperature.Núñez et al. (2022) studied the X-ray emission properties of main-sequence stars and dwarf binaries at temperatures from ∼3000 K to ∼7900 K (corresponding to ∼M6 to late-A type; Cox 2000) of the clusters Praesepe and Hyades.They suggested that single and binary stars have similar distribution characteristics in color-log L X and color-log(L X /L bol ) relations.The former relationship has an overall positive correlation trend while the latter has an overall negative correlation trend.We find that the log T eff − log L X distribution of the main-sequence stars ( 4400 K to 7900 K; K6 to late A type) actually has a local structure, and our observations show that the log L X increases with temperature (log T eff ∼3.64 to 3.73; T eff ∼4460 K to 5400 K) and then decreases (log T eff ∼3.73 to 3.90; T eff ∼5400 K to 7950 K).
Comparing with the color-log L X relationship in Figure 9 of Núñez et al. (2022), one can identify that the X-ray luminosity of two clusters both show initial increasing trends (∼M0 to ∼G1 type; ∼3840K to ∼5860K; Cox 2000) followed by decreasing trends (earlier than G1 type; T eff ⪆ 5860K) as the color index decreases, and the turning point of the trend occurs at the G1 spectral type with the corresponding temperature of ∼5860 K. Owning to the scatter of data, our results on this local trend of the X-ray luminosity of main-sequence stars with temperature are generally consistent with that of Núñez et al. (2022).So we find a peak-like (first up and then down) trend in the log T eff −log L X space for the EBXs with main-sequence components for the first time.

Magnetic Activity versus Effective Temperature
As shown in Figure 4 panels ( 4) and ( 5), for the main-sequence stars in Sample 1, at log T eff ∼ 3.64, the log(L X /L bol ) values of the EBXs distribute around the saturation level -3 of magnetic activity, and we suggest that EBXs at this temperature have the thickest surface convection zones.However, since binaries at this temperature correspond to the lowest mass and radius, the X-ray luminosity is the lowest.Based on the fitting of distribution log T eff − log(L X /L bol ) (the first line in Table 5), the magnetic activity still holds the saturation level ≤ −3.0 until log T eff ∼ 3.70.The increase in temperature and period corresponds to objects with larger mass and radii and, hence, larger surface areas.Therefore, X-ray luminosity increases with increasing temperature.According to the magnetic dynamo model, as the temperature increases, the convection zone becomes thinner, leading to a lower magnetic activity level.However, the larger radius and more X-ray emission area produce more X-ray radiation to compensate for the decrease of X-ray activity, which leads to an X-ray luminosity peak at the temperature log T eff ∼ 3.73.As the temperature continues to increase, it reaches a point where there is insufficient material in the convection zone on its surface to maintain a typical magnetic dynamo, which is primarily powered by convection and rotation.Then, the X-ray luminosity continues to decrease because the weakening of the magnetic activity owing to the thinning of the convection zone cannot be compensated by the increase in X-ray luminosity caused by the larger stellar radius and surface area.
For the sub-giants and giants in Sample 2 shown in Figure 4 panels ( 6) and ( 7), along the direction of the arrow, when the temperature decreases, the log g decreases (color map; corresponding to the period and radius increases).
Conversely, both the X-ray luminosity log L X and activity level log(L X /L bol ) increase monotonically.The above phenomenon can be explained by the decrease in temperature prompts a thickening of the convection zone.Simultaneously, the direction of temperature decrease is also the direction of increasing radius, which increases the area generating X-ray radiation.The combination of these two factors leads to an X-ray radiation trend that steadily increases as temperature decreases.
As shown in Figure 4 panels (3), (5), and (7), component stars of EBXs in different evolutionary stages (the main sequence, sub-giant, and giant stages) within Sample 1 and Sample 2 exhibit similar magnetic activity levels under the same temperature conditions.This may indicate that the magnetic activity in EBXs is related to temperature, and EBXs at different evolutionary stages can have similar magnetic activity levels.
In summary, based on the magnetic dynamo model, we explained the physical mechanism of the two different magnetic activity properties in sub-samples of EBX by elucidating the relationship between changes in magnetic activity level and X-ray luminosity due to variations in convection zone thickness and radiation area.The effective temperature T eff may serve as an indicator of magnetic activity levels.For EBXs, at lower temperatures (log T eff ∼ 3.64), the magnetic activity on the stellar surface reaches saturation.In contrast, at higher temperatures, the convective layer on the stellar surface gradually becomes thinner, leading to weakened magnetic activity.We suggest that the temperature log T eff ∼ 3.73 serves as a crucial threshold for EBXs, indicating the balance between the X-ray luminosity diminishing due to the thinning of the convection zone and the increasing X-ray luminosity caused by the enlargement of the convection zone area.This also demonstrates, from the perspective of stellar structure (convection zone thickness and surface area), that EBX systems are highly significant objects for testing the stellar dynamo model.

