On the migration-induced formation of the 9:7 mean motion resonance

Abstract

We study formation of 9:7 mean motion resonance (MMR) as a result of convergent migration of two planets embedded in a disc. Depending on the migration parameters, initial orbits and planets' masses the system may pass through the resonance or enter it (permanently or temporarily). We illustrate that a stable equilibrium of the averaged system (a periodic orbit of the N-body model) is surrounded by the region of permanent resonance capture, whose size depends on the migration parameters and the planets mass ratio. A system located inside this region tends towards the equilibrium (the capture is permanent), while a system located outside the region evolves away from the equilibrium and leaves the resonance. We verify recent results of Delisle et al. and Xu & Lai where they show that for m 1 m 2 (m 1 , m 2 are the inner and outer planets' masses, respectively) the equilibrium is unstable when the migration is added, so the capture cannot be permanent. We show that for particular migration parameters the situation may be reversed (the equilibria are unstable for m 1 m 2 ). We illustrate that 9:7 MMR consists of two modes separated with a separatrix. The inner one is centred at the equilibrium and the outer one has no equilibrium in its centre. A system located outside the region of stable capture evolves from the inner into the outer mode. The evolution occurs along families of periodic orbits of the averaged system, that play a crucial role in the dynamics after the resonance capture.

On the 9:7 Mean Motion Resonance Capture in a System of Two Equal-mass Super-Earths

“…From our survey, we confirm the general picture of the 9:7 resonant capture presented in previous works (e.g. Quillen 2006;Mustill & Wyatt 2011;Migaszewski 2017;Xu & Lai 2017) and gave a description of this phenomenon as being related to one of the resonance angles having to be located in a particular range as the resonance is entered.…”

“…This type of change does not significantly affect the dynamical parameters of the calculation, but, on account of the change in the time required to reach the resonance, there is expected to be a change in the value of the resonant angle φ 1 when entering into the resonance, which may lead the system to behave in a different manner with regard to the window for resonant capture (see Figure 10). A similar conclusion was obtained by Migaszewski (2017) through N-body simulations.…”

“…However, for a system with comparable masses planets and higher initial eccentricities, they conjecture that trapping should become probabilistic. Finally, Migaszewski (2017) investigated how the formation of 9:7 resonance depends on the migration parameters and initial orbits as a result of the convergent migration of two low-mass planets by using an N-body model with migration parameters. In this work, they also gave conditions for the systems to stay in the 9:7 resonance either permanently or temporarily.…”