Vis-viva equation

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, which requires that the sum of kinetic and potential energy be constant at all points along the 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.

Vis viva equation

For any Kepler orbit (elliptic, parabolic, hyperbolic or radial), the vis viva equation is as follows:

$v^{2}=G(M\!+\!m)\left({{2 \over {r}}-{1 \over {a}}}\right)$

where:

• $v\,\!$ is the relative speed of the two bodies
• $r\,\!$ is the distance between the two bodies
• $a\,\!$ is the semi-major axis (a>0 for ellipses, $a=\infty$ or ${\tfrac {1}{a}}$=0 for parabolas, and a<0 for hyperbolas)
• $G\,\!$ is the gravitational constant
• $M,m\,\!$ are the masses of the two bodies

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

$E={\frac {-GMm}{r}}+{\frac {Mv_{M}^{2}}{2}}+{\frac {mv_{m}^{2}}{2}}$

• $v_{M}\,\!$ is the speed of body M relative to the inertial center of mass of the two bodies.
• $v_{m}\,\!$ 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

$E={\frac {-GMm}{r}}+\mu {\frac {v^{2}}{2}}$

• $v\,\!$ is the relative speed of the two bodies
• $\mu ={\frac {Mm}{M\!+\!m}}$ is the reduced mass

For elliptic and circular orbits, the total energy is given more concisely by

$E={\frac {-GMm}{2a}},$

• $a\,\!$ 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

$\epsilon ={\frac {v^{2}}{2}}-{\frac {G(M\!+\!m)}{r}}$

For elliptic and circular orbits

$\epsilon ={\frac {-G(M\!+\!m)}{2a}}$

Equating the two previous expressions and solving for v yields the vis viva equation:

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

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 $\epsilon \,\!$, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being "suborbital" (a ballistic missile, for example), 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 (as a meteor, for example).