Physical properties and evolutionary status of the W-subtype contact binary V502 Oph with a stellar companion

TL;DR

This study analyzes the contact binary V502 Oph using multi-color photometry, revealing its physical parameters, evolutionary status, and the presence of a tertiary component affecting its orbital period.

Contribution

The paper provides the first detailed modeling of V502 Oph's light curves, including third light detection and analysis of orbital period variations, contributing new insights into its structure and evolution.

Findings

V502 Oph is a W-subtype contact binary with 35.3% fill-out factor.

The orbital period is decreasing at a rate of -1.69×10^{-7} day/year.

A tertiary component with an 18.7-year period influences the system's orbital dynamics.

Abstract

Multi-color ($B$ $V$ $R_c$ $I_c$) CCD photometric light curves of the contact binary V502 Oph are analyzed using the Wilson-Devinney (W-D) program. The solutions reveal that V502 Oph is a W-subtype contact ($f = 35.3\,\%$) binary system. The temperature difference between its two components is $240K$ and the more massive star has a lower surface temperature. A cool spot is added in our model to account for the light curves' asymmetry (O'Connell effect) and third light is detected for the first time in the light curves' modeling. Combining the orbital inclination ($i = 76.4^{\circ}$) with the published mass function of V502 Oph, the absolute physical parameters of the two components are determined, which are $M_1= 0.46(\pm0.02)M_\odot, M_2=1.37(\pm0.02)M_\odot, R_1=0.94(\pm0.01)R_\odot, R_2=1.51(\pm0.01)R_\odot, L_1=1.13(\pm0.02)L_\odot$ and $L_2=2.49(\pm0.03)L_\odot$. The formation and evolutionary status of V502 Oph are discussed. All photoelectric and CCD times of light minimum about V502 Oph are gathered and its orbital period variations are analyzed. The results show that the orbital period of V502 Oph is decreasing continuously at a rate of $dP/dt=-1.69\times{10^{-7}}day\cdot year^{-1}$, which corresponds to a conservative mass transfer rate of $\frac{dM_{2}}{dt}=-3.01\times{10^{-8}}M_\odot/year$. The light-travel time effect (LTTE) is due to the presence of a close-in tertiary component with a period of $P_3=18.7$ years and an amplitute of 0.00402days. V502 Oph is an ideal target to test the formation and evolution theories of binary and multiple systems in which the light curves, $O - C$ curve and spectroscopic observations are comprehensively researched.

Tables

Table 1. Table 1: The original data of V502 Oph

JD(Hel.)	Phase	$\Delta$m	JD(Hel.)	Phase	$\Delta$m	JD(Hel.)	Phase	$\Delta$m
2455376.6160	0.966034	-1.153000	2455376.6400	0.018968	-1.092000	2455376.6720	0.089548	-1.327000
2455376.6190	0.972650	-1.148000	2455376.6470	0.034407	-1.115000	2455376.6760	0.098370	-1.339000
2455376.6210	0.977062	-1.136000	2455376.6490	0.038819	-1.128000	2455376.6790	0.104987	-1.371000
2455376.6230	0.981473	-1.125000	2455376.6520	0.045435	-1.136000	2455376.6820	0.111604	-1.375000
2455376.6260	0.988090	-1.111000	2455376.6540	0.049847	-1.158000	2455376.6940	0.138071	-1.452000
2455376.6280	0.992501	-1.099000	2455376.6560	0.054258	-1.178000	2455376.6970	0.144688	-1.441000
2455376.6310	0.999118	-1.076000	2455376.6590	0.060875	-1.195000	2455376.7010	0.153510	-1.465000
2455376.6330	0.003529	-1.085000	2455376.6640	0.071903	-1.234000	2455376.7040	0.160127	-1.475000
2455376.6350	0.007940	-1.090000	2455376.6660	0.076314	-1.260000	2455376.7070	0.166744	-1.480000
2455376.6380	0.014557	-1.097000	2455376.6690	0.082931	-1.295000	2455376.7100	0.173361	-1.491000

Table 2. Table 2: ( O − C ) 𝑂 𝐶 (O-C) values of light minima for V502 Oph.

