the triple binary star eq tau with an active component - IOPscience

The Astronomical Journal, 147:98 (8pp), 2014 May  C 2014.

doi:10.1088/0004-6256/147/5/98

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

THE TRIPLE BINARY STAR EQ TAU WITH AN ACTIVE COMPONENT 1

K. Li1,2,3 , S.-B. Qian2,3 , S.-M. Hu1 , and J.-J. He2,3

Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute of Space Science and School of Space Science and Physics, Shandong University, Weihai, Weihai 264209, China; [email protected], [email protected], [email protected] 2 Yunnan Observatories, Chinese Academy of Sciences, P.O. Box 110, Kunming 650011, China 3 Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650011, China Received 2013 March 12; accepted 2014 February 10; published 2014 March 24

ABSTRACT New photometric data of EQ Tau observed in 2010 and 2013 are presented. Light curves obtained in 2000 and 2004 by Yuan & Qian and 2001 by Yang & Liu, together with our two newly determined sets of light curves, were analyzed using the Wilson–Devinney code. The five sets of light curves exhibit very obvious variations, implying that the light curves of EQ Tau show a strong O’Connell effect. We found that EQ Tau is an A-type shallow contact binary with a contact degree of f = 11.8%; variable dark spots on the primary component of EQ Tau were also observed. Using 10 new times of minimum light, together with those collected from the literature, the orbital period change of EQ Tau was analyzed. We found that its orbital period includes a secular decrease (dP/dt = −3.63 × 10−8 days yr−1 ) and a cyclic oscillation (A3 = 0.0058 days and P3 = 22.7 yr). The secular increase of the period can be explained by mass transfer from the more massive component to the less massive one or/and angular momentum loss due to a magnetic stellar wind. The Applegate mechanism cannot explain the cyclic orbital period change. A probable transit-like event was observed in 2010. Therefore, the cyclic orbital period change of EQ Tau may be due to the light time effect of a third body. Key words: binaries: close – binaries: eclipsing – stars: individual (EQ Tau) Online-only material: color figures, machine-readable and VO tables

of EQ Tau observed in 2000 October, 2001 December, and 2004 December, respectively, at Yunnan Observatory, China. By employing dark spots on the primary component of EQ Tau, it is found that total spotted area covers 18%, 3%, and 20% of the photospheric surface in the corresponding three sets of light curves. In order to investigate the light curve variation and orbital period change of EQ Tau, we observed two sets of light curves and many new times of minimum light of EQ Tau. Newly determined light curves of EQ Tau were analyzed using the Wilson–Devinney code. By analyzing all the pg, pe, and CCD times of minimum light of EQ Tau, we found that the orbital period of EQ Tau shows a cyclic oscillation superimposed on a long term decrease.

1. INTRODUCTION EQ Tau, which is a W UMa type eclipsing binary, was first discovered by Tsesevich (1954). The first modern orbital period was reported by Whitney (1972). Before 2002, this system was a neglected object, with no complete published light curves. Pribulla & Vanko (2002) published complete photoelectric B and V light curves of EQ Tau, and Yang & Liu (2002) published complete charge-coupled device (CCD) B and V light curves in the same year. After that, EQ Tau was observed and analyzed by Vaˇnko et al. (2004), Zola et al. (2005), Alton (2006, 2009), Hrivnak et al. (2006), and Yuan & Qian (2007). Rucinski et al. (2001) derived the spectroscopic mass ratio to be q = 0.442 ± 0.007 for EQ Tau from their spectroscopic observations and analysis. The orbital period variation of EQ Tau was first analyzed by Qian & Ma (2001), and a secular decrease at a rate of 1.72×10−7 days yr−1 was discovered. The recent detailed investigation on orbital period variation was carried out by Yuan & Qian (2007); no secular change was determined, but a cyclic oscillation was found. They thought that the cyclic oscillation was due to either a third companion or periodic magnetic activity in the primary component. Most W UMa type eclipsing binaries show differences in the brightness at two maxima of their light curves, which is called the O’Connell effect (O’Connell 1951). The O’Connell effect has been of particular interest in understanding light curve variations of W UMa type eclipsing binaries. Such W UMa type eclipsing binaries usually have one or two components that are of spectral type F to K with deep convective envelopes. EQ Tau is a solar-type eclipsing binary with a spectral type of G2. Light curves of EQ Tau show differences at the two maxima, exhibiting a typical O’Connell effect. Yuan & Qian (2007) carried out a detailed study on the variation of the light curves of EQ Tau. They analyzed three sets of light curves

