1. Introduction

A contact binary (CB) is a close binary star system where both components interact strongly, fill out their Roche lobes and share a common envelope (^{Kopal 1959}; ^{Eggleton 2006}). Thermal energy is transferred from the hotter (primary) to the colder (secondary) star mainly through the common envelope leading to the establishment of a similar temperature of the two stars. However, their mass ratio can be rather different.

The observational study of contact binaries allows the testing of theoretical models helping us to further understand of -for example- the merging process of stars and the evolution of their common envelope.

Contact binary systems usually belong to spectral types F, G and K, with orbital periods less than a day. The K-type binaries have been expected to have periods shorter than 0.3 days and show shallow characteristics. The rarity of this kind of binaries makes them very interesting systems for testing the thermal relaxation theory (TRO, ^{Lucy 1976}; ^{Flannery 1976}; ^{Robertson & Eggleton 1977}).

ROTSE J135349.8+305205 (hereafter J135349) was found to be variable during the ROTSE I all-sky survey (^{Akerlof et al. 2000}) as an EW system with an orbital period of 0.24698301 d. After J135349 was discovered it remained a neglected object. Here, the light curves of J135349 are analyzed and presented for the first time.

A light curve for ROTSE J150957.5-115308 = V373 Lib (hereafter J150957) was reported by ^{Lohr et al. (2013)}, which presented the typical EW-type behavior and an orbital period of 0.2290205 d. The official name was assigned in the 82nd name-list of variable stars (^{Kazarovets et al. 2019}).

With no previous studies of both these systems, the aim of the present work is to analyze their light curves using the latest version of the Wilson-Devinney code (^{Wilson & Devinney 1971}; ^{Wilson 1994}; ^{Wilson & van Hamme. 2016}) and to determine the Roche conguration and their orbital parameters.

The outline of the paper is as follows. In § 2 we describe the observations made and their characteristics. In § 3 the times of minima and new ephemeris for our two contact binaries are reported. In § 4 the solution obtained with the Wilson-Devinney code is discussed and presented. In § 5 an estimation of physical parameters using Gaia parallax data is presented. Finally, in § 6 a discussion and final remarks are provided.

2. OBSERVATIONS

Observations done at the San Pedro Martir Observatory with the 0.84-m telescope, the Mexman filter-wheel and the Spectral Instruments 1 CCD detector (an e2v CCD42-40 chip with a gain of 1.39 e−/ADU and readout noise of 3.54 e−). The field of view was 7.6′ 7.6′ and a binning of 2 × 2 was employed during all the observations.

J135349 was observed on April 26 2017 for 7.4h, May 4 2018 for 6.1h and April 4 2021 for 2.2h. Alternated exposures in filters B, V, R_c and I_c, with exposure times of 60, 40, 15 and 15 seconds respectively, were taken in all the observing runs.

J150957 was observed on February 21 2017 for 3.8h, February 23 2017 for 3.6h, March 4 2019 for 3.4h, June 6 2019 for 2.5h, April 11 2021 for 3.2h and April 13 2021 for 6.6h. Alternated exposures in filters B, V and Rc, with exposure times of 60, 35 and 20 seconds respectively, were taken in all the observing runs. Flat field and bias images were also taken during all the nights.

All images were processed using IRAF¹ routines. Images were bias subtracted and flat field corrected before the instrumental magnitudes were computed with the standard aperture photometry method. These fields were also calibrated in the UBV (RI)_c system with the help of some Landolt’s photometric standards.

Based on the previous information, we decided to use star WISEJ135355.54+304735.9 (U = 18.042, B = 16.936, V = 15.839, R_c = 15.148 and I_c = 14.549) as comparison star for J135349 since it has a similar color (making differential extinction corrections negligible). For the case of J150957, star WISEJ150953.24-115045.3 (U = 15.398, B = 14.587, V = 13.589, R_c = 12.931 and I_c = 12.446) was employed. Any part of the data can be provided upon request.

3. Times of Minima and New Ephemeris

From our observations we were able to obtain 4 times of minimum (ToM) for J135349 and 6 for J150957, one ToM has been found in literature. All ToMs are presented in Table 1.

All ToMs are heliocentric and determined by the polynomial fit method. With these data we updated the ephemeris as follows. For J135349 and for J150957

4. Photometric Solution with the W-D Method

The light curves of both systems show clearly the EW behavior with continuous changes in the light. For this reason the Mode 3 of the Wilson-Devinney (W-D) code was used in the calculation.

Using our observations, we were able to determine the color index of both systems and, from the tables of ^{Worthey & Lee (2011)}, the temperature of the primary component; that was xed at 4760 K for J135349, and 4220 K for J150957.