JD(Hel.)	p/s	Epoch	${\left(O-C\right)}_1$	${\left(O-C\right)}_2$	Error	Method	Ref.
(2400000+)
34514.7822	p	-11304	-0.0089	0.0003	0.0010	pe	1
34515.4617	s	-11302.5	-0.0095	-0.0003	0.0008	pe	2
34840.5465	s	-10585.5	-0.0082	0.0001	0.0003	pe	2
34897.4501	p	-10460	-0.0055	0.0026		pe	3
34899.4890	s	-10455.5	-0.0069	0.0013		pe	3
35257.4459	p	-9666	-0.0045	0.0028	0.0004	pe	4
35261.5260	p	-9657	-0.0049	0.0023	0.0009	pe	4
35262.4322	p	-9655	-0.0055	0.0018	0.0005	pe	4
35544.6726	s	-9032.5	-0.0028	0.0039	0.0003	pe	4
35579.5852	s	-8955.5	-0.0016	0.0051	0.0004	pe	4
35582.5274	p	-8949	-0.0064	0.0002	0.0008	pe	4
35587.5148	p	-8938	-0.0064	0.0002	0.0005	pe	4
35617.4408	p	-8872	-0.0044	0.0022	0.0012	pe	4
35621.5205	p	-8863	-0.0052	0.0013	0.0003	pe	4
37394.9740	s	-4951.5	-0.0021	0.0028		pe	5
37404.9460	s	-4929.5	-0.0048	0.0001		pe	5
37405.8546	s	-4927.5	-0.0030	0.0020		pe	5
39631.7829	p	-18	-0.0122	-0.0048		pe	6
39637.9078	s	-4.5	-0.0081	-0.0007		pe	6
39639.9436	p	0	-0.0126	-0.0052		pe	6
39642.8951	s	6.5	-0.0082	-0.0007		pe	6
40719.4714	p	2381	-0.0158	-0.0053		CCD	7
40778.4110	p	2511	-0.0174	-0.0067		CCD	7
41174.2288	p	3384	-0.0125	-0.0004		pe	8
41184.2031	p	3406	-0.0129	-0.0007		pe	8
41194.1799	p	3428	-0.0107	0.0015		pe	8
41214.1326	p	3472	-0.0074	0.0049		pe	8
41491.3772	s	4083.5	-0.0132	0.0002		CCD	9
41802.6320	p	4770	-0.0133	0.0014		pe	10
41833.4620	p	4838	-0.0141	0.0008		CCD	11
41844.3440	p	4862	-0.0135	0.0014		CCD	11
41858.4000	p	4893	-0.0128	0.0022		CCD	11
41860.4400	s	4897.5	-0.0130	0.0020		CCD	11
41888.3200	p	4959	-0.0168	-0.0016		CCD	9
41888.3205	p	4959	-0.0163	-0.0011		pe	9
42570.4558	s	6463.5	-0.0121	0.0063		CCD	12
43238.5240	p	7937	-0.0199	0.0023		CCD	13
43238.5250	p	7937	-0.0189	0.0033		CCD	13
43292.4761	p	8056	-0.0217	0.0008		CCD	13
43665.8463	s	8879.5	-0.0214	0.0034		pe	14

