Forming Orbital Resonances With two SuperEarths During Exoplanetary Migration

On the migration of two planets in a disc and the formation of mean motion resonances

Author:

Migaszewski

Abstract:

We study the dynamics of a system of two super-Earths embedded in a protoplanetary disc. Depending on the disc parameters, planets' masses and positions in the disc, the migration of a planet can be inward or outward and the migration of a two-planet system can be convergent or divergent. The convergent migration means that the period ratio P2/P1 decreases in time. In such a case mean motion resonance (MMR) can be formed when the period ratio reaches a resonant value of a first order MMR (p+1)/p, where p is a small integer. When the divergent migration occurs, P2/P1 increases in time and a system initially close to MMR moves away from the resonance. We build a simple model of an irradiated viscous disc and use analytical prescriptions for the planet-disc interactions. We performed 3500 simulations of the migration of two-planet systems with various masses and initial orbits. We found that approximately half of the systems end up as configurations involved in one of the first order MMRs such as 2:1, 3:2, 4:3 and 5:4. In all these cases both resonant angles of a given MMR librate. The first angle librates around 0, the second around pi, when the period ratio is larger than (p+1)/p. The situation is reversed when P2/P1 less than (p+1)/p. The amplitudes of the librations depend on the period ratio and does not depend on the planets' masses. The amplitudes are minimal and ~0, when P2/P1~(p+1)/p and increase when P2/P1 deviates from the nominal value. We found the range of the period ratios for which the angles of 2:1 MMR librate to be [1.72, 2.12]. The range for 3:2 MMR is [1.4, 1.7]. The upper limit of P2/P1 for which the resonant angles of 4:3 MMR librate is 1.4. The lower limit for this resonance as well as the range for 5:4 MMR could not be determined due to too few solutions with P2/P1 less than 1.33. We found that almost all the systems evolve periodically.