Reanalysis of the Orbital Period Variations of Two DLMR Overcontact Binaries: FG Hya and GR Vir

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1 Research in Astron. Astrophys. Vol.0 (200x) No.0, Research in Astronomy and Astrophysics Reanalysis of the Orbital Period Variations of Two DLMR Overcontact Binaries: FG Hya and GR Vir Xu-Dong Zhang, Yun-Xia Yu, Fu-Yuan Xiang and Ke Hu Department of Physics Xiangtan University, Xiangtan, Hunan, China; Received 2017 May 17; accepted October 9 Abstract We put forward a study of orbital period changes of two deep, low mass ratio (DLMR) overcontact W UMa-type binaries, FG Hya and GR Vir. It is found that the orbital period of FG Hya shows a cyclic change with a period of P mod = yr. The cyclic oscillation may be due to a third body in an eccentric orbit. While the orbital period of GR Vir shows a periodical variation with a period of P mod = yr and an amplitude of A = days. The periodical variation of GR Vir can be interpreted as a result of either the light-time effect of an unseen third body or the magnetic activity circle. Key words: binaries: close binaries: eclipsing stars: individual (FG Hya, GR Vir) 1 INTRODUCTION FG Hya (AN , BD , GSC , Hip ) is a W UMa-type binary with a very low mass ratio and very high degree of overcontact binary (Qian & Yang (2005)). It was discovered by Hoffmeister (1934). After its discovery, this system has been observed by many authors. Firstly, Tsesevich (1949) made the visual observations and classified it as a cluster type. The photoelectric and spectroscopic observations are subsequently performed by Smith(1955, 1963), who suggests that FG Hya is a W UMa-type system with the spectral type of G0. Binnendijk (1963) published the complete light curves in B and V bands, from which Lafta & Grainger (1986) determined the mass ratio to be q = M 2 /M 1 = Yang et al. (1991) performed a photometric study where FG Hya is further identified as an A-type contact system with the fill-out factor of 90%. Lu & Rucinski (1999) presented the spectroscopic observations. They derived a spectroscopic mass ratio of q sp = and the absolute parameters a = 2.32R, M 1 = 1.41M, M 2 = 0.16M. Yang & Liu (2000) made the CCD photometric observations. By comparing their observations with the previous photometric data obtained by Smith(1955), Binnendijk (1963), Mahdy et al. (1985) and Yang et al. (1991), they found that the shape of the light curves of FG Hya shows a long-term light-level changes. Qian & Yang (2005) reported that the light curves showed asymmetries. They interpreted such variations of the light curves as a dark spot on the primary component. Combining their own CCD photometric observations with the spectroscopic elements of Lu & Rucinski (1999), Qian & Yang (2005) improved the absolute parameters (in solar units) which are summarized in Table 1. The orbital period changes of FG Hya were first noted and investigated by Qian et al. (1999). They revealed that its orbital period had undergone five sudden changes from 1950 to 1999, and suggested that these sudden changes are related to the asymmetries of light curves. However, the subsequent analysis of its orbital period performed by Yang & Liu (2000) revealed a secular decrease at a rate of P/P = Qian & Yang (2005) have studied the orbital period changes again. They found that the

