THE CLOSE BINARY PROPERTIES OF MASSIVE STARS IN THE MILKY WAY AND LOW-METALLICITY MAGELLANIC CLOUDS

To model the OGLE eclipsing binaries, we utilize the Z = 0.004, Y = 0.26 stellar tracks from the Padova group (Bertelli et al. 2008, 2009), which correspond to a metallicity between the SMC and LMC mean values. In addition to basic parameters such as radii R(τ) and photospheric temperatures T(τ) as a function of stellar age τ, we also extract the surface gravities g(τ) from the stellar tracks in order to select appropriate model atmospheres in nightfall. We convert stellar radii to Roche-lobe-filling factors according to the volume-averaged formula given by Eggleton (1983). Although nightfall defines the Roche-lobe-filling factor along the polar axis, it is more appropriate to use the Eggleton (1983) approximation in cases where the star fills a large fraction of its Roche lobe and is therefore distorted along this potential. In any case, the volume-averaged Roche lobe radius is only ≈7% larger than the polar Roche lobe radius for systems in our sample, so any systematics due to using the Eggleton (1983) formula as input are small. Based on the numerical calculations performed by Claret (2001) and his comparison to empirical results, we choose an albedo of A = 1.0 for our primary and secondaries hotter than T > 7500 K with radiative envelopes (M₂ ⩾ 1.3 M_☉), and A = 0.75 for low-mass secondaries (M₂ < 1.3 M_☉) at lower temperatures with convective atmospheres.

Because we selected the OGLE samples from a narrow range of absolute magnitudes, we can assume that all eclipsing binaries have the same primary mass. If the luminosity of the primary is dominant, then the median absolute magnitude of M_I ≈ −2.1 in the OGLE samples corresponds to a primary mass of M₁ = 12 M_☉, where we have interpolated the stellar tracks from Bertelli et al. (2009) at half the MS lifetimes and utilized bolometric corrections and color indices from Cox (2000). However, if the typical secondary in the observed eclipsing systems increases the brightness by ΔM_I ≈ 0.3 mag (see Section 3.4.2), then the primary's absolute magnitude of M_I ≈ −1.8 corresponds to M₁ = 10 M_☉. We therefore adopt M₁ = 10 M_☉ for all primaries in our simulations.

We must still consider the systematic error in ${\cal F}_{\rm close}$ due to this single-mass primary approximation. The sample distributions of absolute magnitudes M_I have a dispersion of $\sigma _{M_{\rm I}}$ ≈ 0.4 mag, which implies a dispersion in M₁ of ≈25%. According to the mass–radius relation R ∝ M^0.6 and Kepler's law a ∝ M^1/3, the probability of observing eclipses ${\cal P}_{\rm deep}$ ∝ ${\cal P}_{\rm med}$ $\propto\hspace{-12pt}{\smash{\raise-5pt{{\sim}}}}$ R/a $\propto\hspace{-12pt}{\smash{\raise-5pt{{\sim}}}}$ $M_1^{0.3}$ due to geometrical selection effects is only weakly dependent on M₁. The systematic error in our derived ${\cal F}_{\rm close}$ = ${\cal F}_{\rm deep}/\langle {\cal P}_{\rm deep}\rangle$ = ${\cal F}_{\rm med}/{\langle \cal P}_{\rm med}\rangle$ is therefore only a factor of 7% due to the observed dispersion in primary absolute magnitudes $\sigma _{M_{\rm I}}$ ≈ 0.4 mag. Similarly, the extinction distributions toward young stars in the Magellanic Clouds have a dispersion of $\sigma _{A_{\rm V}}$ ≈ 0.3 mag (Zaritsky 1999; Zaritsky et al. 2002), and the I-band excess distributions from the eclipsing companions have a dispersion of $\sigma _{\Delta {M}_{\rm I}}$ ≈ 0.2 mag (see Section 3.4.2). These effects contribute additional systematic error factors in ${\cal F}_{\rm close}$ of 6% and 4%, respectively. By adding these three sources of uncertainty in quadrature, we find that the total systematic error in ${\cal F}_{\rm close}$ is only a factor of 10% due to our single-mass primary approximation. In our estimate for ${\cal P}_{\rm deep}$ ∝ ${\cal P}_{\rm med}$ $\propto\hspace{-12pt}{\smash{\raise-5pt{{\sim}}}}$ $M_1^{0.3}$, we have assumed that the mass-ratio distributions, and therefore the slopes of the eclipse depth distributions, do not substantially vary across our narrowly selected interval of primary masses. In fact, for the OGLE-III LMC medium eclipse depth sample, we find ${\cal S}_{\Delta {\rm m}}$ ∝ (Δm)^{−1.54 ± 0.12} for the 563 eclipsing binaries brighter than M_I = −2.3, and ${\cal S}_{\Delta {\rm m}}$ ∝ (Δm)^{−1.74 ± 0.11} for the 738 systems fainter than M_I = −2.3. The consistency of these two slopes justifies our approximation, and therefore our assessment of the systematic error in ${\cal F}_{\rm close}$ is valid.

Because we restricted our samples to observed colors V − I < 0.1, i.e., T₁ ≳ 10,000 K once reddening is taken into account, most primaries are relatively unevolved on the MS. For example, a Z = 0.004, M₁ = 10 M_☉ primary evolves from R₁ = 3.3 R_☉, T₁ = 28,000 K on the zero-age MS (luminosity class V) to R₁ = 8.5 R_☉, T₁ = 22,000 K at the top of the MS by age τ_MS = 23 Myr (technically luminosity class III). The star then rapidly expands and cools, passing from R₁ = 9.0 R_☉ to T₁ = 10,000 K in δt ≈ 30,000 yr. Considering δt/τ_MS ∼ 10⁻³, the contamination by the few, short-lived bona fide giants with τ > τ_MS is negligible.

We calculate the I-band light curve at 1% phase intervals across the orbit, where we include the effects of fractional visibility of surface elements computed by nightfall. Because the OGLE eclipsing binary catalogs reported eclipse depths in the I band as the difference between the dimmest and mean out-of-eclipse magnitudes, we set the zero-point magnitude in the nightfall models to the mean value across the phase interval 0.2–0.3. We display some example light curves in Figure 3. The three panels represent orbital periods of P = 2, 6.3, and 20 days, while the colors distinguish various mass ratios q = M₂/M₁. We compute the light curves at inclinations i = 773, 841, and 873 from left to right so that the projected separations a_proj ∝ P^2/3cos i = constant. For spherical stars, the eclipse depths should therefore be identical across these three panels for the same mass ratios. We evaluate these example models at age τ = 17 Myr when the primary reaches an intermediate radius of R₁ = 5.3 R_☉.

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The left panel of Figure 3 with P = 2 days corresponds to primaries filling 60%–80% of their Roche lobes, depending on the mass ratio. The light curves of these close binaries exhibit pronounced ellipsoidal modulations, while the out-of-eclipse magnitudes of systems at longer orbital periods are relatively constant. In the right panel with P = 20 days, the narrow eclipse widths of 4% are just at the detectability limit of eclipsing binary identification algorithms (Söderhjelm & Dischler 2005).

