A closed-form solution φ(r,θ) for Hill's surfaces in the CR3BP is obtained by solving a cubic equation that reproduces tadpole, horseshoe, Roche lobe, and quasi-spherical shapes.
Hill's level surfaces in the circular restricted three-body problem solved
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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.
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astro-ph.IM 1years
2026 1verdicts
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Hill's level surfaces in the circular restricted three-body problem solved
A closed-form solution φ(r,θ) for Hill's surfaces in the CR3BP is obtained by solving a cubic equation that reproduces tadpole, horseshoe, Roche lobe, and quasi-spherical shapes.