2 2 X.-D. Zhang, Y.-X. Yu, F.-Y Xiang, K. Hu Table 1 Summary of Absolute Parameters for FG Hya and GR Vir. Star M 1 M 2 R 1 R 2 a L 1 L 2 T 1 T 2 Reference FG Hya Qian & Yang (2005) GR Vir Qian & Yang (2004) orbital period of FG Hya shows a sinusoidal variation with a period of P mod = 36.4 yr superimposed on a secular period decrease at a rate ofdp/dt = d yr 1. GR Vir (BD , GSC , HD , Hip , SAO ) is another W UMa-type binary with very low mass ratio and very high degree of overcontact. It was discovered by Strohmeier et al. (1965). Its eclipsing nature was identified by Harris (1979). The complete photoelectric light curves of GR Vir were reported almost at the same time by Cereda et al. (1988) and Halbedel (1988). The radial velocity curves of the system were obtained by Rucinski & Lu (1999), who suggested a spectral type of F7/F8. Qian & Yang (2004) made the CCD photometric observations, from which they revealed that GR Vir is an A-type overcontact system with a degree of overcontact of f = 78.6%. Combining their own CCD photometric solutions with the spectroscopic elements of Rucinski & Lu (1999), Qian & Yang (2004) determined the absolute parameters of GR Vir which are compiled in Table 1. Also, Qian & Yang (2004) have analyzed the orbital period changes of GR Vir, where a secular decrease in its orbital period has been revealed the orbital period of GR Vir was variation with a cyclic period of P mod = 19.3 yr superimposed on a secular period decrease at a rate of dp/dt = d yr 1. In the recent decade, a large number of light-minimum times of FG Hya and GR Vir have been published. Unfortunately, these new times cannot be predicted well and even significantly deviate from the nonlinear ephemeris obtained in the previous studies. Therefore, it is necessary to revisit the orbital period changes for these two systems, aiming to uncover the underlying physical processes and provide a useful clue for understanding their evolutionary status. 2 ORBITAL PERIOD ANALYSES 2.1 FG Hya In order to reveal the orbital period changes of FG Hya, we have performed a careful search for all available photoelectric and CCD times of light minimum. Some of them have been compiled by Qian & Yang (2005). Others are listed in Table 2. With the linear ephemeris given by Kreiner (2004), Min.I = E, (1) the(o C) values are calculated and displayed in Figure 1, where solid dots represent the photoelectric and CCD observations. From figure 1, we can see that the orbital period change of FG Hya is continuous, and the (O C) trend shows obviously cyclic variation, which may be caused by the light-time effect of a third body. As shown in the figure, the shape of the the oscillation is not strictly sinusoidal, meaning that the third body is moving in an elliptical orbit. By using the least-squares method, the following equation is obtained, (O C) = a 1 +a 2 E + 3 [b i cos(iωe)+c i sin(iωe)], (2) i=1 where b i, c i, and ω are well-known Fourier constants. The values of the fitted parameters are listed in Table 3. The residuals that have no other variations are displayed in the lower panel of Figure 1, which means our fitting is sufficient. With ω = 360 o P e /P mod, the orbital period of the third body rotating around the eclipsing pair was determined to be P mod = yr. The orbital parameters of the tertiary

3 Reanalysis of the Orbital Period Variations of Two DLMR Overcontact Binaries 3 (O - C) (days) Residuals (days) Epoch Fig. 1 (O C) curve of FG Hya. The dots refer to the photoelectric and CCD data. The solid line in the upper panel represents our fitting curve (equation 2). Residuals of FG Hya with respect to equation (2) are displayed in the lower panel. component were computed with the formulae given by Vinko (1989), e = 4 b 2 3 +c2 3 3 b 2, (3) 2 +c2 2 ω = arctan (c2 1 b 2 1)c 2 +2b 1 b 2 c 1 (b 2 1 c2 1 )b (1 e2 ), (4) 2 +2b 1 c 1 c 2 3 τ = T 0 P b1 h mod 1 1 c arctan 1 g 1 tanω 2π b 1, c 1 + h1 g 1 tanω (5) b 2 a 12 sini = c 1 +c 2 1 h 2 1 +(g2 1 h2 1 )cosω 2, (6) where g 1 = 1 5e2 8,h 1 = 1 3e2 8. The results are listed in Table GR Vir To study the orbital period change of GR Vir in detail, we have collected all available photoelectric and CCD times of light minimum from the literature. Since some of the data have been tabled by Qian & Yang (2004), here we only list the others in Table 5. The (O C) values computed with the linear ephemeris given by Kreiner (2004), Min.I = E, (7)

4 4 X.-D. Zhang, Y.-X. Yu, F.-Y Xiang, K. Hu Table 2 Times of light minimum of FG Hya. HJD Method E Type O C Ref. HJD Method E Type O C Ref CCD I (1) CCD 2751 I (18) CCD I (1) CCD II (18) CCD I (1) CCD 2759 I (17) CCD I (2) CCD II (17) CCD I (3) CCD 2762 I (17) CCD II (3) CCD II (17) CCD I (4) CCD II (17) CCD II (4) CCD II (17) CCD I (5) CCD II (19) CCD II (5) CCD 2777 I (19) CCD II (6) CCD II (20) CCD I (7) CCD II (21) CCD I (7) CCD 3886 I (21) CCD I (2) CCD 3889 I (21) CCD II (2) CCD 3962 I (21) CCD I (8) CCD II (21) CCD II (8) CCD II (22) CCD II (9) CCD II (23) CCD II (9) CCD 6268 I (24) CCD II (9) CCD II (25) CCD II (9) CCD 7289 I (26) CCD -479 I (9) CCD II (27) CCD II (10) CCD 9527 I (28) CCD II (10) CCD II (29) CCD 479 I (11) CCD II (30) CCD 490 I (2) CCD I (31) CCD II (11) CCD I (30) CCD 1580 I (12) CCD II (32) CCD II (13) CCD II (33) CCD II (12) CCD II (33) CCD 1795 I 0.01 (13) CCD I (34) CCD II (14) CCD II (35) CCD II (15) CCD I (35) CCD II (16) CCD I (36) CCD 2707 I (16) CCD I (37) CCD II (17) CCD I (37) CCD II (17) CCD I (37) CCD II (17) CCD I (37) CCD 2747 I (17) CCD II (38) Ref.: (1) Agerer et al. (1991); (2) (3) Agerer et al. (1992); (4) Agerer et al. (1993); (5) Agerer et al. (1994); (6) Agerer et al. (1996); (7) Nagai (2008); (8) Nagai (2002); (9) Nagai (2003); (10) Gerard (2012); (11) Nagai (2004); (12) Nagai (2005); (13) Gerard (2012); (14) Hübscher, J.(2005); (15) Krajci (2005); (16) Dróżdż (2005); (17) Ogloza et al.(2008); (18) Nagai (2006); (19) Dróżdż (2005); (20) Gürol, B. et al. (2007); (21) Nagai (2007); (22) Nelson (2009); (23) Dvorak (2009); (24) Nagai (2009); (25) Diethelm (2009);(26) Diethelm (2009); (27) Nagai (2012); (28) Hübscher, J. et al.(2012); (29) Diethelm (2011); (30) Hübscher, J. et al.(2013); (31) Diethelm (2012);(32) Diethelm (2013); (33) Hübscher, J. et al.(2015); (34) Nagai (2015); (35) Paschke (2015); (36) Hübscher, J.(2016); (37) Nagai (2017); (38) Paschke (2017). are displayed in Figure 2. By using the least-squares method, the following equation is obtained, (O C) = + 0 d.0181(±0.0007)+0 d.1804(±0.0056) 10 5 E + 0 d.0352(±0.0007)sin[0.0120(±0.0000) E 89.8(±1.8)]. (8) The sinusoidal term in Equation (8) suggests a cyclic oscillation with a period ofp mod = yr and an amplitude of A = d. The orbital parameter of the third body was computed to be a 12 sini = 6.10AU with the well-know equation a 12 sini = A c, (9)

5 Reanalysis of the Orbital Period Variations of Two DLMR Overcontact Binaries 5 Table 3 Parameters in the Period Changes of FG Hya. Parameters Values Errors a ± a ± b ± b ± b ± c ± c ± c ± ω (rad) ± (rad) Table 4 Orbital Parameters of the Tertiary Component Star in FG Hya. Parameters Values Unit T day P mod year e 0.40 ω deg a 12sini AU τ day f(m) M where a 12 is the orbital radius of the binary rotating around the common center of mass, i is the inclination of the orbit of the third component, c is the speed of light. In Figure 2, one can clearly see that our fitting is reasonable, and the sum of the squares of the residuals i (O C)2 i = d2 is very acceptable. The residuals from Equation (8) are displayed in the lower panel of Figure 2. 3 MECHANISMS OF THE ORBITAL PERIOD VARIATIONS 3.1 Light-travel Time Effect of Third Companion The orbital period oscillation of FG Hya and GR Vir may be caused by the light-travel time effect of third companion. If it is true, the orbital period of the third body rotating around the eclipsing pair are about yr and yr for FG Hya and GR Vir, respectively. For GR Vir, considering the third body is moving in a circular orbit and inserting the absolute parameters compiled in Table 1 into the well-known equations f(m) = (M 3 sini) 3 (M 1 +M 2 +M 3 ) 2 = 4π2 GPmod 2 a 3 12sin 3 i, (10) a 3 = M 1 +M 2 M 3 a 12, (11) we can compute the third body s lowest limit mass and the upper limit orbital radius to be M 3 = 1.31M, a 3 = 7.17 AU. For FG Hya, the third body is moving in an elliptical orbit. By inserting the absolute parameters of FG Hya (Table 1) and the orbital parameters of the tertiary component star (Table 4) into Equation (10), the minimum mass of the third body can be computed to be M 3 = 2.14M.

6 6 X.-D. Zhang, Y.-X. Yu, F.-Y Xiang, K. Hu Table 5 Times of light minimum of GR Vir. HJD Method E Type O C Ref. HJD Method E Type O C Ref CCD II (1) CCD II (9) CCD II (2) CCD 7107 I (9) CCD 814 I (3) CCD II (9) CCD II (4) CCD II (9) CCD 2855 I (5) CCD II (10) CCD II (5) CCD II (11) CCD 2861 I (5) CCD 8982 I (12) CCD II (5) CCD 9028 I (13) CCD 2875 I (5) CCD II (14) CCD II (5) CCD II (15) CCD II (6) CCD II (16) CCD II (6) CCD II (16) CCD II (6) CCD II (16) CCD II (6) CCD II (17) CCD 3969 I (6) CCD II (17) CCD 6015 I (7) CCD II (1) CCD II (8) CCD II (18) CCD 7064 I (9) CCD I (1) CCD II (9) Ref.: (1) (2) Nagai (2008); (3) Müyesseroğlu et al.(2003); (4) Nagai (2006); (5) Ogloza et al.(2008); (6) Nagai (2007); (7) Nagai (2009); (8) Yilmaz et al.(2009); (9) Nagai (2010); (10) Paschke (2009); (11) Paschke (2010); (12) Hoňková, K. et al.(2013); (13) Hübscher, J. et al.(2012); (14) Paschke (2013); (15) Hoňková, K. et al.(2013); (16) Nagai (2014); (17)Paschke (2014); (18) Nagai (2016) (O - C) (days) Residuals (days) Epoch Fig. 2 Same as Figure 1 but for GR Vir. In the upper panel, the dots refer to the photoelectric and CCD data computed by the linear ephemeris (i.e., Equation 7). The solid line represents our fitting curve. Residuals come from the Equation (8) are displayed in the lower panel.

7 Reanalysis of the Orbital Period Variations of Two DLMR Overcontact Binaries Magnetic Activities Mechanism Since the spectral of GR Vir is G0, it is a late-type binary, and its(o C) curve shows normal sinusoidal variation. This means that the magnetic activity mechanism is also a possibility for the explanation of period cyclic variations in this system. Using the following formula (Lanza et al. 1998): where P P = 9(R a Q )2 MR2, (12) P P = 2πA P mod, (13) we can calculate the variation of the quadruple moment Q 1 = g cm 2 for the primary star and Q 2 = g cm 2 for the secondary star. 4 DISCUSSION AND CONCLUSIONS We have performed the orbital period analysis on FG Hya and GR Vir. It is found that both FG Hya and GR Vir show the period oscillations. We can compare our results with the works of Qian & Yang (2004, 2005). They found that both orbital periods of FG Hya and GR Vir show sinusoidal variations superimposed on a secular period decreases (see Section 1). In our results, we found that each of the orbital periods of these two systems only shows a periodic oscillation. No secular period changes were discovered in either overcontact binary, meaning that FG Hya and GR Vir may be in the transition between the thermal relaxation oscillations (TROs)-controlled and the variable angular momentum loss (AML)-controlled stages (Qian et al. 2005). In Section 3, we have interpreted the orbital period oscillations of FG Hya and GR Vir by the lighttravel time effect of third bodies, and obtained the mass of the third body M M for FG Hya, M M for GR Vir, respectively. If so large third bodies really exist, they should be found photometrically. So far, no third lights have been observed in both systems, suggesting that the third bodies may be the unseen components, e.g., a small black hole and a white dwarf. Since the spectral of GR Vir is G0, the orbital period oscillation was also explained by a magnetic activity on both components of the system. This mechanism is based on the hypothesis that a hydromagnetic dynamo can produce changes of the gravitational quadrupole moment of the active star through a redistribution of the internal angular momentum and/or the action of the Lorentz force in the stellar convective zone (Applegate 1992, Lanza et al. 1998). If the energy for transferring the angular momentum is provided by the luminosity variation of the active star, there must have been L 0.1L. Based on the equation given by Yu et al. (2015), L L = 5G2 M 3 24π 2 σ R 6 ( a RT )4( P)2, (14) P mod we can obtain L 1 /L 1 = 0.083, L 2 /L 2 = 0.028, implying that the mechanism of magnetic activity can be used to explain the cyclic variations of GR Vir. In this equation, G, σ, and T are the gravitation constant, the Stefan-Boltzman constant and the surface temperature of the active star, respectively. All these physical elements are in unit of international units system. In summary, the orbital periods of FG Hya and GR Vir show the period oscillations with the periods of P mod = yr and P mod = yr, respectively. Our study shows that the period variations of FG Hya and GR Vir can be explained by the light-travel time effect of third bodies. For GR Vir, the orbital period oscillation can be also explained by a magnetic activity on both components of the system. In order to check these conclusions, more observations are needed. Acknowledgements This work is supported by thenationalnaturalsciencefoundationofchina ( ), and the Joint Research Funds in Astronomy (U , U and U ) under cooperative agreement between the N ationaln aturalsciencef oundationof China and ChineseAcademyof Sciences.

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