Question

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. Will the cloud collapse?

Answer 1

The mass number of the hydrogen molecule is 2.
Thus the mass of hydrogen molecule is
Now, as given, where is the hydrogen sphere mass and is the mass of the Sun.
Volume of a homogeneous sphere of radius R can be written as

and thus the Mass of the sphere can be written as

where is the mass density.

Given the number density for hydron molecules in the sphere is . So the mass density can be calculated as

Therefore the radius is

In the above expression is the moleculer mass of hydrogen and is the atomic mass of the hydrogen

Now putting all the values we get the radius of the cloud as below

b.
The Jeans length is

is the constant of value  

thus the jeans radius is