1. Introduction

Many W UMa stars consist of solar-type components and are easily recognized by their continuous brightness variations and nearly equal depths of eclipse minima. Their orbital periods are within 0.25 d < P < 0.7 d which means small orbits, synchronized rotation and orbital revolution. The W UMa systems are a result of the evolution of wide binaries by angular momentum loss and mass-ratio reversals (^{Stepien 2006}, ^{Qian 2003}).

It is supposed that components of some W UMa binaries (those with fillout factor around zero) alternate between configurations of full and marginal contact (^{Flannery 1976}; ^{Lucy 1976}; ^{Robertson & Eggleton 1977}). According to the contact model (^{Lucy 1968}), the energy generated in the cores of their components is redistributed to the common gaseous envelope. As a result, the relations between the stellar parameters of W UMa binaries are different from those of detached configurations.

The W UMa binaries are important targets for the modern astrophysics, because they give information about the late stage of the stellar evolution related to the processes of mass and angular momentum loss, merging or fusion of the stars (^{Martin et al. 2011}). Moreover, they are useful tracers of distance and galactic structure due to their period-color-luminosity relation (^{Rucinski 1994}; ^{Rucinski 1996}; ^{Rucinski & Duerbeck 1997}; ^{Klagyivik & Csizmadia 2004}; ^{Gettel et al. 2006}).

This paper presents photometric observations and light curve solutions of four W UMa binaries: CSS J071813.2+505000 (UCAC4 705-045020, 2MASS J07181325+5050000, further CSS J071813.2), NSVS 2459652 (UCAC4 775028323, GSC 04129-01031), NSVS 7178717 (1SWASP J064501.21+342154.9; 2MASS J06450122+3421546; UCAC4 622-035881), NSVS 7377875 (GSC 0249001074; UCAC4 634-043109; 2MASS J08400055+3639287). Table 1 presents the coordinates of our targets and available information about their light variability.

2. Observations

The CCD photometric observations in Sloan g’, i’ bands were carried out at Rozhen Observatory with the 30-cm Ritchey-Chretien Astrograph (located into the IRIDA South dome) using CCD camera ATIK 4000M (2048 × 2048 pixels, 7.4 μm/pixel, field of view 35 x 35 arcmin). The information about our observations is presented in Table 2.

The data were obtained during photometric nights with seeing within 1.1- 1.9 arcsec and humidity below 70 %. The airmass during observations of all targets was within the range 1.01-2.01.

Twilight flat fields were obtained for each filter, dark and bias frames were also taken throughout the run. The frames were combined respectively into a single master bias, dark and flat frames. The standard procedure was used for the reduction of the photometric data (de-biasing, dark frame subtraction and flat-fielding) by software AIP4WIN2.0 (^{Berry & Burnell 2006}).

We used aperture photometry with a radius of 1.5 FWHM of the star image, along with sky background measurements with annuli enclosing a comparable area.

The light variability of the targets was estimated with respect to nearby comparison (constant) stars in the observed field of each target, so called ensemble photometry. A check star served to determine the observational accuracy and to check the constancy of comparison stars. The CCD ensemble photometry calculates the difference between the instrumental magnitude of the target and a comparison magnitude obtained from the mean of the intensities of the chosen comparison stars. The use of numerous comparison stars increases considerably the statistical accuracy of the comparison magnitude (^{Gilliland & Brown 1988}, ^{Honeycutt 1992}).

We performed the ensemble aperture photometry with the software VPHOT. Table 3 presents the coordinates of the comparison and check stars from the catalogue UCAC4 (Zacharias et al. 2010) and their magnitudes from the catalogue APASS DR9 (^{Henden et al. 2016}). The values in brackets correspond to the standard deviations of the standard stars during the observational nights. The choice of comparison and check stars in the same field of view of the targets means practically equal extinctions for all stars.

The transformation of the obtained instrumental magnitudes to standard ones was made manually. For this aim we used the mean color of the ensemble comparison star (g' - i')_comp and transformation coefficients of our equipment (calculated earlier using standard star field M67). Their values applicable to the presented g' i' observations are: T _g',g'i' = −0.002 ± 0.012, T _g',g'i' = −0.061 ± 0.017, T _g',g'i' = 1.063 ± 0.011. They show that our local photometric system is very close to the standard Sloan system (especially in g band). The calculated corrections of the instrumental magnitudes for our targets were from −0.0008 mag to 0.0003 mag in g' filter (within the observational precision) and from −0.0258 mag to 0.0085 mag in i' filter.

3. Light curve solutions

The IRIDA light curves of the targets were solved using the code PHOEBE (PHysics Of Eclipsing BinariEs). It is a modeling package for eclipsing binary stars, built on top of the widely used WD program (^{Wilson & Devinney 1971}) that has undergone many expansions and improvements (^{Wilson & Sofia 1976}; ^{Wilson 1979}, ¹⁹⁹⁰; ^{Milone et al. 1992}; ^{Kallrath et al. 1998}; ^{Van Hamme & Wilson 2003}). PHOEBE was presented firstly in 2005 (^{Prsa & Zwitter 2005}). It retains 100 % WD compatibility, but introduces new computational and physical extensions to WD (proper handling of color indices and therefore temperatures in absolute units; interstellar reddening effects, new minimization schemes aiming stability and convergence improvements). PHOEBE also provides a graphical user interface alongside updated filters (as Sloan ones used in our observations). PHOEBE itself was improved several times and rewritten recently as a whole (^{Prsa et al. 2016}).

