Physical Properties of Three Eclipsing Binaries of V Crt, WY Cnc and CG Cyg with Radio Radiation

Abstract

Radio radiation has been detected across the Hertzsprung Russell diagram. We selected three objects with radio radiation (a semi-detached eclipsing binary V Crt, and two detached binaries WY Cnc and CG Cyg) that show magnetic activity. We made new photometric observations using a SARA 60 cm and NAOC 85 cm optical telescopes. Then, we obtained their orbital and starspot parameters by analyzing our light curves and published radial velocities using the updated Wilson-Devinney program. We revised the ephemeris information for V Crt and WY Cnc by analyzing the orbital minimum times. The orbital periods increased by 2.8 ($\pm 2.1$) ×${10}^{-9}$ d yr${}^{-1}$ for V Crt, which maybe caused by mass transfer. The orbital periods decreased by −8.641 ($\pm 0.004$) ×${10}^{-8}$ d yr${}^{-1}$ for WY Cnc. Orbital period change for CG Cyg was also found and we used a third-body of $M_3$ of 0.14 M${}_{\odot }$ and a period of approximately 59.20 (0.36) yr to explain that. We also analysed the possible second period oscillation of CG Cyg with a cycle about 18.31 (0.06) yr. The long period changes of WY Cnc and CG Cyg might be caused by magnetic activity or stellar wind, rather than mass transfer.

1. Introduction

Radio radiation has been detected in all star types across the Hertzsprung–Russell diagram [1]. There exist approximately 150 eclipsing binaries that displayed radio emission. Stellar radio radiation provides the most direct probes of physical processes, including magnetic activity, particle acceleration, stellar disk and mass transport [2]. Early astronomy studies were based on the observation of visible bands because the study of astrophysics is dependent on the capabilities of the detection equipment. Radio technology, such as Very Large Array (VLA) [3], is improving with the advent technology, which allows the nature of radio to be studied. Early radio observations were influenced by many factors, for example, instrumentation and background radiation. Kellermann and Pauliny-Toth [4] observed that the radio signal of the star named $\alpha$ Ori about 0.11 ± 0.03 Jy in the 2.8 cm band under such condition. Then, Seaquist [5] obtained the radio intensity of four stars in the 2.85 cm band. Güdel [1] presented a radio Hertzsprung-Russell diagram between 1 and 10 GHz based on a radio catalog with 3021 radio stars [6]. Later, Wendker [7] updated it to 3699 radio stars in 2001, and revealed the ubiquity of stellar radio emissions.

Obtaining the parameters of the eclipsing binaries with radio emissions is required for optical and radio observations. We selected three eclipsing binaries and revised their stellar parameters. The introduction of these three binaries (V Crt, WY Cnc, and CG Cyg) with radio radiations is as follows:

V Crt is a conventional semi-detached eclipsing binary with a period of 0.702 d, and was listed as a variable star [8]. Parthasarathy and Sanwal [9] obtained the light curve of photoelectric photometry in the B and V bands. V Crt was detected at a radio intensity close to 25 mJy at 10,600 MHz [10]. Liu [11] has studied more detail and obtained the orbit parameter of V Crt. Sarma and Vivekananda Rao [12] updated its stellar parameters with a mass ratio of q = 0.4, which is smaller than the result of 0.683 [11].

WY Cnc is an RS CVn type detached eclipsing binary with a period of 0.8293686 d [13], which shows strong photospheric starspot activity and chromospheric activity emissions in the H${}_{\alpha }$, and CaII HK lines. It exhibits radio emissions of 0.4 mJy at 4860 MHz and 0.23 mJy at 4885 MHz [7]. It was first defined as an Algol eclipsing system [14]. Rao and Sarma [15] performed preliminary statistics on the total brightness and discovered light curve distortion outside the eclipse. Subsequently, a systematic photometric study was conducted [16,17,18]. Xie et al. [19] analyzed the position of a spot on the photospheric surface, and found a starspot variation outside the eclipse. Helfand et al. [20] obtained a radio intensity of 0.4 to 0.5 mJy using the VLA FIRST survey. Hall et al. [21] observed a decreasing orbital period. Later, Albayrak et al. [22] found that WY Cnc has a third body with a possible light time effect of approximately 85 yr. Recently, many astronomers have obtained more light minimum [23,24,25,26,27,28,29,30,31,32,33,34]. Kjurkchieva et al. [35] revised the orbital parameters based on spectroscopic and photometric observations. Tian et al. [36] analyzed O-C and argued that the periodic change was because of the angular momentum loss of magnetic braking. Chen [37] analyzed the angular momentum loss and suggested that the period change may be due to magnetic braking or circum-binary disk, but also requires too strong magnetic field strength. Long-term photometric monitoring is therefore required to discuss the orbital period of WY Cnc.

CG Cyg is an RS CVn binary consisting of two dwarf components and has a period of 0.6311 d, which was discovered to be a variable star [38]. There is a radio emission of 25 mJy at 10,600 MHz [7]. Naftilan and Milone [39] obtained the spectral type of CG Cyg with G9V and K3V. The remarkable intrinsic variations was observed both outside and within the eclipse [40,41,42,43,44,45,46,47,48,49,50,51], which were explained the starspot models [42,51,52,53,54]. The period of CG Cyg was increased using light curve minima [44]. Later, Albayrak et al. [22] calculated the period variation using a model of a third-body orbit with period of 46.54 yr by fitting the light curve minimum. Afsar et al. [55] revised the period cycle of 51 or 22.5 yrs, which was caused by magnetic activity. Shi et al. [56] estimated that the period increased by 2.48 × 10${}^{-8}$d yr${}^{-1}$.

Because V Crt, WY Cnc and CG Cyg are short period eclipsing binaries with magnetic activity and radio radiation, we performed photometric observations using an optical telescope, and revise their orbital parameters. Further, we also collected the light curve minimum times of eclipsing binaries to analyze their period variation and discussed their physical mechanisms.

