Russell-Ocampo cycler 4.991gG2 (= McConaghy/Longuski/Byrnes 'S1L1' / 'Notable', two-synodic ballistic)

russell-ch4-4.991gG2 · source: literature · validation: V3

Signature

Bodies: E-M
Primary: Sun (default — heliocentric)
Sequence (canonical): E-E-M-M
Sense: outbound
Orbit class: Cycler
strict cycler (infinite returns)
Cycler class: multi-arc
Trajectory regime: ballistic
Maintenance ΔV band: unclassified
Model assumption: circular-coplanar
Idealised: planets on circles, coplanar ecliptic.
Period: 4.270 yr (2 × E-M synodic)
2-synodic E-M cycler per Russell 2004 Table 4.9 row 1. Russell footnote 'a' = 'documented in Ref. 15' (McConaghy/Longuski/Byrnes AIAA 2002-4420).
Maintenance ΔV: 0 km/s — strict ballistic
Priority date: 2002-08-05

V∞ at encounters

E (encounter 1): 4.99 km/s
Russell 2004 Table 4.9 row 1, column v_inf E (km/s) = 4.99. Cycler code 4.991gG2 leads with this value (the '4.991' prefix in the parent-cycler list, Table 5.5, dissertation line 7512).
M (encounter 2): 5.10 km/s
Russell 2004 Table 4.9 row 1, column v_inf M (km/s) = 5.10.

Cycle-level identity (multi-arc invariants)

Aphelion ratio (AR): —
Turn ratio (TR): 2.65
Transit times (days): 150 d, 150 d

Russell 2004 Table 4.9 row 1 tabulates aphel = 1.64 AU. (a, e, peri) are null because the cycler is a piecewise sequence of two generic- return arcs (the leg descriptors are g(1.4612,526.02,Ll) + G(2.8096,651.46,U)), not a single Keplerian ellipse — each leg has its own (a, e). Only the maximum aphelion is well-defined for the cycler as a whole.

Orbit view 2.5D ecliptic projection

spacecraft: idealized coplanar ellipse planets: J2000 osculating ellipse (Standish & Williams Table 1) orientation: coplanar-idealized (no Ω/ω published — in-plane ellipse)

3D view not available for multi-arc trajectories yet. The 3D system can now render a numerically-sampled polyline (n-body or multi-arc), but no sampled trajectory is published for this row — each leg is a separate ellipse with honest gaps where elements are unpublished (the 2D view above shows them). The button will appear here once a sampled path is exported for this trajectory; we never interpolate one from the catalogue's per-leg (a, e).

— — planet orbit (true J2000 ellipse) ··· segment arc ◌ max aphelion ring (sourced)

Not-published gaps (never interpolated):

E → M — segment elements (a, e) not published; no curve drawn.
M → E — segment elements (a, e) not published; no curve drawn.

Real DE440 encounter dates (5)

2031-03-06 V∞ 4.90 / 5.87 km/s
2033-04-14 V∞ 3.17 / 5.10 km/s
2033-05-24 V∞ 5.29 / 3.92 km/s
2035-06-03 V∞ 4.19 / 5.11 km/s
2035-08-02 V∞ 5.15 / 3.11 km/s

Real, verifiable JPL DE440 geometric-match dates (see Launch windows). Positions on the map are idealized; these are the true dates.

Legs (trajectory segments)

E → M: 150 d, 0 revs
Russell 2004 Table 4.9 row 1: t_out = 150 days. Designated leg is the second one G(2.8096,651.46,U) per Russell's convention (uppercase = leg used to compute Mars v_inf and t_out).
M → E: 150 d, 0 revs
Russell 2004 Table 4.9 row 1: t_in = 150 days (symmetric). Generic-return arcs are symmetric, so t_in = t_out always for this row.
E → E: 533.7 d, 1 revs
Intermediate Earth-Earth loop 1 of 1 for this 2-synodic cycler (sequence E-E-M-M). MAPPED (2026-08-09) to the printed non-designated leg g(1.4612,526.02,Ll) of Russell 2004 Table 4.9 row 1: the uppercase G(2.8096,...) leg is the designated E-M transit (out-em + ret-me), so the single lowercase g leg IS this loop. (a,e,n_revs,branch) DERIVED (kind: derive) by the McConaghy/Russell/Longuski 2005 (JSR 42(4), DOI 10.2514/1.8123) generic-transfer Lagrange formulation, Eqs (1)-(11), from the printed (tof=1.4612 yr, transfer angle theta=526.02 deg, epsilon=Ll long-period lower branch): a=1.0512 AU, e=0.1704, coplanar. Cross-checks: emergent V-infinity 4.995 km/s vs the row's independently-printed 4.99 (0.1%); independently reproduced by cyclerfinder.core.lambert.lambert(max_revs=8) n_revs=1/branch=low to 4 decimals in (a,e). The Ll subscript (long-period = larger-a lower-branch root) selects a=1.0512 over the degenerate a=1.0000/e~0 root, consistent with the V-infinity match.