A simple estimate for the eclipse depths can be derived by calculating the bolometric flux in the eclipsed area of the primary assuming spherical stars and no limb darkening. We compare the nightfall models to this simple approximation for the maximum eclipse depth (horizontal dotted lines centered on primary eclipses). For P = 2 days, the actual eclipse depths determined by nightfall are generally deeper than the simple approximations because tidal distortions and reflection effects enhance the light-curve amplitudes. Alternatively, the nightfall results for longer period systems at P = 6.3 and 20 days are typically shallower than the simple approximations because the actual flux eclipsed along grazing angles is less due to the effect of limb darkening.

Because the OGLE eclipsing binary catalogs exclude ellipsoidal variables that did not exhibit genuine eclipses, we consider only systems with inclinations i > i_crit ≡ cos⁻¹([R₁ + R₂]/a). We use nightfall to produce a dense grid of eclipse depths Δm(τ, q, P, i) in our parameter space of stellar ages τ = [0, τ_MS = 23 Myr], mass ratios q = [0.1, 1], orbital periods P(days) = [2, 20], and inclinations i = [i_crit, 90^o]. In the three panels of Figure 4, we plot our simulated Δm as a function of inclination i for the same three orbital periods, various mass ratios indicated by color, and for the same τ = 17 Myr that gives R₁ = 5.3 R_☉.

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The short-period systems in the left panel of Figure 4 are significantly affected by tidal distortions. The twin system with q = 1 observed edge-on at i = 90^o exceeds the maximum eclipse depth limit for spherical stars of Δm = 0.75. Ellipsoidal variables that barely miss eclipses with i = i_crit all have light-curve amplitudes of Δm < 0.05 for this set of parameters (see where curves terminate at bottom left). For systems that do not fill their Roche lobes, all ellipsoidal variables with i = i_crit have amplitudes Δm < 0.09. Granted, some systems with i > i_crit may not have strong enough eclipse features to be included in the catalog of eclipsing binaries. Nevertheless, this transition between ellipsoidal variability and genuine eclipses occurs at Δm ≲ 0.1, so we can be assured that very few eclipsing systems with measured amplitudes Δm > 0.1 have been excluded from the catalogs.

The middle and right panels of Figure 4 represent progressively longer orbital periods where tidal distortions and reflection effects become negligible. Note the smaller range of inclinations that produce observable eclipses, simply due to geometrical selection effects. We display with horizontal dashed lines our adopted intervals for deep eclipses and extension toward medium eclipse depths. Assuming that the middle panel is most representative of close binaries with P = 2–20 days, i > 85^o and q > 0.55 are required to observe deep eclipses. Given random orientations, the correction factor for geometrical selection effects alone is ${\cal C}_{\rm deep,i}$ ≈ 1/cos (85°) ≈ 11. Assuming a uniform mass-ratio distribution over the interval q = [0.1, 1.0], the correction factor for incompleteness toward low-mass companions alone is ${\cal C}_{\rm deep,q}$ ≈ ((1–0.1)/(1–0.55)) ≈ 2. The overall probability of observing a system with a deep eclipse is therefore $\langle {\cal P}_{\rm deep}\rangle$ = (${\cal C}_{\rm deep,i}$ × ${\cal C}_{\rm deep,q}$)⁻¹ ≈ 0.04. Similarly, i > 83^o and q > 0.3 are required to observe eclipses with medium depths, implying ${\cal C}_{\rm med,i}$ ≈ 8, ${\cal C}_{\rm med,q}$ ≈ 1.3, and $\langle {\cal P}_{\rm med} \rangle$ ≈ 0.09. These two overall probabilities imply similar close binary fractions of ${\cal F}_{\rm close}$ = ${\cal F}_{\rm deep}$/$\langle {\cal P}_{\rm deep}\rangle$ = 0.7%/0.04 ≈ 16% and ${\cal F}_{\rm close}$ = ${\cal F}_{\rm med}$/$\langle {\cal P}_{\rm med}\rangle$ = 1.9%/0.09 ≈ 20%. We obtain more precise values in Section 3.3 by fitting the observed eclipse depth and period distributions to constrain the actual binary properties.

In Figure 5, we display simulated eclipse depths from nightfall similar to Figure 4, but for constant P = 2.9 days and three different stages of evolution. The left panel corresponds to zero-age MS systems where the primary radius is R₁ = 3.3 R_☉, the middle panel represents an intermediate-age binary when R₁ = 5.3 R_☉, and the right panel is for the top of the primary's MS with R₁ = 8.5 R_☉. For young systems, q = 0.1 is just at the detectability threshold in our medium eclipse depth samples, which is the primary reason we set the lower limit of our mass-ratio interval to this value. With increasing τ and R₁, the range of inclinations that produce visible eclipses increases due to geometrical selection effects. However, the depths of eclipses for q ≲ 0.9 become smaller because the fractional area of the primary that is eclipsed decreases with increasing primary radius. Therefore, our samples of eclipsing binaries are rather incomplete toward smaller, low-mass companions. For young systems, the probability of observing a low-mass secondary is low, while for older systems the eclipse depths produced by low-mass companions are below the sensitivity of the surveys.

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There is a narrow corner of the parameter space with P ≲ 2.6 days and R₁ ≳ 7.0 R_☉ where the primary overfills its Roche lobe. We assume that either merging or onset of rapid mass transfer causes these systems to evolve outside the parameter space 0.1 < Δm < 0.65. In our Monte Carlo simulations (Section 3.2), we include their contribution toward the close binary fraction but remove these systems as eclipsing binaries when fitting ${\cal O}_{\Delta {\rm m}}$(Δm) and either ${\cal O}_{\rm deep}(P)$ or ${\cal O}_{\rm med}(P)$. A 10 M_☉ star spends 8% of its MS evolution with R₁ > 7.0 R_☉, and (20–30)% of the eclipsing binaries in our samples have orbital periods P < 2.6 days, depending on the survey. Therefore, the systematic error in our evaluation of the close binary fraction due to these few evolved, close, Roche-lobe-filling binaries is only 2%.

For systems that produce eclipse depths Δm > 0.25 and are not filling their Roche lobes, the rms deviation between the detailed nightfall simulations and simple approximations that ignore limb darkening and tidal distortions is 〈δ(Δm)〉 = 0.05 mag. The difference reaches a maximum value of 0.16 mag for a close-period, evolved twin system with q = 1 that nearly fills its Roche lobes. Because of these measurable systematics, it is important that we incorporate the nightfall results instead of relying on the simple estimates.

We repeat our procedure to model eclipse depths Δm for the Hipparcos MW sample of eclipsing binaries, but with some slight modifications. We still assume that all primaries have M₁ = 10 M_☉ because the mean spectral type of our sample is B2, but we implement the solar metallicity Z = 0.017, Y = 0.26 tracks from the Padova group (Bertelli et al. 2008, 2009). A solar-metallicity 10 M_☉ star has a slightly longer lifetime of τ_MS ≈ 25 Myr, and more importantly it is (15–25)% larger depending on the stage of evolution. The primary radius is R₁ = 3.8 R_☉ on the zero-age MS versus R₁ = 3.3 R_☉ for the Z = 0.004 model and reaches R₁ = 10.5 R_☉ at the top of the MS compared to R₁ = 8.5 R_☉ for the low-metallicity track. For the same close binary properties, we actually expect ${\cal F}_{\rm deep}$ in the MW to be 20% higher because the probability of eclipses scales as ${\cal P}$ ∝ (R₁ + R₂). This radius–metallicity relation diminishes the already small 1.2σ difference between the MW and Magellanic Cloud statistics inferred from ${\cal F}_{\rm deep}$. Finally, we evaluate the eclipse depth Δm based on the V-band light curves computed by nightfall, which closely approximates the Hipparcos passband.

