Using the Optical Gravitational Lensing Experiment (OGLE) catalogue of eclipsing binaries, 15 contact binaries were identified towards the Small Magellanic Cloud and the Large Magellanic Cloud at vertical distances from the Galactic plane between 300 pc and 10 kpc. Based on the luminosity function calculated for these contact binaries, we estimated a frequency of occurrence relative to Main Sequence stars in the thick disc at roughly . This estimate suffers from the small number statistics, but is consistent with the value previously found for the solar neighbourhoo

Nef, Pascal D.

Rucinski, Slavek M.

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Mon. Not. R. Astron. Soc. 385, 2239–2242 (2008) doi:10.1111/j.1365-2966.2008.12994.xLuminosity function of contact binaries at high galactic latitudes towardsthe LMC and the SMCPascal D. Nef1 and Slavek M. Rucinski21Swiss Federal Institute of Technology (ETH-Zurich), 8093 Zurich, Switzerland2Department of Astronomy and Astrophysics, University of Toronto, 50 St George Str, Toronto, Ontario M5S 3H4, CanadaAccepted 2008 January 17. Received 2008 January 15; in original form 2007 October 4ABSTRACTUsing the Optical Gravitational Lensing Experiment (OGLE) catalogue of eclipsing binaries,15 contact binaries were identified towards the Small Magellanic Cloud and the Large Mag-ellanic Cloud at vertical distances from the Galactic plane between 300 pc and 10 kpc. Basedon the luminosity function calculated for these contact binaries, we estimated a frequencyof occurrence relative to Main Sequence stars in the thick disc at roughly 1600 . This estimatesuffers from the small number statistics, but is consistent with the value previously found forthe solar neighbourhood.Key words: binaries: eclipsing – stars: luminosity function, mass function.1 I N T RO D U C T I O NDuring the past few decades, several attempts have been made todetermine the local spatial density of contact binaries, with verydiversified results. Rucinski (2002) (where full references are givenand differences discussed) estimated their local spatial density at(1.02 ± 0.24) × 10−5 pc−3. Later, on the basis of the All Sky Au-tomated Survey (ASAS), a relative frequency of occurrence (RFO)of one contact binary among about 500 solar-type stars was derived(Rucinski 2006), which is in full agreement with the mentioned spa-tial density. The RFO for contact binaries at high galactic latitudes,however, has been entirely unknown.Here, we present an estimate for the spatial occurrence of contactbinary systems relative to Main Sequence (MS) stars in the thickdisc of our Galaxy, far from the Galactic plane. In particular, theluminosity function for contact binaries in two conical volumes to-wards the Small Magellanic Cloud (SMC) and the Large MagellanicCloud (LMC) covering parts of the thick disc and the halo is deter-mined. The results are then compared with the luminosity functionfor MS stars in the same region of the sky. The term ‘contact bina-ries’ is used here as a synonym for W UMa-type eclipsing binarieswith orbital periods in a range of 0.22–1 d. In this paper, for rea-sons explained below, we limit ourselves to a subset with periods<0.45 d.2 I D E N T I F I C AT I O N O F S H O RT P E R I O DC O N TAC T B I NA R I E S I N T H EO G L E - C ATA L O G U E O F E C L I P S I N G B I NA R I E SThe Optical Gravitational Lensing Experiment (OGLE) was in-tended to detect dark matter in the Milky Way Galaxy usingthe microlensing technique, with the Magellanic Clouds and theE-mail: pnef@phys.ethz.ch; rucinski@astro.utoronto.caGalactic bulge being the main targets of the survey (Udalski,Szymanski & Kubiak 1997). As a byproduct, OGLE provides high-quality, long-term photometry that can be used to