2. Radio Eclipsing Binaries and Photometric Follow-Up Observations

We found 150 eclipsing binaries from the updated catalogue of 3699 radio stars [7]. We collected their respective stellar parameters, including the coordinate, spectral types, stellar type, radio intensities at different wavelengths (1465 MHz, 1490 MHz, and 4860 MHz) and listed them in

Table 1. Parameter of radio eclipsing binaries [6,7].

Name	RA(J2000)	DEC(J2000)	Companion			Type	1465	1490	2695	4885
	d	d	A	B	C		MHz	MHz	MHz	MHz
RS Cha	130.80083	−79.07010	A5V	A7V		Algol type				0.89∼0
RS Cha	130.80083	−79.07010	A5V	A7V		Algol type				0.89∼0
EX Hya	193.1009	−29.24889	unknown			Dwarf Nova			8∼0	0.2∼0
R CMa	109.86741	−16.39525	F1V	K3IV-V		Algol type			7∼0	0.36∼0
ER Vul	3.31824	27.80734	G0V	G5V		RS CVn binary		1.9∼0.66	8∼0	4.97∼0
Algol	259.33153	40.59976	B8V	K0IV	A7m	Algol type	14.2∼0	300∼4	210∼0
AR Lac	290.41294	45.74225	G2IV	K0III		RS CVn binary	60∼3		30∼3	49∼0
……						…	…	…
RZ Cas	333.34799	69.63429	A2V	G6IV		Algol type	1.1∼0	1.7∼0.9	10∼0	3.73∼0
U Cep	314.30350	81.87558	B8V	G8III		Algol type			7∼0	2.8∼0
EPS UMi	287.95652	82.03726	G5III	A9IV		RS CVn			8∼0	1∼0

Where comp means companion of binary. … means there is no value of flux density in these frequency bands. The flux densities in the 1465 MHz, 1490 MHz, 2695 MHz, and 4885 MHz bands are all in the unit of (mJy).

. The full table will be published in electric format. There are different type eclipsing binaries, such as Algol eclipsing binary, X-ray binaries, RS CVn binary, and so on). V Crt is one of the semi-detached binaries and belonged to the Beta Lyr binary type. We calculated the number of spectral types and binary types using the eclipse binaries exhibiting radio emission. The results are shown in Figure 1. As shown in Figure 1, most of the eclipsing binaries with radio emission are B and A spectral-type stars, whose radio radiation might be explained using the stellar wind. By contrast, the radio radiations of eclipsing binaries with spectral-types FGKM are likely caused by stellar magnetic activity [1,2].

From the radio eclipsing binaries, we selected three objects (V Crt, CG Cyg, and WY Cnc) with short orbital periods to discuss their physical properties. We analyzed their stellar physical parameters via photometric method using an optical telescope. Our photometric observation for V Crt was conducted using the 60 cm optical telescope at the Southeastern Association for Research in Astronomy (SARA) in Chile on 27 February, 28 May, and 2 March 2020. They were conducted in four BVRI bands using a charge-coupled device (CCD) with resolution of a 1024 × 1024 pixels with BVRI filters of a Bessel system with a 13’ × 13’ field. The limit magnitude with an S/N of 10 is approximately 19 mag of V filter in 10 min [57]. For WY Cnc and CG Cyg, the CCD photometric observation data in BVRI were obtained using 85 cm optical telescope at the Xinglong station of the National Astronomical Observatories of China (NAOC). The detector used 1024 × 1024 pixels [58]. The photometric data of WY Cnc was observed on 25 and 30 December 2009, and 4, 6 and 7 January 2010, while CG Cyg was observed on 1, 2, and 3 October 2009. The CCD photometric images were reduced using the IRAF(IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.) package. The photometric data obtained after processing are listed in

Table 2. Some examples of the observational Heliocentric Julian Date (HJD) and different magnitude of V Crt, WY Cnc and CG Cyg.

