= Hydrostatic temperature of an inverse-square halo
{title2=$k_BT=\bar mGM/(2R)$}
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 $P=\rho_gk_BT/\bar m$ give $-2\rho_gk_BT/(\bar m r)=-\rho_gGM/(Rr)$, hence $k_BT=\bar mGM/(2R)$. 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.
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