analyse eclips-ing binary stars. Two online catalogues (Wyrzykowski et al. 2003,2004) were used for this study. They contain data in the stan-dard photometric BVI system for 2850 and 1351 eclipsing bina-ries, and cover an area of 4.6 and 2.4 deg2 of the central partsof the LMC and the SMC, respectively. The OGLE-II survey hasa faint limit of roughly I = 20.5 mag with a corresponding errorof 0.3 mag; the error becomes smaller for decreasing magnitude,reaching a bright limit of the survey at about 13 mag (Wyrzykowskiet al. 2004).The differentiation between contact binaries and other eclipsingbinary types in the two OGLE catalogues was carried out by apply-ing a contact binary criterion based on the Fourier analysis of thelight curves, which was introduced by Rucinski (1997).1 By meansof this criterion and using a visual inspection of the remaining lightcurves, we identified 10 and five contact binaries with orbital peri-ods P < 0.45 d towards the LMC and the SMC, respectively. Thelight curves are plotted later in A1. Furthermore, we used the on-line data base from the Massive Compact Halo Object (MACHO)project (Alcock et al. 1997) to confirm the classification of the con-tact binaries in the OGLE catalogue. A direct comparison with theOGLE catalogue was only possible for the contact binaries towardsthe LMC because there is no MACHO photometry available formost survey fields covering the SMC. The MACHO photometry ofthe 10 contact binaries towards the LMC is in good agreement withthe results obtained from the OGLE survey.21 More detailed explanations for this and other techniques used in this papercan be found in Rucinski (1997, 2006).2 As an exception, the OGLE photometry in V band for the contact bi-nary OGLE050905.22-693315.1 was found to have a gross error of roughly3 mag.C© 2008 The Authors. Journal compilation C© 2008 RAS2240 P. D. Nef and S. M. RucinskiFigure 1. This plot shows the vertical distance from the galactic mid-planefor the 10 and five contact binaries towards the LMC (squares) and the SMC(triangles), respectively. The Sun is assumed to be located 30 pc above theGalactic mid-plane.The contact binaries in our sample have orbital periods in therange of 0.22–0.45 d. The short period limit is a natural cut-off forcontact binaries (Rucinski 2007), whereas the upper period limitwas intentional: on one hand, we wanted to be sure of the contactbinary classification and avoid semidetached binaries, while on theother hand, we wanted to use the simple MV ≡ MV (log P) calibrationand avoid problems with uncertain or missing colour indices. Be-cause the final RFO estimate is done using absolute magnitude binsof the luminosity function, the period limits signify an intentionalrestriction to the low-brightness end of the contact binary sequence.We will explain the details below.3 A B S O L U T E M AG N I T U D E S A N D D I S TA N C E SContact binaries show a correlation between the colour [i.e. (B −V)0] and the orbital period P, which was first observed by Eggen(1967). As was shown by Rucinski & Duerbeck (1997), this corre-lation can be used to determine an absolute magnitude calibrationMV = MV [log P, (B − V)0]; an even simpler version, MV =MV (log P), is applicable (Rucinski 2006) when a restriction to shortperiods is added. In this research, the absolute magnitude calibrationTable 1. The short period contact binaries towards the LMC and the SMC. The table lists the name of the star (OGLE convention),the Magellanic Cloud field, the orbital period, the apparent magnitude in V band, the absolute magnitude MV , the distance alongthe line of sight and the vertical distance above the Galactic mid-plane.Name Field Period (d) V (mag) MV (mag) Distance (kpc) Vertical distance (kpc)OGLE003835.24-735413.2 (S1) SMC 0.26909 15.89 5.3 1.22 ± 0.17 0.82 ± 0.12OGLE005431.85-723510.9 (S2) SMC 0.33125 16.44 4.3 2.59 ± 0.36 1.77 ± 0.26OGLE005846.47-724315.3 (S3) SMC 0.33244 15.61 4.2 1.78 ± 0.25 1.21 ± 0.18OGLE004619.65-725056.2 (S4) SMC 0.37663 13.55 3.6 0.93 ± 0.13 0.62 ± 0.09OGLE004849.85-725554.8 (S5) SMC 0.43328 18.67 2.9 13.78 ± 2.01 9.54 ± 1.44OGLE054003.85-703837.3 (L1) LMC 0.24091 15.48 5.9 0.73 ± 0.10 0.37 ± 0.06OGLE050542.01-691725.9 (L2) LMC 0.27755 16.31 5.2 1.51 ± 0.21 0.79 ± 0.12OGLE051932.28-694633.4 (L3) LMC 0.27871 16.64 5.2 1.78 ± 0.25 0.94 ± 0.14OGLE053540.37-695413.9 (L4) LMC 0.29159 16.30 4.9 1.69 ± 0.23 0.89 ± 0.14OGLE053251.73-700256.2 (L5) LMC 0.29718 17.71 4.8 3.39 ± 0.47 1.82 ± 0.28OGLE053916.98-700903.3 (L6) LMC 0.31132 15.35 4.6 1.28 ± 0.18 0.67 ± 0.10OGLE050905.22-693315.1 (L7) LMC 0.32563 17.00 4.3 3.05 ± 0.42 1.63 ± 0.25OGLE053247.54-694403.1 (L8) LMC 0.33104 17.34 4.3 3.71 ± 0.51 1.99 ± 0.30OGLE053539.86-694759.5 (L9) LMC 0.36844 15.09 3.7 1.70 ± 0.24 0.90 ± 0.14OGLE054701.27-705623.3 (L10) LMC 0.42700 18.18 2.9 10.01 ± 1.41 5.44 ± 0.82MV = −1.5(± 0.8) − 12.0(± 2.0) log P was used, which is applica-ble for P < 0.562 d (Rucinski 2006). While applying this calibrationto the contact binary sample, we arbitrarily assumed an uncertaintyin absolute magnitude of 0.3 mag; the formal uncertainties in thecalibration were larger and probably overestimated. Using this ab-solute magnitude calibration and the relation between the distanceand the apparent and absolute magnitudes, d = dex( V −MV +5−AV5 ),the distance from the Sun and the galactic height for the 15 con-tact binaries were estimated. The extinction towards the LMC andSMC was obtained from Schlegel, Finkbeiner & Davis (1998); weassumed AV = 3.1 EB−V . Fig. 1 illustrates the vertical distancefrom the Galactic mid-plane to the 15 contact binaries as a func-tion of the absolute magnitude, whereas Table 1 lists the numericalresults.4 T H E A M P L I T U D E D I S T R I BU T I O NA N D T H E L U M I N O S I T Y F U N C T I O N F O R T H E1 5 C O N TAC T B I NA R I E SSmall-amplitude systems often remain undetected in variable starsearches when the photometric error is too large for detection ofa small photometric variability. The number of missed small am-plitude systems was estimated by comparing our sample with atheoretical model of the amplitude distribution for contact binaries(Rucinski 2001). This model predicts the amplitude distribution asa function of the degree of contact f and the mass ratio of the twocompanions q = M2M1 for a sample of contact binaries. In this re-search, we assumed f = 0.25, what appears to be the favoured de-gree of contact, and a flat distribution of q. We applied this modelin the same way as in Rucinski (2006), where more informationcan be found. Following this approach, we multiplied the num-ber of contact binaries by factors of 1.6 and 1.5 for the LMC andthe SMC, respectively; both factors have an uncertainty of about20 per cent.In order to determine the luminosity function for the 15 con-tact binaries, we simply used 1-mag wide absolute-magnitude binsby dividing the (corrected) number of stars in each bin by therespective volume V = 4π3 (r 32 − r 31 )C .C denotes the fraction ofthe sphere covered by the survey and is equal to 1.09 × 10−4 forthe LMC and 5.82 × 10−5 for the SMC. The results are given inTable 2.C© 2008 The Authors. Journal compilation C© 2008 RAS, MNRAS 385, 2239–2242Contact binaries at high galactic latitudes 2241Table 2. The luminosity function for the 15 close contact binaries towardsthe LMC and the SMC. The limiting distances are determined from d = dex((V − MV + 5 − AV )/5) with apparent magnitude limits V ∈ (13 ± 0.3,21 ± 0.3). The errors in the number of stars n are