A Detached Eclipsing Binary with a Period Shorter than 0.2 Days in a Triple System

The two cool component stars in W UMa-type eclipsing binaries (EBs) are in contact with each other and sharing a common convective envelope. It is well known that these types of binaries exhibit a sharp period cutoff phenomenon around 0.22 days, i.e., contact binary stars under the short-period limit are very rare (e.g., Rucinski 1992). Recently, a new statistical study reveals that the real short-period limit for W UMa-type stars is about 0.2 days (Qian et al. 2017). The investigation of binary stars below this limit (0.2 days) can provide very valuable information on the formation and evolution of contact binaries; for example, the angular momentum and mass loss, and the final merging of EBs (Qian et al. 2015a; Kjurkchieva et al. 2015). To date, only a few main-sequence EBs with orbital periods shorter than 0.2 days were detected and well investigated, such as SDSS J001641-000925 (P = 0.1985615 days, Davenport et al. 2013; Qian et al. 2015b), BW3 V38 (P = 0.1984 days, Maceroni & Montalbán 2004), and GSC 2314-0530 (P = 0.192636 days, Dimitrov & Kjurkchieva 2010). Both BW3 V38 and GSC 2314-0530 are detached binaries, while SDSS J001641-000925 is the first red-dwarf contact binary. The physical reasons for the short-period limit of contact binaries is still an open question, although several explanations have been proposed (e.g., Rucinski 1992; Stepień 2006; Lohr et al. 2012; Qian et al. 2015a).

On one hand, thanks to the development of several surveys in the world (e.g., SDSS, SuperWASP, and WFCAM Transit Survey), some close binaries with periods near the limit have been discovered and studied, such as GSC 2314-0530 (Dimitrov & Kjurkchieva 2010), 1SWASP J015100.23-100524.2 (Qian et al. 2015a), 1SWASP J200503.05-343726.5 (Zhang et al. 2017a), and 1SWASP J140533.33+114639.1 (Zhang et al. 2018). On the other hand, many short-period EBs are discovered with a close-in companion, which means the tertiary component may play an important role in the origin and evolution of these binary systems. The statistical studies using the photometric database of Kepler EBs suggest 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 the tertiary component can cause a cyclical variation in the times of minimum light, which can be investigated by using the known (O − C) method. Through analyzing the difference of mid-eclipse times between the observed and the computed with a given ephemeris, we can obtain some orbital parameters of the third body (Liao & Qian 2010). Using this method, many successful examples for the detection of the third body around the close binaries have been reported in recent years, such as BI Vul (Qian et al. 2013), CSTAR 038663 (Qian et al. 2014), V1104 Her (Liu et al. 2015), KIC 5513861 (Zasche et al. 2015), KIC 9532219 (Lee et al. 2016), NSVS 01286630 (Zhang et al. 2018a), and NSVS 10441882 (Zhang et al. 2019). Specifically, the stable M-type contact binary system SDSS J001641-000925 with a close-in star companion was also discovered (Davenport et al. 2013; Qian et al. 2015c).

J1155 is an EB system with an orbital period shorter than 0.2 days (P = 0.199724 days). It was observed by several surveys, such as Two Micron All Sky Survey (2MASS), SDSS and Lincoln Near-Earth Asteroid Research (LINEAR), and the g − i color index was derived as 2.54 (Palaversa et al. 2013). Then, J1155 was first confirmed as a M2V EB by using Catalina Surveys (CSS) data since its discovery (Drake et al. 2014). The original research aim of CSS is to confirm the existence of M-dwarf contact binary systems, SDSS J001641-000925 (Qian et al. 2015b), for example. Therefore, J1155 is a newly discovered M-M dwarf binary system, and Drake et al. (2014) listed it as a contact binary candidate. Besides, the obtained optical spectra from 10.4-m Gran Telescopio CANARIAS show obvious H_α, H_β, H_γ, and Ca_II H + K emission lines, which suggests strong magnetic activities in the two components. Its ultrashort orbital period and its extremely cool components make it a very interesting binary for further investigations. However, no modern photometric investigation and orbital period analyses have been published so far. In the present work, we report that J1155 is a special detached red-dwarf EB system with an ultrashort orbital period and a close-in cool third body. The specific observed information is described in Section 2. We present our detailed analysis in Section 3 and Section 4, followed by a discussion and summary in Section 5.

