Eccentricities and Inclinations of Multi-Planet Systems with External Perturbers

Abstract

Compact multi-planet systems containing super-Earths or sub-Neptunes, commonly found around solar-type stars, may be surrounded by external giant planet or stellar companions, which can shape the architechture and observability of the inner systems. We present a comprehensive study on the evolution of the inner planetary system subject to the gravitational influence of an eccentric, misaligned outer perturber. Analytic results are derived for the inner planet eccentricities ($e_i$) and mutual inclination (${\theta }_{12}$) of the "2-planet + perturber" system, calibrated with numerical secular and N-body integrations, as a function of the perturber mass $m_p$, semi-major axis $a_p$ and inclination angle ${\theta }_p$. We find that the dynamics of the inner system is determined by the dimensionless parameter ${\epsilon }_{12}$, given by the ratio between the differential precession rate driven by the perturber and the mutual precession rate of the inner planets. Loosely packed systems (corresponding to ${\epsilon }_{12}\gg 1$) are more susceptible to eccentricity/inclination excitations by the perturber than tightly packed inner systems (with ${\epsilon }_{12}\ll 1$) (or singletons), although resonance may occur around ${\epsilon }_{12}\sim 1$, leading to large $e_i$ and ${\theta }_{12}$. Dynamical instability may set in for inner planet systems with large excited eccentricities and mutual inclinations. We present a formalism to extend our analytical results to general inner systems with $N>2$ planets and apply our results to constrain possible external companions to the Kepler-11 system. Eccentricity and inclination excitation by external companions may help explain the observational trend that systems with fewer transiting planets are dynamically hotter than those with more transiting planets.

arXiv commentcorrected errors in Fig. 1 and 2

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