Because we attempt to address all sources of error, we account for systematics due to eccentric orbits in our determination of the close binary fraction. Unfortunately, the extent of tidal distortions and mutual irradiation continually change in an eccentric orbit, so that all the binary properties in nightfall must be recalculated at each phase of the orbit instead of solely varying the orientation. It would become computationally too expensive if we were to add dimensions of eccentricity e and periastron angle ω to our original grid of models Δm(τ, q, P, i).

Since the eccentricities of close binaries are relatively small due to tidal circularization, we can determine the average error 〈δ(Δm)〉 of a representative eclipsing binary and propagate the uncertainty into our evaluation of ${\cal F}_{\rm close}$. We consider an eclipsing binary with τ = 17 Myr, q = 0.6, P = 4 days, and i = 90^o as our test example, which gives Δm = 0.30 for a circular orbit (see Figure 4). For the 101 systems in the ninth catalog of spectroscopic binary orbits (Pourbaix et al. 2004) with measured eccentricities, orbital periods P = 2–20 days, and primaries with spectral types B0–B3 and luminosity classes III–V, the average eccentricity is only 〈e〉 = 0.17. For the 56 systems with P = 2–5 days, which is more representative of our eclipsing binary sample, the mean eccentricity is even lower at 〈e〉 = 0.11. Using nightfall, we calculate the eclipse depths for our test example at an intermediate value of e = 0.15 and an upper ≈1σ value of e = 0.30, each at varying periastron angles ω.

For e = 0.15, we find that the eclipse depths vary by δ(Δm) < 0.004 mag compared to a circular orbit, with an average value of 〈δ(Δm)〉 = 0.002 mag if we weight uniformly with respect to ω. The error is slightly higher at 〈δ(Δm)〉 = 0.005 mag for e = 0.30. We found in Section 3.1.1 that the average error between the detailed nightfall models and simple estimates ignoring tidal distortions and limb darkening is 〈δ(Δm)〉 = 0.05 mag. We show in Section 3.3 that this would have propagated into a systematic error factor of 20% in our determination of ${\cal F}_{\rm close}$. Since the error in eclipse depths due to eccentric orbits is an order of magnitude smaller, we expect the uncertainty in ${\cal F}_{\rm close}$ due to non-circular orbits to be only a factor of 2%.

A third light source can have a much larger effect on the observed eclipse depth Δm of an eclipsing binary, depending on the luminosity of the contaminant. We first consider wider companions in triple star systems. About 40% of early-type primaries have a visually resolved companion (Turner et al. 2008; Mason et al. 2009). More importantly, most close binaries, such as our eclipsing systems, are observed to be the inner components of triple star systems (Tokovinin et al. 2006). Specifically, this study found that 96% of binaries with P < 3 days have a wider tertiary companion. Assuming that the typical eclipsing secondary increases the brightness by ΔM = 0.3 mag (see Section 3.4), a tertiary companion with q = M₃/M₁ > 0.5 is capable of increasing the system luminosity by ≳10%. The wider companions around early-type primaries are observed to be drawn from a mass-ratio distribution weighted toward lower mass, fainter stars (Abt et al. 1990; Preibisch et al. 1999; Duchêne et al. 2001; Shatsky & Tokovinin 2002). These observations find that only (10–30)% of wide companions have mass ratios q > 0.5. Even if every eclipsing binary has one wider component, we would expect that only ≈20% of tertiaries have large enough luminosities to measurably affect our light-curve modeling.

We also consider third light contamination due to stellar blending in the crowded Magellanic Cloud fields. Based on the OGLE photometric catalogs, there are 4.2 million (Udalski et al. 2000), 12 million (Udalski et al. 2008), and 1.5 million (Udalski et al. 1998) systems with M_I > 1.2 in the OGLE-II LMC, OGLE-III LMC, and OGLE-II SMC footprints, respectively. The median absolute magnitude of these sources is M_I ≈ 0.4, which is 10% the I-band luminosity of our median early B eclipsing binary with M_I ≈ −2.1. The average space densities of stars with M_I > 1.2 are 0.07, 0.03, and 0.05 objects per square arcsecond in the OGLE-II LMC, OGLE-III LMC, and OGLE-II SMC fields, respectively. Given a median seeing of 12–13 during the OGLE observations, we expect only (5–12)% of early B eclipsing binaries to be blended with sources brighter than M_I = 1.2. The probability of stellar blending with a background/foreground source is slightly smaller than the probability of contamination in a triple star system, where in both cases we included third light components ≳10% the luminosity of the eclipsing system.

Because a sizable fraction of eclipsing binaries are affected by third light contamination from stellar blending and triple star systems, we model the third light sources in the eclipsing binary populations using a statistical method. When we conduct our Monte Carlo simulations in the next section, we synthesize distributions of eclipse depths Δm based on our nightfall models, but we assume that a 20% random subset of eclipsing systems have reduced eclipse depths Δm_measured = 0.8 Δm_true. These values approximate the probabilities and representative luminosities of the third light contaminants. By comparing our model fits with and without the third light sources, we can gauge the effect on our derived close binary properties.

We initially fit the observed eclipse depth distribution ${\cal O}_{\Delta {\rm m}}$ only, which primarily constrains the mass-ratio distribution ${\cal U}_{\rm q}$ and the normalization to ${\cal F}_{\rm close}$ according to Equation (2). We determine the best-fit model parameters $\boldsymbol {x}$ = {γ_P, γ_q, ${\cal F}_{\rm twin}$, ${\cal F}_{\rm close}$} by minimizing the $\chi ^2_{\Delta {\rm m}}$($\boldsymbol {x}$) statistic between the observed eclipse depth distribution ${\cal O}_{\Delta {\rm m}}$(Δm) and our Monte Carlo models ${\cal M}_{\Delta {\rm m}}$(Δm, $\boldsymbol {x}$):

$$\begin{equation} \chi ^2_{\Delta {\rm m}}(\boldsymbol {x}) = \sum _k^{N_{\Delta {\rm m}}} \left(\frac{{\cal O}_{\Delta {\rm m}}(\Delta {\rm m}_k) - {\cal M}_{\Delta {\rm m}}(\Delta {\rm m}_k,\boldsymbol {x})}{{\sigma }_{{\cal O}_{\Delta {\rm m}}}(\Delta {\rm m}_k)}\right)^2. \end{equation} \tag{ 3 }$$

We sum over the bins of data displayed in Figure 6 that are complete, specifically the N_Δm = 8 bins across 0.25 < Δm (mag) < 0.65 for the OGLE-II LMC and SMC populations, the N_Δm = 5 bins across 0.10 < Δm < 0.65 for the MW, and the N_Δm = 11 bins across 0.10 < Δm < 0.65 for the OGLE-III LMC sample. In Figure 6, we display the best-fit models ${\cal M}_{\Delta {\rm m}}$(Δm) for each sample, together with the data. Although we have excluded eclipsing binaries with Δm > 0.65 mag, which derive from nearly edge-on twin systems and evolved binaries that have filled their Roche lobes, twins are most likely to have grazing trajectories that produce eclipse depths in our selected parameter space (see Section 3.1.1). For the OGLE Magellanic Cloud samples that have large sample statistics in the interval 0.40 mag < Δm < 0.65 mag, we therefore have sufficient leverage to constrain the excess twin fraction.

