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1. a) A star cluster contains N stars, each of mass m. It has total m.pdf

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A star cluster contains N stars, each of mass m, with total mass M and radius R. The density decreases as 1/r^2 from the center. The document asks to: 1) Derive an expression for the total kinetic energy of the stars in terms of the radial velocity dispersion σR, mass and/or radius. 2) Derive an expression for the internal energy U as a double integral of the mass and radius. 3) Use the results to derive the Virial mass as M=3GR^2/σR^2.

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1. a) A star cluster contains N stars, each of mass m. It has total mass M, radius R, and a density which decreases as 1/r2 out to R, and specified by: =0(Rr)2 where 0 is the density at radius R. Derive an expression for the total kinetic energy of stars in terms of the radial velocity dispersion R, the mass and/or the radius. b) Derive an expression for the internal energy of the cluster U in terms of the cluster mass and radius. Remember, density is dependent on radius in this case, so this is a doubleintegral. c) Using your results from b) and c) derive the Virial mass of this cluster in this case, M=3GR2R

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