Object	HJD_B	Δ Mag_B	HJD_V	Δ Mag_V	HJD_R	Δ Mag_R	HJD_I	Δ Mag_I
	d	mag	d	mag	d	mag	2d	mag
	8906.52006698	−2.007	8906.52139077	−1.672	8906.52059587	−1.491	8906.52039685	−1.390
V Crt	8906.52107439	−2.054	8906.52264576	−1.696	8906.52159315	−1.544	8906.52077154	−1.354
27 February 2020	…	…	…	…	…	…	…	…
	8906.91626556	−1.931	8906.91683844	−1.683	8906.91718293	−1.515	8906.91747364	−1.330
	8907.51665830	−1.320	8907.51723295	−1.074	8907.51119410	−0.993	8907.51786009	−0.843
V Crt	…	…	…	…	…	…	…	…
28 February 2020	8907.91390427	−1.944	8907.91473500	−1.652	8907.90656022	−1.546	8907.91319826	−1.376
	8907.91615277	−1.962	8907.91698951	−1.680	8907.90881480	−1.544	8907.91544725	−1.410
	8910.50631147	−1.936	8910.50715264	−1.698	8910.50781466	−1.573	8910.50815740	−1.420
V Crt	8910.50889491	−1.968	8910.51088698	−1.748	8910.51133995	−1.599	8910.51185182	−1.448
02 March 2020	…	…	…	…	…	…	…	…
	8910.91659417	−1.971	8910.91743337	−1.664	8910.91802435	−1.546	8910.91837180	−1.373
	5191.26970220	−1.089	5191.27036190	−1.347	5191.27083640	−1.505	5191.27832480	−1.824
WY Cnc	5191.27188970	−1.144	5191.27254940	−1.391	5191.27303550	−1.570	5191.27934340	−1.870
25 December 2009	…	…	…	…	…	…	…	…
	5191.43794290	−1.629	5191.43517090	−1.839	5191.43845800	−1.977	5191.43868370	−2.116
	5196.10573310	−1.618	5196.10596460	−1.848	5196.10616720	−1.972	5196.10637550	−2.122
WY Cnc	5196.10752140	−1.620	5196.10778760	−1.861	5196.10799590	−1.987	5196.10819850	−2.107
30 December 2009	…	…	…	…	…	…	…	…
	5196.44852830	−1.661	5196.44875980	−1.840	5196.44808270	−1.982	5196.44915330	−2.149
	5201.11496830	−1.599	5201.11544860	−1.819	5201.11587100	−1.966	5201.11622400	−2.132
WY Cnc	5201.11666960	−1.607	5201.11714990	−1.820	5201.11757240	−1.971	5201.11793700	−2.140
04 January 2010	…	…	…	…	…	…	…	…
	5201.41057010	−1.652	5201.41097520	−1.834	5201.42832470	−1.976	5201.42856780	−2.132
	5203.06410834	−1.668	5203.06444976	−1.843	5203.06473912	−2.006	5203.06499955	−2.181
WY Cnc	5203.06533516	−1.654	5203.06567666	−1.835	5203.06595442	−1.957	5203.06621485	−2.150
06 January 2010	…	…	…	…	…	…	…	…
	5203.44486650	−1.668	5203.44511530	−1.839	5203.44436300	−1.996	5203.44557820	−2.146
	5204.21937330	−1.647	5204.21960480	−1.832	5204.21981310	−1.988	5204.22003300	−2.135
WY Cnc	5204.22028770	−1.638	5204.22051920	−1.850	5204.22165340	−1.995	5204.22094740	−2.150
07 January 2010	…	…	…	…	…	…	…	…
	5204.39721470	−1.618	5204.38325070	−1.863	5204.38936180	−1.981	5204.39688490	−2.107
	5105.95537953	−0.092	5105.95563988	−0.459	5105.95587719	−0.742	5105.95609709	−0.996
CG Cyg	5105.95636332	−0.108	5105.95661215	−0.461	5105.95684938	−0.751	5105.95708089	−0.991
01 October 2009	…	…	…	…	…	…	…	…
	5106.13896750	−0.128	5106.13922780	−0.480	5106.13945350	−0.754	5106.13967920	−1.005
	5106.94128852	−0.081	5106.94154895	−0.430	5106.94178618	−0.713	5106.94201188	−0.974
CG Cyg	5106.94228391	−0.087	5106.94254427	−0.409	5106.94278158	−0.712	5106.94299567	−0.984
02 October 2009	…	…	…	…	…	…	…	…
	5107.20966810	−0.088	5107.20992860	−0.451	5107.20917050	−0.720	5107.21039150	−1.000
	5108.01364665	−0.125	5108.01390701	−0.467	5108.01612350	−0.744	5108.01435841	−1.010
CG Cyg	5108.01463045	−0.142	5108.01489080	−0.494	5108.01710730	−0.746	5108.01535381	−0.992
03 October 2009	…	…	…	…	…	…	…	…
	5108.19186420	−0.069	5108.19312000	−0.430	5108.19335730	−0.709	5108.19357140	−0.974

We only list some part of the observational data. All data are available in the online journal. HJD means Heliocentric Julian Date.

. TYC 6085-1462-1, 2-MASS J11240146-1639367 and NSV 25409 are the comparison stars of V Crt, WY Cnc and CG Cyg, respectively. The $\Delta$ magnitude of the light curves of V Crt, CG Cyg, and WY Cnc obtained from our observations are plotted in Figure 2, Figure 3 and Figure 4, respectively.

3. Period Analyses

It is important to calculate and collect the values of the light minimum time of the eclipsing binaries to determine the period variation and their physical mechanisms. We calculated the new minimum from our observations using the program reported by Nelson et al. [59], which uses polynomial fits [60]. The minimum times are listed in

Table 3. Newly obtained minima times of three eclipsing binaries in BVRI bands.

Object	HJD (B)	HJD (V)	HJD (R)	HJD (I)	HJD (Average)	Type
	8906.81368 ± 0.00055	8906.81360 ± 0.00041	8906.81354 ± 0.00071	8906.81455 ± 0.00059	8906.81384 ± 0.00057	p
V Crt	8907.51201 ± 0.01500	8907.51086 ± 0.01100	8907.51119 ± 0.00900	8907.51148 ± 0.01100	8907.51139 ± 0.01150	p
	8907.87810 ± 0.00221	8907.86710 ± 0.00136	8907.86180 ± 0.00227	8907.86170 ± 0.00495	8907.86718 ± 0.00270	s
	8910.66650 ± 0.00084	8910.67060 ± 0.00047	8910.67150 ± 0.00056	8910.67540 ± 0.00159	8910.67100 ± 0.00087	s
	5196.22863 ± 0.00010	5196.22886 ± 0.00010	5196.22856 ± 0.00010	5196.22865 ± 0.00010	5196.22868 ± 0.00010	p
WY Cnc	5201.20456 ± 0.00010	5201.20452 ± 0.00010	5201.20460 ± 0.00010	5201.20451 ± 0.00010	5201.20455 ± 0.00010	p
	5203.27833 ± 0.00100	5203.27759 ± 0.00100	5203.27782 ± 0.00100	5203.27806 ± 0.00100	5203.27795 ± 0.00100	s
CG Cyg	5107.14424 ± 0.00198	5107.14449 ± 0.00199	5107.14374 ± 0.00200	5107.14396 ± 0.00197	5107.14420 ± 0.00198	p
	5108.09398 ± 0.00199	5108.09424 ± 0.00198	5108.09448 ± 0.00195	5108.09371 ± 0.00193	5108.09410 ± 0.00197	s

. Furthermore, we collected all available light minima times for our objects from the Eclipsing Binaries Minimum Times Database (http://var2.astro.cz/ocgate/ (accessed on 8 June 2016)) and previous articles.

