Physical Parameters of 11,100 Short-period ASAS-SN Eclipsing Contact Binaries

We used the ASAS-SN Variable Stars Database for target selection (Shappee et al. 2014; Jayasinghe et al. 2018, 2019a, 2019b, 2020b, 2021). There are a total of 78,503 EW-type eclipsing binaries in the ASAS-SN Variable Stars Database. We search the short-period (less than 0.5 days) CB candidates under the following conditions. Based on the mean magnitude distribution of ASAS-SN (Jayasinghe et al. 2018), we select targets with a mean V-band magnitude less than 15 mag; this includes the vast majority of targets. We also select targets with amplitudes greater than 0.2 mag because many ellipsoidal variable stars have amplitudes less than 0.2 mag and they are often mistaken for low-amplitude EW-type eclipsing binaries (Dal & Sipahi 2013; Li & Liu 2021; Skarka et al. 2022). The Laflfler–Kinman string length (LKSL) statistic (Lafler & Kinman 1965; Clarke 2002) is a measure of scatter in the light curve of periodic variables. We used an LKSL statistic of ≤0.1 to select the light curves of EW-type eclipsing binaries with good data quality. The class probabilities are given by Jayasinghe et al. (2018, 2019a), which are derived from a random forest classifier. They recommended that a class probability of >0.9 is the best classification. Under these conditions, we selected 11,282 candidate targets. We downloaded ASAS-SN V-band light-curve data from the ASAS-SN Variable Stars Database for these 11,282 targets, of which four targets had no data, and 11,278 targets had valid data.

Due to the development of time-domain photometric survey projects around the world, a large number of CB light curves have been released. It would be clearly impractical to use the Wilson–Devinney (Wilson & Devinney 1971; Wilson 1990; Van Hamme & Wilson 2007; Wilson et al. 2010; Wilson 2012) or PHOEBE (Prša et al. 2016) programs to derive the parameters of all these CBs. Prša et al. (2008) and Ding et al. (2021) used machine-learning methods to derive the parameters of detached binaries and contact binaries, respectively. These methods can speed up parameter derivation but cannot obtain the corresponding parameter errors.

In this work, we use the neural network (NN) and the Markov Chain Monte Carlo (MCMC) algorithm following the work in Ding et al. (2021, 2022) to derive the parameters of contact binaries. The NN model is trained with a set of light curves generated by Phoebe and known input parameters. Then, combined with the MCMC algorithm, the posterior distributions of the parameters can be quickly obtained. We used Phoebe to generate two sets (with or without the third light) of synthetic light curves as samples. The third light is one value for each light curve, representing the effect of the third body on the system as an equivalent result. In this step, the filter band is set to V because this work uses the ASAS-SN V-band data. The number of training light curves without and with the influence of the third light are 346,741 and 344,104, respectively. We select 10% of them as the test set. The model without the influence of the third light achieves a precision of 0.00045 mag in generating the light curves, while the model with the influence of the third light achieves a precision of 0.0002 mag in generating the light curves. Both models achieve a precision in generating the light curves that is less than 0.001 mag. Then, two models are obtained by training on the two sets of sample data. These two trained completed models can generate light curves according to the parameters of the CBs.

We downloaded ASAS-SN V-band light-curve data from the ASAS-SN Variable Stars Database. We also obtained the period and primary eclipse time T₀. Then, we phased these data and turned them into phase-magnitude data. To determine the fundamental photometric parameters of these ASAS-SN short-period CBs, the light curves were analyzed by using the NNs and MCMC algorithm.

