Orbital eccentricity

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

Launching at the correct circular orbital speed but at in-plane angle $\alpha$ from the tangent leaves specific orbital energy $-GM/(2R)$, hence semi-major axis $R$. The specific angular momentum becomes $\sqrt{GMR}\cos \alpha$, so $j^2=GMa(1-e^2)$ gives $e=|\sin \alpha |$. The actual periapsis and apoapsis are $R(1\mp e)$, equally far from the intended circular orbit. The maximum geometric radial separation is therefore $Re$. This is not an equal-time separation between two moving satellites.

An orbit with nonzero orbital eccentricity. A bound Kepler orbit with $0<e<1$ is a noncircular ellipse; $e=1$ and $e>1$ describe the parabolic and hyperbolic unbound cases. A semi-major axis fixed by Kepler's third law need not equal the time average of the orbital radius.

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 $L=L_c\sqrt{1-e^2}$.

For bound Kepler orbits, the Delaunay variables obey $0<G<L$, $-G<H<G$, and three periodic angles. A canonical transformation preserves phase space volume. At fixed $L$, integrating $H$ supplies the measure $2GdG$. Since $G=L\sqrt{1-e^2}$, the fraction with orbital eccentricity less than $e$ is $L^{-2}{\int }_{L\sqrt{1-e^2}}^{L}2GdG=e^2$. Thus the probability density function is $2e$, independently of any normalizable energy-only phase-space distribution function. A Boltzmann distribution is unnecessary. Even the time-dependent density $F(E(L))[1+\eta \cos (l-n(L)t)]$, $|\eta |<1$, has this same marginal because integrating the mean anomaly removes its oscillatory part; an eccentricity marginal alone does not imply equilibrium.

For bound Kepler orbits with gravitational parameter $\mu$, integrating the Delaunay variables over $0<G<L$, $-G<H<G$ and their three full angle periods gives the displayed population. A globally defined thermal eccentricity distribution requires this integral to converge, or an explicit finite action range. Merely having finite $F(E)$ at every energy does not suffice: $F=1$ yields an infinite integral at large $L$. At each fixed energy, or on every finite normalized action band, the cumulative eccentricity fraction remains $e^2$.