SHORT APSIDAL PERIOD OF THREE ECCENTRIC ECLIPSING BINARIES DISCOVERED IN THE LARGE MAGELLANIC CLOUD

Components of newborn close binary systems cannot be considered as point masses. Therefore, tidal torques are established on each component star of the binary system, the stellar rotation axis becomes aligned perpendicular to the orbital plane of the binary system, and the axial-rotational period becomes equal to the orbital period; finally, the system develops a circular orbit and becomes synchronized. The enforced synchronous rotation of both component stars also causes some rotational flattening. These processes are called tidal synchronization. There exist several types of perturbations that can lead to rotation of apsides, such as mutual tidal distortion of the components, distortion of the components due to axial rotation, relativistic effects, presence of a third body, and recession due to resisting circumbinary medium (Landin et al. 2009). The sum of the perturbations produced by each component can be seen in observations of the minimum times of eclipsing binaries (Batten 1973). The motion of the elliptical orbit apsis is a direct consequence of the finite size of the binary star components. It is generally accepted that the apsidal motion rate depends on the internal structure of each component star of the binary system. Studying the apsidal motion provides an important observational test of the theoretical models of stellar structure and evolution (Claret & Willems 2002).

The eclipsing binaries in the Magellanic Clouds are very important as astrophysical laboratories for studying the structure and evolution of massive stars with low metallicity (Ribas 2004). North et al. (2010) presented the apsidal periods of the five eclipsing binary systems located in the Small Magellanic Cloud. They analyzed the full I light curves constructed using the OGLE-II survey data obtained during 4 yr in order to obtain the period of apsidal motion. Michalska & Pigulski (2005) found 14 systems showing apsidal motions in the Large Magellanic Cloud (LMC). They presented times of minimum light obtained from the EROS, MACHO, and OGLE-II data. Zasche & Wolf (2013) analyzed apsidal motions of five LMC detached eclipsing binaries using eclipse timings.

In this paper, three eclipsing binary stars located in the LMC are analyzed for apsidal motion using survey data and our observations. The data and our selection methods are described in Section 2. Section 3 presents the light curve analysis with the third light effect. The apsidal motions are determined from detailed analyses of both the light curves and eclipse timings in Section 4. Finally, Section 5 presents a summary and discussion of this work.

2.1. The Data

2.2. Selections

For the photometric solution, BV and I light curves of EROS 1018 were selected from both our observations in 2006–2007 and the OGLE-III observations in 2002–2009, respectively, and the VI light curves of EROS 1041 and EROS 1054 were selected from the OGLE-III observations in 2002–2009, respectively. The light curves were analyzed using the 2005 version of the Wilson & Devinney (WD) differential correction code (Wilson & Devinney 1971). In order to estimate the initial input temperature of the primary component, we first derived the color excess in the direction to each of our systems as follows. The color excess was determined by all sources within a 2 arcmin radius of our systems from the Magellanic Cloud reddening maps published by Haschke et al. (2011). We then added the mean foreground Galactic reddening E(B − V) of 0.075 in the direction of the LMC by Schlegel et al. (1998), and multiplied this value by a factor of 1.35 given by Schlafly & Finkbeiner (2011) to get an E(V − I) for each system. The color excesses of our targets are listed in Table 2. A statistical error and an additional systematic error for each estimate of the reddening were assigned as 0.02 mag, respectively. The final initial temperature was estimated from the intrinsic color index (V − I)₀ using the relation between V − I color index and effective temperature by Worthey & Lee (2011). The limb-darkening coefficients of the logarithmic law were interpolated from the values in van Hamme (1993). The gravity darkening exponents and bolometric albedos were adopted to be g₁ = g₂ = 1 and A₁ = A₂ = 1 (von Zeipel 1924), because each component should have radiative envelopes. The metallicity 〈[Fe/H]_LMC〉 of the LMC was adopted to be −0.47 using the metallicity of eight eclipsing binaries published by Pietrzyński et al. (2013). The initial parameters were adopted from preliminary solutions. The mass ratios were assumed to be q = m₂/m₁ = 1 for EROS 1018 and EROS 1041, respectively. The mass ratio of EROS 1054 was adopted as q = 0.6 when the light curve analysis gives the minimum χ².

