Tides and Tidal Engulfment in Post-Main Sequence Binaries: Period Gaps for Planets and Brown Dwarfs Around White Dwarfs

The presence of a close, low-mass companion is thought to play a substantial and perhaps necessary role in shaping post-Asymptotic Giant Branch and Planetary Nebula outflows. 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. To investigate this, we couple stellar evolution models (from the zero-age main-sequence through the end of the post-main sequence) to a tidal evolution code. The binary's fate is determined by the initial masses of the primary and the companion, the initial orbit (taken to be circular), and the Reimers mass-loss parameter. 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. The upper end of the gap is the shortest period at which a CE phase is avoided. The lower end is the longest period at which companions survive their CE phase. For binary systems with 1 $M_\odot$ progenitors, we predict no Jupiter-mass companions with periods $\lesssim$270 days. Once engulfed, Jupiter-mass companions do not survive a CE phase. For binary systems consisting of a 1 $M_\odot$ progenitor with a companion 10 times the mass of Jupiter, we predict a period gap between $\sim$0.1 and $\sim$380 days. These results are consistent with both the detection of a $\sim$50 $M_{\rm J}$ brown dwarf in a $\sim$0.003 AU ($\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 $\gtrsim$2.7 AU ($\gtrsim$4 year) orbit around the white dwarf GD66.