APSIDAL MOTIONS OF 27 BINARY SYSTEMS IN THE SMALL MAGELLANIC CLOUD

Apsidal motion in eccentric eclipsing binaries arises mainly from the combined effect of the tidal and rotational distortions in the shape of the stars. The distortions are described as a function of the internal structure of the component stars (Kopal 1978, p. 524). Such apsidal motion can be measured by long-term monitoring observations of the changing positions of eclipses. A good summary of the history of apsidal motion study in close binaries was presented by Giménez (2007).

During the last several decades, the EROS, Optical Gravitational Lensing Experiment (OGLE), and MACHO projects have obtained tens of thousands of light curves of eclipsing binaries in the Magellanic Clouds. Investigation of large samples of extragalactic binaries provides clues about star formation rates and the influence of chemical abundance on stellar evolution and structure (Guinan et al. 2007). In particular, apsidal motion studies of eccentric eclipsing binaries give us useful information on the interior structures of massive stars. In our galaxy, a few hundred apsidal motion eclipsing binaries are known (Bulut & Demircan 2007); however, apsidal motion periods have only been estimated for 13 eclipsing binaries in the Magellanic Clouds.

North et al. (2010) presented the apsidal motion elements and internal structure constants of five eccentric binaries in the SMC using OGLE-II data and spectroscopic observations obtained from the ESO FLAMES facility at the Very Large Telescope (VLT). Zasche & Wolf (2013) presented apsidal motions of five binaries in the LMC. Hong et al. (2014) used observation data and archive data obtained from EROS, MACHO, OGLE-II, and OGLE-III projects to discover and analyze the apsidal periods of three LMC detached binaries using light curve analysis and eclipse timing analysis.

The main aim of this paper is to provide the light curve solutions and accurate apsidal motion elements of 27 eccentric binary stars that were discovered in the large photometric survey data of MACHO and OGLE. In Section 2, we briefly describe the observational data and analysis method of the light curves and eclipse timing diagrams. Section 3 presents the results of the light curve solutions and apsidal motion elements. Finally, Section 4 presents a summary and discussion.

All data used here were obtained from MACHO (1992–1999; Faccioli et al. 2007), OGLE-II (1997–2000; Wyrzykowski et al. 2004), and OGLE-III (2001–2009; Pawlak et al. 2013) with one meter class telescopes. The OGLE-II and OGLE-III survey covered 2.4 and 14 square degrees, respectively, in the SMC (Udalski et al. 1997; Szymański 2005; Udalski et al. 2008). The 32 eccentric binary systems with apsidal motion were listed in the OGLE-III catalog of eclipsing binaries detected in the SMC by Pawlak et al. (2013). The apsidal motion binaries appear to have eccentric orbits with a noticeable apsidal motion in their light curves. Of those systems, we selected the 27 binary systems whose light curves showed relatively small scatter. The basic parameters for these stars are listed in Table 1, in which the coordinates, I, V, and V–I are taken from the OGLE-III eclipsing binaries cataloged by Pawlak et al. (2013).

To find photometric solutions, we have analyzed the VI light curves of the 27 apsidal binary systems obtained from OGLE-III observations using the iteration method applied in the papers of Kang et al. (2012, 2013). The iteration method is a modification of the Wilson & Devinney (WD) differential correction code (Wilson & Devinney 1971) for a huge number of eclipsing binaries. In order to estimate the initial input temperature of the primary component, we derived the color excess using the Magellanic Cloud reddening maps published by Haschke et al. (2011). We then added the mean foreground galactic reddening of E(B–V) = 0.037 in the direction of the SMC derived by Schlegel et al. (1998); we multiplied this value by the factor of 1.35 given by Schlafly & Finkbeiner (2011) to obtain the E(V–I) for each system. The final initial temperature was estimated from $V-I$₀ using the relation between the $V-I$ color index and the effective temperature determined by Worthey & Lee (2011). The color excesses of these stars are listed in Table 2. The statistical error and an additional systematic error for each estimated reddening were assigned the value of 0.02 mag.

