Lambert 1, 2, 3, 4

Last Friday night (properly, Saturday morn) we looked at Lambert problems for the 13/5 Earth synod in Venus years. Since here (unlike Mars) we get a Hohmann at metonic n=5, which is awesome, I am interested in the nonideal solutions where n is 1, 2, 3, or 4. These are such Earth trajectories as might return to Venus before the metonic runs out. Maybe the Earth-Venus stretch is even under the 146 day Hohmann; maybe that's the Venus-Earth stretch, it just cannot be both.

Note that with n=2, double-boarding from Venus is possible as with the Earth-Mars S1L1. This can lower delta-V: lowering my initial launch cost and, on its cycles, how far I must dip into either planet's well. For whatever best-case scenario we might lower launch-cost further from the inner planet's L2 or maybe the outer's L1.

First: uh, what's a Lambert. Johann Heinrich Lambert found in 1776 that time-of-flight aka delta-T, for a trajectory in-system, depends only on its semimajor axis, a few constants, and some vicious mathematics. Per Matthew Peet's pdf: the equation starts with Math.Sqrt(Math.Pow(a,3.0)/μ * (α-β - (Math.Sin(α) - Math.Sin(β)))). Nobody solves Lambert directly. Peet has a graph on slide 19 (of 30), or use Gooding's code. McConaghy et al. are looking to find semi-major or period (by Kepler, the one is had from th'other) such that delta-T is a synod. Best I could do was to get the velocity vectors at r1 and r2. Unfortunately that was enough for my project.

Maybe around 7:00 this morning before work (yeah, I still do that) I tested the solver for n=5 and r=5, which is the five-synod run with five returns. When I got initial velocity near identical to final velocity, I declared victory on my code.

With that, this week, I am evaluating results for the one- and two-synod periods, five revolutions right to five revs left.

V-E DAY 3/3: Well, I'd kicked this post down the road to 1 March and its posting was premature even then. I'd only got aphelion (still progress!); and r=5 was always dumb for Venus' 72°. Real victory was achieved at, like, 6:30 AM, when I wrote a C# Forms app for Earth/Mars and checked it against Longuski-McConaghy's table. And for V/E n=5 and r=10. Let it stand as a promissary note. UPDATE 3/4: promise kept - finally. Only having weeknights and 6 AM wakeups does take a toll.