Are there widespread misconceptions about degeneracy pressure?

[zonde]

Pressure is a diagonal stress-energy tensor. If you look up the definition of the stress-energy tensor, you will see no reference to any collisions anywhere.

Why exactly I should look up definition of the stress-energy tensor? We are talking about pressure (that's kinetic theory) and degenerate matter (that's QM). We are not talking about GR.
And GR is not replacement of either theory. Therefore it does not talk about collisions.

That's another widespread myth about pressure.

You mean that kinetic theory is a myth?

The main thing to get is that pressure gradients produce forces on fluids, which simply means, gradients in momentum fluxes generate momentum deposition when you average over the fluid. The momentum deposition has nothing to do with collisions, it is just how momentum flux gradients work, they yield momentum piling up in a volume. The thing you need collisions for is to keep the fluid behaving nicely, like with locally isotropic distribution functions and so forth (so the stress-energy tensor stays diagonal and pressure takes on its simple meaning). You don't even need collisions off a boundary, the force produced by pressure is perfectly capable of acting on the fluid itself and not anything else.

Why are you bringing into discussion all this "pressure gradient", "momentum flux", "momentum deposition", "momentum flux gradient".
And what the hell do you mean by that "the stress-energy tensor stays diagonal and pressure takes on its simple meaning"? Stress-energy tensor does not describe pressure. Pressure is parameter in stress-energy tensor. You put it into get stress-energy tensor.

Indeed, I think you raise an interesting point, that something quite strange must occur when a degenerate gas encounters a wall. The problem is that you can no longer treat them as being in momentum eigenstates if you have a wall, so the energy eigenstates are not momentum eigenstates any more and life gets a bit complicated, but I presume they induce the normal pressure that an ideal gas would at the same energy density.

You have energy eigenstate for a particle in a potential well. A wall does not create potential well.

 

[Ken G]

What's the difference between increasing the temperature and adding heat?

Well, adding heat will often increase temperature, but they are still two different things. In particular, the issue is, are you tracking the amount of heat added, or are you tracking the temperature rise? Often, in physics, it is wise to follow the energy.

 

[Ken G]

Why exactly I should look up definition of the stress-energy tensor? We are talking about pressure (that's kinetic theory) and degenerate matter (that's QM). We are not talking about GR.

The stress-energy tensor appears in kinetic theory too. What I'm talking about has nothing to do with GR. But let's not worry about the stress-energy tensor, let's just assume the particle distribution functions are isotropic, so pressure is a scalar then anyway. The point is, pressure is a momentum flux density, it is a moment of the particle distribution function. There is never any need to reference anything about collisions when one is determining the pressure in kinetic theory.

You mean that kinetic theory is a myth?

Certainly not, I mean that the myths I refer to are incorrect applications of kinetic theory. I've described the correct way to apply kinetic theory above. A big key is, the pressure of a nonrelativistic gas (ideal or degenerate) is 2/3 the kinetic energy density. This is an elementary result of kinetic theory, are we not in agreement on that fact?

Why are you bringing into discussion all this "pressure gradient", "momentum flux", "momentum deposition", "momentum flux gradient".

Because they are all relevant to the topic of pressure.

And what the hell do you mean by that "the stress-energy tensor stays diagonal and pressure takes on its simple meaning"? Stress-energy tensor does not describe pressure. Pressure is parameter in stress-energy tensor. You put it into get stress-energy tensor.

Do we not agree that the diagonal elements of the stress tensor is the pressure when the particle distribution function is isotropic? (My reference to the stress-energy tensor was just being general, we don't need any relativity here, the stress tensor suffices.)

You have energy eigenstate for a particle in a potential well. A wall does not create potential well.

Huh?

 

[Drakkith]

According to my limited understanding, the following occurs:

1. A star runs out of hydrogen and the core begins to contract.
2. Once the density of the plasma reaches a certain amount the electrons in the core start to become degenerate.
3. This degeneracy slows further contraction because it requires that electrons be forced into higher energy states. The higher the energy of that state is, the higher the pressure must be in order to force an electron into that state. This manifests as a "pressure" or "force" that resists the continued contraction of the core.
4. Throughout this process, the temperature of the ions in the core has been increasing. Once it reaches the point where an appreciable amount of helium fusion is occurring, a "helium flash" occurs.
5. This helium flash occurs because the extra energy released by the fusion events does little to the degenerate electrons. But I'm not quite sure why. (Or maybe it does and I just don't know what happens)
6. The energy that is not given to the electrons is given to the ions, which further increases both the temperature and the fusion rate.
7. This increase in temperature would, in a non-degenerate material, cause an expansion, cooling the gas and regulating the rate of fusion. However, because the majority of the pressure comes from the degenerate electrons, the increased pressure from the ions as they heat up only adds a small amount of total pressure, causing very little expansion even though the temperature has doubled, tripled, etc. So while there is some expansion, it is so much less than normal that we say there is no regulation of fusion, leading to a runaway reaction.
8. This extreme burning rate continues until the temperature is so high that thermal pressure pushes the core out and the entire gas becomes non-degenerate again. This also allows normal regulation of the fusion reaction rate.

