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Braik-Ross common-energy (1,1)a Earth-Moon cycler (CR3BP, C=3.1294)

braik-ross-c11a-cycler-2026 · source: literature · validation: V1

Signature

Bodies

E-Moon

Primary

Earth

Sequence (canonical)

E-Moon

Sense

n/a

Orbit class

Cycler

strict cycler (infinite returns)

Cycler class

non-keplerian (CR3BP)

Trajectory regime

ballistic

Maintenance ΔV band

unclassified

Model assumption

cr3bp

CR3BP — Jacobi-constant-conserved; signature not patched-conic comparable.

Period

— yr (1 × E-Moon synodic)

Period T_PO = 42.140 d (Braik & Ross 2026 Table 2 "(1,1)a-cycler" row, reported in days at the common C_J=3.1294); reproduced in-repo at 42.1405 d (+0.0011%). Encounter cadence: k1=1 perigee + k2=1 perilune pass per period (Ross-RT 2025 Def. 1, family-anchor row ross-rt-em-cycler-11-2025). years: null (CR3BP, rotating-frame periodic orbit; the heliocentric-synodic-multiple convention does not apply).

Priority date

2026-05-29

V∞ at encounters

E (encounter 1)

— (not published)

CR3BP periodic orbit: conserved quantity is the Jacobi constant, not V_inf.

Moon (encounter 2)

— (not published)

Same — Jacobi-constant model.

CR3BP orbit identity

Family

(1,1) prograde Earth-Moon cycler — short-period branch (C11a)

Mass ratio (μ)

0.012150584270572

Jacobi constant

3.1294

Period (non-dim)

9.6911

Stability index

12902.660

Planar CR3BP rotating-frame periodic orbit; Keplerian elements inapplicable. CR3BP identity tuple below.

Definition status

incomplete — core fields missing or known-unknowns tracked below

Known-unknowns (1)

Values we expect to exist but have not yet filled (distinct from "not applicable"). Tracked per upstream docs/spec.md §16.6.4.

• orbit_elements.cr3bp.state_nd [derive]
  Braik & Ross 2026 prints no initial-state vector for any member (only T_PO + sigma in Table 2). state_nd is DERIVED in-repo by the fixed-Jacobi symmetric corrector from the seed region (x0~-0.81164067, ydot0_sign=-1, half-period at the 3rd x-axis crossing) pinned in src/cyclerfinder/search/reachable_representatives.py::_CYCLER_SEEDS and confirmed against the sourced T_PO + (1,1) topology by the reproduction harness.
  source: src/cyclerfinder/search/reachable_representatives.py + docs/notes/2026-06-14-249-unstable-member-recovery-plan.md (commits a19eb24, 4a20243, f608c6b, 325c8a2).

Primary citation

Braik, A. & Ross, S. D. (2026). Orbital Networks in the Three-Body Problem. arXiv preprint arXiv:2605.31543 (29 May 2026).

URL: https://arxiv.org/abs/2605.31543

arXiv submission dated 29 May 2026; PDF dated June 1, 2026 (title page). Table 2 lists thirteen common-energy representative periodic orbits at C_J=3.1294. The (1,1)a-cycler row carries T_PO=42.140 d and instability rate sigma=1.0482 TU^-1 (sigma_d=0.2410 d^-1). The paper does NOT print the initial state.

Corroborating sources

• Ross, S. D. & Roberts-Tsoukkas, M. (2025). Stable, Low-Energy Prograde Earth-Moon Cycler Orbits. AAS/AIAA Astrodynamics Specialist Conference, 2025, paper AAS 25-621. · link
  Family anchor: Ross-RT 2025 defines the (k1,k2)-cycler family numbering (Def. 1) and proves the (1,1) family's stable subfamily at C^stable=3.151; ross-rt-em-cycler-11-2025 is its nu=0 midpoint. C11a is a DIFFERENT member of the same continuous (1,1) family at the lower Jacobi C=3.1294, on the short-period branch of a saddle-center fold inside the family (mining note). Same family identity (k1=1, k2=1, prograde); different orbit (UNSTABLE, T=42.140 d vs the anchor's STABLE T=44.754 d).

Notes

Common-energy (1,1)a-cycler from Braik & Ross 2026 Table 2, the SHORT-PERIOD branch of the (1,1) family at C_J=3.1294. Distinct from the family-anchor row ross-rt-em-cycler-11-2025 (Ross-RT 2025), which is the (1,1) family's per- family STABLE midpoint at C^stable=3.151175879508174 (T=44.754 d, |nu|<1). C11a sits at a different Jacobi (3.1294, below C^stable) and a different branch (short-period side of a saddle-center fold inside the family — the fold geometry is documented in the mining note). At C=3.1294 the (1,1) family is UNSTABLE; Braik-Ross publishes instability rate sigma=1.0482 TU^-1 (sigma_d=0.2410 d^-1, Floquet exponent of the largest-magnitude nontrivial monodromy eigenvalue per Eq. 20), reproduced in-repo as Barden nu=1.29e4 (|lambda|=2.58e4 ~ exp(sigma*T_PO)). REPRODUCTION (2026-06-14): symmetric perpendicular-x-axis-crossing corrector at C=3.1294 from seed x0=-0.81164067 with ydot0_sign=-1 and the 3rd x-axis crossing pinned as half-period -> T=42.140462 d (sourced 42.140, error +0.0011%); Jacobi 3.1294 enforced to machine eps; topology (1,1) prograde confirmed by winding-number diagnostic. Independent Radau integrator at rtol/atol 1e-11 closes the Jacobi (dJ=1.9e-13 over one period); the rotating-frame state closes to ~2.8e-9 (well-conditioned for the moderate instability). Reproduction harness: src/cyclerfinder/search/reachable_representatives.py + docs/notes/2026-06-14-249-unstable-member-recovery-plan.md (final disposition section; commits a19eb24, 4a20243, f608c6b, 325c8a2). MODEL: pure planar CR3BP (PCR3BP). No bicircular / solar-gravity / lunar- eccentricity / ephemeris refinement. Every number here is PCR3BP-only at the Braik-Ross common Jacobi.

Source quotes (per-field provenance)

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

orbit_elements.cr3bp.period_nd

Braik & Ross 2026 Table 2 (p. 12): (1,1)a-cycler T_PO=42.140 d at C_J=3.1294.

orbit_elements.cr3bp.braik_ross_sigma_per_tu

Braik & Ross 2026 Table 2 (p. 12): (1,1)a-cycler instability rate sigma=1.0482 TU^-1.

orbit_elements.cr3bp.braik_ross_sigma_per_day

Braik & Ross 2026 Table 2 (p. 12): (1,1)a-cycler instability rate sigma_d=0.2410 day^-1.

orbit_elements.cr3bp.jacobi_constant

Braik & Ross 2026 Sec. 2.4 / Table 2 header: common Jacobi level C_J=3.1294 used throughout the accessibility study.

orbit_elements.cr3bp.mass_ratio

Braik & Ross 2026 Table 1 (p. 8): Earth-Moon mass parameter mu = 0.012150584270572 (== Ross-RT 2025 p. 3).