Vis-viva Equation - Derivation

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:

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