Magnetic Activity versus Stellar Mass
As shown in Figures 6 and 7, along the direction of mass increasing, both the primary stars and secondary stars exhibit an initial positive correlation followed by a subsequent negative correlation with log L X for Sample 1, and a monotonic decreasing trend for Sample 2. Meanwhile, they all show a consistent negative correlation with log(L X /L bol ).Therefore, we only describe the relationships between the primary star's mass and the magnetic activity.
When the primary star's mass is at its lowest value (Figure 6 panel 3), EBXs in Sample 1 (Figure 6 panel 4) exhibit the highest level of magnetic activity with the thickest convection zone.However, owing to the smallest radius of the star at this point, the X-ray luminosity is at its lowest value.As the mass increases to 1.04 +0.03  −0.04 M ⊙ , the magnetic activity level continues to decrease, indicating a thinning of the convection zone and a decrease in X-ray luminosity.Nonetheless, the increased X-ray radiation due to the enlarged radius and surface area counteracted this part of the decrease, resulting in a peak in X-ray luminosity.With the M 1 further increasing, the increase in X-ray radiation from the expanded surface area is not sufficient to counterbalance the continuous weakening of magnetic activity caused by the ongoing thinning of the convection zone.This results in a sustained decrease in X-ray luminosity.For the Sample 2 in Figure 6 panel ( 5) and ( 6), as the mass of the primary star decreases, the X-ray luminosity increases.This occurs because the increased radius enlarges the radiating surface area (reflected by decreasing log g and the circles in Figure 8), and the thickening convection zone enhances the magnetic activity level (reflected by the increasing magnetic activity level).
Additionally, since Sample 2 comprises components that are sub-giants and giants, as well as objects about to leave the main sequence, their radii are larger than those of main-sequence components in Sample 1 with the same mass and temperature, consequently resulting in higher X-ray radiation intensity.Therefore, in the log T eff − log L X and M − log L X distributions, both demonstrate that the X-ray luminosity of Sample 2 is higher than that of Sample 1.
The correlations between the masses of EBXs and magnetic activity serve as a direct validation of the relationships between temperature and magnetic activity.For X-ray luminosity of Sample 1, the trend of initially increasing and then decreasing with mass mirrors the data distribution in log T eff − log L X .Moreover, the temperature ∼ 5900 +80 −120 K of the mass break-point for the primary star corresponds closely to T eff,break ∼ 5400K within a range of about 3σ, considering the scatter of data.In the case of magnetic activity level for Sample 1, its negative correlation trend with temperature is also replicated by stellar mass.It is evident that the observed phenomena above can be explained by a positive correlation between mass and temperature for the main-sequence components with log g < 4.0 in EBXs.For Sample 2, there is a mutual confirmation relationship between log T eff − log L X and M − log L X , as well as between log T eff − log(L X /L bol ) and M − log(L X /L bol ).More importantly, in this work, stellar mass determination is accomplished through binary spectral fitting, which is independent and unaffected by the temperature used in this study.This independently validates the physical processes discussed in Section 4.4.
The Rossby number R O denotes the characteristic timescale for convective motions occurring within the star's interior.In Figure 9, for R O , the values in Sample 1 are lower than those in Sample 2, possibly because the sources in the former generally have shorter periods than those in the latter, indicating a relatively shorter timescale for convective motions in the star's interior.We also compare the R O -log(L X /L bol ) distribution of binary sample in Núñez et al. (2022) with two sub-samples of EBXs.The saturation level of the whole binary sample in Núñez et al. (2022) is at ∼ −2.98, which is within 2σ compared to the magnetic activity level of Sample 1 (−3.14 ± 0.10).It indicates that the distribution of the magnetic activity levels of Sample 1 and that of binaries in Núñez et al. (2022) are similar.The R O -log(L X /L bol ) distribution of Sample 2 is relatively lower compared to that of Sample 1 by 0.52 dex, possibly because a higher fraction of high-temperature sources in Sample 2 leads to a relatively lower average level of magnetic activity.This makes that the combined distribution of the magnetic activity level of our full sample is lower compared to the binary sample collected by Núñez et al. (2022).
We suggest that the range of orbital sizes could also contribute to differences in the distribution of magnetic activity levels.The differences in the orbital sizes between the samples of Núñez et al. (2022) and our work are evident in their respective distributions of orbital periods.The binaries in Núñez et al. (2022) have a period distribution with 1.183, 7.955, and 13.910 days at the 16th, 50th, and 84th percentiles, respectively, while our sources have the period value of 0.403, 0.597, and 1.364 days at the corresponding percentiles.The former sample has substantially more binaries with wider orbits.Núñez et al. (2022) indeed suggested that the components of their binaries may lack interaction, while there is no doubt that complex material exchanges and transfers occur in EBXs (Yakut & Eggleton 2005;Zhu & Qian 2011).Therefore, we infer that the different degrees of matter transfer or exchange may affect the magnetic activity.Further confirmation of this inference will require the collection of additional samples and detailed analyses of individual systems in the future.