Table 3. Table 2: Continuted

JD(Hel.)	p/s	Epoch	${\left(O-C\right)}_1$	${\left(O-C\right)}_2$	Error	Method	Ref.
(2400000+)
43666.7545	s	8881.5	-0.0200	0.0049		pe	14
43668.7951	p	8886	-0.0197	0.0052		pe	14
43671.7469	s	8892.5	-0.0149	0.0100		pe	14
44370.8684	s	10434.5	-0.0269	0.0027	0.0007	CCD	15
46528.5488	s	15193.5	-0.0482	-0.0010	0.0005	pe	16
46529.4572	s	15195.5	-0.0466	0.0006	0.0005	pe	16
46530.5866	p	15198	-0.0507	-0.0034	0.0004	pe	16
46552.8030	p	15247	-0.0506	-0.0031		pe	17
46555.5244	p	15253	-0.0496	-0.0021	0.0004	pe	16
46582.4954	s	15312.5	-0.0555	-0.0078		pe	18
46584.5391	p	15317	-0.0521	-0.0043		pe	18
46585.4478	p	15319	-0.0502	-0.0024		pe	18
46586.3496	p	15321	-0.0552	-0.0074		pe	18
46586.3496	p	15321	-0.0552	-0.0074		pe	18
46586.5744	s	15321.5	-0.0571	-0.0093		pe	18
46587.4811	s	15323.5	-0.0571	-0.0094		pe	18
46590.4235	p	15330	-0.0618	-0.0140		pe	18
46595.4205	p	15341	-0.0521	-0.0043		pe	18
46640.5356	s	15440.5	-0.0497	-0.0015		pe	17
46915.5165	p	16047	-0.0523	-0.0014	0.0004	pe	16
46961.7750	p	16149	-0.0399	0.0114		pe	19
47707.3612	s	17793.5	-0.0601	-0.0012		pe	20
48094.3284	p	18647	-0.0646	-0.0016		pe	20
48096.3673	s	18651.5	-0.0660	-0.0029		pe	20
48099.3150	p	18658	-0.0653	-0.0023		pe	20
48114.2770	p	18691	-0.0653	-0.0021		pe	20
51266.9174	s	25644.5	-0.0997	0.0030	0.0002	CCD	21
51267.1421	p	25645	-0.1017	0.0010	0.0003	CCD	21
51299.1037	s	25715.5	-0.1044	-0.0012		CCD	22
51299.1040	s	25715.5	-0.1041	-0.0009	0.0011	CCD	23
51319.0560	s	25759.5	-0.1014	0.0020	0.0010	CCD	23
51320.4155	s	25762.5	-0.1021	0.0013	0.0002	CCD	21
51320.6436	p	25763	-0.1007	0.0028	0.0004	CCD	21
51388.8781	s	25913.5	-0.1020	0.0024	0.0005	CCD	21
51389.1042	p	25914	-0.1026	0.0018	0.0002	CCD	21
51666.1193	p	26525	-0.1112	-0.0028		CCD	24
52043.1141	s	27356.5	-0.1134	0.0005		CCD	25
52768.5242	s	28956.5	-0.1336	-0.0086	0.0002	CCD	26
52772.3815	p	28965	-0.1302	-0.0051	0.0002	CCD	27
53189.0452	p	29884	-0.1355	-0.0038		CCD	28

Table 4. Table 2: Continuted

JD(Hel.)	p/s	Epoch	${\left(O-C\right)}_1$	${\left(O-C\right)}_2$	Error	Method	Ref.
(2400000+)
53402.5929	p	30355	-0.1364	-0.0012	0.0003	CCD	29
53405.5412	s	30361.5	-0.1351	0.0001	0.0008	CCD	29
53500.5220	p	30571	-0.1403	-0.0036	0.0009	CCD	30
53524.3252	s	30623.5	-0.1403	-0.0032	0.0004	CCD	30
53537.4733	s	30652.5	-0.1407	-0.0033	0.0003	CCD	31
53863.4527	s	31371.5	-0.1515	-0.0087	0.0002	CCD	32
53905.3952	p	31464	-0.1479	-0.0045	0.0003	CCD	31
54263.1223	p	32253	-0.1487	0.0009		CCD	33
54577.5419	s	32946.5	-0.1577	-0.0028	0.0006	CCD	34
54613.1335	p	33025	-0.1576	-0.0020		CCD	35
54631.0408	s	33064.5	-0.1593	-0.0034		CCD	35
54641.4712	s	33087.5	-0.1570	-0.0009	0.0006	CCD	34
55661.5923	s	35337.5	-0.1723	0.0023	0.0003	CCD	36
55700.1253	s	35422.5	-0.1777	-0.0025		CCD	37
56092.9876	p	36289	-0.1813	0.0014		CCD	38
56428.4975	p	37029	-0.1829	0.0062	0.0003	CCD	36
56447.0859	p	37070	-0.1837	0.0058		CCD	39
56447.0860	p	37070	-0.1836	0.0059		CCD	39
56447.0862	p	37070	-0.1834	0.0061		CCD	39
56452.0720	p	37081	-0.1849	0.0047		CCD	39
56460.4572	s	37099.5	-0.1875	0.0022	0.0006	CCD	36

