Do N-planet systems have a boundary between chaotic and regular motions?

TL;DR

This study investigates the boundary between chaotic and regular motions in equal-mass planetary systems, revealing that for systems with more than two planets, no clear transition boundary exists due to slow velocity diffusion, contrasting with two-planet systems.

Contribution

It demonstrates that for n > 2, there is no distinct transition boundary between chaos and regularity, highlighting the role of slow velocity diffusion in planetary system dynamics.

Findings

No transition boundary for n > 2 planetary systems.

Transition boundary exists for n=2 with separation ~ μ^{2/7}.

Chaotic motions occur well before orbital crossing due to velocity diffusion.

Abstract

Planetary systems consisting of one star and n planets with equal planet masses \mu and scaled orbital separation are referred as EMS systems. They represent an ideal model for planetary systems during the post-oligarchic evolution. Through the calculation of Lyapunov exponents, we study the boundary between chaotic and regular regions of EMS systems. We find that for n > 2, there does not exist a transition region in the initial separation space, whereas for n=2, a clear borderline occurs with relative separation ~ \mu^{2/7} due to overlap of resonances (Wisdom, 1980). This phenomenon is caused by the slow diffusion of velocity dispersion (~ t^{1/2}, t is the time) in planetary systems with n >2, which leads to chaotic motions at the time of roughly two orders of magnitude before the orbital crossing occurs. This result does not conflict with the existence of transition boundary in the full phase space of N-body systems.