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The observed eclipse depth distributions can only constrain ${\cal F}_{\rm close}$, γ_q, and ${\cal F}_{\rm twin}$, which effectively gives ν = N_Δm − 3 degrees of freedom. We report in Table 2 the minimized reduced $\chi ^2_{\Delta {\rm m}}$ statistics, degrees of freedom ν, and probabilities to exceed $\chi ^2_{\Delta {\rm m}}$. We calculate a grid of joint probabilities $p_{\boldsymbol {x}}(\boldsymbol {x})$ ∝ $e^{-\chi ^2_{\Delta {\rm m}}(\boldsymbol {x})/2}$ and then marginalize over the various parameters to calculate the probability density functions $p_{x_i}(x_i)$ for each parameter x_i. In Table 2, we list the average values $\mu _{x_i}$ = $\int x_i\,p_{x_i}(x_i)$ dx_i and uncertainties $\sigma _{x_i}$ = $[\int (x_i - \mu _{x_i})^2\,p_{x_i}(x_i)$ dx_i]^1/2 of the three parameters constrained by ${\cal O}_{\Delta {\rm m}}$ for each of the eclipsing binary samples. Some of the parameters are correlated and have asymmetric probability density distributions, so we display two-dimensional probability contours $p_{x_i, x_j} (x_i, x_j)$ for some combinations of parameters in Figure 7.

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Table 2. Results of Our Monte Carlo Simulations and Fits to the Observed Eclipse Depth Distributions ${\cal O}_{\Delta {\rm m}}$ Only

Sample	$\chi ^2_{\Delta {\rm m}}/{\nu }$	ν	PTE	${\cal F}_{\rm twin}$	γq	${\cal F}_{\rm close}$
MW	0.43	2	0.65	0.16 ± 0.10	−0.9 ± 0.8	0.22 ± 0.06
OGLE-II LMC	0.48	5	0.79	0.10 ± 0.07	−0.6 ± 0.7	0.21 ± 0.08
OGLE-III LMC	0.71	8	0.68	0.04 ± 0.03	−1.0 ± 0.2	0.27 ± 0.05
OGLE-II SMC	0.42	5	0.83	0.08 ± 0.06	−0.9 ± 0.7	0.24 ± 0.08

Notes. For each of the eclipsing binary samples, we list the minimized reduced $\chi ^2_{\Delta {\rm m}}$ statistics, degrees of freedom ν = N_Δm − 3, probabilities to exceed $\chi ^2_{\Delta {\rm m}}$ given ν, and the mean values and 1σ uncertainties of the three model parameters constrained by ${\cal O}_{\Delta {\rm m}}$.

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The higher quality OGLE-III LMC population, with its larger sample size and completeness down to Δm = 0.10, best constrains the model parameters. We find a negligible excess fraction of twins ${\cal F}_{\rm twin}$ = (4 ± 3)%, a mass-ratio distribution weighted toward low-mass companions with γ_q = −1.0 ± 0.2, and a close binary fraction of ${\cal F}_{\rm close}$ = (27 ± 5)% (before corrections for Malmquist bias—see Section 3.4). Based on our Monte Carlo simulations, a uniform mass-ratio distribution would have produced ${\cal S}_{\Delta {\rm m}}$ d(Δm) $\propto\hspace{-12pt}{\smash{\raise-5pt{{\sim}}}}$ (Δm)^−1.0 d(Δm), not as steep as the observed trend ${\cal S}_{\Delta {\rm m}}$ d(Δm) ∝ (Δm)^{−1.65 ± 0.07} d(Δm).

The less complete and/or smaller MW, OGLE-II LMC, and OGLE-II SMC samples do not permit precise determinations of γ_q. Nonetheless, the fitted mean values for these three samples span the range γ_q = −0.9 to −0.6, suggesting that these binary populations also favor low-mass companions. For these populations, our solutions for the model parameters ${\cal F}_{\rm close}$ and γ_q are anti-correlated (see bottom panels of Figure 7). This is because a larger fraction of low-mass secondaries below the threshold of the survey sensitivity implies a higher ${\cal F}_{\rm close}$ given the same ${\cal F}_{\rm deep}$. All four samples are consistent with a close binary fraction of ${\cal F}_{\rm close}$ ≈ 25%, slightly higher than our initial estimate of (16–20)% in Section 3.1. The precise values will decrease slightly once we correct for Malmquist bias (see Section 3.4).

Even though γ_q is not well known for the OGLE-II data, we can still constrain the excess twin fraction to be ${\cal F}_{\rm twin}$ ≈ (4–10)% for all three OGLE Magellanic Cloud samples (see top panels of Figure 7). A dominant twin population would have caused the eclipse depth distribution ${\cal O}_{\Delta {\rm m}}$ to flatten or even rise toward the deepest eclipses Δm > 0.4. Instead, the observed eclipse depth distributions for the three OGLE Magellanic Cloud samples continue with the same power-law ${\cal S}_{\Delta {\rm m}}$ ∝ (Δm)^−1.65. Because there are very few eclipsing binaries with Δm > 0.4 in the MW data, we cannot adequately measure ${\cal F}_{\rm twin}$ for this population, but see our well-constrained estimate of ${\cal F}_{\rm twin}$ ≈ 7% based on spectroscopic observations of early-type stars in the MW (Section 4).

We have reported fitted parameters based on the nightfall models where a 20% random subset have eclipse depths reduced by 20% to account for third light contamination (Section 3.1.4). Because shallower eclipses systematically favor lower mass companions, the fitted mass-ratio distributions would have been shifted toward even lower values, albeit slightly, had we not considered this effect. Specifically, we find that the excess twin fraction would have decreased by $\Delta {\cal F}_{\rm twin}$ = 0.01–0.03 and the mass-ratio distribution exponent would have decreased by Δγ_q = 0.0–0.2, depending on the sample. The close binary fraction would have changed by a factor of (3–6)%, i.e., $\Delta {\cal F}_{\rm close}$ ≈ 0.01, with no general trend on the direction. Hence, third light contamination only mildly affects the inferred close binary properties.