calculated according tothe Poisson statistics, which contribute to the uncertainty in φ as the mainfactor. Furthermore, we arbitrarily set the number of stars equal to one forthe SMC bin centred at MV = 6 with zero star counts; an assumption whichis consistent with the 1σ Poisson error.Cloud MV r1 r2 n φ(mag) (kpc) (kpc) ( starskpc3 )SMC 3 1.19 30.00 1.5 0.23 ± 0.214 0.75 18.93 4.5 2.71 ± 1.715 0.48 11.94 1.5 3.62 ± 3.316 0.30 7.54 1.5 14.39 ± 13.18LMC 3 1.13 28.41 1.6 0.15 ± 0.144 0.71 17.93 4.8 1.82 ± 1.125 0.45 11.31 8.0 12.10 ± 6.596 0.28 7.14 1.6 9.63 ± 8.605 S T E L L A R D E N S I T Y M O D E L A N DL U M I N O S I T Y F U N C T I O N F O R M S S TA R SIn order to establish the relative numbers of contact binaries, inrelation to MS stars, we attempted to derive the predicted numbersof stars in the same search volumes as those of the OGLE andMACHO surveys. According to the pioneering work of Gilmore &Reid (1983) and Bahcall & Soneira (1984), the stellar distributionin the Milky Way can be modelled by a double exponential thinand thick disc and a spheroidal halo. Here, we describe the stellardensity by such a three-component modeln(z, R) = nthin(z, R) + nthick(z, R) + nsp(z, R),where (z, R) are the galacto-centric coordinates. The thick and thethin discs are modelled bynthin(z, R) = n0(1 − cthick − csp)× exp[−|z|zthin]exp[− R − R0hthin]nthick(z, R) = n0cthick× exp[−|z|zthick]exp[− R − R0hthick],where zthin and zthick denote the scaleheights and hthin and hthick thescalelengths for the thin and the thick disc, respectively. Further-more, n0 is the local stellar density, and cthick and csp are the local-density normalizations of the thick disc and the halo relative tothe thin disc. In this research, we use a projected de Vaucouleursspheroid (Young 1976; Bahcall & Soneira 1984) of the formnsp(R′) = n0csp× exp[−10.093(R′R0) 14+ 10.093](R′R0)− 78.Here, R′ is given byR′ =[R2 +( zκ)2] 12,where κ is the axis ratio of the de Vaucouleurs spheroid.In our model, we used zthin = 300 ± 50 pc, zthick = 1000 ± 200 pc,hthin = 2.5 ± 1 kpc, hthick = 3.5 ± 1 kpc, cthick = 0.05 ± 0.03,csp = 0.0015 ± 0.001, κ = 0.55 ± 0.1 and R0 = 8 ± 0.5 kpc. Theseparameters were chosen in consideration of the work of Gilmore &Figure 2. Luminosity function towards the LMC. The continuous line rep-resents the contact binary luminosity function, whereas the dashed line rep-resents the MS luminosity function divided by a factor of 650.Figure 3. Luminosity function towards the SMC. The same as in Fig. 2except for the scaling factor of 600 instead of 650.Reid (1983), Kuijken & Gilmore (1989), Fux & Martinet (1994),Bienayme´ & Se´chaud (1997), Larsen & Humphreys (2003) and Juric´et al. (2005). The rather large uncertainties in the parameters reflectranges in the results obtained in the mentioned papers.Wielen, Jahreiss & Kru¨ger (1983) derived a widely accepted lu-minosity function for MS stars in the solar neighbourhood usinga sample of nearby stars, which we used to determine the stellardensity in the solar neighbourhood n0 as a function of absolutemagnitude. With those results and the stellar distribution n(z, R) asdescribed above, we calculated the mean stellar density of MS starsin each volume corresponding to the four, 1-mag wide, absolutemagnitude bins. By doing so, we were able to compare the derivedcontact binary luminosity function with the expected MS luminosityfunction for the same portion of the sky. The luminosity functionstowards the LMC and the SMC are plotted in Figs 2 and 3, respec-tively. The contact binary luminosity functions were found to bebest approximated by the MS luminosity function, if the latter wasdivided by 650 and 600 for the LMC and the SMC, respectively.6 C O N C L U S I O N SWe conclude that each