Constraining Tidal Migration with the Hot Jupiter Population

Abstract

Hot Jupiters with orbital periods shorter than a few days have probably been affected by tidal orbital migration. We develop an analytical framework for constraining tidal migration from the present-day hot Jupiter period distribution, taking into account the uncertain rate and period distribution of hot Jupiters produced by mechanisms such as high-eccentricity migration or disk-driven migration. Assuming the tidal migration timescale is proportional to $P^{\chi_\tau}$, solutions with $\chi_\tau \simeq 3, 1.7,$ and 5.6 are all compatible with the present-day period distribution. The $\chi_\tau \simeq 3$ solution is consistent with equilibrium tides with suppression of dissipation at short periods, and implies that newly circularized hot Jupiters have periods concentrated near $3-4$ days, as predicted in some high-eccentricity migration models. The $\chi_\tau \simeq 1.7$ solution is also compatible with the $3-4$ day peak but has no clear counterpart in existing tidal theories and is more finely tuned. The $\chi_\tau \simeq 5.6$ solution is compatible with enhanced short-period equilibrium tidal dissipation or weakly nonlinear gravity-wave dissipation, but requires circularization at unexpectedly short periods. Thus, we find the model with $\chi_\tau \simeq 3$ most appealing. Transit timing of individual systems and observational constraints on the rate of hot Jupiter engulfment provide additional constraints, which are presently inconclusive but should improve with future data. Improved measurements of the occurrence of short-period planets as a function of planet mass and system age could also help to sharpen the constraints on tidal migration.