NSVS 01286630: a detached binary with a close-in companion

A lot of close binaries form with a close-in companion, and the tertiary components play an important role in the formation and evolution of these systems (Wolf et al. 2016; Fabrycky & Tremaine 2007). Statistically studying a large sample of eclipsing binaries (EBs) is a useful way to look for evidence of hierarchical triple-star systems. The LAMOST survey released many EB data recently, including some important stellar atmospheric parameters such as effective temperature, gravitational acceleration and metallicity (Qian et al. 2017). Using these data, Qian et al. (2017) investigated the correlations between orbital period and other parameters, and found many EB candidates with a third body. Another search through the photometric database of Kepler EBs suggested that at least 20% of all close binaries have tertiary companions (Gies et al. 2012; Rappaport et al. 2013; Conroy et al. 2014; Borkovits et al. 2015, 2016). For an EB system, the presence of a tertiary component can cause a cyclical variation in the minimum light times, which can be investigated using the known $(O-C)$ method. Through analyzing the difference between observed and computed mid-eclipse times with a given ephemeris, we can obtain some orbital parameters of a third body (Liao & Qian 2010b). In other words, we can discover possible multi-systems by searching for periodic features in the observed $(O-C)$ diagram (Wolf et al. 2016). Using this method, many successful examples of the detection of a third body around close binaries have been reported in recent years, such as V1104 Her (Liu et al. 2015), V401 Cyg (Zhu et al. 2013), V894 Cyg (Li & Qian 2014), KIC 5513861 (Zasche et al. 2015) and KIC 9532219 (Lee et al. 2016). Specifically, the stable M-type contact binary system SDSS J001641-000925 with a close-in star companion was also discovered (Qian et al. 2015b).

NSVS 01286630 (=NSVS 1135262) was first detected as an EB candidate in 2004 (Woźniak et al. 2004), and after 3 yr, the first paper about this target was published (Coughlin & Shaw 2007). The research results from Coughlin & Shaw 2007 suggested that NSVS 01286630 is a detached EB system with big cool starspots in its polar areas, which implies there is strong magnetic activity in this system. The physical masses and radiuses are also calculated based on assigned masses extrapolated from their temperature. The preliminary model seems to support the current findings that low-mass stars have a greater radius and lower effective temperature than the best models predicted (Lacy 1977). Recently, after analyzing the period variations of NSVS 01286630 using the $(O-C)$ method, the existence of a cyclic oscillation with an amplitude of 0.00105 days and a period of about 1317 days was discovered by Wolf et al. (2016). They thought that this period change may be caused by the presence of a third body, and according to their fitting parameters, the lowest mass of the third body was calculated as $0.10\,{M}_{\odot }$. Based on these findings, we monitored this target for several years.

In the present paper, the first four-color (B, V, R_c, I_c) light curve (LC) of NSVS 01286630 is obtained and analyzed. We also use new mid-eclipse times coupled with old data to analyze the $(O-C)$ diagram, which shows a short-term oscillation cycle because of the presence of a third body. Based on the photometric analysis and $(O-C)$ fitting results, the tertiary star, evolution state and magnetic activity of this system are discussed.

Two-sets of LCs in ${{BVR}}_{c}{I}_{c}$ bands of NSVS 01286630 were obtained in 2010 and 2011, respectively. The first set of LCs was observed on 2010 November 2 and 3, and the second set of LCs was acquired on 2011 May 2, 5 and 6, and June 11, 12 and 20. A total of more than 1900 CCD images was recorded using the 85 cm telescope at Xinglong Station of National Astronomical Observatories, Chinese Academy of Sciences. This telescope is equipped with a 1024 × 1024 PI1024 BFT camera with a standard Johnson-Cousins-Bessel multicolor CCD photometric system, the effective field of view of which is 16.5 arcmin by 16.5 arcmin, corresponding to a plate scale of 0.97 arcsec pixel⁻¹ (Zhou et al. 2009). From 2014 to 2015, the telescope had been upgraded by local technicians; detailed information about this process can found in Bai et al. (2018). Its filter system, mounted at the primary focus, can help to lessen optical loss and cut down the exposure time. As a consequence, the integration times in ${{BVR}}_{c}{I}_{c}$ bands are 40 s, 30 s, 20 s and 10 s, respectively. Another two stars near the target with similar brightnesses were chosen as the comparison star and check star, and their coordinates are listed in Table 1.

