This is a short summary of the paper Eclipsing binary statistics - theory and observation (Söderhjelm 2005).

In order to estimate the number of eclipsing binaries to be expected in the upcoming all-sky survey ’Gaia’, the authors executed a large binary population experiment using the rapid binary evolution code (BSE) from Hurley et al. 2002

Population Synthesis

Parameters - SFR, IMF and z

The authors assume a constant Star Formation Rate (0-12Gyr) and a time independent IMF as per (Kroupa 2001) . The metallicity parameter [Fe/H] must be held constant for the duration of a BSE code run (i.e. it is age-independent and uniform) so the authors combined several runs with different upper and lower IMF limits in order to mimic the observed [Fe/H] distributions.

Parameters - orbit-sizes, mass-ratios, eccentricities


For the BSE software runs, the authors made a simplifying assumption that the function \(f(a,q,e)\) is separable and they assigned various analytic formulae to each of: \(f(a)\) - initial orbit-sizes, \(f(q)\) - initial mass ratios and \(f(e)\) - initial eccentricities.

All parameters for the BSE code were kept at their default values and in order to handle brown dwarfs with a mass below \(0.08 M_{\odot}\) the authors used an extra data grid from Baraffe et al. (1998).

For orbital periods above 5 years, each binary component was evolved separately and a binary was made from two single-star models (S+S) using approximations given by Hurley et al. (2000).

The authors used this parallel S+S approximation as a check on the full BSE models and found the S+S approach gave a roughly similar a-distribution but slightly more short-period pairs then the more detailed BSE calculations.

Eclipse Statistics

The authors note it is straightforward to show that the probability of an eclipse at any depth is \(P_e = r_s + r_g\), where \(r_s\) and \(r_g\) are the radii relative to orbit size \((r/a)\), of the smaller and greater stars respectively. But in this study the authors wish to calculate the probability of an eclipse of any specified magnitude depth, which calls for a different approach.

In order to calculate this probability, the function \(i(\Delta m)\) must be derived. Although the maximum primary eclipse depth \(\Delta m\) is a well defined function of the inclination angle i, iteration is needed to find this inverse function \(i(\Delta m)\)

Based on a random distribution of inclinations, the probability of a primary eclipse deeper than \(\Delta m\) is \(cos(i(\Delta m))\) where the authors have not included any effects from limb or gravity darkening.

The mean values for eclipse probabilities are meaningful only for similar sized systems (radii, luminosities and orbit sizes) so the authors have divided the HR diagram into a number of bins, and each bin is further subdivided according to orbital period and \(\Delta m\). The final results are thus histograms giving the mean probability \(P_e(\Delta m, P)\) for each HR bin.

The authors have also included another type of probability which is useful when comparing the synthetic population with observations: namely the fraction of eclipsing binaries relative to all stars (single and binary) in the same HR diagram bin. The authors derive this ’normalized probability’ \(O_e(\Delta m, P)\)


The authors find that the number of observed eclipsing binaries scales directly with the input frequency of short-period orbits. For example, although typically less than 1% of eclipsing FG systems show 1.5 mag eclipses, the ratio at the shortest periods is closer to 20%. The situation is more complex for higher mass sytems although the authors find humps with high probability of deep eclipses at about 2 day and 0.4 day orbital periods.

A sample figure showing eclipse probabilities for FG stars on the main sequence has been reproduced below:

The eclipse probabilities \(P_{\rm e}(\Delta m,P)\) for the FG bin, as a function of the (log)
period in days. Each of the lines (solid from 0.1 to 0.5 mag, dashed from 0.6 to 1.0, and dot-dashed from 1.1 to 1.5 mag) corresponds to a certain maximum eclipse-depth \(\Delta m\).