I Understanding the Formation of Stars: A Look at Gravity, Gas Pressure, and Heat

[Justonequestion]

Specifically how did the stars collapse into themselves. If gravity varies by the distance squared and gas pressure varies by the volume cubed, how could a cloud of hydrogen, or any gas for that matter, collapse into itself? Not to mention the fact that the heat increases which would also increase the gas pressure. Can anybody explain this, or point me to a place that explains this? Thank you.

 

[russ_watters]

Specifically how did the stars collapse into themselves. If gravity varies by the distance squared and gas pressure varies by the volume cubed...

Gas pressure doesn't vary by volume cubed.

 

[Justonequestion]

Gas pressure doesn't vary by volume cubed.

Forget my earlier explanation, let me do better. If you take a gas and reduce the distance between every molecule by half, then the volume goes down to 1/8 of the original volume, so 8 times the pressure, but the gravitation force is multiplied by 4. So how does the gas continue to collapse?

 

[russ_watters]

Forget my earlier explanation, let me do better. If you take a gas and reduce the distance between every molecule by half, then the volume goes down to 1/8 of the original volume, so 8 times the pressure, but the gravitation force merely doubles. So how does the gas continue to collapse?

Well, it takes a lot of collapsing before the pressure even registers at all, much less starts to increase enough to matter.

 

[Justonequestion]

Well, it takes a lot of collapsing before the pressure even registers at all, much less starts to increase enough to matter.

Perhaps but to form a star you need, I thought I remembered 18,000,000 degrees of heat? I may be wrong, but isn't it really hot? That's a lot of pressure. I am going to have to see all the math behind it, I don't know of any video of article or of any other resource that addresses this, at least not in detail. They just say that the gravity is greater, but with no explanation.

 

[snorkack]

Forget my earlier explanation, let me do better. If you take a gas and reduce the distance between every molecule by half, then the volume goes down to 1/8 of the original volume, so 8 times the pressure, but the gravitation force is multiplied by 4. So how does the gas continue to collapse?

Force is indeed multiplied by 4, but the area on which the force is applied is also diminished 4 times. So the pressure increases 16 times, which is why isothermal ideal gas is unstable to collapse.

 

[Drakkith]

Also note that the collapsing gas cloud releases energy, reducing the pressure and allowing it to collapse further. This release of energy is why protostars shine well before they ever ignite fusion in their cores.

 

[Drakkith]

Here's an absolutely excellent resource on stellar formation (or so I think): http://bender.astro.sunysb.edu/oab/star_formation_notes/sfnotes.pdf
It should address all of your questions, but it's pretty math heavy. If you don't know the math just try to understand the general idea of what they're getting at.

 

[snorkack]

Also note that the collapsing gas cloud releases energy, reducing the pressure and allowing it to collapse further. This release of energy is why protostars shine well before they ever ignite fusion in their cores.

If an ideal gas cloud, on contraction to 1/8 its original volume, more than doubles its temperature (the relevant part is that its pressure must increase 16 or more times when density increases 8 times) then the cloud is stable to contraction, and can only contract if and when it loses some heat.
If, however, the pressure increases less than 16 times when density increases 8 times, then the cloud is unstable to contraction and can contract even if it does not lose heat, or faster than it can lose heat.

 

[Ken G]

You're saying the same things from different perspectives. If we neglect the pressure in the surrounding medium, then isothermal contraction is always unstable, and adiabatic contraction is always stable. The latter is why you need to get rid of heat in order to get contraction.

 

[snorkack]

You're saying the same things from different perspectives. If we neglect the pressure in the surrounding medium, then isothermal contraction is always unstable, and adiabatic contraction is always stable. The latter is why you need to get rid of heat in order to get contraction.

False.
If the number of degrees of freedom happens to be 3, the gas consisting of single atoms like unionized He or monoatomic H, then yes, increasing density 8 times will increase temperature 4 times.
But other degrees of freedom do NOT contribute to pressure! If there are 6 degrees of freedom (a nonlinear molecule even with internal vibrations frozen) then increasing density 8 times will only increase temperature 2 times... meaning that the condition of stability is marginal.
And there can be unlimited number of degrees of freedom. Like internal vibrations of solid dust grains. Contribute basically nothing to the pressure of gas between the grains, but take up the heat and prevent the gas from warming on contraction.