Table 5. Table 3: Photometric solutions of V502 Oph

Parameters	Values	Values
	with $l_3$	without $l_3$
$g_1$	0.32(fixed)	0.32(fixed)
$g_2$	0.32(fixed)	0.32(fixed)
$A_1$	0.50(fixed)	0.50(fixed)
$A_2$	0.50(fixed)	0.50(fixed)
q ($M_2/M_1$ )	2.985(fixed)	2.985(fixed)
$T_1\left(K\right)$	6140(fixed)	6140(fixed)
$i{(}^{\circ })$	76.4($\pm 0.6$)	72.8($\pm 0.1$)
${\Omega }_{in}$	6.6030	6.6030
${\Omega }_{out}$	5.9843	5.9843
${\Omega }_1={\Omega }_2$	6.3779($\pm 0.0125$)	6.4384($\pm 0.0056$)
$T_2\left(K\right)$	5900($\pm 8$)	5922($\pm 6$)
$\Delta T\left(K\right)$	240	218
$T_2/T_1$	0.961($\pm 0.001$)	0.964($\pm 0.001$)
$L_1/(L_1+L_2$) ($B$)	0.3285($\pm 0.0033$)	0.3206($\pm 0.0010$)
$L_1/(L_1+L_2$) ($V$)	0.3149($\pm 0.0050$)	0.3085($\pm 0.0009$)
$L_1/(L_1+L_2$) ($R_c$)	0.3088($\pm 0.0049$)	0.3030($\pm 0.0008$)
$L_1/(L_1+L_2$) ($I_c$)	0.3038($\pm 0.0047$)	0.2985($\pm 0.0007$)
$L_1/(L_1+L_2+L_3$) ($B$)	0.2886($\pm 0.0062$)
$L_1/(L_1+L_2+L_3$) ($V$)	0.2742($\pm 0.0076$)
$L_1/(L_1+L_2+L_3$) ($R_c$)	0.2697($\pm 0.0074$)
$L_1/(L_1+L_2+L_3$) ($I_c$)	0.2632($\pm 0.0071$)
$L_3/(L_1+L_2+L_3$) ($B$)	0.1215($\pm 0.0163$)
$L_3/(L_1+L_2+L_3$) ($V$)	0.1294($\pm 0.0192$)
$L_3/(L_1+L_2+L_3$) ($R_c$)	0.1264($\pm 0.0189$)
$L_3/(L_1+L_2+L_3$) ($I_c$)	0.1335($\pm 0.0183$)
$r_1\left(pole\right)$	0.2854($\pm 0.0010$)	0.2805($\pm 0.0004$)
$r_1\left(side\right)$	0.2998($\pm 0.0012$)	0.2939($\pm 0.0005$)
$r_1\left(back\right)$	0.3469($\pm 0.0022$)	0.3360($\pm 0.0009$)
$r_2\left(pole\right)$	0.4619($\pm 0.0009$)	0.4577($\pm 0.0004$)
$r_2\left(side\right)$	0.4992($\pm 0.0012$)	0.4935($\pm 0.0005$)
$r_2\left(back\right)$	0.5306($\pm 0.0016$)	0.5231($\pm 0.0007$)
$f$	$35.3\%$($\pm$2.0 %)	$25.5\%$($\pm$0.9 %)
$\theta {(}^{\circ })$	102.6($\pm 14.4$)	106.0($\pm 10.9$)
$\psi {(}^{\circ })$	116.4($\pm 1.5$)	116.6($\pm 1.5$)
$r$(rad)	0.36($\pm 0.12$)	0.36($\pm 0.12$)
$T_f$	0.87($\pm 0.09$)	0.87($\pm 0.08$)
$\Sigma \omega {\left(O-C\right)}^2$	0.055668	0.066069

Table 6. Table 4: Absolute parameters of the two components in V502 Oph

Parameters	Primary	Secondary
$M$	$0.46\left(\pm 0.02\right)M_{\odot }$	$1.37\left(\pm 0.02\right)M_{\odot }$
$R$	$0.94\left(\pm 0.01\right)R_{\odot }$	$1.51\left(\pm 0.01\right)R_{\odot }$
$L$	$1.13\left(\pm 0.02\right)L_{\odot }$	$2.49\left(\pm 0.03\right)L_{\odot }$

Table 7. Table 5: Photometric solutions of V502 Oph published by other researchers

