Pith Number

pith:OLEXFMAG

pith:2026:OLEXFMAGKQ3473R3JEHMXGVMOS

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

Jean-Marc Hur\'e

Hill's level surfaces in the circular restricted three-body problem admit an exact closed-form expression obtained by inverting the Jacobi integral into a cubic equation.

arxiv:2604.21426 v2 · 2026-04-23 · astro-ph.IM

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Claims

C1strongest claim

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.

C2weakest assumption

That the Jacobi integral level sets in the standard CR3BP effective potential can be inverted exactly into a cubic equation in primary-centric spherical coordinates without loss of information or hidden approximations.

C3one line summary

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.

References

35 extracted · 10 resolved · 1 Pith anchors

[1] Hill's level surfaces in the circular restricted three-body problem solved 2026 · arXiv:2604.21426

[2] V. Szebehely,Theory of orbits. The restricted problem of three bodies.(1982) 1982

[3] A. D. Bruno,The Restricted 3-Body Problem: Plane Pe- riodic Orbits(De Gruyter, Berlin, New York, 1994) 1994

[4] The restricted three- body problem, 2000

[5] J. M. Faidit,Limites et lobes de Roche(Vuibert, 2007) 2007

Cited by

1 paper in Pith

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

Receipt and verification

First computed	2026-05-20T00:01:41.970538Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

72c972b0065437cfee3b490ecb9aac74b10cb9b8cffebf6c867b1248a05bb256

Aliases

arxiv: 2604.21426 · arxiv_version: 2604.21426v2 · doi: 10.48550/arxiv.2604.21426 · pith_short_12: OLEXFMAGKQ34 · pith_short_16: OLEXFMAGKQ3473R3 · pith_short_8: OLEXFMAG

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/OLEXFMAGKQ3473R3JEHMXGVMOS \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 72c972b0065437cfee3b490ecb9aac74b10cb9b8cffebf6c867b1248a05bb256

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "bcbdaf5ed3992948645204e0bc557d2c753b9a05736e823f581e044ec31f9f71",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "astro-ph.IM",
    "submitted_at": "2026-04-23T08:42:22Z",
    "title_canon_sha256": "6fcc69d9dcdd915797fed3e21b5bd48404bdd736ed8b7930efa1640791ae16ab"
  },
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  "source": {
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    "kind": "arxiv",
    "version": 2
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}