Question

Properties of a truncated sphere with a power law density profile Suppose the mass distribution of an elliptical galaxy is spherically symmetric, and its density profile is given by ρ(r)=po(r/rs)-2 a(r) =0 for for r<r, r>rs The important difference compared to the 3D→2D projection problem we considered in class is that this 3D density profile does not extend to infinity, but is truncated at r = rs. (a) Calculate the projected half mass-radius, Re of this mass distribution. First, you will need to calculate the projected surface mass density of the sphere, σ(R), expressed as a function of 2D R. 3D rs, and po-There are a number of ways of calculating the projected density profile, but you are asked to use any one method. One of the m ethods will require the following integral But you need not use that method (and the above integral) to solve the problem. (b) Calculate the potential energy (3D quantity) U of the mass distribution, and express it in terms of the total mass Mt and Re (2D quantity), and hence get the value of α Re

Answer 1

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nsupes 3D ominosil distibution ds