New CCD photometric observations of J1155 in VRI bands were carried out on 2016 March 15 using the 84-cm telescope at the Observatorio Astronómico Nacional (OAN) at Sierra San Pedro Mártir, Mexico. Its filter system is a standard Johnson multicolor CCD photometric system. The integration time of each image for V, R, and I filters were 120 s, 60 s, and 30 s, respectively. Two stars near the target were chosen as the comparison star and the check star, whose coordinates were listed in Table 1.

Table 1.  Coordinates of the J1155, the Comparison Star, and the Check Star

Stars	αj2000	δj2000
J1155	11h55m3339	35°44'399
Comparison	11h55m2112	35°43'446
Check	11h55m2760	35°41'416

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After three days, another whole light curve in W-band (2.25–3.3 μm) was observed by using the 2.12-m Ritchey–Chrétien reflector telescope, which has been used for various astronomical studies in the nearinfrared (NIR) since 2002 (Devaraj et al. 2015). A total of 320 CCD images were obtained, and the integration time of every image was 15 s. Afterwards, we obtained LCs of J1155 in VR_cI_c bands on 2017 March 31, using the 1024 × 1024 PI1024 BFT camera attached to the 85-cm telescope at the Xinglong Station of National Astronomical Observatories of Chinese Academy of Sciences. Its filter system is a standard Johnson-Cousins-Bessel multicolor CCD photometric system built on the primary focus (Zhou et al. 2009). The integration time was 50 s for the V-band, 25 s for R_c-band, and 15 s for I_c-band. After one month, new CCD observations of the system were taken with the Weihai Observatory 1.0-m telescope of Shandong University (WHOT). Observations in R_cI_c-bands were carried out on 2017 April 28 by using the PIXIS 2048B CCD camera (Hu et al. 2014; Li et al. 2015). The integration time was 50 s for the I_c-band, 70 s for R_c-band. Besides, Mr.Drake sent us the Catalina Surveys data of J1155 in V-band (Drake et al. 2013), and these data were obtained from more than 440 CCD images, which were observed during December 8, 2005 to April 16, 2016.

All the data from the 85-cm and 1.0-m telescopes were reduced using the aperture photometric package PHOT (measured magnitudes for a list of stars) in the IRAF, including flat-fielding and bias-fielding correction process. The data from 84-cm and 2.12-m telescopes were reduced by Mr.Michel. It should be noted that the same comparison and the check stars were adopted in this process. After calculating the phase of the observed data with the Equation 2457844.23893 +0.^d199724 × E, the original LCs are displayed in Figure 1. As shown in Figure 1, the magnitude difference between comparison star and check star are nearly a constant, suggesting the authenticity for the changes of the curves for J1155. All observed data using the 84-cm and 85-cm telescopes are listed in Tables 2–4, and the Δm listed in these Tables refer to the magnitude difference between J1155 and the comparison star.

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A detailed analysis of period variations for detached binaries is important for researching their dynamical interaction and evolution. No light minima times of J1155 were published before the present work. We determined 26 times of light minima by a least-squares parabolic fitting method using the observed data, including 13 secondary eclipses. Using the linear ephemeris,

$$\begin{eqnarray}\begin{array}{rcl}{Min}.({HJD}) & = & 2454452.90199(\pm 0.00300)\\ & & +{0}^{d}\,.199724\times E,\end{array}\end{eqnarray} \tag{ 1 }$$

all of the (O − C) values for J1155 were calculated and are listed in the fifth column of Table 5. The (O − C) diagram show a cyclic change, which can be explained by the presence of a third component. Assuming that the third body is moving in a circular orbit, according to the least-squares fitting, the new ephemeris,

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{Min}.\,{\rm{I}} & = & 2454452.89890(\pm 0.00035)\\ & & +\,0{.}^{d}\,199725487(\pm 0.000000304)\times E\\ & & +\,0.0032597(\pm 0.000335)\sin [0\buildrel{\circ}\over{.} 02\times E\\ & & +\,46\buildrel{\circ}\over{.} 25(\pm 5\buildrel{\circ}\over{.} 95)]\end{array}\end{eqnarray} \tag{ 2 }$$

is obtained. Weights of 1/σ² were assigned to data, where σ is the error of the times of light minima. Our analysis suggests that the calculated (O − C) diagram shows a cyclic oscillation with an amplitude of 0.00326(±0.00033) days and a period of 9.84(±0.15) year. The period of the cyclic variation is determined by using the formula

$$\begin{eqnarray}&&{P}_{3}=(360^\circ /\omega )\times P,\end{eqnarray} \tag{ 3 }$$

where ω = 002 is the frequency. The best fitting (O − C) curves are plotted in Figure 2.