Initial logarithmic limb-darkening coefficients were interpolated from the values in van Hamme (1993). The gravity-darkening exponents and bolometric albedos were assumed to be g₁ = g₂ = 1 and A₁ = A₂ = 1 (von Zeipel 1924) because both primary and secondary stars should have radiative envelopes. It is difficult to obtain the mass ratio, q = M₂/M₁, from the analysis of the light curves in detached eclipsing binaries (Wyithe & Wilson 2001). Therefore the mass ratio were assumed to be q = M₂/M₁ = 1.0 for all of the selected systems.

In the WD runs, we used mode 2 (detached system) and adjusted the temperature of the secondary component (T₂), the surface potentials (Ω_1,2), the orbital inclination (i), the eccentricity (e), the longitude of periastron (ω), the rate of periastron advance ($\dot{\omega }$), the epoch (T₀), the orbital period (P), and the luminosity of the primary star (L₁). The lowest values of χ² were also checked according to the fitness. From the light curve solutions determined using the WD, we classified all systems as detached systems because the primary and secondary components fill their Roche lobes to levels of about 20–80%. The quality of the light curves was not sufficient for the detection of any third light. Nevertheless, we tested for the possible presence of a third light (l₃) in the light curves of all systems using the differences of χ² between the third light and the non-third light models. We only found a large contribution of a third light in the VI light curves of OGLE-SMC-ECL-0888, 1298, 1927, 2225, 2251, and 5233. The best solutions for these systems revealed l₃ to contribute about 24–68% to the total luminosity of each system. The light curve solutions and third light contributions are listed in Table 3; uncertainties were determined using the Differential Corrections subroutine. The observations and the theoretical models are presented in Figures 1–3.

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Table 3.  Orbital Parameters

