APSIDAL MOTION AND LIGHT A CURVE SOLUTION FOR EIGHTEEN SMC ECCENTRIC ECLIPSING BINARIES

Draft version December 3, 214 Preprint typeset using L A TEX style emulateapj v. 3/7/7 APSIDAL MOTION AND LIGHT A CURVE SOLUTION FOR EIGHTEEN SMC ECCENTRIC ECLIPSING BINARIES P. Zasche 1 M. Wolf 1 J. Vraštil 1 J. Liška 2 M. Skarka 2 M. Zejda 2 1 Astronomical Institute, Charles University in Prague, Faculty of Mathematics and Physics, CZ-18 Praha 8, V Holešovičkách 2, Czech Republic 2 Department of Theoretical Physics and Astrophysics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic (Dated: Received December 3, 214) Draft version December 3, 214 arxiv:1412.95v1 [astro-ph.sr] 2 Dec 214 ABSTRACT Aims: The Danish 1.54-meter telescope at the La Silla observatory was used for photometric monitoring of selected eccentric eclipsing binaries located in the Small Magellanic Cloud. The new times of minima were derived for these systems, which are needed for accurate determination of the apsidal motion. Moreover, many new times of minima were derived from the photometric databases OGLE and MACHO. Eighteen early-type eccentric-orbit eclipsing binaries were studied. Methods: Their O C diagrams of minima timings were analysed and the parameters of the apsidal motion were obtained. The light curves of these eighteen binaries were analysed using the program PHOEBE, giving the light curve parameters. For several systems the additional third light also was detected. Results: We derived for the first time and significantly improved the relatively short periods of apsidal motion from 19 to 142 years for these systems. The relativistic effects are weak, up to 1 % of the total apsidal motion rate. For one system (OGLE-SMC-ECL-888), the third-body hypothesis was also presented, which agrees with high value of the third light for this system detected during the light curve solution. Subject headings: stars: binaries: eclipsing stars: early-type stars: fundamental parameters Magellanic Clouds 1. INTRODUCTION Other galaxies have become the most prominent battlefields in current astrophysical research, mainly due to the large and long-lasting photometric surveys. These surveys like MACHO or OGLE have discovered thousands of new eclipsing binaries in the Magellanic Clouds, hence, we know only about twice more eclipsing binaries in our own Milky Way than in other galaxies(see Pawlak et al. 213, or Graczyk et al. 211). On the other hand, the chemical composition of the Magellanic Clouds differs from that of the solar neighborhood (e.g. Ribas 24), and the study ofthe massiveand metal-deficient stars in the SMC checks our evolutionary models for these abundances. All eclipsing binaries analysed here have properties that make them important astrophysical laboratories for studying the structure and evolution of massive stars (Ribas 24). Eccentric eclipsing binaries (hereafter EEBs) with an apsidal motion can provide us with an important observational test of theoretical models of stellar structure and evolution. A long-term collection of the times of EEBs minima observed for several years throughout the apsidal motion cycle and a consecutive detailed analysis of the period variations of EEB can be performed, yielding both the orbital eccentricity and the period of rotation of the apsidal line with high accuracy (Giménez 1994). Many different sets of stellar evolution models have been published in recent years, such as for Maeder (1999), or Claret (25); however, to distinguish between them and to test, which one is more suitable, it is still rather difficult. The internal structure constants, as derived from the apsidal motion analysis, could serve as one independent criterion. On the other hand, only stellar param- Electronic address: zasche@sirrah.troja.mff.cuni.cz eters for EEBs with an accuracy of 1% can be used to discriminate between the models. Here, we analyse the observational data and rates of apsidal motion for eighteen SMC detached eclipsing systems. All these systems are early-type objects, having eccentric orbits, which also exhibits an apsidal motion. Similar studies of LMC EEBs have been presented by Michalska & Pigulski (25), by Michalska (27), and recently also by Zasche & Wolf (213). As far as we know, only several eclipsing binaries with apsidal motion were analysed in SMC galaxy until now: SC3 139376, SC5 311566 (Graczyk 23), and nine other systems by North et al. (21). 2. OBSERVATIONS OF MINIMUM LIGHT Monitoring of faint EEBs in external galaxies became almost routine nowadays with quite moderate telescopes of 1-2m class, which are equipped with a modern CCD camera. However, a large amount of observing time is needed, which is usually unavailable at larger telescopes. During the last two observational seasons, we have accumulated 266 photometric observations and derived 29 precise times of minimum light for selected eccentric systems. NewCCDphotometrywasobtainedattheLaSilla Observatory in Chile, where the 1.54-m Danish telescope (hereafterdk154)withtheccdcameraandr filterwas used (remotely from the Czech Republic). All CCD measurements were reduced in a standard way using the bias frames and then the flat fields. The comparisonstarwaschosentobeclosetothevariableone and with similar spectral type. A synthetic aperture photometry and astrometry software developed by M. Velen and P. Pravec Aphot, was routinely used for reducing the data. No correction for differential extinction was applied because of the proximity of the comparison stars

2 Zasche et al. to the variable and the resulting negligible differences in air mass and their similar spectral types. The new times of primary and secondary minima and their respective errors were determined by the classical Kwee-van Woerden (1956) method or by our new approach (see the section 4.2). All new times of minima are given in the appendix Tables 5. 3. PHOTOMETRY AND LIGHT CURVE MODELLING The core of our analysis lies on the huge photometric data sets, as obtained during the Macho (Faccioli et al. 27), Ogle II (Wyrzykowski et al. 24), and Ogle III (Graczyk et al. 211) surveys. These photometric data were used both for minima time analysis and for light curve analysis. The method of how the individual times of minima for the particular system were computed is presented in section 4.2. Our new observations obtained at the Danish 1.54-m telescope were used only for deriving the times of minima for the selected targets. The analysis of the light curves (hereafter LC) for the systems was carried out using the program PHOEBE, ver..31a (Prša & Zwitter 25), which is based on the Wilson-Devinney algorithm (Wilson & Devinney 1971) and its later modifications, but some of the parameters have to be fixed during the fitting process. The albedo coefficients A i remained fixed at value 1., the gravity darkening coefficients g i = 1.. The limb darkening coefficients were interpolated from the van Hammes tables (van Hamme 1993), and the synchronicity parameters (F i ) were also kept fixed at values of F i = 1. The temperature of the primary component was derived from the photometric indices or other sources (see below). The problematic issue of the mass ratio was solved by fixing q = 1 because no spectroscopy for most of these selected systems exists, and for detached eclipsing binaries the LC solution is almost insensitive to the photometric mass ratio (see e.g. Terrell & Wilson 25). 4. METHODS USED FOR THE ANALYSIS 4.1. Apsidal motion analysis For the analysis, we used the approach as presented below. 1. At the beginning, all of the available photometric data were analysed, resulting in a set of minima times. Preliminary apsidal motion parameters were derived (with the assumption i = 9 ). 2. Secondly, the eccentricity (e), argument of periastron (ω), and apsidal motion rate ( ω) that resulted from the apsidal motion analysis were used for the preliminary light curve analysis. 3. As the third step, the inclination (i) from the LC analysis was used for the final apsidal motion analysis. 