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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 $e^2$.

At fixed energy, the circular orbit has $L_c^2=GMa$, whileAn isotropic galactic distribution function has conditional angular-momentum density $n(L)dL\propto LdL$. ThereforeNormalization on $0\le e<1$ gives the thermal eccentricity distribution

Average the phase probability over the thermal eccentricity distribution:Therefore