Photometric investigations on two totally eclipsing contact binaries: V342 UMa and V509 Cam

W UMa contact binaries are comprised of two late-type stars with spectral types from F to K. The two component stars are sharing a common convective envelope and have nearly equal effective temperatures although their masses are very different. The analysis of W UMa contact binaries is very necessary for modern astrophysics because they are probes for understanding tidal interactions, energy exchange, mass transfer and angular momentum loss (AML). The formation, evolution, ultimate fate and magnetic activities of W UMa contact binaries are still debatable issues (e.g., Guinan & Bradstreet 1998; Bradstreet & Guinan 1994; Eggleton & Kisseleva-Eggleton 2006; Fabrycky & Tremaine 2007; Qian et al. 2006, 2007a, 2014, 2017, 2018). To resolve these issues, the determination of physical parameters for a large number of such type binaries is required.

The physical parameters, such as mass ratio, for partially eclipsing contact binaries are poorly estimated (e.g., Pribulla et al. 2003; Terrell & Wilson 2005). In addition, the spectroscopic mass ratio sometimes cannot be reliably derived according to their broadened and blended spectral lines (e.g., Dall & Schmidtobreick 2005; Rucinski 2010). By the study of contact binaries, for which both spectroscopic and photometric mass ratios have been calculated, Pribulla et al. (2003) discovered that the photometric mass ratios of totally eclipsing systems correspond to their spectroscopic ones. Terrell & Wilson (2005) determined a similar result by discussing the relationships between photometric and spectroscopic mass ratios. These results suggest that we can derive very precise and reliable physical parameters for totally eclipsing contact binaries only by the photometric light curves. Thanks to the Gaia mission (Gaia Collaboration et al. 2018), the parallaxes of more than one billion stars have been obtained, which allows researchers to estimate the absolute parameters of contact binaries even if there are no radial velocity observations (e.g., Kjurkchieva et al. 2019a,b). Therefore, we chose two totally eclipsing binaries, V342 UMa and V509 Cam, to analyze their light curves and period variations, and estimate their absolute parameters.

V342 UMa was first discovered as a W UMa type binary by Nelson et al. (2004) during the observations of a nearby star BH UMa. The period of 0.343854 d, the color index of B − V = 0.64 and spectral type of G3 were acquired. Their photometric study revealed that V342 UMa is a low mass ratio (q = 0.331) W-subtype contact binary (the hotter component is the less massive one). It has been 15 years since the discovery and first photometric investigation of V342 UMa, therefore we decided to re-examine the light curves and orbital period changes of this target.

V509 Cam was first identified as an EW type eclipsing binary by Khruslov (2006) during a search for eclipsing binaries in Camelopardalis. He determined the variability amplitude of 0.6mag and the orbital period of 0.35034 d. However, at present, neither the light curve synthesis nor period variation analysis has been carried out for this star. Therefore, we will analyze the light curves and orbital period variations of this target in this paper.

Charge-coupled device (CCD) photometry of V342 UMa and V509 Cam was carried out from 2018 to 2019 using the Weihai Observatory 1.0-m telescope of Shandong University (WHOT, Hu et al. 2014), the Nanshan Onemeter Widefield Telescope (NOWT, Liu et al. 2014) at the Nanshan Station of Xinjiang Astronomical Observatory, the 60-cm Ningbo Bureau of Education and Xinjiang Observatory Telescope (NEXT), and the 85-cm telescope at Xinglong Station of National Astronomical Observatories, Chinese Academy of Sciences (NAOC85- cm). Information about the observations is listed in Table 1. In order to record the observed images, 2k × 2k CCD cameras were utilized for WHOT, NEXT and NAOC85-cm and a 4k × 4k CCD camera was employed with NOWT. The fields of view were 12' × 12' for WHOT, 1.3° × 1.3° for NOWT, 22' × 22' for NEXT and 32' × 32' for NAOC85-cm. The effective subframe of NOWT was 30' × 30' during the observations. The filters we used are standard Johnson-Cousins-Bessel BVR_cI_c systems. The standard IRAF routine was applied to process the observed data including zero and flat calibrations, and aperture photometry, then differential magnitudes between the target and comparison star and those between the comparison and check stars were obtained. The complete light curves of V342 UMa observed by NEXT and WHOT and those of V509 Cam acquired by NOWT and NEXT are illustrated in Figure 1 and Figure 2, respectively. As seen in these two figures, the two targets manifest EW type light curves, and very clear flat primary minima can be located. Based on our observations, six eclipsing minima were derived for V342 UMa, while 10 were obtained for V509Cam. All the minima were calculated by the Kwee-vanWoerden (K-W) method (Kwee & vanWoerden 1956) and are listed in Table 2.

