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

Abstract

We report the closed-form expression for Hill's surfaces in the circular restricted three-body problem. The solution $ϕ(r,θ)$, 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.