Vaquero 3:1 resonant Earth-Moon periodic cycler family, C=3.13 printed-TOF family endpoint (5.04 d) (CR3BP, UNSTABLE, NO lunar SOI entry -- periselene 66,995 km, 1.2% OUTSIDE the lunar SOI, boundary call)

vaquero-31-c313-em-resonant-po-2013 · source: literature · validation: V1

Signature

Bodies: E-Moon
Primary: Earth
Sequence (canonical): E-Moon
Sense: n/a
Orbit class: Resonant PO
resonant periodic orbit (stable, no transport utility)
Cycler class: non-keplerian (CR3BP)
Returns: ∞ — finite for epoch-locked classes
Validity window: — (not published)
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)
T = 6.45496522207971 TU nondim (DERIVED, corrector/continuation -- Vaquero prints NO period for any Sec. 4.4.7 family member; the p:1 interior-resonance construction implies T ~= 2*pi TU, and this member's T/2pi = 1.0273). 28.030557 d is a DERIVED convenience conversion via this project's own DE440-registry t_s (375190.2619 s); years: null (CR3BP, rotating-frame periodic orbit).
Priority date: 2013-08-09

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: Vaquero 2013 Sec. 4.4.7 Earth-Moon 3:1 resonant periodic-cycler family (interior 3:1, cislunar, L1-connected), C in [2.54, 3.13]
Mass ratio (μ): 0.01215058439469525
Jacobi constant: 3.1300
Period (non-dim): 6.4550
Stability index: 6.684

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-family — CR3BP, CCR4BP, or CRNBP) periodic or quasi-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). Whatever identity fields the model supports (Jacobi constant, period, stability — not all apply to every model in this family; see "Model assumption" above) are tabulated above.

model: CR3BP (rotating frame)

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

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]
Vaquero 2013 prints NO digit-grade IC anywhere for the Sec. 4.4.7 cycler families (Fig. 4.44's x0 axis carries no value labels -- digest; a genuinely non-digitizable graphical source). state_nd is DERIVED via the reproduction (Tisserand-matched two-body apoapsis seed + fixed-Jacobi symmetric corrector + continue_family walk across her printed Jacobi range at rtol=atol=1e-13, per-member gauntlet incl. Jacobi conservation <=1e-10 and an independent-Radau cross-check). Family identity rests on her printed prose values (C-range endpoints, all four endpoint Earth->Moon TOFs at 0.10%-1.83% relative, resonance semi-major axis, cislunar/circumlunar geometry, Fig. 4.44 x0 band, stability criterion), not on any printed state. Never a golden.

source: src/cyclerfinder/search/vaquero_em_cyclers.py + docs/notes/2026-08-09-799-vaquero-em-cycler-family-reproduction.md + data/found/799_vaquero_em_cycler_families/results.json.

Primary citation

Vaquero Escribano, T. M. (2013). Spacecraft Transfer Trajectory Design Exploiting Resonant Orbits in Multi-Body Environments. Ph.D. dissertation, Purdue University, School of Aeronautics and Astronautics (advisor K. C. Howell).

Sec. 4.4.7 ("Earth-Moon Periodic Cyclers"), pp.169-172, Figs. 4.43-4.44. Prose-only source: prints the two families' Jacobi-range endpoints and four endpoint Earth->Moon TOFs plus four selection criteria -- NO digit-grade IC/period/stability table exists anywhere in the chapter (Fig. 4.44's x0 axis is unlabeled; digest). No DOI exists for the dissertation; freely available from the author's own Purdue page, corpus copy md5 fdcbf871322b87cd1dd3448059cb2596 (docs/notes/CORPUS_INDEX.md).

Corroborating sources

Casoliva, J. et al. (2010). Two Classes of Cycler Trajectories in the Earth-Moon System. Journal of Guidance, Control, and Dynamics 33(5), pp. 1623-1640. DOI: 10.2514/1.46856
Criteria lineage, NOT a record of these specific orbits: Vaquero's own four cycler-selection criteria are explicitly 'consistent with the criteria for cyclers established by Mondelo and Villac [105, 106]' (p.170), her refs [105]/[106] being this group's own 2008 AIAA / 2010 JGCD papers. the cross-check found NO exact numeric orbit match between her 2:1/3:1 families and Casoliva's own Table 3 rows (different C values on presumably related underlying families -- e.g. her 2:1 range [1.98, 2.66] vs Casoliva 2-1a/b at C=0.4887/1.1964).

