## Virial Mass

Consider 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 as

 (2.121)

for globular clusters and elliptical galaxies, where 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
 (2.122)

where 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 as
 (2.123)

the Virial theorem gives
 (2.124)

if the surface term is ignored. This gives an estimation of mass of astronomical object.
Kohji Tomisaka 2012-10-03