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 $k_{B}T\simeq \mu m_p{v}_c^2/2$.

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.