Table 4. Minimum times and relevant parameters of period variation of V Crt, WY Cnc and CG Cyg.

SYSTEM	HJD(24,)	ERROR	MIN	TYPE	CYCLE	(O-C)I	(O-C)II	REFERENCE
	27,460.5110	-	p	pg	−19,852.0	−0.0066	−0.0073	(1)
	27,460.5250	-	p	vis	−19852.0	0.0074	0.0067	(2)
	…	…	…	…	…	…	…	…
V Crt	57,828.4840		p	ccd	23,405.0	0.0038	0.0008	(17)
	58,166.8686	-	p	ccd	23,887.0	0.0071	0.0040	(2)
	58,208.9898	-	p	ccd	23,947.0	0.0062	0.0031	(2)
	58,208.9948	-	p	ccd	23,947.0	0.0112	0.0081	(2)
	58,210.3930	-	p	ccd	23,949.0	0.0053	0.0022	(2)
	58,242.6893	-	p	ccd	23,995.0	0.0080	0.0049	(2)
	58,254.6227	-	p	ccd	24,012.0	0.0068	0.0037	(2)
	58,562.8162	-	p	ccd	24,451.0	0.0066	0.0034	(2)
	58,906.8138	0.0006	p	ccd	24,941.0	0.0066	0.0034	(18)
	58,907.5114	0.0115	p	ccd	24,942.0	0.0022	−0.0010	(18)
	58,907.8672	0.0027	s	ccd	24,942.5	0.0070	0.0037	(18)
	58,910.6710	0.0009	s	ccd	24,946.5	0.0026	−0.0006	(18)
	26,352.3920	0.0020	p	vis	−34,778.0	−0.0555	0.0134	(19)
	26,396.3520	0.0020	p	vis	−34,725.0	−0.0521	0.0165	(19)
	26,608.6440	0.0020	p	vis	−34,469.0	−0.0784	−0.0110	(19)
	27,125.321	0.0020	p	vis	−33,846.0	−0.0981	−0.0337	(20)
	…	…	…	…	…	…	…	…
WY Cnc	54,923.3667	0.0001	p	V	−329.0	0.0003	−0.0007	(51)
	55,196.2287	0.0002	p	ccd	−0.0	−0.0000	−0.0007	(18)
	55,201.2046	0.0001	p	ccd	6.0	−0.0003	−0.0010	(18)
	55,219.4511	0.0002	p	ccd	28.0	0.0001	−0.0006	(52)
	…	…	…	…	…	…	…	…
	58,502.0890	0.0020	p	ccd	3986.0	−0.0029	0.0013	(2)
	58,530.2865	0.0020	p	ccd	4020.0	−0.0040	0.0002	(2)
	58,544.3867	0.0020	p	ccd	4037.0	−0.0030	0.0012	(2)
	15,320.5250	-	p	pg	−63,039.0	0.0359	0.02126	(2)
	22,967.4270	-	p	pg	−50,923.0	0.0033	0.00443	(59)
	…	…	…	…	…	…	…	…
	55,106.5143	0.0002	p	ccd	−1.0	0.0012	0.00026	(81)
CG Cyg	55,107.1442	0.0002	p	ccd	−0.0	0.0000	−0.00094	(18)
	55,108.0940	0.0002	s	ccd	1.5	0.0031	0.00216	(18)
	55,338.7760	0.0001	p	ccd	367.0	0.0021	0.00079	(2)
	55,380.4305	0.0002	p	R	433.0	0.0012	−0.00017	(82)
	…	…	…	…	…	…	…	…
	58,330.3981	-	p	ccd	5107.0	0.0040	0.00072	(2)
	58,340.4957	-	p	ccd	5123.0	0.0033	0.00003	(2)
	58,687.6249	-	p	ccd	5673.0	0.0036	0.00075	(2)
	58,689.5179	-	p	ccd	5676.0	0.0032	0.00035	(2)

(1). [9]; (2). http://var2.astro.cz/ocgate/ (accessed on 8 June 2016); (3). [61]; (4). https://www.aavso.org/bobnelsons-o-c-files (accessed on 8 June 2016); (5). [62]; (6). [11]; (7). [63]; (8). [64]; (9). [65]; (10). [66]; (11). [67]; (12). [68]; (13). [69]; (14). [70]; (15). [71]; (16). [72]; (17). [73]; (18). This paper (19). [74]; (20). [75]; (21). [76]; (22). [77]; (23). [78]; (24). [79]; (25). [80]; (26). [81]; (27). https://britastro.org/vss/VSSC_archive.htm (accessed on 8 June 2016); (28). [82]; (29). [83]; (30). [84]; (31). [85]; (32). [23]; (33). [24]; (34). [86]; (35). [25]; (36). [26]; (37). [30]; (38). [27]; (39). [34]; (40). [28]; (41). [29]; (42). [87]; (43). [32]; (44). [88]; (45). [33]; (46). [31]; (47). [89]; (48). [90]; (49). [91]; (50). [92]; (51). [93]; (52). [94]; (53). [95]; (54). [65]; (55). [96]; (56). [97]; (57). [98]; (58). [99]; (59). [40]; (60). [41]; (61). [46]; (62). [100]; (63). [101]; (64). [102]; (65). [103]; (66). [104]; (67). [105]; (68). [106]; (69). [107]; (70). [108]; (71). [109]; (72). [110]; (73). [111]; (74). [112]; (75). [113]; (76). [114]; (77). [115]; (78). [116]; (79). [117]; (80). [118]; (81). [119]; (82). [94]; (83). [120]; (84). [121]; (85). [122]; (86). [123]; (87). [124]; (88). [125]; (89). [98]; (90). [126]; (91). [99]. The value in the () mean the number of the reference in the tables. The value in the [ ] means the corresponding number of the reference in our paper. HJD means Heliocentric Julian Date. p means the primary minimum and s means the secondary minimum. vis mean visual observation. ccd mean charge coupled device observation, pg means photograph, and pe means photoelectric observation. (O-C)1 is the difference of the observation and calculated values, which means the residual of the linear fit. (O-C)II is the residual of the third body or parabolic fit.

lists the available light minima times, where the minimum times error is set by the mean uncertainties of other data using the same observational method if there are no errors in the literature.