4. Finally, the resulted e, ω, and ω values from the apsidal motion analysis were used for the final LC analysis. Moreover, this simple approach was a bit complicated because the minima times were also derived using the light curve template (see the AFP method in Section 4.2). Hence, the LC solution from step 2 allows us to derive the better times of minima for the step 3. The whole process run iteratively until the changes are negligible (usually it was enough to run these four steps two times). The Opt. Commun. diagrams of all available times of minima were analysed using the method presented by Giménez & García-Pelayo (1983). This is a weighted least-squares iterative procedure, including terms in the eccentricity up to the fifth order. There are five independent variables (T,P s,e, ω,ω ) determined in this procedure. The periastron position ω is given by the linear equation ω = ω + ω E, where ω is the rate of periastronadvance, E is the epoch, and the position of periastron for the zero epoch T is denoted as ω. The relation between the sidereal and the anomalistic period, P s and P a, is given by P s = P a (1 ω/36 ) and the period of apsidal motion by U = 36 P a / ω. All new precise CCD times of minima were used with a weight of 1 in our computation; some of our less precise measurements were weighted by a factor of five, while the poorly covered minima were given a weight of 1. 4.2. Method of minima fitting We developed and routinely used a method for deriving the times of minima for selected stars observed during the MACHO and OGLE surveys. This semi-automatic fitting procedure (hereafter AFP) has harvested the fact that the number of data points obtained during these two photometric surveys is large (typically thousands of data points) but obtained during many orbital revolutions of the close pair (a so-called sparse photometry). Therefore, we can construct the phased light curve of the eclipsing binary in different time epochs. If the apsidal motion is prominent in the system, the shape of the light curve also slightly varies between the different epochs. The first step is to divide the whole data set of photometry into several different subsets, which are used for constructing the individual light curves. Then, we usually choose the data set closest to the half of the time interval covered with observations and use these data points for constructing the light curve to be analysed. Then, this light curve is analysed using the PHOEBE code, and the theoretical light curve template is being constructed. This LC model is then being used for deriving the individual times of minima easily by fitting this phased light curve to the phased light curves for the individual data sets. The best fit is obtained with the simplex algorithm and the least squares fitting method by only shifting the theoretical and observed light curve in two axis (phase and magnitude). If the star has constant magnitude over the whole time range of our data, there is no need to fit the magnitude shift, and only one free parameter is computed. When we find the best fit, then the times of minima are computed easily according to the ephemerides for a particular data set. Of course,

for eccentric orbit binaries, both primary and secondary minima are being computed separately. For the input, there are the data points, the time intervals, the ephemerides, and also parameters of the method. These are the duration of eclipse(how large portion of the phase curve around minima is being used for computing), minimum number of data points (if lower, the minimum is not computed), and the depth of minima. If 1/5 of the depth of minima is covered with data points, then this particular minimum is being computed. Hence, by using this technique, we can usually obtain both primary and secondary minima for each data subset from an original photometry file. Moreover, this method can also be used in these cases, where the minimum is covered only very poorly, or only a descent to the minimum is covered. In these cases, the classical Kwee-van Woerden method would not work properly, so we can obtain more useful data points. On the other hand, we would like to emphasize that the method is suitable only for systems with low eccentricity, where the shape of the light curve is changing only slightly. Otherwise, we have to construct a separate light curve template for each of the data subset. The whole method is graphically shown in Fig. 1, where an illustrative example of OGLE-SMC-ECL-72 is being presented. All of the derived times of minima are stored in the online-only Tables 5. There are also given the errors of individual minima times, which are being computed also by AFP in the following way. The set of different solutions was computed for a particular minimum with different parameters of the code (length of interval around each minimum used for the analysis, number of data points according to their precision, etc.), yielding a set of times of minima, which is usually more than 1. From these minima data set, an averageand its variance were computed,. The variance is then taken as an approximate error estimation for the particular minimum. 5. NOTES ON INDIVIDUAL SYSTEMS All of the eclipsing systems were analysed using a similar approach, hence we cannot focus on every star in detail. See Table 1 for information and cross-identification of these stars. The abbreviations of the star names were used for all of the systems for a better brevity. That is, OGLE-SMC-ECL-72 was shortened as #72, etc. Only the most important results are summarized below. The final light curve fits, and the O C diagrams are presented in Figs. 2 and 3; the parameters are given in Tables 2 and 3. The whole set of eighteen analysed systems can be divided into a few subsets according to available spectral information. The largest group in our sample of stars comprise these stars, which were never observed spectroscopically, hence no spectral classification or radial velocity study was published so far. These systems are #781, #11, #1298, #147, #2225, #2251, #2524, and #5233. Most of them were discovered as eclipsing binaries by Udalski etal. (1998),Wyrzykowskietal. (24),orFacciolietal. (27). Several of them were mentioned as eccentric ones with apsidal motion in some of the above mentioned papers. Owing to having no information about their spectra, we only roughly estimated the spectral types from the measurements in photometric filters, as seen in Table Apsidal motion and LC solution for 18 SMC eccentric EBs 3 14.4 14.5 14.6 14.7 14.8 14.9 An illustrative example: OGLE-SMC-ECL-72 15 2 25 3 35 4 45 5 Time 14.4 14.5 14.6 14.7 14.8 14.9 d light curve 14.4 14.5 14.6 14.7 14.8 14.9 15.4.2.2.4.6.8 1 15.4.2.2.4.6.8 1 The O C diagram 7.15.1.5.5.1.15 y-shift x-shift OGLE raw data Both minima fitted 2448 245 2452 2454 2456 2458 199 1995 2 25 21 215 22 1 8 6 4 2 2 4 6 8 1 Fig. 1. How the AFP method works..25.2.15.1.5.5.1.15.2.25 1. These observations were usually taken from Massey (22) and from the dereddened photometric indices the spectral types were estimated (Popper 198, Ducati et al. 21, or Cox 2). For some of the systems, there resulted a non-negligible third light contribution (e.g. #2225, #2251). For some of the systems, the spectral types were published, so we can use them for a better primary temperature estimation for a subsequent light curve analysis. These systems are #72, #2534, #3677, #4955, #5422, and #5434 (to this group of stars, two systems #888 and #3951 also belong, but these were given a special focus in the following subsections). These binaries were also discovered by Udalski et al. (1998) and Faccioli et al. (27); for some of them, a short note about their apsidal motion was published. The spectral types for these systems given by Evans et al. (24) and Bonanos et al. (21) are in good agreement with our spectral types that are estimated from the dereddened photometric indices.

4 Zasche et al. 14.4 16.9 15.48 15.5 14.5 17 15.52 14.6 17.1 15.54 14.7 14.8 17.2 17.3 15.56 15.58 15.6 15.62 14.9 #72 15.4.2.2.4.6.8 17.4 #781 17.5.4.2.2.4.6.8 15.64 15.66 #888.4.2.2.4.6.8 18.3 18.4 18.5 18.6 18.7 18.8 18.9 #11.4.2.2.4.6.8 17.2 17.25 17.3 17.35 17.4 17.45 #1298 17.5.4.2.2.4.6.8 16.95 17 17.5 17.1 17.15 17.2 17.25 #147.4.2.2.4.6.8 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 #2186.4.2.2.4.6.8 16.68 16.7 16.72 16.74 16.76 16.78 16.8 16.82 16.84 #2225.4.2.2.4.6.8 16.2 16.25 16.3 16.35 16.4 #2251 16.45.4.2.2.4.6.8 16.5 16.6 16.7 16.8 16.9 17 #2524 17.1.4.2.2.4.6.8 16.5 16.55 16.6 16.65 16.7 16.75 16.8 #2534 16.85.4.2.2.4.6.8 1 16.2 16.3 16.4 16.5 16.6 16.7 #3594.4.2.2.4.6.8 15.5 15.1 15.15 15.2 15.25 15.3 #3677.4.2.2.4.6.8 15.85 15.9 15.95 16 16.5 16.1 16.15 #3951.4.2.2.4.6.8 17 17.5 17.1 17.15 17.2 17.25 17.3 17.35 17.4 17.45 #4955 17.5.4.2.2.4.6.8 15.3 15.35 14.95 15 15.55 15.4 15.45 15.5 15.55 #5233.4.2.2.4.6.8 15.5 15.1 15.15 15.2 #5422 15.25.4.2.2.4.6.8 15.6 15.65 15.7 #5434 15.75.4.2.2.4.6.8 Fig. 2. Light curves of the analysed systems, the data taken from the OGLE III survey, and the I filter.