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Although both V342 UMa and V509 Cam have been identified as variables for more than 10 years, no one has yet analyzed their orbital period variations. Thus, we collected all published eclipsing times for V342 UMa and V509 Cam from literature, and listed them in Table 2. Moreover, the Wide Angle Search for Planets (WASP, Butters et al. 2010) project has observed V342 UMa, and we calculated two minima using the archive data, which are also listed in Table 2. Combining our newly observed ones, we obtained a total of 43 photoelectric or CCD eclipsing times for V342 UMa, and a total of 17 photoelectric or CCD eclipsing times for V509 Cam. Using the least-squares method, the linear ephemeris of V342 UMa taken from Nelson et al. (2004) was corrected to be

$$\begin{eqnarray}\begin{array}{lll}{\rm{Min}}.{\rm{I}} & = & 2453054.83723(\pm 0.00090)\\ & & +{0.34385184}^{{\rm{d}}}(\pm 0.00000010){\rm{E}},\end{array}\end{eqnarray} \tag{ 1 }$$

and the linear ephemeris of V509 Cam referenced from the O − C Gateway¹ was amended to be

$$\begin{eqnarray}\begin{array}{lll}{\rm{Min}}.{\rm{I}} & = & 2451492.27572(\pm 0.00071)\\ & & +{0.35034717}^{{\rm{d}}}(\pm 0.00000004){\rm{E}}.\end{array}\end{eqnarray} \tag{ 2 }$$

All the O − C values calculated by the two equations are listed in Table 2, and the corresponding curves are displayed in Figure 3. We can find that both V342 UMa and V509 Cam manifest a parabolic trend. Then, quadratic ephemerideswere applied to fit the O − C curves of the two targets,

$$\begin{eqnarray}\begin{array}{lll}{\rm{Min}}.{\rm{I}} & = & -0.00043(\pm 0.00095)\\ & & +0.00000069(\pm 0.00000038){\rm{E}}\\ & & -4.81(\pm 2.56)\times {10}^{-11}{{\rm{E}}}^{2},\end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}\begin{array}{lll}{\rm{Min}}.{\rm{I}} & = & -0.00127(\pm 0.00056)\\ & & -0.00000041(\pm 0.00000010){\rm{E}}\\ & & +1.90(\pm 0.43)\times {10}^{-11}{{\rm{E}}}^{2}.\end{array}\end{eqnarray} \tag{ 4 }$$

When removing Equations (3) and (4), the residuals are listed in Table 2 and depicted in the bottom panels of Figure 3. No cyclic variations can be detected in the residuals. According to the coefficients of the quadratic terms in Equations (3) and (4), we determined that the period of V342 UMa is undergoing a secular decrease at a rate of −1.02(± 0.54) × 10⁻⁷ d yr⁻¹ and the period of V509 Cam is continuously increasing at a rate of 3.96(± 0.90) × 10⁻⁸ d yr⁻¹.

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Based on our observations, two sets of complete light curves for V342 UMa and three sets of light curves for V509 Cam were obtained. The Wilson-Devinney (W-D) program (Wilson & Devinney 1971; Wilson 1979, 1990) was used to model these light curves. Gaia Data Release 2 (DR2, Gaia Collaboration et al. 2016, 2018) included these two targets with their determined mean temperatures, T_m = 5741 K for V342 UMa and T_m = 6462 K for V509 Cam. At first, the mean temperature was set as the temperature of the primary, T₁. The bolometric and bandpass limb-darkening coefficients were taken from van Hamme (1993)'s table, and the gravity-darkening coefficients and bolometric albedos were set as g_1,2 = 0.32 and A_1,2 = 0.5 respectively for their convective envelopes (Lucy 1967; Ruciński 1969). Due to the lack of radial velocity curves, the q-search method was applied to determine themass ratios of the two systems.When we obtained the final solutions, the temperatures of the two components were calculated using the following method (Coughlin et al. 2011; Dimitrov & Kjurkchieva 2015),

$$\begin{eqnarray}\begin{array}{lll}{T}_{1} & = & {T}_{m}+\frac{c\Delta T}{c+1},\\ {T}_{2} & = & {T}_{1}-\Delta T,\end{array}\end{eqnarray} \tag{ 5 }$$

where Δ T = T₁ − T₂ and c = L₂/L₁ are derived by the W-D modeling.

