= Delaunay phase-volume proof of the thermal eccentricity distribution
{c}
{title2=$P(e'<e)=e^2$}
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 $2G\,dG$. Since $G=L\sqrt{1-e^2}$, the fraction with <orbital eccentricity> less than $e$ is $L^{-2}\int_{L\sqrt{1-e^2}}^L2G\,dG=e^2$. 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.
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