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Let’s look at some gravitational potentials and the density profiles that can generate them. i) For ρ(r)...

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Physics

Let’s look at some gravitational potentials and the densityprofiles that can generate them.

i) For ρ(r) = ρ◦r ◦^2/r^2 , what is the correspondinggravitational potential?

ii) For Φ(r) = −GM/[b + √ (b ^2 + r ^2)], what is thecorresponding density profile?

iii) For Φ(R, z) = Φ◦ln[ (R ◦^2+R ^2+(z ^2/q^2))/R◦^2 ], what isthe corresponding density profile ρ(R, z)? In this expression, theconstant q controls the axis ratio of the potential, where q = 1would be spherically symmetric and q < 1 means that thepotential is “flattened” a bit along the z-axis. Does somethingstrange happen along the z axis if q < 0.7 ?

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