Parameters	Maceroni et al. (1982)	Zola & Krzesinski (1988)	Rovithis et al. (1988)	Rovithis et al. (1998)	Deb & Singh (2011)
$T_1\left(K\right)$	6200	6140	6140	6200	5940
q ($M_2/M_1$ )	2.65	2.64		2.70	2.985
$i{(}^{\circ })$	71.3($\pm 0.5$)	70.2($\pm 0.2$)	68.5($\pm 0.5$)	71.0($\pm 0.6$)	69.77($\pm 0.21$)
${\Omega }_1={\Omega }_2$	6.004($\pm 0.026$)	6.072($\pm 0.025$)			6.397($\pm 0.013$)
$f$	$23.6\%$	$10.3\%$($\pm$4.1 %)			$22\%$
$T_2\left(K\right)$	5968($\pm 35$)	5750($\pm 30$)	5800	5960	5739($\pm 129$)
$\Delta T\left(K\right)$	232	390	340	240	201
$T_2/T_1$	0.9626($\pm 0.0056$)	0.9365($\pm 0.0049$)	0.9446	0.9613	0.966($\pm 0.022$)
$L_1/\left(L_1+L_2\right)$ ($B$)	0.338($\pm 0.006$)	0.3732($\pm 0.0004$)	0.370($\pm 0.010$)	0.269($\pm 0.010$)
$L_1/\left(L_1+L_2\right)$ ($V$)	0.334($\pm 0.005$)	0.3581($\pm 0.0004$)	0.361$\pm 0.010$)		0.3063($\pm 0.0329$)
$\theta {(}^{\circ })$				90 (star 1), 30 (star 2)
$\psi {(}^{\circ })$				180 (star 1), 270 (star 2)
$r$(rad)				20 (star 1), 40 (star 2)
$T_f$				0.7 (star 1), 0.7 (star 2)

Equations

$$Min.I(HJD)=2456092.9876+0^d.45339\times E.$$

$$Min.I(HJD)=2439639.9562+0^d.45339394\times E,$$

$$\begin{array}[]{lll}Min.I=2439639.94876(\pm 0.00005)+0.453392912(\pm 0.000000005)\times{E}\\ -1.047(\pm 0.002)\times{10^{-10}}\times{E^{2}}\\ +0.00402(\pm 0.00005)\sin[0.0239(\pm 0.0002)\times{E}+263.1(\pm 0.7)]\end{array}$$

$$\begin{array}[]{lll}\frac{dM_{2}}{dt}=-\frac{M_{1}M_{2}}{3P(M_{1}-M_{2})}\times{dP/dt},\end{array}$$

$$\begin{array}[]{lll}a^{\prime}_{12}\sin i^{\prime}=A_{3}\times c,\end{array}$$

$$\begin{array}[]{lll}f(m)=\frac{4\pi^{2}}{GP^{2}_{3}}\times(a^{\prime}_{12}\sin i^{\prime})^{3}=\frac{(M_{3}\sin i^{\prime})^{3}}{(M_{1}+M_{2}+M_{3})^{2}},\end{array}$$

Full text

1 Introduction

V502 Oph (HD 150484, HIP 81703, BD+00 3562, $V$ = $8^{m}.59$) is a W UMa type contact binary system which was discovered to be a variable star by Hoffmeister (1935). In the following decades, many remarkable work have been done on this target. Spectroscopic observations and radial velocity studies have been carried out by Struve & Gratton (1948), Struve & Zebergs (1959), King & Hilditch (1984), Pych et al. (2004); Karami & Mohebi (2007) and Karami et al. (2011). The analysis revealed that the mass ratio was $q=0.335(9)$, and the spectral type were G2 V for the more massive component and F9V for the less massive one. Photometric observations and solutions of V502 Oph have been published by Wilson (1967); Binnendijk (1969); Vader & van der Wal (1973); Polushina (1975); Maceroni et al. (1982); Zola & Krzesinski (1988); Rovithis et al. (1988); Rovithis et al. (1998) and Deb & Singh (2011). The light curves change over time and may indicate variable surface activity, which made it an ideal target to study the physical mechanism of O’Connell effect (Koju & Beaky, 2015). The orbital period of V502 Oph was found to change continuously, according to the investigation by Kwee (1958); Kreiner (1971); Strauss (1976); Maddox & Bookmyer (1979); Hobart et al. (1989); Derman & Demircan (1992); Yüce et al. (2006) and Liu & Yang (2006).

Most W UMa stars consist of solar-type components with orbital periods of $0.25d<P<0.7d$ and share a common envelope between its two components. They are recognized by continuous brightness variations and nearly equal depth for primary and secondary eclipse. W UMa type binaries are superb laboratories for the study of the formation and evolution scenario of close binaries. The research will provide valuable information on the late stage of stellar evolution connected with the processes of mass transfer, angular momentum loss(AML) and merging of stars, etc (Li et al., 2014; Yang & Qian, 2015). In addition, surveys of companions around eclipsing binaries have generally shown that tertiary components are quite common in W UMa systems (Frieboes-Conde & Herczeg, 1973; Mayer, 1990; Chambliss, 1992; Borkovits & Hegedues, 1996; Pribulla & Rucinski, 2006; D’Angelo et al., 2006). The close-in tertiary component may play an important role in the formation and evolution of close binary by shortening the orbital period of the binary system through drawing angular momentum from the central binary system (Tokovinin et al., 2006). And also, many different types of objects have been reported to be companions of binary stars, such as stellar companion (Zhou et al., 2015), circumbinary planet (Qian et al., 2012), brown dwarf (Qian et al., 2009) and black hole (Qian et al., 2008).