Because two or more sets of complete light curves were acquired for both V342 UMa and V509 Cam, and the light curves observed at different times are different, the physical parameters determined by different light curves may be different. Therefore, we chose one set of complete light curves to determine the physical parameters, and the derived physical parameters were set as reference values to model the other light curves. For V342 UMa, the complete light curves observed in 2019 have higher quality comparing to those observed in 2018, so the 2019 light curve was selected. For V509 Cam, the complete light curves observed on 2018 March 06 are symmetric and have the highest precision among the three sets of light curves, so the 201803 light curve was selected. During the modeling, we used Mode 2 (detached configuration) for both targets at first and found that the solutions were quickly convergent at Mode 3 (contact configuration). The adjustable parameters were as follows: the orbital inclination, i, the temperature of the secondary, T₂, the dimensionless potential of the primary Ω₁ and the monochromatic luminosity of the primary, L₁. Then, a series of solutions with fixed values of mass ratio q was generated for them. The weighted sum of squared residuals, $\displaystyle \sum {W}_{i}{(O-C)}_{i}^{2}$, versus mass ratio q for V342 UMa and V509 Cam are, respectively, displayed in the left and right panels of Figure 4. As seen in Figure 4, a very sharp minimum was determined for V342 UMa at q = 2.8, while that derived for V509 Cam was at q = 2.5. These two values were set as initial values, and adjustable parameters and new solutions were calculated. When the solutions were convergent, the physical parameters were obtained. The light curves of V342 UMa were asymmetric, and adding a cool spot on the less massive primary component could reproduce the asymmetric light curves. The derived physical parameters are listed in Table 3, and the corresponding synthetic light curves are plotted in Figure 5 and Figure 6.

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Table 3. Photometric Results of V342 UMa and V509 Cam

Star	V342 UMa		V509 Cam
Parameters	201901	201805	201803	201802	201804
T1(K)	5902 ± 6	5899 ± 8	6523 ± 4	6498 ± 5	6560 ± 5
T2(K)	5662 ± 11	5663 ± 16	6433 ± 9	6446 ± 12	6415 ± 11
q	2.748 ± 0.015	2.748(fixed)	2.549 ± 0.016	2.549(fixed)	2.549(fixed)
i(°)	84.2 ± 0.2	82.9 ± 0.3	82.1 ± 0.1	82.3 ± 0.1	82.8 ± 0.2
$\overline{{L}_{2}/{L}_{1}}$	2.040 ± 0.113	2.024 ± 0.110	2.145 ± 0.037	2.202 ± 0.023	2.073 ± 0.058
Ω1 = Ω2	6.219 ± 0.019	6.164 ± 0.006	5.816 ± 0.021	5.821 ± 0.005	5.828 ± 0.005
r1	0.299 ± 0.001	0.305 ± 0.001	0.321 ± 0.001	0.319 ± 0.001	0.320 ± 0.001
r2	0.476 ± 0.002	0.480 ± 0.001	0.481 ± 0.003	0.481 ± 0.001	0.481 ± 0.001
f	10.0 ± 3.1%	19.0 ± 1.0%	32.1 ± 3.4%	31.3 ± 0.9%	30.1 ± 0.9%
Spot	On Star 1	On Star 1	–	On Star 2	On Star 1
θ(radian)	1.273 ± 0.092	1.759 ± 0.212	–	0.467 ± 0.107	0.583 ± 0.132
ϕ(radian)	5.896 ± 0.064	2.134 ± 0.056	–	1.225 ± 0.057	1.658 ± 0.039
r(radian)	0.258 ± 0.069	0.358 ± 0.085	–	0.298 ± 0.064	0.423 ± 0.086
Tf(Td/T0)	0.832 ± 0.099	0.789 ± 0.124	–	0.811 ± 0.089	0.824 ± 0.078

To model the other light curves, the physical parameters derived earlier were set as reference values and the mass ratio q was fixed. Because the light curves are asymmetric, a spot model was applied. The best fitting results are also displayed in Table 3, and the corresponding fitting curves are respectively displayed in Figure 5 and Figure 6. According to the previous discussion, the physical parameters determined by the 2019 light curve of V342 UMa and the 201803 light curve of V509 Cam should be more reliable. Therefore, the physical parameters determined by the 2019 light curve of V342 UMa and the 201803 light curve of V509 Cam were adopted as the final results. The geometric configurations of the two systems are, respectively, plotted in Figure 7 and Figure 8, and changes in the spot distributions can be clearly ascertained.