Comparison with EWXs
The W Ursa Majoris (W UMa-type) binary, also referred to as an EW-type binary, is characterized by a contact configuration in which both components fill their Roche lobes and jointly share a common envelope.The EW-type binaries with X-ray emission (EWXs; e.g., Stȩpień et al. 2001;Gondoin 2004;Chen et al. 2006;Liu et al. 2019Liu et al. , 2022) ) also constitute an important type of X-ray sources.Additionally, EB-type binaries are generally considered to be precursor stars to EW-type binaries (Yakut & Eggleton 2005).Therefore, it is worth comparing the X-ray radiation properties of EBXs and EWXs.
The orbital period of EBXs in our sample ranges from ∼0.2 to ∼10 days.The X-ray and bolometric luminosities cover ranges of are ∼1.74×10 29to ∼1.32×10 32 erg s −1 , and ∼5.20×10 32 erg s −1 to ∼3.98×10 36 erg s −1 , respectively.The lower limits of luminosity are close to those of the EWXs studied by Liu et al. (2022), while the upper limits are approximately two orders of magnitude higher than those of the EWXs.However, if we limit the EBXs in the same period ranges as the EWXs (0.2 to 0.44 days), these physical parameters of EBXs are almost similar to those of EWXs (Liu et al. 2022).
EBXs and EWXs have similar qualitative correlations between the stellar spectral parameters (temperature, metallicity, and surface gravity) and X-ray emissions (luminosity and activity level), except for the log T eff -log L X relation (see Section 3.3.1 in this work and Section 3.2 of Liu et al. 2022).For this particular relationship, the positive correlation trend of both EBXs and EWXs in the temperature range of 4500 K to 6300 K (Liu et al. 2022) and the range of X-ray luminosity 10 29.5 − 10 30.5 erg s −1 is almost identical.Furthermore, if more EWXs with effective temperatures greater than 6300 K are found, their X-ray luminosity will likely show a downward trend similar to what EBXs have exhibited by EBXs.Both EBXs and EWXs show a negative log T eff -log(L X /L bol ) correlation, that is, objects with low temperatures have a higher magnetic activity level.
The statistical distributions of effective temperatures of both EBXs and EWXs peak at near 5600 K, while the former does not have a significant cutoff at the high temperature (∼6300K) end.The metallicity distribution of EBXs peaks at -0.25 dex, which is lower compared to that of the EWXs (∼ −0.05 dex).For the distributions of surface gravity, both EBXs and EWXs have similar peak locations, while the former has a long tail in the interval from 3.0 dex to 3.75 dex.The EBXs present a better laboratory for testing the magnetic dynamo model because it reflects the magnetic activity characteristics of main-sequence stars and sub-giant and giant stars, and the similarities and differences between them as discussed in the above sections, while the EWXs only mainly include the main-sequence stars as components that have strong positive correlations between period, mass, and temperature at P ≲ 0.44 days.

