Braik-Ross common-energy (3,2) Earth-Moon cycler (CR3BP, C=3.1294)

braik-ross-c32-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 = 78.613 d (Braik & Ross 2026 Table 2 "(3,2)-cycler" row); reproduced in-repo at 78.6126 d (-0.0005%). Encounter cadence: k1=3 perigee + k2=2 perilune passes per period (Ross-RT 2025 Def. 1; family anchor ross-rt-em-cycler-32-2025). Half-period at the 6th x-axis crossing (same crossing-index lesson as Ross-RT 32-2025; see results note §2 there). years: null (CR3BP).
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: (3,2) prograde Earth-Moon cycler
Mass ratio (μ): 0.012150584270572
Jacobi constant: 3.1294
Period (non-dim): 18.0786
Stability index: 126565.000

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

Orbit view 2.5D ecliptic projection

Not renderable from current data. This is a rotating-frame (CR3BP) periodic orbit. A faithful render requires numerical propagation of the synodic-frame state, which the catalogue does not yet publish as a sampled path — so no stand-in ellipse is drawn (drawing a heliocentric ellipse here would misrepresent the dynamics). The CR3BP identity (Jacobi constant, period, stability) is tabulated above.

model: CR3BP (rotating frame)

3D view not available for rotating-frame (CR3BP) orbits.

Definition status

incomplete — core fields missing or known-unknowns tracked below

Known-unknowns (2)

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]
No initial-state vector printed in source. state_nd DERIVED in-repo by the fixed-Jacobi corrector from the seed region (x0~-0.2752115, ydot0_sign=-1, half-period at the 6th x-axis crossing -- same crossing-index lesson as the Ross-RT 2025 (3,2) anchor, see ross-rt-em-cycler-32-2025 results note §2) pinned in _CYCLER_SEEDS and confirmed against the sourced T_PO + (3,2) topology by the reproduction harness.

source: src/cyclerfinder/search/reachable_representatives.py + docs/notes/2026-06-14-249-unstable-member-recovery-plan.md.
orbit_elements.cr3bp.stability_index [uncertain]
Barden half-period monodromy nu=1.27e5 is extreme. The cross-check is consistent: lambda_max from Braik-Ross sigma=0.6886 TU^-1 over one period (T_PO=18.08 TU) is exp(sigma*T_PO)~2.5e5 -- exactly the half-period |lambda| we recover (the Barden reciprocal-pair convention uses the half-period STM so the eigenvalue magnitudes match the full-period Floquet multiplier modulo the (lambda, 1/lambda) reciprocal pairing, which the planar CR3BP factorisation guarantees). For an orbit this unstable the full-state rotating-frame closure ceiling under round-off amplification is ~|lambda|*eps ~ 2.5e-11 amplifying to ~7e-6 over one period (matches the Radau cross-check observation in the V2-blocker reasoning above); only the Jacobi closure (dJ=7.6e-13) is structurally precise.

source: docs/notes/2026-06-14-249-unstable-member-recovery-plan.md.

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

Same source as braik-ross-c11a-cycler-2026. Table 2 row "(3,2)-cycler": T_PO=78.613 d, sigma=0.6886 TU^-1 (sigma_d=0.1583 d^-1). The paper identifies the (3,2)-cycler as the DOMINANT hub/gateway/relay family in the accessibility network (Sec. 5.3, Abstract).

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: same continuous (3,2) family as ross-rt-em-cycler-32-2025 (Ross-RT 2025 Def. 1, Table 3), at a DIFFERENT Jacobi. The anchor is the family's per-family stable midpoint at C^stable=3.182762663084288 (T=77.84 d, |nu|<1, Case 2 -- L2 neck CLOSED). The Braik-Ross member sits at the lower Jacobi C=3.1294, where the (3,2) family is UNSTABLE and the L2 neck is OPEN (Case 3). Same family identity (k1=3, k2=2, prograde); different orbit (UNSTABLE T=78.613 d at low energy vs anchor STABLE T=77.84 d at higher energy).
Leiva, A. M. & Briozzo, C. B. (2008). Extension of fast periodic transfer orbits in the Sun-perturbed Earth-Moon system (quasi-bicircular persistence of a (3,2)-class member). Celestial Mechanics and Dynamical Astronomy 101, pp. 225-245.
Ross-RT 2025 p. 13 notes a member of the (3,2) family was previously identified by Leiva & Briozzo and shown to persist under solar perturbation (Sun-Earth-Moon quasi-bicircular model) as a single unstable orbit. The Braik-Ross C32 -- unstable, at lower C than the per-family stable midpoint -- is plausibly closer in character to the Leiva-Briozzo member than the Ross-RT 2025 stable anchor is. Not the same orbit (different model framing); recorded for cross-reference.

Notes

Common-energy (3,2)-cycler from Braik & Ross 2026 Table 2 at C_J=3.1294, UNSTABLE (sigma=0.6886 TU^-1). Identified in the paper (Sec. 5.3, Abstract) as the DOMINANT hub/gateway/relay family of the cislunar reachable-set network -- the (3,2)-cycler dominates the accessibility hierarchy across much of the sampled DeltaV/ToF budget plane, with the (1,1)a-cycler dominating the low-time-of-flight regime and the 2:1 stable resonant orbit persistently hard-to-access. Distinct from ross-rt-em-cycler-32-2025 (the same family's per-family STABLE midpoint at C=3.183, T=77.84 d, |nu|<1, Case 2). The Braik-Ross member sits at the lower energy C=3.1294 where the family is UNSTABLE and the L2 neck is OPEN (Case 3); this is exactly the regime where the chaotic-sea connectivity argument (Braik-Ross Fig. 4 / Sec. 2.5) applies and the orbit serves as a transport hub. REPRODUCTION (2026-06-14): symmetric corrector at C=3.1294 from seed x0=-0.2752115 (ydot0_sign=-1, 6th x-axis crossing) -> T=78.613 d (sourced 78.613, error -0.0005%); Jacobi enforced to machine eps; topology (3,2) prograde confirmed. Barden nu=1.27e5 (UNSTABLE); independent Radau preserves Jacobi (dJ=7.6e-13) but the full-state rotating-frame closure only reaches ~7e-6 over one period -- the structural ceiling for an orbit with |lambda|~2.5e5 under round-off amplification, NOT a defect (matches Braik-Ross's published sigma exactly: sigma*T_PO=12.45, exp(12.45)=2.5e5). MODEL: pure PCR3BP 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): (3,2)-cycler T_PO=78.613 d at C_J=3.1294.
orbit_elements.cr3bp.braik_ross_sigma_per_tu: Braik & Ross 2026 Table 2 (p. 12): (3,2)-cycler instability rate sigma=0.6886 TU^-1.
orbit_elements.cr3bp.braik_ross_sigma_per_day: Braik & Ross 2026 Table 2 (p. 12): (3,2)-cycler instability rate sigma_d=0.1583 day^-1.
orbit_elements.cr3bp.jacobi_constant: Braik & Ross 2026 Sec. 2.4 / Table 2 header: common Jacobi level C_J=3.1294.
orbit_elements.cr3bp.mass_ratio: Braik & Ross 2026 Table 1 (p. 8): mu = 0.012150584270572.