2. NEW CCD PHOTOMETRIC OBSERVATIONS AND THE CHANGE OF LIGHT CURVES The CCD photometric observations of EQ Tau were first carried out using the 60 cm telescope at Yunnan Observatory from 2009 to 2012, during which some new times of minimum light were determined. Then, EQ Tau was observed using the 1 m telescope at Yunnan Observatory on 2010 October 30 and November 1, and complete V, R, and I light curves were derived. Finally, EQ Tau was observed by using the 85 cm telescope (Zhou et al. 2009) at Xinglong station of the National Astronomical Observatories of China (NAOC) on 2013 January 8 and 11, and complete B, V, R, and I light curves were obtained. All telescopes were equipped with the standard Johnson–Cousin–Bessel UBVRI CCD photometric system. Data reduction was performed using the PHOT (measures magnitudes for a list of stars) aperture photometry package of 1

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Figure 1. Left: complete V, R, and I light curves observed using the 1 m telescope at Yunnan Observatory on 2010 October 30 and November 1 for EQ Tau. Solid lines display theoretical light curves. Open squares, circles, and triangles represent V, R, and I light curves, respectively. Right: differences between observed and theoretical light curves. (A color version of this figure is available in the online journal.)

JD (Hel.)

Errors

Min.

Filter

Telescope

2455148.3213 2455460.3146 2455460.3147 2455500.2505 2455500.2505 2455500.2506 2455500.4222 2455500.4223 2455500.4223 2455511.1719 2455543.0906 2456273.0610 2456273.0612 2456276.1329 2456276.1333 2456301.0512 2456301.0514 2456301.0514 2456301.0516 2456301.2223 2456301.2223 2456301.2224 2456301.2225

0.0002 0.0001 0.0001 0.0002 0.0001 0.0002 0.0003 0.0003 0.0003 0.0001 0.0008 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0005 0.0004 0.0005 0.0003

p p p p p p s s s p s p p p p p p p p s s s s

R V R R V I I V R V R I Clear R I B R V I I R V B

60 cm 60 cm 60 cm 1m 1m 1m 1m 1m 1m 60 cm 60 cm 60 cm 60 cm 60 cm 60 cm 85 cm 85 cm 85 cm 85 cm 85 cm 85 cm 85 cm 85 cm

Δ

Table 1 Newly Determined Times of Minimum Light for EQ Tau

Figure 2. Complete B, V, R, and I light curves observed using the 85 cm telescope at Xinglong station on 2013 January 8 and 11 for EQ Tau. Solid lines display theoretical light curves. Crosses, open squares, circles, and triangles represent B, V, R, and I light curves, respectively.

Based on the light curves derived from the literature, we show the light curve variation of EQ Tau in Figure 3. In Figure 3, V band light curves observed in 2000, 2001, 2004, 2010, and 2013 (2000 and 2004 data are taken from Yuan & Qian 2007; 2001 data comes from Yang & Liu 2002; 2010 and 2013 data are observations from this work) are displayed. The variations of these five sets of light curves can be clearly seen. This may imply that EQ Tau shows strong magnetic activity. The light curve variation of EQ Tau has been analyzed by several investigators (e.g., Hrivnak et al. 2006), and the detailed change was analyzed by Yuan & Qian (2007). The secondary minimum observed in 2010 shows a sudden dimming; we will discuss the reason later.

IRAF.4 During the observations, the comparison and check stars are the same as that used by Yang & Liu (2002). As the comparison is very close to EQ Tau, extinction corrections were not made. When we used the 1 m telescope at Yunnan Observatory on 2010 October 30 and November 1, the typical exposure times for V, R, and I bands were 60 s, 30 s, and 20 s, respectively. When we used the 85 cm telescope at Xinglong Station on 2013 January 8 and 11, the typical exposure times for B, V, R, and I bands were 20 s, 10 s, 6 s, and 4 s, respectively. The newly determined times of minimum light are listed in Table 1. The observed light curves are shown in Figures 1 and 2.

