Mass ratio estimates for overcontact binaries using the derivatives of light curves

We synthesized a total of 117,600 LCs for overcontact binaries using the software package PHOEBE, release 2.4 (Conroy et al., 2020). Here, we adopted the following parameter ranges and steps: the mass ratio $q=M_{\mathrm{s}}/M_{\mathrm{p}}=0.05$–$0.95$ (in steps of 0.1), orbital period $P=0.2$–$1.8$ (0.4) days, primary mass $M_{\mathrm{p}}=0.5$–$2.0$ (0.5) $M_{\odot}$, orbital inclination $i=60^{\circ}$–$90^{\circ}$ ($10^{\circ}$) deg, primary star and secondary star temperatures $T_{\mathrm{p}}$ ($T_{\mathrm{s}}$)$=4000$–$10,000$ (1000) K, and fillout factor $f=0.2$–$0.8$ (0.3). The gravity-darkening coefficient was set either to 0.32 or 1, depending on whether the star’s temperature was lower or higher than 6600 K, respectively. The fluxes were computed at every 0.001 step in phase.

In our method, we first find the numerical derivatives of an LC up to order 3, with respect to time. This work utilizes the second-order central difference for the mean fluxes, each data point of which is 100 phase bins combined. We find a higher-order derivative for each by repeating numerical differentiation.

Figure 1 shows an example of an LC and its numerical derivatives with respect to time. Here, we define the symbol $t_{ij}$ as the time at either a local maximum or minimum. This method requires the following conditions:

1. 1.

   The second derivative of the LC has two local maxima at $t_{21}$ and $t^{\prime}_{21}$ that are symmetric with respect to an eclipse time.

2. 2.

   The third derivative of the LC has a local maximum at $t^{\prime}_{32}$ and a local minimum at $t_{32}$. Both extrema appear around the eclipse time determined by condition (1) and $w^{\prime}_{32}\sim w_{32}$, where $w_{ij}$ is the time interval between $t_{ij}$ and the eclipse time.

3. 3.

   The condition $w_{31}<w_{21}<w_{32}<w_{22}<0.2P$ must be satisfied, where $P$ is the orbital period.

In some cases, the local extrema at $t_{31}$ and $t^{\prime}_{31}$ can be difficult to find. However, as long as a symmetric double peak as described in condition (1) is found, a failure to identify the local extrema does not affect the result.

In this study, a key value is the time interval $t_{32}-t^{\prime}_{32}$. We obtained these two times by finding the zero derivatives using linear interpolation between the derivatives at consecutive phase points. If the shapes of the two peaks at $t_{21}$ and $t^{\prime}_{21}$ are unclear in an LC, the derived mass ratio will be accordingly erroneous.

Next, we compute the following value $W$:

Here, when using the phase instead of the orbital period, we substitute unity for $P$. If a symmetric double peak (i.e., two local maxima at $t_{21}$ and $t^{\prime}_{21}$) is found around both eclipses in the second derivative of an LC as in Figure 1, we use the mean of both the $W$ values and also calculate the uncertainty described in Section 2.4 from both the $W$ values. If an LC has an inadequately small number of data points or insufficient photometric precision, such a double peak may appear even when it does not actually exist or vice versa. As shown in the following sections, this value is closely correlated with the mass ratio and is a key in our method.

In this section, we derive a generic formula for estimating mass ratios. We calculated the $W$ values (Equation 1) for 36,111 binaries out of the 117,600 synthesized LCs. Figure 2 shows the relation between the mass ratio $q$ and calculated $W$ values. The $W$ values of binaries with inclinations of $i=60^{\circ}$, $70^{\circ}$, and $80^{\circ}$ in our synthesized samples were found to be defined for those with mass ratios of $q=0.05$, $q\leq 0.15$, and $q\leq 0.35$, respectively. We found a clear positive correlation between $W$ and $q$. A regression analysis for the correlation yielded

The $W$ values for 99 % of our synthesized LCs were found to fall within a mass ratio range of $q\pm 0.1$ (the two dashed lines in Figure 2). This indicates that our method can predict a mass ratio with an error of less than 0.1 typically and $\lesssim$ 0.12 even in the worst case.

In Figure 2, a dispersion in the $W$ values is observed for a mass ratio. The primary factor for the dispersion is the difference in inclination, although other factors contribute to a lesser degree. Also, the amount of dispersion is observed to be an increasing function of the mass ratio, though its contribution to the dispersion is relatively small. Note that the fillout factor has a much smaller effect on the dispersion. The standard deviation ($\sigma$) of the residuals for $W$ is $\sigma_{W}=$ 0.766, which corresponds to $\sigma_{q}=$ 0.043. This dispersion could not be reduced, and it contributes to the uncertainty of the estimated mass ratio in this method.