Apsidal motion and LC solution for 18 SMC eccentric EBs 5 TABLE 1 Relevant information for the analysed systems. System OGLE II MACHO RA DE I max (B V ) (U B) (B V ) Sp.type Ref. #72 SC3 139376 213.1562.12 h 44 m 8 s.67-73 14 18.5 14 m.47 -.16 -.91 -.26 B(IV) 1, 2 #781 SC3 157218 212.15624.89 h 44 m 39 s.73-72 59 58.5 17 m.3 -.1 -.78 -.23 B2 1 #888 SC4 29231 212.1568.18 h 45 m 3 s.72-73 3 29.7 15 m.52 -.9 -.82 -.25 O9V 1, 2 #11 28.15744.1836 h 46 m 11 s.29-72 35 17.3 18 m.44 -.31 -.58 -.12 late B/early A 3 #1298 SC4 163754 212.15848.1258 h 47 m 52 s.73-73 16 34. 17 m.22 -.4 -.89 -.29 B 1 #147 28.15861.734 h 48 m 19 s.26-72 21 4.2 17 m.2 -.4 -.67 -.21 B3 1 #2186 SC5 311566 28.1683.86 h 51 m 34 s.84-72 45 46.4 16 m.6 -.4 -.79 -.25 B+B-3 1, 4 #2225 SC6 72782 28.1684.117 h 51 m 41 s.8-72 41 6.1 16 m.71 -.128 -.22 B3 5, 6 #2251 SC6 61418 h 51 m 46 s.64-72 51 21.7 16 m.23 -.14 -.91 -.27 B1 1 #2524 SC6 158178 28.16141.6 h 52 m 42 s.32-72 41 27.9 16 m.53 -.176 -.86 -.24 B2 3 #2534 28.16147.22 h 52 m 43 s.85-72 18 8.6 16 m.57 -.4 -.81 -.26 B1 1 #3594 SC7 255621 27.16428.1423 h 57 m 26 s.41-72 36 46.2 16 m.26 -.21 -.72 -.19 B1+B1-3 1, 4 #3677 SC8 52815 27.1649.6 h 57 m 49 s.25-72 16 55.7 15 m.11 -.16 -.84 -.24 B2 1, 7 #3951 SC8 16725 h 59 m 14 s.98-72 11 35.3 15 m.9 -.18 -.84 -.24 B1V 8, 9 #4955 SC1 94636 26.16886.52 1 h 4 m 59 s.18-72 25 29.3 17 m.11 -.16 -.68 -.19 B3 1 #5233 26.1761.14 1 h 7 m 12 s.54-72 11 42. 15 m.34 -.8 -.91 -.28 B 1 #5422 SC11 11197 26.1717.8 1 h 8 m 45 s.74-72 31 22.4 14 m.99 -.22 -.93 -.26 B1 1 #5434 SC11 118966 26.17173.1 1 h 8 m 5 s.47-72 17 26.1 15 m.56 -.11 -.86 -.26 B1 1 Note: [*] - The full name from OGLE II survey should be OGLE SMC-SCn nnnnnn, [**] - Spectral types given in italics were only estimated from the photometric indices for the first time in the present paper. References: [1] - Massey (22), [2] - Evans et al. (24), [3] - Zaritsky et al. (22), [4] - Hilditch et al. (25), [5] - Udalski et al. (1998), [6] - Massey et al. (1995), [7] - Bonanos et al. (21), [8] - Massey et al. (1989), [9] - Massey et al. (212). TABLE 2 Light curve parameters for the analysed systems. System T 1 [K] T 2 [K] i [deg] Ω 1 Ω 2 L 1 [%] L 2 [%] L 3 [%] #72 315 (fixed) 317 (4) 84.57 (.3) 5.524 (.61) 7.356 (.85) 66.87 (.83) 33.13 (.68) #781 231 (fixed) 167 (3) 85.8 (.18) 7.265 (.86) 8.69 (.12) 71.9 (1.25) 28.1 (.98) #888 332 (fixed) 381 (11) 77.77 (.38) 6.119 (.11) 6.128 (.112) 22.1 (1.85) 26.94 (3.46) 51.5 (4.98) #11 11 (fixed) 97 (5) 79.13 (.92) 5.64 (.24) 6.661 (.397) 67.48 (1.37) 32.52 (1.2) #1298 3 (fixed) 199 (7) 74.68 (.45) 6.364 (.138) 6.71 (.174) 7.46 (1.32) 29.54 (.77) #147 19 (fixed) 191 (7) 75.6 (.5) 5.97 (.147) 6.833 (.177) 56.26 (1.6) 39.1 (1.) 4.64 (1.87) #2186 31 (fixed) 285 (3) 87.2 (.22) 6.843 (.58) 8.4 (.97) 62.73 (3.14) 36. (1.25) 4.3 (2.59) #2225 116 (fixed) 71 (2) 8.33 (.53) 5.678 (.1) 11.334 (.49) 6.98 (2.87) 4.58 (.57) 34.45 (1.55) #2251 262 (fixed) 36 (9) 79.62 (.26) 6.149 (.8) 1.184 (.184) 48.35 (3.2) 17.91 (1.26) 33.73 (4.5) #2524 231 (fixed) 246 (6) 83.32 (.37) 6.128 (.85) 7.165 (.112) 56.63 (1.26) 41.53 (2.3) 1.85 (5.76) #2534 262 (fixed) 183 (3) 76.23 (.28) 5.69 (.5) 6.15 (.52) 61.67 (1.79) 28.64 (1.58) 9.69 (2.63) #3594 255 (fixed) 22 (2) 83.6 (.41) 7.95 (.4) 6.668 (.55) 5.5 (1.56) 48. (1.3) 1.49 (1.96) #3677 231 (fixed) 251 (3) 73.6 (.22) 5.29 (.29) 7.833 (.64) 74.51 (1.7) 25.49 (.94) #3951 262 (fixed) 244 (2) 78.5 (.24) 6.699 (.55) 6.789 (.56) 53.76 (.97) 46.24 (1.16) #4955 19 (fixed) 174 (5) 8.1 (.31) 7.868 (.165) 8.591 (.18) 58.97 (.75) 41.3 (.79) #5233 3 (fixed) 294 (3) 8.52 (.21) 8.762 (.98) 7.843 (.93) 44.9 (1.2) 55.91 (.93) #5422 262 (fixed) 24 (3) 79.18 (.33) 7.76 (.98) 7.7 (.12) 54.69 (2.34) 36.51 (4.57) 8.8 (7.82) #5434 262 (fixed) 242 (4) 71.72 (.4) 5.4 (.61) 5.516 (.59) 53.49 (1.14) 43.86 (1.62) 2.65 (2.1) TABLE 3 The parameters of the apsidal motion for the individual systems. System T 24 [HJD] P s [days] e ω [deg/cycle] ω [deg] U [yr] #72 5383.39 (21) 6.52322 (48).62 (16).27 (3) 67.6 (5.) 28.8 (5.5) #781 52745.417 (32) 3.299923 (48).31 (75).31 (37) 244.4 (1.3) 18.1 (15.1) #888 5347.954 (15) 1.918337 (11).143 (34).474 (193) 89.3 (4.7) 39.9 (11.6) #11 539.7643 (35) 1.1621122 (18).72 (14).61 (69) 94.5 (7.8) 19. (2.4) #1298 5351.194 (14) 1.7532121 (99).219 (42).677 (93) 13. (4.2) 25.5 (4.1) #147 5347.497 (13) 2.1755 (11).151 (47).427 (12) 115.9 (3.9) 46.3 (18.) #2186 5347.314 (15) 3.291316 (2).227 (82).258 (42) 91.9 (2.6) 125.7 (24.5) #2225 5389.914 (1) 1.491721 (8).187 (48).351 (121) 291.9 (8.8) 41.9 (21.8) #2251 54179.695 (25) 2.33638 (33).271 (22).484 (132) 99.8 (5.6) 47.6 (17.8) #2524 53471.664 (24) 2.169236 (23).263 (68).475 (92) 81.5 (6.5) 45. (1.8) #2534 53277.1246 (72) 2.2967384 (72).78 (24).357 (57) 265. (3.4) 63.3 (11.9) #3594 5328.315 (41) 4.33333 (8).194 (69).3 (63) 238.1 (9.9) 142.1 (38.9) #3677 53278.36 (52) 5.241539 (117).153 (55).554 (133) 4.2 (4.6) 93.3 (33.6) #3951 53277.2672 (75) 3.14291 (17).92 (2).476 (165) 92.9 (4.) 64.3 (33.9) #4955 5422.771 (52) 2.772183 (53).338 (48).239 (84) 143.3 (5.) 114.4 (51.5) #5233 52746.459 (45) 5.68362 (13).199 (57).1915 (321) 7. (8.7) 26.1 (5.2) #5422 53656.966 (26) 3.4295 (31).199 (56).31 (83) 318.1 (6.) 99.4 (37.9) #5434 53478.7191 (72) 2.886936 (9).51 (16).747 (214) 129.6 (3.4) 38.1 (15.2)