SUMMARY
Based on the AVSD database, we collected the X-ray counterparts of 255 EB-type binaries from the XMM-Newton, RASS and Chandra databases.Correlation analyses of the period and spectral parameters (i.e., effective temperature, metallicity, surface gravity, the masses and radii of component stars) with the X-ray emission properties were performed for the first time for EBXs.Based on the log g − log T eff distribution, we divided the EBXs into Sample 1 with log g > 4.0 and Sample 2 with log g < 4.0.The former, which is primarily composed of main-sequence member stars, as the temperature increases, log g decreases, and the stellar radius increases.On the other hand, the latter, mainly composed of sub-giants, giants, and a portion of stars about to depart from the main sequence, exhibits the characteristic that as the temperature decreases, log g decreases, and the stellar radius increases.The main conclusions are as follows: 1.The X-ray and bolometric luminosity both increase with longer orbital periods.The rates of change are consistent, indicating that the increases in the X-ray and bolometric luminosity of the EBXs are almost synchronous along the period.The X-ray emission may originate from the convection zones of EBXs.Both parameters (log L X and log L bol ) are positively correlated with the surface area of the binary system.
2. Among the atmospheric parameter, surface gravity log g and X-ray luminosity show a strong negative correlation, while the distribution of log g − log(L X /L bol ) can be described using a segmented linear fit, where the level of magnetic activity of the EBXs is proportional to the surface gravity at log g > 4.0, and the opposite at log g < 4.0.log g (in decreasing direction) exhibits consistency with the binary geometric parameter R in the relationships with log L X and log(L X /L bol ).The results of both validate that the X-ray radiation of EBXs likely originates from the convection zone.In addition, metallicity [Fe/H] is almost independent of the X-ray emission.
3. We found that the X-ray emission luminosity of the main-sequence components with log g > 4.0 in EBXs shows an increasing and then decreasing trend with the effective temperature, and confirmed these distributions by comparison with binaries in the Paesepe and Hyadea clusters.
4. We found for the first time the differences in magnetic activity properties for EBXs in different evolution stages (main-sequence, sub-giant, and giant stages).These differences are reflected in the log T eff -log L X , log T efflog(L X /L bol ), log g-log(L X /L bol ), M -log L X , M -log(L X /L bol ) and R-log(L X /L bol ) relationships.Based on the magnetic dynamo model, we used changes in the surface convection zone area and temperature-induced changes in convection zone thickness to explain the physical origin.We suggest log T eff ∼ 3.73 as a crucial temperature value for EBX in testing the magnetic dynamo model because, at this temperature, a balance in X-ray luminosity is achieved due to the combined influence of variations in the thickness and surface area of the convection zone.We found a strong negative correlation between the temperature and the magnetic activity level of log(L X /L bol ).A higher temperature leads to a thinner convection zone, and thus weaker magnetic activity.Furthermore, the magnetic activity levels may be related to the temperature and mass of EBX.
5. We developed the single-and binary-star spectral model to fit the spectra of LAMOST DR9 and then derive the masses and radii for the primary and secondary components in EBXs.The mass versus magnetic activity (log L X and log(L X /L bol )) exhibits distributions similar to those observed in the effective temperature versus magnetic activity.Both mutually support the explanation of the relationship between the thickness of the convection zone and the surface area in EBX's X-ray activity.While mass provides insights into the physical essence, samples with effective temperature are more numerous and statistically significant.
6. Regarding the R O -log(L X /L bol ) relation of EBXs, it is difficult to constrain with a constant plus power-law model.The R O -log(L X /L bol ) distribution for EBXs in Sample 1 is consistent with the binaries in Praesepe and Hyade clusters within a 1σ range.The Sample 1 has a shorter timescale for convective motions compared to Sample 2 and has a higher magnetic activity level, which might be due to the former having more sources with short periods and low temperatures.The overall distribution of EBXs in this relationship is lower than that of binaries in clusters Praesepe and Hyades, which may result from the fact that the EBX sample contains sources with different material exchange or transfer rates.
7. The X-ray luminosity and activity levels of the EBXs were consistent with those of the EWXs at similar periods and temperatures.Because EBXs cover a wider range of periods and spectral parameters, they provide an important laboratory for studying magnetic generator mechanisms.
mission with instruments and contributions directly funded by ESA Member States and NASA.This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).

Figure 1 .
Figure 1.The comparison between observed and binary-star model fitting spectra, as well as the corresponding primary and secondary fitting spectra, for five EBXs.The gray area represents the observational errors in the spectra.

Figure 2 .
Figure 2. The distribution in log T eff − log g space for EBXs.The blue dashed line is located at log g = 4.0.Data points are indicated by circles.The curves are the isochrones with different ages from left to right, ranging from 10 8.8 to 10 10.0 (in red) years in steps of 0.2 dex.

