Dynamics and Transit Variations of Resonant Exoplanets

TL;DR

This paper develops an analytic model for transit timing variations in resonant exoplanet systems, revealing how gravitational interactions amplify TTV signals and providing formulas to interpret observations.

Contribution

It introduces a new analytic model for resonant TTVs, deriving key relationships between TTV periods, amplitudes, and planetary parameters.

Findings

Resonant TTVs have two basic periods and harmonics.

TTV period scales as (m/M_*)^(-2/3).

TTV amplitude scales linearly with libration amplitude.

Abstract

The Transit Timing Variations (TTVs) are deviations of the measured mid-transit times from the exact periodicity. One of the most interesting causes of TTVs is the gravitational interaction between planets. Here we consider a case of two planets in a mean motion resonance (orbital periods in a ratio of small integers). This case is important because the resonant interaction can amplify the TTV effect and allow planets to be detected more easily. We develop an analytic model of the resonant dynamics valid for small orbital eccentricities and use it to derive the principal TTV terms. We find that a resonant system should show TTV terms with two basic periods (and their harmonics). The resonant TTV period is proportional (m/M_*)^(-2/3), where m and M_* are the planetary and stellar masses. For m=10^(-4) M_*, for example, the TTV period exceeds the orbital period by ~2 orders of magnitude. The amplitude of the resonant TTV terms scales linearly with the libration amplitude. The ratio of the TTV amplitudes of two resonant planets is inversely proportional to the ratio of their masses. These and other relationships discussed in the main text can be used to aid the interpretation of TTV observations.