3. PERIOD VARIATION OF EQ TAU The orbital period change of EQ Tau has been analyzed by several investigators. Qian & Ma (2001) first analyzed the period variation of EQ Tau, and a secular decrease rate of 1.72×10−7 days yr−1 was discovered. Then, Pribulla & Vanko (2002) proposed a cyclic change with a period of 50.2 yr, Yang & Liu (2002) presented a cyclic change of about 23 yr,

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IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy under cooperative agreement with the National Science Foundation.

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Li et al. Table 2 Times of Minimum Light for EQ Tau

JD (Hel.) 2430647.2170 2434389.2350 2435093.2750 2435093.4550 2435429.3390 2435476.4210 2436596.2320 2436598.2820 2436599.2810 2436605.2600

Min.

Type

Errorsa

E

(O − C)1

(O − C)2

Residuals

Ref.

II I II I I I II II II I

pg pg pg pg pg pg pg pg pg pg

0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100

−28024.5 −17062 −14999.5 −14999 −14015 −13877 −10596.5 −10590.5 −10587.5 −10570

−0.0066 −0.0138 −0.0037 0.0057 0.0034 −0.0206 −0.0010 0.0009 −0.0242 −0.0188

0.0103 −0.0053 0.0034 0.0128 0.0099 −0.0142 0.0035 0.0054 −0.0197 −0.0143

0.0047 0.0002 0.0093 0.0187 0.0156 −0.0085 0.0074 0.0093 −0.0158 −0.0104

(1) (2) (2) (2) (2) (2) (2) (2) (2) (2)

Notes. a Some references do not show the errors of the times of minimum light. We set the errors of pe and CCD data as 0.0010. References. (1) Sirrah http://sirrah.troja.mff.cuni.cz/mira/variables/lightcurves/lc.cgi?tbl=2.; (2) Romano 1962; (3) Whitney 1972; (4) BAVM 39; (5) BAVM 43; (6) BAVM 50; (7) BAVM 52; (8) BBSAG 90; (9) Benbow & Mutel 1995; (10) O-C gateway; (11) Buckner et al. 1998; (12) Hubscher et al. 2006; (13) Nelson 2001; (14) Yuan & Qian 2007; (15) Pribulla et al. 2001; (16) Agerer & Hubscher 2002; (17) Sarounova & Wolf 2005; (18) Brat et al. 2009; (19) Yang & Liu 2002; (20) Csizmadia et al. 2002; (21) Hrivnak et al. 2006; (22) Diethelm 2003; (23) Muyesseroglu et al. 2003; (24) Nelson 2003; (25) Agerer & Hubscher 2003; (26) VSOLJ 42; (27) Pribulla et al. 2005; (28) Borkovits et al. 2004; (29) Krajci 2005; (30) Cook et al. 2005; (31) Hubscher 2005; (32) Nelson 2004; (33) Hubscher et al. 2005; (34) AAVSO 12; (35) Alton 2006; (36) Biro et al. 2007; (37) Parimucha et al. 2007; (38) Csizmadia et al. 2006; (39) Zasche et al. 2011; (40) Alton 2009; (41) Hubscher & Walter 2007; (42) Liakos & Niarchos 2009; (43) Dogru et al. 2007; (44) Samolyk 2008; (45) Nelson 2008; (46) Br´at et al. 2008; (47) Parimucha et al. 2009; (48) Samolyk 2009; (49) VSOLJ 48; (50) Diethelm 2009; (51) Parimucha et al. 2011; (52) Samolyk 2010; (53) This paper; (54) Nelson 2010; (55) Hubscher & Monninger 2011; (56) Diethelm 2010; (57) Samolyk 2011; (58) Hubscher et al. 2012; (59) AAVSO 151; (60) Diethelm 2011; (61) Nelson 2011; (62) VSOLJ 51; (63) Samolyk 2012; (64) Diethelm 2012; (65) Nelson 2012; (66) VSOLJ 53. (This table is available in its entirety in machine-readable and Virtual Observatory (VO) forms in the online journal. A portion is shown here for guidance regarding its form and content.)

