Consequences of tidal interaction between disks and orbiting protoplanets for the evolution of multi-planet systems with architecture resembling that of Kepler 444

Abstract

We study orbital evolution of multi-planet systems with masses in the terrestrial planet regime induced through tidal interaction with a protoplanetary disk assuming that this is the dominant mechanism for producing orbital migration and circularization. We develop a simple analytic model for a system that maintains consecutive pairs in resonance while undergoing orbital circularization and migration. This model enables migration times for each planet to be estimated once planet masses, circularization times and the migration time for the innermost planet are specified. We applied it to a system with the current architecture of Kepler 444 adopting a simple protoplanetary disk model and planet masses that yield migration times inversely proportional to the planet mass, as expected if they result from torques due to tidal interaction with the protoplanetary disk. Furthermore the evolution time for the system as a whole is comparable to current protoplanetary disk lifetimes. In addition we have performed a number of numerical simulations with input data obtained from this model. These indicate that although the analytic model is inexact, relatively small corrections to the estimated migration rates yield systems for which period ratios vary by a minimal extent. Because of relatively large deviations from exact resonance in the observed system of up to 2%; the migration times obtained in this way indicate only weak convergent migration such that a system for which the planets did not interact would contract by only ~ 1% although undergoing significant inward migration as a whole. We have also performed additional simulations to investigate conditions under which the system could undergo significant convergent migration before reaching its final state. These indicate that migration times have to be significantly shorter and resonances between planet pairs significantly closer during such an evolutionary phase. Relative migration rates would then have to decrease allowing period ratios to increase to become more distant from resonances as the system approached its final state in the inner regions of the protoplanetary disk.