Toward Early-type Eclipsing Binaries as Extragalactic Milestones. I. Physical Parameters of OGLE-LMC-ECL-22270 and OGLE-LMC-ECL-06782

We selected the best systems for this purpose from the OGLE catalog of 26121 eclipsing binaries in the LMC (Graczyk et al. 2011). All of them have very good quality and stable light curves with very similar eclipse depths, and are well detached, with no or little proximity effects present. These properties make them the best targets for the distance determination among all known extragalactic early-type systems. In total we have selected nine such systems and they are all located close to the line of nodes and the center of the LMC galaxy. They are all short-period binaries (periods from 3 to 6 days) and are bright enough to be observed in the LMC, which make them an optimal sample for the purposes of this project.

Although in principle distances to these LMC systems could be measured directly, the expected precision from such direct measurement is not as high as that for the systems composed of late-type giants. For this reason, to calibrate the SBCR we are going to use the known distance to the LMC, which was measured recently with an accuracy of about 1% (Pietrzynski et al. 2019). We will take into account the geometry of the LMC and apply the corresponding corrections to the expected values. Using these distances together with the radii from the modeling we can calculate the angular diameters of the stars of the system. With observed magnitudes and angular diameters we can calculate the surface brightness and obtain the SBCR for early-type stars with unprecedented precision.

We have collected all the photometric data available for the two systems up to now. For both, the most important data (in the sense of the coverage range, data quality, and the use of standard filters) come from the OGLE project. The OGLE observations were carried out during the third and fourth phase of this microlensing survey with the 1.3 m Warsaw telescope at the Las Campanas Observatory. In our analysis we make use of their measurements taken in the I and V bands of the Johnsons-Cousins system.

For the BLMC-01 system in total we collected 821 observations in the I band and 116 data points in the V band from OGLE-III and OGLE-IV, while for BLMC-02 we collected 1004 and 187 data points for the I and V bands, respectively.

The BLMC-01 system was also observed by the EROS-2 survey, where its ID number is lm0031l22987. The survey was carried out with the Marly 1 m telescope at ESO, La Silla from 1996 July to 2003 February. Observations were performed in two (blue and red) wide passbands with central wavelengths at 490 nm (B_EROS) and 670 nm (R_EROS), respectively. The filters used are not standard, and for this reason in our analysis we have only used the R_EROS band, which is centered close to the I_C standard band and can be transformed to a Johnson-Cousins standard system with a precision of ∼0.1 mag, using the simple relation from Tisserand et al. (2007): R_EROS = I_C. We used 421 measurements from this survey, which allowed us to obtain a greater observational coverage in time for the I-band light curve—in total 1242 data points spanning over 17 yr for EROS-2 and OGLE data, together.

This system is also located in one of the fields of the VMC survey, where its ID is J054047.30-692028.25. These survey observations were taken using the VISTA 4.1 m telescope located at the Paranal Observatory in Chile during one of the ESO Public Surveys. In the photometric catalog of the VMC survey there are 14 measurements in the K_S band. Fortunately, both eclipses were covered and we could use these observations in the light-curve modeling.

Unfortunately, BLMC-02 was not observed by the EROS and VMC projects, but additional observations from the MACHO survey are available. MACHO observations were carried out with the 1.27 m telescope at Mount Stromlo Observatory, Australia, in two photometric bands (red and blue), which can be transformed to the standard R_C an V_J bands using relations from Faccioli et al. (2007) with an accuracy of 0.035 mag (Alcock et al. 1999). The star ID in the MACHO database is 23.3908.17.

In Table 2 for both systems we show the numbers of the measurements for all the surveys described above and in Figures 2 and 3 we present a graphical summary of all the photometric data used in the analysis.

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Although our systems are relatively bright for objects located in the LMC, they are still not easy targets for spectroscopic observations and the world's best telescopes are needed to obtain good quality data with reasonable exposure times. We have carried out such observations at two sites in Chile.

The majority of our high-resolution optical spectra were acquired with the Very Large Telescope (VLT) and the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al. 2000) mounted at the Nasmyth B focus of UT2 Kueyen at Paranal Observatory. Observations were taken with the UVES standard configuration DIC1 390+564, which provided coverage of two wavelength ranges 3260–4520 Å and 4580–6690 Å. We used a slit of 07 for the red part and 09 for the blue part of the spectra, obtaining a spectral resolution of R ∼ 50,000. The exposure times per spectrum ranged from 20 to 30 minutes.

