Vis-viva equation

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.<ref name=Logsdon/>Template:Rp 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.

Formulation

For any Keplerian orbit (elliptic, parabolic, hyperbolic, or radial), the vis-viva equation<ref name="Logsdon">Template:Cite book</ref>Template:Rp is as follows:<ref name=lissauer2019>Template:Cite book</ref>Template:Rp <math display="block">v^2 = GM \left({ 2 \over r} - {1 \over a}\right)</math> where:

• Template:Mvar is the relative speed of the two bodies
• Template:Mvar is the distance between the two bodies' centers of mass
• Template:Mvar is the length of the semi-major axis (Template:Math for ellipses, Template:Math or Template:Math for parabolas, and Template:Math for hyperbolas)
• Template:Math is the gravitational constant
• Template:Mvar is the mass of the central body

The product of Template:Math can also be expressed as the standard gravitational parameter using the Greek letter Template:Mvar.<ref name=Logsdon/>Template:Rp

Practical applications

Given the total mass and the scalars Template:Mvar and Template:Mvar at a single point of the orbit, one can compute:

• Template:Mvar and Template:Mvar at any other point in the orbit; and
• the specific orbital energy <math>\varepsilon\,\!</math>, 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 for escape velocity can be obtained from the Vis-viva equation by taking the limit as <math>a</math> approaches <math>\infty</math>: <math display="block">v_e^2 = GM \left(\frac{2}{r}-0 \right) \rightarrow v_e = \sqrt{\frac{2GM}{r}}</math> For a given orbital radius, the escape velocity will be <math>\sqrt{2}</math> times the orbital velocity.<ref name=Logsdon/>Template:Rp

Derivation for elliptic orbits (0 ≤ eccentricity < 1)

Specific total energy is constant throughout the orbit. Thus, using the subscripts Template:Math and Template:Math to denote apoapsis (apogee) and periapsis (perigee), respectively, <math display="block"> \varepsilon = \frac{v_a^2}{2} - \frac{GM}{r_a} = \frac{v_p^2}{2} - \frac{GM}{r_p} </math>

Rearranging, <math display="block"> \frac{v_a^2}{2} - \frac{v_p^2}{2} = \frac{GM}{r_a} - \frac{GM}{r_p} </math>

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 <math> h = r_pv_p = r_av_a = \text{constant}</math>, thus <math>v_p = \frac{r_a}{r_p}v_a</math>: <math display="block"> \frac{1}{2} \left( 1-\frac{r_a^2}{r_p^2} \right) v_a^2 = \frac{GM}{r_a} - \frac{GM}{r_p} </math> <math display="block"> \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} </math>

Isolating the kinetic energy at apoapsis and simplifying, <math display="block">\begin{align}

\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_ar_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{align}</math>

From the geometry of an ellipse, <math>2a=r_p+r_a</math> where a is the length of the semimajor axis. Thus, <math display="block"> \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} </math>

Substituting this into our original expression for specific orbital energy, <math display="block"> \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} </math>

Thus, <math> \varepsilon = - \frac{GM}{2a} </math> and the vis-viva equation may be written <math display="block"> \frac{v^2}{2} - \frac{GM}{r} = -\frac{GM}{2a} </math> or <math display="block"> v^2 = GM \left( \frac{2}{r} - \frac{1}{a} \right) </math>

Therefore, the conserved angular momentum Template:Math can be derived using <math>r_a + r_p = 2a</math> and <math>r_a r_p = b^2</math>, where Template:Mvar is semi-major axis and Template:Mvar is semi-minor axis of the elliptical orbit, as follows: <math display="block">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 </math> and alternately, <math display="block">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 </math>

Therefore, specific angular momentum <math>h = r_p v_p = r_a v_a = b \sqrt{\frac{GM}{a}}</math>, and

Total angular momentum <math>L = mh = mb \sqrt{\frac{GM}{a}}</math>