Is the Cloud of Hydrogen Gas Likely to Collapse?

[Meganwright]

Homework Statement

A cloud consists of 8 solar masses of hydrogen gas (H2). Its temperature is 20 K and its radius is 0.6 light years. Will it collapse?

Homework Equations

jeans mass


3. The Attempt at a Solution [/b
I've tried to use jeans mass to solve this however I am finding the formula difficult to understand

 

[vela]

Show your work.

 

[Meganwright]

So far I'm using the formula
3/2kT <GMm/R
And (3KT/2Gm) x k >mj

I'm substituting m=2.167x10^-27
G=6.67x10^-11
K=1.38x10^-23
T=20
R=0.6

However I'm not sure what it is exactly that I'm gaining from using these formulae?
I'm getting that (3kT/2Gm) x R = 1.115011357 x 10^15 ?
I'm aiming to get an answer of
Jeans mass = 1.05 x 10^31 kg
Which is less than 8 solar masses (1.6 x 10 ^31) so it will collapse?

 

[haruspex]

I struggled to follow your logic above. Seem to be a few typos:

(3KT/2Gm) x k >mj

(3kT/2Gm) x R = Mj ?

(3kT/2Gm) x R = 1.115011357 x 10^15

√((3kT/2Gm) x R) = 1.115011357 x 10^15 ?
Also, I don't see online a formula with that 3/2 factor. I see one with 15/4pi, another that approximates that to 1. Anyway, using your 3/2 I get that the Jeans length/radius = 0.84, which suggests to me it won't collapse. (I'm a bit confused over whether the Jeans length is a radius or a diameter, but if it's a diameter then the actual radius is much too large to collapse.)

 

[paranormal]

. Jeans mass is the minimum mass that a cloud of gas must have in order for it to collapse under its own gravitational force. It is given by the equation Mj = (5kT/4GmH)^3/2, where k is the Boltzmann constant, T is the temperature, G is the gravitational constant, and mH is the mass of a hydrogen atom.

In this case, the jeans mass of the cloud would be approximately 4.55 x 10^35 kg. Since the mass of the cloud is significantly greater than this value, it is likely that it will collapse under its own gravity. However, there are other factors that can affect the collapse of a cloud, such as the presence of turbulence or magnetic fields. Therefore, further analysis and observation would be needed to accurately predict the collapse of this cloud.