6 Zasche et al..15.1.5.5.1.15.6.4.2.2.4 2448 245 2452 2454 2456 2458 199 1995 2 25 21 215 22 #72 1 8 6 4 2 2 4 6 8 1 245 2452 2454 2456 1995 2 25 21 215 #11.25.2.15.1.5.5.1.15.2.25.1.2.3.4.5.6 3 2 1 1 2 3 4 5 245 2452 2454 2456 2458.15.1.5.5.1.4.15 15 1 5 5 1 15 2448 245 2452 2454 2456 2458.2.15.1.5.5.1.15.2 1995 2 25 21 215 #2186 3 25 2 15 1 5 5 1 15 2 25 2448 245 2452 2454 2456 2458.6.3 199 1995 2 25 21 215.2.1.1.2.3.4.3.2.1.1.2.3.4.5.4.3.2.1.5.4.3.2.1.1.2.3.1 199 1995 2 25 21 215 22.8 #2524 #3677 1 8 6 4 2 2 4 6 8 1 2448 245 2452 2454 2456 2458 199 1995 2 25 21 215 #5233 1 8 6 4 2 2 4 6 8 1.6.4.2.2.4.6.8.1.4.2.2.4.8.6.4.2.2.4.6.8.4.3.2.1.1.2.3.15.1.5.5.1.15 2448 245 2452 2454 2456 2458 199 1995 2 25 21 215 22 #781 15 1 5 5 1 15 2 2448 245 2452 2454 2456 2458.1.5.5.1 199 1995 2 25 21 215 #1298 3 2 1 1 2 3 245 2452 2456 2454 1995 2 25 21 215 #2225.1.5.5.1.8.6.4.2.2.4.6.8.2.4.6.8 3 2 1 1 2 3 245 2452 2454 2456.4 1995 2 25 21.8.3.6.4.2.2.4.6.3.8 2 15 1 5 5 1 15 245 2451 2452 2453 2454 2455 2456.1.8.6.4.2.2.4.6.8.15.1.5.5.1.15.2 #2534 1996 1998 2 22 24 26 28 21 212 #3951.8.6.4.2.2.1.1.2.3 1 8 6 4 2 2 4 6 8 1 2448 245 2452 2454 2456 199 1995 2 25 21 #5422 2 15 1 5 5 1.3.2.1.1.2.6.4.2.2.4.6.2.15.1.5.5.1.15.15.1.5.5.1 2448 245 2452 2454 2456 2458 246 199 1995 2 25 21 215 22 #888 3 2 1 1 2 3 245 2452 2454 2456 2458 1995 2 25 21 215 #147 25 2 15 1 5 5 1 15 2 25 2452 2454 2456 2458.25.1 2 25 21 215 22.2.8.15.1.5.5.1.15.2.25.2.15.1.5.5.1.15.2 #2251 15 1 5 5 1 15 2 2448 245 2452 2454 2456 199 1995 2 25 21 215 #3594.1.8.6.4.2.2.4.6.8.6.4.2.2.4.6.6.4.2.2.4.6.8.2.4.25.6 15 1 5 5 1 245 2452 2454 2456.2.15.1.5.5.1 1995 2 25 21.15.6.2.8.25 #4955.1.3 2 15 1 5 5 1 2448 245 2452 2454 2456.8.6.4.2.2.4.6.8 199 1995 2 25 21 2 15 1 5 5 1 Fig. 3. Opt. Commun. diagram for the times of minima for the analysed systems. The continuous and dashed curves represent predictions for the primary and secondary eclipses, respectively. The individual primary and secondary minima are denoted by dots and open circles, respectively. Larger symbols correspond to the measurements, which were given higher weights. 5.1. OGLE-SMC-ECL-2186 Two systems, #2186 and #3594, were even published with their light and radial velocity curves solutions. The first one (#2186) was analysed by Wyithe & Wilson (21), who presented a preliminary light curve solution with an eccentric orbit with e =.68. Graczyk (23) analysed the LC of #2186, yielding an eccentricity of.251, no third light, and the luminosity ratio of L 2 /L 1 =.843. Wyrzykowski et al. (24) presented a note about its apsidal motion but with no estimation of #5434 its period. Concerning the spectral type, Graczyk (23) estimated the types of about O9V+O9V but dealt only with the photometry. Later, Hilditch et al. (25) published its spectral type to be B+B-3 based on 15 spectra of the star. They also analysed the light curve, yielding a value of the eccentricity of the orbit to be.63. However, their LC solution is not very convincing due to poor fit of the secondary minimum. 5.2. OGLE-SMC-ECL-3594.6.4.2.8.6.4.2.2.4.3.2.1.1.2.3

Apsidal motion and LC solution for 18 SMC eccentric EBs 7.8.6.4.2.2.4 2448 245 2452 2454 2456 2458 246 199 1995 2 25 21 215 22.4.3.2.1.1.2 245 2451 2452 2453 2454 2455 2456.3 1996 1998 2 22 24 26 28 21 212.1.8.2.6.1.4.2.2.1.4.2.6.8.3.1 3 2 1 1 2 3 Fig. 4. Opt. Commun. diagram of #888 after subtraction of the apsidal motion term. The second system (#3594) was also studied by Wyithe & Wilson (21), who included this star into their sample of SMC eclipsing binaries with the light curvesolution, which result in orbitalinclination of88.9 and an eccentricity of.144. Hilditch et al. (25) analysed the system in more detail, resulting in an orbital eccentricity of.19(based on photometry and spectroscopy together) and the spectral types of both components as B1+B1-3. 5.3. OGLE-SMC-ECL-888 The object#888 was first mentioned by Wyrzykowski et al. (24), who also noted about its apsidal motion. Its spectral type was derived to be about O9V by Evans et al. (24). We found that the pure apsidal motion is not able to describe the O C diagram in detail, hence anothereffect has alsoto be included. We alsotried to fit the parabolic fit to the ephemerides, with the apsidal motion hypothesis(can be interpreted as a mass transfer between the components, despite improbable for detached binary). However, this fit was also not very satisfactory. Therefore, we used a different code that computes the apsidal motion parameters with the third-body orbit (a so-called light travel time effect), as seen in for example Irwin (1959) or Mayer (199). Ten parameters were fitted (five from the apsidal motion, five from the third body hypothesis); thus, this approach led to an acceptable solution with the lowest sum of squares residuals. The final parameters of the fit are given in Tables 3 and 4; the complete Opt. Commun. diagrams are shown in Figs. 3 and 4. From the third-body parameters, we could also compute the mass function of the distant component, which resulted in f(m 3 ) =.59±.15 M. From this value, one can calculate a predicted minimal mass of the third body (i.e. assuming coplanar orbits i 3 = 9 ), which re- Parameter [Unit] TABLE 4 Third body orbit parameters for #888. Value p 3 [yr] 72.1 ± 28. A 3 [day].3 ±.11 T 3 [HJD] 24549 ± 87 e 3.79 ±.247 ω 3 [deg] 154. ± 15.6 1 8 6 4 2 2 4 6 8 1 Fig. 5. Opt. Commun. diagram of #3951 after subtraction of the apsidal motion term. sulted in m 3,min = 4.9 M. If we propose such a body in the system, one can ask whether it is detectable somehow in the already obtained data. The period is long for continuous monitoring of the radial velocity changes, but detecting the third light in the light curve solution would be promising. Assuming a normal main sequence star, its luminosity would be of about only L 3,min = 1 2% of the total system luminosity. Such a weak third