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
\end{displaymath} (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.
\end{displaymath} (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},
\end{displaymath} (2.123)

the Virial theorem gives
M=\frac{R\sigma_{\rm 3D}^2}{\epsilon G}
\end{displaymath} (2.124)

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