Venerean Hilda's Jacobi

The dubs won around here last weekend (excepting the Texans') started with Laplace's longitudes which had troubled me since, I kid ye not, 2007; when I had a Newton simulator sucking planets into their own sun. That afternoon I figured Jacobi so could apply it for Venus. 'Twas a good couple days. Today I shall apply Jacobi to Venus' customer SV-Hilda.

SV-Hilda shall take Venerean AU. At perihelion she is Y=0. Since any Hilda period is by definition 3:2 we mark that 2/3. Kepler's ratio is 1 in these units so semimajor is Math.Pow(2.0/3.0, 2.0/3.0) which is 0.763143 Venus AU so 0.55175 ours.

I'm doing an ellipse, on account Jupiter's real Hilda likes that. The main constraint I saw in 2020, so now, is Mercury: that aphelion is 0.466697 Earth AU with Solar L2 [+.2roundup] 0.49. 0.49 peri' is Venereal 0.6777 AU. In the planet's direction so X=+0.6777.

Next up: velocity, from that same +0.6777 done on the Y axis only, following Venus also in the positive direction. (Our Hilda's eccentricity maxes 0.1119.) Velocity = √(μ[2/r-1/a]); at r=a=1, v=2π AU per Venereal year; so unit μ=4π². Now we run with r = xy.X:

var xy = new Vector2() { X = 0.6777F, Y = 0F };
var muSmall = 0.815 / 332946;
var dxy = new Vector2() {
  X = 0F,
  Y = 2F * (float)(Math.PI * (Math.Sqrt(2.0 / xy.X - 1.0 / (float)Math.Pow(2.0 / 3.0, 2.0 / 3.0)))) //it'll be fffiiinne
};

J is -61.365... whuuttt. My speed must be too high. Unless it is not . . .

But! the whole frame-of-reference at Venus runs 2π. Y was real, in Venus' units; let it be situational, in Venus' frame. The circle at r (Hilda peri') is 2π x 0.6777 so, is doing that per year. 2F * (float)(Math.PI * (Math.Sqrt(2.0 / xy.X - 1.0 / (float)Math.Pow(2.0 / 3.0, 2.0 / 3.0)) - xy.X)) and J is -10.955.

This is still (magnitudinally) a lot - but we somewhat feared it would be. The need is for a delta-V to get from -10.955 to SVL1 (or L3) at just over 3.