The temperatures of the components of the two systems suggest convective envelopes. Hence, we adopted the following atmospheric parameters: the gravity-darkening coecients g₁ = g₂ = 0.32 (^{Lucy 1967}) and the bolometric albedos A₁ = A₂ = 0.5 (^{Ruciński 1973}) were assigned; the limb-darkening parameters originate from ^{van Hamme (1993)} for log g = 4.0, and solar abundances.

During the dierential correction calculation in the W-D code we left as adjustable parameters the orbital inclination i, the mean surface eective temperature of the secondary component T₂, the dimensionless surface potentials of the primary and secondary stars Ω₁ = Ω₂, the monochromatic luminosity of the primary component L₁ and the third light L₃. In our solutions, we nd that the contribution of the third light is negligible.

The classical q-search method was used to nd the best initial value of the mass ratio to be used during the light curve analysis. The value of the mass ratio q was fixed in each iteration and increased after the sum of residuals showed a minimum number. As one can see from Figure 1, where the sum of squares of residuals (Σ(res)²) versus mass ratio q is shown, the best mass ratio for J135349 is found at q = 0.3 and for J150957 at q = 0.9. These values of q were also treated as free parameters in the successive step of our analysis.

Fig. 1 The relation Σ(res)² versus mass ratio q = M2/M1 in Mode 3 of the W-D code for J135349 and J150957. 

The final results obtained are listed in Table 2, while the obtained fit is shown in Figure 2.

Fig. 2 CCD filtered light curves for the systems at different wavelengths. Points are the original observations while lines represent the theoretical light curves obtained in our modeling. The color figure can be viewed online. 

The values of the mass ratio for both systems indicate that they are typical A-subtype contact binaries in the ^{Binnendijk (1965)} classification. In Figure 3 the graphical representation of the systems and the relative Roche geometries are displayed.

Fig. 3 Model representation of the CBs J135349 (q = 0.3) and J150957 (q = 0.9). The configuration at the primary minimum (left) and the Roche geometry (right) of these systems resulting from our modeling are shown. The color figure can be viewed online. 

By examining Table 2 the following information can be obtained. Both systems are of the A-subtype and in good thermal contact. The temperature of the components suggests that they are of the K spectral type. We note that it is somewhat strange to find a spectral K-type in A-subtype contact systems.

Systems of late spectral type generally belong to the W-subtype of W UMa contact binaries, but there are some exceptions that belong to the A-subtype as 2MASS J11201034-2201340 (^{Hu et al. 2016}), ES Cep(^{Zhu et al. 2014}), NSV 395 (^{Samec et al. 2016}), and AP UMi (^{Awadalla et al. 2016}).

Our two systems, despite having temperatures consistent with late spectral type K, show the characteristics of the subtype-A contact binaries; i.e., T₁ > T₂, transit at primary minimum, and a mass ratio q < 1.

CB J150957, having a mass ratio near unity, can be considered to be a high mass ratio system. High mass ratio systems, proposed firstly by ^{Csizmadia & Klagyivik (2004)} are a subgroup of contact binaries with mass ratio q > 0.72.

For the H-type the rate of energy transfer is less efficient than for other contact binaries at a given luminosity ratio. Having a mass ratio close to unity, less luminosity should be transferred in order to equalize their surface temperature.

Both binary systems show a low fill-out value. The low fill-out value is not a common feature among A-subtype contact binaries, and only a few A-subtype contact binaries are found to have a high mass ratio and a shallow common envelope (see Table 5 of ^{Han et al. 2019}). Contact binary J150957, which shows the same peculiar characteristics, can be added to this short list.

Note that the errors of the parameters given in Table 2 are the formal errors from the W-D code. For a discussion see ^{Barani et al. (2017)}.

5. Estimation of the Physical Parameters with the Gaia Parallax

Physical parameters such as mass, radius and luminosity are very important information for a contact binary system. Hence it is necessary to estimate them. Here we will indicate how we have estimated the physical parameters of J135349 and J150957 without radial velocity curves, using the parallax known by Gaia (^{Gaia Collaboration et al. 2018}).

First, we calculated the Galactic extinction obtained using different methods from which an average value of the A_V (^{Masda et al. 2018}) was extracted; in detail:

Using the parallax from Gaia we calculated the visual absolute magnitude using the relation

and the bolometric magnitude M_bol = M_V + BC_V, where BC_V is the bolometric correction obtained from the ^{Pecaut et al. (2012)} and ^{Pecaut & Mamajek (2013)} tables. This allowed us to obtain the total luminosity of the systems as

and also the individual luminosities of the components.

