Molecular cloud collapsing and fragmentation

[Phigla]

TL;DR Summary

I try to understand the pre-stellar mechanism related to the Jeans law

Good morning,

I read on the internet that a molecular cloud contains denser part, I also read that a molecular cloud start to collapse according the Jeans law
If it's the full cloud collapsing what is happening to these denser parts ? and after collapsing how the fragmentation occurs ?
In fact it is the overall mechanism collapsing-fragmentation I'm not able to understand mathematically using the Jeans law.
Is it a member will have the patience to explain me or to give me a pertinent link where I can find replies to my questions

Thanks in advance

 

[Bandersnatch]

You have the stability criterion of Jeans Mass - if a cloud or certain size and temperature is above this mass, it'll start to collapse.
Collapsing clouds evolve in density and temperature (where $M_J$ scales as $(\frac{T^3}{\rho})^{1/2}$ ). So as the cloud collapses, it may yet end up above or below Jeans Mass depending on how well it can shed the excess temperature.
E.g. if it cools inefficiently, the cloud might end up below Jeans Mass, and stabilise, or remain just above and keep collapsing as a whole.
On the other hand, if it cools efficiently, Jeans Mass might decrease, and the violation of the stability criterion becomes even more severe than what lead to the initial collapse. Meaning, regions less massive than the initial cloud may now be sufficiently massive to collapse themselves.
This is what leads to fragmentation within the cloud.

Each of the new, smaller regions, can evolve similarly until the gas becomes so dense that it turns opaque to radiation, which limits cooling. This is what puts the brakes on fragmentation, so that stars do eventually form.

 

[Phigla]

Dear Bandersnatch,

Thanks for your reply.
I try to go step by step
In a typical molecular cloud, the mean temperature of this cloud is 15-20 K, density few hundred particles per cubic centimeters, but inside some regions have clumps with density much higher 1000-10,000 particles per cubic centimeter.
I understand that is the totality of the molecular gas is going to collapse first, not the clump
Right ?
If it is this way, therefore I've to apply the Jeans formula for the total molecular cloud, not the denser part of the cloud, but then what will happen to these clumps during the collapse ?

Thanks in advance

 

[Bandersnatch]

In a typical molecular cloud, the mean temperature of this cloud is 15-20 K, density few hundred particles per cubic centimeters, but inside some regions have clumps with density much higher 1000-10,000 particles per cubic centimeter.
I understand that is the totality of the molecular gas is going to collapse first, not the clump
Right ?

That depends on how much mass is in those regions. E.g., assuming constant temperature throughout the cloud, and that its mass is just above the collapse criterion, then if the smaller region with 100 times the density contains at least 1/10th of the cloud mass, it will be itself above Jeans mass and fragmentation will occur.
Otherwise the lumps will follow the general behaviour of the gas in the cloud, for as long as they remain below the Jeans mass.

 

[Phigla]

I make progress, with the mass of Jeans,

MJ= Cst1 x square root of T³/Rho
Rj= square root of Cst2 x T/number of atoms per m³ in the cloud.

So what understands is that if a molecular cloud collapses it will reach this mass and this radius.

As the radius decreases, so the volume decreases and the number of atoms in the cloud remains the same, the density of the cloud will increase,
I start the calculation again to see if a new collapse is possible.

I still don’t understand how and when the fragmentation will occurs

I need some help

Thanks in advance