solutions-ps09

solutions-ps09 - Physics 341: Problem Set #9 Solutions 1....

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Unformatted text preview: Physics 341: Problem Set #9 Solutions 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 For a spherical mass distribution, the enclosed mass is M ( r ) = 4 Z r ( u ) u 2 du So for the Plummer density profile we have, M ( r ) = 4 Z r 3 M 4 a 2 ( u 2 + a 2 ) 5 / 2 u 2 du = 3 M Z r/a a 2 ( a 2 x 2 + a 2 ) 5 / 2 ( a 2 x 2 ) adx = 3 M Z r/a a 5 x 2 ( a 2 ) 5 / 2 ( x 2 + 1) 5 / 2 dx = 3 M Z r/a x 2 ( x 2 + 1) 5 / 2 dx = 3 M " x 3 3 ( x 2 + 1) 3 / 2 # r/a = M ( r/a ) 3 [( r/a ) 2 + 1] 3 / 2 = M r 3 a 3 a 2 r 2 + a 2 3 / 2 = Mr 3 ( r 2 + a 2 ) 3 / 2 where in the second step we made the substitution x = u/a , so u = ax and du = adx , and then we used the integral provided. (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 x 4 dx/ ( x 2 + 1) 4 = / 32 . The total potential energy for a spherical mass distribution of radius R is U =- 4 G Z R M ( r ) ( r ) r dr 1 The Plummer model is infinite in size, so we take the upper limit of the integral R , and we have U =- 4 G Z " Mr 3 ( r 2 + a 2 ) 3 / 2 #" 3 Ma 2 4 ( r 2 + a 2 ) 5 / 2 # r dr =- 3 GM 2 a 2 Z r 4 dr ( r 2 + a 2 ) 4 =- 3 GM 2 a 2 Z ( ax ) 4 adx ( a 2 x 2 + a 2 ) 4 =- 3 GM 2 a 2 a 5 a 8 Z x 4 dx ( x 2 + 1) 4 =- 3 GM 2 a 32 =- 3 32 GM 2 a where I again made the substitution x = r/a , so r = ax and dr = adx , and used the integral provided. By the way, to do integrals like these, try searching for integrate from 0 to infinity x^4/(x^2 + 1)^4 dx at http://www.wolframalpha.comhttp://www....
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solutions-ps09 - Physics 341: Problem Set #9 Solutions 1....

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