Issue |
A&A
Volume 558, October 2013
|
|
---|---|---|
Article Number | A51 | |
Number of page(s) | 12 | |
Section | Stellar structure and evolution | |
DOI | https://doi.org/10.1051/0004-6361/201322054 | |
Published online | 03 October 2013 |
Apsidal motion and absolute parameters for five LMC eccentric eclipsing binaries⋆,⋆⋆
Astronomical Institute, Charles University in Prague, Faculty of Mathematics
and Physics, 180 00 Praha 8, V Holešovičkách 2, Czech Republic
e-mail: zasche@sirrah.troja.mff.cuni.cz
Received:
11
June
2013
Accepted:
15
July
2013
Aims. As part of our observational projects at the La Silla Danish 1.54-meter telescope, we aim to measure the precise times of minimum light for eccentric eclipsing binaries in the Large Magellanic Cloud, needed for accurate determination of apsidal motion. Many new times of minima were derived from the photometric databases OGLE and MACHO. Several new minima were also observed. Five early-type and eccentric-orbit eclipsing binaries: HV 982 (P = 5.d34, e = 0.15), HV 2274 (5.d73, 0.17), MACHO 78.6097.13 (3.d11, 0.05), MACHO 81.8881.47 (3.d88, 0.22), and MACHO 79.5377.76 (2.d64, 0.06) were studied.
Methods. The O–C diagrams of the systems were analysed using all reliable timings found in the literature, and new or improved elements of apsidal motion were obtained. Light and radial velocity curves of MACHO 81.8881.47 and MACHO 79.5377.76 were analysed using the program PHOEBE.
Results. We derived for the first time or significantly improved the relatively short periods of apsidal motion of 211 (12), 127 (8), 48 (13), 103 (20), and 42 (19) years, respectively. The internal structure constants, log k2, were found to be –2.37, –2.47, –2.17, –2.02, and –1.86 respectively, under the assumption that the component stars rotate pseudosynchronously. The relativistic effects are weak, up to 6% of the total apsidal motion rate. The masses for MACHO 81.8881.47 resulted in 5.51 (0.21) and 5.40 (0.19) M⊙, while for MACHO 79.5377.76 the masses are 11.26 (0.35) and 11.27 (0.35) M⊙, respectively.
Key words: binaries: eclipsing / stars: early-type / stars: general / stars: fundamental parameters / Magellanic Clouds
Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 68.A-0223(A), and on data collected with the Danish 1.54 m telescope at the ESO La Silla Observatory.
Appendices are available in electronic form at http://www.aanda.org
© ESO, 2013
1. Introduction
Eccentric eclipsing binaries (EEB) with an apsidal motion can provide us with an important observational test of theoretical models of stellar structure and evolution. A long-term collecting the times of minima of EEBs observed throughout the apsidal motion cycle and 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, e.g. Maeder (1999), Claret (2004), or Claret (2006); however, to distinguish between them and to test which one is more suitable 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, to discriminate between the models only stellar parameters for EEBs with an accuracy of 1% can be used.
The Magellanic Clouds are of prime importance in the context of stellar evolution theory. However, the chemical composition of the Magellanic Clouds differs from that of the solar neighborhood (e.g. Ribas 2004) and the study of these massive and metal-deficient stars in the LMC 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 2004).
Here we analyse the observational data and rates of apsidal motion for five LMC detached eclipsing systems. All these systems are early-type objects known to have eccentric orbits and to exhibit apsidal motion. Similar studies of LMC EEBs have been presented by Michalska & Pigulski (2005, hereafter MP05) and Michalska (2007).
Apsidal motion elements for HV 982, HV 2274, MACHO 78.6097.13, MACHO 81.8881.47, and MACHO 79.5377.76.
2. Observations of minimum light
The monitoring of faint EEBs in external galaxies requires only moderate telescopes in the 1–2 m class range equipped with a modern CCD camera. However, a large amount of observing time is needed, which is unavailable at larger telescopes. During past observational seasons, we accumulated over 1600 photometric observations at selected phases during primary and secondary eclipses and derived 16 precise times of minimum light for selected eccentric systems. New CCD photometry was obtained at the La Silla Observatory in Chile, where the 1.54 m Danish telescope (hereafter DK154) with the CCD camera and RI filters was used.
