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.
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:
corresponds to the extrapolated time of the fictitious starting at the center of the central body.
At any time the average speed from is 1.5 times the current speed, i.e. 1.5 times the local escape velocity.
To have 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.
First, he found that the orbits of the planets in our solar system are elliptical, not circular (or epicyclic), as had previously been believed, and that the sun is not located at the center of the orbits, but rather at one focus.
As two objects orbit each other, the periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest from each other.
An open orbit has the shape of a hyperbola (when the velocity is greater than the escape velocity), or a parabola (when the velocity is exactly the escape velocity).
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