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To check whether it is a contact binary and obtain its photometric solutions, the observed LCs of J1155 were analyzed with the 2013 version of the Wilson-Devinney (W-D) code (Wilson & Devinney 1971; Wilson 1990, 2012; Wilson & Van Hamme 2003, 2013).

To obtain the initial parameters, a q-search method was used. This method entails fixing a series of mass ratios and choosing the one which results in the least residuals between the models and the LCs. The result of this method is to indicate the most likely mass ratio (q) from which to then proceed with the final fitting of parameters via differential corrections (Liu et al. 2015; Zhang et al. 2017b). We started our analysis by using the data obtained from the 84-cm telescope, because these LCs are relatively symmetrical. During our calculation, we found that the photometric solution converged at mode 2 (for detached binaries), mode 4 (for semi-detached binaries with primary component exactly filling its limiting Roche lobe, hereafter semi-detached primary), and mode 5 (for semi-detached binaries with secondary component accurately filling its limiting Roche lobe, hereafter semi-detached secondary). Therefore, by adopting three different W-D morphology modes independently, we obtained a first set of q-search curves. Then, using the same method and process, other LCs from the 85-cm telescope were also analyzed to search for a mass ratio. All of the q-search results are displayed in Figure 3. From Figure 3, we found that the Σ value (mean residual for input values) of the detached mode is obviously smaller than those from the semi-detached modes, so we think that J1155 is probably a detached EB (the solution with the smallest residuals is the preferred solution). The q-search curve of detached model using the LCs from 84-cm were plotted in Figure 4 individually, and it can be seen that the lowest value is around q = 0.94.

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In accordance with a spectral type of M2V for J1155 (Drake et al. 2014), we assumed an effective temperature of T₁ = 3600 K for the primary component (the star eclipsed at primary minimum). The bolometric albedo A₁ = A₂ = 0.5 (Rucinski 1969) and the same values of the gravity-darkening coefficient g₁ = g₂ = 0.32 (Lucy 1967) were used for convective stars. Logarithmic functions for limb darkening were adopted (Claret & Gimenez 1990). The adjustable parameters for the final differential corrections were: the mass ratio q; the orbital inclination i; the mean effective temperature of secondary component, T₂; the monochromatic light of star 1, L_1V, L_1R, ${L}_{1{R}_{c}}$, L_1I and ${L}_{1{I}_{c}};$ and the dimensionless potentials of the two components Ω₁ and Ω₂.

A third light was also added in the process of the calculation. The contribution of the tertiary component to the total light in the V, R_c, and I_c bands would roughly be 0.19%, 0.31%, 0.44%, respectively. It reveals that the third companion of the system may be a low-mass, late-type star. Generally, the light curve changes in time (asymmetric LCs) are best explained by changing spots on the photospheres of one or both of the stars. We reanalyzed the LCs from 85-cm telescope by using a detached model with a cool star-spot. According to previous experience, we fixed the spot radius and temperature factor, while making the values of the latitude and longitude adjustable (Applegate 1992; Zhang et al. 2014). The best photometric solutions are obtained with one cool star-spot on the primary component. The spot is located near the polar region, which is in agreement with dynamo mechanism (Durney & Robinson 1982). The obtained mass ratio is q = 0.93, very similar to BW03 V38, and we adopted it as the final solution. It should be noted that the W-band was not in the W-D bandpass list, so we adopted a nearest bandpass K_s (2.159 μm) from the Two Micron All Sky Survey (2MASS) to replace it. We analyzed the LCs observed at WHOT by using the same method as described above. All the photometric solutions are listed in Tables 6 and 7, the corresponding geometrical structure and the equatorial section at 0.75 phase are plotted in Figures 5 and 6, respectively. Besides, the theoretical LCs computed with those photometric elements are plotted in Figures 7 and 8.