Now, it appears that you're issue is with the explanation that a helium flash occurs because expansion doesn't take place. From your earlier post:
Internal energy passes from the electrons to the ions, this is the essential cause of the helium flash that you will basically not find in any textbook because they have all bought off on the myth that the helium flash has something to do with a lack of expansion work being done, which it does not.

However this appears to contradict something else you said:
Assuming it's highly degenerate originally, not much expansion will occur-- you won't have to add much heat to get the temperature to rise. Then to double the temperature, all you have to do is get the degeneracy parameter to drop by a factor of 2, but if it is already very high, it will still be highly degenerate. So it won't change the gas much.

It seems to me that when we take into consideration the non-degenerate nature of the ions, then the fact that very little expansion occurs for a large increase in temperature is pivotal in understanding a helium flash. (In addition to the electrons passing energy to the ions)

Now, a few questions.

If degeneracy pressure is the result of an electron needing to be forced into a higher energy state, then would adding additional energy to an electron through heating mean that the extra pressure it exerts is thermal pressure and not degeneracy pressure? Also, if this particle is now in a higher energy state, does that mean that its previous state has been "opened up" and is available for another electron to take?

Finally, and this just occurred to me, it seems that we have two sources of pressure of degenerate particles. First, you have the pressure due to the actual degeneracy. That is, you need a certain amount of force to make a particle go into a higher energy state, which is where the resisting force comes into play. Then, on top of that, you have the pressure resulting from the extra kinetic energy of the particles above the amount which was given to them by forcing them into a high energy state.

Is any of this correct?

 

[Ken G]

According to my limited understanding, the following occurs:

1. A star runs out of hydrogen and the core begins to contract.

Yes, it contracts because it is losing heat and has no source to replace it.

2. Once the density of the plasma reaches a certain amount the electrons in the core start to become degenerate.

Yes, this begins to drive down the ratio kT/U,where U is the average internal energy per electron.

3. This degeneracy slows further contraction because it requires that electrons be forced into higher energy states.

The degeneracy slows the contraction because it lowers the T, so it reduces the heat transport rate. That's the only reason the contraction slows. This is the point: the virial theorem tells you how much energy is going into the particles, who cares what individual states they are in? The only thing that cares is one thing: the temperature. And the only thing that cares about that is the heat transport rate. All else is utterly insensitive to degeneracy, in particular the pressure.

4. Throughout this process, the temperature of the ions in the core has been increasing. Once it reaches the point where an appreciable amount of helium fusion is occurring, a "helium flash" occurs.

I don't know it the ion temperature increases continuously, that's a rather complicated issue because the average energy per particle is increasing (the virial theorem), but more and more of that increase goes into the electrons as they go degenerate. It is possible that at some point they actually rob the ions of energy, but probably not-- I'd be willing to believe the ion temperature rises continuously.

5. This helium flash occurs because the extra energy released by the fusion events does little to the degenerate electrons.

Now we are getting even deeper into the myths, and I haven't seen a textbook or website that gets it even close to correct. What actually happens is fusion releases heat, which softens the electron degeneracy, which shifts energy from the electrons to the ions, which increases the fusion rate. The shifting only happens because the electrons are degenerate but the ions are not (it wouldn't happen if the ions were equally degenerate, for example), and it is accompanied by expansion that actually reduces the total internal energy of the gas (that's the virial theorem).

6. The energy that is not given to the electrons is given to the ions, which further increases both the temperature and the fusion rate.

Energy is not given to the electrons, energy is lost from the electrons (quite a lot of it, actually). That's one thing the textbooks get completely wrong, they always say or at least suggest that energy is piling up in the gas as a whole, as if the virial theorem for some reason no longer applied!

7. This increase in temperature would, in a non-degenerate material, cause an expansion, cooling the gas and regulating the rate of fusion. However, because the majority of the pressure comes from the degenerate electrons, the increased pressure from the ions as they heat up only adds a small amount of total pressure, causing very little expansion even though the temperature has doubled, tripled, etc.

