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.