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Table 6.  Photometric Solutions for J1155 Using Different Modes

Parameters	Light Curves of 2016	Semi-detached	Semi-detached	Light Curves of 2017	Semi-detached	Semi-detached
	Detached	Primary	Secondary	Detached	Primary	Secondary
g1 = g2	0.32	0.32	0.32	0.32	0.32	0.32
A1 = A2	0.50	0.50	0.50	0.50	0.50	0.50
T1(K)	3600	3600	3600	3600	3600	3600
q	0.929(±0.044)	2.751(±0.082)	0.429(±0.007)	0.926(±0.131)	3.351(±0.129)	0.810(±0.040)
Ω1	4.228(±0.041)	6.285	3.323(±0.018)	4.001(±0.130)	7.077(±0.133)	4.437(±0.053)
Ω2	3.871(±0.106)	8.142(±0.203)	2.736	3.971(±0.360)	9.470(±0.287)	3.434
T2(K)	3587(±6)	3491(±7)	3592(±5)	3555(±18)	3517(±12)	3578(±10)
i(°)	80.383(±0.116)	89.779(±3.24)	89.716(±0.650)	79.447(±0.240)	80.306(±0.283)	88.168(±0.653)
L1/(L1 + L2)(V)	0.5341(±0.0089)	0.4814(±0.0113)	0.5628(±0.0055)	0.5434(±0.0239)	0.4340(±0.0154)	0.3835(±0.0089)
L1/(L1 + L2)(R)	0.5333(±0.0086)	0.4740(±0.0066)	0.5657(±0.0047)
L1/(L1 + L2)(I)	0.5354(±0.0080)	0.4545(±0.0035)	0.5665(±0.0043)
L1/(L1 + L2)(Rc)				0.5422(±0.0230)	0.4278(±0.0148)	0.3867(±0.0082)
L1/(L1 + L2)(Ic)				0.5364(±0.0208)	0.4180(±0.0137)	0.3843(±0.0076)
L3/(L1 + L2)(V)	0.0287(±0.0018)	0.1584(±0.0015)	0.1460(±0.0018)	0.0182(±0.0011)	0.0889(±0.0029)	0.1192(±0.0033)
L3/(L1 + L2)(R)	0.0324(±0.0014)	0.1642(±0.0019)	0.1506(±0.0016)
L3/(L1 + L2)(I)	0.0679(±0.0013)	0.1898(±0.0015)	0.1813(±0.0017)
L3/(L1 + L2)(Rc)				0.0253(±0.0021)	0.1088(±0.0021)	0.1511(±0.0022)
L3/(L1 + L2)(Ic)				0.0529(±0.0026)	0.1315(±0.0013)	0.1763(±0.0013)
r1(pole)	0.3002(±0.0024)	0.2746(±0.0022)	0.3428(±0.0021)	0.3200(±0.0039)	0.2623(±0.0029)	0.2735(±0.0040)
r1(side)	0.3085(±0.0028)	0.2861(±0.0023)	0.3534(±0.0023)	0.3306(±0.0046)	0.2731(±0.0030)	0.2788(±0.0043)
r1(back)	0.3202(±0.0035)	0.3188(±0.0023)	0.3625(±0.0026)	0.3459(±0.0065)	0.3058(±0.0030)	0.2855(±0.0049)
r2(pole)	0.3211(±0.0165)	0.3409(±0.0095)	0.2877(±0.0012)	0.3088(±0.0186)	0.3431(±0.0113)	0.3392(±0.0041)
r2(side)	0.3331(±0.0195)	0.3508(±0.0108)	0.3000(±0.0013)	0.3191(±0.0192)	0.3528(±0.0128)	0.3554(±0.0045)
r2(back)	0.3533(±0.0260)	0.3581(±0.0117)	0.3326(±0.0013)	0.3364(±0.0184)	0.3591(±0.0140)	0.3870(±0.0043)
$\sum (O-C{)}_{i}^{2}$	0.0002384	0.0003076	0.0003645	0.0007235	0.0008188	0.0008382

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Table 7.  Photometric Solutions for J1155 with Cool Star-spot by using Detached Model

