  During post-main-sequence evolution, radial expansion of the primary star,
accompanied by intense winds, can significantly alter the binary orbit via
tidal dissipation and mass loss. The fate of a given binary system is
determined by the initial masses of the primary and companion, the initial
orbit (taken to be circular), the Reimers mass-loss parameter, and the tidal
prescription employed. For a range of these parameters, we determine whether
the orbit expands due to mass loss or decays due to tidal torques. Where a
common envelope (CE) phase ensues, we estimate the final orbital separation
based on the energy required to unbind the envelope. These calculations predict
period gaps for planetary and brown dwarf companions to white dwarfs. In
particular, the lower end of the gap is the longest period at which companions
survive their CE phase while the upper end of the gap is the shortest period at
which a CE phase is avoided. For binary systems with 1 $M_\odot$ progenitors,
we predict no Jupiter-mass companions with periods $\lesssim$270 days. For
binary systems consisting of a 1 $M_\odot$ progenitor with a 10 Jupiter-mass
companion, we predict a close, post-CE population with periods $\lesssim$0.1
days and a far population with periods $\gtrsim$380 days. These results are
consistent with the detection of a $\sim$50 $M_{\rm J}$ brown dwarf in a
$\sim$0.08 day orbit around the white dwarf WD 0137-349 and the tentative
detection of a $\sim$2 $M_{\rm J}$ planet in a $\sim$4 year orbit around the
white dwarf GD66.

Solar and Stellar Astrophysics