All CCD measurements were dark-subtracted and then flat-fielded using sky exposures taken at either dusk or dawn. Several comparison stars were chosen in the same frame as the variables. A synthetic aperture photometry and astrometry software developed by Velen and Pravec called Aphot, was routinely used for data obtained. No correction for differential extinction was applied because of the proximity of the comparison stars 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 errors were determined by the classical Kwee-van Woerden (1956) algorithm. All new times of minima are given in Table A.1, where epochs are calculated from the ephemeris given in Table 1; the other columns are self-evident.
3. Photometry
For all of the systems we harvested the Macho (Faccioli et al. 2007) and Ogle (Graczyk et al. 2011) photometry available online. These photometric data were used both for minima times analysis as well as for light curve analysis.
The analysis of the light curves for two of the systems was carried out using the program PHOEBE, ver. 0.31a (Prša & Zwitter 2005), which is based on the Wilson-Devinney algorithm (Wilson & Devinney 1971) and its later modifications, but some of the parameters had to be fixed during the fitting process. The albedo coefficient remained fixed at value 1.0 and the gravity darkening coefficients g = 1.0. The limb darkening coefficients were interpolated from the van Hamme tables (van Hamme 1993). A problem emerged with the synchronicity parameters (Fi) due to poor coverage of the RV data near the eclipses and low quality of the light curves used for the LC analysis, hence we fixed these values at Fi = 0. The temperature of the primary component was derived from the V − I photometric index, from MP05 and from the resulting masses derived from the combined LC+RV analysis.
4. Spectroscopy
The spectroscopic data for two of the systems (MACHO 81.8881.47, MACHO 79.5377.76) were found in the ESO Archive of the UV-Visual Echelle Spectrograph (UVES) at the Very Large Telescope (VLT). The spectra were obtained during the ESO Period 68 program “Precise distances to the LMC and SMC from double-lined eclipsing binaries”; the PI of the project was A. Clausen. Typical exposure times were about 1500 seconds, while the spectra typically have a signal-to-noise ratio (S/N) of about 50. The original data were reduced using the standard ESO routines. The final radial velocities (hereafter RV) used for the analysis were derived via a manual cross-correlation technique (i.e. the direct and flipped profile of spectral lines manually shifted on the computer screen to achieve the best match) using program SPEFO (Horn et al. 1996; Škoda 1996) on several absorbtion lines in the measured spectral region (usually Hβ to Hϑ). The derived radial velocities are given in Table B.1. We estimate the error of individual data points to be about 5 km s-1.
5. An approach for the analysis
For the analysis we used the approach presented below. For the systems where the inclination of the orbit is known, the first two steps can be skipped.
-
First, all of the available photometric data were analysed,resulting in a set of minima times. Preliminary apsidal motionparameters were derived (with the assumption i = 90°).
-
Second, the eccentricity (e), argument of periastron (ω), and apsidal motion rate (
) derived from the apsidal motion analysis were used for the preliminary light curve (hereafter LC) analysis.
-
Third, the inclination (i) from the LC analysis was used for the final apsidal motion analysis.
-
And finally, the resulting e, ω, and
values from the apsidal motion analysis were used for the final LC + RV analysis.
6. Apsidal motion analysis
The apsidal motion in all eccentric systems was studied by means of an O–C diagram
analysis. All available times of minima, both published in the literature and newly
measured, 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
determined in this procedure. The periastron position ω is given by the
linear equation
where
is the rate of periastron advance, and the position of periastron for the zero epoch
T0 is denoted as ω0. The relation
between the sidereal and the anomalistic period, Ps and
Pa, is given by
and
the period of apsidal motion by
All
new precise CCD times of minima were used with a weight of 10 or 20 in our computation. Some
of our less precise measurements were weighted by a factor of 5, while the earlier visual
and photographic times (esp. the times of the mid-exposure of a photographic plate) were
given a weight of one or nought because of the large scatter in these data.
