2. Photometric observations

The four eclipse binaries were observed using the 60 and 85-cm telescopes at Xinglong station of National Astronomical Observatories of China (NAOC). During the observations, the Andor DZ936 PI2048 CCD photometric system was equipped and the standard Johnson-cousins BVR filters were adopted. The observation information (e.g., date, exposure time, and number of images) are summarised in Table 1. All images are reduced to photometric data using the aperture photometry package from the Image Reduction and Analysis Facility (IRAF Footnote g) software library. From their respective fields of view, we selected two stars with the locations, magnitudes, and colours as close to those of the targets as possible, as their comparison and check stars. By checking the magnitude differences between the comparison and check stars, we confirmed that the photometric luminosity between them is stable throughout the observations. From the Set of Identifications, Measurements and Bibliography for Astronomical Data (SIMBAD) database and the latest data release (DR9) of AVVSO Photometric All Sky Survey (Henden et al. Reference Henden, Levine, Terrell and Welch2015), the coordinates and magnitudes in B and V band were collected. From the Gaia DR2, we extracted the parallaxes. All these information for the targets, comparison, and check stars are compiled in Table 2. The reduced data are presented in the form of different magnitude between the target and comparison stars, and the phases are calculated using the orbital periods (the second column of Table 2) determined by Jayasinghe et al. (Reference Jayasinghe2018). The photometric data, including the Heliocentric Julian Day (HJD), phases, and different magnitudes, are compiled as the Supplementary File (see Tables S1–S4 of Appendix A).

Table 1. Log of photometric observations for four target stars.

Table 2. Coordinates, magnitudes and parallaxes of target, comparison, and check stars.

The phase-folded light curves for each of our targets are displayed in Figure 1. All of the four binary systems show the typical EW-type luminosity variation, indicating they are in contact or near-contact configuration. In addition, the light curves of LQ Com show a very wide flatness around both the primary and secondary eclipses, especially for the secondary eclipse (the constant luminosity sustains about 0.08 phase, i.e., about 40 min). This implies that LQ Com is a total-eclipse binary and the radius of its hot primary component is significantly larger than that of the cool secondary one. The slight and symmetric distortion of the primary-eclipse flatness should mainly attribute to the limb darkening effect of the primary component. Combining the wider flatness around the primary eclipse with the fact that the primary eclipse is deeper than the secondary eclipse, we can infer that LQ Com is an A-subtype and total-eclipse binary. For V805 And, a narrow flatness appears on the deeper primary eclipse, but disappear for the somewhat shallow secondary eclipse. This means that V805 And should be also a total-eclipse binary, but belongs to the W-subtype. The sharp light minima for both V473 And and EG CVn indicate that they should be two partial-eclipse binaries.

Figure 1. Top of each panel: Observed (symbols) and theoretical (solid lines) light curves without spot for four targets: V473 And, V805 And, LQ Com, and EG CVn. Bottom: the corresponding residuals (the residuals for B and R bands are vertically shifted by 0.1 and –0.1 mag, respectively).

3. Photometric solutions and absolute parameters

The Wilson-Devinney (W-D) code (Wilson & Devinney Reference Wilson and Devinney1971; Wilson Reference Wilson1979, Reference Wilson1990; Wilson & Van Hamme Reference Wilson and Van Hamme2014) is one of the most customary programmes to model the observed light and radial velocity curves of various binary systems. It has been revised and improved for many times. Here the latest 2015 version of W-D code is adopted to analyse the multi-band light curves of the four binaries: V473 And, V805 And, LQ Com, and EG CVn. Firstly, we estimate the effective temperatures of their primary components according to two available colour indices ( $B{-}V$ and $J{-}K$ ) and the Gaia spectroscopic observations. The $B{-}V$ and $J{-}K$ colour indices are collected from the AVVSO Photometric All Sky SurveyFootnote h (Henden et al. Reference Henden, Levine, Terrell and Welch2015) and the Two Micron All Sky SurveyFootnote i (Skrutskie et al. Reference Skrutskie2006), respectively. The corresponding temperatures, $T_{\rm BV}$ and $T_{\rm JK}$ , are estimated with the corrected calibrationFootnote j (Pecaut & Mamajek Reference Pecaut and Mamajek2013). The colour indices and the effective temperatures estimated from them and the Gaia spectroscopic observations, are summarised in Table 3. The average values of three temperatures are presented in the last column of Table 3, and the corresponding errors are denoted by the standard deviations. In fact, the average value could not represent exactly the temperature of the primary component, and should be a quadrature temperature of the two components. However, because the temperatures of two components, in general, do not significantly deviate from each other for a typical W UMa-type contact binary, we adopted the average temperature as a preliminary one $T_{\rm pre}$ of the primary and the final temperatures for both primary and secondary would be adjusted according to the following equations (Kjurkchieva & Vasileva Reference Kjurkchieva and Vasileva2015),

(1) \begin{equation}T_1^f= T_{\rm pre} + \frac{c\Delta T}{c+1},\\T_2^f= T_1^f -\Delta T,\end{equation}

where $c = l_2/l_1$ , denotes the relative luminosity ratio of two components, and $\Delta T = T_{\rm pre}-T_2$ . The gravity darkening and bolometric albedos coefficients of the components were set to be $g_1=g_2=0.32$ (Lucy Reference Lucy1967) and $A_1=A_2=0.5$ (Ruciński Reference Ruciński1969), respectively. For the bolometric and bandpass-specific limb darkening law, we selected the square root law (Diaz-Cordoves & Gimenez Reference Diaz-Cordoves and Gimenez1992). The corresponding coefficients were interpolated from the table of Claret & Bloemen (Reference Claret and Bloemen2011). Of course, these coefficients will be modified when the temperatures are adjusted. The four targets show the typical EW-type light curves, suggesting they should be contact binaries. Thus, we selected the mode 3, i.e., the contact model, in the W-D code. Some other parameters, e.g., the mass ratio q, the orbital inclination i, the secondary’s temperature $T_{2}$ , the primary’s monochromatic luminosity $L_{1}$ , and the dimensionless potential $\Omega_{1}=\Omega_{2}$ of two components, are set as the free ones. The secondary’s monochromatic luminosity $L_{2}$ is calculated according to the blackbody radiation model.

Table 3. Colours, and temperatures of V473 And, V805 And, LQ Com, and EG CVn.

