Virial temperature ($T_{\mathrm{vir}}$)

The virial temperature characterizes the thermal energy of gas supported in a gravitational potential. It is of order $\mu m_{p}GM/(k_{B}R)$, with a structural coefficient determined by the density profile. For a uniform self-gravitating gas sphere the virial theorem gives $k_{B}T=GM\mu m_p/(5R)$; a conventional halo circular-speed definition often uses $k_{B}T\simeq \mu m_p{v}_c^2/2$.

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.

Combining a virial-temperature relation with the density of a uniform self-gravitating sphere gives $M\propto T^{3/2}{n}_H^{-1/2}$. Thus contours of constant mass in a density-temperature plane satisfy $n_H\propto T^3/M^2$. Structural coefficients, composition and the distinction between baryonic and total gravitating mass matter when giving numerical labels.