In astrodynamics, the vis-viva equation, also referred to as orbital-energy-invariance law, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which applies when the only force acting on an object is its own weight. Vis viva is a term from the history of mechanics, and it survives in this sole context. It represents the principle that the difference between the total work of the accelerating forces of a system and that of the retarding forces is equal to one half the vis viva accumulated or lost in the system while the work is being done.
Equation
For any Keplerian orbit, the vis-viva equation is as follows: where:
In the vis-viva equation the mass m of the orbiting body is taken to be negligible in comparison to the mass M of the central body. The central body and orbiting body are also often referred to as the primary and a particle respectively. In the specific cases of an elliptical or circular orbit, the vis-viva equation may be readily derived from conservation of energy and momentum. Specific total energy is constant throughout the orbit. Thus, using the subscripts a and p to denote apoapsis and periapsis, respectively, Rearranging, Recalling that for an elliptical orbit the velocity and radius vectors are perpendicular at apoapsis and periapsis, conservation of angular momentum requires specific angular momentum, thus : Isolating the kinetic energy at apoapsis and simplifying, From the geometry of an ellipse, where a is the length of the semimajor axis. Thus, Substituting this into our original expression for specific orbital energy, Thus, and the vis-viva equation may be written or Therefore, the conserved angular momentum L = mh can be derived using and, where a is semi-major axis and b is semi-minor axis of the elliptical orbit, as follows - and alternately, Therefore, specific angular momentum, and Total angular momentum
Practical applications
Given the total mass and the scalars r and v at a single point of the orbit, one can compute r and v at any other point in the orbit. Given the total mass and the scalars r and v at a single point of the orbit, one can compute the specific orbital energy, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being "suborbital", having enough energy to be "orbital", but without the possibility to complete a full orbit anyway because it eventually collides with the other body, or having enough energy to come from and/or go to infinity. The formula for escape velocity can be obtained from the Vis-viva equation by taking the limit as approaches :
