Solution
= Solution
At fixed energy, the circular orbit has $L_c^2=GMa$, while
$$
L=L_c\sqrt{1-e^2}.
$$
An isotropic <galactic distribution function> has conditional angular-momentum density $n(L)\,dL\propto L\,dL$. Therefore
$$
n(e)\,de\propto L\left|\frac{dL}{de}\right|de
=L_c^2e\,de.
$$
Normalization on $0\leq e<1$ gives the <thermal eccentricity distribution>
$$
\boxed{n(e)=2e}.
$$