Object ID	$T_{1}^{{\rm a}}$	T2	e	ω	$\dot{\omega }$	i	Ω1	Ω2	${{({{L}_{2}}/{{L}_{1}})}_{\,V}}$	${{L}_{2}}/{{L}_{1}}{{)}_{I}}$	r1 (volume)c	r2 (volume)c	${\Sigma }$ W $O-C$2
(OGLE-)				(deg)	(deg yr−1)	(deg)			${{l}_{3,V}}^{{\rm b}}$	${{l}_{3,I}}^{{\rm b}}$
SMC-ECL-0781	15593	11643 (589)	0.343 (4)	266 (1)	2.73 (7)	84.2 (1)	8.141 (105)	7.567 (74)	0.613 (14)	0.718 (14)	0.1521 (25)	0.1669 (21)	0.1714
SMC-ECL-0888	33189	27152 (1233)	0.137 (8)	138 (3)	10.82 (79)	84.9 (5)	5.957 (57)	15.023 (326)	0.087 (49)	0.089 (31)	0.2109 (25)	0.0722 (17)	0.0541
	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.258 (46)	0.243 (24)	⋯	⋯	⋯
SMC-ECL-1297	17640	12750 (659)	0.101 (8)	339 (7)	13.1 (1.5)	76.0 (2)	5.727 (89)	5.949 (105)	0.453 (24)	0.527 (23)	0.2194 (44)	0.2090 (47)	0.2231
SMC-ECL-1298	11687	9553 (649)	0.209 (4)	190 (3)	14.40 (56)	76.8 (5)	6.508 (169)	6.409 (141)	0.442 (48)	0.453 (33)	0.1926 (64)	0.1965 (56)	0.2198
	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.321 (60)	0.293 (46)	⋯	⋯	⋯
SMC-ECL-1407	16932	17563 (795)	0.151 (8)	145 (4)	7.32 (65)	75.2 (2)	6.436 (148)	6.026 (110)	1.268 (32)	1.247 (30)	0.1919 (56)	0.2087 (49)	0.1973
SMC-ECL-1927	12081	12210 (518)	0.072 (2)	217 (5)	22.54 (97)	79.2 (1.1)	6.029 (182)	5.365 (108)	1.024 (31)	1.151 (18)	0.2040 (78)	0.2370 (63)	0.2321
	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.484 (51)	0.679 (30)	⋯	⋯	⋯
SMC-ECL-2186	29853	27748 (792)	0.237 (3)	103 (1)	2.66 (5)	86.3 (1)	8.036 (56)	7.711 (46)	0.988 (8)	1.004 (7)	0.1494 (13)	0.1571 (12)	0.1472
SMC-ECL-2194	26510	25149 (472)	0.041 (1)	131 (2)	50.57 (42)	83.5 (1)	4.826(46)	4.514 (37)	0.916 (10)	0.927 (10)	0.2595 (32)	0.2597 (30)	0.1583
SMC-ECL-2225	14436	11948 (540)	0.163 (4)	349 (4)	10.12 (60)	88.5 (1.1)	6.570 (74)	18.62 (3212)	0.065 (43)	0.045 (27)	0.1876 (27)	0.0574 (11)	0.2243
	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.280 (44)	0.289 (27)	⋯	⋯	⋯
SMC-ECL-2251	16129	19452 (884)	0.326 (7)	110 (1)	5.74 (22)	84.1 (7)	6.581 (76)	11.955 (277)	0.134 (35)	0.155 (18)	0.1993 (32)	0.0956 (25)	0.1916
	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.407 (39)	0.484 (20)	⋯	⋯	⋯
SMC-ECL-2524	19564	17428 (370)	0.255 (2)	297 (1)	8.25 (10)	84.5 (3)	7.353 (48)	7.176 (41)	0.855 (8)	0.890 (7)	0.1676 (14)	0.1728 (13)	0.1802
SMC-ECL-2663	16387	14882 (967)	0.488 (1)	239 (1)	1.83 (8)	87.7 (1)	10.895 (203)	10.832 (196)	0.832 (19)	0.866 (20)	0.1123 (26)	0.1131 (25)	0.4365
SMC-ECL-2954	13773	13735 (1290)	0.278 (15)	303 (3)	4.01 (38)	88.4 (1.1)	6.983 (148)	8.660 (305)	0.585 (17)	0.587 (16)	0.1804 (49)	0.1380 (59)	2.8884
SMC-ECL-3407	15096	12328 (928)	0.302 (11)	271 (1)	3.49 (24)	83.3 (3)	7.018 (109)	9.724 (328)	0.285 (14)	0.316 (15)	0.1809 (37)	0.1209 (48)	0.3684
SMC-ECL-3825	12464	13080 (1094)	0.073 (6)	180 (13)	17.2 (2.4)	82.1 (8)	5.166 (85)	8.707 (356)	0.316 (24)	0.307 (22)	0.2492 (55)	0.1314 (62)	1.2264
SMC-ECL-3828	10176	7015 (880)	0.190 (5)	290 (2)	14.48 (89)	87.5 (1.1)	5.703 (82)	12.151 (243)	0.048 (5)	0.070 (4)	0.2276 (44)	0.0917 (20)	0.4764
SMC-ECL-3905	17161	21644 (710)	0.270 (5)	146 (2)	6.39 (24)	83.4 (2)	8.542 (163)	7.474 (121)	2.125 (16)	1.967 (16)	0.1400 (32)	0.1650 (34)	0.3186
SMC-ECL-4651	20114	15945 (440)	0.158 (3)	342 (3)	7.20 (37)	81.2 (2)	9.557 (177)	5.667 (37)	2.268 (13)	2.472 (12)	0.1197 (26)	0.2268 (20)	0.1208
SMC-ECL-4686	15000d	13809 (370)	0.081 (9)	148 (9)	11.9 (1.8)	79.0 (2)	6.243 (125)	6.655 (143)	0.742 (27)	2.472 (5)	0.1957 (49)	0.1808 (47)	0.2500
SMC-ECL-4711	18829	16696 (400)	0.423 (1)	324 (1)	1.43 (4)	86.6 (1)	9.094 (66)	9.370 (79)	0.736 (8)	0.770 (8)	0.1367 (13)	0.1317 (14)	0.1168
SMC-ECL-4805	17765	16481 (699)	0.161 (5)	319 (2)	11.46 (47)	85.5 (1)	6.551 (85)	6.495 (95)	0.880 (15)	0.908 (16)	0.1882 (31)	0.1902 (35)	0.2044
SMC-ECL-5233	21678	20525 (466)	0.209 (1)	272 (1)	13.45 (21)	86.1 (3)	7.741 (89)	12.462 (232)	0.347 (32)	0.193 (29)	0.1552 (22)	0.0894 (19)	0.1134
	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.325 (33)	0.334 (29)	⋯	⋯	⋯
SMC-ECL-5625	17702	16266 (797)	0.126 (9)	103 (2)	9.44 (91)	84.1 (3)	8.707 (154)	5.621 (51)	2.448 (7)	2.521 (7)	0.1326 (27)	0.2267 (27)	0.3590
SMC-ECL-5858	17103	12525 (711)	0.376 (3)	323 (1)	2.85 (13)	84.5 (2)	8.149 (110)	12.318 (291)	0.189 (7)	0.220 (7)	0.1539 (27)	0.0935 (26)	0.2207
SMC-ECL-5923	20955	16427 (528)	0.465 (1)	292 (1)	2.46 (5)	85.6 (1)	9.983(91)	10.396 (84)	0.572 (9)	0.623 (9)	0.1239 (14)	0.1178 (12)	0.0589
SMC-ECL-5977	16764	16126 (824)	0.250 (6)	257 (1)	6.34 (29)	86.7 (3)	6.966 (126)	7.887 (234)	0.681 (22)	0.692 (22)	0.1791 (41)	0.1534 (56)	0.5112
SMC-ECL-6072	29854	29675 (968)	0.393 (5)	97 (1)	3.72 (11)	81.1 (2)	6.797 (43)	9.929 (168)	0.379 (11)	0.379 (10)	0.1982 (18)	0.1212 (25)	0.0616