3.1. Period Analysis of V Crt

The minima were obtained from the light curves of different observational methods, such as visual (vis), photographic (pg) and CCD. Because different observational methods would lead to different observational errors, we generally assigned weighting factors (1 for vis and pg, 5 for pe, and 10 for more accurate CCD data) [127]. The linear ephemeris formula was obtained as follows:

$$\begin{array}{c}\hfill Min.I=HJD2441397.3311(\pm 0.0019)+0.{70203571}^d(\pm 0.00000023)E,\end{array}$$

(1)

where 2441397.3311 is the initial time in HJD format, value of orbital period is 0.70203571 d, and decimal in parentheses are the errors. The residuals (O-C)I values of the linear fit are listed in Table 4, and plotted in Figure 5. There appears to be a very weak upward polynomial variation for V Crt. The following is the quadratic ephemeris:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill Min.I=HJD2441397.3328(\pm 0.0014)+0.70203584(\pm 0.00000004)E\\ \hfill +0.269(\pm 0.206)\times {10}^{-11}{E}^2,\end{array}\end{array}$$

(2)

The period may have been increasing, but this is only a 1.3 sigma result. We calculated the rate of increasing as 2.8 ($\pm 2.1$) ×${10}^{-9}$ d yr${}^{-1}$. The fitting line of V Crt is shown in Figure 5. As the period variation is very weak, more data are required to confirm the obtained results.

3.2. Period Analysis of WY Cnc

The new epoch for WY Cnc was calculated by fitting the light minima as follows by fitting the light minima:

$$\begin{array}{c}\hfill Min.I=HJD2455196.2339(\pm 0.0011)+0.{82936963}^d(\pm 0.00000008)E,\end{array}$$

(3)

We set HJD2441397.3311 as the initial time and then calculated the new orbital period as 0.70203571 d. The (O-C)I values of the linear fit residuals are listed in Table 4, and ploted in Figure 5, where a downward trend was observed. Following is the quadratic ephemeris:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill Min.I=HJD2455196.2294(\pm 0.0007)+0.82936771(\pm 0.00000001)E\\ \hfill -0.831(\pm 0.004)\times {10}^{-10}{E}^2.\end{array}\end{array}$$

(4)

The value of $-0.831(\pm 0.004)\times {10}^{-10}$ is the quadratic coefficient of the period change, which means that the period of WY Cnc is decreasing. Therefore, we obtained a variation period of −8.641($\pm 0.004$) ×${10}^{-8}$ d yr${}^{-1}$.

3.3. Period Analysis of CG Cyg

Using the same method, a new epoch of CG Cyg was obtained as follows:

$$\begin{array}{c}\hfill Min.I=HJD2455196.1392(\pm 0.0016)+0.{63114375}^d(\pm 0.00000009)E,\end{array}$$

(5)

The cyclical variation of the orbital periods can be seen in the (O-C)I curve of CG Cyg in the lower left panel of Figure 5. These were explained using a magnetic activity cycle or light-travel time effects (LITE) of the third-body. The conventional method was used to obtain their parameters,

$$\begin{array}{c}\hfill {\left(\mathrm{O}-\mathrm{C}\right)}_1=\frac{a_{12}sini}{c}\left[\frac{1-e^2}{1+ecosv}sin\left(v+\omega \right)+esin\omega \right],\end{array}$$

(6)

where $a_{12}$ is the orbital semi-major axis in the binary system of a third body system, and $e,i,v$ and $\omega$ is the eccentricity, inclination, true anomaly and longitude of the periastron, respectively. We test for a possible fourth body using the O-C diagram, as shown in the lower right panel in Figure 5. Therefore we fitted the minimum time again. The results are plotted in Figure 5. After fitting, the oscillation cycle of the third-body was 59.20 (±0.36) yr. We found that the cycle of the second oscillation was 18.31 (±0.60) yr with a smaller oscillation amplitude. The values of the two possible cycles of period variation were similar to those results of 51 and 22.5 yrs [55].

4. Orbital Parameters and Starspots

We used the updated 2014 Wilson-Devinney program [128,129,130,131] to revise the orbital parameters of the eclipsing binaries. The Wilson-Devinney program is the most widely used for light curve modeling code to obtain the stellar and starspot parameters of eclipsing binary. This program uses a different correction method for parameter adjustment of the observed light and velocity curves by the lease squares criterion, and produced the corresponding theoretical curves. We set V Crt as a semi-detached binary in the WD program according to previous research [11]. We set 7500 K for the primary component, 5000 K for the secondary component, a mass ratio of the secondary component and primary component (q) of 0.683 as the initial value. Further, we also set the bolometric albedo coefficient $A_1$ = $A_2$ = 0.5 [132] and gravity-darkening coefficients $g_1$ = $g_2$ = 0.32 [133]. The bolometric limb-darkening coefficients $X_{bolo}$ and limb-darkening coefficients for the four BVRI bands are $X_{1bolo}$ = 0.469, $X_{2bolo}$ = 0.429; $X_{1B}$ = 0.558, $X_{2B}$ = 0.864, $X_{1V}$ = 0.471, $X_{2V}$ = 0.725, $X_{1R}$ = 0.375, $X_{2R}$ = 0.599, $X_{1I}$ = 0.294, $X_{2I}$ = 0.489, respectively based on the stellar temperature [134]. We attempted the calculation using several sets of mass ratios to obtain the best rate if the mass ratio was unknown. The temperature was obtained using J-H color index relation [135] or spectral type. We simultaneously adjusted with other parameters: orbital inclination (i), temperature of secondary component ($T_2$), monochromatic luminosity of the primary component ($L_{1B}$, $L_{1V}$, $L_{1R}$ and $L_{1I}$), and dimensionless potentials of the two components (${\Omega }_1$ and ${\Omega }_2$). We adjusted each parameters for converge to obtain the best solution by several steps. We used the weighted sum of squares of the residual between the theoretical and observed light curves to judge the best results.

