gas cloud of uniform density

[lemur]

If you have a large cloud of gas of uniform density far from any gravity well, how much density would be required for gravity-well formation to begin? Presumably the cloud condenses under its own gravity as it cools, but is it possible that some part of it could condense more and begin forming a gravity-well that pulls in other particles from the cloud? Also, if a cloud of hydrogen has to cool to condense to a density where fusion can ignite creating a star, how is there enough energy to ignite the fusion?

 

If this post too much resembles my other post on cooling and gravity, apologies. I'm exploring these issues in slightly different ways. If this post can better be added to the other thread, that's fine too.

[Iggy]

I'm not an expert, but I might be able to answer.

 

If you have a large cloud of gas of uniform density far from any gravity well, how much density would be required for gravity-well formation to begin?

A gas cloud of uniform density is a gravity well. With uniform density it looks like this: http://en.wikipedia.org/wiki/File:GravityPotential.jpg or this showing one spatial dimension where a is the radius of the sphere of uniform density.

 

If you're asking at what point the cloud would collapse, it would depend on the temperature and density of the cloud. When the gravitational potential energy of each gas molecule becomes greater than each molecule's kinetic energy it collapses... and it loses its uniform density at the very center [edited].

 

The Jeans length is a way to calculate. When the Jeans length is smaller than the cloud it starts collapsing.

 

Presumably the cloud condenses under its own gravity as it cools, but is it possible that some part of it could condense more and begin forming a gravity-well that pulls in other particles from the cloud?

Yes, I think that is typically what happens.

 

A molecular cloud won't be perfectly uniform. When a very large nebula in a galaxy starts to collapse, the temperature can initially stay constant or even drop because as it shrinks the cloud becomes more opaque to starlight and cosmic rays that warm interstellar space. If the cloud cools like you say, or even if the collapse causes it to heat up but only by a factor less than 1/R^1/2 where R is the scale of the cloud, then the Jeans length will shrink faster than the collapsing cloud. That means that the collapse itself can cause smaller and smaller portions of the cloud's interior to be able to collapse in on themselves. The areas inside the cloud with the highest density would self-collapse first forming stars in a developing stellar nursery for example.

 

If the cloud were perfectly uniform and symmetrical then I don't believe it would break up into pieces when collapsing -- it would just fall in on its center. With perfect homogeneity every part of the cloud would shrink at the same rate and it would keep its uniformity except at the center where mass gathers [edited]. But, in nature it would never be perfectly uniform.

 

Also, if a cloud of hydrogen has to cool to condense to a density where fusion can ignite creating a star, how is there enough energy to ignite the fusion?

It actually doesn't have to cool. So long as it doesn't heat up by a factor greater than R^-1/2 the Jeans length will remain smaller than the cloud and it will continue to collapse. The increasing temperature from the increasing pressure of the collapsing cloud can be insufficient to halt the collapse until stars start forming and warming up and clearing out the areas that surround them with radiation pressure.

 

If this post too much resembles my other post on cooling and gravity, apologies. I'm exploring these issues in slightly different ways. If this post can better be added to the other thread, that's fine too.

Would you mind posting a link to the other thread. I looked through your recent posts, but didn't see it.

Edited May 22, 2011 by Iggy

[swansont]

Would you mind posting a link to the other thread. I looked through your recent posts, but didn't see it.

 

http://www.scienceforums.net/topic/57203-gravity-and-cooling/

The premise in that thread is that there is no gravity, so there really isn't a reason to combine them.

[lemur]

Thanks for posting the link, Swansont. The point of the thread was to explore whether the effects of gravity could occur purely due to therodynamic processes.

 

The part that confuses me about a condensing gas cloud is that it seems like the condensation/collapse would cause friction and release heat, but Iggy says that this need not result in heating of surrounding gas until star-ignition occurs. It seems like there should be a conflict between condensation-generating-heat and the heat causing gas to expand. I can see how in a star, gravity is strong enough to prevent the heat from expanding the gas despite its very high energy/temperature, but while a cloud is in the process of condensing, why doesn't the heat from the pressurizing core of the cloud cause the cloud to either re-expand or separate into core/atmosphere? Shouldn't there actually be phase-changes between the center and margins of the cloud?