In addition, Table 1 presents the residual standard deviations of the estimated mass ratios using our method for the synthesized LCs with the mass ratios shown in the table. These values offer insights into the extent of the uncertainty in our estimates for the mass ratios of the LCs.

Equation (1) defines that $W$ is a function of $P$, $t_{32}$ ($w_{32}$), and $t^{\prime}_{32}$ ($w^{\prime}_{32}$). Under the assumption that the uncertainty of $P$ is sufficiently small and that the two local extrema in the third derivative of an LC are perfectly symmetric (i.e., $w_{32}=w^{\prime}_{32}$), the standard uncertainty of $W$ is

Considering the uncertainty of $W$ mentioned in Section 2.3 together with $\delta W$, the uncertainty of a mass ratio derived in this method is estimated to be

As shown in Section 4, the above estimate for $\delta q$ is expected to provide a standard uncertainty for $q$. An expanded uncertainty is obtained through multiplication of this value by a coverage factor.

Figure 3 shows a comparison between our mass ratio estimates ($q_{\mathrm{ph}}$) and corresponding spectroscopic mass ratios ($q_{\mathrm{sp}}$), the latter of which are compiled from the relevant papers listed in Latković’s catalog. The standard deviation of $(q_{\mathrm{ph}}-q_{\mathrm{sp}})$ is 0.051, which is 0.008 larger than that for the synthesized LCs. This difference should originate from observational errors, which do not exist in the synthesized LCs. A slight difference in the standard deviation of $(q_{\mathrm{ph}}-q_{\mathrm{sp}})$ is found between the subtypes of W UMa binaries. The W- and A-type systems have standard deviations of 0.048 and 0.053, respectively. One possible factor contributing to this difference should be the variation in the uncertainties of spectroscopic mass ratios: specifically, the mean uncertainties for the W- and A-type systems were 0.015 and 0.017, respectively.

The estimated mass ratios for 95% of the binaries are within the range $q_{\mathrm{sp}}\pm 0.1$, which is $\sim$ four percentage points smaller than that for the synthesized LCs. This decrease should also be due to the observational errors mentioned above. Indeed, the estimated mass ratios for 67% of the binaries are consistent with their spectroscopic values. This percentage is roughly the same as the probability (i.e., 68%) that the true value falls within the confidence interval for the standard uncertainty. Consequently, the application to real binaries demonstrates that our method is practical for estimating the mass ratios of overcontact binaries and provides reasonable uncertainty values.

We also discuss the background of why the proposed method works. It has been demonstrated that photometric mass ratios are accurately determined when overcontact binaries show total annular eclipses (e.g., Pribulla et al., 2003; Terrell & Wilson, 2005). In our synthesized LC samples, condition (1) (Section 2.2) effectively selects binaries with total annular eclipses. In addition, a photometric mass ratio is well determined if the ratio of the radii is known (Terrell & Wilson, 2005). Figure 4 shows a schematic view of a binary from an observer at the key time $t^{\prime}_{32}$. This is a moment immediately after the ratio of change in the luminosity becomes the minimum. In other words, it is a moment immediately after the common chord of the projected surfaces (i.e., circles) of the two stars becomes equal to the diameter of the smaller star, which is the maximum value that the common chord can take. Accordingly, the time interval $t_{32}-t^{\prime}_{32}$ is a decreasing function of the relative size of the smaller star and, as such, embodies information on the ratio of the radii of the stars in a binary. Consequently, our method is expected to satisfy the conditions for accurately determining mass ratios and works well.

The presence of light from the third body decreases the amplitudes of the variations in the LC of the eclipsing binary. In this section, we examine the influence of this third light on our mass ratio estimate.

Figure 5 displays the LCs and their third derivatives for cases in which the ratio ($l_{3}$) of the third light to the total light (i.e., the light from the two component stars in the binary and from the third star) of the binary is 0, 0.2, and 0.8. The top (bottom) figure corresponds to binaries with $i=90^{\circ}$ and $q=0.35$ ($i=70^{\circ}$ and $q=0.15$). The lowermost panel in each figure shows the differences between the third derivatives of LCs with and without a third light. Even though the amplitudes of the variations in the LCs differ between cases with and without third light, their third derivatives show good agreement. Furthermore, when using Equations (2) and (5) to calculate the mass ratio and its uncertainty for all three LCs ($l_{3}=0$, $0.2$, and $0.8$), they all yield the values $0.310\pm 0.043$ and $0.188\pm 0.043$ for the binaries with $q=0.35$ and $0.15$, respectively; that is, all three values are in agreement. Therefore, even when a constant third light contributes to the LC of an eclipsing binary, it does not affect the mass ratio estimate provided by our method.