As for V502 Oph, lines of a third component were detected while spectroscopic observations were carried out (Hendry & Mochnacki, 1998; Pribulla & Rucinski, 2006). The light-travel time effect (LTTE) appearing on the observed-calculated ($O-C$) curve also confirmed the existence of a close-in tertiary component around the binary system (Yüce et al., 2006; Liu & Yang, 2006). Unfortunately, the former photometric solutions had not given any information about third light ($l_{3}$). Thus, new multi-color high accuracy CCD light curves are still need. In the present paper, $B$ $V$ $R_{c}$ and $I_{c}$ bands light curves of V502 Oph are obtained and analyzed using the Wilson-Devinney (W-D) program (Version 2013). More times of light minimum which cover a much longer time scale are collected and fitted. We combine photometric solutions, $O-C$ curve analysis and spectroscopic results together to give out our new research results on V502 Oph, which will shed new insight to our understanding on the formation and evolution theories of binary and multiple systems.

2 New Multi-color Photometric Observations of V502 Oph

The $B$ $V$ $R_{c}$ $I_{c}$ bands light curves (LCs) of V502 Oph were observed over 11 nights between June 29 and July 18, 2010, using a 20-cm Meade LX200GPS telescope at the Truman State University Observatories in Kirksville, Missouri. An SBIG ST-9XE CCD camera with Johnson $BVR_{c}I_{c}$ filters was attached to the telescope (Koju & Beaky, 2015). Astronomers Control Panel (ACP) was used to communicate between the telescope, CCD camera, focuser, and observatory dome. The telescope and CCD camera were controlled by Maxim DL, which was also used to analyze the observed images and obtain the light curves. A few lines of the original light curve data are displayed in Table 1. The resulting differential light curves for V502 Oph in $B$, $V$, $R_{c}$, and $I_{c}$ filters are shown in Fig. 1. The observational time (HJD) were converted to phases with the following linear ephemeris:

[TABLE]

To improve the legibility of the figure, the light curves’ data have been shifted vertically which will make no difference to the solutions of W-D program as differential photometry method is used. As shown in Fig. 1, the light curves vary continuously and have very small magnitude differences between the depth of the primary and secondary minima which indicates that V502 Oph is a typical W UMa type contact binary with EW type light curves.

3 Orbital Period Investigation of V502 Oph

Since it was discovered by Hoffmeister (1935), V502 Oph has been monitored for nearly eighty years. We have abundant light minima for analyzing the $O-C$ (observed - calculated) curve, which will reveal dynamic interactions between its components. As times of light minimum determined by visual observations have very large uncertainty, only those obtained by photoelectric observations and CCD camera are collected in our work. Using the following linear ephemeris determined by Derman & Demircan (1992),

[TABLE]

we obtain the $(O-C)_{1}$ values and list them in the fourth column of Table 2. We adopt a quadratic curve to fit the $(O-C)_{1}$ values, then the residuals ($(O-C)_{2}$ values) are obtained. The $(O-C)_{2}$ values of V502 Oph is listed in the fifth column of Table 2 and shown in the middle panel of Fig. 2, which indicates a cyclic change in its orbital period. Thus, a sinusoidal term is superimposed to the ephemeris. The revised ephemeris is

[TABLE]

According to the quadratic term included in the new ephemeris, we conclude that the period of V502 Oph is decreasing continuously at a rate of $dP/dt=-1.69\times{10^{-7}}day\cdot year^{-1}$. The period of the cyclic change is $P_{3}=18.7$ years, and its amplitude is $A_{3}=0.00402$ days. The residuals from equation (3) are showed in the bottom panel of Fig. 2. The large scatter of the $(O-C)_{2}$ values in Fig. 2 may due to the fact that V502 Oph is an active binary system. However, the tendency of the $(O-C)_{2}$ curve still shows periodic variation obviously and the $(O-C)_{2}$ curve covers over two circles which makes our results more reliable. We just give the most probable fitting result according to the minimum data available in the bibliography. To determine the period of the third component more accurate, more times of light minimum are need in the future.