The probabilities ${\cal P}_{\rm deep}(P)$ and ${\cal P}_{\rm med}(P)$ are defined to be the ratios of systems exhibiting deep (0.25 < Δm < 0.65) and medium (0.10 < Δm < 0.65) eclipses, respectively, to the total number of companions with q > 0.1 at the designated period. These probabilities obviously decrease with increasing orbital period P due to geometrical selection effects. In addition, ${\cal P}_{\rm deep}(P)$ and ${\cal P}_{\rm med}(P)$ depend on the metallicity Z, which determines the radial evolution of the stellar components, and also on the underlying mass-ratio distribution ${\cal U}_{\rm q}$. Mass-ratio distributions that favor lower mass, smaller companions result in lower probabilities of observing eclipses because a larger fraction of the systems have eclipse depths below the sensitivity of the surveys. Because we have constrained ${\cal U}_{\rm q}$ for each of the four eclipsing binary populations, we have already effectively determined these probabilities from our Monte Carlo simulations. We use these more accurately constrained probabilities when we account for Malmquist bias in Section 3.4, as well as to visualize the corrected period distribution in Section 3.5.

Using our solutions for ${\cal U}_{\rm q}$ for each of the four eclipsing binary samples, we display the resulting ${\cal P}_{\rm deep}(P)$ and ${\cal P}_{\rm med}(P)$ in Figure 8. We propagate the fitted errors in γ_q and ${\cal F}_{\rm twin}$, as well as their mutual correlation as displayed in the top panels of Figure 7, to determine the uncertainties in the probabilities. For comparison, we calculate ${\cal P}_{\rm med}(P)$ and ${\cal P}_{\rm deep}(P)$ assuming the low-metallicity Z = 0.004 stellar tracks and a uniform mass-ratio distribution ${\cal U}_{\rm q}$, i.e., γ_q = 0 and ${\cal F}_{\rm twin}$ = 0.

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In the top panel of Figure 8, the probabilities ${\cal P}_{\rm deep}$ for the OGLE Magellanic Cloud samples, which all have fitted values of γ_q that are negative, are systematically lower than the probabilities that assume a uniform mass-ratio distribution. Based on our back-of-the-envelope estimates in Section 3.1.1 where we assumed a uniform mass-ratio distribution, we determined that the correction factor between ${\cal F}_{\rm deep}$ and ${\cal F}_{\rm close}$ due to incompleteness toward low-mass companions alone was ${\cal C}_{\rm deep,q}$ ≈ 2. The fact that the fitted mass-ratio distributions favor more low-mass companions increases this correction factor to ${\cal C}_{\rm deep,q}$ ≈ 3. Therefore, the overall probability of observing deep eclipses at intermediate periods of log P = 0.8 is ${\cal P}_{\rm deep}$ = 0.03, slightly lower than our estimated average in Section 3.1 of $\langle {\cal P}_{\rm deep} \rangle$ = 0.04. Finally, note the intrinsically small probability of observing deep eclipses at long periods, e.g., only ${\cal P}_{\rm deep}$ ≈ 1% of all binaries at P = 20 days are detectable as eclipsing systems with 0.25 < Δm < 0.65.

In the bottom panel of Figure 8, the variations in ${\cal P}_{\rm med}$ are significantly smaller. This is because the probability of observing eclipses becomes less dependent on the underlying mass-ratio distribution as the observations become more sensitive to shallower eclipses. Essentially, the correction factor for incompleteness toward low-mass companions alone is only ${\cal C}_{\rm med,q}$ = 1.5, slightly larger than our original estimate of ${\cal C}_{\rm med,q}$ = 1.3 in Section 3.1.1, but still very close to unity. The MW correction factor ${\cal C}_{\rm med,i}$ for geometrical selection effects is 20% smaller than the OGLE-III LMC values, and therefore the overall probabilities ${\cal P}_{\rm med}$ are 20% larger. This is consistent with our interpretation of the radius–metallicity relation in Section 3.1.2. Söderhjelm & Dischler (2005) calculated the probabilities of observing solar-metallicity eclipsing binaries with Δm > 0.1 as a function of spectral type and period. Because the fraction of systems with Δm > 0.65 is negligible compared to the fraction with 0.1 < Δm < 0.65, we can compare the Söderhjelm & Dischler (2005) results to our ${\cal P}_{\rm med}$(P). We interpolate the probabilities in their Table A.1 for OB stars with 〈M_V〉 = −3.04 and B stars with 〈M_V〉 = −0.55 for our sample's median value of M_V ≈ −2.3. The resulting ${\cal P}_{\rm med}$, which we display in the bottom panel of Figure 8, is consistent with our MW distribution. At log P = 0.8, the OGLE-III LMC value of ${\cal P}_{\rm med}$ = 0.06 is only slightly lower than the uniform mass-ratio distribution value of ${\cal P}_{\rm med}= 0.08$ and our initial estimate in Section 3.1.1 of $\langle {\cal P}_{\rm med} \rangle$ = 0.09.

We now fit the observed eclipsing binary period distributions ${\cal O}_{\rm deep}(P)$ or ${\cal O}_{\rm med}(P)$ only, which constrain the intrinsic period distributions ${\cal U}_{\rm P}$ and the normalizations to ${\cal F}_{\rm close}$ according to Equation (1). We minimize the $\chi ^2_{\rm P}$($\boldsymbol {x}$) statistics between the measured eclipsing binary period distributions ${\cal O}_{\rm deep}$(log P) and our Monte Carlo models ${\cal M}_{\rm deep}$(log P, $\boldsymbol {x}$):

$$\begin{equation} \chi ^2_{\rm P}(\boldsymbol {x}) = \sum _k^{N_{\rm P}} \left(\frac{{\cal O}_{\rm deep}({\rm log}\,P_k) - {\cal M}_{\rm deep}({\rm log}\,P_k, \boldsymbol {x})}{{\sigma }_{{\cal O}_{\rm deep}}({\rm log}\,P_k)}\right)^2. \end{equation} \tag{ 4 }$$

We calculate similar statistics for the medium eclipse depth samples. We sum over the logarithmic period bins of data displayed in Figure 9, specifically the N_P = 10 bins of ${\cal O}_{\rm deep}(P)$ for the OGLE-II LMC and SMC populations, the N_P = 3 bins of ${\cal O}_{\rm med}(P)$ for the MW, and the N_P = 10 bins of ${\cal O}_{\rm med}(P)$ for the OGLE-III LMC sample. The measured period distribution constrains γ_P and ${\cal F}_{\rm close}$, which effectively gives ν = N_P − 2 degrees of freedom. As in Section 3.3.1, we report the $\chi ^2_{\rm P}$ statistics and fitted model parameters in Table 3 and display the two-dimensional probability contour of ${\cal F}_{\rm close}$ versus γ_P in Figure 10.

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Table 3. Results of Our Monte Carlo Simulations and Fits to the Observed Eclipsing Binary Period Distributions ${\cal O}_{\rm deep}(P)$ or ${\cal O}_{\rm med}(P)$ Only

Sample	Eclipse Depths	$\chi ^2_{\rm P}/{\nu }$	ν	PTE	γP	${\cal F}_{\rm close}$
MW	Medium and Deep	0.50	1	0.48	−0.4 ± 0.3	0.22 ± 0.06
OGLE-II LMC	Deep	1.10	8	0.36	−0.3 ± 0.2	0.22 ± 0.08
OGLE-III LMC	Medium and Deep	0.89	8	0.53	−0.1 ± 0.2	0.24 ± 0.05
OGLE-II SMC	Deep	1.02	8	0.42	−0.9 ± 0.2	0.21 ± 0.09

Notes. For each of the eclipsing binary samples, we list whether the deep eclipse ${\cal O}_{\rm deep}(P)$ or extension toward medium eclipse depth ${\cal O}_{\rm med }(P)$ samples were used to fit the period distribution, minimized reduced $\chi ^2_{\rm P}$ statistics, degrees of freedom ν = N_P − 2, probabilities to exceed $\chi ^2_{\rm P}$ given ν, and the mean values and 1σ uncertainties of the two model parameters constrained by ${\cal O}_{\rm deep}(P)$ or ${\cal O}_{\rm med}(P)$.