The corresponding orbital parameters obtained from several calculations are listed in

Table 5. Orbital Parameters of V Crt, WY Cnc and CG Cyg.

	V Crt	WY Cnc	CG Cyg
Parameters	Values	Values	Values
$T_1$(K)	7500 (a)	5500 (a)	5200 (a)
q($M_2/M_1$)	0.637 ± 0.005	0.501 ± 0.002	0.820 ± 0.001
$T_2$(K)	5142 ± 20	3543 ± 20	4610 ± 4
i(∘)	73.630 ± 0.067	89.759 ± 0.002	82.890 ± 0.029
${\Omega }_{in}$	3.131	2.876	3.451
${\Omega }_{out}$	2.761	2.577	2.991
${\Omega }_1$	3.489 ± 0.014	4.561 ± 0.009	4.608 ± 0.011
${\Omega }_2$		4.565 ± 0.016	4.757 ± 0.011
$L_1$/($L_1$+$L_2$)(B)	0.9263 ± 0.0002	0.9919 ± 0.0001	0.7830 ± 0.0005
$L_1$/($L_1$+$L_2$)(V)	0.8721 ± 0.0004	0.9844 ± 0.0001	0.7488 ± 0.0005
$L_1$/($L_1$+$L_2$)(R)	0.8221 ± 0.0006	0.9729 ± 0.0001	0.7147 ± 0.0006
$L_1$/($L_1$+$L_2$)(I)	0.7759 ± 0.0008	0.9490 ± 0.0001	0.6922 ± 0.0006
$r_1$(pole)	0.346 ± 0.002	-	-
$r_1$(side)	0.389 ± 0.003	-	-
$r_1$(back)	0.359 ± 0.002	-	-
$r_1$(average)	0.365 ± 0.002	-	-
$r_2$(pole)	0.319 ± 0.001	-	-
$r_2$(side)	0.454 ± 0.001	-	-
$r_2$(back)	0.334 ± 0.001	-	-
$r_2$(average)	0.369 ± 0.001	-	-
f	-	-	-
a($R_{\odot }$)	4.659 ± 0.348	4.008	3.717
$V_{\gamma }$(kms${}^{-1}$)	-	−0.03 ± 0.05	0.01 ± 0.02
$M_1$($M_{\odot }$)	1.68 ± 0.08	0.842	0.953
$M_2$($M_{\odot }$)	1.07 ± 0.15	0.419	0.782
$R_1$($R_{\odot }$)	1.70 ± 0.13	0.99	0.99
$R_2$($R_{\odot }$)	1.72 ± 0.13	0.60	0.84
$L_1$($L_{\odot }$)	8.19 ± 0.61	0.80 ± 0.10	0.64 ± 0.05
$L_2$($L_{\odot }$)	1.85 ± 0.14	0.05 ± 0.01	0.28 ±0.02
$\Sigma {\omega }_i{(O-C)}_i^2$	6.614	4.665	1.248

(a) means that the temperature of the primary component is assumed and fixed.

. The theoretical and observed light curves are demonstrated in Figure 2. The stellar structure is displayed in Figure 6. An obvious asymmetry was observed in the light curve. We attempted to add a cold spot on the primary or secondary components to explain this. We assumed the spot shape is circular. The spot longitude, radius and temperature are the free parameters. The starspot parameters were appropriately adjusted until they converge. The weighted sum of squares of the residual between the theoretical and observed light curves was used to judge the best results. Before fitting the spot parameters, we must set the initial value, generally, the latitude is fixed at 90${}^{\circ }$ to simplify the model. The spot temperature factor (Tspot/Tstar) was set to 0.8 because the cold spot was approximately 1000 K below the stellar total surface temperature, and the radius was set equal to the phase range where the deviation of the light curves occurs. The spot longitude was determined using the center of the light curve distortion. The spot radius was estimated by the fitting the observed light curve. Because the spot size is correlated with stellar temperature, and the spot latitude is correlated with spot radius. We have to adjust the spot parameters for many runs to determine the converge solutions. We obtained the best solution with the lowest sum of squares of the residual between the theoretical and observed light curves. The details of the fitting of these parameters have been described in the previous paper [136]. Initially, we added a cold spot at the secondary component but the result is not very good. We suppose that that the primary star is gaining mass because the secondary Roche lobe was filled as a semi-detached binary system. Hence, we changed the spot at primary component. Finally, we obtained the two solutions of a spot on the primary and secondary component. The $\Sigma {\omega }_i{(O-C)}_i^2$ was 6.614 for the spot on primary component and 7.0947 for the spot on the secondary component. We found that there are no significantly different for the two solutions, as shown in Figure 2. We also attempted to use two spots to explain the light curve distortion. However, there are no significantly improvements of the residual. For our light curve, there were no significant improvements of the residuals. The longitude of the spot on the primary component was $89.2^{\circ }\pm 4.3^{\circ }$. The spot radius was $18.0^{\circ }\pm 0.9^{\circ }$ and the spot temperature was 6763 K ± 909 K. The spot parameters are listed in

Table 6. Spot parameters of three eclipsing binaries.