Additional spectra were obtained with the red and blue arms of the MIKE spectrograph (Bernstein et al. 2003) on the 6.5 m Magellan Clay Telescope at the Las Campanas Observatory. All the observations were taken with a 07 slit at a resolution of about 40,000 for the red side and about 50,000 for the blue side. The spectra were taken with integration times from 10 to 30 minutes.

A total of 14 high-resolution spectra (10 UVES + 4 MIKE) were obtained for the BLMC-01 system and 12 were obtained (8 UVES + 4 MIKE) for BLMC-02. Typical signal-to-noise ratios near 4000 Å are about 50–60 for UVES spectra and about 30–40 for the MIKE spectra.

For the reduction of the UVES spectroscopic data we used UVES Workflow version 5.8.2 running in the ESO Reflex environment version 2.8.5 (Freudling et al. 2013). This pipeline automatically follows the basic reduction echelle steps and returns the extracted one-dimensional spectrum (blue and red parts). MIKE data were reduced using Daniel Kelson's pipeline available at the Carnegie Observatories software Repository.⁸

To obtain the orbital solution, first radial velocities have to be determined. For that we used the RaveSpan code, which is a user-friendly graphical Python application that allows extracting velocities of the components using such methods as cross-correlation function (CCF), TODCOR, and Broadening Function (BF). We applied the BF technique (Rucinski 1999) because it gives narrower and stronger profiles than the commonly used CCF, which is very important for early-type stars with strong rotational broadening. As spectrum templates we used the spectra of standard stars from the ESO Data Archive. For each system we have selected various templates with a spectral type similar to that estimated for the components. The radial velocity measurements for BLMC-01 are shown in Figure 4 and for BLMC-02 in Figure 5.

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The next step was to fit the obtained RVC. In the model we took into account the following orbital elements of the spectroscopic binary: P (the orbital period), K_A and K_B (the orbital semi-amplitudes), γ (the velocity of the center-of-mass of the system), T₀ (the reference time), e (the orbital eccentricity), and ω (the argument of periastron passage of the primary). Assuming Keplerian motion of the point-like sources, the radial velocities of the components depend on these parameters in the following way:

$\begin{eqnarray*}&&{v}_{i}=\gamma +{K}_{i}\,e\,\cos w+{K}_{i}\,e\,\cos (\nu +w),\end{eqnarray*}$

where ν is the true anomaly. We fitted this relation to the observed RVC of each component of the system and obtained the preliminary orbital solution.

The minimum masses of the components (expressed in the solar mass units) can be calculated as follows:

$\begin{eqnarray*}&&{m}_{A,B}\,{\sin }^{3}i=1.036\times {10}^{-7}{({K}_{A}+{K}_{B})}^{2}{K}_{B,A}\,P\,{(1-{e}^{2})}^{3/2},\end{eqnarray*}$

where i is the orbital inclination and P and K_i are expressed in days and km s⁻¹, respectively. Because ${\sin }^{3}i\leqslant 1$, what we can measure here is just the minimum mass. The mass ratio is inversely proportional to the ratio of the semi-amplitudes, i.e., $q={m}_{B}/{m}_{A}={K}_{A}/{K}_{B}$, and is independent of inclination. The orbital parameters from this analysis for both systems are presented in Table 3.

Table 3.  Orbital Parameters and Rotation from the Ravespan Code

	BLMC-01		BLMC-02
Parameter	Primary	Secondary	Primary	Secondary
orbital period, P [days]	5.4139927		4.2707640
eccentricity, e	0.016 ± 0.005		0.078 ± 0.004
argument of periastron, ω [rad]	6.0 ± 0.3		1.61 ± 0.07
$m{\sin }^{3}i$ [M⊙]	12.38 ± 0.16	13.50 ± 0.24	19.40 ± 0.14	18.86 ± 0.14
orbital semiamplitude, K [km s−1]	187.1 ± 1.5	171.6 ± 0.8	218.7 ± 0.7	224.9 ± 0.7
$a\sin i$ [R⊙]	20.03 ± 0.16	18.40 ± 0.09	18.41 ± 0.06	18.93 ± 0.07
systemic velocity, γ [km s−1]	250.9 ± 0.6		239.6 ± 0.4
mass ratio, $q={m}_{2}/{m}_{1}$	1.091 ± 0.010		0.972 ± 0.004
${v}_{\mathrm{rot}}\,\sin i$ [km s−1]	128 ± 5	115 ± 3	108 ± 3	105 ± 3
rms [km s−1]	3.91	1.49	1.39	2.02

