The most mass of our Milky Way is contained in an inner region close to the core with radius Ro. Because the mass outside this inner region is almost constant, the density distribution can be written as following (assume a flat Milky Way with height zo): { r< Ro r > Ro Po, p(r) = 0, (a) Derive an expression for the mass M(r) enclosed within the radius r. (b) Derive the expected rotational velocity of the Milky Way v(r) at a radius r. (c) Astronomical observations indicate that the rotational velocity follows a different behaviour: Vobs (r) = VGT Po z0 Ro 5/2 1+e-4r/Ro 4 Draw the expected and observed rotational velocity into the plot below:

The most mass of our Milky Way is contained in an inner region close to the core with radius Ro. Because the mass outside this inner region is almost constant, the density distribution can be written as following (assume a flat Milky Way with height zo): { r< Ro r > Ro Po, p(r) = 0, (a) Derive an expression for the mass M(r) enclosed within the radius r. (b) Derive the expected rotational velocity of the Milky Way v(r) at a radius r. (c) Astronomical observations indicate that the rotational velocity follows a different behaviour: Vobs (r) = VGT Po z0 Ro 5/2 1+e-4r/Ro 4 Draw the expected and observed rotational velocity into the plot below: