Lambert's boomerang

On topic of cyclers I am revisiting James M. Longuski 2002 (pdf). For this purpose I am not looking at a permanent cycler - just a trajectory where Planet A boosts cargo that goes by B and comes back.

Longuski starts with these six axiomata, which I abstract out thusly:

1. The [A-B] synodic period S is [some rational fraction of A’s sidereal years].
2. [A]’s orbit, [B’s] orbit, and the cycler trajectory lie in the ecliptic plane.
3. [A] and [B] have circular orbits.
4. The cycler trajectory is conic and prograde (direct).
5. Only [A] has sufficient mass to provide gravity-assist maneuvers.
6. Gravity-assist maneuvers occur instantaneously.

For Earth-Venus, the S fraction is 13/8 Earth years. For Venus years, we need 583.92/224.701 = 2.59865; 13/5 would do it.

Axiom 2 gave Longuski a two-dimensional system, which for Venus and Earth happens only once every 243 Julian years. (For Earth and Mars, his third assumption comes close to killing his project...)

Venus' mass at Venus Express's 250 km is inferior to Earth's at that height. I read #5 as "only Venus has nothing we care about crashing a large artificial asteroid into". If we are using Venus for a gravity-well then my beloved SVL2, although also hitting ecliptic in halo, is a million km high at a right angle to Venus' orbit: pretty darn safe, the safer the closer I huddle Lissajous.

Per Longuski, This is a Lambert problem. Given [natural] n, we want to find a solution R(t) to the two-body problem that connects R₁ = (1, 0) to R₂ = (cos(2πnS), sin(2πnS)) in a time of flight T = nS. R₂ although an oversimplification for E-M is not so bad for V-E.

Each n - number of synods - has multiple solutions. I expect the venerable Hollister cycler for a n=1. And here any astrologer can tell you that n=5 is, exactly, that metonic with the Hohmann orbits: 13 Venus years, 8 Earth years wherein Earth can expect a return on its fourth year. For that what McConaghy and Longuski say about Earth-Mars is equally applicable to Venus-Earth:

Since taxi spacecraft must rendezvous with the cycler spacecraft as it passes A and B, we want the A V_∞ and the B V_∞ to be as small as possible. This typically rules out trajectories with a small number of revs (r) per repeat interval. The orbit that achieves the lowest possible sum of V_∞ at A and V_∞ at B is the Hohmann transfer orbit.

You can see why Hop David loves this one. Some course-correction is needed, so from Earth its first run by Venus has to run into the well; same with its next go 'round Earth. Not, though, notably far down the well; I think even Earth's geostationary satellites might be safe, and Venus just has statites which float anywhere.

SHORTFALL 3/4: I can follow their work now. For circular orbits (admittedly not the case for Mars) I dispute only aphelion. I target outer planet's L1 as if it were a planet too.