Specific Orbital Energy - Equation Forms For Different Orbits

Equation Forms For Different Orbits

For an elliptical orbit, the specific orbital energy equation, when combined with conservation of specific angular momentum at one of the orbit's apsides, simplifies to:

where

• is the standard gravitational parameter;
• is semi-major axis of the orbit.

Proof:

For an elliptical orbit with specific angular momentum h given by

we use the general form of the specific orbital energy equation,

with the relation that the relative velocity at periapsis is

Thus our specific orbital energy equation becomes

and finally with the last simplification we obtain:

For a parabolic orbit this equation simplifies to

For a hyperbolic trajectory this specific orbital energy is either given by

or the same as for an ellipse, depending on the convention for the sign of a.

In this case the specific orbital energy is also referred to as characteristic energy (or ) and is equal to the excess specific energy compared to that for a parabolic orbit.

It is related to the hyperbolic excess velocity (the orbital velocity at infinity) by

It is relevant for interplanetary missions.

Thus, if orbital position vector and orbital velocity vector are known at one position, and is known, then the energy can be computed and from that, for any other position, the orbital speed.