7. Analysis of the systems
7.1. HV 982
The detached eclipsing binary HV 982 (also known as MACHO 82.8043.26, LMV110; Vmax = 14m.65; Sp. B5V) is a relatively well-known LMC binary with an eccentric orbit (e = 0.15) and a moderate orbital period of 5.3 days. It was discovered to be a variable star by Gaposhkin (1970), who published the first photographic light curve.
Pritchard et al. (1994, 1998) in their photometric study derived high surface temperatures
(~28 000 K) and masses (~8 M⊙) of components. They also
derived the apsidal motion period U = 205 ± 7 years. The precise stellar
parameters of components of HV 982 were derived
spectroscopically by Fitzpatrick et al. (2002), who
found components with similar mass and size, The
following linear light elements were given in that paper:
Using
the complete analyses of HV 982 they also found the distance to the center of the LMC
dLMC = 50.7 ± 1.2 kpc. Later, Clausen et al. (2003) presented a new accurate CCD uvby
light curve obtained at the Danish 1.54 m telescope at La Silla and derived the precise
photometric elements with apsidal motion period U = 208 ± 15 years.
All CCD times of minimum light given in Pritchard et al. (1998) and Clausen et al. (2003) were
incorporated into our analysis. Using Macho (Faccioli et al. 2007) and Ogle (Graczyk et al. 2011) photometry, we were able to derive additional
times of minimum light. A total of 68 times of minimum light were used in our analysis
(see Table A.1). The orbital inclination adopted
was , based on the analysis of Fitzpatrick et al. (2002). The computed apsidal motion parameters and their internal errors of the
least-squares fit are given in Table 1. In this
table, Ps denotes the sidereal period,
Pa the anomalistic period, e represents the
eccentricity, and
is the rate of periastron advance (in degrees per cycle and in degrees per year). The zero
epoch is given by T0, and the corresponding position of the
periastron is represented by ω0. The O–C residuals for all
times of minimum with respect to the linear part of the apsidal motion equation
(Min = T0 + P × E) are
shown in Fig. 1. The non-linear predictions,
corresponding to the fitted parameters, are plotted for primary and secondary eclipses.
7.2. HV 2274
![]() |
Fig. 1 O–C diagram for the times of minima of HV 982. 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 photoelectric or CCD measurements that were given higher weights in the calculations. |
The detached and double-lined eclipsing binary HV
2274 (also known as MACHO 19.3577.7, 2MASS J05024076-6824212, LMV 182, FL
3556; Vmax = 14m.13; Sp. B1-2
IV-III) is a relatively bright and well-studied LMC eclipsing binary with an eccentric
orbit (e = 0.17) and a moderate orbital period of 5.7 days. It was
discovered to be a variable star by Leavitt (1908).
Later, Shapley & Nail (1953) recognized its
eclipsing nature and classified it as a β Lyrae type. The eccentric orbit
and apsidal motion of HV 2274 was first announced
by Watson et al. (1992), who obtained the
BVIc CCD photometry at the Mount John
University Observatory. See also the history of HV
2274 in the last paper mentioned. Later Claret (1996) found the masses and evolutionary status of this system:
The
fundamental properties of HV 2274 were most recently given in Ribas et al. (2000) who determined the precise absolute parameters of
both eclipsing components to be
They
also derived the improved apsidal motion period U = 123 ± 3 years and the
value of internal structure constant log
k2,obs = −2.56. Since the above-mentioned
papers were published, new times of minima have been obtained, which allowed us to reduce
the uncertainties in the derived parameters. We collected all times of minimum light given
in the literature together with new ones derived from Macho, Ogle, and
our new photometry obtained in Chile. All of these values are listed in Table A.1. In total, 58 precise times of minimum light were
used in our analysis, including 29 secondary eclipses. The orbital inclination adopted was
, based on the analysis of Ribas et al. (2000).
