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’ \(O_e(\Delta m, P)\)