Jeans instabiity & gravitational collapse

[shounakbhatta]

Hello All,

I am trying to understand how stars are formed and thereby came across Jeans instability. I am new to this subject, however I went through concepts of molecular clouds, nuclear fusions and things like that.

The Virial theorem says E=(1/2)U

As sum of potential energy U and kinetic energy K=0, so we write

K+U=E
K=U=1/2(E)

or 2K+U=0

My question is: if the total atoms is N the equation states that:

K= (N) (3/2kT)

How does this equation comes from?

Thanks

 

[SHISHKABOB]

That's basically the thermal energy of the cloud. All of the particles in the cloud (hydrogen atoms in this case) have their kinetic energy partitioned into the degrees of freedom of the system. Hydrogen atoms have three degrees of freedom because they can only move around in three directions.

Basically, the average energy of a particle in a system of temperature T is

$E_{thermal} = f\frac{1}{2}kT$

where f is the number of degrees of freedom that the particle has, and k is Boltzmann's constant. To get the average energy of a system of N particles, just multiply that number above by N.

This means that the kinetic energy of a cloud of hydrogen is basically the same thing as its thermal energy, which is

$K = N * E_{thermal} = N\frac{3}{2}kT$

I believe that one is able to equate these two things (kinetic energy and thermal energy) because the system as a whole is at rest with respect to the observer. Otherwise there would other parts to the kinetic energy. In other words, it's a fairly simplified situation.

 

[shounakbhatta]

Hello,

Thanks for the reply. What I am looking is how the 3/2 factor came into.

Secondly, is Jeans instability the only factor for gravitational collapse (molecular or otherwise.)

Thanks,

-- Shounak

 

[shounakbhatta]

Hello,

Can anyone please provide the step by step derivation of Jeans law?

Thanks,

-- Shounak