We apply the two obtained models (with or without the third light) to obtain the parameters of the phased ASAS-SN light curves. Before using the models, we calculate the temperature for these CBs based on the given period–temperature relationship by Jayasinghe et al. (2020a) as follows:

$$\begin{eqnarray}&&{T}_{\mathrm{eff}}=6598+5260\,* \,{\mathrm{log}}_{10}(P/0.5){\rm{K}}.\end{eqnarray} \tag{ 1 }$$

We use the parameter of the goodness of fit (R²) to evaluate the relevant results. The goodness of fit (R²) is calculated based on the light curves generated by Phoebe and the original phased ASAS-SN light curves. If one target fits better under the third-light model and the luminosity ratio of the third light

$$\begin{eqnarray}&&{L}_{3}/({L}_{1}+{L}_{2}+{L}_{3})\geqslant 0.2,\end{eqnarray} \tag{ 2 }$$

we consider the third light present and use the third-light-fitting model. The non-third-light model was used for the rest of the cases.

We have listed four example targets; the observations and best-fitting solutions of the light curves are displayed in Figure 1. We select all targets with a goodness of fit (R²) greater than 0.8, with a total of 11,145 short-period ASAS-SN CBs. The sky distribution of the short-period CBs in ASAS-SN, colored by their period, is shown in Figure 2.

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Based on the analysis with the machine-learning method (NN and MCMC algorithm model), we obtained the system parameters of ASAS-SN short-period CBs. The initial effective temperature of the first substar (T₁) is calculated by the period–temperature relationship (Jayasinghe et al. 2020a). The orbital inclination (incl), mass ratio (q), temperature ratio (T₂/T₁), fill-out factor (f), third light ratio (l₃), and corresponding errors are derived by the machine-learning method. The relative radii of the two substars (r₁, r₂) and luminosity ratio (L₂/L₁) are determined by Phoebe.

We define mass ratio as values between 0 and 1, with the more massive star being the primary star and the less massive star being the secondary star. Thus, when the derived mass ratio (q) is greater than 1, we take the reciprocal as the final mass ratio. Similarly, the reciprocal of the temperature ratio T₂/T₁ and luminosity ratio L₂/L₁ should be T_s /T_p and L_s /L_p , respectively. The radius r₁, r₂ should be exchanged as R_s , R_p , and the rest of the parameters do not need to be changed.

We crossmatched the CBs with Gaia DR3 (Gaia Collaboration et al. 2023) using a matching radius of 50 to obtain the temperature of the primary stars (T_p ). Finally, we matched a total of 11,111 targets, of which 9621 could obtain the temperatures (T_eff) from Gaia DR3. For the remaining 1490 targets, we still use the calculated temperature by the period–temperature relationship (Jayasinghe et al. 2020a) as the primary star temperature. The final parameters for these CBs are given in the attached table in catalog format. The information listed in Table 1 is an introduction to the contents of Catalog 1.

Table 1. Contents of Catalog 1 with 11,111 CBs

No.	Label	Units	Description
1	Name	⋯	ASAS-SN name
2	R.A.	deg	R.A. (J2000)
3	Decl.	deg	Decl. (J2000)
4	Period	Day	Orbital period
5	Tp	K	Effective temperature of the primary star
6	incl	deg	Orbital inclination
7	σi	deg	Uncertainty in incl
8	q	⋯	Mass ratio, $q=\tfrac{{m}_{s}}{{m}_{p}}\leqslant 1$
9	σq	⋯	Uncertainty in q
10	f	⋯	Fill-out factor, $f=\tfrac{{{\rm{\Omega }}}_{\mathrm{in}}-{\rm{\Omega }}}{{{\rm{\Omega }}}_{\mathrm{in}}-{{\rm{\Omega }}}_{\mathrm{out}}}$,
			where Ωin and Ωout are the inner and outer Lagrangian surface potential values,
			surface potential Ω = Ω1 = Ω2.
11	σf	⋯	Uncertainty in f
12	Ts /Tp	⋯	Temperature ratio
13	Ts	K	Effective temperature of the secondary star
14	Type	⋯	CBs' subtype classification (A or W)
15	Rp	⋯	Relative radius of the primary star
16	Rs	⋯	Relative radius of the secondary star
17	Ls /Lp	⋯	Luminosity ratio
18	l3	⋯	Third light ratio l3 = L3/(L1 + L2 + L3)
19	R2	⋯	Goodness of fit
20	DR3 Name	⋯	Gaia DR3 ID
21	Teff	K	Effective temperature from Gaia DR3