We used traditional convention MinI (phase 0.0) to be the deeper light minimum and the star that is eclipsed at MinI to be a primary component.

Target temperatures T_m were determined in advance (see Table 5 further) on the basis of their infrared color indices (J-K) from the 2MASS catalog and the calibration color-temperature of ^{Tokunaga (2000)}.

The initial runs revealed that all targets are overcontact systems. Hence, we applied mode "Overcontact binary not in thermal contact” of the code. The fit quality was estimated by the X² value.

Firstly, we fixed T₁ = T_m and varied the initial epoch T₀ and period P to search for fitting the phases of light minima and maxima. After that we fixed their values and varied simultaneously secondary temperature T, orbital inclination i, mass ratio q and potential Ω to search for reproducing the whole light curves. The data in i’ and g’ bands were modelled simultaneously.

We adopted coefficients of gravity brightening g₁ = g₂ = 0.32 and reflection effect A₁ = A₂ = 0.5 appropriate for late-type stars while the linear limb-darkening coefficients for each component and each color were updated according to the tables of ^{Van Hamme (1993)}. Solar metallicity was assumed for the targets, because they consist of late stars from the solar vicinity.

In order to reproduce the light curve anomalies we used cool spots, whose parameters (longitude λ, angular size α and temperature factor κ) were adjusted within reasonable ranges (but not varied simultaneously along other configuration parameters). Due to the ambiguous solution of this inverse problem we chose equatorial spots on the side surfaces of the primaries, because they have the smallest size and temperature contrast required to fit a given light curve distortion.

After reaching the best solution, we varied together all parameters (including period P and initial epoch T ₀ ) around the values from the last run and obtained the final values of the fitted quantities: T2PH, i, q, Ω, T ₀ and P.

In order to obtain the stellar temperatures T ₁ and T ₂ around the target value T _m we used the formulae (^{Kjurkchieva et al. 2016}):

where c = l ₂ /l ₁ (luminosity ratio) and ∆T = T_m - T2PH were taken from the final PHOEBE fitting.

Although PHOEBE (as WD) works with potentials, it has the possibility to calculate directly all values (polar, point, side, and back) of the relative radius r _i = Ri/a of each component (R _i is linear radius and a is orbital separation). In the absence of radial velocity curves we put as default a = 1. Moreover, PHOEBE yields as output parameters bolometric magnitudes Mboli of the two components in conditional units (when radial velocity data are bol not available). But their difference Mbol2-Mbol1 determines the true luminosity ratio c = L ₂ /L ₁ = l ₂ /l ₁ . Fillout factor f=Ω-ΩcritL1ΩcritL2-ΩcritL1 can be also calculated from the output parameters of PHOEBE solution (ΩcritL1 and ΩcritL2 are the Roche potentials at inner/outer Lagrangian point L ₁ and L ₂ ).

Table 4 contains final values of the fitted stellar parameters and their PHOEBE uncertainties: initial epoch T ₀ ; period P; mass ratio q; inclination i; potential Ω; secondary temperature T2PH. Table 5 exhibits the calculated parameters: stellar temperatures T _1,2; relative stellar radii r _1,2; fillout factor f ; ratio of the relative stellar luminosities l ₂ /l ₁ . The errors were determined from the uncertainties of the output parameters used for their calculations. Table 6 gives information for the spot parameters and the steps of their adjusting (in brackets).

The synthetic light curves corresponding to our solutions are shown in Figs. 1-2 as continuous lines. The mean (g’, i’) residuals for the final fittings are: (0.0196, 0.0267) for CSS J071813.2; (0.0125, 0.0126) for NSVS 2459652; (0.0346, 0.0288) for NSVS 7178717; (0.0220, 0.0188) for NSVS 7377875.

Fig. 1 The folded light curves of CSS J071813.2 and NSVS 2459652 with their fits and the corresponding residuals (shifted vertically by different number to save space). The observational data are accessible at: http://www.iridaobservatory.org/Observations/IRIDA-7.zip. 

Fig. 2 The same as Fig. 2 for NSVS 7178717 and NSVS 7377875. 

In order to check the effect of the correlation between the mass ratio and the orbital inclination (suspected from the correlation matrix) we carried out a q-search analysis (^{Kjurkchieva et al. 2016}). For this aim we fixed all configuration parameters except i and q. The last ones were varied by a fine grid and the corresponding normalized X₂ values were calculated. Figure 3 illustrates the q-search procedure for the target NSVS 7377875. It reveals the correctness of the obtained uncertainties of i and q.

Fig. 3 Illustration of the q-search analysis for NSVS 7377875: the different isolines circumscribe the areas whose normalized χ₂ are smaller than the marked values; the empty circle corresponds to the final values of the mass ratio and orbital inclination given in Table 4. 

4. Conclusions

The main results of the light curve solutions of our photometric data are as follows.

Fig. 4 From the left to the right 3D configurations of CSS J071813.2, NSVS 2459652, NSVS 7178717, NSVS 7377875 

This study adds four new systems to the family of W UMa binaries with estimated parameters. They could help to develop and improve our empirical knowledge about these stars and statistical relations between their parameters.