light would be hardly detectable in our poor-quality photometric data, but it is worth of try. Hence, we performed a new light curve solution with a special focus on the value of the third light for a LC solution. The value was really obtained, and its value is not negligible at all. As one can see from the parameters presented in Table 2, the third light represents about one half of the total light. This finding naturally explains why both the eclipses are so shallow. On the other hand, one can ask to which body the estimated spectral type of O9V belongs. If the third body is the dominant source, this is probably the O9V component, but the primary temperature of 332 K was assumed using the O9V primary, which now seems to be incorrect. However, having no other relevant information about the individual spectral types, one cannot easily assume a different primary temperature. Thus, we can conclude that the third body is probably present and orbits around the EB pair on orbit which is mildly inclined from the originally assumed 9. It is hard to say anything more about such a body because of the high errors of the parameters (period, third light, etc.). More precise photometry or radial velocities would be very welcome for a final confirmation of our hypothesis. 5.4. OGLE-SMC-ECL-3951 The object #3951 is a part of the SMC open cluster NGC 346. Its eclipsing nature and orbital period was first presented by Udalski et al. (1998). Later, the star was classified as B1V by Massey et al. (212). From the period analysis, a weak quasi-periodic signal also on the residuals after subtraction of the apsidal motion hypothesis (see Fig. 5) resulted. However, the variation is still too spurious for any final confirmation yet and we have not even try to fit the data with any additional variation, as in the previous case. 6. DISCUSSION AND CONCLUSIONS Our study provides the parameters of the apsidal motion for eighteen early-type binary systems located in the SMC. For most of the binaries, this is the first attempt

8 Zasche et al. to estimate the apsidal motion rates, and the light curve solution. In our own Galaxy there are a few hundreds of apsidal motion eclipsing binaries known; however, in other galaxies their number is still very limited. Hence this study still presents an important contribution to the topic. However, for only three systems from our sample (#11, #1298, and #5233), the apsidal motion was derived from adequately large data set covering almost one apsidal period. The relativistic effects for the selected systems are weak, being up to 1% of the total apsidal motion rate. For the system #888, the third body hypothesis was also presented and discussed. The apsidal motion in EEBs has been used for decades to test evolutionary stellar models. Thus, one can ask whether our results can be used for deriving the internal structure constants for these stars in SMC. However, dealing with no radial velocities for most of the systems and with rather poor data coverage during the apsidal motion period, the parameters are too uncertain and affected by large errors. For these systems where the apsidal period is well-covered, a detailed spectroscopic analysis is missing, and vice versa, for systems where the radial velocity study was performed, the apsidal period has yet to be only poorly covered with data. However, for any testing of the stellar structure models or for the general relativity tests, the quality of the input data has to be an order of magnitude better (which implies longterm collection of the observations and data that covers the whole apsidal period in the following decades). Some of the systems are bright enough for a spectral monitoring, hence we encourage the observers to obtain new, high-dispersion, and high-s/n spectroscopic observations. With such data, methods, like spectral disentangling, can help us construct the radial velocity curves of both components, confirm the apsidal motion hypothesis, test the stellar structure models, or detect the third bodies, as indicated from our analysis. We do thank the MACHO and OGLE teams for making all of the observations easily public available. This work was supported by the Czech Science Foundation grant no. P29/1/715, by the grant UNCE 12 of the Charles University in Prague, and by the grant LG121 of the Ministry of Education of the Czech Republic. We are also grateful to the ESO team at the La Silla Observatory for their help in maintaining and operating the Danish telescope. The following internet-based resources were used in research for this paper: the SIM- BAD database and the VizieR service operated at the CDS, Strasbourg, France, and the NASA s Astrophysics Data System Bibliographic Services. Bonanos, A. Z., Lennon, D. J., Köhlinger, F., et al. 21, AJ, 14, 416 Claret, A. 25, A&A, 44, 647 Cox, A. N. 2, in Allen s Astrophysical Quantities, 4th ed., ed. Arthur N. Cox (Springer Verlag, New York) Ducati, J. R., Bevilacqua, C. M., Rembold, S. B., & Ribeiro, D. 21, ApJ, 558, 39 Evans, C. J., Howarth, I. D., Irwin, M. J., Burnley, A. W., & Harries, T. J. 24, MNRAS, 353, 61 Faccioli, L., Alcock, C., Cook, K. et al. 27, AJ, 134, 1963 Giménez, A. 1994, Experimental Astronomy, 5, 91 Giménez, A., & García-Pelayo, J.M. 1983, Ap&SS, 92, 23 Graczyk, D. 23, MNRAS, 342, 1334 Graczyk, D., Soszyński, I., Poleski, R., et al. 211, AcA, 61, 13 Hilditch, R. W., Howarth, I. D., & Harries, T. J. 25, MNRAS, 357, 34 Irwin, J. B. 1959, AJ, 64, 149 Kwee, K.K., & van Woerden, H. 1956, Bull. Astron. Inst. Netherlands, 12, 327 Maeder, A. 1999, A&A, 347, 185 