Stars & galaxies, weekly problems

Problem 2.

Short questions

1. What is the origin of the Hydrogen 21-cm line? How is it used to measure rotation curves of spiral galaxies? [2 marks]

2. What is the ‘winding’ problem of spiral galaxies? How is this problem resolved? [2 marks]

Long question

The speed, V_{c}(R), of a particle on a circular orbit with radius R, enclosing a mass M(< R) is V_{c}^{2}(R) = GM(< R)/R.

1. Demonstrate that a constant rotation speed, V_{c} = V_{c,0}, requires a density distribution ρ(r) = V_{c,0}^{2}/4πGR^{2}. Assume this is the dark matter distribution in the Milky Way. [2 marks]

2. The surface density of stars in the Milky Way’s disk falls with distance R to the centre as σ(R) = σ_{0} exp(−R/R_{0}). Derive an expression for the disk mass, M_{d}(< R), enclosed in a sphere of radius R. [2 marks]

3. In the Milky Way galaxy, both the dark halo and the disk itself contribute to the rotation speed. Use your previous results to compute V_{c}(R) at distance R = 1 kpc and R = 10 kpc. Assume σ_{0} = 2 × 10^{4} M_{⊙} pc^{−2}, R_{0} = 0.3 kpc and V_{c,0} = 200 km s^{−1}. [2 marks]

1 pc = 3.09 × 10^{16} m, M_{⊙} = 2.0 × 10^{30} kg, G = 6.67 × 10^{−11} m^{3} kg^{−1} s^{−2}