This is where it gets subtle. It is true that the temperature spikes immediately, if the gas is highly degenerate. But that doesn't require much energy input, it causes a shift of energy from electrons to ions, and happens with little expansion. But again, it's not much heat input yet-- as the flash proceeds, and a lot of heat is added, the gas will of course expand exactly like an ideal gas would, because that's what the virial theorem says it will do. The real point is, expansion, or the imagined lack thereof, has nothing to do with the helium flash, itis a complete red herring. The cause of the flash is the weird temperature behavior, what the total energy is doing is a mundane application of the virial theorem and invokes no contrasts between ideal and degenerate gas.

8. This extreme burning rate continues until the temperature is so high that thermal pressure pushes the core out and the entire gas becomes non-degenerate again. This also allows normal regulation of the fusion reaction rate.

The loss of degeneracy does indeed allow stabilization, because the weird temperature behavior of degeneracy was the cause of the instability.

Now, it appears that you're issue is with the explanation that a helium flash occurs because expansion doesn't take place.

Again, the key point here is what expansion is doing. In either degenerate or ideal gas, if you add heat, the gas expands, and the internal energy drops. So if you are tracking energy, you don't see anything at all different about expansion. So expansion has nothing to do with the helium flash, it's just not the important physics. The important physics is how the temperature of a degenerate gas behaves, even as it is expanding, even as it is doing work, and even as its total internal energy is dropping, all while heat is being added by runaway fusion. If that picture does not come through in the textbooks, it is because they are completely missing the mark. They are not helping us understand what degeneracy does, they are just propagating a set of myths that, at best, obfuscate the real physics, and at worst, make statements that are just demonstrably wrong. The most common wrong statements are those that violate the virial theorem.

However this appears to contradict something else you said:
Assuming it's highly degenerate originally, not much expansion will occur-- you won't have to add much heat to get the temperature to rise. Then to double the temperature, all you have to do is get the degeneracy parameter to drop by a factor of 2, but if it is already very high, it will still be highly degenerate. So it won't change the gas much.

There is no contradiction. If you track the temperature, you can make correct statements about the helium flash, but if you are not tracking the energy, then you really don't understand the helium flash, because tracking energy is at the core of all good physics.

It seems to me that when we take into consideration the non-degenerate nature of the ions, then the fact that very little expansion occurs for a large increase in temperature is pivotal in understanding a helium flash.

No, expansion is irrelevant, because what causes the helium flash is about energy being transferred from the electrons to the ions. Expansion does not play any role in that transfer, and indeed the expansion is both present, and causing the total energy to drop, exactly as it does in an ideal gas. Expansion just is the wrong thing to focus on, it is not a player in the helium flash, it's just a routine application of the virial theorem and the helium flash isn't about the virial theorem.

If degeneracy pressure is the result of an electron needing to be forced into a higher energy state, then would adding additional energy to an electron through heating mean that the extra pressure it exerts is thermal pressure and not degeneracy pressure?

Degeneracy pressure is not caused by that, because there's really no such thing as degeneracy pressure. There's just gas pressure, and it is always caused by the same thing-- adding energy to the gas. If you add energy to a gas, and make its internal energy density rise, it's pressure rises. If the gas expands and the kinetic energy density drops, then the pressure drops. This is called the virial theorem, and it has nothing to do with degeneracy. Degeneracy is about heat transport, and what people call "degeneracy pressure" is just a value that the garden variety gas pressure reaches when the temperature of a degenerate gas reaches zero. Degeneracy is all about temperature, but since it will drive the temperature to zero at some finite pressure (for given density), that fact naively gets called "degeneracy pressure." It's just a name, like the "Chandrasekhar mass", but it is not a type of pressure any more than the Chandrasekhar mass is a type of mass.

Then, on top of that, you have the pressure resulting from the extra kinetic energy of the particles above the amount which was given to them by forcing them into a high energy state.
Is any of this correct?

It's affected by the myths that degeneracy pressure is a different kind of pressure. It's just kinetic energy density, that's all gas pressure ever needs to know.

 

[Drakkith]

If I have any hope of understanding this, I'm going to need you to explain what the particles are doing when they are degenerate and where exactly this gas pressure comes from.

 

[Ken G]

Where gas pressure comes from is just one place: the momentum flux of the particles. That means, pressure is the rate that momentum crosses any imaginary surface, per area and per time. That's all gas pressure ever is, and that's exactly what "degeneracy pressure" is. There's no quantum mechanics at all in where the pressure comes from. Quantum mechanics, and degeneracy, appears when you want to know what the temperature is, in cases where you already know the density and energy of the particles some other way, like you have a virial theorem or you have been tracking where the energy is going. There could be other scenarios, like if the gas is in thermal contact with a reservoir of known temperature, but that's not how white dwarfs work-- they work by the virial theorem, and they have a history of contraction, and together that's what determines the pressure. But the history of contraction is going to need to know the heat transfer rate, and that is going to require knowing the temperature, and so that's where degeneracy comes in-- determining the temperature, given the density and energy that come from the history of contraction.