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Two sets of complete BVR_cI_c light curves for V789 Her and three sets of complete BVR_cI_c light curves for V509 Cam were obtained and analyzed. We discovered that both of these two systems are W-subtype contact binaries, with q = 2.748 ± 0.015 for V342 UMa and q = 2.549 ± 0.016 for V509 Cam. V342 UMa is a shallow contact binary with a fill-out factor of f = 10.0 ± 3.1%, and V509 Cam is a medium contact binary with a contact degree of f = 32.1 ± 3.4%. The two systems show totally eclipsing primary minima, their inclinations are higher than 82° and the q-search curves exhibit very clear sharpness around the bottom. This indicates that the photometric solutions derived only by the photometric light curves are reliable (e.g., Pribulla et al. 2003; Terrell & Wilson 2005; Zhang et al. 2017). Our photometric results of V342 UMa, such as the less massive component being the hotter one, and the reciprocal of ourmass ratio is 1/q ∼ 0.364, are similar with those determined by Nelson et al. (2004). By collecting all of the available eclipsing times for V342 UMa and V509 Cam, we examined the period changes and found that the period of V342 UMa manifests a secular decrease at a rate of −1.02(± 0.54) × 10⁻⁷ d yr⁻¹ and the period of V509 Cam is continuously increasing at a rate of 3.96(± 0.90) × 10⁻⁸ d yr⁻¹.

5.1. Absolute Parameter Estimation

5.2. The Secular Period Changes

The period of V342 UMa exhibits a secular decrease at a rate of −1.02(± 0.54) × 10⁻⁷ d yr⁻¹. Usually, the long-term period decrease is produced by conservative mass transfer or AML. If it is caused by conservative mass transfer, the mass transfer rate can be determined to be dM₁/dt = 3.01(± 1.59) × 10⁻⁸ M_⊙ yr⁻¹ by using the following equation,

$$\begin{eqnarray}&&\frac{\dot{P}}{P}=-3\mathop{{M}_{1}}\limits^{.}(\frac{1}{{M}_{1}}-\frac{1}{{M}_{2}}).\end{eqnarray} \tag{ 6 }$$

The positive sign indicates that the less massive primary component is receiving mass. Assuming the angular momentum and total mass are constant and the more massive component transfers its mass on a thermal timescale, τ_th = 3.39 × 10⁷ yr can be calculated using the relation ${\tau }_{{\rm{th}}}=\frac{G{M}_{2}^{2}}{{R}_{2}{L}_{2}}$. Then, the mass transfer rate can be roughly derived to be M₂/τ_th = 3.81 × 10⁻⁸ M_⊙ yr⁻¹, which coincides with the result determined by Equation (6). Another possibility is AML due to magnetic stellar winds. An approximation for calculating the period decrease rate was given by Guinan & Bradstreet (1998) as follows,

$$\begin{eqnarray}\begin{array}{lll}\frac{dP}{dt} & \approx & -1.1\times {10}^{-8}{q}^{-1}{(1+q)}^{2}{({M}_{1}+{M}_{2})}^{-5/3}{k}^{2}\\ & & \times ({M}_{1}{R}_{1}^{4}+{M}_{2}{R}_{2}^{4}){P}^{-7/3},\end{array}\end{eqnarray} \tag{ 7 }$$

where k² is the gyration constant. Taking k² = 0.1 from Webbink (1976) for low-mass main sequence stars, we inferred that the period decrease rate caused by AML is − 0.71 × 10⁻⁷ d yr⁻¹, which is similar to the observed value. Therefore, both conservative mass transfer and AML can explain the long-term period decrease of V342 UMa. At present, we cannot know which one is dominant only based on the observed period variation.

The period of V509 Cam is continuously increasing at a rate of 3.96(± 0.90) × 10⁻⁸ d yr⁻¹. The long-term period increase is generally attributed to conservative mass transfer. Using Equation (6), we concluded that the mass transfer rate is dM₁/dt = −1.07(± 0.24) × 10⁻⁸ M_⊙ yr⁻¹. The negative sign signifies that the less massive primary component is transferring mass to the more massive secondary one. V509 Cam is a late type contact binary, and AML due to magnetic stellar winds can also happen in V509 Cam. Therefore, the determined mass transfer rate should be considered as a minimal value.