Then we adjusted the temperature of the secondary component (T₂), surface potentials (Ω_{1, 2}), orbital inclination (i), eccentricity (e), longitude of periastron (ω), the rate of periastron advance ($\dot{\omega }$), epoch (T₀), orbital period (P), and luminosity of the primary star (L₁) using the iteration analysis method applied in the paper of Kang et al. (2012, 2013). We also checked lowest value of the χ² by fitness. The solutions without third light for each system are listed as Model 1 in Table 3. The uncertainties were calculated with the differential corrections subroutine. The observations and the theoretical models are presented in Figure 1.

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Table 3. Photometric Solutions of Three Eclipsing Binary Systems

	EROS 1018				ER0S 1041				EROS 1054
Parameter	Model 1		Model 2		Model 1		Model 2		Model 1		Model 2
	(without Third Light)		(with Third Light)		(without Third Light)		(with Third Light)		(without Third Light)		(with Third Light)
	Star 1	Star 2	Star 1	Star 2	Star 1	Star 2	Star 1	Star 2	Star 1	Star 2	Star 1	Star 2
T0 (HJD)	2,453,537.58335 (50)		2,453,537.58343 (48)		2,453,565.07895 (47)		2,453,565.07889 (47)		2,453,540.2241 (93)		2,453,540.22720 (91)
P (day)	1.4419670 (8)		1.4419670 (8)		2.688219 (1)		2.688219 (1)		3.6282776 (37)		3.6282774 (38)
q	1.0		1.0		1.0		1.0		0.6		0.6
e	0.021 (2)		0.023 (2)		0.105 (3)		0.107 (4)		0.036 (3)		0.037 (3)
ω (deg)	176 (5)		173 (4)		44 (2)		46 (2)		141 (5)		136 (5)
$\dot{\omega }$ (deg yr−1)	16.7 (1.8)		16.8 (1.8)		4.9 (3)		4.7 (3)		5.3 (1.1)		5.1 (1.1)
i (deg)	74.1 (2)		79.5 (1.0)		85.9 (1)		86.9 (4)		85.9 (6)		85.8 (5)
T (K)	18250a	18710 (720)	18250a	18780 (710)	16680a	16280 (350)	16680a	16150 (350)	20710a	17020 (920)	20710a	17060 (1210)
Ω	5.324 (77)	5.322 (72)	5.222 (76)	5.710 (112)	5.931 (68)	5.786 (54)	6.003 (78)	5.686 (53)	3.940 (19)	6.641 (41)	3.916 (21)	6.375 (80)
Xbola, Ybola	0.783, 0.093	0.791, 0.103	0.783, 0.093	0.791, 0.102	0.772, 0.073	0.770, 0.069	0.772, 0.073	0.769, 0.069	0.744, 0.074	0.759, 0.063	0.744, 0.074	0.769, 0.063
xBa, yBa	0.509, 0.277	0.499, 0.273	0.509, 0.277	0.499, 0.274
xVa, yVa	0.432, 0.231	0.424, 0.227	0.432, 0.231	0.425, 0.227	0.457, 0.242	0.462, 0.244	0.457, 0.242	0.464, 0.245	0.447, 0.244	0.465, 0.247	0.447, 0.244	0.465, 0.246
xIa, yIa	0.280, 0.156	0.273, 0.151	0.280, 0.156	0.273, 0.151	0.296, 0.165	0.300, 0.167	0.296, 0.165	0.301, 0.167	0.306, 0.184	0.308, 0.175	0.306, 0.184	0.307, 0.175
l/(l1 + l2 + l3)B	0.486 (21)	0.514	0.341 (30)	0.290
l/(l1 + l2 + l3)V	0.488 (20)	0.512	0.299 (27)	0.252	0.496 (13)	0.504	0.447 (13)	0.480	0.917 (13)	0.083	0.833 (24)	0.083
l/(l1 + l2 + l3)I	0.490 (20)	0.510	0.339 (27)	0.283	0.494 (13)	0.506	0.459 (13)	0.500	0.911 (12)	0.089	0.848 (19)	0.090
l3Bb	0.0a		0.368 (53)
l3Vb	0.0a		0.448 (48)		0.0a		0.070 (20)		0.0a		0.084 (23)
l3Ib	0.0a		0.377 (48)		0.0a		0.038 (19)		0.0a		0.062 (18)
r (pole)	0.2310 (41)	0.2311 (38)	0.2365 (42)	0.2123 (50)	0.2065 (29)	0.2128 (24)	0.2036 (32)	0.2175 (25)	0.2989 (17)	0.1109 (8)	0.3011 (19)	0.1166 (18)
r (point)	0.2402 (48)	0.2403 (45)	0.2469 (51)	0.2187 (57)	0.2134 (33)	0.2208 (28)	0.2102 (37)	0.2263 (29)	0.3190 (22)	0.1116 (9)	0.3220 (25)	0.1175 (19)
r (side)	0.2339 (43)	0.2340 (40)	0.2398 (45)	0.2144 (52)	0.2083 (30)	0.2149 (25)	0.2054 (33)	0.2198 (26)	0.3056 (18)	0.1111 (9)	0.3080 (21)	0.1169 (18)
r (back)	0.2383 (46)	0.2384 (43)	0.2446 (48)	0.2175 (55)	0.2119 (32)	0.2190 (27)	0.2088 (35)	0.2243 (28)	0.3135 (20)	0.1115 (9)	0.3162 (23)	0.1174 (18)
r (volume)c	0.2358	0.2359	0.2420	0.2157	0.2100	0.2169	0.2070	0.2220	0.3093	0.1113	0.3118	0.1171
ΣW(O − C)2	0.1135		0.0967		0.1248		0.1215		0.0678		0.0674
Absolute dimensions by assuming the distance to the LMC
a (R☉)	14.4 (9)				15.7 (1.1)				25.0 (1.5)
Mass (M☉)	9.6 (2.0)	9.6 (2.0)			3.6 (8)	3.6 (8)			10.0 (1.9)	6.0 (1.1)
Radius (R☉)	3.4 (3)	3.4 (3)			3.6 (3)	3.5 (3)			7.8 (6)	2.9 (6)
log g (cgs)	4.35 (16)	4.45 (16)			3.96 (17)	3.91 (17)			3.65 (14)	4.31 (17)
Luminosity (M☉)	1140 (343)	1280 (389)			757 (206)	711 (195)			10136 (3426)	591 (242)
Mbol (mag)	−2.90 (29)	−3.01 (29)			−2.44 (26)	−2.39 (26)			−5.26 (32)	−2.22 (32)
Mv (mag)	−1.21 (23)	−1.26 (23)			−0.96 (22)	−0.98 (22)			−3.29 (23)	−0.68 (23)

