Keplerian number-theory

If you are of a mathematical bent, the first thing you'll likely notice about the Earth-Venus Hohmann orbit is that it looks, also, resonant with the already-near-resonant Earth:Venus orbits. Namely, 8:13 with 10 shuttling in between. I wonder what other orbital resonances a:c where a and c are natural numbers (or near-enough) can have natural Hohmann b between them.

At stake here is how we can float two satellites in a high orbit and a low orbit, with some service to both - without stopping. Ballistic cyclers, baby! Not every satellite-pair lets you have one. Earth/Mars doesn't - unless you're like McConaghy and Longuski, a genius (S1L1).

There are a couple of equations we're dealing with. One is Bronze Age aphelion-to-perihelion. That's from the highest height at one end (say, Earth); to its other height at its other end, opposite (Venus). Add these values together and that is semi-major axis, doubled. Diameter if a circle which obviously Hohmann is not. Major Axis, perhaps.

The other equation is Kepler's Ratio of time² and semimajor³. We are all sharing the same centre of orbit and Newton's gravitational constant - so this ratio is unity. Time will be based on the innermost body in orbit, here Venus. So Earth will be 8/13 and Hohmann, 10/13, after a single Venus year.

What I get are (8/13)^-2/3 + 1 against 2 * (10/13)^-2/3. In C#: 0.25 * Math.Pow(169, 1.0/3) - 2 * Math.Pow(1.69, 1.0/3) + 1. This is -8.3E-05.

I was doing double-precision, not 32-bit "float". This is not Microsoft failing to do a zero. Too bad for number-theorists: if Kepler does have a Ratio to beat out Pythagoras' 3-4-5, 8-10-13 ain't it. Too bad for orbital planners also. But orbital planners between Venus and Earth were already having to put up with those orbits not being perfect 8:13. They were already pondering solar sails (and, in my case, different cyclers). An error below the myriadth range is zero enough, for them.

To be looked for: natural {a,b,c} such that a<b<c and (c/a)^2/3 + 1 - 2 * (c/b)^2/3 between (-0.0001 , 0.0001). If anyone can surpass {8, 10, 13}, tell me.