Using population synthesis tools we create a synthetic Kepler Input Catalogue (KIC) and subject it to the Kepler Stellar Classification Program (SCP) method for determining stellar parameters such as the effective temperature Teff and surface gravity
g. We achieve a satisfactory match between the synthetic KIC and the real KIC in the log g vs log Teff diagram, while there is a significant difference between the actual physical stellar parameters and those derived by the SCP of the stars in the synthetic sample. We find a median difference Delta Teff=+500K and Delta log g =-0.2dex for main-sequence stars, and Delta Teff=+50K and Delta log g =-0.5dex for giants, although there is a large variation across parameter space. For a MS star the median difference in g would equate to a ~3% increase in stellar radius and a consequent ~3% overestimate of the radius for any transiting exoplanet. We find no significant difference between Delta Teff and Delta log g for single stars and the primary star in a binary system. We also re-created the Kepler target selection method and found that the binary fraction is unchanged by the target selection. Binaries are selected in similar proportions to single star systems; the fraction of MS dwarfs in the sample increases from about 75% to 80%, and the giant star fraction decreases from 25% to 20%.
We reconstruct the common envelope (CE) phase for the current sample of observed white dwarf-main sequence post-common envelope binaries (PCEBs). We apply multi-regression analysis in order to investigate whether correlations exist between the CE eje
ction efficiencies, alpha_CE, inferred from the sample, and the binary parameters: white dwarf mass, secondary mass, orbital period at the point the CE commences, or the orbital period immediately after the CE phase. We do this with and without consideration for the internal energy of the progenitor primary giants envelope. Our fits should pave the first steps towards an observationally motivated recipe for calculating alpha_CE using the binary parameters at the start of the CE phase, which will be useful for population synthesis calculations or models of compact binary evolution. If we do consider the internal energy of the giants envelope, we find a statistically significant correlation between alpha_CE and the white dwarf mass. If we do not, a correlation is found between alpha_CE and the orbital period at the point the CE phase commences. Furthermore, if the internal energy of the progenitor primary envelope is taken into account, then the CE ejection efficiencies are within the canonical range 0<alpha_CE<=1, although PCEBs with brown dwarf secondaries still require alpha_CE>=1.
We apply population synthesis techniques to calculate the present day population of post-common envelope binaries (PCEBs) for a range of theoretical models describing the common envelope (CE) phase. Adopting the canonical energy budget approach we co
nsider models where the ejection efficiency, $alpha_{rmn{CE}}$ is either a constant, or a function of the secondary mass. We obtain the envelope binding energy from detailed stellar models of the progenitor primary, with and without the thermal and ionization energy, but we also test a commonly used analytical scaling. We also employ the alternative angular momentum budget approach, known as the $gamma$-algorithm. We find that a constant, global value of $alpha_{rmn{CE}} ga 0.1$ can adequately account for the observed population of PCEBs with late spectral-type secondaries. However, this prescription fails to reproduce IK Pegasi, which has a secondary with spectral type A8. We can account for IK Pegasi if we include thermal and ionization energy of the giants envelope, or if we use the $gamma$-algorithm. However, the $gamma$-algorithm predicts local space densities that are 1 to 2 orders of magnitude greater than estimates from observations. In contrast, the canonical energy budget prescription with an initial mass ratio distribution that favours unequal initial mass ratios gives a local space density which is in good agreement with observations, and best reproduces the observed distribution of PCEBs. Finally, all models fail to reproduce the sharp decline for orbital periods, $P_{rmn{orb}} ga 1$ d in the orbital period distribution of observed PCEBs, even if we take into account selection effects against systems with long orbital periods and early spectral-type secondaries.