Notes. All quoted uncertainties of the photometric solutions were calculated from the Differential Corrections subroutine of the Wilson–Devinney Code. All of these estimations assumed a mass ratio of 1.0.

^a Fixed parameter.

^b Value at 0.25 phase.

^c Mean volume radius.

^d Assumed value.

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For the eclipse timing analysis, the time of minimum light was determined from the full light curve combined with the observation data, with intervals of 1 yr. Each light curve was analyzed using the iteration method. The light curves obtained from the MACHO, OGLE-II, and OGLE-III surveys are plotted with best-fit models in Figures 4–6.

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The apsidal motions of the selected systems were analyzed by both light curve analysis and eclipse timing analysis. The elements from the former were determined using the rate of periastron advance ($\dot{\omega }$) from the light curve solutions; those from the latter used all eclipse times obtained from the light curve analyses of MACHO, OGLE-II, and OGLE-III observations using the ephemeris-curve equation presented by Giménez & Bastero (1995). The individual primary and secondary minima are listed in Table 5. The $O-C$ eclipse timing diagrams of 27 binary systems with respect to the apsidal motion equation are plotted in Figures 7–9.

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We verified the results for apsidal motion using two different analysis methods. An accurate apsidal motion elements were measured by light curve analysis and eclipse timing analysis. For all systems, a strong correlation between the apsidal motion periods estimated using both types of analysis is shown. The resulting apsidal motion elements and their errors are provided in Table 4. The blue and red continuous lines in Figures 7–9 are the apsidal motion curves obtained from the light curve analysis and eclipse timing analysis of 27 eclipsing binaries.