Because neither spectroscopic nor photometric mass ratios have been reported for these four binaries so far, we employed the $q-$ search method to estimate their initial mass ratios. In the numerical scheme, we calculated Chi square $\chi^{2}=\frac {1} {n}\sum_{i}$ $w_{i}(O_i-C_i)^2$ for the optimal fit of light curves at a series of fixed mass ratios q, with the values ranging from 0.20 to 5.0 in step of 0.1. For V473 And and LQ Com, finer grid search at increment of 0.01 for the value of q was conducted near the approximate value to obtain the more accurate values of their initial mass ratios. Figure 2 displayed the relation between Chi square $\chi^{2}$ and mass ratio $q=M_2/M_1$ . Clearly, there are two minima for LQ Com, and one minimum for the other three targets. The first minimum for LQ Com appears at $q=0.26$ , and the second minimum is at $q=4.2$ . As has been mentioned, we could determine that LQ Com is an A-subtype contact binary from the light-curve morphology. Moreover, $\chi^{2}$ for the first minimum is smaller than that for the second minimum. Thus the mass ratio of LQ Com should be about 0.26. According to the criteria for the reliability of photometric mass ratio proposed by Zhang et al. (Reference Zhang, Qian, Han and Wu2017b), the sharper of the minimum in the $q-$ search curve, the more reliable the photometric mass ratio. Both V473 And and LQ Com show a very sharper minimum in their $q-$ search curves, but the minima for V805 And and EG CVn are relatively flat. This implies that the photometric mass ratios for V473 And and LQ Com are more reliable than those of V805 And and EG CVn. However, according to the statistical analysis of Pribulla et al. (Reference Pribulla, Kreiner and Tremko2003) and the numerical simulations of Terrell & Wilson (Reference Terrell and Wilson2005), photometric mass ratio should be reliable for total-eclipse contact binaries. Thus, in spite of the flat minimum of the $q-$ search curve for V805 And, its mass ratio determined from the photometric light curves, would be still reliable. Comparatively, the mass ratio of EG CVn should be unreliable due to both the partial eclipse and the flat minimum in the q-search curve.

Figure 2. Relations between $\chi^{2}$ and mass ratio q of four targets: V473 And, V805 And, LQ Com, and EG CVn.

Based on the above settings of parameters and the initial mass ratios, we run the W-D code to simultaneously solve the multi-band light curves. After achieving the optimal converging solutions via enough iterations, we modified the temperatures of the primary and secondary components according to Equation (1), and rerun the W-D code to obtain the final and self-consistent converging solutions. The final solutions of these four targets are listed in Table 4 and the corresponding theoretical light curves are displayed in Figure 1 as solid lines. The theoretical light curves could well fit the observation data for V473 And, but slightly deviate from the observational ones around the out-of-eclipse maxima for V805 And, LQ Com, and EG CVn.

Table 4. Photometric solutions of four contact binaries.

It is well known that many W UMa-type contact binaries show asymmetries in two out-of-eclipse maxima of their light curves, which is known as the O’Connell effect (O’Connell Reference O’Connell1951). Usually, it is very difficult to identify the slight asymmetry (i.e., the weak O’Connell effect) by a visual inspection. In order to justify weather the light curves of our four targets are symmetrical, or not, we employed two measurements to quantify their possible asymmetries in their light curves. The simplest and conventional measurement is the magnitude difference ( $\Delta m_{\rm max}$ ) between two out-of-eclipse maxima

(2) \begin{equation}\Delta m_{\rm max}= m_{\rm max.II}-m_{\rm max.I}.\end{equation}

Another one, called the O’Connell Effect Ratio (OER), was proposed by McCartney (Reference McCartney1997). It is the ratio of the areas beneath the two maxima of a phased light curve. In the numerical scheme, one may first divide the phased light curve into n equally wide bins, where n is chosen to sufficiently sample the light curve. Then, the weighted mean magnitude in each bin is calculated and normalised by subtracting the mean magnitude in the bin including the primary minimum. The OER is finally calculated according to the definition

(3) \begin{equation}OER=\frac {\sum^{n/2}_{k=1}\! (m_{k}-m_{1})} {\sum^{n}_{k=n/2+1}\! (m_{k}-m_{1})},\end{equation}

where $m_{k}$ represents the weighted mean magnitude in the kth bin. Obviously, OER can more thoroughly detect the asymmetries of a light curve than $\Delta m_{\rm max}$ . If $\Delta m_{\rm max}>0$ or $OER>1$ , the O’Connell effect is positive, or it will be negative.

In view of the high precision of TESS photometry, the wide-band (approximately 600–1000 nm) and long-cadence (30 min) observations of three targets are also used to estimate the possible O’Connell effect and derive photometric solutions. The data product of TESS includes the Single Aperture Photometry (SAP) and the Pre-search Data Conditioning light curves. The latter has been processed to make it more suitable for detecting shallow transits of extrasolar planets, and not intended for eclipse binaries which show large variability amplitudes. Thus, we adopted only the SAP data in the present work. The SAP flux data are downloaded from Mikulski Archive for Space Telescopes (MAST) portalFootnote k and converted into magnitude units. We excluded the data points with a nonzero flag for the QUALITY parameter.

Table 5 summarised both $\Delta m$ and OER of the light curve of each band for all four targets. From Table 5, we can conclude that the light curves of V473 And, obtained from both our observations and TESS photometry, are almost perfect symmetrical, while there are slight asymmetries (i.e., weak O’Connell effects) for the other three targets. It should be noted that although both two measurements for the ground-based observations of LQ Com, suggest the existence of positive O’Connell effect, we cannot decide it because the values of $\Delta m_{\rm max}$ are close and even smaller to the corresponding mean errors. But fortunately, the observation dates of TESS for LQ Com were just very close to those of our ground-based observations. Moreover, TESS light curve suggests a significant and the same positive O’Connell effect as our ground-based observations. Thus, we come to the conclusion that there exists a positive O’Connell effect for LQ Com.

Table 5. Measurements of O’Connell effects for the four targets.

${}^{\rm a}$ In this column, GB-B, V and R denote the ground-based observation with B, V and R band, respectively. TESS-W refers to the wide-band observations of TESS.

${}^{\rm b}$ The errors are the mean error of the corresponding photometric data.

