Calculating Degenerate Remnants and Main Sequence Lifetime for a Cloud Collapse

[AnniB]

Homework Statement

1. Consider a cloud that consists entirely of molecular hydrogen. The temperature is a uniform 15 K and the number density of particles is 5x10⁹ m-3. Assuming a 100% efficient collapse, subsequent star birth and normal evolution, what type of degenerate remnant will result? Justify your answer with appropriate calculations.



2. For the same cloud in problem 1, assume that the cloud is able to form multiple stars of various masses. Calculate the initial mass of a star whose main sequence lifetime will equal the free fall timescale for the cloud.

Homework Equations

N=M_cloud/m_H
M=(4/3)$\pi$r³$\rho$
r=(3M/(4$\pi$$\rho$))^1/3
Jean's Mass and Radius (?)
t_ff=$\sqrt{}3\pi/32Gρ$

The Attempt at a Solution

The first thing I did, since I was given the number density N, was to solve for the mass of the cloud by rearranging the first equation that I gave up above. However, when I do that, I get a really low number for the mass of the star, and I don't think it's actually possible. The only thing I can think of is that it's actually ρ, not N, but that doesn't seem right. Also, if I do that it looks like there will be a lot of messy calculation to acquire the mass and radius, and I'm not sure if I need to do all that to get the answer.

 

[_coyote_]

For the second problem, I'm not sure where to start. I know that I need to find the mass of the star whose main sequence lifetime will equal the free fall timescale for the cloud, but I'm not sure how to go about doing this.