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# Parabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is an orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit.
Under standard assumptions a body traveling along an escape orbit will coast to infinity, with velocity relative to the central body tending to zero, and therefore will never return. Parabolic trajectory is a minimum-energy escape trajectory.

## Velocity

Under standard assumptions the orbital velocity ($v,$) of a body traveling along parabolic trajectory can be computed as:
$v=sqrt\left\{2muover\left\{r\right\}\right\}$
where:
• $r,!$ is radial distance of orbiting body from central body,
• $mu,!$ is standard gravitational parameter.

At any position the orbiting body has the escape velocity for that position.
If the body has the 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 ($v,$) 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:
$v=sqrt\left\{2\right\}cdot v_O$
where:
• $v_O,!$ is orbital velocity of a body in circular orbit.

## Equation of motion

Under standard assumptions, for a body moving along this kind of trajectory an orbital equation becomes:
$r=$
where:
• $r,$ is radial distance of orbiting body from central body,
• $h,$ is specific angular momentum of the orbiting body,
• $theta,$ is a true anomaly of the orbiting body,
• $mu,$ is standard gravitational parameter.

## Energy

Under standard assumptions, specific orbital energy ($epsilon,$) of parabolic trajectory is zero, so the orbital energy conservation equation for this trajectory takes form:
$epsilon=\left\{v^2over2\right\}-\left\{muover\left\{r\right\}\right\}=0$
where:
• $v,$ is orbital velocity of orbiting body,
• $r,$ is radial distance of orbiting body from central body,
• $mu,$ is standard gravitational parameter.