Rall 1970 periodic orbit M5-1D (eccentric-inclined, App F)

rall-1970-m5-1d · 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: analytic-ephemeris
Mid-fidelity: real eccentricity / inclination retained; not full N-body.
Period: 10.675 yr (5 × E-M synodic)
Rall App F M5-1D is the eccentric-inclined (Model III.B) version of the M5 family; k x 2.135 yr is an approximate synodic label for a multi-arc periodic orbit. Absolute span not stated here (App-F date encoding unresolved, note §1).
Maintenance ΔV: 0 km/s — strict ballistic
Priority date: 1970-01-01

V∞ at encounters

E (encounter 1): 5.24 km/s
Rall App F M5-1D (p.197): first Earth hyperbolic-excess speed 0.176 EMOS x 29.77 km/s/EMOS = 5.24 km/s (EMOS = 29.77 km/s per Rall; mining-note Table 3-1). Original EMOS preserved.
M (encounter 2): 9.55 km/s
Rall App F M5-1D: Mars hyperbolic-excess speed varies per encounter over 0.252.0.39 EMOS (7.5.11.61 km/s) across 6 Mars encounters (note §4). Stored value is the range MIDPOINT 9.55 km/s as a representative; full per-encounter list is in the transcription note. NOT a single sourced point — see data_gaps.

Cycle-level identity (multi-arc invariants)

Aphelion ratio (AR): —
Turn ratio (TR): —
Transit times (days): 107 d

Rall App F is the eccentric-inclined Model III.B (constant mean elements from a published series). Per-arc (a,e,i) of the spacecraft not tabulated; only per-encounter date/speed/turn/passing-distance/type.

Orbit view 2.5D ecliptic projection

spacecraft: analytic-ephemeris (real e/i retained) 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no sampled trajectory is published for this row — each leg is a separate ellipse with honest gaps where elements are unpublished (the 2D view above shows them).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.

Legs (trajectory segments)

E → M: (TOF not published)
Rall App F M5-1D Earth->Mars leg. ToF null: depends on the App-F last-5-digits-with-hyphen JD reconstruction whose rule carries a documented ambiguity (note §1); not converted here.
M → E: (TOF not published)
Rall App F M5-1D Mars->Earth leg. ToF null for the same App-F date-encoding reason.

Definition status

incomplete — core fields missing or known-unknowns tracked below

Family: Rall 1970 M5 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[out-em].tof_days [unknown]
App-F per-leg ToFs require resolving the App-F date encoding (last 5 JD digits with a hyphen; reconstruction rule carries a documented ambiguity, note §1). Resolution path: anchor App-F tokens to the App-E clean JD-2440000 integers for the same orbit family, then take consecutive-encounter date differences.

source: Rall 1970 TE-34 App F p.197; App E JD-2440000 anchors (note §1, §7)
vinf_kms_at_encounters[1].vinf_kms [uncertain]
Mars V_inf stored as the midpoint of the per-encounter range 0.252.0.39 EMOS over 6 encounters; the full per-encounter list is in the transcription note §4. A future pass may materialise per-encounter Mars rows.

source: Rall 1970 TE-34 App F p.197 (per-encounter Mars rows, note §4)

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 F (Eccentric Inclined Earth-Mars Periodic Orbits, p.197), orbit M5-1D — a converged eccentric-inclined Model III.B version.

orbit source rall-1970-te34 orbit fidelity analytic-ephemeris V∞ source rall-1970-te34 V∞ fidelity analytic-ephemeris

Notes

Rall 1970 (MIT TE-34, Hollister adv.) free-fall periodic orbit M5-1D, Appendix F eccentric-inclined Model III.B (one of the 11 orbits that converged in the eccentric-inclined model: M4-1a, M5-1a.e, M5-2a.e). These versions a.e are re-phasings (different oppositions) of the same family, not different families (note §6). Units: V_inf in EMOS (= 29.77 km/s per Rall) converted to km/s with EMOS preserved in notes; turn angles in degrees (SOURCED for the first Earth encounter); passing distances in local planetary radii. Mars V_inf varies per encounter; the stored value is the per-encounter range midpoint (full list in the transcription note §4). DATE HAZARD: App-F dates use the last-5-digits-of-JD hyphen encoding whose reconstruction rule carries a documented ambiguity (note §1). Absolute App-F dates and the per-leg ToFs that depend on them are NOT ingested (data_gap); the safe per-encounter V_inf/turn/passing-distance (EMOS/deg/radii) ARE. 1969-era patched-conic mean-element dynamics; modern free-return families supersede the dynamics. Ingested 2026-06-07.

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 F M5-1D (p.197): first Earth V_inf 0.176 EMOS; x 29.77 = 5.24 km/s.
flyby_mechanics[0].turning_angle_deg: Rall 1970 TE-34 App F M5-1D: first Earth encounter turn angle 76.9 deg (App F column 4, degrees).