Notes. All quoted uncertainties of the photometric solutions were calculated from the differential corrections subroutine of the Wilson–Devinney code. All of these estimations used the recommended values of fundamental parameters as published by Harmanec & Prša (2011). The errors of the absolute dimension were determined from the mean value of the line of sight depth for the LMC central bar region by Subramanian & Subramaniam (2009). ^aFixed parameter. ^bValue at 0.25 phase. ^cMean volume radius.

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The conventional method for determining the absolute dimensions is to measure the radial velocity of an orbit of a binary star using spectroscopic observations. Unfortunately, the spectroscopic observations were not available. Therefore, the semi-major-axis (a) was estimated using the luminosity of the binary system and the relative photometric parameters, by adopting the methods of Graczyk (2003). The semi-major-axis (a) of a binary orbit can be expressed as

$$\begin{equation} {a} = \left(\frac{\mbox{Luminosity of the binary system}}{4\pi \sigma \left(r_{1}^{2}T_{1}^{4}+r_{2}^{2}T_{2}^{4}\right)}\right)^{1/2}, \end{equation} \tag{ 1 }$$

where the luminosity of a binary system of Equation (1) is converted by the bolometric magnitude (M_{bol, system}) of the binary system and the bolometric magnitude (M_{bol, ☉} = 4.75) of the Sun. The bolometric magnitude (M_{bol, 1, 2}) of each component was determined from the absolute visual magnitude (M_{V, 1, 2}) by using our light ratios and assuming the mean distance modulus of the LMC of V − M_V = 18.493 (Pietrzyński et al. 2013) and the interstellar extinction of A_V = 3.1E(B − V), and the bolometric corrections table by Flower (1996).