In general, the O’Connell effect could be interpreted as a result of the spots on the surface of the component stars. According to the above calculations, the light curves of both V805 And and LQ Com show a positive O’Connell effect, while the light curves of EG CVn presented a weak and negative O’Connell effect. The positive/negative O’Connell effect could be caused by any one of the following four spot models: (1) a hot spot on the primary star (HS1); (2) a cool spot on the primary star (CS1); (3) a hot spot on the secondary star (HS2); (4) a cool spot on the secondary star (CS2). With the four different spot models, we reapplied the W-D code to fit the photometric data and obtained the photometric solutions with different spot models, which are compiled in Table 6. Among those spot models, the model with a cool spot on the primary component (CS1) seems to be the optimal one to reproduce the observational light curves for V805 And, LQ Com and EG CVn, because its chi-square value is smallest. However, it should be noted that the differences of $\chi^2$ among four different spot models are rather negligible for the three targets, especially for V805 And and EG CVn.

Table 6. Photometric solutions with spot for V805 And, LQ Com and EG CVn derived from our ground-based observations.

In order to determine the optimal solutions, we also fitted the TESS light curves with each of those spot models using the W-D code. During the analyses of TESS light curves, the photometric solutions without spot derived from our ground-based observations are set as the initial parameters. The limb darkening coefficients for TESS light curves were taken from Claret (Reference Claret2017). The corresponding photometric solutions are summarised in Table 7. Although $\chi^2$ of four different spot models did not also significantly deviate from each other, the optimal light-curves solutions tended to the CS1. The results are consistent with those derived from our ground-based observations. Thus, we think that the CS1 is the optimal one among the four possible spot models for there three targets. The theoretical light curves corresponding to the optimal photometric solutions for both TESS and our ground-based observations, were calculated and showed as the solid lines in Figure 3.

Table 7. Photometric solutions without/with spot for V473 And, LQ Com and EG CVn obtained from TESS’s observations.

Figure 3. Observed (symbols) light curves from TESS (left panels) and our ground-based observations (right panels). Theoretical light curves corresponding to their respective optimal solutions are plotted as solid lines. Bottom of each panel: the corresponding residuals (the residuals for B and R bands are vertically shifted by 0.1 and –0.1 mag, respectively).

Because the O’Connell effect is positive for V805 And and LQ Com, but be negative for EG CVn, the cool spots for V805 And and LQ Com could be observed around phase 0.25, while the cool spot for EG CVn could be around phase 0.75 (see the three upper panels of Figure 4). In addition, except from the spot’s parameters, these photometric solutions with different spot models did not significantly deviate from each other. Thus, we would adopt the photometric solutions of CS1 for both V805 And, LQ Com, and EG CVn in the following analysis.

Figure 4. Upper three panels display the geometrical configurations of V805 And and EG CVn for the best photometric solutions with one spot. Bottom four panels display the geometrical configurations of four binary systems at phase 0.0 or 0.5 for visually checking whether are they the total-eclipse contact binaries or not.

According to the photometric solutions, the orbital inclinations for V805 And and LQ Com are larger than $80^{\circ}$ , confirming the total-eclipse inference from their light-curves morphology. For V473 And and EG CVn, the photometric solutions suggest that they are the partial-eclipse binaries, but very close to the total eclipse. The total- or partial-eclipse characteristics for the four targets could be checked visually from the geometrical configurations shown in the bottom panels of Figure 4. In addition, we noted that in the photometric solutions with different spot models, the primary components of both V805 And and EG CVn have lower masses and higher temperatures than their secondary components, implying that they are two W-subtype contact binaries.

Although the radial-velocity curves for the four binaries are absent so far, we can still estimate their absolute physical parameters according to the following two different methods: (1) the scheme based on the Gaia distance (Kjurkchieva et al. Reference Kjurkchieva, Popov, Eneva and Petrov2019), and (2) the empirical relations (Qian Reference Qian2003; Gazeas Reference Gazeas2009; Yu et al. Reference Yu, Li, Huang, Hu and Xiang2022). The standard procedure for the former was proposed by Kjurkchieva et al. (Reference Kjurkchieva, Popov, Eneva and Petrov2019) and developed by Liu et al. (Reference Liu, Qian, Li, He, Li, Zhao and Li2020) and Li et al. (Reference Li2021). In our calculations, the V-band magnitudes $V_{\rm ASASSN}$ at maximum brightness were determined from the phase-folded light curves of ASAS-SN observations, the interstellar extinctions $A_{\rm V}$ were derived with the S&F method (Schlafly & Finkbeiner Reference Schlafly and Finkbeiner2011) from the IRAS database.Footnote l The bolometric corrections $BC_{\rm V}$ were interpolated from the calibration proposed by Pecaut & Mamajek (Reference Pecaut and Mamajek2013). These parameters are summarised in Table 8. For the later, we adopted the up-date empirical relation between the semi-major axis and orbital period derived by Yu et al. (Reference Yu, Li, Huang, Hu and Xiang2022),

(4) \begin{equation}\log{a} = 0.856(18)\log{P} + 0.784(7).\end{equation}

Table 8. Gaia distances, V-band magnitudes at maximum brightness, interstellar extinctions, bolometric corrections, Absolute V-band magnitude, and absolute bolometric magnitudes of Four Target Binaries.

The values in parentheses represent the standard error which are calculated according to the error propagation rule.

Combining our photometric solutions with the semi-major axis determined from the two distinct methods, we calculated the absolute parameters of the four targets which are summarised in Table 9. For V473 And, LQ Com, and EG CVn, two sets of parameters calculated from two distinct methods are almost equal within the error range. However for V805 And, the absolute parameters calculated from the Gaia distance are significantly deviated from those determined from the empirical relation. As has been pointed out by Li et al. (Reference Li2021), the absolute parameters for most of the contact binaries (more than 80% of contact binaries) can be well estimated from the Gaia distance. But for a friction of contact binaries, the absolute parameters determined from the Gaia distance may be unreliable and even unrealistic due to many reasons, such as too far away, the third-light effect, inaccurate interstellar extinction, and inaccurate V-band magnitude at maximum brightness. Therefore, in the following analysis, we adopted the weighted mean parameters for V473 And, LQ Com, and EG CVn, and the parameters determined by the empirical relation for V805 And.

Table 9. Absolute physical parameters of four target binaries calculated from two different methods and their corresponding weighted means.