Notes

Vaquero 2013 (Purdue Ph.D. dissertation) Sec. 4.4.7 'Earth-Moon Periodic Cyclers', 3:1 interior-resonant family member at C=3.13 -- cislunar, naturally L1-connected (UNSTABLE, Barden nu=6.6836, |lambda|=13.292). Her own printed 3:1 family high-energy endpoint (C=3.13, printed Earth->Moon TOF 5.04 d, recovered 5.1320 d, 1.83% relative). No digit-grade IC/period/stability table exists anywhere in her chapter (Fig. 4.44's x0 axis is unlabeled) -- the prose C-ranges/endpoint TOFs are her only printed numbers, so reproduced the families by direct continuation against those prose values; this row writes back the digit-grade, Radau-cross-checked IC. REPRODUCTION of a published result, nothing novel claimed (precedent: the reproduction target IS the published record; literature_check.py not in play). SELECTION: one of the four printed-TOF family endpoints -- the only members whose admission is anchored by a digit-grade Vaquero print (her 5.04 d Earth->Moon TOF at C=3.13, recovered at 5.1320 d, 1.83% relative). The four endpoints + the 2:1 stability-transition bracket (vaquero-21-c246/c247) are the family's first-documented member's own minimum representative subset of the 129-member record; the full record stays in data/found/799_vaquero_em_cycler_families/results.json. ORBIT_CLASS = resonant_po, NOT cycler (per-member check -- SOI-MARGINAL, the closest call in this writeback): closest lunar approach 66995.2 km = 1.012x the lunar SOI (66,182.9 km), 1.089x the Hill radius (61,524.1 km) -- outside both, by only ~812 km (1.2%). Under the project's own SOI discriminator (em-cycler-21-3d-spatial-2026) that is resonant_po; the call is grid-conditional (the 3:1 family exits the SOI between C=3.12 at 65,802 km and this C=3.13 top endpoint -- this is the ONLY 3:1 member outside the SOI in the 60-member record) and is stated, not hidden. Vaquero's own criterion 2 for this family is an L1-LPO connection (approach = 1.155x the Moon-L1 distance here), satisfied without SOI entry -- her 'cycler' label is her own usage. Cross-checked two independent ways: the dense DOP853 propagation and the standalone Radau re-derivation, agreeing to <0.1 km. Honesty note: this row is UNSTABLE (|lambda|=13.29 -- 'small unstable modes' in her own sense); resonant_po's descriptive 'stable' prose does not fit, but only epoch_locked=false/n_returns=infinite is schema-enforced (tests/data/test_schema_v47_orbit_class.py), same honesty-note pattern as casoliva-1-2d-em-resonant-po-2010. MODEL: pure planar CR3BP (PCR3BP), this project's own DE440-registry mu/l*/t* throughout (mu=0.01215058439469525, l*=384,400 km, t*=375,190.26 s). Vaquero's own mu for Sec. 4.4 is UNSTATED (her Ch. 2 gives only the generic definition); registry mu per project policy + the lineage evidence. Her Purdue/Howell frame convention (Earth at (-mu,0), Moon at (1-mu,0)) matches this project's own -- x0 compares directly, NO coordinate flip (unlike the Casoliva rows). No bicircular/solar-gravity/lunar-eccentricity/ephemeris refinement. REPRODUCTION (evidence reused unmodified): src/cyclerfinder/search/vaquero_em_cyclers.py -- Tisserand-matched two-body APOAPSIS seed (NOT the 4x-documented-failing naive two_body_resonant_seed), fixed-Jacobi symmetric corrector, continue_family walk across the family's full printed Jacobi range at rtol=atol=1e-13 (1e-12 measurably leaks the 1e-10 Jacobi-conservation gate at the families' ~7,000-8,500 km low-C perigees), both branches terminating on jacobi_bound at her printed endpoints. Every member passes corrector convergence, period bounds, dedup, Jacobi conservation <=1e-10, and an independent-Radau cross-check (this member: crossing residual 1.95e-12, radau dJ=5.62e-13). Family identity verified on ALL constraints (discipline): printed C-range endpoint-to-endpoint, all four printed endpoint TOFs (0.10%-1.83%), two-body resonance a within 2-4% of p^(-2/3) (this member: a=0.4882 LU vs 3^(-2/3)=0.4807), cislunar (apogee inside the Moon's orbit), L1-connected geometry, Fig. 4.44 x0 band, and her stability criterion. the family's first-documented member's own independent DERIVE re-verification (not merely inherited): standalone Radau rtol=atol=1e-12 dense propagation (inline CR3BP EOM, 20k-sample grid + parabolic refinement, no cyclerfinder imports) confirms periselene/perigee/apogee to <0.1 km and closes the period-map to <=3.1e-12 -- the evidence this row's orbit_class determination rests on.

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.jacobi_constant: Vaquero 2013, Sec. 4.4.7 (pp.170-171): "Similarly, the arc from the Earth to the Moon along the 3:1 cycler orbits ranges from 4.90 days for a cycler with C = 2.54 to 5.04 days for a cycler with C = 3.13." -- C=3.13 is her own printed 3:1 family high-energy endpoint.
orbit_elements.cr3bp.vaquero_printed_tof_days: Vaquero 2013, Sec. 4.4.7 (pp.170-171): "Similarly, the arc from the Earth to the Moon along the 3:1 cycler orbits ranges from 4.90 days for a cycler with C = 2.54 to 5.04 days for a cycler with C = 3.13."
orbit_elements.cr3bp.stability_index: Vaquero 2013, Sec. 4.4.7 (p.170): "only resonant cyclers that are stable or possess small unstable modes are considered in this investigation" (her criterion 4 -- the family-level stability constraint; she prints NO per-member stability value, Fig. 4.44's nu-arrows are graphical only).
orbit_elements.cr3bp.mass_ratio: cyclerfinder.core.cr3bp.cr3bp_system("Earth","Moon").mu = 0.01215058439469525, re-verified this task against data/found/799_vaquero_em_cycler_families/results.json's own recorded mu. Vaquero's own mu for Sec. 4.4 is UNSTATED -- see the field's own comment.
orbit_class: data/README.md/catalogue.schema.json v4.9 resonant_po definition ('it never encounters the secondary') + this row's own periselene_km (66995.2 km = 1.012x SOI, 1.089x Hill), and cross-checked, mirroring em-cycler-21-3d-spatial-2026/the reasoning.