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4.1.1. The Virial Theorem

Because the virial theorem is of central importance in dynamical mass estimates it is worthwhile to derive it in a rigorous manner using the moment of Inertia of a system of N-particles. For this system, we define the moment of inertia I as

Equation 1   (1)

After one dynamical timescale, the time derivative of I is constant so the second derivative is zero. Hence we can write

Equation 2   (2)
Equation 3   (3)

The first term on the right hand side is sumi=1n MiVi2 which is twice the kinetic energy of the system or 2T. The second term consists of a spatial coordinate times its second derivative which is displacement times force which is an energy. We identify the second term as sumi=1n Rvector Fvector where Fvector is the total Force. This term is the potential energy of the system or W. We thus have

Equation 4   (4)
Equation 5   (5)

which is the well known virial theorem in which the total energy of a system is zero.

For a self-gravitating N-body system

Equation 6a   (6a)

If we assume that each particle as the same mass then Mi = Mj + Mt / N where Mt = total system mass. This yields

Equation 6b   (6b)

For large N, this reduces to

Equation 6c   (6c)

where Rhms is the harmonic mean separation between the system of N particles. The kinetic energy of this system is

Equation 6d   (6d)

The time average of Vi is defined to be the r.m.s velocity dispersion of the system, sigmav. By the virial theorem we then have

Equation 6e   (6e)

or

Equation 6f   (6f)

which indicates that under the virial theorem, masses can be derived by measuring characteristic velocities over some characteristic scale size. In general, the virial theorem can be applied to any gravitating system after one dynamical timescale has elapsed.

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