Knowing the temperatures of the first and second component of each system we obtained their radii, and finally the total mass of the system; by using the value of the mass ratio obtained from the Wilson-Devinney analysis, the single masses M₁ and M₂ as shown in Table 3 were obtained.

We used the absolute elements of the primary and secondary components of both systems (Table 3) to estimate their evolutionary status by means of the log T_eff - log L (i.e. HertzsprungRussell) diagram on the evolutionary tracks of ^{Girardi et al. (2000)}. The results are shown in Figure 4.

Fig. 4 Components of our binary systems plotted in the HR diagram. Zero age main sequence (ZAMS), terminal age main sequence (TAMS), evolutionary tracks and isochrones were taken from ^{Girardi et al. (2000)} for a solar chemical composition. The numbers denote initial masses. The color figure can be viewed online. 

It is possible to see from Figure 4 that both the primary and the secondary components of J150957 are undermassive, with a luminosity comparable to that of a zero age main sequence (ZAMS) star.

For J135349 the primary component is located in the region between the ZAMS and TAMS (terminal age main sequence) near the evolutionary track of 0.6, but it is underluminous given its mass. The secondary component, located under the ZAMS, is overmassive and slightly underluminous.

These results suggest that both systems consist of two stars of similar surface brightness, but in dierent evolutionary stages.

According to ^{Flannery (1976)} the stability parameter ℑ for the mass-exchange in a CB can dened as:

where R_p refers to the primary’s radius and R_s to the secondary. If ℑ =0 no mass transfer occurs; if ℑ >0 an unbalanced pressure gradient will force gas from the primary to secondary, and vice versa if ℑ <0.

In our case we obtain ℑ =0.074 for J150957; hence there is mass transfer from primary to the secondary; the contrary is true for J135349 were we obtain ℑ = 0.019. In both CBs the value of indicates a poor mass exchange between the components.

6. Discussion and Final Remarks

The results of our analysis lead to two contact binary systems of the A-subtype that are in good thermal contact and have a shallow degree of contact between their components.

J150957, with a mass ratio q = 0.905, belongs to the high mass ratio type contact binaries (i.e. an H-type).

The spectral type K for all the components of the binary systems is somewhat strange for the A-subtype, but there are few A-subtype systems of K-type.

In our analysis we found no signicant evidence of spots on the surfaces of the two components, the OConnell effect. This fact could be explained by a period of magnetic quiescence in the CBs (^{Zhang et al. 2011}).

Using the absolute elements provided in Table 3 the dynamical evolution of contact binaries can be inferred via the determination of the orbital angular momentum J₀ (^{Eker et al. 2006}).

In Figure 5 it is possible to see the position our systems occupy in the log J₀ log M diagram. The curved borderline separates the detached from the contact region and provides a check of the Roche configurations of J150957 and J130349. The values of log J₀ place J150597 near the borderline of this diagram, while J130349 is in the well defined region of contact systems.

Fig. 5 Position of J150957 and J135349 in the log J ₀ − log M diagram. Symbols are described in Figure 1 of the original paper of ^{Eker et al. (2006)}. The color figure can be viewed online. 

An exhaustive characterization of contact binaries, via the period-temperature relation, was recently conducted by ^{Qian et al. (2020)}. In Figure 6 (Figure 4 in the original paper) we show the position of J150957 and J135349 in the period-temperature plot. Systems near the lower line are marginal contact systems while systems near the upper line are deep-contact ones. Between the two lines there are normal contact systems.

Fig. 6 Correlation between the orbital period P (days) and temperature T (K), based on parameters of 8510 contact binaries from ^{Qian et al. (2020)}. The position J150957 is marked in blue, and that of J135349 in red. The color figure can be viewed online. 

J150957 is near the lower boundary at the beginning of the evolutionary stage of contact binary evolution (Figure 6). It is also shown in the log J₀ log M plot where the system is slightly under the borderline, in the contact region. This assumption is endorsed by the high mass ratio (q = 0.9) and the low fill-out (15.6%).

The other system, J135349 is well inside the boundaries for normal EW (Figures 5 and 6) and, with its small mass ratio (q = 0.38), its fill-out value (20.9%) and the almost equal temperature of the components, it follows that it is approaching the final evolutionary stage of the contact binary evolution.

The total mass determined for J150957 is over the minimum total mass limit for W UMa systems of 1.0 - 1.2 M_⊙ (^{Stępień 2006}), while J135349, with its total mass M_tot = 0.813 M_⊙, is under this limit. This means a mass loss, and may imply a late evolutionary stage of this contact binary system.