4 Photometric Solutions of V502 Oph

In the present paper, a simultaneous multi-color light curves’ analysis of V502 Oph is performed with the Wilson-Devinney (W-D) program (Version 2013) (Wilson & Devinney, 1971; Van Hamme & Wilson, 2007; Wilson et al., 2010) to derive its physical parameters. According to the Radial Velocity (RV) studies published by Pych et al. (2004), the mass ratio (q) of V502 Oph is fixed at $q=2.985$. The effective temperature of the hotter component ($T1$) in V502 Oph is fixed to $T1=6140K$, which was used by Rovithis et al. (1988) and Zola & Krzesinski (1988) corresponding to its spectral type of F9V (Struve & Zebergs, 1959). V502 Oph is a late-type W-subtype W UMa contact binary system, so convective envelopes is assumed. The gravity-darkening coefficients and bolometric albedo adoptable for convective envelopes are used, which were $g_{1}=g_{2}=0.32$ (Lucy, 1967) and $A_{1}=A_{2}=0.5$ (Ruciński, 1969). And we choose the limb darkening coefficients from Van Hamme (1993)’s table in logarithmic law.

While modeling the light curves of V502 Oph, mode 3 for contact system is chosen. The adjustable parameters are: the orbital inclination of V502 Oph ($i$); the surface temperature for the secondary star ($T_{2}$); the monochromatic luminosity of the primary star ($L_{1B}$, $L_{1V}$, $L_{1R}$ and $L_{1I}$); the dimensionless surface potential of the primary and secondary star ($\Omega_{1}=\Omega_{2}$ for contact binary ). Since both spectroscopic observations and orbital period variations confirmed the existence of a tertiary component around V502 Oph, the third light ($l_{3}$) is also set as an adjustable parameter in our solutions. It has to be mentioned that we use a spot model in the calculating as the O’Connell effect appears in the light curves. The final parameters of the solutions are list in Table 3. The theoretical light curves are displayed in Fig. 1.

5 Discussions and Conclusions

We have present multi-color ($B$ $V$ $R_{c}$ $I_{c}$) CCD photometric solutions of the contact binary V502 Oph. The light curves have a long-lived asymmetric phenomenon (Koju & Beaky, 2015) in the placement of the light maxima called O’Connell effect (Milone, 1969). We use a spot model to fit the light curves in the W-D program. The solutions indicate that V502 Oph is a W-subtype contact binary with the contact degree of $f=35.3\,\%(\pm 2.0\,\%)$. The temperature difference between its two components is $240K$ and the more massive star has a lower surface temperature. Combining the orbital inclination ($i=76.4^{\circ}\pm 0.6$) of ours and the mass function given by Pych et al. (2004): $(M_{1}+M_{2})sin^{3}i=1.679\pm 0.022M_{\odot}$, the absolute physical parameters of the two components in V502 Oph are determined and listed in Table 4. The orbital semi-major axis of V502 Oph is calculated to be $a=3.04(\pm 0.02)R_{\odot}$. We have also checked the degeneracy of the third light, and the results are also in Table 4. It is found that degeneracy of the third light result in very little difference on parameters such as $i$, $T_{2}$, $\Omega_{1}$. However, the solutions with the third light give out a much smaller fitting residual.

The light curves of V502 Oph have been investigated by many other researchers (Table 5). However, there is not any information about third light $(l_{3})$ in their solutions. As shown in Table 5, Deb & Singh (2011) assumed that the hotter component is a G0V type star and set the temperature to be $T1=5940K$. Only Rovithis et al. (1998) used spot model in their solutions. In our solutions, we set the temperature to be $T1=6140K$ according to its spectral type of F9V and add a spot in our model considering the light curves’ asymmetry. Compared with the solutions listed in Table 5, we obtain a higher orbital inclination and contact degree, which may be caused by the contribution of third light in our solutions.

W UMa type contact binaries have been observed and analyzed for over half century. However, there is not a generally accepted formation and evolution scenario for them. Numerical calculations suggest that W UMa type contact binaries are formed from initially detached binary systems. The initially detached systems are mainly controlled by two mechanisms in the pre-contact phase, namely nuclear evolution of the initially massive star and angular momentum loss of the initially lighter star (Stȩpień, 2006, 2011; Yıldız, 2014). Considering the physical parameters listed in Table 4 and the method established by Yıldız & Doğan (2013), the initial mass of the two components in V502 Oph are calculated to be $M_{1i}=1.87(\pm 0.01)M_{\odot}$ and $M_{2i}=0.86(\pm 0.04)M_{\odot}$. The initial mass of the present primary one ($M_{2i}$) indicates that it is a late-type star, which has played a very important role in early angular momentum evolution of close binaries as magnetic braking is supposed to be the main mechanism for such stars with convective envelope. The mass loss of the initially massive star is calculated to be $1.41M_{\odot}$ and the mass gained by the present massive star is $0.49M_{\odot}$. About one-third of mass lost by the initially massive star is transferred to the present massive star.

The $O-C$ analysis shows that the orbital period of V502 Oph is decreasing at a rate of $dP/dt=-1.69\times{10^{-7}}day\cdot year^{-1}$. It may be caused by the mass transfer from the more massive component to the less massive one. By assuming a conservative mass transfer case and using the well-known equation

[TABLE]

the mass transfer is determined to be $\frac{dM_{2}}{dt}=-3.01\times{10^{-8}}M_{\odot}/year$. However, the real mass transfer rate may be quite different while the contribution of angular momentum loss (AML) is considered.

As shown in Fig. 2, after the downward parabolic variation in the $(O-C)_{1}$ curve is subtracted, the $(O-C)_{2}$ residuals suggest that there is a cyclic variation with a period of 18.7 years. It is due to the the light-travel time effect that caused by the gravitational influence of a tertiary component. By assuming a circular orbit, the projected radius ($a^{\prime}_{12}\sin i^{\prime}$) of the orbit that the eclipsing binary rotates around the barycenter of the triple system is calculated with the following equation,

[TABLE]

where $A_{3}$ is the amplitude of the periodic change and $c$ is the speed of light. The values is calculated to be $a^{\prime}_{12}\sin i^{\prime}=0.69(\pm 0.01)AU$. The mass function and the mass of the tertiary companion are computed with the following equation,

[TABLE]

where $G$ is the gravitational constant and $P_{3}$ is the period that the tertiary component orbits around V502 Oph. The mass function are calculated to be $f(m)=0.00096M_{\odot}$. The $i^{\prime}$-$M_{3}$ and $i^{\prime}$-$a_{3}$ diagram are displayed in Fig. 3. The red star in Fig. 3 are the values of the tertiary component when the orbits of the three components are coplanar.

During the photometric solutions, the third light was also included as an adjustable parameters, and our photometric solutions confirm the existence of the third light ($l_{3}$) in the binary system. Based on the amounts of the third light listed in Table 3, the $B-V$ color of the light source is calculated to be 0.61, which may correspond to be a G1 type main sequence star with its mass to be nearly $1.0M_{\odot}$. As shown in Fig. 4, the semi-major axes of the tertiary component is less than 8.11 AU and its mass is larger than $0.16M_{\odot}$. The tertiary component confirmed by the $O-C$ curve analysis is a stellar type component and it can not be outside the measured area on the CCD image considering the distance to V502 Oph of 86($\pm 8$) pc determined by Hipparchos (ESA, 1997). Therefore, the tertiary detected by the $O-C$ curve analysis may possibly be the third light source. In this case, the orbital inclination of the tertiary component will less than 20 degrees in Fig. 3.

Although tertiary components are quite common in W UMa systems, the formation theory of triple systems and the effects of the tertiary component on the close binary systems are not clear. The tertiary component around V502 Oph may accelerate the orbital evolution of the binary system by removing angular momentum from it (Qian et al., 2013). As for V502 Oph, its close-in companion has been confirmed and comprehensively researched through light curves’ solutions, $O-C$ curve analysis and spectroscopic observations, which make it an important target for testing formation theories of W UMa-type binaries, and dynamical interaction and angular momentum evolution of multiple systems.

This work is supported by the Chinese Natural Science Foundation (Grant No. 11133007, 11325315, 11203066 and No. 11403095), the Strategic Priority Research Program “The Emergence of Cosmological Structure” of the Chinese Academy of Sciences (Grant No. XDB09010202) and the Science Foundation of Yunnan Province (Grant No. 2012HC011 and 2014FB187). We thank Koju, Vijay and Beaky, Matthew M. for the high accuracy light curves’ data they observed using the 20-cm Meade LX200GPS telescope at the Truman State University Observatory in Kirksville, Missouri.