The values in parentheses represent the standard error which are calculated according to the error propagation rule.

4. Orbital period variations and their physical origins

4.1. Orbital period variations

The orbital period variations of close binaries have an underlying implication to various astrophysical processes and dynamical evolution of binary systems. In order to uncover the period variation of the four targets: V473 And, V805 And, LQ Com, and EG CVn, we performed an extensive search for all available eclipse timings, and obtained 107 data points from the literature and the eclipse timings database (i.e., the O-C gatewayFootnote m). From our observations, we calculated 16 eclipse timings with the K-W method (Kwee & van Woerden Reference Kwee and van Woerden1956). In addition, from several surveys (e.g., NSVS, WASP, and ASAS-SN, ZTF, TESS, etc.), we extracted the photometric data of the four targets and calculated the phase-folded light curves. With the method of Borkovits et al. (Reference Borkovits, Rappaport, Hajdu and Sztakovics2015), we derived 114 eclipse timings. Finally, a total of 237 eclipse timings, including those collected from the literature and calculated from our observations and surveys’ ones, are compiled in Table 10.

Table 10. Eclipse Timings of four targets collected from literature and calculated from our observations or surveys’ data.

References: 1. This paper; 2. Kuzmin (Reference Kuzmin2008); 3. Banfi et al. (Reference Banfi2012); 4. Corfini et al. (Reference Corfini2014); 5. Diethelm (Reference Diethelm2013); 6. Hubscher (Reference Hubscher2014); 7. (Vrastak Reference Vrastak2015); 8. Yao et al. (Reference Yao2015); 9. GCVS; 10. Paschke (2019); 11. Diethelm (Reference Diethelm2001); 12. Paczyński et al. (Reference Paczyński, Szczygieł, Pilecki and Pojmański2006); 13. Nelson (Reference Nelson2008); 14. Brat et al. (Reference Brat2011); 15. Hubscher et al. (Reference Hubscher, Lehmann, Monninger, Steinbach and Walter2010); 16. Hubscher & Monninger (Reference Hubscher and Monninger2011); 17. Hoňková et al. (Reference Hoňková2013); 18. Hubscher, Lehmann, & Walter (Reference Hubscher, Lehmann and Walter2012); 19. Hubscher et al. (Reference Hubscher, Braune and Lehmann2013); 20. Honková et al. (Reference Honková2014); 21. Honkova et al. (Reference Honkova2015); 22. Hubscher (Reference Hubscher2016); 23. Juryšek et al. (Reference Juryšek2017); 24. Hubscher (Reference Hubscher2017); 25. Pagel (Reference Pagel2018); 26. Pagel (Reference Pagel2020); 27. Blattler & Diethelm (Reference Blattler and Diethelm2002); 28. Nelson (Reference Nelson2002); 29. Diethelm (Reference Diethelm2003); 30. Diethelm (Reference Diethelm2004); 31. Diethelm (Reference Diethelm2005); 32. Diethelm (Reference Diethelm2006); 33. Diethelm (Reference Diethelm2007); 34. Nelson (Reference Nelson2009); 35. Diethelm (Reference Diethelm2009); 36. Diethelm (Reference Diethelm2011); 37. Nelson (Reference Nelson2019).

${}^{\rm a}$ These times were derived from the photometric data of NSVS.

${}^{\rm b}$ These times were derived from the photometric data of WASP.

${}^{\rm c}$ These times were derived from the photometric data of ASAS-SN.

${}^{\rm d}$ These times were derived from our observations.

${}^{\rm e}$ These times were derived from the photometric data of ZTF.

${}^{\rm f}$ These times were derived from the photometric data of TESS.

${}^{\rm g}$ These times were derived from the photometric data of BRNO.

${}^{\rm h}$ These times were derived from the photometric data of ASAS DR3.

${}^{\rm i}$ These times were derived from the photometric data of AAVSO.

In order to build the $O-C$ diagrams, we constructed the following linear ephemerides

(5) \begin{equation}{\rm Min.I}=T_{0}+P_{0}\times E,\end{equation}

where $T_{0}$ is the primary eclipse timing derived from our observations and $P_{0}$ is the orbital period determined by Jayasinghe et al. (Reference Jayasinghe2018). With the linear ephemerides, we calculated the $O-C$ values for their eclipse timings which are displayed in Figure 5. Clearly, the $O-C$ diagrams for V473 And, V805 And, and EG CVn, exhibit a complex variation. However, the variation of the $O-C$ curve for LQ Com seems to be not observable due to the large scatters. In order to uncover the secular period changes, we first adopt the quadratic function, i.e.,

(6) \begin{equation}O-C=\Delta T_{0}+\Delta P_{0}\times E + \gamma E^{2},\end{equation}

to fit their $O-C$ values. Among the four targets, LQ Com should be not undergoing a secular period variation because the coefficient of its quadratic term is 3 orders of magnitude smaller than the typical values for contact binaries (Hu et al. Reference Hu, Jiang, Yu and Xiang2018). Thus, for LQ Com, we adopted a linear fit. After subtracting the linear or quadratic fit, the residuals $(O-C)_{1}$ still exhibit the obvious periodic variation. By introducing the sinusoidal term into the quadratic function, i.e.,

(7) \begin{equation}O-C=\Delta T_{0}+\Delta P_{0}\times E + \gamma E^{2} + A\sin\!(\omega E+\unicode{x03D5}),\end{equation}

we can achieve the sufficiently good fits.

Figure 5. $O-C$ diagrams (symbols) and fitting curves (dotted or solid lines) for four targets: V473 And (upper left), V805 And (upper right), LQ Com (left bottom), and EG CVn (right bottom). Top of each panel: $O-C$ curves and the fitting curves of the linear/quadratic function (dotted line) and the linear/quadratic function superimposed a sinusoidal term (solid line). Middle: residuals $(O-C)_{1}$ (symbols) after subtracting the linear/quadratic fit and the fitting curve of the sinusoidal function (solid line). Bottom: the final residuals.

By assigning the reciprocals of the normalised square errors as the weights and using the weighted least-squares method, we derived the fitting parameters which were summarised in Table 11. As should be mentioned, although the errors for two eclipse timings of V805 And are absent, we performed a weighted fit by cutting the two data points. The quadratic term implied that V473 and, V805 And and EG CVn are undergoing a secular period change. The sinusoidal term revealed a cyclic Eclipse-Timing Variation (ETV) for all four targets. Based on the fitting parameters, we calculated the rates of secular period changes and the modulation periods of cyclic ETVs (see the third and fourth lines from the bottom of Table 11).

