Let’s look at some gravitational potentials and the densityprofiles that can generate them.
i) For Ï(r) = Ïâ—¦r â—¦^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 â—¦^2+R ^2+(z ^2/q^2))/Râ—¦^2 ], what isthe corresponding density profile Ï(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 ?
