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Vis-viva equation

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Concept in gravitational orbital mechanics

Astrodynamics
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Inastrodynamics, thevis-viva equation is one of the equations that model themotion of orbiting bodies. It is the direct result of the principle ofconservation 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 surroundinggravitational 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 byIsaac Newton.^[1]^: 30 It represents the principle that the difference between the totalwork of theaccelerating forces of asystem and that of the retarding forces is equal to one half thevis viva accumulated or lost in the system while the work is being done.

Formulation

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For anyKeplerian orbit (elliptic,parabolic,hyperbolic, orradial), thevis-viva equation^[1]^: 30 is as follows:^[2]^: 30 $v^{2}=GM\left({2 \over r}-{1 \over a}\right)$ where:

v is the relative speed of the two bodies
r is the distance between the two bodies' centers of mass
a is the length of thesemi-major axis (a > 0 forellipses,a = ∞ or1/a = 0 forparabolas, anda < 0 forhyperbolas)
G is thegravitational constant
M is the mass of the central body

The product ofGM can also be expressed as thestandard gravitational parameter using the Greek letterμ.^[1]^: 33

Practical applications

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Given the total mass and the scalarsr andv at a single point of the orbit, one can compute:

r andv at any other point in the orbit; and
thespecific orbital energy $\varepsilon \,\!$ , 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).

The formula forescape velocity can be obtained from the Vis-viva equation by taking the limit as $a {\displaystyle a}$ approaches $\infty$ : $v_{e}^{2}=GM\left({\frac {2}{r}}-0\right)\rightarrow v_{e}={\sqrt {\frac {2GM}{r}}}$ For a given orbital radius, the escape velocity will be ${\sqrt {2}}$ times the orbital velocity.^[1]^: 32

Derivation for elliptic orbits (0 ≤ eccentricity < 1)

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Specific total energy is constant throughout the orbit. Thus, using the subscriptsa andp to denote apoapsis (apogee) and periapsis (perigee), respectively, $\varepsilon ={\frac {v_{a}^{2}}{2}}-{\frac {GM}{r_{a}}}={\frac {v_{p}^{2}}{2}}-{\frac {GM}{r_{p}}}$

Rearranging, ${\frac {v_{a}^{2}}{2}}-{\frac {v_{p}^{2}}{2}}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}$

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 $h=r_{p}v_{p}=r_{a}v_{a}={\text{constant}}$ , thus $v_{p}={\frac {r_{a}}{r_{p}}}v_{a}$ : ${\frac {1}{2}}\left(1-{\frac {r_{a}^{2}}{r_{p}^{2}}}\right)v_{a}^{2}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}$ ${\frac {1}{2}}\left({\frac {r_{p}^{2}-r_{a}^{2}}{r_{p}^{2}}}\right)v_{a}^{2}={\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}$

Isolating the kinetic energy at apoapsis and simplifying, ${\begin{aligned}{\frac {1}{2}}v_{a}^{2}&=\left({\frac {GM}{r_{a}}}-{\frac {GM}{r_{p}}}\right)\cdot {\frac {r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}}\\{\frac {1}{2}}v_{a}^{2}&=GM\left({\frac {r_{p}-r_{a}}{r_{a}r_{p}}}\right){\frac {r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}}\\{\frac {1}{2}}v_{a}^{2}&=GM{\frac {r_{p}}{r_{a}(r_{p}+r_{a})}}\end{aligned}}$

From the geometry of an ellipse, $2a=r_{p}+r_{a}$ wherea is the length of the semimajor axis. Thus, ${\frac {1}{2}}v_{a}^{2}=GM{\frac {2a-r_{a}}{r_{a}(2a)}}=GM\left({\frac {1}{r_{a}}}-{\frac {1}{2a}}\right)={\frac {GM}{r_{a}}}-{\frac {GM}{2a}}$

Substituting this into our original expression for specific orbital energy, $\varepsilon ={\frac {v^{2}}{2}}-{\frac {GM}{r}}={\frac {v_{p}^{2}}{2}}-{\frac {GM}{r_{p}}}={\frac {v_{a}^{2}}{2}}-{\frac {GM}{r_{a}}}=-{\frac {GM}{2a}}$

Thus, $\varepsilon =-{\frac {GM}{2a}}$ and the vis-viva equation may be written ${\frac {v^{2}}{2}}-{\frac {GM}{r}}=-{\frac {GM}{2a}}$ or $v^{2}=GM\left({\frac {2}{r}}-{\frac {1}{a}}\right)$

Therefore, the conservedangular momentumL =mh can be derived using $r_{a}+r_{p}=2a$ and $r_{a}r_{p}=b^{2}$ , wherea issemi-major axis andb issemi-minor axis of the elliptical orbit, as follows: $v_{a}^{2}=GM\left({\frac {2}{r_{a}}}-{\frac {1}{a}}\right)={\frac {GM}{a}}\left({\frac {2a-r_{a}}{r_{a}}}\right)={\frac {GM}{a}}\left({\frac {r_{p}}{r_{a}}}\right)={\frac {GM}{a}}\left({\frac {b}{r_{a}}}\right)^{2}$ and alternately, $v_{p}^{2}=GM\left({\frac {2}{r_{p}}}-{\frac {1}{a}}\right)={\frac {GM}{a}}\left({\frac {2a-r_{p}}{r_{p}}}\right)={\frac {GM}{a}}\left({\frac {r_{a}}{r_{p}}}\right)={\frac {GM}{a}}\left({\frac {b}{r_{p}}}\right)^{2}$

Therefore, specific angular momentum $h=r_{p}v_{p}=r_{a}v_{a}=b{\sqrt {\frac {GM}{a}}}$ , and

Total angular momentum $L=mh=mb{\sqrt {\frac {GM}{a}}}$

References

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^^a ^b ^c ^dLogsdon, Thomas S.; Logsdon, Tom (1998).Orbital mechanics: theory and applications. A Wiley-Interscience publication. New York, NY: Wiley.ISBN 978-0-471-14636-0.{{cite book}}: CS1 maint: date and year (link)
^Lissauer, Jack J.; de Pater, Imke (2019).Fundamental Planetary Sciences : physics, chemistry, and habitability. New York, NY, USA: Cambridge University Press. pp. 29–31.ISBN 9781108411981.

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