Free-return arcs (Russell decomposition)

Russell's Earth-to-Earth free-return arcs (spec §16.7.7) — a decomposition distinct from the encounter legs above. A single arc spans what the catalogue models as two or more encounter segments.

Arc type	Resonance	TOF (yr)	Raw descriptor
generic	—	1.4612	g(1.4612,526.02,Ll)
generic	—	2.8096	G(2.8096,651.46,U)

Definition status

incomplete — core fields missing or known-unknowns tracked below

Family: Russell 2-synodic Earth-Mars ballistic · Russell-Ocampo cycler code

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.

trajectory.segments[out-em].a_au [unknown]
Per-segment (a,e) of the outbound generic-return arc not tabulated by Russell 2004 (orbit_elements.a_au/e are null for the same reason — each generic-return arc has its own (a,e); only the aggregate aphelion is well-defined). [ 2026-08-09: the loop-ee gap on this row is CLOSED (descriptor-derived, see the segment note); out-em/ret-me are halves of the designated G(2.8096,651.46,U) arc split by the Mars encounter, so their per-segment elements need the Mars-phase treatment, not the loop solve.]

source: Russell 2004 dissertation Table 4.x leg descriptors g(...)/G(...)
trajectory.segments[ret-me].a_au [unknown]
Per-segment (a,e) of the return generic-return arc not tabulated by Russell 2004; same source gap as out-em.

source: Russell 2004 dissertation Table 4.x leg descriptors g(...)/G(...)

Primary citation

McConaghy, T. T. et al. (2002). Analysis of a Broad Class of Earth-Mars Cycler Trajectories. AIAA/AAS Astrodynamics Specialist Conference, AIAA 2002-4420.

DOI: 10.2514/6.2002-4420

orbit source Russell 2004, Tables 4.9 / 4.13 orbit fidelity circular-coplanar V∞ source Russell 2004, Tables 4.9 / 4.13 V∞ fidelity circular-coplanar

Corroborating sources

McConaghy, T. T. et al. (2006). Notable Two-Synodic-Period Earth-Mars Cycler. Journal of Spacecraft and Rockets, Vol. 43, No. 2, pp. 456-465. DOI: 10.2514/1.15215
The 'Notable' paper. Catalogue entry 2 (mcconaghy-2006-em-k2) carries the McConaghy-convention attribution of the same cycler with V_inf 4.7/5.0 km/s and 153-d ToF (realistic-ephemeris optimised) vs Russell's 4.99/5.10 and 150 d (circular-coplanar simple model).
Russell, R. P. (2004). Global Search and Optimization for Free-Return Earth-Mars Cyclers. PhD dissertation, UT Austin. · link
Table 4.9 row 1; parent cycler 4.991gG2 (no. 83). Cross-reference to McConaghy 'S1L1' at dissertation lines 7416, 5476, 8008.

Notes

SUPERSEDES NUMERICALLY (for orbital/V_inf data) catalogue entry 2 (mcconaghy-2006-em-k2) — same physical cycler, different nomenclature framing. Both entries preserved: entry 2 carries the McConaghy SnLm-convention citation (V_inf 4.7/5.0 from the JSR paper's realistic-ephemeris optimisation, 153-d ToF), this entry carries Russell's circular-coplanar simple-model values from Table 4.9 (V_inf 4.99/5.10, 150-d ToF, aphel 1.64 AU). Discrepancy analysis: - V_inf E: McConaghy 2006 = 4.7 vs Russell 2004 = 4.99 (Δ = 0.29 km/s) - V_inf M: McConaghy 2006 = 5.0 vs Russell 2004 = 5.10 (Δ = 0.10 km/s) - E-M ToF: McConaghy 2006 = 153 d vs Russell 2004 = 150 d (Δ = 3 d) The most plausible explanation per Russell's own text (dissertation line 7416): McConaghy 2006 reports ephemeris-optimised values for a realistic launch, while Russell Table 4.9 reports the circular- coplanar simple-model reference values. Russell explicitly states this cycler "is essentially ballistic for all launch dates. This is consistent with the findings in Ref. 15" (McConaghy 2002 et al). CORRECTED (2026-08-12): the prior claim here that M7 matching tolerances "should be wide enough (~0.5 km/s on V_inf, ~10 d on ToF) to collapse these two entries under one signature" was wrong about the spec. Section 16.2's canonical signature bins V_inf to 0.05 km/s, an order of magnitude tighter — the pair's V_inf gaps (0.29/0.10 km/s) sit several bins apart, so the exact hash-match branch never fires and no merge is spec-mandated. ran the full section 16.3 decision (at most probable-match-NEEDS-HUMAN) and resolved it: KEEP BOTH ROWS, the same physical cycler carried as two sourced realizations. See docs/notes/2026-08-12-831-mcconaghy-russell-4991gg2-duplicate-disposition.md. The leg-descriptor "g(1.4612,526.02,Ll) G(2.8096,651.46,U)" — generic return Ll patched with generic return U — remains the shared structural fingerprint identifying them as one object. Russell's Chapter 5 parent-cycler list (Table 5.5, line 7512) gives a 7-cycle realistic-ephemeris optimisation result for no. 83 4.991gG2: launch Jun-25, total delta-V = 0 m/s, avg E-M transit 165 d, avg V_inf E = 5.37, avg V_inf M = 5.48 km/s. These are different again from both the McConaghy 2006 abstract values and the Table 4.9 simple-model values — they characterise the ephemeris-optimal trajectory, not the simple-model parent.

