Specific orbital energy
In the gravitational two-body problem, the specific orbital energy ε {\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, ε p {\displaystyle \varepsilon _{p}} , and their kinetic energy, ε k {\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: ε = ε k + ε p = v 2 2 − μ r = − 1 2 μ 2 h 2 ( 1 − e 2 ) = − μ 2 a {\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 v {\displaystyle v} is the relative orbital speed; r {\displaystyle r} is the orbital distance between the bodies; μ = G ( m 1 + m 2 ) {\displaystyle \mu ={G}(m_{1}+m_{2})} is the sum of the standard gravitational parameters of the bodies; h {\displaystyle h} is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass; e {\displaystyle e} is the orbital eccentricity; a {\displaystyle a} is the semi-major axis.