Amáltheia to Io (with maths)

Let's Jacobi that Amáltheia outbound. Our units are Io's, pretending that the other moons and Jove's own tides aren't messing this up for us. And I'm pretending the small moon orbits circle-y. Wasting no time:

var xy = new Vector2() { X = 181300F / 421600F, Y = 0F };
var muSmall = 0.014960 / 317.8; //neglect 1.1988e-06
var dxy = new Vector2() {
  X = 0F,
  Y = 2F * (float)(Math.PI * (Math.Sqrt(1.0 / xy.X) - xy.X))
};

J is -42.492. Check out the reciprocal-root! But I am in no hurry.

Meanwhile let's look at JIL1. This system being Laplacian, there really isn't one... instead I'll use direct-reports of Io's Hill-Sphere. This is 5.8 Io's radius.

var xy = new Vector2() { X = 1F - 5.8F * 1821.6F / 421600F, Y = 0F };
var dxy = Vector2.Zero;

Target J is 3.00557659591. Looks about par for these. Amáltheia by contrast is not par with the SV-Hilda; it looks more like a Mercury, deep down there.