From the denominator of Equation (1), "σ" is the Stefan–Boltzmann constant, and the fractional radii (r_p and r_s) of the primary and secondary components were obtained from the light curve solutions, respectively. As a result, we obtained the semi-major-axis (a) from Equation (1). The resulting absolute dimensions are listed as Model 1 in Table 3. Note, the systematic uncertainties of absolute dimensions came from the mean value of the line of sight depth (3.95 ± 1.42 kpc) for the LMC central bar region by Subramanian & Subramaniam (2009).

We tested the possible presence of a third light in our light curves, including the third light parameter (l₃) as an adjustable variable. We found a large contribution of a third light in the BVI light curves of EROS 1018, and detected small contributions in the VI light curves of EROS 1041 and EROS 1054. Note, EROS 1041 and 1054 have a similar value of χ² between Model 1 without third light and Model 2 with third light, respectively. The results are given as Model 2 of Table 3 and plotted as red lines in Figure 1. Figure 1 shows the considerable difference between Model 1 and Model 2 of EROS 1018 as compared with the other systems.

The analysis can be divided into "light curve analysis" and "eclipse timing analysis (O − C)." During the "light curve analysis," we used Model 2 of the light curve solutions in Section 3. In the "eclipse timing analysis (O − C)," we used all of the new times of minimum obtained from the light curve analyses of EROS, MACHO, OGLE-II, OGLE-III, and our observations. Then we confirmed the final period of apsidal motion.

4.1. Light Curve Analysis

4.2. Eclipse Timing Analysis

The I light curves were constructed for three binary systems using the EROS, MACHO, OGLE-II, OGLE-III, and our observations taken between 1991 and 2009 in Figure 2. The light curves were analyzed with the modified WD code. We only adjusted the epoch (T₀) and longitude of periastron (ω₀) to find the times of minimum light from each of the light curves with the third light. Then we derived the new times of primary and secondary minima from the theoretical models, and their error with each of the light curve solutions.

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The apsidal motions of three eccentric binaries were investigated using the mean of the eclipse timing analysis (O − C). The apsidal motion can be described using five apsidal elements: the zero epoch (T₀), sidereal period (P_s), anomalistic period (P_a), rate of periastron advance in degree per cycle ($\dot{\omega }$), and eccentricity (e). In order to determine the elements of the apsidal motion of the three binary systems, an ephemeris model was used with the ephemeris-curve equation presented by Giménez & Bastero (1995); this equation includes sixth order terms of eccentricity. The Levenberg–Marquardt minimization algorithm (Press et al. 1992) was applied to solve the five independent variables. The initial input values for the algorithm are important for its application. First, the fixed values of an orbital eccentricity and orbital inclination were derived by analyzing the light curve solutions with third light in Section 3. The initial input values of sidereal period (P_s), anomalistic period (P_a), and the zero epoch (T₀) were adopted from the catalog of eclipsing binary systems in the LMC by Graczyk et al. (2011).

4.2.1. EROS 1018

The detached eclipsing binary EROS 1018 (MACHO 80.6345.4601, 2MASS J05194962-6924579; V = 16.78) is an eccentric system with a period of about 1.44 days. EROS 1018 was discovered to be a variable star by Grison et al. (1995). The best solution for this system was obtained with a third light contribution, included in Table 3 and Figure 1. The solution in Model 2 reveals l₃ to contribute about 35%–44% to the total luminosity of this system.

Fifteen full light curves were also constructed for EROS 1018 using the EROS, OGLE-II, OGLE-III, and our observations (SEJONG) taken between 1991 and 2009. The system was not identified as an eclipsing binary in the MACHO database by Faccioli et al. (2007). The light curves obtained from EROS, MACHO, OGLE-II, and OGLE-III surveys are plotted with best fitting models in Figure 2. Thirty times of minimum light, including our measurements in the B band, were obtained for EROS 1018 and listed in Table 4. The analysis provided the following light elements:

