Virial Mass

Consider $\gamma =5/3$ gas, the gravitational energy is equal to twice of the kinetic+thermal energy [eq.(2.120)] in the equilibrium state. This gives a definition of Virial mass. The kinetic energy is obtained from the 3D velocity dispersion $\sigma_{\rm 3D}$ as

$$U=\frac{1}{2}M\sigma_{\rm 3D}^2$$ (2.121)

for globular clusters and elliptical galaxies, where $M$ is the mass of the object. For molecular cloud this is estimated from the width of the emission line. Equation (E.7) of Appendix E gives

$$\sigma_{\rm 3D}^2=\frac{3}{2\ln 2}v_{x,\rm HWHM}^2.$$ (2.122)

where $v_{x,\rm HWHM}$ is the half width of the half maximum of the emission line (if the radiation is optically thin; see Fig.E.1). Since the gravitational energy is proportional to $M^2$ as

$$W=-\epsilon \frac{GM^2}{R},$$ (2.123)

the Virial theorem gives

$$M=\frac{R\sigma_{\rm 3D}^2}{\epsilon G}$$ (2.124)

if the surface term is ignored. This gives an estimation of mass of astronomical object.