How to measure tidal dissipation in long resonant chains

Abstract

Context. Resonant chains are systems with three or more planets caught in a succession of two- and three-planet mean-motion resonances (2P-MMRs and 3P-MMRs). Most of the observed chains show significant amounts of separation from the nominal commensurabilities. These are lower energy states and therefore suggestive of a process of long-scale dissipation. The most frequently invoked mechanism is active tides affecting the innermost planets, produced by the star. Aims. Simulations of tidal separation are expensive and generally impractical for extensive parameter explorations. Therefore, it is essential to have access to analytical tools that would allow us to inspect tidally separated chains, as probing these systems can give valuable insight into the physical parameters involved in dissipation. Methods. We extended an existing analytical model of the tidal separation of resonant chains with adjacent first-order 2P-MMRs that is meant to be applicable to longer N-planet chains. We have demonstrated how this approach can be used to constrain those parameters involved in the tidal evolution, such as the frequently unresolved Q' factors. Results. We show how this tool can be used to place meaningful bounds over the effective planetary Q' value of long resonant chains, even in the realistic case where the system is poorly characterized, lacking measurements of parameters such as the stellar age or one of the planetary masses. We also show how the magnitude of separation in a resonant chain is specially sensitive to the mass of certain planets. In particular, a more massive second planet will boost tidal separation, while a more massive last planet will inhibit it.