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): -{ p(r) Po, r Ro (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: 5/2 Vobs (r) = G Pozo Ro 1+e-4r/R Draw the expected and observed rotational velocity into the plot below: 1) 2.0 1.5 Rotational Velocity v [G#pe2 Re] 1.0 0.5 0.0 8 9 10 2 4 5 6 Radius from Center (R) (d) Scientists believe the reasons for the difference to be dark matter: Determine the rotational velocity due to dark matter Upm(r) from Ro and draw it into the plot above. (e) Derive the dark matter mass Mom(r) enclosed in r and explain its distributed.

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): -{ p(r) Po, r Ro (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: 5/2 Vobs (r) = G Pozo Ro 1+e-4r/R Draw the expected and observed rotational velocity into the plot below: 1) 2.0 1.5 Rotational Velocity v [G#pe2 Re] 1.0 0.5 0.0 8 9 10 2 4 5 6 Radius from Center (R) (d) Scientists believe the reasons for the difference to be dark matter: Determine the rotational velocity due to dark matter Upm(r) from Ro and draw it into the plot above. (e) Derive the dark matter mass Mom(r) enclosed in r and explain its distributed.