Parabolic trajectory

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.

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.

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.

) 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

of a parabolic trajectory:[1] where: More generally, the time (epoch) 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.

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.

The green path in this image is an example of a parabolic trajectory.
A parabolic trajectory is depicted in the bottom-left quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the parabolic trajectory is shown in red. The height of the kinetic energy decreases asymptotically toward zero as the speed decreases and distance increases according to Kepler's laws.