Note. Formal errors from the fitting procedure are given. Orbital periods were fixed at photometric values. The argument of periastron for a mean epoch of spectroscopic observations of a system is given.

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Early-type stars are known for their fast rotation, which is responsible for the strong broadening of the absorption lines in the spectra. In the BF method, which we used for the radial velocity determination, the rotation velocity of the components is also measured. We have determined these values for various spectra taken close to the orbital quadratures, where the profiles are well separated, and measured the average (projected) rotation velocity of each component of the system (${v}_{\mathrm{1,2}}\sin i$). They are given in Table 3.

We performed a preliminary light-curve modeling of the system using a Python wrapper for the JKTEBOP code (Southworth et al. 2007), which is based on the EBOP code (Popper & Etzel 1981). In JKTEBOP the stars are represented as biaxial spheroids, but the spherical approximation is used for the calculation of the light lost during the eclipse. Using our wrapper the determination of apsidal motion is also possible with this code.

Because of small deformations present in the light curve that come from the tidal distortions of the components, JKTEBOP could be used only to obtain a first model approximation. However, it is extremely fast when compared to the WD code that we use later to improve the binary models, especially for eccentric systems. It let us scan relatively quickly a large parameter space and look for a solution that corresponds to a global minimum. A common problem in using differential corrections implemented in the WD code is that the solutions tend to stack at any local minimum that is first found.

We modeled all the photometric data sets simultaneously, which, in the case of BLMC-01 are the I, V, and K bands, and in the case of BLMC-02 are the I, V, and R bands. All the photometric data sets were first detrended and cleaned of significantly (higher than 5σ) outlying points.

In the model we fitted the following: the orbital period, the reference time, the eccentricity and argument of periastron of the primary represented in the code as $e\sin \omega$ and $e\cos \omega$, relative stellar radii, the inclination, and the surface-brightness ratio and third light in each band. In this model the component temperature is only used to calculate the limb-darkening coefficients. At the beginning the temperature was estimated from the dereddened system color (see Section 4.5), and later updated according to the color of the components once a stable solution was found.

To find the best model we started from a scan of a wide parameter space containing all possible solutions for these type of binaries. Then we used the Markov Chain Monte Carlo (MCMC) method starting from all the local minima found, looking for the solution with the lowest χ² value.

After a closer examination of the light-curve residuals we found evidence of the apsidal motion and also added the change of the position of the periastron ($\dot{\omega }$) to the list of fitted parameters. We repeated the modeling with the apsidal motion taken into account. The best solution from this analysis was then used as a starting point for the final analysis with a more complex modeling.

In the analysis of variable stars and in the modeling of binary systems in particular, it is very important to have a long baseline, which improves the determination of the period and let us also study additional secular effects. In our case we have almost 20 yr long coverage, which for orbital periods of 4–5 days can give a very high precision in the determination of the time-related parameters, like period and times of minima. This in turn let us detect and measure the precession of the line of apsides, which is a very common phenomena in massive binary systems (Garcia et al. 2014; Pablo et al. 2015; Rauw et al. 2016; Zasche & Wolf 2019).

Because of low eccentricity, the effect for BLMC-01 could be barely noticed looking at a light curve from one data set, although the time span of the observation was long (7.5 yr in case of OGLE-III). During the analysis of the light curves we have noticed, however, that in data sets from different surveys the position of eclipses was slightly different. We have identified this change as evidence of apsidal motion in this system. The apsidal motion of BLMC-02, which is more eccentric, is much stronger and significantly affects the position of the eclipses even within the time span of one data set.

