Russell-Ocampo cycler 6.4.1.+6 (6-synodic, ballistic)

russell-ocampo-6.4.1+6 · source: literature · validation: V0

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: 12.810 yr (6 × E-M synodic)
6 x Earth-Mars synodic period (2.135 yr) per Russell 2004 dissertation Table 3.10 row 6.4.1.+6.
Maintenance ΔV: 0 km/s — strict ballistic
Priority date: 2003-02-09

V∞ at encounters

E (encounter 1): 4.70 km/s
Russell 2004 dissertation Table 3.10 row 6.4.1.+6: Earth v_inf = 4.7 km/s.
M (encounter 2): 6.00 km/s
Russell 2004 dissertation Table 3.10 row 6.4.1.+6: Mars v_inf = 6.0 km/s.

Cycle-level identity (multi-arc invariants)

Aphelion ratio (AR): 1.13
Turn ratio (TR): 1.33
Transit times (days): 142 d

Aphelion derived from Russell 2004 Table 3.10 Aphelion Ratio AR = 1.13 x 1.524 AU (Mars semi-major axis) = 1.722 AU. Other elements (a, e, perihelion) not directly tabulated in Tables 3.9-3.11.

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no sampled trajectory is published for this row — each leg is a separate ellipse with honest gaps where elements are unpublished (the 2D view above shows them).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 (5)

2031-03-09 V∞ 5.24 / 6.31 km/s
2033-04-07 V∞ 3.28 / 6.28 km/s
2033-05-07 V∞ 4.23 / 4.39 km/s
2035-05-27 V∞ 4.84 / 6.16 km/s
2035-07-26 V∞ 4.68 / 3.44 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: 142 d, 0 revs
Russell 2004 dissertation Table 3.10 row 6.4.1.+6: Earth->Mars (or aphelion) time = 142 days.
M → E: (TOF not published)
Return Mars->Earth leg. Russell 2004 Table 3.4/4.x tabulates only the E->M (or-aphelion) transit time for this row, not the return arc ToF / rev count / per-segment (a,e); tracked in data_gaps.
E → E: (TOF not published)
Intermediate Earth-Earth loop arc(s) that fill the cycler period between Mars encounters (sequence E-E-M-M). Russell's required-geocentric-turn column tabulates 3 Earth flyby(s) for this row, implying 2 intermediate Earth-Earth loop(s). Multi-rev arc(s) — tof_days / n_revs / branch / (a,e) to be set by the multi-rev Lambert solver; represented here as a single structural slot, exact multi-loop topology tracked as uncertain in data_gaps[].

Definition status

incomplete — core fields missing or known-unknowns tracked below

Family: Russell 6-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 3.4/4.x give only the E->M transit time for this row).

source: Russell 2004 dissertation Table 3.4 / 4.x
trajectory.segments[loop-ee] [derive]
Intermediate Earth-Earth loop arc(s) tof_days / n_revs / branch / (a,e) are computable once the multi-rev Lambert solver closes the arc(s) to the published 4.7/6 km/s V-infinity; not yet derived. Russell tabulates only the transit time + aggregate aphelion, not the loop arc(s).

source: multi-rev Lambert solver; Russell 2004 Table 3.4/4.x leg descriptors
trajectory.segments [uncertain]
Segment topology is provisional. sequence_canonical E-E-M-M is the lex-min canonical form; the actual cycle has 3 Earth flyby(s) (Russell required-geocentric-turn column) implying 2 intermediate Earth-Earth loop(s) over the 12.81-yr period. Only a single representative loop-ee slot is materialised here; the full multi-loop decomposition must be verified against Russell 2004 / closed by the solver.

source: Russell 2004 dissertation Table 3.4 / 4.x; multi-rev Lambert solver
trajectory.segments[out-em].a_au [unknown]
Per-segment (a,e) of the outbound transit arc not tabulated by Russell 2004 (orbit_elements.a_au/e are null for the same reason — only the aggregate aphelion ratio is published).

source: Russell 2004 dissertation Table 3.4 / 4.x 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 Russell-Ocampo cycler catalogue (geometric analysis branch). The conference precursor (Russell/Ocampo AAS 03-145, Feb 2003) establishes priority.

orbit source Russell 2004, Tables 3.9 / 3.11 orbit fidelity circular-coplanar V∞ source Russell 2004, Tables 3.9 / 3.11 V∞ fidelity circular-coplanar

Corroborating sources

Russell, R. P. & Ocampo, C. A. (2004). Systematic Method for Constructing Earth-Mars Cyclers Using Free-Return Trajectories. Journal of Guidance, Control, and Dynamics, Vol. 27, No. 3, pp. 321-335. DOI: 10.2514/1.1011
The companion JGCD paper that introduced the free-return construction method.
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 3.10 row 6.4.1.+6: AR=1.13, TR=1.33, E->M days=142, V_inf,E=4.7, V_inf,M=6.0, turning angles [71, 71, 71] deg. Strictly ballistic.

Notes

Strictly ballistic per Russell's strict criterion (AR>=1.0 AND TR>=1.0). Russell 2004 Tables 3.9-3.11 catalogue 5- and 6-synodic-period cyclers found at the wider ARMIN=0.9 / TRMIN=0.85 net (vs the strict AR>=1.0 / TR>=1.0 net of Table 3.4). 6-synodic period. Required Earth-flyby turning angles per Russell Table 3.10 row 6.4.1.+6: [71, 71, 71] deg. Note Russell's nomenclature: the trailing signed index (+6) denotes the Lambert-branch / phasing index of the generic-return solution.

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 dissertation Table 3.10 row 6.4.1.+6: Earth v_inf (km/s) = 4.7.
vinf_kms_at_encounters[1].vinf_kms: Russell 2004 dissertation Table 3.10 row 6.4.1.+6: Mars v_inf (km/s) = 6.0.
legs[0].tof_days: Russell 2004 dissertation Table 3.10 row 6.4.1.+6: Earth->Mars (or aphelion) Time (days) = 142.
aphelion_au: Russell 2004 dissertation Table 3.10 row 6.4.1.+6: Aphelion Ratio = 1.13 (Russell's ratio to Mars sma 1.524 AU); aphelion = 1.13 x 1.524 AU = 1.722 AU.
period.years: 6 x Earth-Mars synodic 2.135 yr = 12.81 yr; spec.md §9 convention.