[Iggy]

The part that confuses me about a condensing gas cloud is that it seems like the condensation/collapse would cause friction and release heat

The collapse does indeed work to raise the temperature of the cloud. Other factors work to cool it -- explained by this site:

 

A molecular cloud's temperature is set by the balance of heating from ultraviolet radiation and cosmic rays against cooling by the emission of infrared radiation. When a cloud contracts, its gas density rises, which increases the rate at which it generates infrared radiation. At the same time, the higher density of dust within the cloud blocks more starlight, which slows the heating of the cloud. These effects cause the temperature of a cloud to drop as the cloud shrinks from the 100°K seen in the cool interstellar medium and in the outer layers of molecular clouds to the 10°K found in the cores of molecular clouds. This drop in temperature as the molecular cloud shrinks ensures that the Jeans length remains less than the scale of the cloud; the cloud must collapse if only gas pressure provides support against the force of gravity. The nature of this collapse, the inability of gas pressure to rise faster than the force of gravity as the cloud collapses, is similar to the collapse of a massive star to a neutron star or a black hole.

http://www.astrophysicsspectator.com/topics/milkyway/MolecularCloudCollapse.html

 

But, as it continues to collapse the increased pressure will eventually start to increase the temperature.

 

, but Iggy says that this need not result in heating of surrounding gas until star-ignition occurs.

I'm not sure how much molecular clouds heat up before they start making a lot of protostars, but they certainly are allowed to heat up yet continue collapsing.

 

When it gets small enough, the strength of gravity trying to collapse the cloud is overcome by the increased temperature and pressure. At that point the Jeans length is equal to the radius of the cloud and the kinetic energy of the constituent parts are equal to their gravitational potential.

 

It seems like there should be a conflict between condensation-generating-heat and the heat causing gas to expand. I can see how in a star, gravity is strong enough to prevent the heat from expanding the gas despite its very high energy/temperature, but while a cloud is in the process of condensing, why doesn't the heat from the pressurizing core of the cloud cause the cloud to either re-expand or separate into core/atmosphere? Shouldn't there actually be phase-changes between the center and margins of the cloud?

 

The pressure of the gas, which increases as the volume decreases and temperature increases, tries to expand the cloud. The strength of gravity tries to collapse it. If the pressure is stronger then the cloud will expand. If gravity is stronger then it will collapse. It stops when it finds and equilibrium. To find out if a uniform gas cloud wants to collapse or expand calculate:

 

[math]\lambda_J\approx\sqrt{\frac{k_B Tr^3}{GM \mu}}[/math]

 

where [math]k_B[/math] is Boltzmann's constant, T is the temperature, r is the radius, G is the gravitational constant, M is the cloud's mass and [math]\mu[/math] is the mass per particle.

 

If the answer is smaller than r then it will want to collapse. Larger than r and it will want to expand. Molecular clouds can have a small enough T, yet large enough M/r³, for it to want to collapse.

Edited May 22, 2011 by Iggy

[lemur]

I'm not sure how much molecular clouds heat up before they start making a lot of protostars, but they certainly are allowed to heat up yet continue collapsing.

So multiple protostars can form within a still-condensing cloud. Does that suggest that condensation of the protostar-pockets is increasing while the area between these pockets is expanding due to the heat? In other words, is density-differentiation overcoming entropy between the dense and less-dense areas of the cloud?

 

When it gets small enough, the strength of gravity trying to collapse the cloud is overcome by the increased temperature and pressure. At that point the Jeans length is equal to the radius of the cloud and the kinetic energy of the constituent parts are equal to their gravitational potential.

Could you sum up what "Jeans length" means in a simple sentence or two? I opened your link about it but it was lots of math without any explanation I could directly follow. The concept sounds a little like that of a Schwarzschild radius but different.

 

The pressure of the gas, which increases as the volume decreases and temperature increases, tries to expand the cloud. The strength of gravity tries to collapse it. If the pressure is stronger then the cloud will expand. If gravity is stronger then it will collapse. It stops when it finds and equilibrium. To find out if a uniform gas cloud wants to collapse or expand calculate:

The interesting part is the entropy conflict going on, imo. If density increases gravity and strengthens the centripetal force on the particles (causing them to do more work as they condense), but heat must also be creating convection currents that result in fragmentation of the cloud into denser and less-dense parts, no?

