Velocity dispersion method
(for ellipticals)
Assuming a spherical distribution
with radius R of N stars, each with mass m,
the mass of the galaxy is M=Nm.
From the virial theorem
- 2 ∑i=1,N
(1/2 mivi2 ) = U
- m/N ∑i=1,N
vi2 = U/N
and assuming random motions
<v2> = <vr2> + <vθ2> +
<vφ2> = 3<vr2>
1/N ∑i=1,N
vi2 = <v2> = 3<vr2>
= 3 σr2
(<vr>=0)
σr2 =
- U/3mN = GM2/(5RmN) (U = - 3GM2/5R , for constant density)
=> M = 5Rσr2/G
This is the virial mass.
The virial mass of an elliptical galaxy is usually found to be much larger than
its luminous mass.