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

 

[pmacias]

for your question. Jeans instability is a fundamental concept in the study of star formation. It refers to the collapse of a molecular cloud under its own gravity, leading to the formation of a protostar.

To understand this, let's first look at the equation you mentioned: K = (N) (3/2kT). This equation comes from the equipartition theorem, which states that in a system in thermal equilibrium, the total energy is equally distributed among all the degrees of freedom in that system. In this case, the degrees of freedom are the particles (atoms) in the molecular cloud.

The equation essentially tells us that the average kinetic energy of each particle is directly proportional to the temperature and the number of particles in the system. This is why as the number of particles (N) increases, so does the kinetic energy (K).

Now, going back to Jeans instability, we know that a molecular cloud is held together by its own gravity. However, there is also thermal pressure from the high temperature of the cloud pushing outwards. When the cloud becomes too massive, the gravitational force overcomes the thermal pressure and the cloud begins to collapse.

The collapse increases the density and temperature of the cloud, leading to an increase in kinetic energy (K). This increase in kinetic energy eventually balances out the gravitational potential energy (U), as seen in the Virial theorem. This is what leads to the formation of a protostar.

I hope this helps to clarify the connection between Jeans instability and the Virial theorem. Keep exploring and learning about these concepts, and you will gain a deeper understanding of how stars are formed. Best of luck on your journey!