Parameters	VRcIc-bands	W-band	CSSV-band	RcIc-bands	Mean Value
	85-cm Telescope	2.12-m Telescope	70-cm Telescope	1.0-m Telescope
g1 = g2	0.32	0.32	0.32	0.32	0.32
A1 = A2	0.5	0.5	0.5	0.5	0.5
T1(K)	3600	3600	3600	3600	3600
q	0.9287(±0.0936)	0.9054(±0.0943)	0.8118(±0.0779)	0.91(±0.09)	0.9042(±0.085)
T2(K)	3546(±15)	3378(±24)	3531(±14)	3591(±7)	3572(±13)
i(°)	79.572(±0.274)	85.153(±0.921)	80.400(±0.490)	81.843(±0.312)	81.777(±0.40)
L1/(L1 + L2)(V)	0.5676(±0.0195)		0.6650(±0.0021)		0.6163(±0.0108)
L1/(L1 + L2)(Rc)	0.5658(±0.0184)			0.5435(±0.0165)	0.5565(±0.0180)
L1/(L1 + L2)(Ic)	0.5590(±0.0170)			0.5312(±0.0175)	0.5451(±0.0173)
L1/(L1 + L2)(Ks)		0.6910(±0.0117)			0.6910(±0.0117)
L3/(L1 + L2 + L3)(V)	0.0257(±0.0015)		0.0225(±0.0021)		0.0241(±0.0018)
L3/(L1 + L2 + L3)(Rc)	0.0341(±0.0018)			0.0210(±0.0030)	0.0276(±0.0024)
L3/(L1 + L2 + L3)(Ic)	0.0594(±0.0017)			0.0442(±0.0045)	0.0518(±0.0031)
L3/(L1 + L2 + L3)(Ks)		0.0594(±0.0017)			0.0594(±0.0017)
Ω1	3.9812(±0.1038)	3.6347(±0.1348)	3.6474(±0.1115)	3.7275(±0.0692)	3.7410(±0.0930)
Ω2	4.0616(±0.2703)	4.3275(±0.3431)	4.0318(±0.2591)	4.6925(±0.0114)	4.4164(±0.1511)
r1(pole)	0.3221(±0.0035)	0.3590(±0.0060)	0.3475(±0.0052)	0.3328(±0.0013)	0.3379(±0.0031)
r1(point)	0.3650(±0.0072)	0.4606(±0.0123)	0.4069(±0.0076)	0.4037(±0.0038)	0.4073(±0.0064)
r1(side)	0.3333(±0.0039)	0.3767(±0.0064)	0.3619(±0.0057)	0.3469(±0.0015)	0.3521(±0.0034)
r1(back)	0.3497(±0.0049)	0.4051(±0.0056)	0.3824(±0.0057)	0.3713(±0.0021)	0.3752(±0.0038)
r2(pole)	0.3046(±0.0361)	0.2728(±0.0354)	0.2706(±0.0309)	0.2620(±0.0046)	0.2724(±0.0194)
r2(point)	0.3427(±0.0676)	0.2940(±0.0516)	0.2936(±0.0469)	0.2765(±0.0059)	0.2933(±0.0307)
r2(side)	0.3142(±0.0414)	0.2789(±0.0390)	0.2768(±0.0342)	0.2667(±0.0050)	0.2784(±0.0216)
r2(back)	0.3297(±0.0521)	0.2883(±0.0456)	0.2872(±0.0408)	0.2731(±0.0055)	0.2874(±0.0259)
θs(°)	4.77 (±0.15)	4.36(±0.25)	5.37 (±0.34)	6.15 (±0.23)	5.16
ψs(°)	277.25(±2.35)	270.22(±4.33)	257.24 (±3.47)	260.33(±3.15)	266.26
rs(°)	14.32 (±1.02)	20.30 (±2.15)	24.33 (±2.45)	16.32 (±1.15)	18.82
Tf	0.85	0.85	0.85	0.85	0.85
$\sum (O-C{)}_{i}^{2}$	0.0006635	0.003781	0.00006561	0.0006805

