Vis-viva equation

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In astrodynamics, the vis viva equation, also referred to as orbital energy conservation equation, is one of the fundamental and useful equations that govern the motion of orbiting bodies.

It is the direct result of the law of conservation of energy, where the sum of kinetic and potential energy is constant as a satellite moves about its orbit.

Vis viva (Latin for "live force") is a term from the history of mechanics, and it survives in this sole context. It represents the principle that the difference between the aggregate 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.

[edit] Vis viva equation

For any two-body motion (elliptic orbits, parabolic trajectories, and hyperbolic trajectories), the vis viva equation is as follows:

$v^2=\mu\left({{2 \over{r}} - {1 \over{a}}}\right)$

where:

$v\,\!$ is the velocity of orbiting body
$r\,\!$ is the distance from orbit's focus
$a\,\!$ is the semi-major axis ( $a > 0$ for ellipses, $a=\infty$ or $\frac{1}{a} = 0$ for parabolas, and $a < 0$ for hyperbolas)
$\mu\,\!$ is the gravitational constant times the mass of the body the satellite is orbiting, $G M\,$ . This is also known as the standard gravitational parameter

[edit] Derivation

Let m be the mass of the satellite. The total energy of the satellite in its orbit is the sum of its kinetic and potential energies:

$E = \frac{1}{2} m v^2 + \frac{- GM m}{r}.$

For orbits that are circular or elliptical, the total energy is also given by

$E = \frac{-G M m}{2 a},$

where a is the semi-major axis of the ellipse (or the radius, in the case of a circle). Equating these two expressions for the energy and then moving kinetic energy to one side,

$\begin{align} \frac{1}{2} m v^2 & = \frac{G M m}{r} + \frac{-G M m}{2 a} \\ v^2 & = \frac{2}{m} \left( \frac{G M m}{r} + \frac{-G M m}{2 a} \right). \end{align}$