Table 1.  Coordinates of NSVS 01286630, the Comparison Star and the Check Star

Star	${\alpha }_{j2000}$	${\delta }_{j2000}$	${V}_{\mathrm{mag}}$
NSVS 01286630	18h47m08.59s	$+78^\circ {42}^{{\prime} }{29.2}^{{\prime\prime} }$	13.09
Comparison	18h46m 05.37s	+$78^\circ {38}^{{\prime} }{30.2}^{{\prime\prime} }$	13.28
Check	18h47m23.27s	$+78^\circ {39}^{{\prime} }{46.3}^{{\prime\prime} }$	12.76

All of the observed CCD images were reduced with IRAF¹, including flat-fielding and bias correction. By calculating the phase of the observations with a new equation, the observed LCs are plotted in Figure 1. In order to avoid the phase shift, the phase of those observations displayed in Figure 1 was calculated with our new linear ephemeris

$$\begin{eqnarray}\mathrm{Min}.(\mathrm{HJD})= & & 2455504.0113(\pm 0.00052)\\ & & +0.38392787(\pm 0.00000001{)}^{{\rm{d}}}\times E.\end{eqnarray} \tag{ 1 }$$

In addition, new CCD times of light minima for NSVS 01286630 were also observed and determined. These are listed in Table 2.

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Table 2.  New CCD Mean Times of Light Minima for NSVS 01286630

JD (Hel.)	Error (d)	Filter	Telescope
2455503.05012	±0.00040	B	85 cm
2455503.05015	±0.00034	Ic	85 cm
2455503.05024	±0.00035	Rc	85 cm
2455503.05010	±0.00040	V	85 cm
2455504.01103	±0.00039	B	85 cm
2455504.01142	±0.00061	Ic	85 cm
2455504.01138	±0.00056	Rc	85 cm
2455504.01137	±0.00050	V	85 cm
2455684.26348	±0.00043	B	85 cm
2455684.26361	±0.00049	Ic	85 cm
2455684.26382	±0.00039	Rc	85 cm
2455684.26245	±0.00041	V	85 cm
2455688.29501	±0.00018	B	85 cm
2455688.29517	±0.00024	Ic	85 cm
2455688.29491	±0.00019	Rc	85 cm
2455688.29509	±0.00025	V	85 cm
2455733.20970	±0.00063	B	85 cm
2455733.21128	±0.00045	Ic	85 cm
2455733.21062	±0.00048	Rc	85 cm
2455733.21239	±0.00057	V	85 cm
2455503.05040	±0.00037	${{BVR}}_{c}{I}_{c}$	85 cm
2455504.01130	±0.00052	${{BVR}}_{c}{I}_{c}$	85 cm
2455684.26334	±0.00043	${{BVR}}_{c}{I}_{c}$	85 cm
2455688.29505	±0.00022	${{BVR}}_{c}{I}_{c}$	85 cm
2455733.21100	±0.00063	${{BVR}}_{c}{I}_{c}$	85 cm

Long-term modern photometric observations with the goal of accumulating minimum light times are necessary when using the O − C method to analyze the period variation of EBs. Generally, the short-period detached EBs have deep and symmetric eclipses, which can help us to obtain individual minimum light times of mid-eclipses with high accuracy (Wolf et al. 2016).

The EB NSVS 01286630 is a newly identified EB with a precise orbital period of 0.38392787 days. Changes in eclipsing times of NSVS 01286630 were analyzed using all the minimum times published until now. During our analysis, the following linear ephemeris was adopted

$$\begin{eqnarray}\mathrm{Min}.(\mathrm{HJD}) & = & 2454272.7532(\pm 0.00008)\\ & & +0.38392787(\pm 0.00000001{)}^{{\rm{d}}}\times E.\end{eqnarray} \tag{ 2 }$$

There are altogether 91 CCD times of light minimum that were used in our analysis, with 42 primary eclipses among them. New CCD eclipse-times of NSVS 01286630 from our observations were determined using the least-squares parabolic fitting method (Kwee & van Woerden 1956; Li et al. 2017), whose uncertainties were determined by the error propagation formula. The same minimum light times in different bands have been averaged, and only the mean values are adopted. Weights of 1/${\sigma }^{2}$ were assigned to data, where σ is error in the times of light minima. We tried several different cases for fitting the $(O-C)$ diagram, including a single sinusoidal function as well as a parabolic curve with a sinusoidal function. To fit the $(O-C)$ diagram well, we considered cyclic variation with eccentricity in the orbit. By considering an eccentric orbit (Irwin 1952), the fitting $(O-C)$ curve was described by the following equations:

