Planetary migration and the origin of the 2:1 and 3:2 (near)-resonant population of close-in exoplanets

Abstract

We present an analytical and numerical study of the orbital migration and resonance capture of fictitious two-planet systems with masses in the super-Earth range undergoing Type-I migration. We find that, depending on the flare index and proximity to the central star, the average value of the period ratio, P 2 /P 1 , between both planets may show a significant deviation with respect to the nominal value. For planets trapped in the 2:1 commensurability, offsets may reach values on the order of 0.1 for orbital periods on the order of 1 day, while systems in the 3:2 mean-motion resonance (MMR) show much smaller offsets for all values of the semimajor axis. These properties are in good agreement with the observed distribution of near-resonant exoplanets, independent of their detection method. We show that 2:1-resonant systems far from the star, such as HD82943 and HR8799, are characterized by very small resonant offsets, while higher values are typical of systems discovered by Kepler with orbital periods approximately a few days. Conversely, planetary systems in the vicinity of the 3:2 MMR show little offset with no significant dependence on the orbital distance. In conclusion, our results indicate that the distribution of Kepler planetary systems around the 2:1 and 3:2 MMR are consistent with resonant configurations obtained as a consequence of a smooth migration in a laminar flared disk, and no external forces are required to induce the observed offset or its dependence with the commensurability or orbital distance from the star.

The K2-138 System: A Near-resonant Chain of Five Sub-Neptune Planets Discovered by Citizen Scientists

“…and Batygin & Morbidelli (2013) investigate the suggestion of tidal dissipation as a mechanism for keeping individual pairs of planets just outward of the resonance; they note that in systems with more than two planets, where the planets can inhabit multiple resonances, the planets can remain close to resonance despite tidal dissipation. Recently, Ramos et al (2017) analytically derived the expected offset from a first-order resonance for a pair of planets due to Type I migration. Their Figure 3 shows that for periods shorter than ∼10 days, the resonance period ratio is 1.505-1.525, depending on the mass of the inner planet and the mass ratio of the two planets, with higher period ratios expected as the mass ratio approaches unity.…”

Capture into first-order resonances and long-term stability of pairs of equal-mass planets

“…This advantageous reduction can be used to obtain a general description of the dynamics (e.g. ], [Ramos et al(2017)]). However it is insufficient when one confronts even qualitatively the prediction of this theoretical model with results from numerical simulations, as we will see in the next Section, where we compute the locus of equilibrium points (i.e.…”

“…This is an efficient method to obtain planets deeply in mutual mean motion resonance (e.g. [Matsumoto et al(2012)], [Ramos et al(2017)]). We start with two planets of equal mass, m 1 = m 2 = m, typically m/M * = 10 −5 − 10 −2 , on coplanar orbits, embedded in a protoplanetary disk.…”

“…where τ a = τ mig /2 and p 2 for small e. It is customary to introduce the quantity K = τ a /τ e which we call K-factor (cfr. [Ramos et al(2017)]). Note that, K = τ a τ e 0.780 2.7 + 1.1 h −2 : (3.9)…”

“…In order to insure convergent migration and resonant capture, we need to stop the migration of the inner planet, since two equally massive planets would migrate inward at roughly the same rate and resonant capture would not occur (e.g. [Ramos et al(2017)]). To do this, we simulate the effect of a disk edge, which corresponds to a sharp drop in Σ as r decreases.…”