Specific orbital energy

In the gravitational two-body problem, the specific orbital energy (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy () and their total kinetic energy (), divided by the reduced mass. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time: where is the relative orbital speed; is the orbital distance between the bodies; is the sum of the standard gravitational parameters of the bodies; is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass; is the orbital eccentricity; is the semi-major axis. It is expressed in or . For an elliptic orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity (parabolic orbit). For a hyperbolic orbit, it is equal to the excess energy compared to that of a parabolic orbit. In this case the specific orbital energy is also referred to as characteristic energy. For an elliptic 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. 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.