What degeneracy is doing, and the way it sets the temperature, is altering the way the known amount of energy is distributed among the particles. In an ideal gas, the energy is distributed via a Maxwell-Boltzmann distribution, and in a degenerate gas, it is by a Fermi-Dirac distribution. But that only affects the temperature and the entropy of the gas-- not the pressure, the pressure is still set by the kinetic energy density, regardless of how the energy is partitioned among the particles. However, as I keep stressing, the history of contraction that gives rise to the kinetic energy density and the pressure is affected by the temperature, so it is affected by the degeneracy. Degeneracy is about temperature, that is the takeaway message, because the way energy is partitioned among the particles is what sets the thermodynamical properties of the gas, including the temperature and entropy. Pressure and energy density are mechanical properties, and can often be known with no reference to the thermodynamics, such as if you measure the mass and radius of a star (as was done for Sirius B, the first white dwarf ever discovered). The pressure in Sirius B, and where it came from (momentum flux density, which can be inferred from kinetic energy density) could have been determined long before there was anything called quantum mechanics.

 

[Drakkith]

Isn't the momentum of the electrons so high because they are degenerate though?

 

[Ken G]

The momentum is high because the electrons have energy. Where does the energy come from? How is that any different for an ideal gas? Recall the virial theorem.

 

[Drakkith]

The momentum is high because the electrons have energy. Where does the energy come from? How is that any different for an ideal gas?

To my understanding it comes from the force of gravity. Thus adding mass to degenerate matter increases the force which then increases the number of states the electrons can occupy, reducing the size of the object and increasing its density.

The difference between an ideal gas and a degenerate gas, to my understanding, was that it isn't possible to get rid of this energy. There are no states for the electrons to drop into and they cannot move to another location in space.

Recall the virial theorem.

I know nothing of the virial theorem other than what I briefly read on wiki.

 

[Ken G]

To my understanding it comes from the force of gravity. Thus adding mass to degenerate matter increases the force which then increases the number of states the electrons can occupy, reducing the size of the object and increasing its density.

Yes, the energy comes from gravitational contraction, just like for an ideal gas. The energy comes from the same place, so the pressure comes from the same place. So it is no kind of special type of pressure.

The difference between an ideal gas and a degenerate gas, to my understanding, was that it isn't possible to get rid of this energy. There are no states for the electrons to drop into and they cannot move to another location in space.

Exactly, it's not a different type of pressure, nor a different type of energy, nor does the energy have a different source. All that is the same, what is different is what can happen to the energy-- the door is closed on letting the energy leave. That is a temperature effect, not a pressure effect.

 

[Drakkith]

So you're saying that the pressure from both degenerate and non-degenerate gasses comes from the kinetic energy of the particles and because of this degenerate pressure isn't "special"?

I guess I can see what you're getting at.

 

[Ken G]

Right. And note this is not, by a long shot, the only big misconception that appears when people talk about degeneracy pressure. Most of the rest could be summarized by saying that many textbooks and course websites suggest that degeneracy pressure can somehow suspend the virial theorem, and thus avoid expansion when heat is added and so forth.

 

[zonde]

The point is, pressure is a momentum flux density, it is a moment of the particle distribution function. There is never any need to reference anything about collisions when one is determining the pressure in kinetic theory.

and similar quote ...

Where gas pressure comes from is just one place: the momentum flux of the particles. That means, pressure is the rate that momentum crosses any imaginary surface, per area and per time. That's all gas pressure ever is, and that's exactly what "degeneracy pressure" is.

Now I got it. If you define pressure that way then the other things you say make sense.

But I am not sure that it is good idea to redefine term that already has very well established classical definition. You will just increase confusion. Why don't you say then "momentum flux density" instead of "pressure"? Or that "pressure" and "degeneracy pressure" is exactly the same if we look only at "momentum flux density"?

Apart from that. The thing about high increase in temperature when little heat is added and vice versa. Wouldn't it be like phase change? Say like between fluid and superfluid.

 

[Chronos]

How do degenerate matter stars cool? They obviously emit a large amount of intrinsic energy [unlike black holes]. Do 'old' degenerate matter stars expand or collapse as they cool?

 

[Ken G]

Now I got it. If you define pressure that way then the other things you say make sense.

It isn't me who defines pressure that way, that's just what pressure is in any fluid model of a gas. It's what has to go into the momentum conservation equation.

But I am not sure that it is good idea to redefine term that already has very well established classical definition.

Again, if you would like to use a momentum conservation equation, which I presume you do, in any fluid model of a gas, then you are going to be forced to use my definition, as there just isn't any other that is going to actually work.

