Hydrostatic temperature of an inverse-square halo ($k_{B}T=\bar{m}GM/(2R)$)

Suppose an isothermal gas with ${\rho }_g\propto r^{-2}$ is supported in a spherical halo with $M(r)=Mr/R$. Then hydrostatic equilibrium and $P={\rho }_g{k}_{B}T/\bar{m}$ give $-2{\rho }_g{k}_{B}T/(\bar{m}r)=-{\rho }_{g}GM/(Rr)$, hence $k_{B}T=\bar{m}GM/(2R)$. Here $\bar{m}$ is the dimensionful mean particle mass. The factor two follows from the density slope and isothermal pressure response; it is not the uniform-sphere virial coefficient. The singular profile and finite boundary are idealizations.