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Figure 3 shows the distribution of orbital periods (p), temperature (T_p , T_s ), temperature ratio (T_s /T_p ), orbital inclination (incl), fill-out factor (f), mass ratio (q), and relative radius (R_p , R_s ) for the 11,111 ASAS-SN short-period CBs. The period distribution of ASAS-SN short-period CBs peaks at approximately 0.35 days, which is slightly shifted from the typical distribution of short-period EW-type eclipsing binaries with a peak at 0.31 days (Jayasinghe et al. 2020a; Qian et al. 2020). This may be caused by the fact that some short-period ellipsoidal variable stars are usually classified as EW-type eclipsing binaries. Ellipsoidal variable stars and low-amplitude EW-type eclipsing binaries are not easily distinguished directly by light curves (Dal & Sipahi 2013; Li & Liu 2021; Skarka et al. 2022). However, in this paper, the selection of our sample with amplitudes greater than 0.2 mag has led to the early exclusion of many ellipsoidal variable stars that have been misclassified as low-amplitude EW-type eclipsing binaries.

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The temperature distribution of ASAS-SN short-period CBs peaks at approximately 5750 K (the red short dashed lines) for the primary star and 6000 K (the green short dashed lines) for the secondary star. The peak of the secondary star temperature is higher than the peak of the primary star temperature, as can also be seen from the temperature ratio distribution. Traditionally, CBs have been classified into two subtypes based on temperature: A type and W type (Binnendijk 1970). In the A type, the more massive star is hotter, while in the W type, the more massive star is cooler. Our sample contains 2399 A-type and 8712 W-type CBs.

The orbital inclination distribution indicates that most CBs have orbital inclinations larger than 50°, which is likely due to selection effects, and CBs with high orbital inclination are more easily observed. Another reason is that our method is more reliable when used to fit targets with orbital inclinations greater than 50° (Ding et al. 2021). The fill-out factor distribution of ASAS-SN short-period CBs indicates that CBs generally have shallow fill-out. The mass ratio distribution indicates that both extremely low-mass ratio (q ≤ 0.1) and large mass ratio (q ≥ 0.7) CBs are less frequent.

Very interestingly, according to the distribution of these short-period CB parameters, we find that only the distributions of the mass ratio and the fill-out factor obey log-normal distributions, while the rest of the parameters roughly obey normal distributions. We are confident that this is related to the evolution of contact binaries, but more simulation work is needed to confirm this.

Figure 4 is a 2D kernel density graph. There is a clear correlation between the orbital period and the temperature of the primary star. The period–temperature relationship is actually the well-known PLC relations (Eggen 1967; Rucinski1994; Chen et al. 2016, 2018b; Mateo & Rucinski 2017; Qian et al. 2017; Jayasinghe et al. 2020a; Qian et al. 2020). Most CBs are located in a banded region rather than a simple linear PLC. This is related to the evolution of the common envelope of CBs, where the fill-out factor affects the PLC (Qian et al. 2020).

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We plot the 2D kernel density graph between the orbital period and the temperature ratio, as shown in Figure 5. The period–temperature ratio seems to have some trend of correlation, as shown by the black line with arrows in Figure 5. As shown in Figure 5, there are far more W-type samples than A-type samples in the short-period CBs. For A-type CBs, there is no significant relationship between the temperature ratio and period, similar to a random distribution; however, for W-type CBs, the temperature ratio tends to increase as the orbital period decreases. This suggests that the evolution of A-type CBs is different from that of W-type CBs. Perhaps this could indicate that A-type CBs may be less evolved and suggest that W-type CBs evolved from A-type CBs (Li et al. 2020; Zhang et al. 2020). Currently, the formation and evolution of CBs are still open issues (Maceroni & van't Veer 1996; Awadalla & Hanna 2005; Yakut & Eggleton 2005; Eker et al. 2006; Gazeas & Niarchos 2006; Li et al. 2007, 2008; Gazeas & Stȩpień 2008; Stȩpień 2009; Yildiz & Doğan 2013; Yıldız 2014; Qian et al. 2017, 2018; Jiang 2020).