Massey, P., Parker, J. W., & Garmany, C. D. 1989, AJ, 98, 135 Massey, P., Lang, C. C., Degioia-Eastwood, K., & Garmany, C. D. 1995, ApJ, 438, 188 REFERENCES APPENDIX TABLES OF MINIMA Massey, P. 22, ApJS, 141, 81 Massey, P., Morrell, N. I., Neugent, K. F., et al. 212, ApJ, 748, 96 Mayer, P. 199, BAICz, 41, 231 Michalska, G. 27, IBVS No. 5759 Michalska, G. & Pigulski, A. 25, A&A, 434, 89 North, P., Gauderon, R., Barblan, F., & Royer, F. 21, A&A, 52, A74 Pawlak, M., Graczyk, D., Soszyński, I., et al. 213, AcA, 63, 323 Popper, D. M. 198, ARA&A, 18, 115 Prša, A., & Zwitter, T. 25, ApJ, 628, 426 Ribas, I. 24, New Astronomy Reviews, 48, 731 Terrell, D., & Wilson, R. E. 25, Ap&SS, 296, 221 Udalski, A., Soszynski, I., Szymanski, M., et al. 1998, AcA, 48, 563 van Hamme, W. 1993, AJ, 16, 296 Wilson, R. E., & Devinney, E. J. 1971, ApJ, 166, 65 Wyrzykowski, L., Udalski, A., Kubiak, M., et al. 24, AcA, 54, 1 Wyithe, J. S. B., & Wilson, R. E. 21, ApJ, 559, 26 Zaritsky, D., Harris, J., Thompson, I. B., Grebel, E. K., & Massey, P. 22, AJ, 123, 855 Zasche, P. & Wolf, M. 213, A&A, 558, 51

Apsidal motion and LC solution for 18 SMC eccentric EBs 9 TABLE 5 List of the minima timings used for the analysis. Star JD Hel.- Error Type Filter Source / 24 [day] Observatory OGLE-SMC-ECL-72 4871.29443.741 Prim B+R MACHO OGLE-SMC-ECL-72 4874.29533.922 Sec B+R MACHO OGLE-SMC-ECL-72 49548.649.1197 Prim B+R MACHO OGLE-SMC-ECL-72 49551.64897.1236 Sec B+R MACHO OGLE-SMC-ECL-72 49851.19352.1915 Prim B+R MACHO OGLE-SMC-ECL-72 49854.28789.2131 Sec B+R MACHO OGLE-SMC-ECL-72 522.21473.924 Prim B+R MACHO OGLE-SMC-ECL-72 525.3226.2286 Sec B+R MACHO OGLE-SMC-ECL-72 5547.16343.657 Prim B+R MACHO OGLE-SMC-ECL-72 555.33647.16 Sec B+R MACHO OGLE-SMC-ECL-72 5849.7628.929 Prim B+R MACHO OGLE-SMC-ECL-72 5852.9735.98 Sec B+R MACHO OGLE-SMC-ECL-72 51497.3432.1915 Prim B+R MACHO OGLE-SMC-ECL-72 515.5913.1798 Sec B+R MACHO OGLE-SMC-ECL-72 555.35962.592 Sec I OGLE II OGLE-SMC-ECL-72 5553.18661.439 Prim I OGLE II OGLE-SMC-ECL-72 5111.127.486 Sec I OGLE II OGLE-SMC-ECL-72 5113.92761.22 Prim I OGLE II OGLE-SMC-ECL-72 517.32442.2 Sec I OGLE II OGLE-SMC-ECL-72 5173.1399.89 Prim I OGLE II OGLE-SMC-ECL-72 52199.4921.391 Prim I OGLE III OGLE-SMC-ECL-72 5222.68144.649 Sec I OGLE III OGLE-SMC-ECL-72 52574.65616.1128 Prim I OGLE III OGLE-SMC-ECL-72 52577.9814.526 Sec I OGLE III OGLE-SMC-ECL-72 52925.7455.698 Prim I OGLE III OGLE-SMC-ECL-72 52928.9335.34 Sec I OGLE III OGLE-SMC-ECL-72 533.96643.535 Prim I OGLE III OGLE-SMC-ECL-72 5334.15675.26 Sec I OGLE III OGLE-SMC-ECL-72 53652.492.113 Prim I OGLE III OGLE-SMC-ECL-72 53655.16657.666 Sec I OGLE III OGLE-SMC-ECL-72 53997.3938.49 Prim I OGLE III OGLE-SMC-ECL-72 54.1147.739 Sec I OGLE III OGLE-SMC-ECL-72 54348.9727.1183 Prim I OGLE III OGLE-SMC-ECL-72 54351.12853.173 Sec I OGLE III OGLE-SMC-ECL-72 5482.4975.811 Prim I OGLE III OGLE-SMC-ECL-72 5485.2717.4 Sec I OGLE III OGLE-SMC-ECL-781 4865.5217.589 Prim B+R MACHO OGLE-SMC-ECL-781 48651.61198.61 Sec B+R MACHO OGLE-SMC-ECL-781 49498.6339.758 Prim B+R MACHO OGLE-SMC-ECL-781 49499.71447.766 Sec B+R MACHO OGLE-SMC-ECL-781 49851.6741.639 Prim B+R MACHO OGLE-SMC-ECL-781 49852.8913.995 Sec B+R MACHO OGLE-SMC-ECL-781 521.46365.429 Prim B+R MACHO OGLE-SMC-ECL-781 522.62334.27 Sec B+R MACHO OGLE-SMC-ECL-781 5551.24174.636 Prim B+R MACHO OGLE-SMC-ECL-781 5552.4287.43 Sec B+R MACHO OGLE-SMC-ECL-781 5851.52298.449 Prim B+R MACHO OGLE-SMC-ECL-781 5852.72368.32 Sec B+R MACHO OGLE-SMC-ECL-781 5151.5999.867 Prim B+R MACHO OGLE-SMC-ECL-781 5152.82123.378 Sec B+R MACHO OGLE-SMC-ECL-781 595.51118.632 Prim I OGLE II OGLE-SMC-ECL-781 5951.72688.346 Sec I OGLE II OGLE-SMC-ECL-781 5165.6399.269 Prim I OGLE II OGLE-SMC-ECL-781 51651.3417.33 Sec I OGLE II OGLE-SMC-ECL-781 5221.1237.245 Prim I OGLE III OGLE-SMC-ECL-781 5222.45512.33 Sec I OGLE III OGLE-SMC-ECL-781 52574.67.23 Prim I OGLE III OGLE-SMC-ECL-781 52575.35665.219 Sec I OGLE III OGLE-SMC-ECL-781 52923.77283.191 Prim I OGLE III OGLE-SMC-ECL-781 52925.16354.283 Sec I OGLE III OGLE-SMC-ECL-781 53299.94961.223 Prim I OGLE III OGLE-SMC-ECL-781 5331.3689.39 Sec I OGLE III OGLE-SMC-ECL-781 53649.71827.569 Prim I OGLE III OGLE-SMC-ECL-781 53651.16977.287 Sec I OGLE III OGLE-SMC-ECL-781 53999.48473.358 Prim I OGLE III OGLE-SMC-ECL-781 54.98147.319 Sec I OGLE III OGLE-SMC-ECL-781 54349.25336.15 Prim I OGLE III OGLE-SMC-ECL-781 5435.7866.575 Sec I OGLE III OGLE-SMC-ECL-781 5481.3227.152 Prim I OGLE III OGLE-SMC-ECL-781 5482.88593.47 Sec I OGLE III OGLE-SMC-ECL-781 56637.7177.52 Sec R DK154 OGLE-SMC-ECL-781 56675.55172.225 Prim R DK154 OGLE-SMC-ECL-888 4865.12114.495 Prim B+R MACHO OGLE-SMC-ECL-888 48651.22335.58 Sec B+R MACHO OGLE-SMC-ECL-888 49499.946.87 Prim B+R MACHO OGLE-SMC-ECL-888 4951.5324.649 Sec B+R MACHO OGLE-SMC-ECL-888 49849.7573.144 Prim B+R MACHO OGLE-SMC-ECL-888 4985.2619.93 Sec B+R MACHO OGLE-SMC-ECL-888 52.13984.54 Prim B+R MACHO OGLE-SMC-ECL-888 521.26861.575 Sec B+R MACHO OGLE-SMC-ECL-888 5549.27599.811 Prim B+R MACHO

