# A Criterion for the Onset of Chaos in Systems of Two Eccentric Planets

## TL;DR

This paper develops a new criterion to predict when chaos begins in two-planet systems with eccentric orbits, extending previous work to include all resonance orders and confirming results with simulations.

## Contribution

It introduces a generalized chaos criterion for eccentric two-planet systems, expanding beyond near-circular assumptions and incorporating all resonance orders.

## Key findings

- Derived a simple expression for critical eccentricity triggering chaos.
- Validated the criterion with detailed numerical simulations.
- Explored the link between chaos onset and planetary collisions.

## Abstract

We derive a criterion for the onset of chaos in systems consisting of two massive, eccentric, coplanar planets. Given the planets' masses and separation, the criterion predicts the critical eccentricity above which chaos is triggered. Chaos occurs where mean motion resonances overlap, as in Wisdom (1980)'s pioneering work. But whereas Wisdom considered only nearly circular planets, and hence examined only first order resonances, we extend his results to arbitrarily eccentric planets (up to crossing orbits) by examining resonances of all orders. We thereby arrive at a simple expression for the critical eccentricity. We do this first for a test particle in the presence of a planet, and then generalize to the case of two massive planets, based on a new approximation to the Hamiltonian (Hadden, in prep). We then confirm our results with detailed numerical simulations. Finally, we explore the extent to which chaotic two-planet systems eventually result in planetary collisions.