Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev2.2) Automated analysis of eclipsing binary lightcurves with EBAS. II. Statistical analysis of OGLE LMC eclipsing binaries 6 0 0 2 T. Mazeh,1⋆ O. Tamuz1 and P. North2 n a 1School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, TelAviv University,TelAviv, Israel J 0 2Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Laboratoire d’Astrophysique, 1 Observatoire, CH-1290 Sauverny, Switzerland 1 v 1 5February2008 0 2 1 0 ABSTRACT 6 0 / h p InthefirstpaperofthisserieswepresentedEBAS,anewfullyautomated - o algorithmtoanalysethelightcurvesofeclipsingbinaries,basedonthe EBOP r t code. Here we apply the new algorithm to the whole sample of 2580 binaries s a : found in the OGLE LMC photometric surveyand derive the orbitalelements v i for1931systems.Toobtainthestatisticalpropertiesoftheshort-periodbina- X r riesoftheLMCweconstructawelldefinedsubsampleof938eclipsingbinaries a with main-sequence B-type primaries. Correcting for observational selection effects, we derive the distributions of the fractional radii of the two compo- nentsandtheir sum,the brightnessratiosandthe periodsofthe short-period binaries. Somewhat surprisingly, the results are consistent with a flat distri- bution in log P between 2 and 10 days. We also estimate the total number of binariesin the LMC with the same characteristics,and notonly the eclipsing binaries, to be about 5000. This figure leads us to suggest that (0.7±0.4)% of the main-sequence B-type stars in the LMC are found in binaries with pe- riods shorter than 10 days. This frequency is substantially smaller than the fractionofbinaries foundby smallGalacticradial-velocitysurveysof B stars. Onthe other hand,the binary frequency foundby HST photometric searches within the late main-sequence stars of 47 Tuc is only slightly higher and still consistent with the frequency we deduced for the B stars in the LMC. Keywords: methods:dataanalysis-binaries:eclipsing-MagellanicClouds. 2 T. Mazeh, O. Tamuz & P. North 1 INTRODUCTION A number of large photometric surveys have recently produced unprecedentedly large sets of high S/N stellar lightcurves (e.g., Alcock et al. 1997). One of these projects is the OGLE study ofthe SMC (Udalski et al.1998)and theLMC (Udalski et al.2000),which has already yielded (Wyrzykowski etal.2003)afewthousandlightcurves ofeclipsing binaries.These two sets of lightcurves enable a statistical analysis of a large sample of short-period extragalactic binaries, for which the absolute magnitudes of all systems are known to a few percent. No such sample is known in our Galaxy. One such statistical study was the analysis of North & Zahn (2003), who derived the orbital elements and stellar parameters of 153 eclipsing binaries discovered by OGLE in the SMC. North & Zahn examined the eccentricities of the eclipsing binaries as a function of the ratio between the stellar radii and the binary separation, and compared their result with a similar dependence derived from the elements of the binaries discovered by Alcock et al. (1997) in the LMC. They then considered the implication of their findings for the theory of tidal circularization of early-type primaries in close binaries (e.g., Zahn 1975, 1977; Zahn & Bouchet 1989). Inafollow-uppaperNorth&Zahn(2004)analyzedanothersetof510lightcurvesselected from the 2580 eclipsing binaries discovered in the LMC by the OGLE team (Wyrzykowski et al. 2003).Again, they derived the ratio between the stellar radii and the binary separation for these systems, and examined the dependence of the binary eccentricity on this ratio. The previous analyses used only a small fraction of the sample of eclipsing binaries found in the OGLE data. The goal of the present study is to analyze the whole sample of 2580 lightcurves discovered by OGLE in the LMC, and to derive some statistical properties of the short-period binaries, after correcting for observational selection effects. Such a correction is essential for the derivation of the period distribution, for example, because the probability of detecting an eclipsing binary is a strong function of the orbital period. In order to apply an appropriate correction one needs a complete homogeneous data set of lightcurves, all discovered by the same photometric survey of a