$${\rm Min.\ I} = {\rm HJD} 2{,}453{,}537.5831(6) + 1.441969(6)E.$$

Table 4. List of Minimum Timings of EROS 1018

JD Hel.	Error	Cycle	Min	Source
(+2,400,000)	(day)
48649.3116	0.00177	−3390.0	I	EROS R
48688.2542	0.00172	−3363.0	I	EROS R
50495.0163	0.00087	−2110.0	I	OGLE-II
50805.0429	0.00235	−1895.0	I	OGLE-II
51195.8138	0.00202	−1624.0	I	OGLE-II
51553.4272	0.00191	−1376.0	I	OGLE-II
52251.3561	0.00191	−892.0	I	OGLE-III
52643.5682	0.00199	−620.0	I	OGLE-III
53008.3924	0.00142	−367.0	I	OGLE-III
53334.2709	0.00248	−141.0	I	OGLE-III
53719.2824	0.00228	126.0	I	OGLE-III
54052.3739	0.00220	357.0	I	OGLE-III
54131.6822	0.00013	412.0	I	SEJONG B
54460.4583	0.00319	640.0	I	OGLE-III
54822.3812	0.00139	891.0	I	OGLE-III
48650.0325	0.00177	−3389.5	II	EROS R
48688.9627	0.00172	−3362.5	II	EROS R
50495.7585	0.00087	−2109.5	II	OGLE-II
50805.7714	0.00235	−1894.5	II	OGLE-II
51196.5519	0.00202	−1623.5	II	OGLE-II
51554.1579	0.00191	−1375.5	II	OGLE-II
52252.0663	0.00191	−891.5	II	OGLE-III
52644.2750	0.00199	−619.5	II	OGLE-III
53009.0928	0.00142	−366.5	II	OGLE-III
53334.9781	0.00248	−140.5	II	OGLE-III
53719.9883	0.00228	126.5	II	OGLE-III
54053.0733	0.00220	357.5	II	OGLE-III
54132.3876	0.00009	412.5	II	SEJONG B
54461.1634	0.00319	640.5	II	OGLE-III
54823.0896	0.00139	891.5	II	OGLE-III

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Table 5. Apsidal Motion Elements of Three Eclipsing Binary Systems

Parameters	Unit	EROS 1018	EROS 1041	EROS 1054
		From the eclipse timing analysis
Epoch (T0)	HJD	2,453,537.5831 (6)	2,453,565.0781 (5)	2,453,540.2255 (9)
Sidereal period (Ps)	days	1.441969 (6)	2.688220 (4)	3.628281 (14)
Anomalistic period (Pa)	days	1.442314 (6)	2.688476 (4)	3.628694 (14)
Longitude of periastron (ω0)	deg	187 (14)	46 (3)	137 (18)
Rate of periastron advance ($\dot{\omega }$)	deg yr−1	21.78 (4.08)	4.62 (13)	4.13 (44)
Apsidal motion period (U)	yr	16.5 (3.8)	77.9 (3.3)	87.2 (10.4)
		From the light curve analysis
Apsidal motion period (U)	yr	21.5 (2.6)	76.6 (6.1)	73.5 (20.6)
		Adopted apsidal motion period (weighted mean)
Apsidal motion period (U)	yr	19.9 (2.2)	77.6 (2.9)	84.4 (9.3)

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The rate of periastron advance ($\dot{\omega }$) was determined to be 21.78 ± 4.08 deg yr⁻¹ using the eclipse timing analysis (O − C). A very short apsidal motion period (U) was also determined to be 16.5 ± 3.8 yr.

The red continuous lines represent the complete apsidal motion curves obtained from the eclipse timing analysis (O − C) of EROS 1018 in Figure 3. The final apsidal motion elements are given in the third column of Table 5.

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4.2.2. EROS 1041

The detached eclipsing binary EROS 1041 (MACHO 78.6826.298, OGLE J052224.82-693622.6, OGLE-LMC-ECL-13981; V = 17.05) is an early-type binary and has a period of about 2.69 days. The variability of EROS 1041 was discovered in the EROS observations (Grison et al. 1995).

Twenty-two full light curves were determined for EROS 1041 using the EROS, MACHO, OGLE-II, and OGLE-III observations between 1991 and 2009. The light curve was analyzed using the WD code. The light curves with the fitted models are plotted without the MACHO data between 1997 and 1999 in Figure 2. Forty-four times of minimum light were obtained from the models and are listed in Table 6.