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Figure 6 shows the correlation between the orbital period and the temperature, mass ratio, orbital inclination, and fill-out factor. This shows that they do not have a good correlation with the period, except for temperature. In addition, we find that the temperature, mass ratio, orbital inclination, and fill-out factor are independently distributed from each other. There is no correlation among them.

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Figure 7 shows the correlation between the mass ratio and fill-out factor of ASAS-SN short-period CBs. As shown in Figure 3 and discussed in Section 3.1, we find that the distributions of the mass ratio and the fill-out factor obey log-normal distributions. The correlation between the mass ratio and the fill-out factor is weak, as shown in Figure 7. However, we can also find much useful information in the q−f diagram.

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In recent decades, there have been several studies on deep, low-mass ratio contact binaries (DLMRCBs) with mass ratios and fill-out factors of q < 0.25 and f > 50% (Yang & Qian 2015; Zhou et al. 2016; Li et al. 2021, 2022), corresponding to the upper left region in Figure 7. Model calculations suggest that these DLMRCBs will finally merge into single stars, such as blue stragglers or FK-Com-type stars (Stepien 2006; Stępień 2011). V1309 Sco, the only observed sample, was a DLMRCB before the merge and was recently found as a blue straggler (Tylenda et al. 2011; Zhu et al. 2016; Ferreira et al. 2019). Recently, Gazeas et al. (2021) and Loukaidou et al. (2022) led a project focusing on DLMRCBs. They find that low-mass ratio systems seem more likely to eventually merge into single fast-rotating stars. Our DLMRCB samples are located in the upper left region in Figure 7, and they should also be candidates for merging.

As shown in Figure 7, there is no clear correlation between the mass ratio and the fill-out factor. However, it shows the upper limit of the fill-out factor increases as the mass ratio decreases, as shown by the green solid line in the figure, which is consistent with our previous results (Li et al. 2020). In the previous study we had a sample of only 380 Kepler CBs, this time we have a much larger sample.

We can also see in Figure 7 that there are very few samples in the upper right region that contain CBs with a deep fill-out factor and high-mass ratio (q > 0.25 and f > 50%). The common convective envelope-dominated mechanism (Liu et al. 2018) can explain the presence of a short period, deep fill-out factor, and high-mass ratio CBs. According to the CCE-dominated mechanism, P will become very short when f is large at a high value of q.

The mass–radius relation (M–R) and mass–luminosity relation (M−L) of the primary and secondary stars of CBs have been studied by many authors (Yakut & Eggleton 2005; Li et al. 2008; Yildiz & Doğan 2013; Li et al. 2020; Latković et al. 2021). The M–R relation of a CB system is significantly different from that of a single zero main-sequence star. An approximate relation between the radii and masses of the components is $\tfrac{{R}_{s}}{{R}_{p}}={q}^{0.46}$ (Kuiper 1941), while for the zero main-sequence star, the relation is $\tfrac{{R}_{s}}{{R}_{p}}={q}^{0.6}$ (Lucy 1968a).

We used the obtained data to fit the q–R_s /R_p relation, and the results are shown as the red solid line in Figure 8 as R_s /R_p = q^0.435, which is very close to the results in Kuiper (1941). The relationship between R_s /R_p and q is monotonically increasing overall, but with dispersion at all q-value locations. We fit the upper and lower bounds of this region, as shown in the black and magenta solid lines in Figure 8. The black solid line has a relation of R_s /R_p = q^0.38, while the magenta solid line has R_s /R_p = q^0.48. We find that the upper bound region is the deep fill-out CBs and the lower bound region is the shallow fill-out CBs. This suggests that the radius ratio is related to the fill-out factor of CBs for the same mass ratio (Yakut & Eggleton2005).

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