2024AJ....167..112W

An increasing number of compact planetary systems with multiple planets in a resonant chain have been detected. The resonant chain must be maintained by convergent migration of the planets due to planet-disk interactions if it is formed before the dispersal of the protoplanetary gas disk. For type I migration in an adiabatic disk, we show that an analytic criterion for convergent migration can be developed by requiring that any part of the resonant chain should be convergently migrating toward the remaining part. The criterion depends primarily on the logarithmic gradients α and β of the surface density and temperature profiles of the disk, respectively, and it is independent of the absolute values of the surface density and temperature. The analytic criterion is applied to the Kepler-60, Kepler-80, Kepler-223, TOI-178, and TRAPPIST-1 systems. Due to the variation of planetary masses within the resonant chains, we find that convergent migration typically requires rather extreme values of (α, β) that have little or no overlap with common disk models. Finally, we show that there is an empirical relationship between the distance of the innermost planet from the central star and the stellar mass for the observed resonant chain systems, which supports the idea that the resonant chains are formed and maintained by stalling the migration of the innermost planet near the inner edge of the disk truncated by the magnetic fields of the protostar.