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As the solutions listed in Tables 6 and 7 show, different LCs have different photometric solutions. Because of strong spot activity of the target, the shape of observed LCs is changing with time. Just as Kang et al. (2002) discussed, the main reason for that is the variation of spot location and size. For clear comparison, the observed LCs in V-band are displayed in Figure 9, and one can find an obvious scatter of the light curve observed using 85-cm telescope in its out-eclipsing, which will affect photometric analysis of the LCs. Remarkably, this variation is very obvious in the light curve obtained from 2.12-m telescope. Hence, we think that our main results are reliable and acceptable, although these solutions have some small difference due to the distortion of the LCs. It should be noted that the errors listed in Tables 6 and 7 are calculated by the W-D code, the true uncertainties of the system parameters may be three to five times larger than those listed (Wilson & Van Hamme 2013; Popper 1984). The reason is that there is a strong correlation among those parameters and the non-normal distribution of measurement errors (Maceroni & Rucinski 1997).

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Based on our multi-sets of LCs, the photometric solutions of the extremely short-period EB J1155 were derived using the 2013 version of the Wilson-Devinney code. Our photometric solutions reveal that J1155 is a detached red-dwarf binary system with a high-mass ratio of 0.90 and a small temperature difference between the two components. The LCs are asymmetric and their shapes change with time. The spectroscopic observations obtained by using 10.4-m Gran Telescopio CANARIAS indicate that J1155 shows strong magnetic activities. Therefore, the asymmetric LCs are explained by the presence of a cool star-spot on the primary component.

Our solutions reveal that the configuration of J1155 is similar to those of BW3 V38 and GSC 2314-0530. All three binaries are detached systems with period shorter than 0.2 days. By using the formula ${r}_{i}=\sqrt[3]{{r}_{i({pole})}\cdot {r}_{i({side})}\cdot {r}_{i({back})}}$ and the well-known relation given by Eggleton (1983),

$$\begin{eqnarray}&&{R}_{L}=\displaystyle \frac{0.49{q}^{\tfrac{2}{3}}}{0.6{q}^{\tfrac{2}{3}}+{ln}(1+{q}^{\tfrac{1}{3}})},\end{eqnarray} \tag{ 4 }$$

the mean relative radius, r_i, and Roche lobe radius, R_L, can be calculated for the component stars. It reveals that the primary and secondary components are filling 90% and 84.8% of their critical Roche lobes, respectively. The primary component is closer to the critical Roche lobe which is similar to the situations of BW3 V38 and GSC 2314-0530, where the primary components are almost filling their Roche lobe (Maceroni & Montalbán 2004; Dimitrov & Kjurkchieva 2010). The temperature of the more massive component is about 3600 K (M2V) which lies between the values of BW03 V38 and GSC 2314-0530. Assuming that it is a main-sequence star, its mass could be estimated as M₁ = 0.475 M_⊙ (Cox 2000; Maceroni & Montalbán 2004; Dimitrov & Kjurkchieva 2010). Then, the mass of the secondary component can be estimated as M₂ = 0.441 M_⊙ by using the derived value of q. These physical parameters of the system calculated by using the empirical relation given by Demircan & Kahraman (1991) are listed in Table 8.

As plotted in Figure 2, the (O − C) diagram shows a cyclic period oscillation that could be explained by the Applegate mechanism (Applegate 1992) and the LTTE via the presence of a third body. However, a recent detailed investigation has shown that the Applegate mechanism may not suffice to produce the observed variations in close binary systems. Therefore, the cyclic variation in the O – C curve was explained by LTTE, indicating that J1155 is a triple system. According to the fitting parameters we obtained, the projected radius of the orbit that the eclipsing binary rotates around at the barycenter of the triple system 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 light, A₃ is the amplitude of the (O − C) oscillation, and i₃ is the orbital inclination of the third body, i.e., ${a}_{12}\sin {i}_{3}=0.56(\pm 0.11)\,\mathrm{AU}$. And then, the mass function of the tertiary companion is computed with

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

where P₃ and G are the period of the (O − C)₂ oscillation and the gravitational constant, and M₃ is the mass of the third body. All of the parameters of the third body are listed in Table 9. Similar close-in companions have also been reported in the system of SDSS J001641-000925 (Qian et al. 2015b) by analyzing the LTTE, which is also a short-period, red-dwarf EB. Moreover, third light is also found in GSC 2314-0530, the luminosity contributions of the third body are 17.1%, 22.2%, and 29.8% in V, R, and I-bands (Dimitrov & Kjurkchieva 2010), respectively. The results indicate the presence of a cool third body in the system.

