# On the classification of orbits in the three-dimensional Copenhagen problem with oblate primaries

## TL;DR

This study classifies various orbit types in a three-dimensional restricted three-body problem with oblate primaries, revealing how initial conditions and oblateness influence orbital behavior and collision probabilities.

## Contribution

It provides a detailed numerical classification of orbit types in the 3D oblate primaries problem, including the impact of oblateness on collision orbits.

## Key findings

- Orbits are classified into four types based on initial conditions and energy.
- Collision orbit fraction depends algebraically on oblateness coefficient.
- Color-coded diagrams effectively visualize orbit distributions.

## Abstract

The character of motion for the three-dimensional circular restricted three-body problem with oblate primaries is investigated. The orbits of the test particle are classified into four types: non-escaping regular orbits around the primaries, trapped chaotic (or sticky) orbits, escaping orbits that pass over the Lagrange saddle points $L_2$ and $L_3$, and orbits that lead the test particle to collide with one of the primary bodies. We numerically explore the motion of the test particle by presenting color-coded diagrams, where the initial conditions are mapped to the orbit type and studied as a function of the total orbital energy, the initial value of the $z$-coordinate and the oblateness coefficient. The fraction of the collision orbits, measured on the color-coded diagrams, show an algebraic dependence on the oblateness coefficient, which can be derived by simple semi-theoretical arguments.