Hill's level surfaces in the circular restricted three-body problem solved

TL;DR

This paper presents a closed-form solution for Hill's surfaces in the circular restricted three-body problem, accurately reproducing classic patterns such as tadpole, horseshoe, and peanut shapes.

Contribution

It provides the first explicit closed-form expression for Hill's surfaces derived from a cubic equation in the primary-centric spherical coordinate system.

Findings

Exact reproduction of classic Hill's surface patterns

Derivation from a cubic equation with at most two roots per side of a separatrix

Solution applicable in primary-centric spherical coordinates

Abstract

We report the closed-form expression for Hill's surfaces in the circular restricted three-body problem. The solution $\phi(r,\theta)$, derived in the primary-centric spherical coordinate system, is deduced from a cubic equation delivering at most two roots on each side of a separatrix. The famous patterns (tadpole, horseshoe and peanut shapes, Roche lobes and Hill's quasi-spheres) are exactly produced.