Derivation
The total orbital energy is the sum of the shared potential energy, the kinetic energy of body M, and the kinetic energy of body m
- is the speed of body M relative to the inertial center of mass of the two bodies.
- is the speed of body m relative to the inertial center of mass of the two bodies.
The orbital energy can also be calculated using only relative quantities
- is the relative speed of the two bodies
- is the reduced mass
For elliptic and circular orbits, the total energy is given more concisely by
- is the semi-major axis.
Dividing the total energy by the reduced mass gives the vis-viva energy, more commonly known in modern times as the specific orbital energy
For elliptic and circular orbits
Equating the two previous expressions and solving for v yields the vis viva equation:
Read more about this topic: Vis-viva Equation