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

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

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

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

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

Expert Solution

genius_generous answered 1 year ago

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