linear ephemeris as the O − C gateway,5 Min.I = 2440213.325 + 0.3413478E,

(1)

the (O − C)1 values were calculated and are listed in Column 6 of Table 2. The corresponding (O − C)1 curve is shown in the upper panel of Figure 4; we find that a simple second-order polynomial cannot make a satisfactory fit to the (O − C)1 curve, and an additional periodic term is required (e.g., Irwin 1952; Paparo et al. 1988), β (O − C)1 = T0 + ΔT0 + (P0 + ΔP0 )E + E 2 2   sin(ν + ω) + e sin ω + A (1 − e2 ) (1 + e cos ν) β = T0 + ΔT0 + (P0 + ΔP0 )E + E 2 2  + A[ (1 − e2 ) sin E ∗ cos ω + cos E ∗ sin ω], (2)

Figure 3. Comparison of light curves in V bands for EQ Tau. Different years are distinguished by different symbols.

where T0 is the initial epoch and P0 is the eclipsing period. Other parameters are taken from Irwin (1952). During the fitting process, the weight of the pg data is 1 and that of the pe and CCD data is 10. The parameters and the derived values are shown in Table 3. ΔT0 is the correction on the initial epoch, ΔP0 is the correction on the initial period, β is the rate of secular decrease, A is the semi-amplitude, e is the eccentricity, ω is the longitude of the periastron passage, P3 is the cyclic oscillation period, and TP is the time of passage through the periastron. After the secular decrease was removed from the (O − C)1 curve, the (O − C)2 values are displayed in the middle panel of Figure 4. The residuals are displayed in the

Alton (2006) inferred an increase rate of 0.02 s yr−1 , Hrivnak et al. (2006) considered a discrete period change around 1974, Yuan & Qian (2007) found a cyclic period change with a period of 48.5 yr, and Alton (2009) determined an increase rate of 0.015 s yr−1 . In summary, different investigators derived different results. In order to obtain a more precise orbital period change of EQ Tau, we collected all pg, pe, and CCD times of minimum light. All the pg, pe, and CCD times of minimum light collected from the literature are listed in the first column of Table 2. Combining our newly determined CCD times of minimum light, we reanalyzed the period change of EQ Tau. Using the same

5

3

http://var.astro.cz/ocgate/

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Figure 4. O − C diagram of EQ Tau. The (O − C)1 values are calculated using all pg, pe, and CCD data. Open circles show the pg times of minimum light, solid circles represent the pe and CCD times of minimum light, solid lines display the fit of Equation (2), and red open circles represent the secondary minimum observed using the 1 m telescope in 2010. The (O − C)1 curve is plotted in the upper panel. It shows a secular decrease and periodic variation. After the secular decrease is removed from the (O − C)1 curve, the (O − C)2 values are displayed in the middle. After both the secular decrease and the periodic variation are removed, the residuals are displayed in the lower panel. The small insert figure in the upper panel is an enlargement of the light minima observed by us in 2013. (A color version of this figure is available in the online journal.) Table 3 Parameters for the Fit of the Times of Minimum Light Parameters ΔT0 (days) ΔP0 (days) β (days yr−1 ) A (days) e P3 (days) ω(◦ ) TP (HJD)

Values

Errors

0.0005 3.83 × 10−7 3.63 × 10−8 0.0058 0.47 8304.6 87.9 2447445.5

±0.0005 ±0.34 × 10−7 ±0.37 × 10−8 ±0.0003 ±0.06 ±86.6 ±9.5 ±182.9

lower panel of Figure 4 after both the secular decrease and the periodic variation were removed; no regularity is exhibited in the residuals. According to Figure 4 and Table 3, the orbital period of EQ Tau shows a cyclic change with a period of 22.7 yr and an amplitude of 0.0058 days. The results are very different from what the other investigators determined.

Figure 5. Radial velocity fit of EQ Tau.

limb-darkening coefficients of the two components were taken from Van Hamme’s (1993) table. The fit of the radial velocity is plotted in Figure 5. Table 4 shows the different magnitude values for each of the five sets of light curves at the two maxima. In order to compare with the results derived by Yuan & Qian (2007), we supposed that the spots are on the primary component. The solutions for the five sets of light curves are listed in Table 5 (the errors are formal errors only and have no real meaning about the uncertainties). The fitted light curves of 2010 are shown in Figure 1, and of 2013 in Figure 2. Other fitted light curves are similar to that determined by Yuan & Qian (2007), so we did not display them. The spot parameters in different years are listed in Table 6 and geometrical structures of EQ Tau in different years are displayed in Figure 6. In Figure 6, we can clearly find the change in the dark spots on the primary component of EQ Tau. We chose the solutions derived using the 2013 data as our final

4. LIGHT CURVE SYNTHESIS The light curves of EQ Tau reveal that it is an A-type W UMa contact binary and shows an obvious O’Connell effect. We analyzed the light curves of EQ Tau with spot models using the fourth version of the W-D program (Wilson & Devinney 1971; Wilson 1990, 1994; Wilson & Van Hamme 2003). The five sets of light curves together with the radial velocity data of Rucinski et al. (2001) were used to obtain the solutions. During the solutions, the effective temperature of the primary component of EQ Tau was set as T1 = 5800 ± 100 K according to Rucinski et al. (2001), then the gravity-darkening coefficients of the two components were taken to be 0.32 and bolometric albedo coefficients of the two components were fixed at 0.5 for convective atmosphere following Lucy (1976) and Ruci´nski (1969). The bolometric and bandpass 4

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However, these fits are almost the same as the previous ones. We think that the sudden dimming (not continuous change) in the secondary minimum led to the bad fit. Thus, we deleted the points at the secondary minimum and carried out new photometric solutions with one dark spot on the primary component. This time, the theoretical light curves make a better fit at the primary minimum, but cannot give a good fit at the secondary minimum. We will discuss the reason for this in the next section.

Table 4 Different Magnitude Values for Each of the Five Sets of Light Curves at the Two Maxima Year 2000a 2001b 2004a 2010 2013

B band

V band

Max.I

Max.II

Max.I

Max.II

0.273 0.234 0.245 ··· 0.288

0.280 0.266 0.316 ··· 0.302

0.710 0.651 0.663 0.677 0.673

0.706 0.680 0.712 0.713 0.684

R band

I band

Max.I

Max.II

Max.I

Max.II

0.942 0.947

··· ··· ··· 0.972 1.233 0.957 1.232

1.257 1.240

5. RESULTS AND DISCUSSIONS

Notes. a The 2000 and 2004 data are observed by Yuan & Qian (2007). b The 2001 data are taken from Yang & Liu (2002).

Light curves and the orbital period change of EQ Tau have been analyzed. We found that EQ Tau is an A-type shallow contact binary with a contact degree of f = 11.8%. Combining the photometric solutions and spectroscopic elements (Rucinski et al. 2001), the absolute physical parameters of EQ Tau were determined as follows: a = 2.48 ± 0.03 R , M1 = 1.22 ± 0.04 M , M2 = 0.54 ± 0.02 M , R1 = 1.14 ± 0.01 R , R2 = 0.79 ± 0.01 R , L1 = 1.32 ± 0.03 L , and L2 = 0.63±0.02 L . The O − C curve of EQ Tau shows that its orbital period changes as a secular period decrease superimposed on a cyclic oscillation. The orbital period of EQ Tau is decreasing at a rate of dP/dt = 3.63(±0.37) × 10−8 days yr−1 . If the long term period decrease is due to conservative mass transfer from the more

result. The reason is that this is the first time complete B, V, R, and I light curves have been obtained for EQ Tau, and the magnitude difference at the two maxima is small, so we can determine more precise physical parameters of EQ Tau. According to the right panel of Figure 1, we find that the theoretical light curves do not make a good fit to the observed light curves at the primary and secondary minima. In addition, the secondary minimum derived in 2010 shows a sudden dimming. Normally, changing the spots on the primary component can lead to a good fit (see Kang et al. 2002).

Table 5 Photometric Solutions for EQ Tau Parameters

Photometric Element 2000a

g1 = g2 A1 = A2 x1bol = x2bol y1bol = y2bol x1B = x2B y1B = y2B x1V = x2V y1V = y2V x1R = x2R y1R = y2R x1I = x2I y1I = y2I T1 (K) q(M2 /M1 ) Ωin Ωout T2 (K) i L1 /(L1 + L2 )(B) L1 /(L1 + L2 )(V ) L1 /(L1 + L2 )(R) L1 /(L1 + L2 )(I ) Ω1 = Ω2 r1 (pole) r1 (side) r1 (back) r2 (pole) r2 (side) r2 (back) f

0.839 0.145

2001b

2004a 0.32 0.5 0.648 0.207 0.839 0.145 0.762 0.232

0.839 0.145

··· ··· ··· ···

5768 ± 104 86.4 ± 0.3 0.685 ± 0.001 0.683 ± 0.001 2.739 ± 0.002 0.429 ± 0.001 0.457 ± 0.001 0.486 ± 0.001 0.295 ± 0.001 0.308 ± 0.001 0.343 ± 0.001 8.8% ± 0.9%

5800 ± 100 0.442 ± 0.004 2.763 2.495 5740 ± 112 85.1 ± 0.5 0.689 ± 0.002 0.686 ± 0.002

5743 ± 106 85.3 ± 0.4 0.688 ± 0.001 0.685 ± 0.001 ··· ··· 2.715 ± 0.003 0.433 ± 0.001 0.463 ± 0.002 0.494 ± 0.002 0.299 ± 0.001 0.313 ± 0.002 0.352 ± 0.002 17.7% ± 1.3%

2.717 ± 0.004 0.433 ± 0.002 0.463 ± 0.002 0.493 ± 0.002 0.299 ± 0.002 0.313 ± 0.002 0.351 ± 0.003 17.0% ± 1.6%

Notes. a The 2000 and 2004 data are observed by Yuan & Qian (2007). b The 2001 data are taken from Yang & Liu (2002).

5

2010

2013

··· ···

0.839 0.145

0.670 0.250 0.576 0.244

0.670 0.250 0.576 0.244

5844 ± 106 87.1 ± 0.4 ··· 0.668 ± 0.001 0.669 ± 0.001 0.670 ± 0.001 2.709 ± 0.002 0.434 ± 0.001 0.465 ± 0.001 0.496 ± 0.002 0.300 ± 0.001 0.315 ± 0.001 0.355 ± 0.002 20.2% ± 1.0%

5819 ± 103 86.5 ± 0.2 0.672 ± 0.001 0.674 ± 0.001 0.674 ± 0.001 0.675 ± 0.001 2.731 ± 0.001 0.430 ± 0.001 0.459 ± 0.001 0.489 ± 0.001 0.296 ± 0.001 0.310 ± 0.001 0.346 ± 0.001 11.8% ± 0.5%

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Figure 6. Geometrical structures of EQ Tau between 2000 and 2013; the change of the dark spots on the primary component is shown.

as fast as the Sun, so it should exhibit magnetic activity. Using the equations (Lanza & Rodon`o 2002; Rovithis-Livaniou et al. 2000)

Table 6 Spot Parameters for EQ Tau Data

Spot Parameters

θ (radian)

φ(radian)

r(radian)

Tf (Td /T0 )

2000 2001 2004 2004 2010 2013

Spot 1 Spot 1 Spot 1 Spot 2 Spot 1 Spot 1

0.3482 1.0582 2.1674 0.7234 1.1595 1.0140

2.9092 1.5004 5.7122 1.9230 1.5018 1.6411

0.2449 0.2219 0.3694 0.4595 0.3385 0.1666

0.7475 0.6077 0.8629 0.7552 0.8890 0.8298

ΔP ΔQ = −9 , P Ma 2  ΔP = [1 − cos(2π P /P3 )] × A,

(4) (5)

we derived the required quadrupole moments as ΔQ1 = 8.9 × 1048 g cm2 for the primary component and ΔQ2 = 1.7 × 1048 g cm2 for the secondary. However, the typical values of the required quadrupole moments rang from 1051 to 1052 g cm2 for active close binaries. So, the Applegate mechanism is possibly not suitable to explain the cyclic period change of EQ Tau. The other possible explanation of the cyclic variation is that the light time effect of a tertiary component causes this phenomenon. Using the mass function

massive component to the less massive one, by combining the absolute parameters of EQ Tau with the well-known equation   P˙ 1 1 ˙ , (3) = −3M1 − P M1 M2 the mass transfer, at a rate of dM1 /dt = 3.43(±0.35) × 10−8 M yr−1 , was determined. The timescale of mass transfer is τ ∼ (M1 /M˙ 1 ) ∼ 3.55 × 107 yr, which is approximate to the thermal timescale of the more massive component ((GM 2 /RL) ∼ 1.98 × 107 yr). This indicates that thermal conservative mass transfer from the more massive component to the less massive one can explain the secular decrease of the orbital period. However, we cannot exclude the possibility of angular momentum loss due to a magnetic stellar wind. The cyclic period change of EQ Tau can be caused in two main ways, the Applegate mechanism and the light time effect. The Applegate mechanism (Applegate 1992) suggested that magnetic activity-driven variations in the quadrupole moment of solar-type components can explain cyclic period oscillation. The primary component of EQ Tau is a solar-type star (G2) with a deep convective envelope, and it rotates about 80 times

f (m) =

(m3 sin i)3 4π = × (a12 sin i)3 , 2 (m1 + m2 + m3 ) GP32

(6)

where a12 sin i = A × c (where c is the speed of light), f (m) = 1.96(±0.30) × 10−3 M for the third body was determined. Assuming the third body and the central system are coplanar (i3 = 86.◦ 5), the mass of the tertiary companion should be m3 = 0.20(±0.02) M with a septation of 10.7(±1.1) AU. No spectroscopic line of the tertiary was reported by the spectroscopic study of Rucinski et al. (2001). During the photometric solutions, we also searched for the contribution of the third body to the total light of the system (L3 ), but the value of L3 was always negative. Thus, the third companion may 6

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the O − C curve. A transit-like event was observed in 2010. The cyclic orbital period change of EQ Tau could be explained by the light time effect of a third companion. Δ

This work is partly supported by Chinese Natural Science Foundation (Nos. 11203066, 11133007, and 11103074) and the Open Research Program of Key Laboratory for the Structure and Evolution of Celestial Objects (No. OP201303). New observations of the system were based on the 60 cm and 1 m telescopes at Yunnan Observatory and the 85 cm telescope at Xinglong station of the NAOC. We are especially indebted to the anonymous referee who provided very useful comments and suggestions, which helped us to greatly improve the paper. REFERENCES

Figure 7. Upper panel: three color 1 m light curves observed in 2010 between phases 0.4 and 0.6 and the 60 cm data in the R band observed on 2010 December 12. Open squares, circles, and triangles represent V, R, and I 1 m light curves, respectively. Solid circles show the 60 cm R light curve. Lower panel: comparison between observed and theoretical light curves.

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be a very faint main sequence star or a compact object such as a white dwarf. When we analyzed the light curve variation of EQ Tau, we found that the secondary minimum observed in 2010 shows a sudden dimming. If the dimming is a continuous change, we can model the light curves with changing spots on the primary component. Since the dimming at the secondary minimum is a sudden change, we cannot make a good fit. From the right panel of Figure 1, we find that the difference between the observed and theoretical light curves is probably due to transit-like tertiary signals. Actually, we observed two secondary minima of EQ Tau in 2010. One was observed by the 1 m telescope and the other was observed by the 60 cm telescope on December 12. Three color 1 m light curves between phases 0.4 and 0.6 and the 60 cm data in the R band are shown in the upper panel of Figure 7. The lower panel shows the comparison between observed and theoretical light curves. The difference between the observed and theoretical light curve of the R band 60 cm data also seems to be a transit-like event, but the time of duration is shorter than that of the transit-like event observed by the 1 m telescope. The light curve and C–CH curve of the R band 60 cm data is much more dispersive than that of the 1 m data. Also, it was cloudy on 2010 December 12, in Kunming, Yunnan. Therefore, we believe the transit-like event observed by the 60 cm telescope is caused by the dispersion of the light curve. Based on Figure 7, the transit-like event observed using the 1 m telescope is more reliable. Then, we marked the secondary minimum observed using the 1 m telescope in 2010 with red open circles in the O − C curve of EQ Tau (Figure 4). The red open circles are very close to the maximum of the O − C curve. Considering the error of the period of the cyclic change, this may imply that the additional companion was between EQ Tau and the telescope when we observed the binary EQ Tau using the 1 m telescope in 2010. Therefore, the suddenly dimming secondary minimum derived using the 1 m telescope in 2010 was possibly caused by the transit of a third body. In summary, EQ Tau is a shallow contact binary with a total eclipse secondary minimum, which allows us to derive very precise photometric elements of it. Long term period investigation reveals that the period of EQ Tau shows a cyclic modulation superimposed on a secular period decrease. The Applegate mechanism cannot explain the periodic oscillation of 7

The Astronomical Journal, 147:98 (8pp), 2014 May

Li et al.

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