Name	Position (P/N)	Latitude (${}^{\circ }$)	Longitude (${}^{\circ }$)	Radius (${}^{\circ }$)	Temperature (K)
V Crt	P	90 (a)	89.2 ± 4.3	18.0 ± 0.9	6764 ± 909
V Crt	S	90 (a)	83.9 ± 17.6	79.7 ± 21.1	4998 ± 13
WY Cnc	P	90 (a)	289.9 ± 1.7	15.9 ± 0.8	5176 ± 26
CG Cyg	P	70.5 ± 2.2	283.4 ± 1.6	33.3 ± 0.9	362 ± 81
CG Cyg	S	90 (a)	284.2 ± 0.7	80.1 ± 2.1	4284 ± 24

(a) mean the parameters are not adjusted, which indicate that the spot center is on the equator of the component. P means spot on primary component and S means spot on secondary component.

.

We used a similar method to obtain orbital and starspot parameters for WY Cnc and CG Cyg. We set the temperature of the primary component of WY Cnc to 5500 K and set the secondary temperature to 4000 K [35]. Based on the temperatures of both components, we determined the bolometric limb-darkening coefficients $X_{bolo}$ and limb-darkening coefficients for BVRI four bands to be $X_{1bolo}$ = 0.520, $X_{2bolo}$ = 0.429; $X_{1B}$ = 0.781, $X_{2B}$ = 0.817, $X_{1V}$ = 0.644, $X_{2V}$ = 0.705, $X_{1R}$ = 0.533, $X_{2R}$ = 0.612, $X_{1I}$ = 0.438, $X_{2I}$ = 0.476, respectively. For CG Cyg, we set the temperature of the primary component to 5200 K and that of the secondary component to 4400 K [51]. The corresponding bolometric limb-darkening coefficients $X_{bolo}$ and limb-darkening coefficients for the BVRI bands were $X_{1bolo}$ = 0.530, $X_{2bolo}$ = 0.532; $X_{1B}$ = 0.822, $X_{2B}$ = 0.944, $X_{1V}$ = 0.684, $X_{2V}$ = 0.795, $X_{1R}$ = 0.566, $X_{2R}$ = 0.658, $X_{1I}$ = 0.464, $X_{2I}$ = 0.532, respectively. We collected radial velocity values of WY Cnc [35] and CG Cyg [51]. We analyzed the published radial velocities and light curve to obtain orbital parameters using the WD program. The resultant light curves are shown in Figure 3 and Figure 4 for WY Cnc and CG Cyg. The orbital parameters are listed in Table 5 and the spot parameters are listed in Table 6. Further, we plotted the theoretical and observational radial velocity curves in Figure 7. The starspot structures of WY Cnc and CG Cyg are displayed in Figure 6.

5. Discussions and Conclusions

The photometric orbital parameters of V Crt were obtained using the WD program. The mass ratio and orbital inclination were 0.637 ± 0.004 and 73.629${}^{\circ }$± 0.067, respectively, which are similar to the results of the mass ratio of 0.683 ± 0.002 and inclination of 73.08${}^{\circ }$± 0.05 [11]. The luminosity ratio of the primary component and total intensity of V Crt in the difference BVRI bands were 0.9263 ± 0.0026, 0.8721 ± 0.0035, 0.8221 ± 0.0042 and 0.7759 ± 0.0046, respectively. The dimensionless potential of V Crt was ${\Omega }_1$ = ${\Omega }_2$ = 3.489 ± 0.014. The mass of primary component of V Crt was obtained as $M_1$ = 1.68 M${}_{\odot }$ based on the relationship between mass and color index. The mass of the secondary component $M_2$ was 1.07 M${}_{\odot }$. The semi-major axis a was obtained using the Kepler’s third law ($M_1+M_2=0.0134a^3/p^2$). We calculated the radius (R) of each component using $R_{1,2}=a\times r_{1,2\left(mean\right)}$, where r${}_{\left(mean\right)}$ is the weighted average value of r${}_{\left(pola\right)}$, r${}_{\left(side\right)}$ and r${}_{\left(back\right)}$. The luminosity was obtained by $L_{1,2}$ = ($R_{1,2}$/${\mathrm{R}}_{\odot }{)}^2$($T_{1,2}$/${\mathrm{T}}_{\odot }{)}^4$, where T${}_{\odot }$ = 5770 K. We calculated the period change of dP/dt = 2.80 ($\pm 2.14$) ×${10}^{-9}$ d yr${}^{-1}$ using the change of parabolic fit with quadratic coefficient terms greater than zero, which may means that the mass was transferred mass from the secondary component to the primary component [137] or magnetic breaking [138]. Because of the semi-detached system, we calculated the mass transfer rate using the following formula [139]:

$$\begin{array}{c}\hfill \frac{d{M}_1/dt}{M_1}=\frac{q}{3(1-q)}\frac{dp/dt}{P}\end{array}$$

(7)

Finally, we obtained the rate of mass transfer of d$M_1$/dt = 3.92 ($\pm 3.00$) ×${10}^{-9}$ M${}_{\odot }$ yr${}^{-1}$.

We obtained a rate of mass decrease for WY Cnc as d$M_1$/dt= −7.31 ($\pm 0.35$) ×${10}^{-9}$$M_{\odot }$ yr${}^{-1}$. Because WY Cnc is a detached binary system, we opine that the period variation can not explained by the mass transfer. If the change in period contributes to magnetic breaking, we can calculate the angular momentum loss as follows:

$$\begin{array}{c}\hfill \frac{\Delta J}{J}=\frac{\Delta P}{3P}+\frac{\Delta M_1}{M_1}+\frac{\Delta M_2}{M_2}+\frac{\Delta M_{Total}}{3M_{Total}}\end{array}$$

(8)

where the angular momentum is:

$$\begin{array}{c}\hfill J=\frac{2\pi a^2{M}_1{M}_2}{M_{Total}P}=3.8\times {10}^{51}\mathrm{g}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}\end{array}$$

(9)

where G is the gravitational constant. We calculated $\Delta$J as $-3.54\times {10}^{36}\mathrm{g}\mathrm{c}{\mathrm{m}}^2/{\mathrm{s}}^2$, which is similar to the result reported by Chen [37]. The period change of WY Cnc may be caused by the magnetic braking effect [37].