Table 11. Linear ephemerides and fitting parameters of $O-C$ curves for the four binaries: V473 And, V805 And, LQ Com and EG CVn.

4.2. Physical origins of orbital period variations

4.2.1. Secular period changes

The above orbital period investigations suggested that V473 And, V805 And, and EG CVn shows the secular orbital period decrease and increase, respectively. The secular orbital period variations could be, in general, interpreted as a result of the mass transfer and/or the mass and angular momentum loss. According to the photometric solutions, these three binaries are not in the relatively deep contact configuration ( $f<50\%$ ), i.e., their surface potentials are significantly lower than those of their outer Lagrange points. Thus, it should be difficult and even impossible to loss their mass via Roche-lobe overflow from the outer Lagrange point. In addition, the typical mass-loss rate due to stellar wind for an active dwarf star is about $10^{-10}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ (Mullan et al. Reference Mullan, Doyle, Redman and Mathioudakis1992) or even lower (Lim & White Reference Lim and White1996). It contributes the decrease rate of orbital period at the level of ${\sim}10^{-10}$ ${\rm d}\cdot {\rm yr}^{-1}$ , which is two or three orders of magnitude smaller than the observed ones (typical observed value is $10^{-7}$ ${\rm d}\cdot {\rm yr}^{-1}$ , Hu et al. Reference Hu, Jiang, Yu and Xiang2018). Thus, the effect of mass loss on orbital period decrease may be neglected for V473 And, V805 And, and EG CVn.

When considering the mass conservation and the circular orbit of a contact binary, its orbit evolution can be dominated by

(8) \begin{equation}\frac {\dot J} {J}=\frac {\dot P} {3P} + \dot M_{1}\left(\frac {1} {M_{1}} - \frac {1} {M_{2}}\right),\end{equation}

where $J=\sqrt[3]{\frac{G^{2}M_{1}M_{2}P}{2\pi (M_{1}+M_{2})}}$ is the orbital angular momentum. Equation (6) suggests explicitly that both the conservative mass transfer and the angular momentum loss could yield the secular period changes.

In regard of the angular momentum loss, the most plausible mechanism is the magnetic breaking due to stellar wind. From the standard magnetic breaking model (Weber & Davis Reference Weber and Davis1967), Guinan & Bradstreet (Reference Guinan and Bradstreet1988) derived the period decrease rate caused by the angular momentum loss via magnetic stellar wind

(9) \begin{equation}\dot P_{\rm aml}\approx -1.1\times 10^{-8} \cdot \frac {(1+q)^{2}} {q} \cdot \frac {k^{2}\left(M_{1}R^{4}_{1}+M_{2}R^{4}_{2}\right)} {(M_{1}+M_{2})^{5/3}P^{7/3}}.\end{equation}

In the formula, the gyration constant $k^{2}$ is roughly 0.1 estimated from the main sequence stars. By putting the absolute physical parameters into Equation (7), we estimated the orbital period decrease rates of $-1.59\times 10^{-7}$ ${\rm d}\cdot {\rm yr}^{-1}$ , $-7.92\times 10^{-8}$ ${\rm d}\cdot {\rm yr}^{-1}$ and $-1.18\times 10^{-7}$ ${\rm d}\cdot {\rm yr}^{-1}$ for V473 And, V805 And and EG CVn, respectively. For V473 And and V805 And, the orbital period decrease rates due to the angular momentum loss are smaller than the observed values, implying the angular momentum loss via magnetic stellar wind could not fully yield the observed orbital period decrease. For EG CVn, the orbital period is undergoing a secular increase at the rate of $2.69\times 10^{-7}$ ${\rm d}\cdot {\rm yr}^{-1}$ , but the angular momentum loss due to magnetic stellar wind can only lead to secular orbital period decrease. Thus, in order to yield the observed orbital period increase/decrease for V473 And, V805 And, and EG CVn, the conservative mass transfer seems to be required.

Combining Equation (8) with (9), we may write the mass-transfer rate as

(10) \begin{equation}\dot M_{1}=-\dot M_{2}=\frac {M_{1}M_{2}} {3P(M_{1}-M_{2})}\cdot (\dot P_{\rm obs}-\dot P_{\rm aml}),\end{equation}

where $\dot P_{\rm obs}-\dot P_{\rm aml}$ denotes the secular orbital period variation yielded by the conservative mass transfer. Thus, if the secular orbital period changes of V805 And and EG CVn are yielded by a combination of the angular momentum loss and the mass transfer, the conservative mass transfer rates would be $\dot M_{2}=-\dot M_{1}=5.89\times 10^{-9}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ for V473 And, $\dot M_{1}=-\dot M_{2}=2.58\times 10^{-7}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ for V805 And, and $\dot M_{2}=-\dot M_{1}=1.73\times 10^{-7}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ for EG CVn. If the secular orbital period changes are caused separately by the conservative mass transfer (i.e., $\dot P_{\rm aml}=0$ ), the required mass transfer rates are $\dot M_{2}=6.86\times 10^{-8}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ for V473 And, $\dot M_{1}=3.11\times 10^{-7}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ for V805 And, and $\dot M_{2}=1.20\times 10^{-7}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ for EG CVn. We may estimate the ranges of mass-transfer rate as $[0.59\sim 6.86]\times 10^{-8}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ , $[2.58\sim 3.11]\times 10^{-7}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ and $[1.20\sim 1.73]\times 10^{-7}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ for V805 And and EG CVn, respectively.

According to the thermal timescale of the donor star ( $\tau_{\rm th}\sim \frac {GM_{\rm donor}^{2}} {R_{\rm donor}L_{\rm donor}}$ , Paczyński Reference Paczyński1971) of V473 And, V805 And and EG CVn, we estimated the mass transfer rates on the thermal timescale $\frac {M_{donor}} {\tau_{\rm th}}=-1.05\times 10^{-8}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ , $-5.23\times 10^{-8}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ and $4.89\times 10^{-8}$ ${\rm M}_{\odot}\cdot {\rm yr}^{-1}$ . For V473 and, the mass transfer rate on the thermal timescale is significantly larger than the upper limit of the mass transfer rate estimated from Equation (10), implying that its secular period decrease should be involved with some additional mechanisms, e.g., the mass/angular momentum loss and unstable mass transfer. For V805 And and EG CVn, the thermal-timescale mass transfer rates are significantly smaller than the lower limits of the mass transfer rates determined from Equation (10), implying that the conservative mass transfer could yield the observed period decrease and increase.

