Jacobi at TLL3

The general Jacobi formula is centripetal potential + gravitational potential - velocity-squared. My aim is to universalise this for any generic system where the semimajor and the period are unit. The outer planet is pinned to that, so: eccentricity zero.

Per wiki n²(x²+y²) + 2(μ₁/r₁ + μ₂/r₂) - v². Here the μ's are gravitational-parameters, or GMs. There exists a sidereal version; we'd replace that first term with 2n(ξδη+ηδξ).

If we take the frame-of-reference to unit period and unit astro'units, μ₁ = 4π²; and μ₂ is a fraction of that. Further: let's y to zero, and at our planet x can approximate 1 on one side or the other. 4π² = 39.4784176. n=2π/T.

I am curious about the Jacobi when the craft is at a Libration focus. At the planet's orbit, which means the planet or else Libration focus, v is of course semimajor/year. For the planet or L3-5 that's the unit circumference 2π. For the planet and L3, x²=1. For the corotating Cartesian n² is (also) 4π²; for sidereal ξδη+ηδξ = 2π at the planet or else at L3, tangent to the orbit (so up or down). So the first and third terms cancel in corotation.

On to gravitational potential. μ₂ for Terra/Luna is 0.4856.

Now we'd need x as L*'s distance from the Earth in system units. At L3, r₁=1 but x=-1, r₂=2(ish).

The constant seems high to me - and nonnegative.

Okay let's try Wolfram which simplifies 2Ω, where Ω is a function of the "μ" which is the Moon's fraction of the Earth-Moon total mass = 0.01215. Distance-squared minus velocity-squared is 1-4π². The middle term here is 2([1-μ]/r₁ + μ/r₂); at L3, 2[1-μ] + μ = 2-μ. Dominated by the negative velocity-squared.

12/25: Skroo dis my units be bunk. Gereshes has some real examples. Let's follow along, System.Numerics in C#:

var xy = new Vector2() { X = .1F, Y = .491F };
var dxy = new Vector2() { X = -.7F, Y = .7F };
var sat = new Vector2() { X = 1F, Y = 0F };
var muSmall = 0.012 / 1.012;
var twoOmega = xy.LengthSquared() 
  + 2 * ((1-muSmall) / (xy - Vector2.Zero).Length() + muSmall / (xy - sat).Length());
Console.Write(twoOmega - dxy.LengthSquared());
Console.ReadLine();

Gereshes claimed 3.2178; I got 3.23(etc). I'm calling that he's misrounded his values especially v² which he's made 1. So I win, and the distance-measure is the local AU. J=3.1883 is where Gereshes says TLL1 opens up but I'll need to doublecheck this.

Velocity poses more Problems, based on some time-unit I don't know. I want it to be the period, here sidereal month. So the velocity of the Moon would be 2π Moon-AU per sidereal-month. I get expected 3.18 Jacobi at {0.99,0} AU only approaching {-1.495,0}; more than that, J gets too low. That last looks like half-pi. To that: in our frame of reference, the Moon's velocity is actually... zero. A body smack in a Libration point would also be zero. So something beneath GEO would indeed appear to orbit as it does "in real life" relative to Earth's rotation: faster. Also faster than our Moon orbits.

So - on assumption Wiki is right that T/L L1 be 1-.1509 from the barycentre:

var xy = new Vector2() { X = 1F - .1509F, Y = 0F };
var dxy = new Vector2() { X = 0F, Y = 0F };
var sat = new Vector2() { X = 1F, Y = 0F };
var muSmall = 0.012 / 1.012;

J=3.2056. For L3 xy.X is -38168 / 38440 so J=2.988. þe lower the J at these keyholes þe more the energy, from the inner region(s), needed to get to the point - therefore, to escape out.