We have used a very simple method to measure this effect in the JKTEBOP code. We have added just one parameter to the model, a derivative of the longitude of periastron ($\dot{\omega }$), which is used to calculate a constant ω for every light curve. Then, for every survey we have calculated an average time of observation $\langle {T}_{i}\rangle$. The longitude of periastron for a given light curve is then calculated as

$\begin{eqnarray*}&&\omega ={\omega }_{0}+\dot{\omega }\times (\langle {T}_{i}\rangle -{T}_{0}),\end{eqnarray*}$

where ${\omega }_{0}=\omega ({T}_{0})$.

As a result, we have a model for each light curve that shares all but one parameter, with the only difference being ω, which influences the position of the eclipses. This way we obtain just one solution that fits all the light curves without the need to fit different models for different epochs and then average them. The values of ω₀ and $\dot{\omega }$ from the best model were used as input values for the WD model. The final values of the apsidal motion for each system are given in the next section.

For BLMC-01 the apsidal motion is weak, but the proximity effects (deformation of the stars) are significant; for BLMC-02 we have the opposite situation—strong apsidal motion and practically insignificant proximity effects. Because neither of these phenomena are natively included in the JKTEBOP, we decided to model these systems using the widely used and well-tested WD code, which takes these effects into account.

This code works much slower, especially in the case of eccentric systems, which makes it very important to have an approximate solution and a well-studied parameter space to avoid the traps of local minima in the modeling process. Our solution from the JKTEBOP code meets these conditions. As initial input parameters for the WD code we have used the parameters of the model with the lowest global χ² found during the Monte Carlo analysis using the JKTEBOP code.

We modeled photometric and spectroscopic data simultaneously, so this time the model included more parameters: the orbital period (P), the reference time (T₀), the semimajor axis (a), the mass ratio (q), the systemic velocity (γ), the eccentricity (e), and argument of periastron of the primary (hotter) component (ω), the first derivative of the argument of periastron ($\dot{\omega }$), the inclination (i), surface potentials (Ω₁ and Ω₂), the third light (l3) and the temperature of a less luminous component (${T}_{\mathrm{eff},i}$). The temperature of a more luminous component was estimated from its dereddened (preferentially $(V-K)$) color using a transformation from Worthey & Lee (2011). At the beginning the color from the preliminary solution was used. Later it was updated according to the most recent solution if the change was higher than ∼20% of the estimated error. Please also note that the used color–temperature transformation was obtained for solar metallicity stars, while the metallicity of the LMC is about half the solar.

To look for the solution, we first used the differential correction method implemented in the code, which we repeated until the convergence was obtained. Then we ran a Monte Carlo simulation, calculating about 50,000 models to find the χ² minimum and estimate the errors in a reliable way. Eventually, we took the model with the lowest χ² value as the final solution.

The final light curve models for BLMC-01 and BLMC-02 are shown in Figures 6 and 7, respectively. It is not easy to present the light curve for a binary system with non-zero apsidal motion. To solve this problem we corrected the observational data for $\dot{\omega }$ from the model (black points) and plotted the model light curve for a fixed epoch (T₀). The correction was performed by applying to every measured magnitude a shift between the model magnitude at T₀ and the model magnitude at the time of observation.

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To demonstrate the effect of apsidal motion on the light curve, in Figure 8 we show the same model and data without applying the correction. The difference is especially noticeable at eclipses, where the observations are shifted to the left or right of the model, in the opposite direction for the primary and secondary eclipses. In the bottom panel of the figure (residuals) one can see the effect is very significant. For BLMC-01 the effect is much weaker, but also visible and detected with high confidence (9σ detection) using all sets of data. The full apsidal motion cycle takes 144 yr for BLMC-01 and 118.7 yr for BLMC-02.

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For BLMC-01 we have detected a negligible third light in the OGLE I-band light curve (<0.5%) and a very small negative third light in the OGLE V-band (∼1%), which may result from the uncertainties in the flux calibration process. Only the EROS light curve was significantly affected, with additional flux of the level of almost 6%. The additional/missing flux was taken into account in calculating the component magnitudes in each band. Because of the low number of points, the third light for the K-band light curve was not fitted. For BLMC-02 no third light was detected in any band.

The best fits to the RVC from the WD code for the BLMC-01 and BLMC-02 systems are shown in Figures 4 and 5, respectively.

