Rall 1970 periodic orbit M6-2 (circular-coplanar, App E)

rall-1970-m6-2 · source: literature · validation: V0

Signature

Bodies: E-M
Primary: Sun (default — heliocentric)
Sequence (canonical): E-E-M-E-E-M-E
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)
Rall App E M6-2 spans encounters JD-2440000 534.5214 (~12.81 yr); labelled 6-synodic-class (M6 family). k x 2.135 yr is an approximate synodic label for a multi-arc periodic orbit.
Maintenance ΔV: 0 km/s — strict ballistic
Priority date: 1970-01-01

V∞ at encounters

E (encounter 1): 7.06 km/s
Rall App E M6-2 (p.185): first Earth hyperbolic-excess speed 0.237 EMOS x 29.77 km/s/EMOS = 7.06 km/s (EMOS = Earth Mean Orbital Speed per Rall; mining-note Table 3-1). Original units (EMOS) preserved here.
M (encounter 2): 9.35 km/s
Rall App E M6-2: Mars hyperbolic-excess speed 0.314 EMOS x 29.77 = 9.35 km/s (speeds at both Mars encounters equal by symmetry, Rall legend).
E (encounter 3): 5.51 km/s
Rall App E M6-2: second Earth hyperbolic-excess speed 0.185 EMOS x 29.77 = 5.51 km/s.

Cycle-level identity (multi-arc invariants)

Aphelion ratio (AR): —
Turn ratio (TR): —
Transit times (days): 157 d

Rall App E is the circular-coplanar Model I.B (e=i=0 for the planets). Per-orbit (a,e) of the spacecraft arcs are not tabulated; only encounter dates, speeds, and passing distances.

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no sampled trajectory is published for this row — each leg is a separate ellipse with honest gaps where elements are unpublished (the 2D view above shows them).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

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)

2026-10-31 V∞ 6.53 / 10.66 km/s
2028-11-29 V∞ 6.29 / 11.30 km/s
2030-12-29 V∞ 6.18 / 11.18 km/s
2033-02-06 V∞ 6.27 / 9.86 km/s
2035-03-18 V∞ 8.39 / 9.08 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: 157 d, 0 revs
Rall App E M6-2: Earth->Mars leg = App-E date difference 2760-2603 = 157 d (JD-2440000, the CLEAN encoding; note §1).
M → E: 600 d, 0 revs
Rall App E M6-2: Mars->Earth leg = App-E date difference 3360-2760 = 600 d (JD-2440000).

Definition status

incomplete — core fields missing or known-unknowns tracked below

Family: Rall 1970 M6 free-fall periodic Earth-Mars orbit · Rall (Hollister adv.) 1970 TE-34 periodic-orbit 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 [uncertain]
Rall App E gives a COLLAPSED 7-encounter listing for M6-2 (only Earth encounters adjacent to Mars). The full multi-arc decomposition (direct-return loops between Mars encounters, scheme symbols e.g. 3FR/S1SR per App D) is present in the source but not materialised as segments here; out-em/ret-me are the load-bearing E-M-E round-trip legs.

source: Rall 1970 TE-34 App E p.185; App D direct-return combinations
orbit_elements.a_au [unknown]
Per-arc (a,e) not tabulated by Rall App E (circular-coplanar Model I.B lists encounter dates/speeds/passing-distances only).

source: Rall 1970 TE-34 App E

Primary citation

Rall, C. S. (1970). Free-Fall Periodic Orbits Connecting Earth and Mars. Sc.D. thesis, MIT Dept. of Aeronautics and Astronautics, report TE-34 (NASA Grant NGR 22-009-010, Measurement Systems Laboratory), Oct 1969 (pub. 1970).

Appendix E (Circular Coplanar Earth-Mars Periodic Orbits, p.185), orbit M6-2. The 'a' version exists for the best approximations used.

orbit source rall-1970-te34 orbit fidelity circular-coplanar V∞ source rall-1970-te34 V∞ fidelity circular-coplanar

Notes

Rall 1970 (MIT TE-34, Hollister adv.) free-fall periodic orbit M6-2, Appendix E circular-coplanar Model I.B. 6-synodic period; circular-coplanar exists (App E). No App-F eccentric per-encounter member data (not listed). Existence (Rall verbatim): "The 'a' version exists for the best approximations used." Units: V_inf in EMOS (Earth Mean Orbital Speed = 29.77 km/s per Rall; mining-note Table 3-1) — converted to km/s in vinf_kms_at_encounters with the EMOS values preserved in each note. Passing distances are in local planetary radii (Earth radii at E, Mars radii at M). Dates are Julian Date minus 2440000 (App E clean encoding; note §1); per-leg ToFs are date differences. 1969-era patched-conic mean-element dynamics; modern Byrnes/Russell/Casalino free-return families supersede the dynamics, but Rall's dates/identifiers remain the primary-source member record. ET-vs-UT for the 1969 JD usage is unstated by Rall (conversion caveat). Ingested 2026-06-07 from docs/notes/2026-06-07-rall-1970-appendices-transcription.md.

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: Rall 1970 TE-34 App E M6-2 (p.185): Earth V_inf 0.237 EMOS; x 29.77 km/s/EMOS = 7.06 km/s.
vinf_kms_at_encounters[1].vinf_kms: Rall 1970 TE-34 App E M6-2: Mars V_inf 0.314 EMOS; x 29.77 = 9.35 km/s.
legs[0].tof_days: Rall 1970 TE-34 App E M6-2: E->M = JD-2440000 dates 2760-2603 = 157 d.