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 $m_1 \lesssim m_2$ ($m_1, m_2$ are the inner and outer planets' masses) 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 $m_1 \gtrsim m_2$). 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.