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Vaquero 2:1 resonant Earth-Moon periodic cycler family, C=2.46 last linearly stable member (stability transition) (CR3BP, STABLE, genuine lunar encounter -- periselene 49,700 km, inside lunar SOI)

vaquero-21-c246-em-cycler-2013 · 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)

T = 6.02801832920549 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 = 0.9594). 26.176548 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 2:1 resonant periodic-cycler family (interior 2:1, circumlunar, L2-connected), C in [1.98, 2.66]

Mass ratio (μ)

0.01215058439469525

Jacobi constant

2.4600

Period (non-dim)

6.0280

Stability index

-0.927

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 (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]
  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.
• orbit_elements.cr3bp.jacobi_constant [derive]
  Vaquero prints NO per-member values inside the family interior. This member's C sits on the family's first-documented member's own dC=0.01 continuation grid, selected this task as the last linearly STABLE member bracketing the 2:1 family's stability transition (between C=2.46 and C=2.47) -- a family-structure feature Vaquero's own criterion 4 prose and Fig. 4.44's nu-arrows describe qualitatively but never quantify. The transition C itself is a DERIVED, grid-resolution-conditional (dC=0.01) value, not a sourced one.
  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', 2:1 interior-resonant family member at C=2.46 -- circumlunar, naturally L2-connected (STABLE, Barden nu=-0.9274). An interior member selected as part of the 2:1 stability-transition bracket (see SELECTION below). 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: the last linearly STABLE member of the 2:1 family on the dC=0.01 grid -- with its successor vaquero-21-c247 it brackets the family's stability transition (linearly stable through C=2.46, |lambda|<=5.73 above -- the note), the family-structure feature Vaquero's own criterion 4 ('stable or possess small unstable modes') makes load-bearing and her own free-transfer prose (p.171-172) implicitly requires (unstable 2:1 members must exist in the [2.54, 2.66] overlap band -- and do, above this transition). The four printed-TOF endpoints + this bracket are the family's first-documented member's own minimum representative subset of the 129-member record. ORBIT_CLASS = cycler (per-member check): closest lunar approach 49700.3 km = 0.751x the lunar SOI (66,182.9 km) and 0.808x the Hill radius (61,524.1 km) -- a genuine lunar flyby every period ('transports between encounters', data/README.md schema v4.7). Determined per member, not per family: the 2:1 family only enters the SOI for C >= ~2.30 (its C=1.98 end approaches no closer than 86,911 km) and the 3:1 family exits it at its C=3.13 endpoint -- each of the six rows carries its own determination. Cross-checked two independent ways: the family's first-documented member's own dense DOP853 member_report propagation and the standalone Radau re-derivation (inline CR3BP EOM, no cyclerfinder imports), agreeing to <0.1 km. 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 7.44e-15, radau dJ=4.30e-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=0.6469 LU vs 2^(-2/3)=0.6300), circumlunar (apogee beyond the Moon), L2-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.family

Vaquero 2013, Sec. 4.4.7 (pp.170-171): "The time-of-flight along the 2:1 cyclers from the vicinity of the Earth to the vicinity of the Moon ranges from 4.91 days for a cycler with C = 2.66 to 6.39 days for a cycler with C = 1.98." -- the printed family C-range this interior member sits inside.

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). Vaquero 2013, Sec. 4.4.7 (pp.171-172): "a free-transfer between two periodic orbits may exist if both orbits are unstable and possess the same value of Jacobi constant. As illustrated in Figure 4.44, a cost-free transfer between a 2:1 and a 3:1 cycler is available at energy levels such that 2.66 < C < 2.54 approximately." -- her own prose REQUIRES unstable 2:1 members in the [2.54, 2.66] overlap band (the printed range is reversed, a likely typesetting slip flagged by the digest).

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 schema v4.7 cycler definition ('transports between encounters') + this row's own periselene_km (49700.3 km = 0.751x SOI), and cross-checked.