Vector Equation

Vector Equation#

The vector equation (aka the vis-viva equation) is given by:

v = \sqrt{2 (\frac{μ}{r} + ϵ)}

In terms of the semi-major axis $a$ by replacing $ϵ = - μ / 2 a$ ,

v = \sqrt{μ (\frac{2}{r} - \frac{1}{a})}

Using the relationship of $a$ in ellipse equation of motion gives us a form that is computationally more useful when dealing with parabolic orbits,

v = \sqrt{\frac{μ}{r} (2 - \frac{1 - e^{2}}{1 + e \cos ν})}

For a circle, $r = a$ so

v_{circ} = \sqrt{\frac{μ}{r}}

ϵ = \frac{v^{2}}{2} - \frac{μ}{r}

Solving for velocity gives,

v = \sqrt{2 (\frac{μ}{r} + ϵ)}