Table 6. List of Minimum Timings of EROS 1041

JD Hel.	Error	Cycle	Min	Source
(+2,400,000)	(day)
48653.6151	0.00184	−1827.0	I	EROS R
48685.8700	0.00265	−1815.0	I	EROS R
48933.1851	0.00146	−1723.0	I	MACHO r
49368.6818	0.00220	−1561.0	I	MACHO r
49734.2830	0.00219	−1425.0	I	MACHO r
50099.8758	0.00175	−1289.0	I	MACHO r
50468.1632	0.00156	−1152.0	I	MACHO r
50831.0741	0.00175	−1017.0	I	MACHO r
51191.2921	0.00209	−883.0	I	MACHO r
51460.1172	0.00300	−783.0	I	MACHO r
50495.0412	0.00080	−1142.0	I	OGLE-II
50796.1206	0.00132	−1030.0	I	OGLE-II
51196.6643	0.00112	−881.0	I	OGLE-II
51551.5142	0.00141	−749.0	I	OGLE-II
52255.8383	0.00142	−487.0	I	OGLE-III
52640.2583	0.00117	−344.0	I	OGLE-III
53008.5479	0.00099	−207.0	I	OGLE-III
53339.2079	0.00185	−84.0	I	OGLE-III
53723.6226	0.00110	59.0	I	OGLE-III
54051.5955	0.00170	181.0	I	OGLE-III
54457.5214	0.00128	332.0	I	OGLE-III
54890.3320	0.00078	493.0	I	OGLE-III
48655.1344	0.00184	−1826.5	II	EROS R
48687.3928	0.00265	−1814.5	II	EROS R
49370.2019	0.00220	−1560.5	II	MACHO r
49735.8035	0.00219	−1424.5	II	MACHO r
48934.7097	0.00146	−1722.5	II	MACHO r
50101.3986	0.00175	−1288.5	II	MACHO r
50469.6875	0.00156	−1151.5	II	MACHO r
50832.5944	0.00175	−1016.5	II	MACHO r
51192.8166	0.00209	−882.5	II	MACHO r
51461.6342	0.00300	−782.5	II	MACHO r
50496.5639	0.00080	−1141.5	II	OGLE-II
50797.6424	0.00132	−1029.5	II	OGLE-II
51198.1868	0.00112	−880.5	II	OGLE-II
51553.0345	0.00141	−748.5	II	OGLE-II
52257.3425	0.00142	−486.5	II	OGLE-III
52641.7523	0.00117	−343.5	II	OGLE-III
53010.0381	0.00099	−206.5	II	OGLE-III
53340.6862	0.00185	−83.5	II	OGLE-III
53725.0921	0.00110	59.5	II	OGLE-III
54053.0512	0.00170	181.5	II	OGLE-III
54458.9647	0.00128	332.5	II	OGLE-III
54891.7608	0.00078	493.5	II	OGLE-III

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We obtained the values of orbital inclination (i) of 86.9 ± 0.4 deg and an orbital eccentricity (e) of 0.1072 ± 0.0038 from Model 2 of the light curve solutions in Section 3. The orbital inclination (i) and an orbital eccentricity (e) of this system were previously studied to be 84.1 deg and 0.108 by Michalska & Pigulski (2005). Our photometric solutions are very close to theirs.

A total of 44 times of minimum light from the EROS, MACHO, OGLE-II, and OGLE-III data were analyzed. The linear ephemeris was determined as follows:

$${\rm Min.\ I} = {\rm HJD} 2{,}453{,}565.0780(5) + 2.688220(4)E.$$

The O − C residuals for all times of minimum light with respect to the linear part of the apsidal motion equation are plotted in Figure 4. As one can see, only about 22% has been covered. The rate of periastron advance ($\dot{\omega }$) and the period of apsidal motion (U) were determined to be 4.62 ± 0.13 deg yr⁻¹ and 77.9 ± 3.3 yr, respectively. The computed apsidal motion elements and their errors are listed in Table 5.

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4.2.3. EROS 1054

The detached eclipsing binary EROS 1054 (2MASS J05164937-6932460; V = 15.42) is an early-type binary with a period of about 3.62 days. The variability of EROS 1054 was discovered in the EROS observations (Grison et al. 1995).

