Three body resonances in close orbiting planetary systems: Tidal dissipation and orbital evolution

TL;DR

This paper investigates the orbital evolution of three-planet systems with super-Earth masses under stellar tidal effects, developing an analytic model and confirming it with numerical simulations, to constrain tidal dissipation parameters.

Contribution

It introduces a simple analytic solution for orbital evolution near three-body commensurability and applies it to constrain tidal dissipation in planetary systems.

Findings

Analytic model describes system expansion from commensurability over time.

Numerical simulations confirm the scaling laws predicted by the analytic solution.

Constraints on tidal dissipation parameter Q' based on system's resonant state.

Abstract

We study the orbital evolution of a three planet system with masses in the super-Earth regime resulting from the action of tides on the planets induced by the central star which cause orbital circularization. We consider systems either in or near to a three body commensurability for which adjacent pairs of planets are in a first order commensurability. We develop a simple analytic solution, derived from a time averaged set of equations, that describes the expansion of the system away from strict commensurability as a function of time, once a state where relevant resonant angles undergo small amplitude librations has been attained. We perform numerical simulations that show the attainment of such resonant states focusing on the Kepler 60 system. The results of the simulations confirm many of the scalings predicted by the appropriate analytic solution. We go on to indicate how the results can be applied to put constraints on the amount of tidal dissipation that has occurred in the system. For example, if the system has been in a librating state since its formation, we find that its present period ratios imply an upper limit on the time average of 1/Q', with Q' being the tidal dissipation parameter. On the other hand if a librating state has not been attained, a lower upper bound applies.