Orbital eccentricity

Orbital eccentricity distinguishes circular, elliptic, parabolic, and hyperbolic conics.

For an attractive Kepler orbit with gravitational parameter $GM$, velocity $\mathbf{v}$ and specific angular momentum $\mathbf{h}=\mathbf{r}\times \mathbf{v}$, the eccentricity vector isIts magnitude is the orbital eccentricity, and it points toward periapsis for a noncircular orbit. It is a normalized attractive Laplace-Runge-Lenz vector; it remains constant under the pure inverse-square force.

If the relative acceleration is $\ddot{\mathbf{r}}=-GM\mathbf{r}/r^3+\mathbf{f}$, with constant masses, then the specific orbital energy $\epsilon$, specific angular momentum $\mathbf{h}$ and eccentricity vector obeyThe inverse-square-force terms cancel in each derivative. These identities express perturbing work and torque, and track the instantaneous change of the ellipse's shape and orientation.

An isotropic population of bound Kepler orbits has eccentricity probability density $n(e)=2e$ on $0\le e<1$. It follows from the fixed-energy angular-momentum measure $n(L)dL\propto LdL$ and $L=L_c\sqrt{1-e^2}$.