Self-consistent Models of Orbital Migration of Earths and Super-Earths

Data from the Kepler mission indicate that the occurrence rate of planets orbiting solar-type stars, with radii between ~2 and ~4 Rearth and periods less than 50 days, is about 0.13 (Howard et al., 2011, arXiv:1103.2541). A similar rate is provided by radial velocity surveys for planet less than 10 Mearth. In situ formation of all these planets is deemed unlikely. Instead, it is argued that most of them formed farther away from the star and then moved toward smaller radii. One mechanism, capable of delivering to small radii planets formed elsewhere in a gaseous disk, is orbital migration driven by tidal interaction with the disk. We implemented time-dependent models of the orbital evolution of Earth-mass and Super-Earth-mass planets in a protoplanetary disk. The models take into account, in a self-consistent fashion, the following processes: 1) viscous evolution of the gaseous disk; 2) photo-evaporation of the disk gas originating from Far-Ultraviolet, Extreme Ultraviolet, and X-ray photons emitted by a solar-mass star; 3) tidal torque exchange between the disk and the planet. The viscous evolution model is based on the solution of a modified form of the 1D equation that regulates conservation of angular momentum in the disk and accounts for dissipation produced by photo-evaporation and perturbations induced by tidal torques. The photo evaporation rates are calculated by solving the 1+1D radial-vertical structure of the disk (Gorti et al., 2009, Astrophys. J., 705, 1237). The tidal torques exerted by the planet on the disk and vice versa are embedded in the disk evolution equation via a torque density term, derived from 3D hydrodynamics calculations of disk-planet interactions (D'Angelo and Lubow, 2010, Astrophys. J., 724, 730). We performed calculations in which the disk extends in radius from 0.02 to 1000 AU and has an initial surface density distribution applicable to a minimum-mass solar nebula (Davis, 2005, Astrophys. J. Lett., 627, 153). The X-ray and EUV luminosities are ∼1e-3 Lsun, while the FUV luminosity is between 1e-4 and 0.1 Lsun, depending on the time-dependent accretion rate. Various viscosity regimes were investigated, assuming a kinematic viscosity nu=nu1*sqrt(R1), where R1=R/(1 AU), and nu1 is between 1e-6 and 2e-5 in units of sqrt(G*Msun*AU). The planet grows from 0.3 Mearth to Miso, the isolation mass, within a time Tiso, at an oligarchic growth rate. The values of Miso and Tiso were varied randomly in the ranges between 1 and 10 Mearth and between 1e4 and 1e5 local orbital periods. The initial orbital radius of the planetary core was randomly chosen between 1 and 10 AU. The final distributions of orbital radii show a dependence on disk viscosity. For nu1~1e-6, most of the planets are delivered within ~0.3 AU of the star (periods less than ~60 days). For nu1~1e-5, about 30% of the planets orbit within 1 AU, whereas ~90% of them orbit beyond 1 AU when nu1~2e-5. Overall, planets within ~0.07 AU (periods less than ~7 days) are rare. Additionally, a large fraction of the total population resides inside of 1 AU. In our sample of the parameter space, we did not identify any radial region significantly devoid of planets.