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By making simple approximations in Section 2, we showed that all four eclipsing binary samples were skewed toward shorter periods relative to Öpik's prediction of ${\cal S}_{\rm deep}(P)$ d(log P) ∝ ${\cal S}_{\rm med}(P)$ d(log P) ∝ P^−2/3 d(log P). We confirm this result with our more robust light-curve modeling and Monte Carlo simulations, where we find fitted mean values of γ_P that are negative for all four main samples. However, the OGLE-III LMC value of γ_P = −0.1 ± 0.2 is still consistent with Öpik's law of γ_P = 0, while the OGLE-II SMC population is significantly skewed toward shorter periods with γ_P = −0.9 ± 0.2. These two values for γ_P are discrepant at the 2.4σ level. This is similar to our K-S test in Section 2 between the OGLE-II SMC and OGLE-III LMC unbinned ${\cal O}_{\rm deep}(P)$ data, which gave a probability of consistency of p_KS = 0.01.

As discussed in Section 2, it is possible that long-period systems P > 10 days with moderate eclipse depths Δm = 0.25–0.30 mag have remained undetected in the OGLE-II SMC sample because their members are systematically 0.5 mag fainter. If we only use the OGLE-II SMC data with P = 2–10 days and Δm = 0.30–0.65 mag to constrain our fit, then we find γ_P = −0.7 ± 0.4, which is more consistent with the LMC result. In any case, whether the slight discrepancy is intrinsic or due to small systematics, the best-fitting period exponent for the MW of γ_P ≈ −0.4 is between the LMC and SMC values. We confirm this intermediate value based on spectroscopic radial velocity observations of nearby early-type stars (see Section 4). Although there is a strong indication that the SMC period distribution is skewed toward shorter periods compared to the LMC data, there is no clear trend with metallicity. Moreover, the MW, SMC, and LMC samples are all mildly consistent, i.e., less than 2σ discrepancy, with the intermediate value of γ_P ≈ −0.4.

The close binary fractions ${\cal F}_{\rm close}$ are not well constrained by fitting the observed eclipse depth and period distributions separately. For example, the 1σ errors in the close binary fractions from only fitting ${\cal O}_{\Delta {\rm m}}$ were $\delta {\cal F}_{\rm close}$ ≈ 0.05–0.08, depending on the sample (see Table 2), while the errors from only fitting ${\cal O}_{\rm deep}(P)$ or ${\cal O}_{\rm med}(P)$ were $\delta {\cal F}_{\rm close}$ ≈ 0.05–0.09 (Table 3). To measure ${\cal F}_{\rm close}$ most precisely, we now fit ${\cal O}_{\Delta {\rm m}}$ and either ${\cal O}_{\rm deep}(P)$ or ${\cal O}_{\rm med}(P)$ simultaneously by minimizing χ² = $\chi ^2_{\Delta {\rm m}}$ + $\chi ^2_{\rm P}$. For each sample, we sum over the same bins of eclipse depths and orbital periods that are complete as reported in Sections 3.3.1 and 3.3.3, respectively. This combined fit gives ν = N_Δm + N_P − 4 degrees of freedom since all four model parameters are constrained. In Table 4, we report the fitting statistics as well as the means and 1σ uncertainties for ${\cal F}_{\rm close}$ only because this combined method does not alter our previous estimates of γ_q, ${\cal F}_{\rm twin}$, and γ_P. The χ²/ν values are all close to unity, and the probabilities to exceed are in the 1σ range 0.16–0.84, demonstrating that our models are sufficient in explaining the data.

Table 4. Results of Our Fits to the Observed Eclipse Depth Distributions ${\cal O}_{\Delta {\rm m}}$ and Observed Eclipsing Binary Period Distributions ${\cal O}_{\rm deep}(P)$ or ${\cal O}_{\rm med}(P)$

Sample	Eclipse Depths	χ2/ν	ν	PTE	${\cal F}_{\rm close}$
MW	Medium and Deep	0.44	4	0.76	0.22 ± 0.04
OGLE-II LMC	Deep	0.89	14	0.58	0.21 ± 0.06
OGLE-III LMC	Medium and Deep	1.02	17	0.39	0.28 ± 0.02
OGLE-II SMC	Deep	0.81	14	0.68	0.23 ± 0.06

Notes. For each sample, we list whether the deep or extension toward medium eclipse depth samples were used to simultaneously fit the eclipse depth and period distributions. We also report the minimized reduced χ² = $\chi ^2_{\Delta {\rm m}}$ + $\chi ^2_{\rm P}$ statistics, degrees of freedom ν = N_Δm + N_P − 4, probabilities to exceed χ² given ν, and the mean values and 1σ uncertainties of the close binary fractions ${\cal F}_{\rm close}$ before correcting for Malmquist bias and propagating systematic errors.

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In order to fit ${\cal O}_{\Delta {\rm m}}$ and either ${\cal O}_{\rm deep}(P)$ or ${\cal O}_{\rm med}(P)$ simultaneously, we have assumed that Δm and P are independent so that p ∝ $e^{-\chi ^2_{\Delta {\rm m}}/2}$ × $e^{-\chi ^2_{\rm P}/2}$ = $e^{-(\chi ^2_{\Delta {\rm m}}+\chi ^2_{\rm P})/2}$ = $e^{-\chi ^2/2}$. For all four samples of eclipsing binaries, the Spearman rank correlation coefficients between Δm and P are rather small at |ρ| < 0.15 across the eclipse depth intervals that are complete. These small coefficients justify our procedure for fitting the eclipsing binary period and eclipse depth distributions together in order to better constrain ${\cal F}_{\rm close}$. Moreover, the probability of observing medium eclipses ${\cal P}_{\rm med}$(P) determined in Section 3.3.2 only marginally depends on the underlying mass-ratio distribution ${\cal U}_{\rm q}$. Therefore, any trend between mass ratios and orbital periods will not affect the fitted close binary fractions beyond the quantified errors.

If we had used simple prescriptions for eclipse depths instead of the detailed nightfall light-curve models, our fitted values for ${\cal F}_{\rm close}$ would have been a factor of (10–20)% different, i.e., $\Delta {\cal F}_{\rm close}$ ≈ 0.02–0.04 depending on the sample with no general trend on the direction. This would have been a dominant source of error, especially for the OGLE-III LMC data, so it was imperative that we implemented the more precise nightfall simulations. Before we comment further on our measurements of ${\cal F}_{\rm close}$ in the different environments, we must first correct for Malmquist bias.