Figure 3 .
Figure 3.The relationships between orbital period log P and X-ray luminosity log LX, bolometric luminosity log L bol , and X-ray activity level log(LX/L bol ).In the middle panel, the error bars of log L bol are smaller than the size of the symbols.The color map on the right illustrates the surface gravity values.

Figure 4 .
Figure 4.The distribution of T eff of our sample is plotted in panel (1) with the best-fit Gaussian profile.Panels (2) and (3) illustrates the log T eff -log LX and log T eff -log(LX/L bol ) relationships, respectively, with the symbols color-coded by log g (see the right colorbar).Panels (4) and (6) are the distributions for Sample 1 and Sample 2 in log T eff -log LX relation, respectively.Panels (5) and (7) are the distributions for Sample 1 and Sample 2 in log T eff -log(LX/L bol ) relation, respectively.The black lines are the (segmented) linear fitting results.The dashed lines in all panels represent 95% uncertainty ranges of the MCMC fitting.The arrows show the direction of the decreasing log g.

Figure 5 .
Figure 5.The relationships for the metallicity [Fe/H] and surface gravity log g versus X-ray luminosity log LX and activity level log(LX/L bol ).Panels (1) and (4) are the distributions of [Fe/H] and log g with Gaussian fittings.The dashed lines in all panels represent 95% uncertainty ranges of the MCMC fitting.

Figure 6 .
Figure 6.The relationships for the primary components masses M1 versus X-ray luminosity log LX (panel 1) and activity level log(LX/L bol ) (panel 2).Panels (3) and (4) are for the Sample 1, while panels (5) and (6) are for the Sample 2. The dashed lines in all panels represent 95% uncertainty ranges of the MCMC fitting.

Figure 7 .
Figure 7.The relationships for the secondary components masses M2 versus X-ray luminosity log LX (panel 1) and activity level log(LX/L bol ) (panel 2).Panels (3) and (4) are for the Sample 1, while panels (5) and (6) are for the Sample 2. The dashed lines in all panels represent 95% uncertainty ranges of the MCMC fitting.

Figure 8 .
Figure8.The relationships for the radii of primary components R1 (panel 1 and 2), secondary components R2 (panel 3 and 4) and the binary system equivalent radii R1+2 (panel 5 and 6) versus X-ray luminosity log LX and magnetic activity level log(LX/L bol ).The objects from Sample 1 and Sample 2 are represented by triangles and circles, respectively.The dashed lines in all panels represent 95% uncertainty ranges of the MCMC fitting.

Figure 9 .
Figure 9. RO versus log(LX/L bol ) for EBXs.The red and blue open circles are objects from Sample 1 and Sample 2, respectively, while the red and blue solid lines are the MCMC fitting results, respectively.The dashed lines are the corresponding 1 σ fitting error.The black open circles are binary candidates collected from Núñez et al. (2022).

Figure 11 .
Figure 11.Marginalized posterior probability distributions from the MCMC fitting for the log g-log(LX/L bol ) relationship.The parameter values are the peaks of the one-dimensional distributions, while the vertical dashed lines are located at the 16th, 50th, and 84th percentiles.

Figure 12 .
Figure 12.Marginalized posterior probability distributions from the MCMC fitting for the M1-log LX (left) and M2-log LX (right) relationships.The parameter values are the peaks of the one-dimensional distributions, while the vertical dashed lines are located at the 16th, 50th, and 84th percentiles.

Table 2 .
The Masses and Radii of the components of 40 EBXs

Table 3 .
The parameters of linear fitting (log(LX/L bol ) = a × log T eff + b) and Kendall's τ test for Sample 1 and Sample 2 in the relation log T eff − log(LX/L bol ).

Table 5 .
The parameters of linear fitting (log(LX/L bol ) = a × M1 + b) and Kendall's τ test for Sample 1 and Sample 2 in the relation M1 − log(LX/L bol ).

Table 6 .
The parameters of linear fitting (log(LX/L bol ) = a × M2 + b) and the Kendall's τ test for Sample 1 and Sample 2 in the relation M2 − log(LX/L bol ).

Table 6 ,
we listed the linear fitting of the M 2 − log(L X /L bol ) for Sample 1 and Sample 2.

Table 7 .
The parameters of least-square fitting and Kendall's τ test for R1, R2 and R1+2 versus X-ray luminosity log LX.