Surface structure of an invariant manifold of a halo orbit

We extract the surface structure of the unstable invariant manifold tube projected into position space, of a halo orbit near L ₂ in the circular restricted three body model. We do this by using transversal planes to intersect trajectories that approximate the tube. From these intersection points we construct spline-interpolated cross section curves which give a good idea of the structure of the tube. For example, we show that, for the value of μ we use, the tube pinches, develops a self-intersection, develops loop-inside-tube structure, pinches some more, and so on. We also construct surfaces made of quadrilaterals and triangles from these cross-sections. The transversal planes are obtained by taking planes orthogonal to a curve that follows the general shape of the tube. One such curve we use, is the unstable invariant manifold of the equilibrium point K ₂ itself. In another example, we take a circle that follows the tube, as the curve for finding planes transversal to the tube. We also show that tubes of different energies, that start out in certain ordering, do not obey the ordering after a while. Our method is complementary to the method of taking cross-sections of constant time (the isochronous method), as used by some other researchers. The isochronous method is good at revealing the temporal structure of trajectories on a tube. However, due to the unequal speeds of different trajectories, it is harder to use for long length surface extraction. In contrast, using our method, we show cross-sections of the tube through an angular extent of nearly n during which the tube becomes extremely convoluted. Our work is motivated by applications to space mission design.