Why don't you say then "momentum flux density" instead of "pressure"?

Because people don't say "degeneracy momentum flux density," they say "degeneracy pressure." And when they say that, they are always talking about the momentum flux density of the fermions.

Or that "pressure" and "degeneracy pressure" is exactly the same if we look only at "momentum flux density"?

If they are physically the same thing, then they are physically the same thing period, no matter what we are "only looking at." But I will agree with you that, to get past the misconceptions, we must also get past the misconceptions about what people think the language means.

Apart from that. The thing about high increase in temperature when little heat is added and vice versa. Wouldn't it be like phase change? Say like between fluid and superfluid.

It's not strictly a phase change, those have particular definitions that are not met. But the analogy isn't bad, I think it helps to see that connection.

 

[Ken G]

How do degenerate matter stars cool?

First we must clarify what you mean by "cool", because that term gets used in two very different ways, causing lots of confusion. People who tend to automatically associate temperature with energy per particle will use "cool" interchangeably to mean either a drop in temperature, or a drop in energy per particle. So we must first recognize that these are not at all the same thing, and establish which meaning you take here. I will presume you are taking the official meaning of "cool" as "drop in temperature."

They obviously emit a large amount of intrinsic energy [unlike black holes]. Do 'old' degenerate matter stars expand or collapse as they cool?

The main thing to get about degenerate matter is that the degeneracy is acting to lock up huge amounts of internal kinetic energy into modes that are not thermally accessible, and cannot be lost from the system as radiated heat. This is actually the reason that white dwarfs are generally quite dim, it's because they hang on so steadfastly to their energy. Since they lose heat only slowly, they evolve only slowly.

 

[zonde]

If they are physically the same thing, then they are physically the same thing period, no matter what we are "only looking at." But I will agree with you that, to get past the misconceptions, we must also get past the misconceptions about what people think the language means.

Well but we are interested in other things related to pressure. First of all at what speed pressure change will travel across gas. If we say that two pressures are the same thing we would assume that related things are similar too. But I believe that degenerate matter is much better carrier of "momentum flux density" change than non-degenerate matter.
Another thing is when we have more complex gas consisting of different types of particles (electrons and ions for example). In ordinary matter you expect that homogeneous mixture of particles will stay that way when expanding. But in mixture of degenerate particles and non-degenerate particles "momentum flux density" will be different for both kinds. So it seems posible that they might separate a bit.

 

[Ken G]

First of all at what speed pressure change will travel across gas.

Yes, we are interested in the sound speed, which is the square root of dP/drho. So we are interested in how P depends on rho, which is the same way P depends on rho in all garden variety forms of gas pressure, which degeneracy pressure is.

But I believe that degenerate matter is much better carrier of "momentum flux density" change than non-degenerate matter.

As I have been stressing, the differences between degenerate gas and ideal gas are not mechanical, and have nothing to do with pressure, they are thermodynamic, and have everything to do with heat transport. So no difference in "carrying momentum flux", but a lot of difference in carrying heat. They are very good conductors of heat.

Another thing is when we have more complex gas consisting of different types of particles (electrons and ions for example). In ordinary matter you expect that homogeneous mixture of particles will stay that way when expanding. But in mixture of degenerate particles and non-degenerate particles "momentum flux density" will be different for both kinds. So it seems posible that they might separate a bit.

It is indeed very important that degenerate electrons can mix with ideal ions. This is just another reason why it is important to really understand what degeneracy does, and what it does not do that is often attributed to it, so we can actually understand what happens when you mix degenerate and ideal gases. That's how you get past all the baloney that is said about helium flashes and so forth.

 

[Hornbein]

what speed pressure change will travel across gas.

You are asking about the speed of sound. In superfluid neutron star cores it is half the speed of light. I don't know about white dwarfs, but I imagine the speed of sound is quite high.

 

[Ken G]

One way to estimate it is to realize that typical white dwarfs have enough energy per ion to fuse helium but not carbon, so that should mean a few thousand km/s for the ion speeds, roughly, maybe 1% of c. It will depend on the mass of the white dwarf, but that's pretty fast, especially over those small distances. The sound crossing time might be a few seconds, though much less as the mass approaches the Chandrasekhar limit.

 

[blaisem]

As part of my chemistry stat. mech. course, I was trying to figure out what the Fermi temperature signifies when I stumbled upon this thread. I have a couple questions if you are still around.

1.

So you're saying that the pressure from both degenerate and non-degenerate gasses comes from the kinetic energy of the particles and because of this degenerate pressure isn't "special"?

Right.