Analysing the available data using the apsidal motion hypothesis, we found an additional variation superimposed on the apsidal motion. Hence, we used a different code computing the apsidal motion parameters together with the third-body orbit (a so-called light travel time effect), see e.g. Irwin (1959). Altogether, ten parameters were fitted (five from apsidal motion, five from the third body hypothesis), thus this approach led to an acceptable solution. The resulting parameters of the fit are given in Tables 1 and 2, the complete O–C diagrams are shown in Figs. 2 and 3. From the third-body parameters we were also able to compute the mass function of the distant component, which resulted in f(m3) = 0.053 ± 0.008 M⊙. From this value, we calculated the predicted minimal mass of the third body (i.e. assuming coplanar orbits i3 = 90°), which resulted in m3,min = 3.4 M⊙. If we propose this body in the system, one can ask whether it is detectable somehow in the already-obtained data. The period is rather long for continuous monitoring of the radial velocity changes, but detecting the third light in the light curve solution is difficult. Assuming a normal main sequence star, its luminosity is about only 1% of the total system luminosity. Such a weak third light could be detectable only in extremely precise photometric data for the light curve solution.
7.3. MACHO 78.6097.13
Third-body orbit parameters for HV 2274.
![]() |
Fig. 2 O–C diagram for the times of minima of HV 2274. See legend to Fig. 1. The final fit is composed from two effects: the apsidal motion together with the third-body hypothesis. |
![]() |
Fig. 3 O–C residuals for HV 2274 after subtraction of the apsidal motion term. Only the third body effect is plotted. |
The detached eclipsing binary MACHO 78.6097.13 (also known as OGLE J051804.81-694818.9, LMC MP#5; Vmax = 14m.37, sp O9V+O9V) is an eccentric binary system (e = 0.046) with a short orbital period (P = 3.1 d) discovered by MP05 in the Ogle field LMC-SC7.
Our analysis gives the following ephemeris: The
most detailed analysis of the system was published by González et al. (2005), which was based on the ESO spectral observations
together with the MACHO data. They also derived the masses and radii of both components
that were used in the present analysis. Both components are of O9 spectral types, which
makes this system the earliest in our sample.
Our new times of minimum light, as well as the timings derived from Macho and Ogle photometry are given in Table A.1. All of these data (31 minima times) were used in our calculations. Using the parameters presented by González et al. (2005), we analysed the system on apsidal motion, deriving the parameters given in Table 1. As one can see, this system has the lowest eccentricity in our sample and the apsidal advance is rather fast: almost one half of the period has been covered with observations so far.
7.4. MACHO 81.8881.47
The detached eclipsing binary MACHO 81.8881.47
(also known as OGLE J053517.75-694318.7, LMC MP#7; Vmax =
14m.9; Sp. B) is a relatively bright binary system with an
eccentric orbit (e = 0.2) and a moderate orbital period
P ≃ 3.9 days. Its variability was discovered by MP05 in the OGLE field
LMC-SC16. Most recently, Graczyk et al. (2011)
included this star in their catalogue of eclipsing binaries in the LMC. They also gave the
preliminary ephemeris: Using
Ogle and Macho photometry, we were also able to derive additional
times of minimum light; two more minima were derived from our observations. Only 30 times
of minimum light were used in our analysis (see Table A.1). As one can clearly see from the shape of the light curve and the different
duration of both the primary and secondary eclipse, this is the system with the highest
value of eccentricity in our sample.
We used the ESO data for deriving the radial velocities of the components. Ten UVES spectra were found, of which nine were usable for the analysis. The derived RVs of both components are given in Table B.1. Together with these radial velocities, the MACHO light curve was used for the subsequent combined LC+RV analysis. The final parameters of the solution are given in Table 3, while the fits are plotted in Figs. 5 and 6. Our results show the rather noticeable property that the more massive component (the primary) is smaller and less luminous (i.e. has a lower temperature). However, this is still a preliminary result based on only nine RVs and a poor fit. Although the scenario is possible, the more probable explanation is that the mass ratio is inverse, and the primary and secondary components are interchanged. This can be allowed for within the uncertainties of the mass ratio. A more detailed analysis is needed, based on more precise spectral observations. The orbital inclination was about , which was later used for the apsidal motion analysis. The resulting parameters of apsidal motion are found in Table 1, and the current O–C diagram is shown in Fig. 7. As one can see, the period is rather long and only about 1/5 has been covered so far.
Light and radial velocity curve fit parameters for MACHO 81.8881.47 and MACHO 79.5377.76.
