The Effect of Tidal Inflation Instability on the Mass and Dynamical Evolution of Extrasolar Planets with Ultrashort Periods

Abstract

We investigate the possibility of substantial inflation of short-period Jupiter-mass planets, as a result of their internal tidal dissipation associated with the synchronization and circularization of their orbits. We employ the simplest prescription based on an equilibrium model with a constant lag angle for all components of the tide. We show the following: (1) In the low-eccentricity limit, the synchronization of the planets' spin with their mean motion is established before tidal dissipation can significantly modify their internal structure. (2) However, above a critical eccentricity, which is a function of the planets' semimajor axis, tidal dissipation of energy during the circularization process can induce planets to inflate in size before their eccentricity is damped. (3) For moderate eccentricities, the planets adjust to stable thermal equilibria in which the rate of their internal tidal dissipation is balanced by the enhanced radiative flux associated with their enlarged radii. (4) For sufficiently large eccentricities, the planets swell beyond two Jupiter radii and their internal degeneracy is partially lifted. Thereafter, their thermal equilibria become unstable and they undergo runaway inflation until their radii exceed the Roche radius. (5) We determine the necessary and sufficient condition for this tidal inflation instability. (6) These results are applied to study short-period planets. We show that for young Jupiter-mass planets, with a period of less than 3 days, an initial radius of about 2RJ, and an orbital eccentricity greater than 0.2, the energy dissipated during the circularization of their orbits is sufficiently intense and protracted to inflate their sizes up to their Roche radii. (7) We estimate the mass-loss rate, the asymptotic planetary masses, and the semimajor axes for various planetary initial orbital parameters. The possibility of gas overflow through both inner (L1) and outer (L2) Lagrangian points for the planets with short periods or large eccentricities is discussed. (8) Planets with more modest eccentricity ( 0.03-0.04 AU) lose mass via Roche lobe overflow mostly through the inner Lagrangian (L1) point. As a result of the conservation of total angular momentum, these mass-losing planets migrate outward, such that their semimajor axes and Roche radii increase while their mass, eccentricity, and tidal dissipation rate decrease until the mass loss is quenched. (9) Based on these results, we suggest that the combined effects of self-regulated mass loss and tidally driven orbital evolution may be responsible for the apparent lack of giant planets with ultrashort periods 3 days. (10) Mass loss during their orbital circularization may also have caused the planets with periods in the range ~3-7 days to be less massive than long-period planets, which are not affected by the star-planet tidal interaction. (11) The accretion of the short-period planets' tidal debris can also lead to the surface layer contamination and metallicity enhancement of their host stars. (12) Among the planets with periods of 1-3 weeks today, some may have migrated outward and attained circular orbits while others may have preserved their initial eccentricity and semimajor axis. Therefore, planets with circular orbits are expected to coexist with those with eccentric orbits in this period range. (13) Gross tidal inflation of planets occurs on the timescale ~106 yr after their formation for a brief interval of ~105 yr. The relatively large sizes of their classical and weak-line T Tauri host stars increase the planets' transit probability. The inflated sizes of the tidally heated planets also increase the eclipse depth of such transit events. Thus, the tidal inflation and disruption of planets may be directly observable around classical and weak-line T Tauri stars.