Specific orbital energy

In the gravitational two-body problem, the specific orbital energy $\varepsilon$ (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy ( $\varepsilon _{p}$ ) and their total kinetic energy ( $\varepsilon _{k}$ ), 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:

{\begin{aligned}\varepsilon &=\varepsilon _{k}+\varepsilon _{p}\\&={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=-{\frac {1}{2}}{\frac {\mu ^{2}}{h^{2}}}\left(1-e^{2}\right)=-{\frac {\mu }{2a}}\end{aligned}}

where

$v$ is the relative orbital speed;
$r$ is the orbital distance between the bodies;
$\mu ={G}(m_{1}+m_{2})$ is the sum of the standard gravitational parameters of the bodies;
$h$ is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass;
$e$ is the orbital eccentricity;
$a$ is the semi-major axis.

It is typically expressed in ${\frac {\text{MJ}}{\text{kg}}}$ (megajoule per kilogram) or ${\frac {{\text{km}}^{2}}{{\text{s}}^{2}}}$ (squared kilometer per squared second). 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.

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