Conditions for Convergent Migration of N-Planet Systems

Abstract

Given an N-planet system with coplanar, low eccentricity orbits, and under the effects of an external non-conservative force, we derive necessary conditions for the orbital migration to undergo global convergent migration and allow the formation of a mean-motion resonance chain. Since the conditions are expressed in terms of the time-derivatives of the semimajor axes, the method is applicable to any non-conservative force, including disk–planet interactions and tides. Although the number of possible conditions increase exponentially with the number of bodies, the calculations may be enormously simplified adopting a tree diagram for the decision process. We deduce explicit expressions for N= 3 and N= 4 together with applications to Kepler-60 and Kepler-223 assuming a simple prescription for Type-I migration and no inner disk edge. This disk model is chosen for simplicity and to highlight the applicability of the method. We also present all the necessary steps for the implementation of an algebraic tree algorithm for the general N-planet case, and discuss possible implications for the migration history of the TOI-178 and TRAPPIST-1 systems.