The point I took from your proposal, Ken G, is that the distinction between "degeneracy pressure" and "thermal pressure" is arbitrary because both can be defined by the same term, the kinetic energy density, which is dependent on the temperature.

Using seemingly credible thermo/quantum dynamic definitions, one can derive a formula for the degenerate pressure that is proportional only to the density of the gas. There are no velocity or temperature parameters in the result or the derivation. How can "degeneracy pressure" be related to the kinetic energy density? Or am I misunderstanding the result? I know the particles are still moving, which implies kinetic energy, but the math doesn't state a correlation between the kinetic energy and the pressure that I can see.

For my reference for the derivation (not sure how to clearly express the math here; unfortunately, this link has some of the exponents written upside down):

http://people.duke.edu/~ad159/files/p112/28.pdf

Summary: Assuming T = 0, the energy of all the states up to the Fermi energy is summed, then the derivative taken with respect to the volume.

2.

Another question occurred to me if you have time, but I suspect this one is most likely me missing some basic thermodynamic facts:

How does the temperature rise if the heat isn't "piling up"? (Not really even sure what that means)

It means the internal energy of the gas is dropping throughout the helium flash, the way it is normally modeled. The reason it is dropping is exactly the process that is often said is not happening-- expansion work. The temperature rises because that's what happens when degenerate gas is heated, expands, and has its internal energy drop.

Why must the temperature rise? The expansion work results in a decrease in internal energy, but according to the Clausius Theorem, isn't it possible to add heat and effect a change in entropy, not temperature? Or is the keyphrase "degenerate gas," so the entropy increase is hindered due to limited access to states; as a result, the temperature must rise?

Thanks a lot for your time and insight. I have enjoyed trying to follow along with this thread, and I appreciate patience if my questions appear too uninformed of general knowledge.

 

[Ken G]

The point I took from your proposal, Ken G, is that the distinction between "degeneracy pressure" and "thermal pressure" is arbitrary because both can be defined by the same term, the kinetic energy density, which is dependent on the temperature.

Correct, except that I would have put it a little differently at the end there, I would have said that the temperature is dependent on the kinetic energy density and the particle statistics.

Using seemingly credible thermo/quantum dynamic definitions, one can derive a formula for the degenerate pressure that is proportional only to the density of the gas. There are no velocity or temperature parameters in the result or the derivation.

Yes, and this is what leads to the misconceptions right there. The assumption you make when you do that derivation is that the temperature is zero. Then, amazingly, the result ends up not depending on temperature! People make great hay out of this lack of temperature dependence, seemingly forgetting they they already put the temperature in. What they should really say is that degenerate gas, unlike ideal gas, still has finite pressure at zero temperature, but that is not the same thing as pressure being independent of temperature. A fully degenerate gas will also have its first derivative of pressure with respect to temperature be zero, so we could say that the temperature dependence locally vanishes at T=0 in that case, but this is not true in most situations. Generally, dP/dT is not zero, even at T=0, say for astrophysical plasmas or normal metals, because of the presence of ions.

How can "degeneracy pressure" be related to the kinetic energy density? Or am I misunderstanding the result?

There are also derivations that will give the same answer you get, except from the perspective of kinetic energy density. It's not controversial.

I know the particles are still moving, which implies kinetic energy, but the math doesn't state a correlation between the kinetic energy and the pressure that I can see.

You should be able to find it, just calculate the kinetic energy density. The pressure will be 2/3 of that, if the gas is nonrelativistic. It makes no difference if it is degenerate or ideal.

Summary: Assuming T = 0, the energy of all the states up to the Fermi energy is summed, then the derivative taken with respect to the volume.

Right, assuming T=0. That is always done. I wonder why it is also then concluded that the result is independent of T? It's just a mistake, but a subtle one-- and a common one.

Why must the temperature rise?

Because fusion is adding heat, which is breaking the degeneracy of the electrons. It is that degeneracy that was causing T to be so low, and robbing the ions of their kinetic energy. Lifting that degeneracy causes T to rise, even though the average kinetic energy of the particles is falling (as per the virial theorem).

The expansion work results in a decrease in internal energy, but according to the Clausius Theorem, isn't it possible to add heat and effect a change in entropy, not temperature?

Adding heat certainly raises the entropy, but in this case, that also raises the temperature. The temperature of a completely degenerate gas is zero, and its entropy is therefore minimal. Anything that adds heat to something at zero temperature will raise its temperature, and even if the temperature isn't exactly zero, it still rises if the gas is highly degenerate.

Or is the keyphrase "degenerate gas," so the entropy increase is hindered due to limited access to states; as a result, the temperature must rise?

Both the entropy and the temperature rise. Don't worry, your questions are very good, they are just what you should be wondering about.

