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In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an escape orbit, otherwise a capture orbit. It is also sometimes referred to as a C3 = 0 orbit (see Characteristic energy).
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-energy hyperbolic trajectories from negative-energy elliptic orbits.
The orbital velocity () of a body travelling along a parabolic trajectory can be computed as:
where:
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:
where:
For a body moving along this kind of trajectory the orbital equation is:
where:
Under standard assumptions, the specific orbital energy () of a parabolic trajectory is zero, so the orbital energy conservation equation for this trajectory takes the form:
where:
This is entirely equivalent to the characteristic energy (square of the speed at infinity) being 0:
Barker's equation relates the time of flight to the true anomaly of a parabolic trajectory:[1]
where:
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
then
With hyperbolic functions the solution can be also expressed as:[2]
where
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:
where
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.