Understand Bonnar-Ebert Model: Complement to Jeans Law?

[Phigla]

TL;DR Summary

Difference between Bonnar-Ebert and Jeans laws

Good morning,
I continue to try to understand the collapse and fragmentation of a molecular cloud.
I was focused on the Jeans law.
I would like, now to understand the difference between this Jeans law and the Bonnar-Ebert model.
Are they complementary ?
Thanks for your help

 

[masher]

I'll not be a great help but your question deserves an answer. They are complementary and both are concerned with simplistic assumptions about self-gravitation but from different perspectives. Jeans gets you in the ballpark by considering the dispersion of sound in a homogenous medium initially in equilibrium. BE was developed 60 yrs later and considers a sphere initially in equilibrium with a reservoir. I don't think BE describes fragmentation but gives a more accurate critical mass or pressure. When both work they give results within a an order of magnitude from each other. Neither considers spin, charge, or turbulence. If you get stumped on the algebra first consider that someone may have made substitutions without showing you what they did because they assume you know or should know. I'm thinking you really need to see an instructor during their office hours for this stuff. I know I would.

 

[Phigla]

Dear Masher,

Thanks trying to help me.
Since I posted this thread I did not made a lot of progress and I'm still stuck with the fragmentation step.
I hope the light will come soon.
You are the only one having sent me few words, Thanks for that

 

[Astronuc]

What is one's primary text? Does one have secondary or supplemental references?

For example, Peter H. Bodenheimer, Principles of Star Formation, Springer, 2011.

Fragmentation is associated with turbulence in the cloud. It seems both models assume some form of hydrostatic equilibrium, and certainly large amounts of matter are rarely in static, or hydrostatic equilibrium. As such, a static model would not address turbulence, but only indicate a stability criterion.

Perhaps this text will be helpful - http://www.ifa.hawaii.edu/users/jpw/classes/ism/reading/klessen_glover_saasfe.pdf (search on fragmentation)

Klessen and Glover write "some of these fluctuations may exceed the critical mass for gravitational collapse to set in. The presence of turbulence thus leads to the break-up into smaller units. The core fragments to build up a cluster of stars with a wide range of masses rather than forming one single high-mass star. We call this process gravoturbulent fragmentation, because turbulence generates the distribution of clumps in the first place, and then gravity selects a subset of them for subsequent star formation."

https://en.wikipedia.org/wiki/Jeans_instability
https://en.wikipedia.org/wiki/Bonnor–Ebert_mass
https://www2.mpia-hd.mpg.de/homes/ppvi/posters/1B086.pdf

Some information on Jeans and Bonnor-Ebert models.

Here is Jeans's original article - The stability of a spherical nebula
https://royalsocietypublishing.org/doi/10.1098/rsta.1902.0012

"The Bonnor-Ebert sphere actually does not have uniform density but requires a density gradient to stay in equilibrium."

https://books.google.com/books?id=3en405A82WgC&pg=PA116&lpg=PA116&dq=Bonnor-Ebert+model+compared+with+Jean+model&source=bl&ots=ZvdR1y2MpC&sig=ACfU3U3c-QfFh6o08h90Kk_wOxs6OpyOAQ&hl=en&sa=X&ved=2ahUKEwjfptrT4K71AhVukokEHclgBogQ6AF6BAgMEAM#v=onepage&q=Bonnor-Ebert model compared with Jean model&f=false

Perhaps this might help - Cloud Equilibrium and Stability
https://onlinelibrary.wiley.com/doi/10.1002/9783527618675.ch9

and Multiple Star Formation
https://onlinelibrary.wiley.com/doi/10.1002/9783527618675.ch12

ALMA Observations of Massive Clouds in the Central Molecular Zone: Jeans Fragmentation and Cluster Formation
https://iopscience.iop.org/article/10.3847/2041-8213/ab8b65

HIERARCHICAL GRAVITATIONAL FRAGMENTATION. I. COLLAPSING CORES WITHIN COLLAPSING CLOUDS
https://iopscience.iop.org/article/10.1088/0004-637X/814/1/48Edit/update: Something to ponder while studying star formation.
1,000-light-year wide bubble surrounding Earth is source of all nearby, young stars
https://phys.org/news/2022-01-light-year-wide-earth-source-nearby.html

 

[Phigla]

Dear Astronuc,
Thanks for your reply trying to help me for the understanding of the collapse and fragmentation of a molecular cloud leading to proto-star.
Everything started when I read an article on the internet ( I attach illustration of this article) and this lead me to search for the process including Jeans mass and Bonnar-Ebert models.
I defined an imaginary molecular cloud and made calculations applying these two models, it leads me nowhere.
I hope the readings you sent me will clarify the process to me, I will keep posted here my progress

Thanks a lot

 

[Phigla]

Hi,
To try to understand this process I tried to make an exercise with an imaginary molecular cloud
Under is the way I thought to use for that

Initial molecular cloud: Mass M0, Density ρ0 = μnmH, Temp. T

1. I compute R0, V0, initial radius and volume considering a spherical molecular cloud having a number of particles N= nV0
2. Using the Jeans model I compute the Jeans length R1. the Jeans mass M1 and the free fall time. As Jeans instability are met I consider this R1 and M1 the values reached after collapse.
3. I compute the volume after collapse V1.
The number of particles remaining the same, I compute the number of particles in the cloud after collapse n1 = N/V1
4. I compute the density after collapase ρ1 = μn1mH
5. I redo the Jeans Criterion with the new values and these ones are met again, so the cloud will collapse again

BUT to verify if my logic is correct I compute M1= V1ρ1,
According me I would have to find the Jeans mass M1 with the new density value ρ1
NO ?

It’s not the case where is my mistake ?

Thanks if a member can highlight my mistake, I'm lost

Best regards