The extinction in the direction of our system was estimated in two independent ways. First, we used the reddening map for the Magellanic Clouds obtained from the analysis of red clump stars (Haschke et al. 2011). We calculated the color excess using the equation

$\begin{eqnarray*}&&E(B-V)=\displaystyle \frac{E(V-I)}{1.3}+0.04\ \mathrm{mag},\end{eqnarray*}$

where $E(V-I)$ is the color excess from the reddening map and a denominator of 1.3 is adopted from Bessell et al. (1998). As the zero-point of this map is not well-defined to obtain absolute values of $E(B-V)$, we used a shift of 0.04 mag, which results from the determination of the reddening-free mean Red Clump color $(V-I{)}_{0}^{\mathrm{RC}}$ given by Pawlak (2016).

We obtained the reddening toward BLMC-01 E(B − V) = 0.198 ± 0.029 mag, which is higher than an average for LMC, but it was expected as the object is located in the 30 Dor region. The reddening to the BLMC-02 system was estimated in the same way to be 0.077 ± 0.023 mag.

The second method used to estimate the reddening was measuring the equivalent widths of Na i sodium doublet lines (around 5893 Å) and using the calibration of Munari & Zwitter (1997). This method works well for early-type stars because virtually no other lines are present in the vicinity, making the measurement more robust compared to late-type stars with many metal lines. Nevertheless, the sodium line profiles can be quite complex, as different components with different wavelength shifts may contribute to their overall shape.

To estimate the contribution of each component we fitted multiple Gaussians to the Na i profiles. The fit was performed simultaneously to several best quality spectra from different orbital phases. The final value of $E(B-V)$ was then calculated as a sum of reddenings of individual components.

In Figures 9 and 10 we show the spectra in the region of the Na i lines and the profile fits for each system. As can be seen, stellar lines (that should vary with the orbital phase) are not present in this region. In the case of BLMC-01 there are two main Na i components related to the Galaxy and a complex structure of four components related to the LMC, where one component of D1 is mixed with the Galactic components of the D2 line. In the case of BLMC-02, which is positioned on the other side of the LMC and farther from the center, there is only one weak component related to that galaxy and the majority of the extinction comes from the Milky Way (MW).

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Using the method described above we determined the reddening $E(B-V)$ of 0.187 ± 0.031 mag for BLMC-01 and 0.088 ± 0.019 mag for BLMC-02, which are consistent with the values obtained from the reddening map. Eventually, we used the average of both determinations as final values, i.e., $E(B-V)$ of 0.193 ± 0.021 mag for BLMC-01 and 0.082 ± 0.015 mag for BLMC-02. We then used these values to deredden all the colors and to calculate absorption (A_λ) in each band using the standard reddening law relations from Cardelli et al. (1989) and the ratio of total-to-selective extinction R_V = 3.1.

For the BLMC-02 system, the temperature estimated from the dereddened $(V-I)$ color is, however, very low (∼22,000), which is not consistent with the observed spectral line features. Moreover, it also leads to an unrealistically short distance to the LMC. The problem may partly come from the uncertain $T-(V-I)$ calibration (Worthey & Lee 2011), which is not well-defined for hot stars, or with the calibration of the photometry, as a difference of 0.03 mag in V − I color translates to a difference of about 4000 K. We also cannot exclude that for some reason the reddening determinations for BLMC-02 are wrong, although two independent methods gave a similar result.

For BLMC-01 we obtain a good consistency between the temperature from the dereddened $(V-K)$ color (Worthey & Lee 2011) and the observed spectral features. The determined value of $E(B-V)$ also leads to a distance to LMC consistent with the accurate value obtained by Pietrzynski et al. (2019).

We checked that for BLMC-02, to obtain a consistent distance (DM ∼ 18.5 mag) and spectral type (O-type) and assuming our reddening estimation is correct, the temperature of the stars should be around 34,000–35,000 K. However, if we assume that the photometry and temperature-color calibration are correct, to obtain a consistent distance we would need a reddening of $E(B-V)=0.18$ mag, which would imply a temperature of about 42,000 K. Such high values are rather excluded, so for the moment we accept the first option as the more probable one. In the final modeling we used T₁ = 35,000 K as the temperature of the primary (e.g., for calculation of the limb-darkening coefficients).

