01
Dec
05

Halo Orbit

In 1772, Joseph Lagrange showed that a small satelite moving in the Earth-Moon system can have 5 balance points, called Lagrange points. Here is a small halo orbit around Lagrange point L4. This is interesting because L4 is off to the side, so it looks like the satelite is orbiting nothing! The velocity components have been represented by the third axis and color.
(* runtime = 6 seconds *)
Clear[x, y, t, p, v]; mu = 0.0123001; x1 = -mu; x2 = 1 - mu; r1 = Sqrt[(x - x1)^2 + y^2]; r2 = Sqrt[(x - x2)^2 + y^2];
{x0, y0} = {x, y} /. Last[NSolve[{-x == -x2(x - x1)/r1^3 + x1(x - x2)/r2^3, -y == -(x2/r1^3 - x1/r2^3) y}, {x, y}]];
JacobianMatrix[p_, q_] := Outer[D, p, q];
vlist = Eigenvectors[JacobianMatrix[{xdot, -x2(x - x1)/r1^3 + x1(x - x2)/r2^3 + 2ydot + x, ydot, -(x2/r1^3 - x1/r2^3)y - 2xdot + y}, {x, xdot, y, ydot}] /. {x -> x0, xdot -> 0, y -> y0, ydot -> 0}];
r1 = Sqrt[(x[t] - x1)^2 + y[t]^2]; r2 = Sqrt[(x[t] - x2)^2 + y[t]^2];
tmax = 355; p0 = {x0, y0, 0, 0} + Re[0.5 vlist[[1]] + vlist[[3]]]/3.84400*^5;
p[t_] = {x[t], y[t]} /. NDSolve[{x''[t] - 2y'[t] - x[t] == -x2(x[t] - x1)/r1^3 + x1(x[t] - x2)/r2^3, y''[t] + 2x'[t] - y[t] == -(x2/r1^3 - x1/r2^3)y[t], x[0] == p0[[1]], y[0] == p0[[2]], x'[0] == p0[[3]], y'[0] == p0[[4]]}, {x[t], y[t]}, {t, 0, tmax}, MaxSteps -> 100000, AccuracyGoal -> 10][[1]];
v[t_] = D[p[t], t];
ParametricPlot3D[Append[p[t], v[t][[1]]], {t, 0, tmax}, Compiled -> False, PlotPoints -> 1000, BoxRatios -> {1, 1, 1}, ViewPoint -> {0, 10, 0}]

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