For CG Cyg, a cyclic variation of O-C was observed, as shown in the lower left panel of Figure 5, which might be explained by the light-time effect with a third body. Firstly, the semi-axis $a_{12}$ of the binary must be calculated:

$$\begin{array}{c}\hfill a_{12}^{\prime }sin{i}_3^{\prime }=A_3\times c\end{array}$$

(10)

where i is the orbital inclination, A${}_3$ is the amplitude, and c is the speed of light. The amplitude of the oscillation of the systemic velocity can be calculated as follows [140]:

$$\begin{array}{c}\hfill K_{RV}=\frac{2\pi }{P_3}\frac{a_{12}sini}{\sqrt{1-e^2}}\end{array}$$

(11)

where $K_{RV}$, $a_{12}$, and $P_3$ are the velocity (km/s), distance from the sun to the earth (Au), and period of third body (yr), respectively. Subsequently, we can calculate the mass ($M_3$) of the third-body as followings:

$$\begin{array}{c}\hfill f\left(m\right)=\frac{4{\pi }^2}{G{P_3}^2}\times {(a_{12}^{\prime }sin{i}_3^{\prime })}^3=\frac{{(M_{3}sin{i}_3^{\prime })}^3}{{(M_1+M_2+M_3)}^2}\end{array}$$

(12)

The value of the third body was smallest when i = 90, as illustrated in Figure 8. Therefore, we obtained the minimum mass of $M_3$ as 0.14 M${}_{\odot }$, which is close to the result of 0.12 M${}_{\odot }$ [55]. The parameters of the periodic oscillation are listed in

Table 7. Parameters of the additional bodies of CG Cyg.

Parameters	Values (CG Cyg)	Values (Second Oscillation)
A (d)	0.0066(±0.0010)	0.0010 (±0.0001)
P3 (yr)	59.20(±0.36)	18.31 (± 0.6)
e	0.38(±0.01)	0.185 (±0.012)
$T_0\left(HJD\right)$	2455107.1440	2455107.1440
$a_{12}$sini (AU)	1.15(± 0.01)	0.18 (±0.01)
$K_{RV}$ (km s${}^{-1}$)	0.625(±0.0.007)	0.30 (±0.01)
$f\left(m\right)\left(M_{\odot }\right)$	0.0004(±0.0001)	0.00002(±0.00001)

A represents the semi-amplitude of the light time effect and d is the unit of day. f(m) is the mass of the third body.

.

Except for the third body, the period oscillation in lower left panel of Figure 5 might be caused by magnetic cycles. The change in quadrupole moment due to magnetic activity in the internal constitution can also cause cyclic oscillations ($\Delta$ P/P). Therefore, we considered the magnetic activity cycles of this system, as following:

$$\begin{array}{c}\hfill △P/P=A_3\times \sqrt{2[1-cos2\pi \times P/P_3)]}/P\end{array}$$

(13)

where $P_3$ is the period of magnetic activity. The $\Delta$P/P of P${}_3$ of 59.20 yr was calculated to be 1.918 ×${10}^{-6}$. This value is larger than the traditional value of $\Delta$P/P−${10}^{-8}$∼${10}^{-7}$ for magnetic activity in RS CVn and Algol binaries [141]. Therefore, we can conclude that the periodic oscillations might have been caused by the LITE of a third body, not the magnetic cycle. For the second cycle of P${}_3$ of 18.31 yr, the $\Delta$P/P was calculated to be 9.398 × ${10}^{-7}$. It is in the range of traditional values, which might be caused by the magnetic activity. There is no direct evidence for the third based on our observation of the telescopes. Therefore, we prefer the magnetic activity cycle than the third body to explain the cyclic variations. However, we cannot rule out that these variations might be caused by the third body. New telescope imaging technology are required to check that in the future.

WY Cnc and CG Cyg are detached and RS CVn eclipsing binaries. Using the obtained light curves, we revised the orbital parameters using the WD program. The derived mass ratio of 0.501 ± 0.002 and orbit inclination of 89.759 ± 0.002${}^{\circ }$ are similar to that obtained in the previous results in regions for a mass ratio of 0.38–0.59 and for an inclination of 86–90${}^{\circ }$ for WY Cnc [35]. The mass ratio of 0.820 ± 0.001 for CG Cyg is similar to 0.862 [53] and 0.825 [51]. The orbit inclination of 82.890 ± 0.029${}^{\circ }$ for CG Cyg is similar to the previous results of 83 ± 0.2${}^{\circ }$ [51] and 82.602 ± 0.099${}^{\circ }$ [35]. We obtained the starspot parameters of WY Cnc and CG Cyg at our observational times, which were different from the previous starspot parameters [35]. We confirm that variation in the light curves variations owing to varying starspot in different seasons [53,142]. Further photometric and spectroscopic observations are required to determine the magnetic activity cycle.

In Figure 9, we plotted the $log(L/L_{\odot })$ vs. $T_{eff}$ diagrams with many theoretical isochrone [143], and indicated the positions of both primary and second components of V Crt, WY Cnc and CG Cyg. We found that the parameters of the primary and secondary components of WY Cnc and CG Cyg are in agreement with the 0.030 Gyr tracks of the theoretical model. The secondary component of V Crt are in the region of 0.03–0.05 Gyr. Because V Crt is a semi-detached eclipsing binary and the primary component of V Crt is approximately 7500 K, it is difficult to determine its age. However, its age can be determined by investigating additional observations and theoretical models.