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Virial Theorem

3-D Vectors

The Virial Theorem states that in an assembly of particles in stable statistical equilibrium under its own gravitation, the potential energy of the system due to the mutual attraction of its members must equal to twice the total kinetic energy of the particles due to their random motions, that is, PE = 2 x KE. For the case of cluster of galaxies, each galaxy is considered as one particle. It follows that the total mass of the system M can be derived from the formula:

M ~ (2RV2)/G

where R ~ the radius of the system, V ~ the velocity dispersion of the particles, and G = 6.67x10-8 cm3/sec2-gm is the gravitational constant.

Figure 01 3-D Vectors
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Following is the mathematical derivation of the Virial Theorem. It starts with a rather complicated formulation according to the Newtonian mechanics. The final result appears in a much simpler form with some assumptions and time averaged entities. The bold-faced alphabets represent vectors in the equations (see Figure 01 for a visual description of the various vectors).



NGC5139 The Globular Cluster, Omega Centauri (NGC 5139) has been in existence for 12 billion. It should be old and stable enough to apply the Virial Theorem. Since all its physical parameters are known : M = 4x106 Msun, R = 86 light years, and V = 12.20 km/s, it is particularly important for checking out the accuracy of the Virial Theorem, which yields M = 1.8x106 Msun using the known values of R and V - the computed value is off by a factor of 2. Anyway, the Virial Theorem has been applied to systems such as the cluster of galaxies, or galaxies etc. when M or R is not subjected to measurement.

Figure 02 NGC5139