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Unformatted text preview: Physics 341: Problem Set #9 due November 11 1. The Plummer model for a spherical star cluster is given by the density profile ( r ) = 3 M 4 a 2 ( r 2 + a 2 ) 5 / 2 where M is the total mass, and a is a core radius. (a) Show that the enclosed mass in the Plummer model is M ( r ) = Mr 3 ( r 2 + a 2 ) 3 / 2 You may need to make an appropriate substitution and use the fact that Z x 2 dx ( x 2 + 1) 5 / 2 = x 3 3( x 2 + 1) 3 / 2 + constant (b) Now calculate the total potential energy of the Plummer mass distribution. Hint: The answer should depend only on G , M , and a . After a substitution, you should find the following integral useful: R 1 x 4 dx/ ( x 2 + 1) 4 = / 32. (c) If the mass distribution is in equilibrium, what is the total kinetic energy? What is the total energy? Again, your answers should depend only on G , M , and a . (d) The globular cluster ! Centauri can be described by a Plummer model with a total mass M = 5 10 6 M and core radius a = 4 . 5 pc. Use your derived total kinetic5 pc....
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