 

[Ken G]

False.
If the number of degrees of freedom happens to be 3, the gas consisting of single atoms like unionized He or monoatomic H, then yes, increasing density 8 times will increase temperature 4 times. But other degrees of freedom do NOT contribute to pressure! If there are 6 degrees of freedom (a nonlinear molecule even with internal vibrations frozen) then increasing density 8 times will only increase temperature 2 times... meaning that the condition of stability is marginal.

So the stability is marginal. So it's stable. So the answer was "True," not "False," if we are talking about molecules or atoms. Which we are, given that the topic is star formation. As for dust, there's not enough of it to matter to the heat capacity. (I suspect even pure dust would be adiabatically stable, by the way, because the kinetic temperature of the dust would likely rise before it equilibrated with the dust surface temperature, so would re-expand anyway.)

 

[anorlunda]

can only contract if and when it loses some heat.

The OP should pay close attention to that point. If it can't get rid of the energy, no collapse occurs.

The ability to shed energy is related to the reason that asteroids are potato shaped, planets and stars are spherical, galaxies are disc shaped, and that dark matter near galaxies doesn't collapse. I think that is fascinating.

Edit: See this delightful paper, The Potato Radius: a Lower Minimum Size for Dwarf Planets

 

[Drakkith]

The OP should pay close attention to that point. If it can't get rid of the energy, no collapse occurs.

Indeed. Most people think of stars as "generating" energy when in reality the star is simply losing its own internal energy and mass over time as it slowly collapses. Fusion of course throws some curve balls and let's stars become giants at various points in their lives, but the end result is a very dense collapsed object incapable of releasing any more energy.

To the OP, note that this same process happens to gas giants as well. The difference is that they are much less massive and cannot ignite fusion in their cores. So they start out very large and over billions of years they slowly shrink as they lose energy. I believe both Jupiter and Saturn release more energy from their own collapse process than they receive from the Sun.

See here: https://en.wikipedia.org/wiki/Kelvin–Helmholtz_mechanism

The ability to shed energy is related to the reason that asteroids are potato shaped, planets and stars are spherical, galaxies are disc shaped, and that dark matter near galaxies doesn't collapse. I think that is fascinating.

Seconded.

See this delightful paper, The Potato Radius: a Lower Minimum Size for Dwarf Planets

That's perhaps the most amusing title for a paper I've ever seen!

 

[snorkack]

So the stability is marginal. So it's stable. So the answer was "True," not "False," if we are talking about molecules or atoms.

Only if the molecules have no degrees of freedom for vibration... or certain other stuff.

Which we are, given that the topic is star formation. As for dust, there's not enough of it to matter to the heat capacity.

While dust, with small amount of it, has little effect expected in practice, it is a simple example where the internal vibrations have no effect on pressure, it is not the only one.
Hydrogen happens to also have a degree of freedom for vibration.
And dissociation has an effect on heat capacity.
It is true that dissociation adds particles to the system. However, dissociation, like dihydrogen molecules to hydrogen atoms, then hydrogen atoms to protons and electrons, then helium atoms... also slows down the warming of the remaining particles.
How much volume does an unit of heat add to a gas that is undergoing dissociation, compared to gas that is not?

(I suspect even pure dust would be adiabatically stable, by the way, because the kinetic temperature of the dust would likely rise before it equilibrated with the dust surface temperature, so would re-expand anyway.)

The dust would contract at the rate of heat transfer to grains, not at the rate of heat escape from cloud.

 

[anorlunda]

While dust, with small amount of it, has little effect expected i

Hmm,now I have a question of my own.

Heavy elements, supernova remnants, in what form did they get transported to sol's precursor cloud? Atomic? Molecular? Dust?
Do we know the elemental fractions of the precursor cloud?

 

[Ken G]

Only if the molecules have no degrees of freedom for vibration... or certain other stuff.

Whenever answering a question, we can always delve into all the hypotheticals that are not actually relevant in practice. Or, we can just answer the question, as it applies to the real world. Both approaches have value, you and I simply chose different ones.

While dust, with small amount of it, has little effect expected in practice, it is a simple example where the internal vibrations have no effect on pressure, it is not the only one.

Yes, that's what I mean.

How much volume does an unit of heat add to a gas that is undergoing dissociation, compared to gas that is not?

No doubt every individual situation has complexities and subtleties, it's a matter of whether or not that is really what is being asked. But you're not wrong here.

The dust would contract at the rate of heat transfer to grains, not at the rate of heat escape from cloud.