well defined sample of stars. These require- ments were exactly met by the OGLE data set for the LMC, enabling such analysis for a large sample of eclipsing binaries for the first time. A completely automated algorithm is needed to analyse the large set of eclipsing binaries ⋆ E-mail:[email protected] Analysing lightcurves of eclipsing binaries in LMC with EBAS 3 at hand. Two such codes were developed recently. Wyithe & Wilson (2001) have constructed an automatic scheme based on the Wilson-Divenney (=WD) code to analyze the OGLE lightcurves detected in the SMC in order to find eclipsing binaries suitable for distance measurements. At the last stages of writing this paper another study with an automated lightcurve fitter — DEBiL, was published (Devor 2005). DEBiL was constructed to be quick andsimple, andtherefore has itsown lightcurve generator, which does not account for stellar deformation and reflection effects. This makes it especially suitable for detached binaries. We will use in this work our EBAS, which was presented in Paper I (Tamuz, Mazeh & North 2005) and is based on the EBOP code. The complexity of EBAS is in between DEBiL and the automated WD code of Wyithe & Wilson. To facilitate the search for global minima in the convolved parameter space, EBAS performs two parameter transformations. Instead of the radii of the two stellar components of the binary system, measured in terms of the binary separation, EBAS uses the total radius, which is the sum of the two relative radii, and their ratio. Instead of the inclination we use the impact parameter — the projected distance between the centres of the two stars in the middle of the primary eclipse, measured in terms of the total radius. The set of parameters of the EBAS version used here includes the bolometric reflection of the two stars, A and A . When A = 1 the primary star reflects all the light cast on it p s p by the secondary. Together with the tidal distortion of the two components, which is mainly determined by the mass ratio of the two stars, thereflection coefficients A and A determine p s the light variability of the system outside the eclipses. Paper I discussed the reliability of the values of these two parameters as found by EBAS. To simplify the analysis, the present version of EBAS assumes there is no contribution of light from a third star and that the mass ratio is unity. Paper I discusses the implication of these choices on the parameters’ values, showing that the values of the total radius and the surface brightness ratio are only slightly modified by these two assumptions. The only parameter which is systematically modified by the third-light assumption is the impact parameter, which is directly associated with the orbital inclination of the binary. Note, however, that we are not interested in one specific system but aim, instead, at deriving the gross characteristics of the short-period binaries. As the analysis of this paper does not use the inclinations of the eclipsing binaries for the derivation of the statistical features of the short-period binaries, we regard the resulting distributions as probably correct. PaperIintroducesanew’alarm’statistic,A,toreplacehumaninspectionoftheresiduals. 4 T. Mazeh, O. Tamuz & P. North EBAS uses the new statistic, which is sensitive to the correlation between neighbouring residuals to decide automatically whether a solution is satisfactory. In this work we consider only the 1931 systems that yielded solutions with low enough alarm value. To check the reliability of our results we compare our geometrical elements with those derived by Michalska & Pigulski (2005, hereafter MiP05) with the WD code for 85 binaries, based on EROS, MACHO and OGLE data. The comparison is reassuring, as it shows that the geometrical parameters of EBAS are close to the ones derived by the more sophisticated WD approach. To derive the orbital distributions of the short-period binaries we had to trim the sam- ple, to get a homogeneous sample which we could correct for selection effects. We were left with 938 main-sequence binaries with periods shorter than 10 days and system magni- tudes between 17 and 19 in the I band, most of which are binaries with B-type primaries. This makes the range of the derived period distribution quite narrow. Nevertheless, the data yielded somewhat surprising distributions, which might have implications on binary population studies. Section 2 presents the resulting orbital elements of the OGLE LMC eclipsing binaries and compares the derived elements with those of MiP05. Section 3 details the procedure to focus on a well defined homogeneous subsample and Section 4 derives the statistical features of the short-period binaries, after correcting for the observational selection effects. Section 5 discusses the new findings, and Section 6 summarizes the paper. 