Population synthesis studies of close binary systems using a variable common envelope efficiency parameter. I. Dependence on secondary mass

We perform population synthesis calculations of present-day post-CE binaries ( PCEBs) and zero-age cataclysmic variables (ZACVs) using a common envelope (CE) efficiency parameter, alpha(CE), that is a function of secondary mass, M-s. We investigate three basic possibilities: (1) a standard constant alpha(CE) prescription, with alpha(CE) ranging from 0.05 to 1.0, to provide a baseline for comparison; (2) a power-law dependence, alpha(CE) (M-s)(n), with n = 0.5; 1.0, and 2.0; and (3) a dependence in which alpha(CE) approaches 1 for large M-s and alpha(CE) = 0 below some assumed cutoff mass, alpha(CE) = 1 - M-cut/M-s, where M-cut = 0.0375; 0.075, and 0.15 M.. We find that if alpha(CE) less than or similar to 0.2 in our constant alpha(CE) sequence, the predicted present-day ZACV population is significantly modified compared with our standard model (alpha(CE) = 1.0). All prior population synthesis calculations of the formation of CVs only considered values of alpha(CE) >= 0.3 and found that their model populations were not strongly dependent upon the value of alpha CE. Our results indicate that a much wider range of values for alpha(CE), including very low values, must be considered in order for a dependence to be seen. In our variable alpha(CE) sequences for ZACVs, we find that for models in which alpha(CE) decreases very rapidly for small M-s, the orbital period distribution below the period gap differs significantly from our standard model. In particular, the fraction of CVs forming with P < 2 hr is reduced significantly, the fraction forming in the gap is increased significantly, and both the short-period peak and the minimum period shift to considerably longer periods compared with our standard model. We suggest that the observed scarcity of CVs with P < 77 minutes may possibly provide evidence that progenitor binaries with M-s less than or similar to 0.10 M. are unable to avoid merger within the CE. We also suggest that if alpha(CE) decreases rapidly for small M-s, it is possible that the lower edge of the period gap could be, in part, an imprint of the ZACV population.