 

[virgil1612]

I read with real excitement this thread about degeneracy pressure. It is important for me as I’m teaching elements of stellar structure and evolution, so once a year I have to present my students (in a qualitative way, no formulae) the gradual shift between H burning and He burning and the helium flash. I can confirm that everywhere I read I find the same description of the helium flash as a heat build up (because there is no expansion initially), finally producing a runaway fusion reaction.

This is what I understood following this thread:

1. The He core contracts, heats up and becomes more and more dense. At this point a partial degeneracy for the electrons starts to develop. Gradually, with compression, the degeneracy parameter increases, lowering more and more the electron temperature. While T_e decreases, the temperature of the He ions increases because they form a classical gas, they are non-degenerate.

There is something here that I don’t understand. How can we talk about two temperatures? For the electrons and for the He ions? Maybe I’m missing something? I would love to understand more about this.

2. When He burning begins the electrons are degenerate. Heat coming from He fusion goes to the electrons and He ions. As a result the electron temperature increases, as the degeneracy lessens. From an energy point of view, the core starts to expand, as required by the virial theorem and heat flows from the electrons to the ions (but such that globally the internal energy decreases, as required by virial theorem). This heat flowing from the electrons to the ions is in fact responsible for the He flash.

If what I say is true, I wonder if Ken G could offer me a link to some kind of equations explaining the heat flow between electrons and ions. Something quantitative but not quite the full-fledged treatment, I tried to read some physics of partially degenerate gazes and it’s just too difficult.

Thanks in advance for comments and critics.

Virgil.

 

[Ken G]

I can confirm that everywhere I read I find the same description of the helium flash as a heat build up (because there is no expansion initially), finally producing a runaway fusion reaction.

Yes, they always say there is no expansion, which is wrong. If you put kinetic energy into a gas (as fusion certainly does), it expands, period. It makes no difference at all if the gas is degenerate, degeneracy is a thermodynamic effect not a mechanical one.

1. The He core contracts, heats up and becomes more and more dense. At this point a partial degeneracy for the electrons starts to develop. Gradually, with compression, the degeneracy parameter increases, lowering more and more the electron temperature. While T_e decreases, the temperature of the He ions increases because they form a classical gas, they are non-degenerate.

Not quite, one would normally assume the temperatures of He ions and electrons is equilibrated, so they both rise. The rising degeneracy just means that the kT of the electrons is way less than the average kinetic energy of each electron. That's why the kinetic energy is in the degenerate electrons, not the ideal-gas ions, a standard effect of electron degeneracy.

There is something here that I don’t understand. How can we talk about two temperatures? For the electrons and for the He ions? Maybe I’m missing something? I would love to understand more about this.

Just one temperature, the key is that kT only reflects the kinetic energy of the ions, it is way less than the kinetic energy of each electron. That's what you mean by a rising degeneracy parameter.

2. When He burning begins the electrons are degenerate. Heat coming from He fusion goes to the electrons and He ions.

Remarkably, it all goes into the ions-- essentially none goes into the electrons. This is the key to the whole business. The reason for this is that putting heat in reduces the degeneracy parameter, which passes energy from the electrons to the ions. It works out to be exactly the right amount so that the electron kinetic energy does not change due to the added heat. I worked this out myself, I don't know where else it is worked out but it is an elementary result, it certainly should be in a lot of places (instead of the incorrect idea that expansion does not occur). By the way, the same thing happens when you put heat into a metal spoon-- the heat goes into the ions, even though the electrons have most of the kinetic energy in there.

As a result the electron temperature increases, as the degeneracy lessens.

Right, the causation there is that as a result of the reducing degeneracy (adding heat, as opposed to doing compression work, always reduces degeneracy), the electron temperature increases. That's what keeps it matched to the rising ion temperature. But the electron kinetic energy does not rise-- only its temperature. In fact, the electron kinetic energy will fall, because it will do expansion work, and that will come from the electrons. But the temperature rises even as the kinetic energy falls. This is the crucial thing about a falling degeneracy parameter, and is what actually leads to the helium flash.

From an energy point of view, the core starts to expand, as required by the virial theorem and heat flows from the electrons to the ions (but such that globally the internal energy decreases, as required by virial theorem).

Yes, exactly, this is just what you never see explained correctly.

This heat flowing from the electrons to the ions is in fact responsible for the He flash.

Here there is a little freedom in what you say the heat is doing, it's like following money in a complicated bank transaction. But I would say the simplest way to look at it is what happens in the net-- in the net, when fusion initiates, heat is added strictly to the ions. This adds to the pressure, causing expansion, which causes the electrons to do expansion work, causing the electron kinetic energy to drop. So what the ions and electrons are doing is largely decoupled in the net-- you dump heat in, it all goes into the ions, causing the fusion rate to run away. The gas expands normally, causing the electrons to lose kinetic energy, but the ions (unlike in the Sun) are unaffected, as they are not asked to provide any of that expansion work. So the runaway is not because there is no expansion, it is because the ions don't care about the expansion (except to the extent that the density drops, but this is of little consequence given the extreme temperature sensitivity of fusion).