2 ANALYSIS OF THE OGLE LMC ECLIPSING BINARIES The LMC OGLE-II photometric campaign (Udalski et al. 2000) was carried out from 1997 to 2000, during which between 260 and 512 measurements in the I band were taken for 21 fields (Zebrun et al. 2001). Wyrzykowski et al. (2003) searched the photometric data base and identified 2580 binaries. We analysed those systems with EBAS and found 1931 acceptable solutions. Three types of binaries were excluded: • Four binaries were found to appear twice in the list of binaries, with the same period (except for a factor of 2) and with very close positions. Apparently, they originated from the overlap between the different fields of OGLE, and evaded the scrutiny of Wyrzykowski et al. (2003). • EBAS found 376 solutions with alarm too high — A > 0.5. Visual inspection showed Analysing lightcurves of eclipsing binaries in LMC with EBAS 5 that most of these systems are contact binaries that EBOP can not model properly. Some might be ellipsoidal variables (see Wyrzykowski et al. 2003) or other types of periodic vari- ables, other than eclipsing binaries. • EBAS found 269 solutions that yielded sum of radii r = r + r too large. Following t p s the Roche lobe radius calculation in Eggleton (1983): 0.49q2/3 R /a = , (1) RL 0.6q2/3 +ln(1+q1/3) which reduces for q = 1 to R /a = 0.379, we did not accept solutions with RL r > 0.65(1−ecosω) , (2) t assuming the EBOP model can not properly account for the deformation of the two stars if one or two stars are too close to their Roche-lobe limit, at least at the periastron passage. We were thus left with 1931 acceptable solutions. 2.1 Solution examples In Fig 1 we plot 10 representative lightcurves derived by the EBAS. Their elements are given in Table 1 and Table 2. Table 1 gives some global information on each lightcurve and the goodness-of-fit of its model. It lists the number of measurements, the averaged observed I-magnitude and the rms of the scatter of the OGLE lightcurve. It also lists the χ2, the alarm A of the fit and the detectability measure D (see below), and whether this binary was included in the trimmed sample (+), as detailed in the next section. Table 2 lists the derived elements of EBAS and their estimated uncertainties. This includes the I-magnitude of the system at quadrature (the SFACT parameter of EBOP, see Table 2 of Paper I), the period, P, in days, the sum of radii, r , the ratio of radii, k, the surface brightness ratio, t J , the impact parameter, x, and the eccentricity e, multiplied by cosω and sinω, where ω s is the longitude of periastron. Note that the mag parameter in Table 2 is the brightness of the system out of eclipse, and therefore is different from the observed averaged mag given in Table 1. Since its formal error is very small, we give it to 4 or 5 decimal figures because this might be useful for comparison purposes, even though we are well aware that systematic errors — whether in the original photometric data or in their fit — are far larger than that level of accuracy. Let us recall that some parameters were held fixed in the fit, as explained in Paper I: the linear limb-darkening coefficients, with a value typical of main-sequence B stars (u = u = 0.18), the gravity darkening coefficients (y = y = 0.36), the mass ratio p s p s 6 T. Mazeh, O. Tamuz & P. North Table 1.Thetenbinaries:observations andgoodness-of-fit OGLEName N mag rms χ2 A logD Sample 052100.97-692526.3 482 17.44 0.12 397 -0.3 15015 + 052101.19-694725.0 478 18.73 0.09 453 -0.2 688 + 052101.72-693505.0 479 18.12 0.07 381 0.1 1606 052101.94-692834.5 472 17.70 0.06 335 -0.1 1969 + 052104.75-691842.6 479 18.64 0.16 425 -0.0 3658 + 053008.33-693438.0 507 17.69 0.11 410 0.4 12503 + 052105.63-694533.3 477 