Specific orbital energy

In the gravitational two-body problem, the specific orbital energy ${\displaystyle \varepsilon }$ (or specific vis-viva energy) of two orbiting bodies is the constant quotient of their mechanical energy (the sum of their mutual potential energy, ${\displaystyle \varepsilon _{p}}$, and their kinetic energy, ${\displaystyle \varepsilon _{k}}$) to their reduced mass.

According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time: $${\displaystyle {\begin{aligned}\varepsilon &=\varepsilon _{k}+\varepsilon _{p}\\&={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=-{\frac {1}{2}}{\frac {\mu ^{2}}{h^{2}}}\left(1-e^{2}\right)=-{\frac {\mu }{2a}}\end{aligned}}}$$ where

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

It is a kind of specific energy, typically expressed in units of ${\displaystyle {\frac {\text{MJ}}{\text{kg}}}}$ (megajoule per kilogram) or ${\displaystyle {\frac {{\text{km}}^{2}}{{\text{s}}^{2}}}}$ (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.