1 Zasche et al. TABLE 6 List of the minima timings used for the analysis. Star JD Hel.- Error Type Filter Source / 24 [day] Observatory OGLE-SMC-ECL-888 555.471.1648 Sec B+R MACHO OGLE-SMC-ECL-888 585.4648.33 Prim B+R MACHO OGLE-SMC-ECL-888 5851.59112.1159 Sec B+R MACHO OGLE-SMC-ECL-888 515.78388.681 Prim B+R MACHO OGLE-SMC-ECL-888 5151.8963.55 Sec B+R MACHO OGLE-SMC-ECL-888 52199.928.528 Prim I OGLE III OGLE-SMC-ECL-888 522.13742.884 Sec I OGLE III OGLE-SMC-ECL-888 526.2846.631 Prim I OGLE III OGLE-SMC-ECL-888 5261.6977.736 Sec I OGLE III OGLE-SMC-ECL-888 52949.18978.333 Prim I OGLE III OGLE-SMC-ECL-888 5295.1952.191 Sec I OGLE III OGLE-SMC-ECL-888 533.25949.116 Prim I OGLE III OGLE-SMC-ECL-888 5331.2384.399 Sec I OGLE III OGLE-SMC-ECL-888 53674.34361.49 Prim I OGLE III OGLE-SMC-ECL-888 53675.28319.115 Sec I OGLE III OGLE-SMC-ECL-888 545.35279.313 Prim I OGLE III OGLE-SMC-ECL-888 5451.26885.38 Sec I OGLE III OGLE-SMC-ECL-888 54374.55955.38 Prim I OGLE III OGLE-SMC-ECL-888 54375.44152.39 Sec I OGLE III OGLE-SMC-ECL-888 548.4418.414 Prim I OGLE III OGLE-SMC-ECL-888 5481.28789.95 Sec I OGLE III OGLE-SMC-ECL-888 56648.59424.55 Sec R DK154 OGLE-SMC-ECL-888 56699.59546.175 Prim R DK154 OGLE-SMC-ECL-11 4885.21555.273 Prim B+R MACHO OGLE-SMC-ECL-11 4885.79998.69 Sec B+R MACHO OGLE-SMC-ECL-11 49849.63212.151 Prim B+R MACHO OGLE-SMC-ECL-11 4985.2364.839 Sec B+R MACHO OGLE-SMC-ECL-11 5199.41863.345 Prim B+R MACHO OGLE-SMC-ECL-11 52.2376.438 Sec B+R MACHO OGLE-SMC-ECL-11 555.36477.44 Prim B+R MACHO OGLE-SMC-ECL-11 555.98731.361 Sec B+R MACHO OGLE-SMC-ECL-11 585.18196.182 Prim B+R MACHO OGLE-SMC-ECL-11 585.81887.465 Sec B+R MACHO OGLE-SMC-ECL-11 51499.8461.227 Prim B+R MACHO OGLE-SMC-ECL-11 515.44253.341 Sec B+R MACHO OGLE-SMC-ECL-11 52199.4475.254 Prim I OGLE III OGLE-SMC-ECL-11 522.2416.93 Sec I OGLE III OGLE-SMC-ECL-11 52199.4475.254 Prim I OGLE III OGLE-SMC-ECL-11 52575.37135.282 Sec I OGLE III OGLE-SMC-ECL-11 52924.5829.7 Prim I OGLE III OGLE-SMC-ECL-11 52925.16579.325 Sec I OGLE III OGLE-SMC-ECL-11 53299.95187.42 Prim I OGLE III OGLE-SMC-ECL-11 533.5237.219 Sec I OGLE III OGLE-SMC-ECL-11 53649.7411.6 Prim I OGLE III OGLE-SMC-ECL-11 5365.3821.44 Sec I OGLE III OGLE-SMC-ECL-11 53999.557.15 Prim I OGLE III OGLE-SMC-ECL-11 54.8782.11 Sec I OGLE III OGLE-SMC-ECL-11 5435.5194.14 Prim I OGLE III OGLE-SMC-ECL-11 54351.5432.7 Sec I OGLE III OGLE-SMC-ECL-11 548.26261.178 Prim I OGLE III OGLE-SMC-ECL-11 548.7859.223 Sec I OGLE III OGLE-SMC-ECL-11 56641.6734.122 Sec R DK154 OGLE-SMC-ECL-11 56673.54658.453 Prim R DK154 OGLE-SMC-ECL-11 5672.626.2 Prim R DK154 OGLE-SMC-ECL-1298 487.8975.568 Prim B+R MACHO OGLE-SMC-ECL-1298 4871.85571.73 Sec B+R MACHO OGLE-SMC-ECL-1298 49549.3431.65 Prim B+R MACHO OGLE-SMC-ECL-1298 4955.42315.753 Sec B+R MACHO OGLE-SMC-ECL-1298 49849.1317.1288 Prim B+R MACHO OGLE-SMC-ECL-1298 4985.24535.998 Sec B+R MACHO OGLE-SMC-ECL-1298 5199.77445.3 Prim B+R MACHO OGLE-SMC-ECL-1298 52.89616.724 Sec B+R MACHO OGLE-SMC-ECL-1298 555.42479.531 Prim B+R MACHO OGLE-SMC-ECL-1298 5551.54191.447 Sec B+R MACHO OGLE-SMC-ECL-1298 585.23337.688 Prim B+R MACHO OGLE-SMC-ECL-1298 5851.33824.1225 Sec B+R MACHO OGLE-SMC-ECL-1298 515.69514.657 Prim B+R MACHO OGLE-SMC-ECL-1298 5151.76815.126 Sec B+R MACHO OGLE-SMC-ECL-1298 5749.64318.432 Sec I OGLE II OGLE-SMC-ECL-1298 575.29216.168 Prim I OGLE II OGLE-SMC-ECL-1298 5136.43853.65 Prim I OGLE II OGLE-SMC-ECL-1298 51361.49925.1471 Sec I OGLE II OGLE-SMC-ECL-1298 522.28714.19 Prim I OGLE III OGLE-SMC-ECL-1298 5221.2277.3 Sec I OGLE III OGLE-SMC-ECL-1298 526.4481.253 Prim I OGLE III OGLE-SMC-ECL-1298 526.88875.71 Sec I OGLE III OGLE-SMC-ECL-1298 5295.72224.129 Prim I OGLE III OGLE-SMC-ECL-1298 52951.52767.227 Sec I OGLE III OGLE-SMC-ECL-1298 53299.62553.384 Prim I OGLE III OGLE-SMC-ECL-1298 533.37165.3122 Sec I OGLE III OGLE-SMC-ECL-1298 53674.84148.361 Prim I OGLE III

Apsidal motion and LC solution for 18 SMC eccentric EBs 11 TABLE 7 List of the minima timings used for the analysis. Star JD Hel.- Error Type Filter Source / 24 [day] Observatory OGLE-SMC-ECL-1298 53675.54332.274 Sec I OGLE III OGLE-SMC-ECL-1298 545.434.84 Prim I OGLE III OGLE-SMC-ECL-1298 545.7148.8333 Sec I OGLE III OGLE-SMC-ECL-1298 54374.4371.178 Prim I OGLE III OGLE-SMC-ECL-1298 54375.5651.324 Sec I OGLE III OGLE-SMC-ECL-1298 548.44849.472 Prim I OGLE III OGLE-SMC-ECL-1298 5481.8349.9458 Sec I OGLE III OGLE-SMC-ECL-1298 56397.5345.545 Prim R DK154 OGLE-SMC-ECL-1298 5658.67593.131 Sec R DK154 OGLE-SMC-ECL-1298 5668.7669.287 Sec R DK154 OGLE-SMC-ECL-1298 56673.62977.118 Sec R DK154 OGLE-SMC-ECL-1298 5672.56995.786 Prim R DK154 OGLE-SMC-ECL-147 48649.17421.323 Prim B+R MACHO OGLE-SMC-ECL-147 4865.4889.669 Sec B+R MACHO OGLE-SMC-ECL-147 49499.99436.697 Prim B+R MACHO OGLE-SMC-ECL-147 4951.21343.811 Sec B+R MACHO OGLE-SMC-ECL-147 4985.83227.611 Prim B+R MACHO OGLE-SMC-ECL-147 49852.2643.71 Sec B+R MACHO OGLE-SMC-ECL-147 5199.56787.147 Prim B+R MACHO OGLE-SMC-ECL-147 52.7458.719 Sec B+R MACHO OGLE-SMC-ECL-147 555.398.31 Prim B+R MACHO OGLE-SMC-ECL-147 5551.563.742 Sec B+R MACHO OGLE-SMC-ECL-147 585.81538.361 Prim B+R MACHO OGLE-SMC-ECL-147 5851.95315.58 Sec B+R MACHO OGLE-SMC-ECL-147 51499.9612.528 Prim B+R MACHO OGLE-SMC-ECL-147 5151.7678.138 Sec B+R MACHO OGLE-SMC-ECL-147 52199.541.251 Prim I OGLE III OGLE-SMC-ECL-147 522.59766.496 Sec I OGLE III OGLE-SMC-ECL-147 52575.58581.118 Prim I OGLE III OGLE-SMC-ECL-147 52576.61588.614 Sec I OGLE III OGLE-SMC-ECL-147 52924.32346.232 Prim I OGLE III OGLE-SMC-ECL-147 52925.31293.14 Sec I OGLE III OGLE-SMC-ECL-147 533.37128.145 Prim I OGLE III OGLE-SMC-ECL-147 5331.33887.32 Sec I OGLE III OGLE-SMC-ECL-147 53649.1211.177 Prim I OGLE III OGLE-SMC-ECL-147 5365.6528.44 Sec I OGLE III OGLE-SMC-ECL-147 53999.94323.416 Prim I OGLE III OGLE-SMC-ECL-147 54.8555.555 Sec I OGLE III OGLE-SMC-ECL-147 5435.77939.263 Prim I OGLE III OGLE-SMC-ECL-147 54351.6932.463 Sec I OGLE III OGLE-SMC-ECL-147 548.3516.327 Prim I OGLE III OGLE-SMC-ECL-147 5481.23674.381 Sec I OGLE III OGLE-SMC-ECL-147 5664.64137.119 Prim R DK154 OGLE-SMC-ECL-147 5676.628.212 Sec R DK154 OGLE-SMC-ECL-2186 4871.9378.523 Prim B+R MACHO OGLE-SMC-ECL-2186 4872.98665.172 Sec B+R MACHO OGLE-SMC-ECL-2186 4955.25484.192 Prim