18.98 0.10 376 0.1 660 + 052105.71-693143.7 482 18.71 0.10 401 -0.2 1540 + 052106.82-693136.4 478 17.69 0.04 299 0.0 1353 052106.83-694632.3 481 17.59 0.04 353 0.2 1704 + Table 2.Thetenbinaries:orbitalelements OGLEName mag P rt k Js x ecosω esinω Ap As 052100.97 17.39743 3.1227063 0.2968 0.80 0.934 0.233 0.05902 0.0041 0.73 1.00000 -692526.3 ±0.00095 ±0.0000061 ±0.0038 ±0.26 ±0.015 ±0.018 ±0.00066 ±0.0093 ±0.18 ±0.00026 052101.19 18.7058 9.156860 0.1788 1.16 0.97 0.339 -0.0038 0.005 0.9999 0.99993 -694725.0 ±0.0025 ±0.000051 ±0.0086 ±0.37 ±0.13 ±0.062 ±0.0014 ±0.057 ±0.0013 ±0.00090 052101.72 18.0867 1.4570012 0.674 1.31 0.65 0.696 0.0016 0.004 0.75 0.916 -693505.0 ±0.0015 ±0.0000070 ±0.024 ±0.26 ±0.11 ±0.016 ±0.0042 ±0.012 ±0.20 ±0.099 052101.94 17.6753 1.3626777 0.560 1.02 0.954 0.607 0.0003 0.011 1.000 0.998 -692834.5 ±0.0015 ±0.0000025 ±0.015 ±0.10 ±0.069 ±0.019 ±0.0029 ±0.017 ±0.013 ±0.054 052104.75 18.5378 1.7568736 0.534 1.65 0.895 -0.035 0.0002 -0.018 0.55 0.11 -691842.6 ±0.0023 ±0.0000041 ±0.016 ±0.48 ±0.038 ±0.094 ±0.0032 ±0.028 ±0.31 ±0.28 053008.33 17.6360 4.136428 0.4594 4.86 0.065 0.736 0.0006 -0.061 0.01 0.945 -693438.0 ±0.0010 ±0.000015 ±0.0092 ±0.51 ±0.018 ±0.026 ±0.0052 ±0.028 ±0.16 ±0.099 052105.63 18.9528 6.7992460 0.1450 0.80 0.757 0.275 0.0018 -0.002 1.0000 0.990 -694533.3 ±0.0029 ±0.0000035 ±0.0074 ±0.26 ±0.076 ±0.047 ±0.0021 ±0.066 ±0.0011 ±0.038 052105.71 18.6663 8.000289 0.1147 1.8 0.987 0.22 0.2069 -0.060 1.00 0.095 -693143.7 ±0.0022 ±0.000029 ±0.0053 ±1.8 ±0.089 ±0.13 ±0.0012 ±0.042 ±0.20 ±0.040 052106.82 17.6869 1.6991553 0.591 1.20 0.362 0.780 0.0015 0.031 0.988 0.820 -693136.4 ±0.0011 ±0.0000086 ±0.017 ±0.12 ±0.094 ±0.018 ±0.0079 ±0.022 ±0.022 ±0.087 052106.83 17.5775 4.796268 0.379 1.40 0.64 0.686 0.0029 -0.038 0.99 0.79 -694632.3 ±0.0011 ±0.000024 ±0.010 ±0.24 ±0.22 ±0.035 ±0.0024 ±0.055 ±0.15 ±0.20 (q = 1), the tidal lead/lag angle (t = 0) and the third light (L = 0). A and A are the 3 p s bolometric reflection coefficients, the value of which is between 0 and 1; they determine (together with the tidal distorsion of the components) the out-of-eclipse variability. They are adjusted in order to fit the variation outside eclipses, but since reflection effects are only crudely modeled by EBOP, one has to keep in mind that their value may have very limited physical significance. Of the ten stars, the lightcurve of OGLE052101.72-693505 yielded r too high, and the t primary of OGLE052106.82-693136.4 is probably not a main-sequence star (see below). Therefore both stars were not included in the final statistical analysis. Because the system 053008.33-693438.0 has a very shallow secondary minimum, its ratio of radii is poorly con- strained, and it probably has undergone mass exchange. Because of the very small surface brightness ratio, the reflection coefficient is meaningful for the secondary but not for the Analysing lightcurves of eclipsing binaries in LMC with EBAS 7 OGLE052100.97−692526.3 OGLE052101.19−694725.0 17.4 18.6 18.8 17.6 19 17.8 19.2 19.4 18 19.6 19.8 18.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 OGLE052101.72−693505.0 OGLE052101.94−692834.5 18 18.2 17.8 18.4 18 18.6 18.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 OGLE052104.75−691842.6 OGLE053008.33−693438.0 17.6 18.5 17.8 19 18 18.2 19.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 OGLE052105.63−694533.3 OGLE052105.71−693143.7 18.6 19 18.8 19 19.5 19.2 19.4 20 19.6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 OGLE052106.82−693136.4 OGLE052106.83−694632.3 17.7 17.6 17.8 17.7 17.9 17.8 17.9 18 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 1.Tenderivedlightcurves 8 T. Mazeh, O. Tamuz & P. North primary, which remains unaffected by the presence of its much cooler companion. Thus the very small A value has no real meaning. p The elements of all 1931 binaries are given in Table 3 and Table 4 with exactly the same format as in Table 1 and Table 2. Tables 3 & 4 appear only in electronic version of the journal in postscript format, while their machine-readable ASCII version is available in our web site1. 