Dynamical Instability in Multi-Orbiter Systems with Gas Friction

TL;DR

This study investigates how gas friction influences the dynamical stability of closely-packed multi-planet systems and black hole systems in AGN discs, revealing conditions under which friction stabilizes or fails to stabilize the systems.

Contribution

It provides a quantitative analysis of the stabilizing effects of gas friction on multi-orbiter systems using $N$-body simulations and a linear map, highlighting the dependence on damping timescales and orbital configurations.

Findings

Friction stabilizes systems if instability growth time exceeds damping timescale.

Stability requires $t_{inst} extgreater au$ for two planets, and $t_{inst} extgreater 10 au - 100 au$ for three.

Instability timescales grow exponentially with orbital spacing, with a transition zone between stable and unstable regimes.

Abstract

Closely-packed multi-planet systems are known to experience dynamical instability if the spacings between the planets are too small. Such instability can be tempered by the frictional forces acting on the planets from gaseous discs. A similar situation applies to stellar-mass black holes embedded in AGN discs around supermassive black holes. In this paper, we use $N$-body integrations to evaluate how the frictional damping of orbital eccentricity affects the growth of dynamical instability for a wide range of planetary spacing and planet-to-star mass ratios. We find that the stability of a system depends on the damping timescale $\tau$ relative to the zero-friction instability growth timescale $t_{\rm inst}$. In a two-planet system, the frictional damping can stabilise the dynamical evolution if $t_{\rm inst}\gtrsim\tau$. With three planets, $t_{\rm inst} \gtrsim 10\tau - 100\tau$ is needed for stabilisation. When the separations between the planetary orbits are sufficiently small, $t_{\rm inst}$ can be less than the synodic period between the planets, which makes frictional stabilisation unlikely to occur. As the orbital spacing increases, the instability timescale tends to grow exponentially on average, but it can vary by a few orders of magnitude depending on the initial orbital phases of the planets. In general, the stable region (at large orbital spacings) and unstable region (at small orbital spacings) are separated by a transition zone, in which the (in)stability of the system is not guaranteed. We also devise a linear map to analyse the dynamical instability of the "planet + test-mass" system, and we find qualitatively similar results to the $N$-body simulations.