Unresolved binaries, including eclipsing systems, are systematically brighter than their single star counterparts. For a magnitude-limited sample within our MW, more luminous binaries are probed over a larger volume than their single star counterparts, which causes the binary fraction to be artificially enhanced. This classical Malmquist bias is sometimes referred to as the Öpik (1923) or Branch (1976) effect in the context of binary stars.

Of the ${\cal N}_{\rm med}$ = 31 eclipsing binaries in our medium eclipse depth MW sample with 〈H_P〉 < 9.3, only four systems are fainter than 〈H_P〉 > 8.8 (Lefèvre et al. 2009). One of these systems, V2126 Cyg, has a moderate magnitude of 〈H_P〉 = 9.0 and shallow eclipse depth of ΔH_P = 0.13. This small eclipse depth indicates a faint, low-mass companion, although the less likely scenario of a grazing eclipse with a more massive secondary is also feasible. The remaining three systems, IT Lib, LN Mus, and TU Mon, all have fainter system magnitudes 〈H_P〉 > 9.1 and deeper eclipses ΔH_P > 0.18, suggesting that their primaries alone do not fall within our magnitude limit of 〈H_P〉 < 9.3. If we remove this excess number of ${\cal N}_{\rm ex}$ = 3–4 eclipsing binaries from both our eclipsing binary sample ${\cal N}_{\rm med}$ and the total number of systems ${\cal N}_{\rm B}$, then the eclipsing binary fraction with medium eclipse depths ${\cal F}_{\rm med}$ = ${\cal N}_{\rm med}$/${\cal N}_{\rm B}$ would decrease by a factor of ≈11%, i.e., $\Delta {\cal F}_{\rm med}$ ≈ −0.002.

However, we must also remove from the denominator ${\cal N}_{\rm B}$ other binaries with luminous secondaries that have primaries that fall below our magnitude limit. These include close binaries that remain undetected because they have orientations that do not produce observable eclipses. Based on the correction factor ${\cal C}_{\rm med,i}$ = 9 ± 2 for geometrical selection effects alone for the MW sample (see Section 3.3.2), we expect a total of ${\cal N}_{\rm med}$ × ${\cal C}_{\rm med,i}$ ≈ 30 binaries with P = 2–20 days that should be removed from ${\cal N}_{\rm B}$.

Additional systems that contaminate ${\cal N}_{\rm B}$ consist of binaries with luminous secondaries outside of our period range of P = 2–20 days. To estimate their contribution toward Malmquist bias, we calculate the ratio ${\cal R}_{\rm P}$ between the frequency of massive secondaries across all orbital periods and the frequency of massive secondaries with P = 2–20 days. Spectroscopic observations of O- and B-type stars in the MW reveal 0.16–0.31 companions with q > 0.1 per decade of orbital period at log P ≈ 0.8 (Garmany et al. 1980; Levato et al. 1987; Abt et al. 1990; Sana et al. 2012; see also Section 4). At longer orbital periods of log P ≈ 6.5, photometric observations of visually resolved binaries give a lower value of ≈0.10–0.16 companions with q ≳ 0.1 per decade of orbital period (Duchêne et al. 2001; Shatsky & Tokovinin 2002; Turner et al. 2008; Mason et al. 2009). Using these two points to anchor the slope of the period distribution, we integrate from log P = 0.1 to the widest, stable orbits of log P ≈ 8.5. We find that there are 6.4 ± 1.3 as many total companions as there are binaries with P = 2–20 days. However, longer period binaries with P > 20 days may have a mass-ratio distribution that differs from our sample at shorter orbital periods. For example, Abt et al. (1990) and Duchêne et al. (2001) suggest random pairings of the initial mass function for wide binaries so that γ_q ≈ −2.3, while the distribution of Preibisch et al. (1999) indicates a more moderate value of γ_q ≈ −1.5. Shatsky & Tokovinin (2002) give γ_q ≈ −0.5 for visually resolved binaries, which is consistent with the values inferred from our close eclipsing binary samples of γ_q ≈ −1.0–−0.6. Assuming γ_q = −1.5 ± 0.5 for binaries outside our period range, there are 2.3 ± 1.1 times fewer binaries with q > 0.6 relative to the mass-ratio distribution constrained for our close eclipsing binaries. Since we are primarily concerned with massive secondaries that contribute toward Malmquist bias, ${\cal R}_{\rm P}$ ≈ (6.4 ± 1.3)/(2.3 ± 1.1) = 2.8 ± 1.4.

The eclipsing binary fraction for the MW sample after correcting for classical Malmquist bias is then

$$\begin{equation} {\cal F}_{\rm med} = \frac{{\cal N}_{\rm med} - {\cal N}_{\rm ex}}{{\cal N}_{\rm B} - {\cal N}_{\rm ex} {\cal C}_{\rm med, i} {\cal R}_{\rm P}} = (1.83\,\pm \,0.38)\%, \end{equation} \tag{ 5 }$$

where we propagated the uncertainties in ${\cal C}_{\rm med, i}$ and ${\cal R}_{\rm P}$ and the Poisson errors in ${\cal N}_{\rm med}$ and ${\cal N}_{\rm ex}$. Note that removing non-eclipsing binaries with luminous secondaries that remain undetected mitigates the effects of Malmquist bias. Specifically, we find the reduction factor to be ${\cal C}_{\rm Malm}$ = 0.94 ± 0.05 instead of the factor of ${\cal C}_{\rm Malm}$ = 0.89 determined above when we only removed ${\cal N}_{\rm ex}$ eclipsing systems. Although these two competing effects in the numerator and denominator of the above relation have been discussed in the literature (e.g., Bouy et al. 2003), the removal of binaries with luminous secondaries that remain undetected is typically neglected. The inferred close binary fraction for the MW will also decrease by a factor of ${\cal C}_{\rm Malm}$ = 0.94, so that the corrected value is only slightly lower at ${\cal F}_{\rm close}$ = 21% (see Section 3.5).

In the case of the Magellanic Clouds at fixed, known distances, classical Malmquist bias does not apply. Nonetheless, our absolute magnitude interval of $\overline{M}_{\rm I}$ = [−3.8, −1.3] contains binaries with primaries that are lower in intrinsic luminosity and stellar mass relative to single stars in the same magnitude range. Some binaries in our sample have primaries that are fainter than our magnitude limit of M_I = −1.3, while some systems have primaries in the range we want to consider but are pushed beyond M_I = −3.8 because of the excess light added by the secondary. Since the number of primaries dramatically increases with decreasing stellar mass and luminosity, the net effect is that the binary fractions are biased toward larger values. Hence, our statistics are affected by Malmquist bias of the second kind because two classes of objects, e.g., binaries and single stars, are surveyed to a certain depth down their respective luminosity functions (Teerikorpi 1997; Butkevich et al. 2005).

For example, Mazeh et al. (2006) used OGLE-II data of the LMC to identify 938 eclipsing binaries on the MS with apparent magnitudes 17 < I < 19 and periods 2 < P (days) < 10. Instead of normalizing these eclipsing binaries to the total number of ≈330,000 MS systems with 17 < I < 19, they assumed that the average eclipsing binary was 〈ΔM_I〉 = 0.5 mag brighter than the primary component alone, and therefore normalized to the ≈700,000 MS systems with 17.5 < I < 19.5. Their correction for Malmquist bias of the second kind lowered the inferred close binary fraction by a factor of 2.1, i.e., ${\cal C}_{\rm Malm}$ = 0.48.

