Escape orbit

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An escape orbit (also known as C3 = 0 orbit) is the high-energy parabolic orbit around the central body. A body in this orbit has at each position the escape velocity with respect to this central body, for this position. If this energy were further increased the orbit would turn to a hyperbolic trajectory.

[edit] Position as function of time

Finding the position as function of time corresponds to solving a differential equation. In the theoretical case of a straight escape trajectory there is a rather simple expression for the solution:

$r=(4.5\mu t^2)^{1/3}\!\,$

where

μ is the standard gravitational parameter
$t=0\!\,$ corresponds to the extrapolated time of the fictitious starting at the center of the central body.

At any time the average speed from $t=0\!\,$ is 1.5 times the current speed, i.e. 1.5 times the local escape velocity.

To have $t=0\!\,$ at the surface, apply a time shift; for the Earth (and any other spherically symmetric body with the same average density) as central body this time shift is 6 minutes and 20 seconds; seven of these periods later the height above the surface is three times the radius, etc.