2.2 Comparison with the analysis of Michalska & Pigulski (2005) In Paper I we compared the elements of four systems derived by EBAS with those obtained by Gonz´alez et al. (2005). Here we wish to compare our results with the more extensive work ofMiP05 publishedvery recently. UsingdatafromEROS,MACHO andOGLE,MiP05fitted 98 LMC binaries with the WD code. Of those, 85 were included in the OGLE catalogue and 81 were solved by EBAS with low A. Because of the differences between WD and EBOP, we wish to compare only the geo- metric parameters of the solutions (see Paper I). Plotted in Fig 2 are our values for the 81 binaries versus MiP05’s, for the sum of radii, inclination and eccentricity. The total radius andtheeccentricity panelsshowquitesmallspreadaroundthestraightlines,whichrepresent the locus of equal values of the two solutions. Only the inclination shows a large scatter and a slight bias. However, as noted above, our statistical analysis does not use the inclinations of the eclipsing binaries for the derivation of the characteristics of the short-period binaries. Therefore, the comparison with MiP05’s solutions supports our assessment that while indi- vidual EBAS solutions might be inferior to WD solutions, the statistical interpretation of the entire sample remains valid. 3 TRIMMING THE SAMPLE The very large sample of 1931 short-period binaries enables us to derive some statistical features of the population of short-period binaries in the LMC. However, the sample suffers from serious observational selection effects, which affected the discovery of the eclipsing binaries. To be able to correct for the selection effects we need a well-defined homogeneous sample. We therefore trim the sample before deriving some parameter distributions in the next section. 1 http://wise-obs.tau.ac.il/∼omert/ Analysing lightcurves of eclipsing binaries in LMC with EBAS 9 r 0.6 t 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5 0.6 90 i 85 S A 80 B E 75 70 70 75 80 85 90 0.5 e 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 Michalska & Pigulski Figure2.ComparisonwiththeworkofMiP05:Elementsfor81systems.Panelsareforsumofradii,inclinationandeccentricity The most important observational selection effect is associated with the weak signal of the eclipse relative to the noise of the measurements. To apply a correction for this selection effect we need a distinct criterion that defines the systems that could have been detected by the OGLE LMC survey. Such a criterion is the detectability parameter D, which we define to be the number of points in the lightcurve, N, times the ratio of the variance of the ideal but variable signal, to the variance of the residuals: 10 T. Mazeh, O. Tamuz & P. North 250 200 150 100 50 0 2 3 4 5 6 Log D Figure 3.Distributionofthedetectabilityparameter logD forthe1931binarieswithacceptable solutions. var(m ) i D = N . (3) var(r ) i The m ’s are the values of the EBAS model at the times of observation, and the r ’s are the i i residualsofthemeasurementsrelativetothemodel.Oneseesthatsystemswithdeepeclipses, which are more easily detected, have larger D, and even more so when many measurements are concentrated within phases of eclipses. Fig 3 shows the distribution of logD for all 1931 solved systems. The histogram shows that there are only very few systems with logD below 2.6. We therefore set our detection limit, somewhat arbitrarily, at logD = 2.6, assuming that any binary below this limit could not have been detected. As pointed out by the referee, this limit translates into D = 400, which implies a 1σ variation in a lightcurve having 400 points — a rather typical number. We consider the few systems below this limit as exceptions. In order to get a homogeneous sample we ignore the binaries below this limit, and consider only the 1875 systems with logD > 2.6. Fig 4 shows the total magnitude of the binaries left in the sample as a function of their orbital periods. We are witnessing a lack of systems in the lower right corner of the plot. This is due to the fact that the probability of having an eclipse is approximately equal to the sum of fractional radii, r . The two absolute radii determine the total luminosity of their t system, while the binary separation determines the orbital period. Therefore, for a given period, the probability of having an eclipse is smaller for faint systems, with smaller stellar radii, than for brighter systems, with larger stars. In addition, the detectability D is smaller