 

 

[Iggy]

So multiple protostars can form within a still-condensing cloud.

Absolutely. Star clusters are the remnants of collapsing molecular clouds so you can imagine that quite a few stars form out of a collapsing cloud.

 

Does that suggest that condensation of the protostar-pockets is increasing while the area between these pockets is expanding due to the heat?

I wouldn't put it quite like that, but the gas and dust between the collapsing fragments is eventually boiled away and scattered from the star's radiation.

 

In other words, is density-differentiation overcoming entropy between the dense and less-dense areas of the cloud?

I'd again put it differently. A collapsing protostar will lower in entropy when it gets small enough. The heat it radiates raises the entropy of the surroundings to more than make up for the difference.

 

Could you sum up what "Jeans length" means in a simple sentence or two? I opened your link about it but it was lots of math without any explanation I could directly follow. The concept sounds a little like that of a Schwarzschild radius but different.

It is something like a schwarzschild raidus. A very small volume of a gas at a certain temperature and density won't hold itself together gravitationally. It will disperse. A very large volume of the same gas at the same temperature and density will hold itself together. The Jeans length is the minimum size needed for a gas at a certain temperature and density to hold itself together gravitationally. If the volume of gas is larger than the Jeans length it will collapse. If it is smaller, it will expand.

Edited May 22, 2011 by Iggy

[lemur]

I wouldn't put it quite like that, but the gas and dust between the collapsing fragments is eventually boiled away and scattered from the star's radiation.

What's odd about this to me is that the cloud began as a more or less homogenized array of particles and then as it condenses under its own gravity, it reaches a point where it begins fusing, which generates a separation between the part of the cloud held together as a (proto)star and the part that dissipates due to the energy being radiated. It seems like there should be some relationship between Jeans length and the portion of a condensing cloud that doesn't get blown away when fusion begins. Is there?

[Iggy]

What's odd about this to me is that the cloud began as a more or less homogenized array of particles and then as it condenses under its own gravity, it reaches a point where it begins fusing, which generates a separation between the part of the cloud held together as a (proto)star and the part that dissipates due to the energy being radiated.

Why odd?

It seems like there should be some relationship between Jeans length and the portion of a condensing cloud that doesn't get blown away when fusion begins. Is there?

Probably not exactly because the Jeans length doesn't take into account radiation pressure and stellar winds and things like that. It is a better approximation before fusion becomes a significant factor.

Edited May 24, 2011 by Iggy

[lemur]

Why odd?

Because an entropic system of thermal differentiation emerges from a cloud in apparent thermal equilibrium (though perhaps not gravitational).

[Iggy]

Because an entropic system of thermal differentiation emerges from a cloud in apparent thermal equilibrium (though perhaps not gravitational).

Most things lower in temperature when they give up heat. If everything in a system is the same temperature then the system will not spontaneously segregate its temperature.

 

A gravitationally bound system is different. It has a negative heat capacity -- if it gives up heat then its temperature can rise as gravitational potential energy is converted to kinetic energy. For this reason, a gas cloud that is homogeneous in temperature, but not density, can spontaneously turn heterogeneous in temperature without it being a violation of the second law.

[lemur]

Most things lower in temperature when they give up heat. If everything in a system is the same temperature then the system will not spontaneously segregate its temperature.

 

A gravitationally bound system is different. It has a negative heat capacity -- if it gives up heat then its temperature can rise as gravitational potential energy is converted to kinetic energy. For this reason, a gas cloud that is homogeneous in temperature, but not density, can spontaneously turn heterogeneous in temperature without it being a violation of the second law.

This is what I find interesting. So the particles are "falling" toward the center of gravity that has formed, and that converts gravitational potential into kinetic energy and therefore heat. But at the same time that is occurring, the 'fallen' particles are building up density and thus increasing gravitational potential of further particles. So it's like gravity is increasing as is kinetic energy of 'falling particles,' which is causing the inner particles to both heat and condense simultaneously, and so the heat has to be radiated out and that causes outer gasses to heat up and "boil away" as you put it. It almost sounds like that odd situation where a highly pressurized liquid can become superheated without expanding into a gas.