# Homework Help: Deriving Jeans' Mass for Gravitational Collapse

1. Oct 8, 2012

### jkrivda

I need to show that the critical (Jeans') mass for a hydrogen cloud of uniform density to begin gravitational collapse can be expressed as:

M=(v^4)/((P^.5)(G^1.5))

Where v is the isothermal sound speed, and P is the pressure associated with the density ρ and temperature T.

I don't really know where to start. I have found a lot of derivations for the Jeans' Mass, however, none of them relate to the isothermal speed of sound. I assume I have to do some algebraic manipulations, I just need some help getting started.

Thanks!

2. Oct 8, 2012

### SHISHKABOB

my textbook tells me that the isothermal sound speed is

$v_{T} = (\frac{kT}{\mu m_{H}})^{1/2}$

and that the Jeans mass is

$M_{J} = (\frac{5kT}{G\mu m_{H}})^{3/2} (\frac{3}{4\pi \rho})^{1/2}$

but my text also tells me that this was derived while neglecting an external pressure on the cloud due to the surrounding interstellar medium.

Does that help?

3. Oct 8, 2012

### bcrowell

Staff Emeritus
If this is homework, it belongs in Homework and Coursework.

4. Oct 8, 2012

### jkrivda

yes, that helps. with that info, i can derive the required jeans' mass, with some arbitrary coefficient before the variable terms.

my textbook hasn't arrived by mail yet, and i am worried about not giving enough reasoning behind my derivation. is there any way you can give me a little insight as to how your textbook arrives at those conclusions? as in, how your textbook arrived at what you gave for the jeans' mass and isothermal sound speed formulae?

5. Oct 8, 2012

### SHISHKABOB

my textbook is Carroll and Ostlie's Introduction to Modern Astrophysics, and they do not actually give the derivation for the equation that you've got. They call it the Bonnor-Ebert mass.

6. Oct 9, 2012

### clamtrox

I guess the derivations you have seen use the virial theorem and temperature then?

Do you know what the relevant physics here is? Why does a clump of matter collapse if its mass is bigger than Jeans mass? Why it does not if the mass is smaller? That would be a good place to start :-)

7. Oct 11, 2012

### jkrivda

thanks for the help, guys! i referenced the carrol/ostlie text and managed to get a near-perfect score on my problem set. =]