Arenstorf periodic Earth-Moon free-return orbit (CR3BP)

arenstorf-em-figure8-1963 · source: literature · validation: V0

Signature

Bodies: E-Moon
Sequence (canonical): E-Moon
Sense: n/a
Period: — yr (1 × E-Moon synodic)
Arenstorf's 1963 paper proves existence of one-parameter analytic families of "synodically closed" periodic solutions of the planar restricted three-body problem with small mass ratio (Earth-Moon mu ≈ 0.012). The closing period for the most-cited member used as the Apollo free-return analogue is on the order of one Earth-Moon synodic period (~14 d in the spacecraft's frame, or ~6 d in the Earth-centered patched-conic projection); the original paper does not tabulate a single period because it presents a family. Left `years: null` because the catalogue's period convention is heliocentric-synodic-multiple and does not naturally express the sidereal-lunar / synodic-lunar ambiguity in this CR3BP setting.
Priority date: 1963-01-01

V∞ at encounters

E (#1): — (not published)
The Arenstorf orbit lives in the CR3BP, where the conserved quantity is the Jacobi constant, not patched-conic V_infinity. Quoting a V_inf at Earth would be model-mismatched. Left null with a model-mismatch flag per spec §16.1 honesty rules.
Moon (#2): — (not published)
Same as above — Jacobi-constant model, not patched-conic.

Orbit elements (heliocentric)

Semi-major axis a: — AU
Eccentricity e: —
Perihelion: — AU
Aphelion: — AU
Inclination: —°

Heliocentric Keplerian orbit elements are inapplicable: the Arenstorf orbit is a periodic solution in the rotating Earth-Moon synodic frame, not a Kepler ellipse around the Sun. Left null throughout per honesty rule.

Primary citation

Arenstorf, R. F. (1963). Periodic Solutions of the Restricted Three Body Problem Representing Analytic Continuations of Keplerian Elliptic Motions. American Journal of Mathematics, Vol. 85, No. 1, pp. 27-35.

DOI: 10.2307/2373181

URL: https://www.jstor.org/stable/2373181

Also issued as NASA Technical Note NASA-TN-D-1859 (May 1963), NTRS accession 19630005545, URL https://ntrs.nasa.gov/citations/19630005545 . Arenstorf was at NASA Marshall Space Flight Center at the time. The Apollo programme later adopted free-return trajectories of this family; Arenstorf is credited with the rescue trajectory used by Apollo 13 (secondary attribution).

Corroborating sources

Cook, J. D. (2020). Arenstorf orbit. John D. Cook blog (johndcook.com), 2020-02-08. · link
Modern explainer; useful for the standard mu = 0.012277471 Earth-Moon mass ratio commonly used in numerical CR3BP demos of the Arenstorf orbit.

Notes

MODEL MISMATCH FLAG: this is a CR3BP (circular restricted three-body problem) periodic orbit, not a patched-conic + gravity-assist cycler. The V_infinity + bend-angle abstraction the rest of this catalogue uses does not apply to it. The entry exists for two reasons: 1. Historical / pedagogical priority: Arenstorf 1963 is the foundational lunar free-return analytic family; the Apollo missions used trajectories of this kind. 2. Future scope flag: if cyclerfinder ever extends to CR3BP / invariant-manifold modelling (see spec §2 stretch goals and docs/known-cyclers.md §H), this entry is the first known-prior art the matcher would need to handle. Until such an extension exists, M7 novelty matching MUST treat any finder hit purely on `primary == "Earth"` + bodies ["E", "Moon"] as a candidate for human review against this entry rather than an automatic prior-art match; the numerical signature comparison the rest of the catalogue performs is not meaningful here.

Source quotes (per-field provenance)

Every numerical value in this entry traces to a verbatim or paraphrased quote from a cited source.

first_published.title: NASA NTRS metadata for accession 19630005545: "PERIODIC SOLUTIONS OF THE RESTRICTED THREE BODY PROBLEM REPRESENTING ANALYTIC CONTINUATIONS OF KEPLERIAN ELLIPTIC MOTIONS" — Arenstorf, R. F., NASA-TN-D-1859, May 1963.
first_published.venue: Standard library citation: American Journal of Mathematics, Vol. 85, No. 1 (Jan., 1963), pp. 27-35; JSTOR stable URL https://www.jstor.org/stable/2373181 .
notes: Cook 2020 blog explainer: "Richard Arenstorf discovered a stable periodic orbit between the Earth and the Moon which was used as the basis for the Apollo missions." Multiple secondary sources independently corroborate the Apollo provenance.