B+R MACHO OGLE-SMC-ECL-2186 49552.13771.1594 Sec B+R MACHO OGLE-SMC-ECL-2186 49849.77123.794 Prim B+R MACHO OGLE-SMC-ECL-2186 49851.647.2195 Sec B+R MACHO OGLE-SMC-ECL-2186 5198.6644.176 Prim B+R MACHO OGLE-SMC-ECL-2186 52.532.638 Sec B+R MACHO OGLE-SMC-ECL-2186 555.8462.851 Prim B+R MACHO OGLE-SMC-ECL-2186 5552.66865.1982 Sec B+R MACHO OGLE-SMC-ECL-2186 585.36453.127 Prim B+R MACHO OGLE-SMC-ECL-2186 5852.16776.89 Sec B+R MACHO OGLE-SMC-ECL-2186 51498.7649.26 Prim B+R MACHO OGLE-SMC-ECL-2186 515.5416.316 Sec B+R MACHO OGLE-SMC-ECL-2186 5748.32719.435 Prim I OGLE III OGLE-SMC-ECL-2186 575.1197.5 Sec I OGLE III OGLE-SMC-ECL-2186 51699.53127.179 Prim I OGLE III OGLE-SMC-ECL-2186 5171.28957.242 Sec I OGLE III OGLE-SMC-ECL-2186 52199.83143.657 Prim I OGLE III OGLE-SMC-ECL-2186 5221.54632.741 Sec I OGLE III OGLE-SMC-ECL-2186 52575.611.45 Prim I OGLE III OGLE-SMC-ECL-2186 52576.73683.513 Sec I OGLE III OGLE-SMC-ECL-2186 52923.96294.88 Prim I OGLE III OGLE-SMC-ECL-2186 52925.6931.59 Sec I OGLE III OGLE-SMC-ECL-2186 53299.1785.23 Prim I OGLE III OGLE-SMC-ECL-2186 533.8791.373 Sec I OGLE III OGLE-SMC-ECL-2186 53651.34727.639 Prim I OGLE III OGLE-SMC-ECL-2186 53652.96525.485 Sec I OGLE III OGLE-SMC-ECL-2186 54.22926.325 Prim I OGLE III OGLE-SMC-ECL-2186 541.83984.567 Sec I OGLE III OGLE-SMC-ECL-2186 54349.12883.566 Prim I OGLE III OGLE-SMC-ECL-2186 5435.69183.641 Sec I OGLE III OGLE-SMC-ECL-2186 548.463.45 Prim I OGLE III OGLE-SMC-ECL-2186 5481.58628.743 Sec I OGLE III OGLE-SMC-ECL-2186 56669.56854.91 Prim R DK154 OGLE-SMC-ECL-2186 56677.5822.66 Sec R DK154 OGLE-SMC-ECL-2225 48699.873.966 Prim B+R MACHO

12 Zasche et al. TABLE 8 List of the minima timings used for the analysis. Star JD Hel.- Error Type Filter Source / 24 [day] Observatory OGLE-SMC-ECL-2225 487.4557.67 Sec B+R MACHO OGLE-SMC-ECL-2225 4955.14572.366 Prim B+R MACHO OGLE-SMC-ECL-2225 4955.73392.56 Sec B+R MACHO OGLE-SMC-ECL-2225 49849.97432.275 Prim B+R MACHO OGLE-SMC-ECL-2225 4985.58771.696 Sec B+R MACHO OGLE-SMC-ECL-2225 52.5277.631 Prim B+R MACHO OGLE-SMC-ECL-2225 521.12738.641 Sec B+R MACHO OGLE-SMC-ECL-2225 5549.57873.1289 Prim B+R MACHO OGLE-SMC-ECL-2225 555.2266.979 Sec B+R MACHO OGLE-SMC-ECL-2225 5849.39455.747 Prim B+R MACHO OGLE-SMC-ECL-2225 585.4827.596 Sec B+R MACHO OGLE-SMC-ECL-2225 51499.76771.276 Prim B+R MACHO OGLE-SMC-ECL-2225 515.4454.253 Sec B+R MACHO OGLE-SMC-ECL-2225 52199.34712.252 Prim I OGLE III OGLE-SMC-ECL-2225 522.163.175 Sec I OGLE III OGLE-SMC-ECL-2225 52575.2568.215 Prim I OGLE III OGLE-SMC-ECL-2225 52576.2555.544 Sec I OGLE III OGLE-SMC-ECL-2225 52924.29983.319 Prim I OGLE III OGLE-SMC-ECL-2225 52925.171.252 Sec I OGLE III OGLE-SMC-ECL-2225 533.261.534 Prim I OGLE III OGLE-SMC-ECL-2225 5331.2592.359 Sec I OGLE III OGLE-SMC-ECL-2225 53649.24349.271 Prim I OGLE III OGLE-SMC-ECL-2225 5365.1421.27 Sec I OGLE III OGLE-SMC-ECL-2225 53999.78851.337 Prim I OGLE III OGLE-SMC-ECL-2225 54.67118.923 Sec I OGLE III OGLE-SMC-ECL-2225 5435.35287.291 Prim I OGLE III OGLE-SMC-ECL-2225 54351.22518.15 Sec I OGLE III OGLE-SMC-ECL-2225 54799.33928.355 Prim I OGLE III OGLE-SMC-ECL-2225 548.2431.34 Sec I OGLE III OGLE-SMC-ECL-2225 56641.62568.99 Prim R DK154 OGLE-SMC-ECL-2225 56677.435.213 Prim R DK154 OGLE-SMC-ECL-2225 5676.66141.932 Sec R DK154 OGLE-SMC-ECL-2225 56727.5726.856 Sec R DK154 OGLE-SMC-ECL-2225 56739.49729.448 Sec R DK154 OGLE-SMC-ECL-2251 575.22871.1111 Prim I OGLE II OGLE-SMC-ECL-2251 5751.76132.99 Sec I OGLE II OGLE-SMC-ECL-2251 5171.3998.534 Prim I OGLE II OGLE-SMC-ECL-2251 5172.49569.466 Sec I OGLE II OGLE-SMC-ECL-2251 522.98659.419 Prim I OGLE III OGLE-SMC-ECL-2251 5222.35316.76 Sec I OGLE III OGLE-SMC-ECL-2251 52574.7784.435 Prim I OGLE III OGLE-SMC-ECL-2251 52576.9912.1312 Sec I OGLE III OGLE-SMC-ECL-2251 52925.18554.564 Prim I OGLE III OGLE-SMC-ECL-2251 52926.45681.2329 Sec I OGLE III OGLE-SMC-ECL-2251 53298.99286.46 Prim I OGLE III OGLE-SMC-ECL-2251 533.22628.2132 Sec I OGLE III OGLE-SMC-ECL-2251 53649.4932.442 Prim I OGLE III OGLE-SMC-ECL-2251 5365.58269.382 Sec I OGLE III OGLE-SMC-ECL-2251 53999.8346.299 Prim I OGLE III OGLE-SMC-ECL-2251 54.94196.673 Sec I OGLE III OGLE-SMC-ECL-2251 5435.26571.468 Prim I OGLE III OGLE-SMC-ECL-2251 54351.32752.3865 Sec I OGLE III OGLE-SMC-ECL-2251 5481.13489.56 Prim I OGLE III OGLE-SMC-ECL-2251 5482.18335.1739 Sec I OGLE III OGLE-SMC-ECL-2251 56639.69916.113 Prim R DK154 OGLE-SMC-ECL-2251 56675.56777.287 Sec R DK154 OGLE-SMC-ECL-2251 56689.57815.124 Sec R DK154 OGLE-SMC-ECL-2524 48749.5117.48 Prim B+R MACHO OGLE-SMC-ECL-2524 4875.47954.177 Sec B+R MACHO OGLE-SMC-ECL-2524 49749.7351.122 Prim B+R MACHO OGLE-SMC-ECL-2524 4975.5676.382 Sec B+R MACHO OGLE-SMC-ECL-2524 525.189.171 Prim B+R MACHO OGLE-SMC-ECL-2524 5251.6117.254 Sec B+R MACHO OGLE-SMC-ECL-2524 5749.1199.28 Prim B+R MACHO OGLE-SMC-ECL-2524 575.53425.654 Sec B+R MACHO OGLE-SMC-ECL-2524 51499.6973.381 Prim B+R MACHO OGLE-SMC-ECL-2524 5151.7663.18 Sec B+R MACHO OGLE-SMC-ECL-2524 522.4699.311 Prim I OGLE III OGLE-SMC-ECL-2524 5221.7247.847 Sec I OGLE III OGLE-SMC-ECL-2524 52575.718.258 Prim I OGLE III OGLE-SMC-ECL-2524 52576.9628.417 Sec I OGLE III OGLE-SMC-ECL-2524 52924.96844.138 Prim I OGLE III OGLE-SMC-ECL-2524 52926.17872.242 Sec I OGLE III OGLE-SMC-ECL-2524 533.26282.111 Prim I OGLE III OGLE-SMC-ECL-2524 5331.42517.349 Sec I OGLE III OGLE-SMC-ECL-2524 53649.5314.122 Prim I OGLE III OGLE-SMC-ECL-2524 5365.64581.225 Sec I OGLE III OGLE-SMC-ECL-2524 54.95733.455 Prim I OGLE III OGLE-SMC-ECL-2524 542.2888.174 Sec I OGLE III OGLE-SMC-ECL-2524 5435.23684.791 Prim I OGLE III OGLE-SMC-ECL-2524 54351.24487.24 Sec I OGLE III