![]() |
Fig. 5 Radial velocity curve for MACHO 81.8881.47. |
![]() |
Fig. 6 Light curve for MACHO 81.8881.47. |
7.5. MACHO 79.5377.76
The detached eclipsing binary MACHO 79.5377.76
(also known as OGLE J051323.98-692249.2, Vmax =
15m.8; Sp. B) is a fairly neglected binary system with a
moderate eccentric orbit (e = 0.06) and a short orbital period
(P ≃ 2.64 day): We
used the MACHO and OGLE photometry together with our new observations from Chile to derive
the times of minima, the ESO spectral observations to derive the radial velocities, and
the MACHO photometry to model the light curve of the system. Our results are plotted in
Figs. 8 and 9.
The resulting parameters are given in Table 3. As
one can see, both components are of equal mass and their spectral type was estimated to be
about B0-B1. Radial velocities used for our analysis are also given in Table B.1. The coverage of the radial velocity curve is
better than for MACHO 81.8881.47; however, it was rather difficult to derive the
synchronicity parameters Fi, and so we fixed
them at Fi = 0. A total of 34 reliable times
of minimum light were used in our analysis including 16 secondary eclipses (see tables in
the Appendices). The final apsidal motion elements are given in Table 1, and the O–C graph is shown in Fig. 10. The apsidal period is the shortest among systems
studied here, only 42 years.
![]() |
Fig. 8 Radial velocity curve for MACHO 79.5377.76. |
![]() |
Fig. 9 Light curve for MACHO 79.5377.76. |
Basic physical properties of HV 982, HV 2284, MACHO 78.6097.13, MACHO 81.8881.47, and MACHO 79.5377.76 and their internal structure constants.
8. Discussion
The detection of apsidal motion in EEB provides the opportunity to test models of stellar
internal structure. The internal structure constant (ISC)
k2,obs is related to the variation in the
density inside the star and can be derived using the expression
(1)where
c21 and c22 are functions of the
orbital eccentricity, fractional radii, the masses of the components, and the ratio between
rotational velocity of the stars and Keplerian velocity (Kopal 1978). We also assume that the component stars rotate pseudosynchronously
with the same angular velocity as the maximum orbital value at periastron (see e.g. Kopal
1978). Another possible approach is to use the
value of vsini as derived from the combined LC+RV analysis
published earlier. However, there could be a problem with the inclination of the rotation
axis (as in the case of DI Her) and, moreover, the error of the internal structure constants
is by far dominated by the term
in the
equations. In addition to the classical Newtonian contribution, the observed rate of
rotation of the apses includes the contribution from General Relativity (Giménez 1985),
(2)where
Mi denotes the individual masses of the
components in solar units and P is the orbital period in days.
The values of
and the resulting mean internal structure constants
k2,obs for the systems studied are given in
Table 4. Theoretical values
k2,theo according to available theoretical
models for the internal stellar structure computed by Claret (2006) for given masses of components are presented in Table 4.
![]() |
Fig. 11 Position and systemic velocities (see the colour scale on the right) of eclipsing binaries located in the LMC and its vicinity, see Table 5. The larger the symbol, the higher the precision. The background image of the LMC is used with the permission of the author Robert Gendler (http://www.robgendlerastropics.com). |
We tried to compare our resulting systemic velocities with other published values of eclipsing binaries in the LMC, see Table 5 and Fig. 11. There are still only a few such systems studied in detail (i.e. LC+RV analysis). Therefore, reliable analysis of different velocities within the LMC is still very difficult. We can only compare our Fig. 11 with other kinematic studies of the LMC published earlier, e.g. that by Reid & Parker (2006) or Rohlfs et al. (1984), that were based on much larger radial-velocity data sets. Nevertheless, we can conclude that our two new systemic velocities roughly fit the overall picture and the total velocity dispersion within the LMC is more than 30 km s-1.
Radial velocities of eclipsing binaries in the LMC derived from various analyses.
