= Two-dimensional self-gravitating rod equilibrium
{title2=$kT_{\min}=GMm/2$}
For a mean-field gas of parallel infinite rods with total mass per unit length $M=Nm$ confined to radius $R_e$, put $q=GM\beta/2$, $\beta=m/(kT)$ and $F=1-q+qR^2/R_e^2$. The isothermal solution is $\psi=\psi(R_e)-2\log F/\beta$ with <mass density> $\rho=M(1-q)/(\pi R_e^2F^2)$ and enclosed line mass $M(<R)=M(R^2/R_e^2)/F$. Direct differentiation verifies the cylindrical <Poisson equation for Newtonian gravity>. Regularity requires $q<1$, or $T>T_{\min}$; equality is a singular core-concentration limit, not a regular equilibrium.
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