# Orbital dynamics of circumbinary planets

## TL;DR

This study uses numerical simulations to analyze the orbital dynamics of circumbinary planets, focusing on stationary inclination states, libration versus circulation, and the effects of binary and planetary parameters.

## Contribution

It provides a detailed numerical validation of analytic models for circumbinary planet dynamics, exploring prograde and retrograde orbit behaviors and stationary inclination conditions.

## Key findings

- Prograde planets can librate around stationary tilted states at high initial inclinations.
- Stationary angles decrease monotonically with the ratio of planet to binary angular momentum.
- Librating orbit properties differ significantly between prograde and retrograde orbits.

## Abstract

We investigate the dynamics of a nonzero mass, circular orbit planet around an eccentric orbit binary for various values of the binary eccentricity, binary mass fraction, planet mass, and planet semi--major axis by means of numerical simulations. Previous studies investigated the secular dynamics mainly by approximate analytic methods. In the stationary inclination state, the planet and binary precess together with no change in relative tilt. For both prograde and retrograde planetary orbits, we explore the conditions for planetary orbital libration versus circulation and the conditions for stationary inclination. As was predicted by analytic models, for sufficiently high initial inclination, a prograde planet's orbit librates about the stationary tilted state. For a fixed binary eccentricity, the stationary angle is a monotonically decreasing function of the ratio of the planet--to--binary angular momentum $j$. The larger $j$, the stronger the evolutionary changes in the binary eccentricity and inclination. We also calculate the critical tilt angle that separates the circulating from the librating orbits for both prograde and retrograde planet orbits. The properties of the librating orbits and stationary angles are quite different for prograde versus retrograde orbits. The results of the numerical simulations are in very good quantitative agreement with the analytic models. Our results have implications for circumbinary planet formation and evolution.