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, rS R 0, 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 u(r) at a radius r. (c) Astronomical observations indicate that the rotational velocity follows a different behaviour: 5/2 1+e-/Ro Draw the expected and observed rotational velocity into the plot below: 2.0 1.5 0.5 0.0 10 Radius from Centerr [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 Mpar(r) enclosed in r and explain its distributed. (f) Explain briefly three theories that provide explanations for dark matter. Rotational Velocity [yG )

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, rS R 0, 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 u(r) at a radius r. (c) Astronomical observations indicate that the rotational velocity follows a different behaviour: 5/2 1+e-/Ro Draw the expected and observed rotational velocity into the plot below: 2.0 1.5 0.5 0.0 10 Radius from Centerr [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 Mpar(r) enclosed in r and explain its distributed. (f) Explain briefly three theories that provide explanations for dark matter. Rotational Velocity [yG )