In the future we plan to determine the temperature and $E(B-V)$ for BLMC-02 directly from the spectra. This detailed spectral analysis will be published in a follow-up paper.

During the binary evolution, due to the mutual tidal influence, the orbits of the components are slowly circularized and their rotational velocities are synchronized. Knowing the physical properties of the stars we can compare the rotational angular velocity of components with the synchronous (Ω_sync) and pseudo-synchronous angular velocity (Ω_ps) for each system, to check whether they are synchronized or not.

We obtain Ω_sync = 1.161 radian per day for BLMC-01 and Ω_sync = 1.471 day⁻¹ for BLMC-02, while using equations from Hut (1981) we obtain ${{\rm{\Omega }}}_{\mathrm{ps}}=1.162\,{\mathrm{day}}^{-1}$ and Ω_ps = 1.529 day⁻¹ for BLMC-01 and BLMC-02, respectively.

Using the radii of the stars from our models we can calculate the angular rotational velocity of the components of BLMC-01: 1.91 ± 0.07 day⁻¹ and 1.12 ± 0.03 day⁻¹, which means that the primary is still rotating super-synchronously, while the secondary is already synchronized, probably because of its larger radius resulting from its more advanced evolutionary state (it is more massive than the primary).

For BLMC-02 the angular velocities of the components are 1.51 ± 0.04 day⁻¹ and 1.66 ± 0.05 day⁻¹. This means that the larger and more massive primary is already rotating synchronously, while the smaller secondary has not yet synchronized its rotation.

Here we will present two approaches for the determination of the distance to a system in the LMC. One is a direct measurement using the bolometric flux scaling method, and in the other one the distance is inferred from the assumed LMC distance.

In the first measurement a DM is calculated using the absolute V-band magnitude of a system (M_V) and the observed magnitude corrected for extinction determined in Section 4.5 (${V}_{0}={V}_{\mathrm{obs}}-{A}_{V}$) using the formula $\mathrm{DM}={V}_{0}-{M}_{V}$. The value of M_V is calculated using the derived physical parameters of the components and applying a bolometric correction (BC_V), as obtained for the observed colors and surface gravities from the calibration of Worthey & Lee (2011): ${M}_{V}={M}_{\mathrm{bol}}-{\mathrm{BC}}_{V}$. The final equation is

$\begin{eqnarray*}&&{\mathrm{DM}}_{\mathrm{bol}}={V}_{\mathrm{obs}}-{M}_{\mathrm{bol}}+{\mathrm{BC}}_{V}-{A}_{V}.\end{eqnarray*}$

Following this procedure for BLMC-01, we obtain a distance modulus of DM${}_{\mathrm{bol}}=18.47\pm 0.15$ mag.

For BLMC-02, if using the measured reddening and transforming the resulting $(V-I)$ color to temperature, we would obtain a DM about 17.5 mag, which is clearly wrong for a system that is expected to belong to the LMC. As described before, to obtain a consistent distance and use the measured reddening, a temperature of the primary around 34,000–35,000 K is needed.

An alternative way to reliably estimate the distance to both systems is to use the known distance to the LMC and apply the geometrical correction (GC) for the position of the system in the plane of the LMC (see Section 3) We used the LMC distance (DM_LMC) determined from the analysis of binary systems composed of late-type giants by Pietrzynski et al. 2019) and applied the GC based on the "PMs+Young" LMC model of van der Marel & Kallivayalil (2014):

$\begin{eqnarray*}&&{\mathrm{DM}}_{\mathrm{gc}}={\mathrm{DM}}_{\mathrm{LMC}}+\mathrm{GC}.\end{eqnarray*}$

Using this method we determined the expected distance moduli (DM_gc) to the systems to be 18.45 ± 0.04mag for BLMC-01, and 18.51 ± 0.04 mag for BLMC-02. To account for the depth of the LMC, the quoted uncertainties include both the total error of the LMC distance (0.028 mag), and the scatter of 20 individual distance measurements (0.026 mag) used by Pietrzynski et al. (2019). For BLMC-01, this distance is consistent within errors with the DM determined with the first method (DM_bol).