The issue is adiabatic stability, the rate of heat escape is an issue that would play out over much longer timescales. If you had a cloud of pure dust, self-gravitating, and you asked if it was adiabatically stable to contraction, in normal situations the answer is going to be yes. This despite all those internal degrees of freedom for storing heat. The reason is, the pressure balance in a dust cloud is kinematic, it relates to the kinetic energy of motion of the dust-- the heat content within the dust will be irrelevant. If the dust cloud were to contract slightly, the released gravitational energy would go into the kinetic energy of motion of the dust, not the dust temperature. Given enough time, the two would equilibrate, but that's just what you don't have-- enough time. Instead, the dust cloud would simply re-expand, it would be adiabatically stable without ever exciting the internal degrees of freedom in the dust. It's a timescale issue.

 

[snorkack]

Whenever answering a question, we can always delve into all the hypotheticals that are not actually relevant in practice. Or, we can just answer the question, as it applies to the real world. Both approaches have value, you and I simply chose different ones.

And breakdown of adiabatic stability is common in practice and thus highly relevant to real world. It is just that the real mechanism is slightly complex.

Yes, that's what I mean.No doubt every individual situation has complexities and subtleties, it's a matter of whether or not that is really what is being asked. But you're not wrong here.

And dissociation of dihydrogen molecules into hydrogen atoms by effects of heat is highly common process in star formation.

The issue is adiabatic stability, the rate of heat escape is an issue that would play out over much longer timescales. If you had a cloud of pure dust, self-gravitating, and you asked if it was adiabatically stable to contraction, in normal situations the answer is going to be yes. This despite all those internal degrees of freedom for storing heat. The reason is, the pressure balance in a dust cloud is kinematic, it relates to the kinetic energy of motion of the dust-- the heat content within the dust will be irrelevant. If the dust cloud were to contract slightly, the released gravitational energy would go into the kinetic energy of motion of the dust, not the dust temperature. Given enough time, the two would equilibrate, but that's just what you don't have-- enough time. Instead, the dust cloud would simply re-expand, it would be adiabatically stable without ever exciting the internal degrees of freedom in the dust. It's a timescale issue.

Yes, but here we have different timescales.
The timescale at which heat escapes the whole cloud into radiation (or particles) escaping into infinity. And the timescale at which heat is conducted from kinetic energy of dust grains into heat capacity of grains, either by collisions between grains or else collisions of grains with gas molecules between grains.
The latter is dictated by local conditions. And the cloud will contract at the shorter of the two timescales... unless it is shorter than timescale of free fall.

 

[Ken G]

And dissociation of dihydrogen molecules into hydrogen atoms by effects of heat is highly common process in star formation.

That's exactly why I said you need to "get rid of the heat" in order to have contraction. There are multiple ways to get rid of heat. I should have clarified what I meant by getting rid of heat, but note that the phases where heat goes into internal changes would never even happen were there not a prevailing process of heat exchange with the environment. So we have a need for heat loss as Drakkith said, and most of the time this is by heat exchange with the environment, but in isolated instances during the process, the heat can go into internal changes like dissociation or ionization. The issue is, what dog is wagging that tail, and the answer is, heat loss.

Yes, but here we have different timescales.

Yes I know, that was my point.

The timescale at which heat escapes the whole cloud into radiation (or particles) escaping into infinity.

The timescale that matters at the tiny densities we find in the real universe is the timescale to share the kinetic energy with the heat content in the dust. That's what is long.

And the timescale at which heat is conducted from kinetic energy of dust grains into heat capacity of grains, either by collisions between grains or else collisions of grains with gas molecules between grains.

Yup, that's the slow one. Much faster is the sound crossing time, which is why a pure dust cloud would be stable unless there was a way to remove kinetic energy from the dust on the sound crossing time. I'm saying I doubt there is unless one used a pathologically high dust density in a cloud that already has a pathologically high dust content. In short, it's a non-starter. There may be some value in pointing out that it is hypothetically possible, but almost anything is.

 

[snorkack]

Yes I know, that was my point.The timescale that matters at the tiny densities we find in the real universe is the timescale to share the kinetic energy with the heat content in the dust. That's what is long.
Yup, that's the slow one. Much faster is the sound crossing time, which is why a pure dust cloud would be stable unless there was a way to remove kinetic energy from the dust on the sound crossing time. I'm saying I doubt there is unless one used a pathologically high dust density in a cloud that already has a pathologically high dust content. In short, it's a non-starter. There may be some value in pointing out that it is hypothetically possible, but almost anything is.