$$\begin{eqnarray}(O-C) & = & {\rm{\Delta }}{{JD}}_{0}+{\rm{\Delta }}{P}_{0}E\\ & & +A\left[\sqrt{1-{e}_{3}^{2}}\sin {E}_{3}\cos {\omega }_{3}\right.\\ & & \left.+\cos {E}_{3}\sin {\omega }_{3}\right],\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{M}_{3}={E}_{3}-{e}_{3}\sin {E}_{3}=\displaystyle \frac{2\pi }{{P}_{3}}(t-{T}_{3}).\end{eqnarray} \tag{ 4 }$$

In this case, the fitting residual is the smallest, and most $(O-C)$ values can be described well using these newly obtained equations, hence we accepted it as the final solution. More detailed explanation and application of equations used in the present paper can found in Liao & Qian (2010a). Two possible mechanisms can be applied to interpret this cyclical variation, the light-travel time effect (LTTE) of a third body and a magnetic activity cycle for a system with an active late-type secondary star (Applegate 1992). However, if the Applegate mechanism was working here, the mean value of period variation should be around 40–50 yr for active EBs (Wolf et al. 2016). Recently, a study that used the Kepler photometric database suggested that at least 8% of close binaries have tertiary companions with ${P}_{\mathrm{trip}}\le 7\,\mathrm{yr}$, which implies that multiplicity may be a common phenomenon among close EBs (Rappaport et al. 2013). The effect of apsidal motion in eccentric EBs can also result in a quasi-sinusoidal $(O-C)$ diagram, but the resulting $(O-C)$ curves are anti-correlated between the primary minima and secondary minima (Borkovits et al. 2016). As discussed by Liao & Qian (2010b), the presence of a tertiary component around close binary systems should mainly be responsible for those variations (Zhou et al. 2016), so we think that the LTTE is the primary reason.

According to the $(O-C)$ fitting parameters, the orbital projected radius is calculated with the equation

$$\begin{eqnarray}&&{a}_{12}\sin {i}_{3}={A}_{3}\times c,\end{eqnarray} \tag{ 5 }$$

where c is the speed of the light and A₃ is the amplitude of the $(O-C)$ oscillation, i.e., ${a}_{12}\sin {i}_{3}=0.17(\pm 0.01)\,\mathrm{AU}$. The mass and mass function of the tertiary companion are computed with

$$\begin{eqnarray}f(m) & = & \displaystyle \frac{4{\pi }^{2}}{{{GP}}_{3}^{2}}\times ({a}_{12}\sin {i}_{3}{)}^{3}\\ & = & \displaystyle \frac{({M}_{3}\sin {i}_{3}{)}^{3}}{({M}_{1}+{M}_{2}+{M}_{3}{)}^{2}},\end{eqnarray} \tag{ 6 }$$

where P₃ is the period of the $(O-C)$ oscillation and G is the gravitational constant. Finally, the best LTTE parameters and their errors are listed in Table 3. From Table 3, you can find that most parameters we obtained are very similar to the published results except for orbital eccentricity. The main reason for that is the data adopted in the analysis; during our fitting, the data with errors more than 0.00005 days were not used, however, about 112 reliable times of minimum light are included in Wolfʼs analysis, and some of them are not CCD measurements. It should be noted that the errors of these parameters are unbiased standard errors, and they were calculated using the degrees of freedom and the covariance matrix (Qian et al. 2015a). The $(O-C)$ diagram and residuals for all data with respect to the above-mentioned linear ephemeris are shown in Figure 2.

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Table 3.  The LTTE Parameters for NSVS 01286630

Element	Unit	Value (Present work)	Value (Wolf et al. 2016)
Zero epoch, T0	HJD	2454272.75324(± 0.00005)	2454272.75319(± 0.00008)
Sidereal period, Ps	d	0.383927887(± 0.000000013)	0.383927874(± 0.000000012)
Period of third body, P3	d	1322(± 13)	1317(± 15)
Eccentricity, e3	—	0.08(± 0.10)	0.17(± 0.05)
${\omega }_{3}$	deg	342.23(± 1.3 )	326.4(± 2.6 )
Amplitude, A	d	0.00097(± 0.00006)	0.00105(± 0.00005)
Periastron passage, T3	HJD	2457144.34662(± 0.00010)	2454450(± 15)
${P}_{s}^{2}/{P}_{3}$	d	0.000115	0.000 11