of the two samples yields the relative fre-quency of one contact binary among about 600 MS stars in the twoconical volumes towards the SMC and the LMC at galactic latitudesof −44◦ and −33◦, respectively. These estimates have uncertaintiesof roughly 50 per cent due to the large uncertainties arising fromC© 2008 The Authors. Journal compilation C© 2008 RAS, MNRAS 385, 2239–22422242 P. D. Nef and S. M. Rucinskismall number statistics and to the still uncertain parameters of thespheroidal component of the Galaxy. This component has been ex-plicitly accounted for in the previous section and is estimated tocontribute to the total number of stars in the SMC and the LMCsearch volumes at levels of about 30 and 20 per cent, respectively.The contribution varies with the MV . Specifically, for the SMC, itchanges from 38 per cent for the MV bin with the largest distances to17 per cent for the MV bin centred at 6 mag; for the LMC, it changesfrom 29 to 10 per cent in the same range of MV .The results on the relative frequency of contact binary stars atlarge galacto-centric distances is consistent with estimates for thesolar neighbourhood in previous work (Rucinski 2006). These arethe very first data available on the contact binary distribution at highgalactic latitudes.AC K N OW L E D G M E N T SThis work was done during the stay of PDN as an Exchange Under-graduate Student at the Department of Astronomy and Astrophysicsof the University of Toronto. Thanks to the members of the depart-ment for hospitality and support. PDN would also like to thank BryceCroll and Mirza Ahmic whose suggestions on how to improve thepaper have been most helpful. Thanks are due to the OGLE andMACHO teams for making their results available for direct onlineaccess. Support from the Natural Sciences and Engineering Councilof Canada to SMR is acknowledged with gratitude.R E F E R E N C E SAlcock C. et al. (The MACHO Collaboration), 1997, ApJ, 486, 697Bahcall J. N., Soneira R. M., 1984, ApJS, 55, 67Figure A1. The light curves of the 15 close contact binaries.This paper has been typeset from a TEX/LATEX file prepared by the author.Bienayme´ O., Se´chaud N., 1997, A&A, 323, 781Eggen O. J., 1967, Mem. RAS, 70, 111Fux R., Martinet L., 1994, A&A, 287, 21Gilmore G., Reid N., 1983, MNRAS, 202, 1025Juric´ M. et al., 2005, preprint (astro-ph/0510520)Kuijken K., Gilmore G., 1989, MNRAS, 239, 605Larsen J. A., Humphreys R. M., 2003, AJ, 125, 1958Rucinski S. M., 1993, PASP, 105, 1433Rucinski S. M., 1997, AJ, 113, 407Rucinski S. M., 2001, AJ, 122, 1007Rucinski S. M., 2002, PASP, 114, 1124Rucinski S. M., 2006, MNRAS, 368, 1319Rucinski S. M., 2007, MNRAS, 382, 393Rucinski S. M., Duerbeck H. W., 1997, PASP, 109, 1340Schlegel D. J., Finkbeiner D. P., Davis M., 1998, AJ, 500, 525Udalski A., Kubiak M., Szyman´ski M., 1997, Acta Astron., 47, 319Wielen R., Jahreiss H., Kru¨ger R., 1983, in Philip A. G. D., Upgren A.R., eds, IAU Colloq. 76, The Nearby Stars and the Stellar LuminosityFunction. L. Davis Press, Schenectady NYWyrzykowski L. et al., 2003, Acta Astron., 53, 1Wyrzykowski L. et al., 2004, Acta Astron., 54, 1Young P. J., 1976, AJ, 81, 807A P P E N D I X A : T H E L I G H T C U RV E S O F T H E1 5 C O N TAC T B I NA R I E SIn Fig. A1 we show the light curves of the 15 contact binaries.The photometric data are taken from the two OGLE catalogues(Wyrzykowski et al. 2003, 2004). The fitted curves were obtainedaccording to the approach of Rucinski (1993), i.e. using only thefirst five cosine and the first sine Fourier coefficients.C© 2008 The Authors. Journal compilation C© 2008 RAS, MNRAS 385, 2239–2242

Luminosity function of contact binaries at high galactic latitudes towards the LMC and the SMC

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