Thirteen full light curves were constructed for EROS 1054 using the EROS, OGLE-II, and OGLE-III survey observations taken between 1991 and 2009. The light curve observed in 2000 of the OGLE-II could not be used due to its few data points. The 13 light curves and theoretical curves are plotted in Figure 2. Twenty-six times of minimum light were determined for EROS 1054, and those minima are listed in Table 7. The linear light elements were obtained as below:

$${\rm Min.\ I} = {\rm HJD} 2{,}453{,}540.2246(9) + 3.628289(14)E.$$

Table 7. List of Minimum Timings of EROS 1054

JD Hel.	Error	Cycle	Min	Source
(+2,400,000)	(day)
48649.2991	0.00151	−1348.0	I	EROS R
48685.5871	0.00239	−1338.0	I	EROS R
50510.6217	0.00157	−835.0	I	OGLE-II
50797.2563	0.00191	−756.0	I	OGLE-II
51410.4350	0.00123	−587.0	I	OGLE-II
52255.8351	0.00252	−354.0	I	OGLE-III
52647.6964	0.00203	−246.0	I	OGLE-III
53010.5281	0.00265	−146.0	I	OGLE-III
53337.0668	0.00675	−56.0	I	OGLE-III
53718.0394	0.00230	49.0	I	OGLE-III
54055.4820	0.00346	142.0	I	OGLE-III
54465.4780	0.00232	255.0	I	OGLE-III
54831.9338	0.00259	356.0	I	OGLE-III
48651.1303	0.00151	−1347.5	II	EROS R
48687.3955	0.00239	−1337.5	II	EROS R
50512.4176	0.00157	−834.5	II	OGLE-II
50799.0477	0.00191	−755.5	II	OGLE-II
51412.2153	0.00123	−586.5	II	OGLE-II
52257.6104	0.00252	−353.5	II	OGLE-III
52649.4566	0.00203	−245.5	II	OGLE-III
53012.2814	0.00265	−145.5	II	OGLE-III
53338.8347	0.00675	−55.5	II	OGLE-III
53719.7887	0.00230	49.5	II	OGLE-III
54057.2242	0.00346	142.5	II	OGLE-III
54467.2135	0.00232	255.5	II	OGLE-III
54833.6781	0.00259	356.5	II	OGLE-III

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The O − C diagram was plotted using the times of minimum in Figure 5. As one can see, only about 30% has been covered. The rate of periastron advance ($\dot{\omega }$) was obtained as 4.13 ± 0.44 deg yr⁻¹. The period of apsidal motion (U) was obtained as 87.2 ± 10.4 yr. The resulting apsidal motion elements are provided in the fifth column of Table 5.

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• 1.  
  We made 14 full light curves of 79 eclipsing binaries discovered in the LMC by Grison et al. (1995) using EROS, OGLE II, and OGLE-III survey observations. It was confirmed that 19 of 79 eclipsing binaries exhibited apsidal motions and period variations. We selected three binary systems, EROS 1018, 1041, and 1054, whose times of minimum light covered their apsidal motion period relatively well.
• 2.  
  The apsidal motion in all three eccentric systems was studied by the detailed analyses of light curves and eclipse timings. All of the new times of minimum were analyzed by the method of Giménez & Bastero (1995). The rate of periastron advance ($\dot{\omega }$) and the period of apsidal motion U were determined with the five independent variables (T₀, P_s, P_a, e, ω₀).

We provide basic stellar parameters of three early-type binary systems with apsidal periods of less than 100 yr—EROS 1018, EROS 1041, and EROS 1054—from the light curve and apsidal motion analyses. The apsidal periods are derived from light curve analysis and eclipse timing analysis. The weighted mean values of apsidal periods are U = 19.9 ± 2 yr for EROS 1018, U = 77.6 ± 2.9 yr for EROS 1041, and U = 84.4 ± 9.3 yr for EROS 1054, respectively.

The light contribution of third body is prominent for EROS 1018, while for EROS 1041 and EROS 1054 it is little. However, it is too difficult to detect the third light in the light curves with large scatter. All of the systems need photometric and spectroscopic observations to confirm the third light.

The authors wish to thank Dr. J. W. Lee for his careful reading and comments. We thank the EROS, MACHO, and OGLE teams for all of the observations. This research was supported in part by the General Researcher Program 2011-0010696 and Mid-career Researcher Program 2011-0028001, respectively, through the National Research Foundation (NRF) of Korea funded by the Ministry of Education.