Backflip

Back to the h>3 thing: let's look at the paper which introduced h to the cycler calculus. Let us read Byrne et al. 2002 (pdf).

This is contemporary with McConaghy et al.. In fact it's the same trio doing these papers at about the same time.

They are looking at the two-synod space, with of course one symmetric return such that "symmetric return" is assumed. So, these are 2-h-1-i and they start with 2L3 which will be 2-0-1-4. This, they dislike for its V_∞ and turn-ratio, and for only making it to 1.51 AU which means half the time the "cycler" isn't cycling to Mars. Here is where they bring in h, for 2-1-1-5 and 2-3-1-5.

h is from CW Uphoff 1989: the Back-Flip. The idea is that the cycler left Earth and passed Mars, and then goes around the Sun for a synod (15/7 years), and then part of a second one. After 2 11/14 of its own orbits, which is 3 11/14 Earth years, it comes back to Earth. But a half year early: Earth is 180° (π radians) behind its, what, +103° at two-synod end. The 2-1-1-5, we can already assume, is coming at a slight angle to the ecliptic. The cycler here positions itself that it changes its orbit to shadow ours. It is not dragging itself at a statite above Earth. It's now going at an inclined orbit "above" or "below" (depending on frame-of-reference). It then meets Earth once more, a half year later. Earth can shoot it off to Mars again. Synod III, cycler repeats.

Although the turnratio is right for 2-1-1-5, it's like unlamented 2-0-1-4 in one important respect... it only gets to 1.45 AU. That's even worse. After all that dangerous mucking about in Earth low orbit, changes are less than a coin-flip they'll be back to Mars again.

2-3-1-5 means it gets back to Earth earlier still: 1 11/14 orbits, 2 11/14 years. Then comes Uphoff's backflip. Or, a full-rev loop - basically, any orbit of semimajor 1 AU. Either order will do, as long as the half-years add to 3. Its problem is that it's lost turnratio again - it's not ballistic. Although it's in the right ballpark, as it were. Oberth could slug this cycler out again.

Russell and Ocampo note 3-1-1-17 (3 revs) as the sweet spot, if no-one minds a three-synod span with lots of downtime and an extra cycler-pair.