What is the minimum mass that a cloud with these properties needs in order to form a star?

Question

Isolated molecular clouds can have a temperature as low as 11.0 and a particle density as great as 1.10×105 per cubic centimeter.

Estimate the size of such a cloud.(For the purposes of this estimate, you can assume the cloud is perfectly spherical with a volume and that it is made entirely of hydrogen molecules of mass 3.34×10−24 )

Can someone answer both questions? I have an answer for the size of the cloud and the minimum mass but I'm not sure.

Answer 1

The minimum mass that is needed depends on the metallicity and is estimated to be about 50 to 100 times the mass of Jupiter. Your pure hydrogen cloud would be at the high end.

Now for the calculations: 100 times the mass of Jupiter is 100 x 2 x 10^27 = 2 x 10^29 kg. (about 10% of the mass of the Sun).

1 hydrogen molecule has a mass of 3.34 x 10^-27 kg. Divide the two to find that we need 6x10^55 molecules.

Divide by the density of 1.1 x 10^5 particles/cm^3 to get 5.4 x 10^50 cm^3.

Such a volume corresponds to a radius of 5.1 x 10^16 cm or about 3000 AU.

Answer 2

This is a problem applying the Jean's mass. This is the minimum mass required before it will collapse under its own gravity (see Wikipedia link).

I don't have a calculator with me so I won't compute it, but knowing the temperature you calculate the sound speed (c_s in the Wikipedia entry). You already have the density, n so you would plug them into M_J to get the mass.

Since you have the size and the density (and the mass density, rho is hydrogen mass*n) you can estimate the radius if it is spherical and thus obtain the volume using just M=4pi/3 rho * R^3