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Vaquero 3:1 resonant Earth-Moon periodic cycler family, C=2.54 printed-TOF family endpoint (4.90 d) (CR3BP, UNSTABLE, genuine lunar encounter -- periselene 33,258 km, inside lunar SOI)

vaquero-31-c254-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 = 6.269604424886022 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 = 0.9978). 27.225631 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

2.5400

Period (non-dim)

6.2696

Stability index

5.675

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

Manifold connections

Recorded transport evidence between orbits (verified heteroclinic/homoclinic transfers). A connection does not change either orbit's validation level — see the endpoints' own V-levels. What this means →

• heteroclinic · ballistic (0 km/s) · C = 2.54 · Earth-Moon planar CR3BP
  Vaquero 2013 Sec 4.4.7 Earth-Moon 2:1 Periodic Cycler family member (not a catalogued orbit) — departs along its unstable manifold → this orbit — arrives on its stable manifold
  verified by this project's own numerical battery (no published transfer state exists to compare against)
  Round trip demonstrated: the reverse direction Wu(3:1) to Ws(2:1) is independently verified at this same C=2.54 and this row's own registry mass ratio (see em-vaquero-hetero-wu31c254-ws21c254-2026). That reverse connection's own crossing matches the CR3BP time-reversal-symmetry prediction from THIS forward crossing to about 6e-9, confirming it is genuinely the mirror-image manifold intersection of this same connection, not a coincidentally nearby different one.
  verification numbers

  full_state_gap

  0.000004483722190016192

  ydot_signs_match

  true

  ghost_distance_from

  0.24275809881390115

  ghost_distance_to

  1.7999326727722387

  radau_gap

  0.0000018053763121228701

  backward_reapproach

  1.0216784882076572e-9

  forward_reapproach

  0.00038053502861397056

  t_u_nd

  38.852089547100014

  t_s_nd

  -27.627817271177918

  jacobi_drift

  2.7355895326763857e-12, 1.9741097645464833e-11

  dv_kms

  0

  Source: data/found/822_vaquero_em_free_transfer/results.json (sweep C=2.54)

• heteroclinic · ballistic (0 km/s) · C = 2.54 · Earth-Moon planar CR3BP (Kumar et al. 2026's own printed mu, NOT this project's registry mu)
  this orbit — departs along its unstable manifold → Kumar-Rawat-Rosengren-Ross 2026 Table 6 Earth-Moon 2:1 family member (their own printed mu) (not a catalogued orbit) — arrives on its stable manifold
  reproduces a state published in the literature
  The forward direction Wu(2:1) to Ws(3:1) at this same node exists in this project's own registry mass ratio (see em-vaquero-hetero-wu21c254-ws31c254-2026 above), a different model from THIS entry's own Kumar mass ratio (delta mu about 1.24e-10). A same-model round trip at this row's own registry mass ratio is now separately demonstrated (see em-vaquero-hetero-wu31c254-ws21c254-2026), but that is still a DIFFERENT model from THIS entry's own Kumar mass ratio -- this entry's own round trip, within Kumar's own mass ratio specifically, remains not demonstrated.
  verification numbers

  full_state_gap

  4.6513318085564455e-7

  ydot_signs_match

  true

  ghost_distance_from

  0.592829534095866

  ghost_distance_to

  1.6996467513384976

  radau_gap

  4.584414112651359e-7

  backward_reapproach

  4.965745036947741e-8

  forward_reapproach

  0.00009264828886692251

  t_u_nd

  20.360431324628056

  t_s_nd

  -13.4503643835325

  jacobi_drift

  2.1882051726151985e-11, 2.9749536167855695e-12

  dv_kms

  0

  Source: data/found/827_kumar_table5_reproduction/results.json (C=2.54 row); independently re-verified fresh via keh.reproduce_table5_intersection(system, 2.54), seeded only from Kumar's printed state, never from results.json

• heteroclinic · ballistic (0 km/s) · C = 2.54 · Earth-Moon planar CR3BP
  this orbit — departs along its unstable manifold → Vaquero 2013 Sec 4.4.7 Earth-Moon 2:1 Periodic Cycler family member (not a catalogued orbit) — arrives on its stable manifold
  verified by this project's own numerical battery (no published transfer state exists to compare against)
  Round trip demonstrated: the forward direction Wu(2:1) to Ws(3:1) at this same C=2.54 and this row's own registry mass ratio is already verified (see em-vaquero-hetero-wu21c254-ws31c254-2026). This entry's own verified crossing matches the CR3BP time-reversal-symmetry prediction from that forward crossing to about 6e-9, confirming it is genuinely the mirror-image manifold intersection of the same forward connection, not a coincidentally nearby different one.
  verification numbers

  full_state_gap

  0.0000033703459348666194

  ydot_signs_match

  true

  ghost_distance_from

  1.7999326702997418

  ghost_distance_to

  0.24275809473080914

  radau_gap

  0.00000134526141373117

  backward_reapproach

  4.404466549909856e-8

  forward_reapproach

  0.0017654933650994179

  t_u_nd

  25.901339218718725

  t_s_nd

  -37.057354011499974

  jacobi_drift

  2.0730528405010773e-11, 2.149835864884153e-12

  dv_kms

  0

  Source: data/found/840_em_reverse_round_trip/results.json (C=2.54 sweep row)

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=2.54 -- cislunar, naturally L1-connected (UNSTABLE, Barden nu=5.6750, |lambda|=11.261). Her own printed 3:1 family low-energy endpoint (C=2.54, printed Earth->Moon TOF 4.90 d, recovered 4.9050 d, 0.10% 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 4.90 d Earth->Moon TOF at C=2.54, recovered at 4.9050 d, 0.10% 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 = cycler (per-member check): closest lunar approach 33258.0 km = 0.503x the lunar SOI (66,182.9 km) and 0.541x 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 1.03e-12, radau dJ=2.10e-12). 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=0.4664 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=2.54 is her own printed 3:1 family low-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 schema v4.7 cycler definition ('transports between encounters') + this row's own periselene_km (33258.0 km = 0.503x SOI), and cross-checked.