Theoretically, the continuous mass transfer could yield the impacting spots on the surface of the mass gaining component (Flannery Reference Flannery1975; Zhou & Leung Reference Zhou and Leung1990; Zhai & Fang Reference Zhai and Fang1995; Zhu et al. Reference Zhu, Qian, Zola and Kreiner2009; Hu et al. Reference Hu, Chen, Xiang, Yu and Zhao2019). For V805 And, the more massive secondary is slightly cooler than the primary. Moreover, due to the contact configuration, the surface potentials of two components are equal so that there is not accreting energy to heat the transferred material. Thus, the spot yielded by mass transfer should be a cool one. Under the effect of Coriolis force, this cool spot can move to a small longitude and be observed around phase 0.75. These theoretical predictions seem to be just consistent with the best photometric solutions (CS1), indicating that the cool spot might be indeed yielded by the mass transfer. However, for EG CVn, the optimal photometric solution (CS1) cannot match the theoretical prediction of impacting spot, which means that the cool spot of EG CVn may be originated from the magnetic activity of its primary component.

4.2.2. Cyclic period changes

The above analyses of $O-C$ curves suggested that all four targets: V473 And, V805 And, LQ Com, and EG CVn, show periodic ETVs. The periodic ETVs are, generally, regarded as the cyclic period variation of an eclipsing binary. However, for some cases, they are not the embodiment of the cyclic period variations. This is dependent on the causes yielding the cyclic ETVs. So far, there are at least three different causes of cyclic ETVs: (1) the apsidal motion; (2) the light-time effect due to the third body; (3) the cyclic magnetic activity. For the cyclic ETVs of the three targets, we can exclude the apsidal motion because the primary and secondary eclipse timings do not follow a reversed-phase variation. Thus, we would focus on the light-time effect due to a third body and the cyclic magnetic activity.

4.2.2.1. Light-time effect due to third body

If the cyclic ETVs of V473 And, V805 And, LQ Com, and EG CVn, are caused by the light-time effect of the third body (Irwin Reference Irwin1952), the orbits of their third body should be or at least close to circular one because of the good sinusoidal fits for the $O-C$ curves. By inserting the modulation period $P_{\rm mod}$ and amplitude A of cyclic ETVs into the following equation

(11) \begin{equation}f(M_{3})=\frac {4\pi^{2}} {GP^{2}_{\rm mod}} \cdot (a_{12}\sin i')^{3},\end{equation}

where $a_{12}\sin i'=A\times c$ ( $a_{12}$ and i ^′ denote the orbital radius and inclination of eclipse binary pair or third body around the barycentre of triple system, and c is light velocity), we may determine the mass function $f(M_{3})$ of the third body. Based on the equation

(12) \begin{equation}f(M_{3})=\frac {M^{3}_{3} \sin^{3} i'} {(M_{1}+M_{2}+M_{3})^{3}},\end{equation}

we could then obtain the relation between the mass $M_{3}$ and orbital inclination i ^′ of the third body. Figure 6 shows the relations between the mass and orbital inclination of the third bodies for these four targets. If these third bodies move just in a coplanar orbit around their corresponding host binaries, their masses are equal to $0.184$ , $0.336$ , $0.066$ and 307 ${\rm M}_{\odot}$ , respectively. When $i'=90^{\circ}$ , the third bodies of four binary systems would reach their minimum masses, i.e., $0.178$ , $0.334$ , $0.066$ and $0.298$ ${\rm M}_{\odot}$ , respectively. For LQ Com, the minimum mass is about 70 times of mass of Jupiter. If there really exists the third body orbiting LQ Com, it should be a brown dwarf. For V473 And, V805 And, and EG CVn, the minimum masses of third bodies are comparable to those of the less massive components of their hosting binaries. Their third bodies might be a stellar companion. Thus, it can be expected to detect the spectra and light contributions of the third bodies. Unfortunately, the spectroscopic observations for these targets are absent so far. For the light contributions of the third bodies, our photometric studies did not suggest any significant third lights. Of course, it should be very difficult, or even unrealistic to detect the third light from light curves.

Figure 6. Relations between the mass and orbital inclination of the third body for V473 And (upper left), V805 And (upper right), LQ Com (left bottom), and EG CVn (right bottom).

The most direct way to confirm the existence of the third body orbiting a binary system is to detect the transiting circumbinary events (Carter et al. Reference Carter2011; Zhang, Qian, & He Reference Zhang, Qian and He2017a; Socia et al. Reference Socia2020) of the third body from the photometric light curves with high resolution, or the radial-velocity change of binary system (i.e., the so-called $\gamma$ -velocity variation, Van Hamme & Wilson Reference Van Hamme and Wilson2007). However, the transiting circumbinary events are very rare because of both strict limitation to the orbital geometry and the low frequency of occurrence. For the $\gamma$ -velocity variation, as has been mentioned, the spectroscopic observations for these targets are still absent. In addition, the gravitational interaction of the third body may typically result in precession of orbital plane of binary system, which could be observable as a change of eclipsing depth on a long timescale. But for our targets, both the low-precision observations from early surveys and the disturbances of magnetic activity spots did not allow us to precisely detect any significant eclipse-depth variations on a long timescale. Of course, besides the Røemer delay (the light-time effect), the third body may also yield the dynamical delay on eclipse timings (Soderhjelm Reference Soderhjelm1975; Borkovits et al. Reference Borkovits, Érdi, Forgács-Dajka and Kovács2003, Reference Borkovits, Csizmadia, Forgács-Dajka and Hegedüs2011, Reference Borkovits, Rappaport, Hajdu and Sztakovics2015, Reference Borkovits, Hajdu, Sztakovics, Rappaport, Levine, Bró and Klagyivik2016). Because of the short orbital periods ( $P<1$ d) of the targeting binaries and the relatively long modulation periods ( $P_{\rm mod}\sim 10$ yr) of the third bodies, the classical Røemer delay will dominate the ETVs and the dynamical delay may be negligible (Rappaport et al. Reference Rappaport, Deck, Levine, Borkovits, Carter, El Mellah, Sanchis-Ojeda and Kalomeni2013). In view of these facts, the follow-up spectroscopic and photometric observations for these binaries should be performed to confirm the existence of their third bodies.