There are fairly high density dust clouds in the world. Even in solar system... such as rings of Saturn.
Over the formation of Sun, the composition of Solar System must have somehow changed from gas with small admixture of dust (solar composition) to still mostly gas but with appreciable enrichment of dust (so that the heat capacity of dust grains and latent heat of evaporation of methane snow became appreciable contributors to total heat capacity) to mostly dust but with still appreciable amount of gas (contributing to viscosity and friction) to almost pure dust with little gas left (as we now see in asteroid belt or rings of Saturn).

 

[Ken G]

There are fairly high density dust clouds in the world. Even in solar system... such as rings of Saturn.

So you think the rings of Saturn have a dust temperature that is the same as the kinetic temperature of the orbits? I doubt that.

Over the formation of Sun, the composition of Solar System must have somehow changed from gas with small admixture of dust (solar composition) to still mostly gas but with appreciable enrichment of dust (so that the heat capacity of dust grains and latent heat of evaporation of methane snow became appreciable contributors to total heat capacity) to mostly dust but with still appreciable amount of gas (contributing to viscosity and friction) to almost pure dust with little gas left (as we now see in asteroid belt or rings of Saturn).

That's not the formation of the Sun, it's the formation of what orbits the Sun, such as the formation of the Earth. But even so, I doubt there are any situations where the equilibration timescale between the temperature of the dust, and the kinetic temperature of the dust motion, ever competes with the dynamical timescale. So in none of those situations would adiabatic stability ever be challenged by the internal degrees of freedom in the dust.

 

[sophiecentaur]

The Energy Consideration.:
I'm not sure that the point has been made explicitly here that the total Gravitational Potential Energy of the cloud was there all the time - even when the vast gas cloud was highly dispersed. Each atom heads to the position (region) of minimum Potential Energy and that, as with anything that 'drops' ends up transferring some energy from PE to KE (temperature of the gas). Luckily for us, the end product of the collapse was a high enough temperature and pressure to initiate Fusion, which involved a significant increase in the amount of energy around and produced all the elements we are made of.

 

[anorlunda]

Don't forget angular momentum and rotational kinetic energy.

A star can spin very fast, but a galaxy can't. That potato radius paper linked above discusses that. Galaxies have a hard time shedding rotational energy.

We have the plane of the ecliptic. Doesn't that suggest that our primordial cloud became disc shaped before collapse?

 

[sophiecentaur]

We have the plane of the ecliptic. Doesn't that suggest that our primordial cloud became disc shaped before collapse?

I'm having difficulty with that one. A primordial cloud has only H He and Li (I believe) as it is the result of the big bang. A primordial cloud involved in our origin would have had to form our Galaxy, in which the first stars and Novae created all the heavy elements. I guess that would / could explain the saucer shape of our particular galaxy. But the galactic plane is not parallel with the Ecliptic. (just as well or the lovely Milky Way wouldn't be as visible)
Edit / addition
I guess the irregularities in the original clouds could have been expected to have random values of angular momentum about some mean (zero or otherwise). In classical terms, the way Moment of Inertia works, the rotation would increase proportional to 1/r² where r is the radius of the cloud / galaxy so there could be an embarrassing amount or rotation after just a bit of asymmetry to start with.

 

[anorlunda]

Think a bit more local. Think of those pictures of the nebula in Orion. Every star must begin with the localized collapse of a region in that nebula. That is what I mean by Sol's primordial cloud.

 

[sophiecentaur]

Don't forget angular momentum and rotational kinetic energy.

Yes indeed. Funny how we all go along happily with the idea of Conservation of Momentum but we rarely concern ourselves with the actual mechanism (it's more a sort of religion) whereas we always try to find and explain energy transfers.

 

[sophiecentaur]

Sol's primordial cloud.

Then your conclusion has to be right.
However, I always understood 'primordial' necessarily referred to the product of the big bang. You are referring to a nebula, consisting of non-primordial stuff, I think. I did a quick search and all the links I found seemed to use primordial as big bang related.
Apparently, there's still some of the stuff around, https://stardate.org/radio/program/primordial-clouds. Somehow, I find that a tad creepy.

 

[anorlunda]

Whoops, maybe precursor cloud would be a better word.