Parabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is an orbit with the eccentricity equal to 1. Under standard assumptions a body traveling along this trajectory will coast to infinity, arriving there with zero velocity relative to the central body, and therefore will never return. Therefore paraobolic trajectory is a class of escape trajectory and in fact is a minimum-energy escape trajectory.

Velocity

Under standard assumptions the orbital velocity ($v\,$) of a body traveling along parabolic trajctory can be computed as:

$v={\sqrt {2\mu \over {r}}}$

where:

• $r\,$ is radial distance of orbiting body from central body,
• $\mu \,$ is standard gravitational parameter.

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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:

$v={\sqrt {2}}v_{O}$

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:

$r={{h^{2}} \over {\mu }}{{1} \over {1+\cos \theta }}$

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 kind of trajectory takes form:

$\epsilon ={v^{2} \over 2}-{\mu \over {r}}=0$

where:

• $v\,$ is orbital velocity of orbiting body,
• $r\,$ is radial distance of orbiting body from central body,
• $\mu \,$ is standard gravitational parameter.

See also

• Orbit
• Orbital equation