Vector Equation#
The vector equation (aka the vis-viva equation) is given by:
\[
v = \sqrt{2\left(\frac{\mu}{r} + \epsilon \right)}
\]
In terms of the semi-major axis \(a\) by replacing \(\epsilon = -\mu/2a\),
\[
v = \sqrt{\mu\left(\frac{2}{r} - \frac{1}{a} \right)}
\]
Using the relationship of \(a\) in ellipse equation of motion gives us a form that is computationally more useful when dealing with parabolic orbits,
\[
v = \sqrt{\frac{\mu}{r} \bigg( 2 - \frac{1-e^2}{1 + e \cos\nu} \bigg)}
\]
For a circle, \(r=a\) so
\[
v_\text{circ} = \sqrt{\frac{\mu}{r}}
\]
Proof - Energy Integral#
Recall the energy integral equation,
\[
\epsilon = \frac{v^2}{2} - \frac{\mu}{r}
\]
Solving for velocity gives,
\[
v = \sqrt{2\left(\frac{\mu}{r} + \epsilon \right)}
\]