4.2.2.2. Cyclic magnetic activity

Regarding the explanation of the cyclic magnetic activity, Applegate (Reference Applegate1992) proposed a theoretical model to explain the cyclic period oscillations for contact binaries. It has been developed lately by Lanza, Rodono, & Rosner (Reference Lanza, Rodono and Rosner1998) and Lanza & Rodonò (Reference Lanza and Rodonò2002). According to this theory, strong magnetic activities below the surface of one or both components play major roles in producing the cyclic period variations. When the magnetic field distribution of an active component changes during its magnetic cycle, the distribution of angular momentum will be also adjusted. This sequentially yields variations of the quadrupole moment, resulting in variations of the star’s oblateness and radial differential rotation. Variations in shape of the star would be gravitationally coupled to the orbit, modulating the orbital period. Meanwhile, the luminosity of the active component would be variable with variations of its shape.

Using the modulation period and amplitude of cyclic period variation, we may firstly calculate the amplitude $\Delta P$ and relative amplitude $\Delta P/P$ of orbital period modulation with the formula

(13) \begin{equation}\Delta P = 2\pi A \frac {P} {P_{\rm mod}}.\end{equation}

The other model parameters, i.e., the change of quadruple moment $\Delta Q$ , the transferred angular momentum $\Delta J$ , the relative variation of angular velocity $\Delta \Omega/\Omega$ , the required energy $\Delta E$ , the relative luminosity change $\Delta L/L$ , and the required strength of magnetic field B, can be obtained from the following formulas derived by Applegate (Reference Applegate1992)

(14) \begin{equation}\Delta Q = \frac {1} {9}\,{\rm Ma}^{2} \frac {\Delta P} {P},\end{equation}

(15) \begin{equation}\Delta J = \frac {GM^{2}} {R} \left(\frac {a} {R}\right)^{2} \frac {\Delta P} {6\pi},\end{equation}

(16) \begin{equation}\frac {\Delta \Omega} {\Omega} = \frac {M} {M_{\rm shell}} \frac {GM} {3R^{3}} \left(\frac {a} {R}\right)^{2} \left(\frac {P} {2\pi}\right)^{2} \frac {\Delta P} {P},\end{equation}

(17) \begin{equation}\Delta E = 2\frac {(\Delta J)^{2}} {I_{\rm shell}},\end{equation}

(18) \begin{equation}\frac {\Delta L} {L} = \pi \frac {\Delta E} {P_{\rm mod}L},\end{equation}

(19) \begin{equation}B^{2} \sim 10 \frac {GM^{2}} {R^{4}} \left(\frac {a} {R}\right)^{2} \frac {\Delta P} {P_{\rm mod}}.\end{equation}

In these formulas, G is the gravitational constant, $M_{\rm shell}$ and $I_{\rm shell}$ denote the mass and the moment of inertia of the shell for the active component, respectively. M, R, and L are the mass, radius, and luminosity of the active component.

By insetting the absolute physical parameters and modulation period of three targets into the above formulas, we calculated the model parameters needed to yield the observed cyclic period variations for the three targets, which are listed in Table 12. In the calculations, the mass of convective shell was assumed to be typically equal to one-tenth of the mass of the corresponding active component, i.e., $M_{\rm shell}= 0.1M_{\rm active}$ . Also, our calculations were performed for both the primary and secondary components because both two components of these binaries can, in principle, satisfy the condition to yield strong magnetic field. In addition, the typical values of those model parameters required by Applegate’s mechanism are presented in the penultimate column of Table 12 for a reference. Clearly, our four binaries are observed to undergo orbital period modulation with amplitude $\Delta P/P\sim 10^{-6}$ , over time scales of $P_{\rm mod}\sim 10$ yr. These modulations can be yielded by the gravitational coupling of the orbit to variations in the shape of a magnetically active component in these systems. In order to yield the variable deformation of their active stars, the distribution of angular momentum needed to be variable at a level of $\Delta J\sim 10^{47}$ g $\cdot{\rm cm}^{2}\cdot{\rm s}^{-1}$ , to produce the variation of the quadrupole moment at a level of $\Delta Q\sim 10^{49}$ g $\cdot{\rm cm}^{2}$ . Meanwhile, due to the variation of shape of the active component, the relative luminosity variation should be at the $\Delta L/L\simeq 0.01$ level, and the active component would be differentially rotating at the $\Delta \Omega/\Omega\simeq 0.01$ level. The torque needed to redistribute the angular momentum should be exerted by a mean subsurface magnetic field of several kilogauss. Obviously, the variations of those model parameters and the mean subsurface magnetic field, well followed the typical values required by Applegate’s mechanism. These results indicated that the cyclic period changes for all four targets may be interpreted as a result of the cyclic magnetic activity.

Table 12. Model parameters for magnetic activity of three binaries: V473 And, V805 And, LQ Com, and EG CVn.

For the energy required by Applegate’s model, Brinkworth et al. (Reference Brinkworth, Marsh, Dhillon and Knigge2006) proposed another quantised criterion, i.e., whether the active component can provide enough energy to support Applegate’s mechanism, or not. According to the method of Brinkworth et al. (Reference Brinkworth, Marsh, Dhillon and Knigge2006), we calculated the available energiesFootnote n generated by the components for the four targets. Because the energy required by Applegate’s mechanism is very sensitive to the assumed shell mass, the required energies with various different shell masses are calculated and showed in Figure 7. From Figure 7, it can be seen that except for the less massive primary components of V805 And, the available energies provided by the components for all four targets may significantly exceed the required ones for a certain range of shell mass, i.e., these components can generate enough energy to drive Applegate’s mechanism. In another word, the observed cyclic period variations for the four targets may be caused by the cyclic magnetic activity of at least one component of these binaries. Of course, some observable evidences should be further investigated to confirm the explanation of cyclic magnetic activity. For instance, the luminosity variation and any other indicator of magnetic activity (starspot activity, coronal X-ray luminosity, emission cores in CaII or MgII lines, etc.) should also show the same period as the orbital period modulation (Kim et al. Reference Kim, Jeong, Demircan, Muyesseroglu and Budding1997; Hu et al. Reference Hu, Yu, Zhang and Xiang2020). However, these evidences require a long-term (the timescale of several decades, or at least one period of cyclic period variation) photometric and/or spectroscopic observations.

