1. Begin with a spherical hydrogen molecular cloud (H₂) of 8 solar masses, temperature of 10 K, and number density n_H2 of 10¹⁰ m^-3. (a) Considering that density is mass/unit volume, if the density of the cloud is uniform, show that the radius of the cloud is R = [3 M_sun/(pm_Hn_H2)]^1/3. Putting in the numbers, what will be the radius of the cloud? Express your answer in AU. (b) Compare to the Jeans length (equation 12.16 of the text). Will the cloud collapse?

2. If the cloud in problem 1 collapses, (a) How long will the collapse take, if given by the free-fall time t_ff (equation 12.26)? (b) If the cloud starts with a tiny rotation velocity of 0.5 m/s, how fast will it be rotating by the time it reaches a radius of 10¹⁰ m? Compare with the rotation velocity of the surface of the Sun. [Hint, use the Sun's radius and its rotation rate of 1 rotation every 28 days to get the rotation velocity.] (c) If the cloud starts with a tiny magnetic field strength B_i = 0.1 nT, what will be its magnetic field strength by the time it reaches a radius of 10¹⁰ m? Compare this magnetic field strength with that of a typical sunspot, 4000 G. [Hint, 1 T = 10,000 G; G = gauss]. The Sun and most stars are thought to lose their rotation rate and magnetic field by interactions of the magnetic field with the protoplanetary disk. The magnetic field acts as a break to the central object, and speeds up the disk rotation.