The first photometric solutions of NSVS 01286630 in V, R and I bands were published in 2007. In order to check these photometric elements and understand this starʼs evolution state, we intend to analyze the present new LCs in ${{BVR}}_{c}{I}_{c}$ bands with the 2013 version of the Wilson-Devinney (W–D) code (Wilson & Devinney 1971; Wilson 1979, 2012). Before using the W–D code, we set the initial values for some fixed parameters. The possible effective temperature for the primary component (star 1), ${T}_{1}=4290\,{\rm{K}}$, was fixed according to average color index. The same value of gravity-darkening coefficients is taken, i.e., ${g}_{1}={g}_{2}=0.32$ (Claret 2000), and the bolometric albedos for both components were set as ${A}_{1}={A}_{2}=0.5$ (Ruciński 1969). The orbital eccentricity e = 0, since when the orbital period is less than 1 day, close binary orbits become circularized (Zahn 1989). The adjustable parameters are: mean effective temperature of the secondary component, T₂; mass ratio, q; monochromatic light of star 1, ${L}_{1B}$, ${L}_{1V}$, ${L}_{1{R}_{c}}$ and ${L}_{1{I}_{c}};$ orbital inclination, i; dimensionless potentials of the two components, ${{\rm{\Omega }}}_{1}$ and ${{\rm{\Omega }}}_{2}$.

As is well known, a reliable mass ratio of EBs can be directly constrained by high quality radial velocities (Becker et al. 2008). The magnitude of NSVS 01286630 in the V-band is about 13.1 mag (Hoffman et al. 2008). Fundamental parameters can be determined with high signal-to-noise ratio ($\gt 10$) spectra, but this will require at least a 4 m class telescope, which is difficult for us now. Besides, because there are no spectroscopic observations for NSVS 01286630 published up to now, we used a q-search method (with fixed q) to obtain initial input parameters. During this process, we added a third light considering the fitting results of the $(O-C)$ diagram. From Figure 1, we can see that the LCs observed in 2010 are of better quality, so we started the analysis with this set of LCs. Using the grid computing method, a range from 0.5 to 3 (with a step of 0.1) was explored, however, when the value of mass ratio was less than 0.5 we could not find a convergent solution. During the calculation, it was found that the solutions converged at mode 2 (for detached binaries) in the end. The q-search result for NSVS 01286630 is shown in Figure 3, and the optimal mass ratio we found was q=1.5. After that, we treated q as a free parameter and set its initial value to be 1.5. Then, one set of solutions would be derived when running the (W–D) code over and over again until all free parameters converged, although when adopting the optimal mass ratio, the fitting results are still not good when the star is outside of eclipses. In view of this case, a cool starspot model was adopted to fit the observed LCs (Zhang et al. 2018). After many attempts, we found that one spot on the surface of the primary component and another one on the surface of the secondary component could fit the LCs pretty well, reducing the residual by 60%. This means that the two components of NSVS 01286630 are active stars with cool starspots. This case is similar to NSVS 11868841 the study results of which were published several years ago (Çakirli et al. 2010). At the same time, it should be noted that both of these cool starspots are near the polar field.

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According to the same process as stated above, we analyzed the LCs from 2011 using the (W–D) code again. A similar spotted scenario is adopted in the final solutions and the ${\rm{\Sigma }}-q$ curves are generally the same as the former. However, the solutions are found to be have some differences as compared to the former, which can be caused by changes in the shape of observed LCs. Finally, after a detailed comparison, the best photometric solutions are listed in Table 4. However it should be noted that the listed errors calculated with the (W–D) code are only probable errors (Liu et al. 2015). These errors are obtained in nonlinear situations, and because of complex correlations among the full set of parameters (Wilson & Biermann 1976), the real parameter uncertainties may be three to five times larger than what the (W–D) code provided (Popper 1984; Abubekerov et al. 2009). The theoretical LCs are plotted in Figure 4. The geometrical structure of the system at different phases is displayed in Figure 5.

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Table 4.  Photometric solutions for NSVS 01286630

Parameter	This paper	This Paper
	2010	2011
${g}_{1}={g}_{2}$	0.32	0.32
${A}_{1}={A}_{2}$	0.5	0.5
T1 (K)	4290	4290
q	1.12(± 0.10)	1.15(± 0.15)
${T}_{2}/{T}_{1}$	0.9697(± 23)	0.9655(± 58)
i (°)	89.814(± 0.167)	89.917(± 0.058)
${L}_{1}/({L}_{1}+{L}_{2})(B)$	0.5678(± 0.0040)	0.5509(± 0.0150)
${L}_{1}/({L}_{1}+{L}_{2})(V)$	0.5544(± 0.0035)	0.5357(± 0.0121)
${L}_{1}/({L}_{1}+{L}_{2})({R}_{c})$	0.5448(± 0.0038)	0.5256(± 0.0122)
${L}_{1}/({L}_{1}+{L}_{2})({I}_{c})$	0.5353(± 0.0042)	0.5146(± 0.0126)
${L}_{3}/({L}_{\mathrm{all}})(B)$	0.0048(± 0.0002)	0.0044(± 0.0002)
${L}_{3}/({L}_{\mathrm{all}})(V)$	0.0062(± 0.0002)	0.0058(± 0.0002)
${L}_{3}/({L}_{\mathrm{all}})({R}_{c})$	0.0082(± 0.0003)	0.0078(± 0.0003)
${L}_{3}/({L}_{\mathrm{all}})({i}_{c})$	0.0295(± 0.0004)	0.0361(± 0.0006)
${{\rm{\Omega }}}_{1}$	4.077(± 0.025)	4.183(± 0.037)
${{\rm{\Omega }}}_{2}$	4.204(± 0.030)	4.201(± 0.028)
r1 (pole)	0.3315(± 0.0010)	0.3226(± 0.0038)
r1 (point)	0.4026(± 0.0030)	0.3826(± 0.0091)
r1 (side)	0.3455(± 0.0012)	0.3353(± 0.0044)
r1 (back)	0.3700(± 0.0016)	0.3574(± 0.0058)
r2 (pole)	0.3375(± 0.0040)	0.3455(± 0.0028)
r2 (point)	0.3936(± 0.0052)	0.4094(± 0.0068)
r2 (side)	0.3509(± 0.0044)	0.3602(± 0.0033)
r2 (back)	0.3710(± 0.0060)	0.3820(± 0.0042)
${\theta }_{s}$ (°)	P:292.574(± 2.349)	P:27.275(± 5.157)
	S:129.90(± 2.865)	S:117.465(± 6.016)
${\psi }_{s}$ (°)	P:114.94(± 1.490)	P:71.338(± 4.240)
	S:277.447(± 0.974)	S:94.946(± 1.489)
${r}_{s}$ (°)	21.6(± 0.12), 24.87(± 0.17)	23.32(± 0.25), 24.92(± 0.28)
${T}_{s}/{T}_{* }$	0.86, 0.85	0.86, 0.85
$\sum (O-C{)}_{i}^{2}$	0.00178	0.00410

Notes: ${L}_{\mathrm{all}}={L}_{1}+{L}_{2}+{L}_{3}$, P refers to the primary component, S means the secondary one.

First, four-color LC solutions for NSVS 01286630 are obtained using the 2013 version of the (W–D) code. Due to the asymmetry of these LCs, a cool starspot model is used to analyze the LCs for better determination of basic orbital parameters. The photometric solutions are relatively reliable considering an inclination of nearly 90°. The difference between two sets of photometric solutions may be caused by the evolution of the cool starspots. The solutions from both sets of LCs suggest that NSVS 01286630 is an active detached EB system with a small temperature difference. Moreover, the photometric solutions show that the value of mass ratio $q(\tfrac{{M}_{2}}{{M}_{1}})$ is more than 1, which is a very different value compared with other late-type detached binaries (Torres & Ribas 2002; Morales et al. 2009). The main reason for this may be the presence of big cool starspots on the secondary component of NSVS 01286630. Our photometric solutions suggest that the coverage of cool starspots on the surface of the secondary component is about 5% (2010) and 11% (2011). Usually, late-type stars with a deeper convective envelope and faster rotation will, in turn, produce a strong magnetic field (Claret. 2000; Zhang et al. 2014). With a strong magnetic field and sufficient internal convection, the radius of these late-type low-mass stars could inflate while at the same time the surface effective temperature decreases (Chabrier & Baraffe 2000; Morales et al. 2010).