To compare our results with the existing surface-brightness–color relations we need $(V-K)$ color and the surface brightness in the V band (S_V). As already mentioned, for BLMC-01 the K-band light curve is available covering both eclipses. From the light-curve solution, the individual K-band magnitudes of each component were derived and then their infrared $(V-K)$ colors were calculated. Knowing the distance to this system, we can also calculate S_V of the components. To do this we used the distance modulus DM_gc, which provides much better precision than the DM determined using the bolometric flux scaling method. In the future we are going to use the same methodology for the whole sample. The following equations were used to obtain the surface brightness:

$\begin{eqnarray*}\begin{array}{rcl}\theta [\mathrm{mas}] & = & 1.337\times {10}^{-5}\times R[\mathrm{km}]/D[\mathrm{pc}]\\ {S}_{V} & = & {m}_{0}+5\,\mathrm{log}\,\theta [\mathrm{mas}].\end{array}\end{eqnarray*}$

In Figure 12 we show a comparison of our measurements for the components of the BLMC-01 system with the SBCR of Challouf et al. (2014). The solid line is the fit of the relation for stars of all spectral types (blue points) given in Equation (10) by these authors. Our measurements are consistent with their results. Once we have similar data for all 18 components of the 9 systems from the sample, we will be able to improve the calibration of the blue part of this relation, which now suffers from a lack of precise measurements.

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The evolution of early-type stars is still not well understood and discrepancies between the observations and models are often found. One example is the mass discrepancy problem. It was first noticed by Herrero et al. (1992), where the evolutionary masses were systematically higher than the ones determined with spectroscopy, and is still being raised by other authors (e.g., Pavlovski et al. 2018). There are also not many systems composed of the early-type components for which accurate physical parameters are known. This makes a proper calibration of theoretical models harder for this type of stars.

In this study we have obtained precise and accurate physical parameters for the components of the BLMC-01 system. Below we will test how well our measurements are described by modern evolutionary models.

First, we compared our results with the evolutionary tracks of Choi et al. (2016). We used a set models for the LMC metallicity and masses from 13 to 16 M_⊙ and also two interpolated tracks for the masses of our components (see Figure 13). All these models were calculated with the overshooting value of ${f}_{\mathrm{ov},\mathrm{core}}=0.016$, which is an equivalent of α = 0.2 in the step overshooting scheme.

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The tracks are consistent within the errors with our results, but there are few points that have to be noted. First, these models suggest that the secondary has already left the main sequence, which would be interesting, but because of the fast phase of evolution, it would be a rare coincidence and would have to be confirmed independently. Second, the uncertainty of the temperature determination is high but strongly correlated between the two components, so any shift would move both systems in the same direction in the HR diagram. If we adjust the temperature of the secondary for it to lie exactly on the track, the primary would be overluminous, and if we do the same for the primary, the secondary would be underluminous. The age of the system in this model is 13.8 million years.

To check the dependence on the model and the assumed parameters, we also used the grid of evolutionary tracks of Brott et al. (2011). This grid covers masses from 5 to 60 M_⊙ and were calculated for three different metallicities (MW, LMC, and the Small Magellanic Cloud) and a wide range of surface rotation velocities (0–400 km s⁻¹). For all models the same overshooting was used (α = 0.335). We chose models with an LMC metallicity initial velocity of ∼140 km s⁻¹, and interpolated them to a given mass when necessary.

As can be seen in Figure 14 our results are consistent with these evolutionary models and in this case the secondary is still on the main sequence, which is a more likely scenario. This is probably the effect of higher overshooting, which extends the main sequence toward lower temperatures. Contrary to the models of Choi et al. (2016) both components lie close to their corresponding evolutionary tracks and at roughly the same distance from them. The model rotational velocities of the components are also consistent with our measurements. In general we find the models of Brott et al. (2011) to be in better agreement with our data, which points to higher overshooting values for early-type stars. The age of the system in this model is 11.7 million years.

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Both presented grids were calculated for single-star evolution, which should work reasonably well for our system. The stars are still small enough not to interact directly with each other, although the interaction through tidal forces has probably taken place—the larger components have already synchronized their rotation and orbital velocities. However, as the stars started their evolution on the main sequence being much smaller than they are now, for a significant part of their life tidal forces were negligible.