Deriving Jeans' Mass for Gravitational Collapse

[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!

 

[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?

 

[bcrowell]

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

 

[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?

 

[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.

 

[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 :-)

 

[jkrivda]

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