Jacobi's escape-velocity

I found a constant over the weekend, which I may as well backdate to "today" on account all I can really do is summarise it. This is a constant in a three-body problem, involving escape energy for the third body.

In a two-body system where one body exerts all the gravity, the second body has an Escape Velocity. Depending upon where the second body starts - usually the surface, so 6000-some kilometers from the barycentre - a given Delta-V will get that body off the system entirely. This doesn't depend upon the second body's mass. It's just a speed; plus the angle, which is the right-angle to the planet. For better results we prefer Vinf but still: a constant.

How nice if you are in your own orbit around the Sun. Over Earth, however, arise some complications. Most bodies we care about are the other planets, in our ecliptic. To these... something rolls in our way: our Moon.

This 19th-century mathematician Jacobi turned Escape Velocity around. In a three-body system, with that third body being the rocket, is a total energy. Say the rocket drifts in a cislunar ellipse around the Earth without propellant. At any point in its trajectory, the constant is constant: say, -1.7. It'll be there forever.

Yes that's a minus-sign. This explains it: much like I_sp measured in seconds but really a forceratio times seconds; we have here Ω - v². Or, it's negative H². Anyway we can measure it with [minus-](m/s)² not so different from Escape Velocity; the Ω massratio is a ratio of the two major bodies, such that the third - craft's - mass doesn't figure.

So: let's dock with the hulk as to transfer some propellant to it, or just to give it a push. The craft remains stuck down here until/unless -1.6 is reached. At this point the rocket can (potentially) get across the TLL1 boundary, to enter a Lunar orbit. This still isn't escape-velocity for the whole system but maybe we WANT to-the-Moon (...Alice). Over there we might request more propellant (or a cushion) not to rake regolith; or else to escape a new permanent orbit around that other body - but hey, progress.

Over the -1.5's we can get to the moon and swing by it, skating close to L1 thence, on the farside, L2. In practice we might open more of L2's region by heisting some Lunar momentum to assist us to those other planets; but this isn't in Jacobi's scope. In the -1.5's something else interesting happens: although the potential energy exists to run through that L1-Moon-L2 keyhole, it won't get us to any other Libration like - say - TLL4/L5 Trojans. L3 opens up at -1.51ish and we only get to L4/5 slightly over -1.50. The (thick) walls around our range are Earth's "Hill’s Region [of constant J]"; not to confuse with the Moon's Hill Sphere relative to Earth which is just L1/L2.

UNITS 12/25: Gereshes wins on account I was able - on this day of our Lord's birth, not 20 November nor 10 December - to get my code to work. So: never mind the orbital-mechanics.space negative values, which values I couldn't verify. Instead the values will be circling around the number three (3). Happy Hanukkah!