 Two-dimensional self-gravitating rod equilibrium

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 $F=1-q+q{R}^2/R_e^2$. The isothermal solution is $\psi =\psi (R_e)-2\log F/\beta$ with mass density $\rho =M(1-q)/(\pi R_e^2{F}^2)$ 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.