A sample hyperbolic trajectory of an asteroid that makes a tangential collision with the planet Jupiter, shown as the semicircle of unit radius centered on the origin. The straight line is the asymptote of the incoming hyperbola as defined by the velocity at infinity and the parameter . In this example , , , , and the asymptote intersects the axis of symmetry at . All dimensions in this diagram are in units of the planet’s radius.
A family of four encounter trajectories to the planet Jupiter for . The four approach asymptotes that are parallel and in the same plane, with , 6.30, 4.20, and 2.10, respectively, from right to left. The left-most trajectory results in a collision with the planet, but is drawn as though the entire mass of the planet were concentrated at its center. All dimensions in the diagram are in units of the planet’s radius as in Fig. 1. Note that the axes of symmetry of the several trajectories rotate counterclockwise as increases.
Data for several planets and calculated values of .
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