Figure 7. Required energy for the primary (red solid line) and secondary (blue solid line) of V473 And (upper left), V805 And (upper right), LQ Com (left bottom), and EG CVn (right bottom) as a function of mass ratio between shell mass and total mass of the active star. The dashed lines represent the available energies provided by their primary (red dotted line) and secondary (blue dotted line).

5. Evolution of contact binaries

Based on the absolute physical parameters, we located the four binaries at the mass-radius and mass-luminosity diagrams (Figure 8) to discuss their evolutionary states. For comparison, 179 contact binaries collected by Yu et al. (Reference Yu, Li, Huang, Hu and Xiang2022) were also added in Figure 8. In the diagrams, the Zero-Age Main Sequence (ZAMS) and Terminal-Age Main Sequence (TAMS) lines were calculated based on the PARSEC modelsFootnote o (Bressan et al. Reference Bressan, Marigo, Girardi, Salasnich, Dal Cero, Rubele and Nanni2012). All the contact binaries are classified into two subtypes: A and W subtype. Similar to other contact binaries, the more massive components of our four targets are mainly located on the main sequence band and fairly close to the ZAMS line with relatively high metallicity. The less massive components exhibit the over-sized and over-luminous characteristics relative to the single star with the same mass. They have radii 3-4 times larger than expected for the corresponding ZAMS masses. Both Lucy (Reference Lucy1968) and Moses (Reference Moses1976) suggested the over-size and over-luminous phenomena for the less massive components are caused by the energy transfer from the more massive component to the less massive one. However, it seems physically hard to inflate a ZAMS star to such size purely by the energy transfer from its more massive companion (Stępień Reference Stępień2004). So, the less massive components should be more evolved with hydrogen depleted in their centres.

Figure 8. Mass-Radius (left panel), and Mass-Luminosity (right panel) diagrams. The ZAMS lines with different metallicities and TAMS line are calculated for single main-sequence stars using PARSEC models (Bressan et al. Reference Bressan, Marigo, Girardi, Salasnich, Dal Cero, Rubele and Nanni2012). The filled and open squares denote the primary and secondary components of A-subtype samples, respectively. The filled and open circles denote the primary and secondary components of W-subtype samples, respectively. The primary and secondary components for our four targets are depicted as the corresponding red symbols with somewhat larger size.

Although both A- and W-subtype systems belong to the same family of low-mass contact binaries, the evolutionary relation between two subtypes is still a matter of controversy and debate. Following the original idea of Lucy (Reference Lucy1976), Hilditch King & McFarlane (Reference Hilditch, King and McFarlane1988) and Hilditch (Reference Hilditch1989) suggested that the A-subtype contact binaries are more evolved than the W-subtype systems from the mass-radius and colour-luminosity diagrams. However, by further analysing the mass-luminosity and period-angular momentum diagrams, both Maceroni & van’t Veer (Reference Maceroni and van’t Veer1996) and Yakut & Eggleton (Reference Yakut and Eggleton2005) pointed out that the conclusion of Hilditch et al. (Reference Hilditch, King and McFarlane1988) might be incorrect. Moreover, from a statistical standpoint, because the total mass and angular momentum of A-subtype systems are significantly larger than those of W-subtype systems, the A-subtype contact binaries should be less evolved than the W-subtype (Gazeas & Niarchos Reference Gazeas and Niarchos2006).

Combining the catalog of Yu et al. (Reference Yu, Li, Huang, Hu and Xiang2022) with our four targets, we here systematically calculated the statistical averages of physical parameters for two subtypes of contact binaries (Table 13). Clearly, the W-subtype contact binaries have shorter orbital periods and lower temperatures than the A-subtype systems. The two statistical features are similar with those of Rucinski (Reference Rucinski1974) and Smith (Reference Smith1984), indicating that the W-subtype systems are more evolved than A-subtype ones. Moreover, similar to the statistical results of Gazeas & Niarchos (Reference Gazeas and Niarchos2006), our calculations also show that both the total mass and orbital angular momentum of W-subtype systems are lower than those of A-subtype ones. In addition, we found that the temperature difference between two components of a W-subtype system is significantly smaller than that of an A-subtype one, implying that W-subtype systems should be closer to the thermal equilibrium state than A-subtype systems. These statistical evidences show formation of W-subtype systems from A-subtype systems is possible, but the opposite direction is not strongly supported. Usually, the evolution from A-subtype to W-subtype systems may be accompanied by a simultaneous mass and angular momentum loss, as well as mass and energy exchanges between two components of a contact binary. Thus, the opposite evolutionary direction seems to be impossible, because it requires an increase of both total mass, orbital angular momentum, and temperature difference.

Table 13. Statistical averages of physical parameters for two subtypes of contact binaries.

The units of orbital period and temperature are day and Kelvin, respectively. The orbital angular momentum is in a unit of $\times 10^{51}$ ${\rm cm}^{2}\cdot {\rm g}\cdot {\rm s}^{-1}$ . The other parameters are in the solar unit. The values in the third and fifth lines denote the standard derivations of the corresponding parameters.

An additional evidence for the evolutionary direction is that binary systems with shorter orbital periods are, in general, kinematically older (age 8 Gyr) than those with longer periods (age 2 Gyr) (Bilir et al. Reference Bilir, Karataş, Demircan and Eker2005). Of course, there is an alternative evolutionary path for the two different subtypes. For instance, the close detached binaries with different initial states may evolve independently into either A- or W-subtype contact binaries (Zhang, Qian, & Liao Reference Zhang, Qian and Liao2020). Also, the close detached binaries evolve in a similar way but with different mass/energy transfer rate and/or different mass/angular momentum loss rate, leading to different evolutionary outcomes (Gazeas & Stępień Reference Gazeas and Stępień2008). Moreover, we cannot exclude the possibility that some A-subtype systems with small angular momentum have evolved from W-subtype systems, while others (with relatively large angular momentum) have evolved directly from near-contact binaries (Yakut & Eggleton Reference Yakut and Eggleton2005).