The most striking feature of this binary system is its starspot activity. Therefore, the photometric solutions with cool starspots was adopted. The surface cool starspots can make the observed LCs distorted, especially in shorter wavelengths (U-, B- and V-bands), such as what is observed in CU Cnc (Qian et al. 2012). Generally, the cool starspots tend to occur around some longitude, forming active longitude belts (Lanza et al. 2002; Çakirli et al. 2003). The main features of these longitudes are permanent activity and continuous migration (Berdyugina & Tuominen 1998; Berdyugina 2005). As previously emphasized, the cool starspots we obtained were near polar regions, which are listed at the bottom of Table 4. According to the dynamo mechanism, the magnetic field of a late-type star is similar to a magnetic dipole, which manifests a stronger magnetic field intensity near the polar regions than other locations considering its rotation (Durney & Robinson 1982). Therefore, stars that are fast rotators are more likely to exhibit cool starspots at high latitudes and polar regions (Schuessler & Solanki 1992). More importantly, the tidal effects can result in evolution of cool starspots along with the active longitudes and emerge on the surface (Zhang et al. 2014). Tran et al. (2013) suggest that a cool starspot is continuously visible around the orbit and slowly changes its longitude on timescales of weeks to months. This is the reason why the two sets of parameters for cool starspots we obtained are somewhat different. Similar late-type detached binaries include NSVS 02502726 (Lee et al. 2013); NSVS 10653195 (Zhang et al. 2015); NSVS 6507557 (Ćakırlı & Ibanoǧlu 2010) and NSVS 07453183 (Zhang et al. 2014).

The $(O-C)$ diagram of NSVS 01286630 shows typical cyclical variation, which may be caused by the presence of a third body or the magnetic activity from one or both components (Applegate 1992). If such magnetic activity is operating, variations in the gravitational quadrupole moment (${\rm{\Delta }}Q$) may result in observed oscillations. By using the formula ${\rm{\Delta }}P/P=9{\rm{\Delta }}Q/{{Ma}}^{2}=2\pi A/{P}_{\mathrm{mod}}$ (Lanza & Rodonò 2002), where M is the mass of the active component and a is the separation between both components (Yang et al. 2012), we can compute the values as follows: ${\rm{\Delta }}{Q}_{1}=9.64\times {10}^{48}\,{\rm{g}}\,{\mathrm{cm}}^{2}$ and ${\rm{\Delta }}{Q}_{2}=1.14\times {10}^{49}\,{\rm{g}}\,{\mathrm{cm}}^{2}$. These are evidently smaller than the typical values of 10⁵¹–10⁵² g cm² for close binaries (Lanza & Rodonò 1999). Therefore, we can rule out Applegateʼs mechanism for interpreting the cyclic variations of NSVS 01286630 (Yang et al. 2014). Hence, NSVS 01286630 may be a triple system. Based on fitting parameters from the $(O-C)$ diagram, we can estimate the mass of the third body to be ${M}_{3}\sin ({i}_{3})=0.11\,{M}_{\odot }$, and the distance of the binary system to the barycenter of the triple system is calculated to be ${a}_{12}\sin ({i}_{3})=0.17(\pm 0.01)\,\mathrm{AU}$. In this case, the distance of the third body to the barycenter of the triple system can be calculated to be ${a}_{3}\sin ({i}_{3})=2.38\,\mathrm{AU}$. As yet, however, the data adopted in our analysis just span 10 yr, so further observations are required to verify the oscillation observed by us.

NSVS 01286630 is a detached close EB in which neither component is filling the critical Roche lobe now. It will evolve into a short-period W UMa-type binary until the components fill their critical Roche lobe. So far, the formation and evolution of short-period W UMa-type binaries are still an unsolved problem in astrophysics, and NSVS 01286630 offers as good an example as DV Psc (Zhang & Zhang 2007) for studying this process. Some researchers (Liao & Qian 2010b; Torres & Ribas 2002) surveyed many EB systems with modern photometric analysis and found that many close binaries have additional companions. The spectral type of these binaries ranged from B to M, and more than half of them are late-type EBs. However, their origin is still questionable in several cases. One theory suggests that an intensive dynamic interaction process between the binary system and these close-in stellar companions may change the formation and evolutional path of these stars. For example, the third body might extract angular momentum from the central binary system during early dynamical interaction or late evolution (Qian et al. 2014). As a result, the third body may help to shorten the time of orbital evolution for these central binary systems (Fabrycky & Tremaine 2007; Zhou et al. 2016).

This work is partly supported by the National Natural Science Foundation of China (Nos. 11573063, 11611530685) and the Key Science Foundation of Yunnan Province (No. 2017FA001). We acknowledge support from the staff of the Xinglong 85 cm telescope, and this work was partially supported by the Open Project Program of the Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences.