Under standard assumptions a body traveling along an escape orbit will coast along a parabolic trajectory to infinity, with velocity relative to the central body tending to zero, and therefore will never return. Parabolic trajectories are minimum-energy escape trajectories, separating positive-energyhyperbolic trajectories from negative-energy elliptic orbits.
The orbital velocity () of a body travelling along parabolic trajectory can be computed as:
At any position the orbiting body has the escape velocity for that position.
If a body has an escape velocity with respect to the Earth, this is not enough to escape the Solar System, so near the Earth the orbit resembles a parabola, but further away it bends into an elliptical orbit around the Sun.
This velocity () is closely related to the orbital velocity of a body in a circular orbit of the radius equal to the radial position of orbiting body on the parabolic trajectory:
More generally, the time between any two points on an orbit is
Alternately, the equation can be expressed in terms of periapsis distance, in a parabolic orbit :
Unlike Kepler's equation, which is used to solve for true anomalies in elliptical and hyperbolic trajectories, the true anomaly in Barker's equation can be solved directly for . If the following substitutions are made
With hyperbolic functions the solution can be also expressed as:
Radial parabolic trajectoryedit
A radial parabolic trajectory is a non-periodic trajectory on a straight line where the relative velocity of the two objects is always the escape velocity. There are two cases: the bodies move away from each other or towards each other.
There is a rather simple expression for the position as function of time:
corresponds to the extrapolated time of the fictitious starting or ending 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.