Table 9.  Parameters of the Tertiary Component in J1155

Parameters	Values	Units
P3	9.84(±0.15)	Years
A3	0.00326(±0.00033)	Days
${a}_{12}^{{\prime} }\sin ({i}_{3})$	0.56 ± 0.11	A.U.
f(m)	1.90(±0.37) × 10−3	M⊙
M3sin(i3)	0.127(±0.020)	M⊙
a3(i = 90°)	4.04 (±0.80)	A.U.

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To date, there are four well-studied red-dwarf EBs which are listed in Table 10. Three of them, GSC 2314, J001641, and J1155, were found to have a cool third body. To better understand the effect of the third body to the central system, J1155, we calculated the orbital angular momentum of the central system by using the following expression (Popper & Ulrich 1977),

$$\begin{eqnarray}&&{J}_{{rel}}={M}_{1}{M}_{2}{\left(\displaystyle \frac{P}{{M}_{1}+{M}_{2}}\right)}^{\tfrac{1}{3}},\end{eqnarray} \tag{ 7 }$$

with P in days and M_i in solar units. The obtained value of J1155 is logJ_rel = −0.899, which is very similar to those of BW03 V38 (logJ_rel = −0.954) and GSC 2314-0530 (logJ_rel = −1.078). They are smaller than the orbital angular momentums of RS CVn binaries (logJ_rel ≥ +0.08) and contact systems (logJ_rel ≥ −0.5), but larger than that of short-period CVs of SU UMa type (Dimitrov & Kjurkchieva 2010). Since the timescale of AML for the M dwarfs is very long, the third body may play an important role in this case (Stepień 2006, 2011). It is suggested that these close-in stellar companions might extract the angular momentum from the central binary system during the early dynamical interaction or late evolution, and then shorten the time of orbital evolution for these EBs (Liao & Qian 2010; Qian et al. 2013; Zhou et al. 2016). As a result, the red-dwarf EBs formed through this way to have a low angular momentum and a very short orbital period (Qian et al. 2015b). For J1155, further observations are necessary in the future (such as radial velocity curves with high accuracy). Fundamental parameters (mass, radius) of component stars can be determined with high signal-to-noise ratio (more than 10) spectra, while this may require at least a 4.0-m class telescope, which is difficult for us now.

Table 10.  Eclipsing Binaries with Periods Shorter than 0.2 days

Name	Period	M1,2	R1,2	logJrel	qsp	Ref.
	Days	M⊙	R⊙		$\tfrac{{M}_{2}}{{M}_{1}}$
GSC 2314	0.192636	0.51, 0.26	0.55, 0.29	−1.078	0.519	(1)
BW03 V38	0.1984	0.44, 0.41	0.51, 0.44	−0.947	0.95	(2)
J001641	0.19856	0.54, 0.340	0.68, 0.58	−0.9517	0.62	(3)
J1155	0.199724	0.475, 0.441	0.516, 0.491	−0.899	0.90	(4)

References: (1) Dimitrov & Kjurkchieva (2010), (2) Maceroni & Montalbán (2004), (3) Davenport et al. (2013), (4) The present work.

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We thank the anonymous referee for useful comments and suggestions that have improved the quality of the manuscript. Many thanks to Mr.Andrew J.Drake for his kindly sending us the Catalina Surveys data of J1155. This work is partly supported by Chinese Natural Science Foundation (No.11611530685, 11573063, 11503077, 11565010, U1731238, and U1831120), the Key Science Foundation of Yunnan Province (No. 2017FA001), Guizhou Provincial Joint Fund (20177349), Doctor Starting Up Foundation of Guizhou Normal University (0516134), and the Joint Research Fund in Astronomy (grant No. U1631108) under cooperative agreement between the National Natural Science Foundation of China (NSFC) and Chinese Academy of Sciences (CAS). The original data in 2016 were observed by Mr.Valeri Orlov and Mr.Raul Michel at OAN, Mexico. New CCD photometric observations of the system were obtained with the 1.0-m telescope at the Yunnan Observatories, and the 1.0-m telescope at Weihai Observatory of Shandong University. We acknowledge the support of the staff of the Xinglong 85-cm telescope. This work was partially supported by the Open Project Program of the Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences.