1 Introduction
FG Hya (AN 1934.0334, BD +03°1979, GSC 0201.01844, Hip 041437) is a W UMatype binary with a very low mass ratio and very high degree of overcontact (Qian & Yang
2005
). It was discovered by Hoffmeister (
1934
). After its discovery, this system has been observed by many authors. Firstly, Tsesevich (
1949
) made visual observations and classified it as a cluster type. Photoelectric and spectroscopic observations were subsequently performed by Smith (Smith
1955
, Smith
1963
), who suggested that FG Hya is a W UMatype system with the spectral type of G0. Binnendijk (
1963
) published complete light curves in the B and V bands, from which Lafta & Grainger (
1986
) determined the mass ratio to be
. Yang et al. (
1991
) performed a photometric study in which FG Hya was further identified as an Atype contact system with a fillout factor of 90%. Lu & Rucinski (
1999
) presented spectroscopic observations. They derived a spectroscopic mass ratio of
and the absolute parameters
,
and
. Yang & Liu (
2000
) acquired CCD photometric observations. By comparing their observations with the previous photometric data obtained by Smith (
1955
), Binnendijk (
1963
), Mahdy et al. (
1985
), Yang et al. (
1991
), they found that the shape of the light curves of FG Hya shows longterm changes in the light level changes. Qian & Yang (
2005
) reported that the light curves showed asymmetries. They interpreted such variations in 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.
Star 
M
_{1}

M
_{2}

R
_{1}

R
_{2}

a

L
_{1}

L
_{2}

T
_{1}

T
_{2}

Reference 
FG Hya 
1.44 
0.16 
1.41 
0.59 
2.34 
2.16 
0.41 
5900 
6012 
Qian & Yang (
2005
) 
GR Vir 
1.37 
0.17 
1.42 
0.61 
2.40 
2.87 
0.48 
6300 
6163 
Qian & Yang (
2004
) 

Table 1
Summary of Absolute Parameters for FG Hya and GR Vir

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 asymmetries in the light curves. However, subsequent analysis of its orbital period performed by Yang & Liu (
2000
) revealed a secular decrease at a rate of
. Qian & Yang (
2005
) studied the orbital period changes again. They found that the orbital period of FG Hya shows a sinusoidal variation with a period of
superimposed on a secular period decrease at a rate of
.
GR Vir (BD −06°4068, GSC 4998.00885, HD 129903, Hip 072138, SAO 140120) is another W UMatype 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
) & 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 Atype 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
) analyzed the orbital period changes of GR Vir, where a secular decrease in its orbital period has revealed the orbital period of GR Vir varies with a cyclic period of
superimposed on a secular period decrease at a rate of
.
In the most recent decade, a large number of times of light minimum for 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 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
),
the
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
trend shows obvious cyclic variation, which may be caused by the lighttime 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 leastsquares method, the following equation is obtained,
where
b
_{
i
},
c
_{
i
} and
ω are wellknown 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
, the orbital period of the third body rotating around the eclipsing pair was determined to be
. The orbital parameters of the tertiary component were computed with the formulae given by Vinko (
1989
),
where
and
. The results are listed in Table
4.
HJD.2400000+ 
Method 
E

Type 

Ref. 
HJD.2400000+ 
Method 
E

Type 

Ref. 
48271.4960 
CCD 
−12899 
I 
0.0675 
[1] 
53402.0060 
CCD 
2751 
I 
0.00362 
[1] 
48290.5064 
CCD 
−12841 
I 
0.06368 
[1] 
53402.1700 
CCD 
2751.5 
II 
0.0037 
[1] 
48358.3682 
CCD 
−12634 
I 
0.06416 
[1] 
53404.6285 
CCD 
2759 
I 
0.00346 
[1] 
48500.3210 
CCD 
−12201 
I 
0.06567 
[1] 
53404.7926 
CCD 
2759.5 
II 
0.00364 
[1] 
48625.5493 
CCD 
−11819 
I 
0.06207 
[1] 
53405.6126 
CCD 
2762 
I 
0.00406 
[1] 
48683.4109 
CCD 
−11642.5 
II 
0.06129 
[1] 
53405.7800 
CCD 
2762.5 
II 
0.00755 
[1] 
49004.5203 
CCD 
−10663 
I 
0.0591 
[1] 
53406.7616 
CCD 
2765.5 
II 
0.00565 
[1] 
49004.6796 
CCD 
−10662.5 
II 
0.05443 
[1] 
53409.7104 
CCD 
2774.5 
II 
0.00396 
[1] 
49393.3247 
CCD 
37899 
I 
0.02473 
[1] 
53410.3672 
CCD 
2776.5 
II 
0.0051 
[1] 
49416.4349 
CCD 
−9406.5 
II 
0.05249 
[1] 
53410.5292 
CCD 
2777 
I 
0.00318 
[1] 
49772.4535 
CCD 
−8320.5 
II 
0.04532 
[2] 
53445.4428 
CCD 
2883.5 
II 
0.00265 
[1] 
51192.1180 
CCD 
−3990 
I 
0.03248 
[1] 
53764.0943 
CCD 
3855.5 
II 
0.00125 
[1] 
51216.0490 
CCD 
−3917 
I 
0.03173 
[1] 
53774.0928 
CCD 
3886 
I 
0.00087 
[1] 
51485.8480 
CCD 
−3094 
I 
0.02483 
[1] 
53775.0773 
CCD 
3889 
I 
0.00187 
[1] 
51908.2566 
CCD 
−1805.5 
II 
0.02164 
[1] 
53799.0056 
CCD 
3962 
I 
−0.00158 
[1] 
51950.0564 
CCD 
−1678 
I 
0.02283 
[1] 
53829.0034 
CCD 
4053.5 
II 
−0.00042 
[1] 
51958.0953 
CCD 
−1653.5 
II 
0.02984 
[1] 
54499.7477 
CCD 
6099.5 
II 
−0.0008 
[1] 
52297.0621 
CCD 
−619.5 
II 
0.01815 
[1] 
54529.5808 
CCD 
6190.5 
II 
−0.00043 
[1] 
52299.0260 
CCD 
−613.5 
II 
0.01505 
[1] 
54554.9858 
CCD 
6268 
I 
−0.00243 
[1] 
52300.0136 
CCD 
−610.5 
II 
0.01916 
[1] 
54829.8831 
CCD 
7106.5 
II 
0.00757 
[1] 
52341.9745 
CCD 
−482.5 
II 
0.01754 
[1] 
54889.7014 
CCD 
7289 
I 
−0.00351 
[1] 
52343.1236 
CCD 
−479 
I 
0.01922 
[1] 
55593.0691 
CCD 
9434.5 
II 
0.00021 
[1] 
52347.5530 
CCD 
−465.5 
II 
0.02289 
[3] 
55623.3952 
CCD 
9527 
I 
0.00183 
[1] 
52629.8087 
CCD 
395.5 
II 
0.01506 
[3] 
55632.7353 
CCD 
9555.5 
II 
−0.00129 
[1] 
52657.1851 
CCD 
479 
I 
0.01748 
[1] 
55979.5835 
CCD 
10613.5 
II 
0.00045 
[1] 
52660.7860 
CCD 
490 
I 
0.01222 
[1] 
56000.7286 
CCD 
10678 
I 
0.00037 
[1] 
52696.0303 
CCD 
597.5 
II 
0.01456 
[1] 
56003.3548 
CCD 
10686 
I 
0.00391 
[1] 
53018.1229 
CCD 
1580 
I 
0.01202 
[1] 
56298.8936 
CCD 
11587.5 
II 
0.00198 
[1] 
53025.8222 
CCD 
1603.5 
II 
0.00727 
[3] 
56743.4426 
CCD 
12943.5 
II 
0.01052 
[1] 
53055.0030 
CCD 
1692.5 
II 
0.011 
[1] 
56745.4071 
CCD 
12949.5 
II 
0.00803 
[1] 
53088.6048 
CCD 
1795 
I 
0.01 
[3] 
57010.1411 
CCD 
13757 
I 
0.01752 
[1] 
53094.3444 
CCD 
1812.5 
II 
0.01254 
[1] 
57048.3330 
CCD 
13873.5 
II 
0.01697 
[1] 
53105.1573 
CCD 
1845.5 
II 
0.00697 
[1] 
57048.4950 
CCD 
13874 
I 
0.01506 
[1] 
53387.4199 
CCD 
2706.5 
II 
0.00605 
[1] 
57097.3480 
CCD 
14023 
I 
0.02106 
[1] 
53387.5812 
CCD 
2707 
I 
0.00343 
[1] 
57415.0261 
CCD 
14992 
I 
0.02976 
[1] 
53387.7496 
CCD 
2707.5 
II 
0.00792 
[1] 
57415.0262 
CCD 
14992 
I 
0.02986 
[1] 
53388.7329 
CCD 
2710.5 
II 
0.00772 
[1] 
57415.0278 
CCD 
14992 
I 
0.03146 
[1] 
53389.7165 
CCD 
2713.5 
II 
0.00783 
[1] 
57415.0281 
CCD 
14992 
I 
0.03176 
[1] 
53400.6934 
CCD 
2747 
I 
0.00235 
[1] 
57498.4590 
CCD 
15246.5 
II 
0.02936 
[1] 
Notes: References [1] http://var2.astro.cz/ocgate/; [2] Agerer & Hubscher (
1996
); [3] Samolyk (
2012
).

Table 2
Times of Light Minimum for FG Hya

2.2 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 tabulated by Qian & Yang (
2004
), here we only list the others in Table 5. The
values computed with the linear ephemeris given by Kreiner (
2004
),
are displayed in Figure
2. By using the leastsquares method, the following equation is obtained,
The sinusoidal term in Equation (
8) suggests a cyclic oscillation with a period of
and an amplitude of
A = 0.0352 d. The orbital parameter of the third body was computed to be
with the wellknow equation
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 and
c is the speed of light.
HJD.2400000+ 
Method 
E

Type 

Ref. 
HJD.2400000+ 
Method 
E

Type 

Ref. 
48500.069 
CCD 
−11528.5 
II 
0.02516 
[1] 
54961.0042 
CCD 
7092.5 
II 
0.02938 
[1] 
50594.002 
CCD 
−5493.5 
II 
−0.00664 
[1] 
54966.0339 
CCD 
7107 
I 
0.02802 
[1] 
52782.511 
CCD 
814 
I 
−0.01179 
[1] 
54976.9647 
CCD 
7138.5 
II 
0.02926 
[1] 
53489.1241 
CCD 
2850.5 
II 
−0.00338 
[1] 
54978.0003 
CCD 
7141.5 
II 
0.02395 
[1] 
53490.6846 
CCD 
2855 
I 
−0.00425 
[2] 
55004.375 
CCD 
7217.5 
II 
0.02891 
[1] 
53492.594 
CCD 
2860.5 
II 
−0.00319 
[2] 
55384.316 
CCD 
8312.5 
II 
0.03761 
[1] 
53492.7665 
CCD 
2861 
I 
−0.00417 
[2] 
55616.6148 
CCD 
8982 
I 
0.0399 
[1] 
53496.7582 
CCD 
2872.5 
II 
−0.00263 
[2] 
55632.5836 
CCD 
9028 
I 
0.04808 
[4] 
53497.624 
CCD 
2875 
I 
−0.00425 
[2] 
56100.481 
CCD 
10376.5 
II 
0.05624 
[1] 
53497.7983 
CCD 
2875.5 
II 
−0.00344 
[2] 
56404.4268 
CCD 
11252.5 
II 
0.05622 
[1] 
53846.1555 
CCD 
3879.5 
II 
−0.00426 
[1] 
56419.0047 
CCD 
11294.5 
II 
0.06135 
[1] 
53854.1363 
CCD 
3902.5 
II 
−0.00377 
[1] 
56421.0878 
CCD 
11300.5 
II 
0.06251 
[1] 
53861.0808 
CCD 
3922.5 
II 
0.00133 
[1] 
56426.9863 
CCD 
11317.5 
II 
0.06267 
[1] 
53877.0415 
CCD 
3968.5 
II 
0.0014 
[1] 
56455.435 
CCD 
11399.5 
II 
0.05979 
[1] 
53877.2173 
CCD 
3969 
I 
0.00371 
[1] 
56456.479 
CCD 
11402.5 
II 
0.06288 
[1] 
54587.1354 
CCD 
6015 
I 
0.02091 
[1] 
56805.5413 
CCD 
12408.5 
II 
0.07324 
[1] 
54647.3362 
CCD 
6188.5 
II 
0.02239 
[3] 
57140.0262 
CCD 
13372.5 
II 
0.0789 
[1] 
54951.1149 
CCD 
7064 
I 
0.02873 
[1] 
57187.7353 
CCD 
13510 
I 
0.07961 
[1] 
54954.0645 
CCD 
7072.5 
II 
0.02908 
[1] 
Notes: References [1] http://var2.astro.cz/ocgate/; [2] Ogloza et al. (
2008
); [3] Yilmaz et al. (
2009
); [4] Hubscher et al. (
2012
).

Table 5
Times of Light Minimum for GR Vir

In Figure 2, one can clearly see that our fitting is reasonable, and the sum of squares of the residuals
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 LightTime Effect of a Third Companion
The orbital period oscillation of FG Hya and GR Vir may be caused by the lighttime effect of a third companion. If this is true, the orbital period of the third body rotating around the eclipsing pair is about 54.44 yr and 28.56 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 wellknown equations
we can compute the lower limit on the third body’s mass and the upper limit on orbital radius to be
and
respectively.
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
.
3.2 Magnetic Activity Mechanism
Since the spectral type of GR Vir is G0, it is a latetype binary, and its
curve shows normal sinusoidal variation. This means that the magnetic activity mechanism is also a possibility for explaining cyclic variations of the period in this system. Using the following formula (Lanza et al.
1998
)
where
we can calculate variation of the quadruple moment
for the primary star and
for the secondary star.
4 Discussion and Conclusions
We have analyzed the orbital period of FG Hya and GR Vir. It is found that both FG Hya and GR Vir show period oscillations. We can compare our results with the works of Qian & Yang (
2004
;
2005
). They found that orbital periods of both FG Hya and GR Vir show sinusoidal variations superimposed on secular period decreases (see Sect. 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 thermal relaxation oscillation (TRO)controlled and variable angular momentum loss (AML)controlled stages (Qian et al.
2005
).
In Section 3, we interpreted the orbital period oscillations of FG Hya and GR Vir by the lighttime effect of third bodies, and obtained the mass of the third body
for FG Hya and
for GR Vir. If such large third bodies really exist, they should be found photometrically. So far, no third lights have been observed in either system, suggesting that the third bodies may be unseen components, e.g., a small black hole or a white dwarf.
Since the spectral type of GR Vir is G0, the orbital period oscillation can also be explained by magnetic activity on both components of the system. This mechanism is based on the hypothesis that a hydromagnetic dynamo can produce changes in the gravitational quadrupole moment of an active star through a redistribution of the internal angular momentum and/or action of the Lorentz force in the stellar convective zone (Applegate
1992
; Lanza et al.
1998
). If energy for transferring angular momentum is provided by luminosity variation of the active star, there must be
. Based on the equation given by Yu et al. (
2015
),
we can obtain
and
, implying that the mechanism of magnetic activity can be used to explain cyclic variations of GR Vir. In this equation,
G,
σ and
T are the gravitational constant, StefanBoltzman constant and surface temperature of the active star, respectively. All these physical elements are in the international system of units.
In summary, the orbital periods of FG Hya and GR Vir show periodic oscillations with periods of
and
, respectively. Our study demonstrates that periodic variations in FG Hya and GR Vir can be explained by the lighttime effect of third bodies. For GR Vir, the orbital period oscillation can be also explained by magnetic activity on both components of the system. In order to check these conclusions, more observations are needed.
Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 11703020) and Joint Research Funds in Astronomy (U1531108, U1731106 and U1731110) under cooperative agreement between the National Natural Science Foundation of China and Chinese Academy of Sciences.