The secular period decrease rate, − 1.02(± 0.54) × 10⁻⁷ d yr⁻¹, of V342 UMa is very common in W-subtype contact binaries, such as − 1.69 × 10⁻⁷ d yr⁻¹ for V502 Oph (Zhou et al. 2016), −0.62 × 10⁻⁷ d yr⁻¹ for GU Ori (Zhou et al. 2018) and −1.78 × 10⁻⁷ d yr⁻¹ for V1007 Cas (Li et al. 2018). With decreasing period, V342 UMa will evolve from the present shallow contact configuration to high fill-out contact state. The long-term period increase rate, 3.96(± 0.90) × 10⁻⁸ d yr⁻¹, of V509 Cam is also very common in W-subtype contact binaries, such as 5.09 × 10⁻⁸ d yr⁻¹ for EP And (Lee et al. 2013), 7.7 × 10⁻⁸ d yr⁻¹ for UX Eri (Qian et al. 2007b) and 6.5 × 10⁻⁸ d yr⁻¹ for LR Cam (Yang & Dai 2010).With increasing period, V509 Cam may evolve into a broken-contact binary. There is another possibility that the long-term period variations of the two systems are only part of very long period periodic variations that result from a distant third body (Liao & Qian 2010), and continuous observations of the two targets are required to confirm this in the future.

5.3. Light Curve Variations of the Two Targets

5.4. The Evolutionary Status

To study the evolutionary status of the two stars, the Hertzsprung-Russell (H-R) diagram and the color-density (C-D) diagram were constructed and are shown in the left and right panels of Figure 9, respectively. The zero age main sequence (ZAMS) and terminal age main sequence (TAMS) in the H-R diagram were taken from Girardi et al. (2000), while the ZAMS and TAMS in the C-D diagram come from Mochnacki (1981). In order to compare with other W-subtype contact binaries, the W-subtype low mass contact binaries (LMCBs) listed in Yakut & Eggleton (2005) are also displayed in Figure 9. The horizontal axis of the right panel in Figure 9 is color index, and we converted the temperatures of the components of all systems including V342 UMa and V509 Cam to color index based on table 5 of Pecaut & Mamajek (2013). The mean densities of the components were calculated using the relation provided by Mochnacki (1981),

$$\begin{eqnarray}&&\begin{array}{l}\overline{{\rho }_{1}}=\frac{0.079}{{V}_{1}(1+q){P}^{2}}{{\rm{gcm}}}^{-3},\\ \overline{{\rho }_{2}}=\frac{0.079q}{{V}_{2}(1+q){P}^{2}}{{\rm{gcm}}}^{-3}.\end{array}\end{eqnarray} \tag{ 8 }$$

In Figure 9, the ZAMS and TAMS are labeled as solid and dotted lines, respectively, and the circles refer to the more massive components (p), while the triangles represent the less massive ones (s). Both the H-R diagram and C-D diagram demonstrate that the components of V342 UMa and V509 Cam are consistent with those of other W-subtype contact systems. The less massive components are close to the ZAMS, meaning that they are main-sequence or little evolved stars, while themoremassive ones are located near the TAMS, indicating that they are at an advanced evolutionary stage. The evolutionary status of the components of W-subtype contact systems is similar with that of the A-subtype ones. Therefore, the W-subtype phenomenon is still an open question. More observations and investigations of these two subtype contact binaries are needed.

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In this paper, investigations of the light curves and period variations of V342 UMa and V509 Cam are presented. We found that both of the two systems are Wsubtype contact binaries and manifest very strong light curve changes. The period change analysis reveals that V342 UMa displays long-term period decrease which can be caused by conservative mass transfer or AML due to magnetic stellar winds and that V509 Cam exhibits longterm period increase which can be attributed to conservative mass transfer. The absolute parameters of the two binaries were obtained based on the Gaia distances. To identify their cyclic period changes, further observations are required.

This work is supported by the National Natural Science Foundation of China (No. 11703016), the Joint Research Fund in Astronomy (No. U1431105) under cooperative agreement between the National Natural Science Foundation of China and the Chinese Academy of Sciences, the program of the Light in China's Western Region (No. 2015-XBQN-A-02), the Natural Science Foundation of Shandong Province (Nos. ZR2014AQ019 and JQ201702), the Young Scholars Program of Shandong University, Weihai (Nos. 20820162003 and 20820171006), the program of Tianshan Youth (No. 2017Q091), and the Open Research Program of Key Laboratory for the Structure and Evolution of Celestial Objects (No. OP201704). Great thanks go the referee for very helpful comments and suggestions that improved our manuscript.

We acknowledge the support of the staff of the Xinglong 85-cm telescope, NOWT, WHOT and NEXT. This work was partially supported by the Open Project Program of the Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences.

This paper makes use of data from the DR1 of the WASP data (Butters et al. 2010) as provided by the WASP consortium, and the computing and storage facilities at the CERIT Scientific Cloud, reg. no. CZ.1.05/3.2.00/08.0144 which is operated byMasaryk University, Czech Republic.

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). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.