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Concept
Parabolic trajectory
Summary
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 parabolic trajectory can be computed as:
where:
is the radial distance of orbiting body from central body,
is the standard gravitational parameter.
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:
is orbital velocity of a body in circular orbit.
For a body moving along this kind of trajectory an orbital equation becomes:
where:
is radial distance of orbiting body from central body,
is specific angular momentum of the orbiting body,
is a true anomaly of the orbiting body,
is the standard gravitational parameter.
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:
is orbital velocity of orbiting body,
is radial distance of orbiting body from central body,
is the standard gravitational parameter.
Official source
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