Russell-Ocampo cycler 3.1.3.+0 (3-synodic)

russell-ocampo-3.1.3+0 · source: literature · validation: V1

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: 6.405 yr (3 × E-M synodic)
3 x Earth-Mars synodic period (2.135 yr) per Russell 2004 §3.8 row 3.1.3.+0; n_flybys = 4.
Priority date: 2003-02-09

V∞ at encounters

E (encounter 1): 5.10 km/s
Russell 2004 Table 3.4 row 3.1.3.+0: Earth v_inf = 5.1 km/s.
M (encounter 2): 9.10 km/s
Russell 2004 Table 3.4 row 3.1.3.+0: Mars v_inf = 9.1 km/s.

Cycle-level identity (multi-arc invariants)

Aphelion ratio (AR): 1.43
Turn ratio (TR): 0.93
Transit times (days): 123 d

Aphelion derived from Russell 2004 Table 3.4 Aphelion Ratio AR = 1.43 x 1.52 AU (Mars sma) = 2.174 AU. Other elements (a, e, perihelion) not tabulated in Table 3.4.

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no sampled trajectory is published for this row — each leg is a separate ellipse with honest gaps where elements are unpublished (the 2D view above shows them).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) ◌ 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.
E → E — segment elements (a, e) not published; no curve drawn.

Real DE440 encounter dates (4)

2029-01-28 V∞ 6.82 / 9.59 km/s
2031-02-27 V∞ 4.88 / 9.45 km/s
2033-03-28 V∞ 4.49 / 9.54 km/s
2035-05-17 V∞ 6.57 / 9.04 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: 123 d, 0 revs
Russell 2004 Table 3.4 row 3.1.3.+0: Earth->Mars (or aphelion) time = 123 days.
M → E: (TOF not published)
Return Mars->Earth leg. Russell 2004 Table 3.4 tabulates only the E->M transit time for this row, not the return arc ToF / rev count / (a,e); tracked in data_gaps.
E → E: (TOF not published)
Intermediate Earth-Earth loop arc(s) filling the cycler period between Mars encounters (sequence E-E-M-M). Russell tabulates 4 Earth flyby(s) for this row, implying 3 intermediate Earth-Earth loop(s). Multi-rev arc(s) to be set by the multi-rev Lambert solver; a single structural slot here, topology tracked as uncertain in data_gaps[].

Definition status

incomplete — core fields missing or known-unknowns tracked below

Family: Russell 3-synodic Earth-Mars ballistic · Russell-Ocampo cycler code

Known-unknowns (4)

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[ret-me].tof_days [unknown]
Return M->E leg time-of-flight not tabulated by Russell 2004 (Table 3.4 gives only the E->M transit time for this row).

source: Russell 2004 dissertation Table 3.4
trajectory.segments[loop-ee] [derive]
Intermediate Earth-Earth loop arc(s) tof_days / n_revs / branch / (a,e) computable once the multi-rev Lambert solver closes the arc(s) to the published 5.1/9.1 km/s V-infinity; not yet derived. Russell tabulates only the transit time + aggregate aphelion.

source: multi-rev Lambert solver; Russell 2004 Table 3.4 leg descriptors
trajectory.segments [uncertain]
Segment topology provisional. sequence_canonical E-E-M-M is the lex-min canonical form; the actual cycle has 4 Earth flyby(s) (Russell required-geocentric-turn column) implying 3 intermediate Earth-Earth loop(s) over the 6.405-yr period. Only one representative loop-ee slot materialised; full decomposition to be closed by the solver.

source: Russell 2004 dissertation Table 3.4; multi-rev Lambert solver
orbit_elements.a_au [unknown]
Per-segment / per-orbit (a,e) not tabulated by Russell 2004 Table 3.4 (only the aggregate aphelion ratio is published).

source: Russell 2004 dissertation Table 3.4 leg descriptors

Primary citation

Russell, R. P. & Ocampo, C. A. (2005). Geometric Analysis of Free-Return Trajectories Following a Gravity-Assisted Flyby. Journal of Spacecraft and Rockets, Vol. 42, No. 1, pp. 138-152.

DOI: 10.2514/1.5571

URL: https://arc.aiaa.org/doi/abs/10.2514/1.5571

Peer-reviewed venue for the ballistic cycler catalogue (geometric analysis branch). Conference precursor Russell/Ocampo AAS 03-145 (Feb 2003) establishes priority.

orbit source Russell 2004, Table 3.4 orbit fidelity circular-coplanar V∞ source Russell 2004, Table 3.4 V∞ fidelity circular-coplanar

Corroborating sources

Russell, R. P. (2004). Global Search and Optimization for Free-Return Earth-Mars Cyclers. Ph.D. dissertation, University of Texas at Austin, Department of Aerospace Engineering. · link
Table 3.4 row 3.1.3.+0: AR=1.43, TR=0.93, E->M days=123, V_inf,E=5.1, V_inf,M=9.1, 4 flyby turning angles [95, 95, 16, 16] deg.

Notes

Near-ballistic per Russell's Table 3.4 (passes the AR_MIN=0.9 / TR_MIN=0.85 documentation gate but not strictly ballistic: AR=1.43, TR=0.93). 3-synodic period. 4 required Earth flybys (turning angles [95, 95, 16, 16] deg per Table 3.4 last column). Where AR<1 the cycler does not quite reach Mars / where TR<1 a small powered turn is required; Russell does not tabulate the per-cycle DV (left null). Russell nomenclature p.h.s.i: the trailing index (+0) is the higher-energy multiple-revolution Lambert branch of the generic-return solution (sign: + fast, - slow). Circular-coplanar simple-model values (Russell 2004 Ch.3); modern ephemeris families (App.C / DE405) refine the dynamics. Ingested 2026-06-07 from docs/notes/2026-06-07-russell-2004-member-tables-transcription.md (Table 3.4, p.83).

Source quotes (per-field provenance)

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

vinf_kms_at_encounters[0].vinf_kms: Russell 2004 Table 3.4 row 3.1.3.+0: Earth v_inf (km/s) = 5.1.
vinf_kms_at_encounters[1].vinf_kms: Russell 2004 Table 3.4 row 3.1.3.+0: Mars v_inf (km/s) = 9.1.
legs[0].tof_days: Russell 2004 Table 3.4 row 3.1.3.+0: Earth->Mars (or aphelion) Time (days) = 123.
aphelion_au: Russell 2004 Table 3.4 row 3.1.3.+0: Aphelion Ratio = 1.43; aphelion = 1.43 x 1.52 AU = 2.174 AU.