4 Simulation of separability in binary systems

In order to interpret our results with respect to the different evolutionary scenarios we simulate binary systems containing main sequence (MS) companions and sdBs with period distributions found for normal main sequence binaries (Duquennoy & Mayor 1991). Assuming that the sdB mass is 0.5$M_{\odot}$ and the MS companion mass is 1$M_{\odot}$ we convert the period distribution published by Duquennoy & Mayor (1991) to physical separations using Kepler's Harmonic law. The orientation of the axis of the system is then chosen to be random in space and the projected separation, or $a \sin i$, is calculated, given the distance to the system which is found from the apparent and absolute brightness of the system. The orbits are assumed to be circular.

Based on the spectroscopic distances derived above (see Table 4) we then simulate a huge number of such binary systems. For three stars (HE 0430-2457, PG 0942+461, and HE 2213-2212) the magnitude ratio of the components could not be determined and therefore the distances are unknown. We adopted the mean value of the other stars ($\Delta R$ = $\rm 1\hspace{-0.25em}\stackrel{m}{.} \hspace{-1.0mm}1$), which is consistent with their spectral appearance (see Fig. A.1). The numerical simulation predicts a mean value of $a \sin i$ = 0 $.\!\!{\hbox{$^{\prime\prime}$ }}$04 and that, out of the 19 observed systems, we should resolve six systems at a resolution limit of 0 $.\!\!{\hbox{$^{\prime\prime}$ }}$1, one of which should show a separation greater than 1 $.\!\!{\hbox{$^{\prime\prime}$ }}$0.

Since the orbital motion for an eccentric orbit is lower during phases of large separation, the time averaged distance is larger than the semi major axis. Thus eccentric orbits would increase the detectability. Duquennoy & Mayor (1991) also provide a distribution of ellipticities for normal stars. If the sdB systems did not experience phases of binary interaction, the distribution of eccentricity should correspond to that of normal stars. We used Duquennoy & Mayor's distribution corrected for selection effects. For each eccentricity the ratio of the time averaged distance to a was calculated and finally the mean over the Duquennoy & Mayor distribution was computed. We find the average distance of the companions to increase by 17%. Another mechanism that tends to increase the separation of the components in a sdB binary is mass loss during post-main sequence evolution in order to reduce the mass of the sdB progenitor to its present value of half a solar mass. Assuming that the sdB evolved from a 1$M_{\odot}$ main sequence progenitor it must have lost 0.5$M_{\odot}$ due to a stellar wind during its post-main sequence evolution. Assuming that the wind emanates in a spherical symmetric manner and does not interact with the companion the increase in separation can be calculated according to $\frac{\dot{a}}{a}=-\frac{\dot{M_{\rm s}}}{M_{\rm s}+M_{\rm c}}$ (Pringle 1985), with a being the separation and $M_{\rm s}$ and $M_{\rm c}$ the masses of the sdB progenitor and that of the cool star, respectively. As a result the separation increases by 33%.

We repeated the Monte Carlo simulations for increased separations. Even when we consider both elliptical orbits and evolution of the orbits due to a stellar wind as described above the prediction increased only slightly to 7 resolvable stars in our sample.

Hence we predict that 6 to 7 stars should be resolvable in our sample if the systems have separations consistent with the Duquennoy & Mayor (1991) distribution.