Table 4.  Apsidal Motion Elements of 27 Eclipsing Binary Systems

Object ID	T0	Ps	Pa	ω0	$\dot{\omega }$	Ua	Ub	Uc
(OGLE-)	(+2450000)	(day)	(day)	(deg)	(deg yr−1)	(yr)	(yr)	(yr)
SMC-ECL-0781	5002.5654 (6)	3.299930 (10)	3.300156 (10)	266 (3)	2.73 (14)	131.9 (6.4)	131.9 (3.3)	131.9 (2.9)
SMC-ECL-0888	5001.8295 (5)	1.918341 (3)	1.918649 (3)	139 (5)	10.98 (68)	32.8 (1.9)	33.1 (2.3)	32.9 (1.5)
SMC-ECL-1297	5000.7329 (12)	1.569614 (37)	1.569854 (36)	337 (8)	12.77 (93)	28.2 (1.9)	27.6 (2.8)	28.0 (1.6)
SMC-ECL-1298	5000.1977 (8)	1.753216 (3)	1.753562 (3)	191 (5)	14.76 (44)	24.4 (7)	25.0 (9)	24.6 (6)
SMC-ECL-1407	5001.9505 (9)	2.100758 (3)	2.100990 (3)	144 (10)	6.92 (38)	52.0 (2.7)	49.2 (4.0)	51.1 (2.2)
SMC-ECL-1927	5001.5492 (8)	1.849169 (8)	1.849724 (8)	207 (4)	21.32 (88)	16.9 (7)	16.0 (7)	16.4 (5)
SMC-ECL-2186	5000.7808 (2)	3.291324 (1)	3.291547 (1)	103 (6)	2.70 (4)	133.3 (1.9)	135.3 (2.5)	134.1 (1.5)
SMC-ECL-2194	5001.0416 (3)	1.302943 (2)	1.303604 (2)	136 (3)	51.17 (36)	7.0 (1)	7.1 (1)	7.1 (0.1)
SMC-ECL-2225	5000.8083 (6)	1.491723 (4)	1.491911 (4)	355 (7)	11.11 (71)	32.4 (1.9)	35.6 (2.0)	34.0 (1.4)
SMC-ECL-2251	5001.9848 (12)	2.336046 (9)	2.336283 (9)	109 (8)	5.70 (15)	63.2 (1.6)	62.7 (2.3)	63.0 (1.3)
SMC-ECL-2524	5002.0635 (2)	2.169241 (1)	2.169536 (1)	297 (2)	8.26 (7)	43.6 (4)	43.6 (5)	43.6 (3)
SMC-ECL-2663	5001.8519 (6)	3.640218 (2)	3.640395 (2)	239 (3)	1.75 (5)	205.7 (5.7)	196.7 (8.2)	202.8 (4.7)
SMC-ECL-2954	5000.4638 (14)	1.519361 (3)	1.519434 (3)	303 (4)	4.15 (21)	86.7 (4.2)	89.8 (7.8)	87.4 (3.7)
SMC-ECL-3407	5001.7037 (9)	2.409294 (2)	2.409446 (2)	271 (4)	3.45 (17)	104.3 (4.9)	103.2 (6.6)	103.9 (3.9)
SMC-ECL-3825	5000.5006 (11)	1.210991 (20)	1.211201 (20)	187 (15)	18.86 (1.94)	19.1 (1.8)	20.9 (2.6)	19.7 (1.5)
SMC-ECL-3828	5000.8594 (11)	1.570171 (2)	1.570444 (2)	290 (6)	14.55 (32)	24.7 (5)	24.9 (1.4)	24.8 (5)
SMC-ECL-3905	5000.1850 (27)	2.333636 (16)	2.333900 (16)	146 (3)	6.36 (31)	56.6 (2.6)	56.3 (2.0)	56.4 (1.6)
SMC-ECL-4651	5001.8460 (9)	2.437438 (11)	2.437757 (11)	342 (5)	7.05 (32)	51.1 (2.2)	50.0 (2.4)	50.6 (1.6)
SMC-ECL-4686	5001.4466 (11)	1.982344 (66)	1.982708 (65)	149 (6)	12.16 (1.30)	29.6 (2.9)	30.2 (4.0)	29.8 (2.3)
SMC-ECL-4711	5004.7179 (6)	5.029545 (3)	5.029818 (3)	324 (3)	1.42(5)	253.5 (8.6)	251.7 (6.9)	252.2 (5.4)
SMC-ECL-4805	5000.5295 (3)	1.861807 (3)	1.862111 (3)	319 (2)	11.55 (38)	31.2 (1.0)	31.4 (1.2)	31.3 (8)
SMC-ECL-5233	5004.4191 (10)	5.068369 (5)	5.071138 (5)	273 (8)	14.16 (17)	25.4 (0.3)	26.8 (0.4)	25.9 (0.2)
SMC-ECL-5625	5000.1767 (8)	1.714506 (25)	1.714715 (25)	103 (3)	9.35 (70)	38.5 (2.7)	38.1 (3.4)	38.3 (2.1)
SMC-ECL-5858	5002.2200 (10)	2.479043 (3)	2.479180 (3)	323 (4)	2.93 (12)	122.9 (4.8)	126.3 (5.5)	124.4 (3.6)
SMC-ECL-5923	5001.7139 (5)	4.024876 (2)	4.025179 (2)	292 (3)	2.46 (11)	146.3 (6.3)	146.3 (2.9)	146.3 (2.6)
SMC-ECL-5977	5000.9772 (7)	2.023014 (3)	2.023216 (3)	257 (4)	6.48 (22)	55.6 (1.8)	56.8 (2.5)	56.0 (1.5)
SMC-ECL-6072	5000.6635 (7)	3.001977 (2)	3.002287 (2)	97 (5)	4.52 (49)	101.2 (12.4)	92.8 (2.6)	93.2 (2.5)

