Vis-viva equation

Remarks

The specific energy $E = \frac{1}{2} v^2 - \mu / \rho$ is constant along the orbit, by conservation of energy. At periapsis and apoapsis the velocity is orthogonal to $r$, and the distances are $a(1 - e)$ and $a(1 + e)$. Using the conservation of $\norm{r \times r'}$ at these two points and solving gives $E = -\mu / (2a)$, which is the equation. For a hyperbolic orbit the same formula holds with $a < 0$, and for a parabolic orbit with $1/a = 0$. Example: on a circular orbit, $\rho = a$, so $v = \sqrt{\mu / a}$. Letting $a \to \infty$ gives $v = \sqrt{2 \mu / \rho}$, the escape velocity.

Going deeper

For a student

The vis-viva equation is the working tool for manoeuvre planning. For a tangential burn, the change of speed needed to go from one orbit to another at a given point is the difference of two vis-viva speeds. The Hohmann and bi-elliptic transfers are computed this way. It also shows that the energy of a Kepler orbit depends only on $a$, which, together with Kepler's third law, gives the period $2\pi \sqrt{a^3 / \mu}$ from the energy.