9. Conclusions
The apsidal motion in EEB has been used for decades to test evolutionary stellar models. This study provides accurate information on the apsidal motion rates of five main-sequence early-type binary systems in the LMC: HV 982, HV 2274, MACHO 78.6097.13, MACHO 81.8881.47, and MACHO 79.5377.76. In our Galaxy there are known a few hundreds of apsidal motion EEBs, however in the LMC there are still only a few dozen of these systems (Michalska & Pigulski 2005). Hence this study still presents an important contribution to the topic. The relativistic effects are weak, being up to 6% of the total apsidal motion rate. For the systems MACHO 79.5377.76 and MACHO 81.8881.47, their light and radial velocity curves were analysed for the first time yielding the stellar parameters of both components given in Table 4. Moreover, when the MACHO 79.5377.76 internal structure constant and the theoretical model are compared the system appears to be very young (~2 × 106 yr).
In spite of the considerable amount of observational data that has been collected for decades, the absolute dimensions of massive binary components are only known with an accuracy of about 1–3% (e.g. Clausen 2004). More detailed study based on more precise radial velocities would be very profitable to derive the physical properties of components with higher accuracy. The most promising system for further detailed analysis seems to be HV 2274 because of the putative third component in the system. We still know only a few such systems nowadays, see e.g. Bozkurt & Deǧirmenci (2007), while HV 2274 is the first to be discovered out of our Galaxy. Only precise spectral observations and their disentangling should reveal its true nature.
Online material
Appendix A: Table of minima
List of the minima timings used for the analysis.
Appendix B: Table of radial velocities
List of the radial velocities used for the analysis.
Acknowledgments
We do thank the MACHO and OGLE teams for making all of the observations easily and publicly available. This work was supported by the Research Program MSM0021620860 Physical Study of objects and processes in the Solar System and in Astrophysics of the Ministry of Education of the Czech Republic, by the Czech Science Foundation grant no. P209/10/0715, by the grant UNCE 12 of the Charles University in Prague, and by the grant LG12001 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 authors would like to thank M. Zejda, J. Liška, J. Janík, and M. Skarka for their important help with photometric observations. G.Michalska is also acknowledged for sending us the unpublished times of minima for selected binaries. The following internet-based resources were used in research for this paper: the SIMBAD database and the VizieR service operated at the CDS, Strasbourg, France, the NASA Astrophysics Data System Bibliographic Services, and the ESO Science Archive Facility.
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All Tables
Apsidal motion elements for HV 982, HV 2274, MACHO 78.6097.13, MACHO 81.8881.47, and MACHO 79.5377.76.
Light and radial velocity curve fit parameters for MACHO 81.8881.47 and MACHO 79.5377.76.
Basic physical properties of HV 982, HV 2284, MACHO 78.6097.13, MACHO 81.8881.47, and MACHO 79.5377.76 and their internal structure constants.
Radial velocities of eclipsing binaries in the LMC derived from various analyses.
All Figures
![]() |
Fig. 1 O–C diagram for the times of minima of HV 982. 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 photoelectric or CCD measurements that were given higher weights in the calculations. |
In the text |
![]() |
Fig. 2 O–C diagram for the times of minima of HV 2274. See legend to Fig. 1. The final fit is composed from two effects: the apsidal motion together with the third-body hypothesis. |
In the text |
![]() |
Fig. 3 O–C residuals for HV 2274 after subtraction of the apsidal motion term. Only the third body effect is plotted. |
In the text |
![]() |
Fig. 4 O–C graph for the times of minimum of MACHO 78.6097.13. See legend to Fig. 1. |
In the text |
![]() |
Fig. 5 Radial velocity curve for MACHO 81.8881.47. |
In the text |
![]() |
Fig. 6 Light curve for MACHO 81.8881.47. |
In the text |
![]() |
Fig. 7 O–C graph of MACHO 81.8881.47. See legend for Fig. 1. |
In the text |
![]() |
Fig. 8 Radial velocity curve for MACHO 79.5377.76. |
In the text |
![]() |
Fig. 9 Light curve for MACHO 79.5377.76. |
In the text |
![]() |
Fig. 10 O–C diagram for MACHO 79.5377.76. See legend to Fig. 1. |
In the text |
![]() |
Fig. 11 Position and systemic velocities (see the colour scale on the right) of eclipsing binaries located in the LMC and its vicinity, see Table 5. The larger the symbol, the higher the precision. The background image of the LMC is used with the permission of the author Robert Gendler (http://www.robgendlerastropics.com). |
In the text |
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