Notes.

^a Determined from the eclipse timing analysis.

^b Determined from the light curve analysis.

^c Computed the weighted mean from U^a and U^b.

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3.1. System with a Period Less than 10 yr

3.2. Systems with Periods between 10 and 100 yr

3.3. Systems with Longer Periods

We first presented the light curve parameters and the apsidal motion elements for 27 apsidal motion binary systems in the SMC. All of the selected systems were studied here for the first time by eclipse timing analysis $O-C$. Figure 10 provides a histogram of the apsidal periods of these 27 binaries; also included is information on 91 detached binaries in our galaxy from the catalog of eccentric EBs by Bulut & Demircan (2007). The apsidal period distribution of the 27 binary systems is in the range of 7–255 yr, with most values falling in the category of 10–100 yr in the time interval log U = 0–5; the 91 eclipsing binaries in our galaxy are in the range of 20–50,000 yr, with the majority of examples falling in the category of 100–1000 yr. The binaries that exhibited very long periods of apsidal motion in the SMC do not appear in this figure because of the selection effect.

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For OGLE-SMC-ECL-0888, 1298, 1927, 2194, 2225, 3825, 3828, and 5233, the apsidal motion periods were very accurately determined because more than half of the apsidal period is already covered by the observations. The newly discovered apsidal motion binary systems will provide possibilities for determining the structures and methods of evolution of massive stars outside our galaxy.

Figure 11 shows that most of the binary systems in our sample exhibit a correlation between the orbital period and its apsidal motion period. However, the orbital period of OGLE-SMC-ECL-5233 has a short apsidal period of 25.9 yr in spite of this being the longest orbital period, at 5.068 days, in our sample. For a detailed study of this system, we need more information from spectroscopic observations.

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We determined the basic stellar parameters of the selected systems by light curve analysis. However the analysis still has several problems. For all systems except OGLE-SMC-ECL-2194, the spectral type of the components and their mass ratio are not known at this time. Specifically, high-dispersion, high signal-to-noise ratio spectroscopic observations are needed to obtain the temperatures and the radial velocities of both components for these systems.

We are grateful to the MACHO and OGLE teams for making their excellent photometric data base publicly available. This work was supported by KASI (Korea Astronomy and Space Science Institute) grant 2015-1-850-04.