Stars & galaxies, weekly problems

Problem 1.

The expression for the Jeans mass in a gas of temperature T and density ρ is

M_{J} = (\frac{3k_{B}T}{2m_{p} G})^{3/2} (\frac{3}{4πρ})^{1/2} ,

where k_{B} = 1.38 × 10^{−23}J K^{−1} is Boltzmann’s constant, m_{p} = 1.67 × 10^{−27}kg is the proton mass and G = 6.67 × 10^{−11} m^{3} kg^{−1} s^{−2} Newton’s constant.

Show that

M_{J} = 1.5 × 10^{4} M_{⊙}(n/cm^{−3})^{−1/2} (T/100K)^{3/2} ,

when n = ρ/m_{p} is the particle density, and M_{⊙} = 1.99 × 10^{30} kg is the solar mass. [2 marks]

Suppose two clouds with density n = 1cm^{−3}, T = 100K and M = 10^{4}M_{⊙} collide. Due to the shock, the density n increases by a factor of four. Compute the maximum T below which the new cloud will be Jeans unstable. [4 marks]

Determine whether the cloud is stable if the gas is isothermal. [2 marks]

Determine whether the cloud is stable if the gas behaves adiabatically, (T ∝ ρ^{2/3}). [2 marks]