In astrodynamics, the vis-viva equation 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 which is the gravitational force determined by the product of the mass of the object and the strength of the surrounding gravitational field.
Vis viva (Latin for "living force") is a term from the history of mechanics and this name is given to the orbital equation originally derived by Isaac Newton.
[1]: 30 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.
For any Keplerian orbit (elliptic, parabolic, hyperbolic, or radial), the vis-viva equation[1]: 30 is as follows:[2]: 30
where: The product of GM can also be expressed as the standard gravitational parameter using the Greek letter μ.
[1]: 33 Given the total mass and the scalars r and v at a single point of the orbit, one can compute: The formula for escape velocity can be obtained from the Vis-viva equation by taking the limit as
For a given orbital radius, the escape velocity will be
times the orbital velocity.
[1]: 32 Specific total energy is constant throughout the orbit.
Thus, using the subscripts a and p to denote apoapsis (apogee) and periapsis (perigee), respectively,
Rearranging,
{\displaystyle {\frac {v_{a}^{2}}{2}}-{\frac {v_{p}^{2}}{2}}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}
Recalling that for an elliptical orbit (and hence also a circular orbit) the velocity and radius vectors are perpendicular at apoapsis and periapsis, conservation of angular momentum requires specific angular momentum
{\displaystyle {\frac {1}{2}}\left(1-{\frac {r_{a}^{2}}{r_{p}^{2}}}\right)v_{a}^{2}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}
{\displaystyle {\frac {1}{2}}\left({\frac {r_{p}^{2}-r_{a}^{2}}{r_{p}^{2}}}\right)v_{a}^{2}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}}
Isolating the kinetic energy at apoapsis and simplifying,
{\displaystyle {\begin{aligned}{\frac {1}{2}}v_{a}^{2}&=\left({\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}\right)\cdot {\frac {r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}}\\{\frac {1}{2}}v_{a}^{2}&=GM\left({\frac {r_{p}-r_{a}}{r_{a}r_{p}}}\right){\frac {r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}}\\{\frac {1}{2}}v_{a}^{2}&=GM{\frac {r_{p}}{r_{a}(r_{p}+r_{a})}}\end{aligned}}}
From the geometry of an ellipse,
where a is the length of the semimajor axis.
{\displaystyle {\frac {1}{2}}v_{a}^{2}=GM{\frac {2a-r_{a}}{r_{a}(2a)}}=GM\left({\frac {1}{r_{a}}}-{\frac {1}{2a}}\right)={\frac {GM}{r_{a}}}-{\frac {GM}{2a}}}
Substituting this into our original expression for specific orbital energy,
{\displaystyle \varepsilon ={\frac {v^{2}}{2}}-{\frac {GM}{r}}={\frac {v_{p}^{2}}{2}}-{\frac {GM}{r_{p}}}={\frac {v_{a}^{2}}{2}}-{\frac {GM}{r_{a}}}=-{\frac {GM}{2a}}}
{\displaystyle \varepsilon =-{\frac {GM}{2a}}}
and the vis-viva equation may be written
{\displaystyle {\frac {v^{2}}{2}}-{\frac {GM}{r}}=-{\frac {GM}{2a}}}
Therefore, the conserved angular momentum L = mh can be derived using
, where a is semi-major axis and b is semi-minor axis of the elliptical orbit, as follows:
Therefore, specific angular momentum
, and Total angular momentum