Source quotes (per-field provenance)

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

delta_v_kms: Derived from Russell 2004's tabulated AR / TR via the M2 powered-flyby surrogate (src/cyclerfinder/core/flyby.py): ΔV ≈ (V∞_E + V∞_M) × max(0, 1 − TR) + (1 − AR) × (V∞_E + V∞_M) × 0.025 [AR-correction term active only when AR < 1]. Upper bound; biases near-ballistic entries toward 'requires nudge'. Strict-ballistic rows (AR ≥ 1 AND TR ≥ 1) give ΔV = 0 exactly. AR/TR primary values: Russell 2004 Table 4.9 row 1 (4.991gG2): aphel=1.64 AU, TR=2.65, v∞E=4.99, v∞M=5.10.
vinf_kms_at_encounters[0].vinf_kms: Russell 2004 dissertation Table 4.9 (page 127), row 1: column "v_inf E (km/s)" = 4.99. Confirmed by parent-cycler code "4.991gG2" (dissertation Table 5.5 line 7512, Appendix C line 8600).
vinf_kms_at_encounters[1].vinf_kms: Russell 2004 Table 4.9 row 1, column "v_inf M (km/s)" = 5.10.
legs[0].tof_days: Russell 2004 Table 4.9 row 1, column "t_out (days)" = 150.
legs[1].tof_days: Russell 2004 Table 4.9 row 1, column "t_in (days)" = 150 (symmetric with t_out for generic-return cyclers).
orbit_elements.aphelion_au: Russell 2004 Table 4.9 row 1, column "aphel. (AU)" = 1.64.
period.years: Derived: 2 x Earth-Mars synodic period 2.135 yr = 4.27 yr. Russell dissertation does not separately tabulate the cycler period in years; the "2" suffix in 4.991gG2 indicates 2 synodic periods per Russell §4.7 and §5.5 (dissertation line 7375).
name: Russell 2004 dissertation line 7416: "cycler 4.991gG2 (no. 83)... Also known as the 'S1L1' cycler". Line 5476: "notable 'S1L1' cycler... discovered first by McConaghy et al. in Ref. 15". Line 8008: "the S1L1 cycler (4.991Gg2), 8.049gGf2, and the Aldrin cycler".
trajectory.segments[loop-ee].a_au: DERIVED (kind: derive, 2026-08-09) from the printed leg descriptor g(1.4612,526.02,Ll) (Russell 2004 Table 4.9 row 1) via the McConaghy/Russell/Longuski 2005 (JSR 42(4), DOI 10.2514/1.8123) Lagrange generic-transfer solve, Eqs (1)-(11). Emergent V-infinity 4.995 km/s vs the printed 4.99 (independent cross-check); independently reproduced by cyclerfinder.core.lambert (n_revs=1, branch=low). See docs/notes/2026-08-09-794-loop-ee-descriptor-closure.md.
trajectory.segments[loop-ee].e: DERIVED (kind: derive), see trajectory.segments[loop-ee].a_au.
trajectory.segments[loop-ee].tof_days: SOURCED: 1.4612 yr printed in the leg descriptor (Russell 2004 p.126: first number in the parenthesis is ToF in years) = 533.7 d.
trajectory.segments[loop-ee].n_revs: DERIVED: floor(526.02/360) = 1 (McConaghy/Russell/Longuski 2005 Eq (2)); branch per cyclerfinder.core.lambert branch vocabulary (low), matching the Ll (lower-curve long-period) printed subscript.