Eccentricities and Inclinations of Multi-Planet Systems with External Perturbers

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.