Instead of adding systems below our lower magnitude limit as done by Mazeh et al. (2006), we remove binaries with luminous secondaries within our magnitude interval $\overline{M}_{\rm I}$ = [−3.8, −1.3] as described above for the MW. To determine the average fraction $\langle \delta {\cal F}_{\rm I} \rangle$ of eclipsing binaries that should be removed from our Magellanic Cloud samples, we use the OGLE photometric catalogs (Udalski et al. 1998, 2000, 2008) to compute the observed fractional decrease $\delta {\cal F}_{\rm I}$ in the total number of MS systems as a function of incremental I-band magnitude ΔM_I. Quantitatively,

$$\begin{equation} \delta {\cal F}_{\rm I} (\Delta {M}_{\rm I}) = 1 - \frac{{\cal N}(\overline{M}_{\rm I} - \Delta {M}_{\rm I})}{{\cal N}(\overline{M}_{\rm I})}, \end{equation} \tag{ 6 }$$

where ${\cal N}(\overline{M}_{\rm I})$ = ${\cal N}_{\rm B}$ is our original total number of MS systems and ${\cal N}(\overline{M}_{\rm I} - \Delta {M}_{\rm I})$ is the number of systems with colors V − I < 0.1 in the interval M_I = [−3.8, −1.3 − ΔM_I]. We display $\delta {\cal F}_{\rm I}$ in the top panel of Figure 11 for the three OGLE Magellanic Cloud samples. We only show the fractional decreases $\delta {\cal F}_{\rm I}$ across the interval 0 < ΔM_I < 0.75 because binary companions can only contribute a luminosity excess in this range. The three distributions of $\delta {\cal F}_{\rm I}$ are similar among the three populations due to the consistency of the stellar mass function in the different environments. The total number of systems is approximately halved, i.e., $\delta {\cal F}_{\rm I}$ = 0.5, at ΔM_I ≈ 0.5, consistent with the result of Mazeh et al. (2006).

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Instead of assuming an average value for the magnitude difference 〈ΔM_I〉 = 0.5 mag between a single star and eclipsing binary with the same primary, we use the OGLE eclipsing binary data and our Monte Carlo simulations to model an I-band excess probability distribution p_I (ΔM_I) d(ΔM_I). Using the best-fit models for each of the three OGLE samples, we synthesize distributions of secondary masses that produce observable eclipses, i.e., systems with eclipse depths 0.25 < Δm < 0.65 for our deep samples and 0.1 < Δm < 0.65 for our extension toward medium eclipse depths (OGLE-III LMC only). We then use the stellar tracks of Bertelli et al. (2009), as well as color indices and bolometric corrections of Cox (2000), to convert the distribution of secondary masses that produce observable eclipses into a distribution of secondary absolute magnitudes in the I band. We can then easily determine the system luminosity, the luminosity of the primary alone, and the I-band excess ΔM_I between the two for each eclipsing binary. In the bottom panel of Figure 11, we display our results for the I-band excess probability distribution p_I (ΔM_I) d(ΔM_I), which is normalized so that the distribution integrates to unity.

The I-band excess probability distributions p_I for the three OGLE samples exhibiting deep eclipses are all quite similar. This is because they have similar eclipse depth distributions ${\cal O}_{\Delta {\rm m}}$ and therefore similar mass-ratio distributions ${\cal U}_{\rm q}$. Very few low-mass, low-luminosity secondaries with ΔM_I < 0.1 mag are capable of producing deep eclipses with 0.25 < Δm < 0.65. However, many of these faint secondaries are included in the OGLE-III LMC medium eclipse depth sample. The median I-band excess is only 〈ΔM_I〉 = 0.35 and 〈ΔM_I〉 = 0.20 mag for the deep and medium samples, respectively, which are lower than the value of 〈ΔM_I〉 = 0.5 used by Mazeh et al. (2006). Note that these values of 〈ΔM〉 = 0.2–0.5 mag are the reason we excluded the ${\cal N}_{\rm ex}$ = 3–4 eclipsing binaries in the MW sample (Section 3.4.1) that were within 0.2–0.5 mag of our magnitude limit of 〈H_P〉 = 9.3.

We can now compute the average fraction $\langle \delta {\cal F}_{\rm I}\rangle$ of eclipsing binaries that should be removed from our samples by weighting $\delta {\cal F}_{\rm I}$ with the I-band excess probability distribution, i.e., $\langle \delta {\cal F}_{\rm I}\rangle$ = $\int \delta {\cal F}_{\rm I}$(ΔM_I) p_I(ΔM_I) d(ΔM_I). We find $\langle \delta {\cal F}_{\rm I} \rangle$ = 0.38 ± 0.11 and $\langle \delta {\cal F}_{\rm I} \rangle$ = 0.35 ± 0.10 for the OGLE-II LMC and SMC deep eclipse samples, respectively, and $\langle \delta {\cal F}_{\rm I} \rangle$ = 0.23 ± 0.08 for the OGLE-III LMC medium eclipse sample. These values are lower than the estimate of $\langle \delta {\cal F}_{\rm I} \rangle$ = 0.52 by Mazeh et al. (2006) because the modeled I-band excess probability distributions are weighted more toward fainter companions.

Instead of only removing this average fraction $\langle \delta {\cal F}_{\rm I}\rangle$ of eclipsing binaries, i.e., assuming ${\cal C}_{\rm Malm}$ = 1 − $\langle \delta {\cal F}_{\rm I} \rangle$, we must also account for the other binaries with luminous secondaries outside our parameter space of eclipse depths and orbital periods. Using a similar format as in Equation (5), we derive

$$\begin{equation} {\cal C}_{\rm Malm} = \frac{1 - \langle \delta {\cal F}_{\rm I} \rangle }{1 - {\cal F}_{\rm med} \langle \delta {\cal F}_{\rm I} \rangle {\cal C}_{\rm med, i} {\cal R}_{\rm P}}, \end{equation} \tag{ 7 }$$

where ${\cal F}_{\rm med}$ = 1.87% is the uncorrected eclipsing binary fraction in Table 1, ${\cal C}_{\rm med,i}$ = 11 ± 2 is the correction factor for geometrical selection effects alone (see Section 3.3.2) for the OGLE-III LMC medium sample, and ${\cal R}_{\rm P}$ = 2.8 ± 1.4 has the same definition as in Section 3.4.1. We calculate similar values for the OGLE-II LMC and SMC deep eclipse samples, where ${\cal F}_{\rm deep}$ = 0.70% and ${\cal C}_{\rm deep,i}$ = 14 ± 3. We find the overall correction factors for Malmquist bias of the second kind to be ${\cal C}_{\rm Malm}$ = 0.73 ± 0.16, 0.91 ± 0.12, and 0.76 ± 0.15 for the OGLE-II LMC, OGLE-III LMC, and OGLE-II SMC samples, respectively. Because the OGLE-III LMC survey was sensitive to shallow eclipses that systematically favored low-luminosity companions with 〈ΔM_I〉 ≈ 0.2 mag, the correction for Malmquist bias for this population is nearly negligible.