If what I say is true, I wonder if Ken G could offer me a link to some kind of equations explaining the heat flow between electrons and ions. Something quantitative but not quite the full-fledged treatment, I tried to read some physics of partially degenerate gazes and it’s just too difficult.

I cannot cite a refereed reference that displays my argument. I can link you to the calculation I did, that shows everything I just explained. Indeed, I attempted to get this published in the American Journal of Physics, but they did not feel that the helium flash had a broad enough appeal. The calculation can be found in equations (20) through (26) of http://astro.physics.uiowa.edu/~kgg/research/degeneracy/gaspressure.pdf . I would prefer to cite a published paper, as per the requirements of this forum, but I thought you would want to know the truth of the situation, so just work through those equations. That I don't know where else this explanation is published is pretty much the problem, and the source of my disappointment with AJP.

 

[virgil1612]

Thank you for the link to that document. It is exactly what I was looking for.

Not quite, one would normally assume the temperatures of He ions and electrons is equilibrated, so they both rise. The rising degeneracy just means that the kT of the electrons is way less than the average kinetic energy of each electron. That's why the kinetic energy is in the degenerate electrons, not the ideal-gas ions, a standard effect of electron degeneracy.
Just one temperature, the key is that kT only reflects the kinetic energy of the ions, it is way less than the kinetic energy of each electron. That's what you mean by a rising degeneracy parameter.

kT is no longer measuring the kinetic energy of the electrons? So because electrons are degenerate, there's another equation for calculating their kinetic energy? It was said more than once that degeneracy lowers the temperature of the electrons. While this happens, T of the ions increases because of the compression. And now you say there is only one temperature. I really don't understand. Maybe after I read that paper...

 

[Ken G]

kT is no longer measuring the kinetic energy of the electrons? So because electrons are degenerate, there's another equation for calculating their kinetic energy?

Yes, that's precisely what is different with degeneracy. It's a thermodynamic effect, relating to temperature, not a mechanical effect, relating to pressure. It only ends up connecting to the pressure indirectly, because temperature influences heat transport. There are so many places that promote the misconception that degeneracy is a type of pressure.

It was said more than once that degeneracy lowers the temperature of the electrons.

Yes, in the sense of lowering it compared to E/k I mean-- not necessarily lowering it compared to what it was before. The problem is that we often use complete degeneracy as a kind of benchmark, to get approximate results, but formally, complete degeneracy means T=0. So that benchmark isn't actually achieved, since T tends to keep rising, so complete degeneracy is just a useful signpost.

While this happens, T of the ions increases because of the compression. And now you say there is only one temperature. I really don't understand. Maybe after I read that paper...

It is certainly a subtle point. As the core loses heat and contracts, its degeneracy rises. So the kT of the electrons goes way below their E, so much so that we can approximate the situation by setting T=0. However, this won't work for the ions, we could not understand why they undergo helium fusion at all. So we keep track of the ion T, know that it is the same as the electron T, but only use it for the ions-- for the electrons, we approximate the situation with T=0 to get the overall mass-radius relationship and so on. The latter is just a benchmark-- the actual electron T matches the ion T. These are tricks of approximation. The key subtlety is that as T rises, the electron pressure is only increased by a fractional amount of order (kT/E)², which is negligible. This is why so many sources incorrectly say the pressure does not rise-- what they mean is that the electron pressure does not rise. But that's only because the heat goes into the ions, which is the whole point of what is going on there. The total pressure rises completely normally, it's mechanical not thermodynamic. I have tried this argument on half a dozen referees already, none seem able to grasp it sadly.

 

[virgil1612]

I know that you can only define a temperature in a system in equilibrium. Could it be that when electrons go degenerate you can no longer talk about thermal equilibrium between them and the ions?

 

[Ken G]

I know that you can only define a temperature in a system in equilibrium. Could it be that when electrons go degenerate you can no longer talk about thermal equilibrium between them and the ions?

It is OK for the electrons and ions to have a temperature, indeed it is crucial that they have the same temperature. This is what regulates the amount of kinetic energy in each, so is what is involved in the helium flash. For example, let us imagine the opposite limit of no thermal contact at all between electrons and ions. Then when helium fusion initiates, the heat will go into the electrons (it is largely released as gamma rays, which interact more with electrons than ions). If the ion T did not need to equilibrate with the electron T, there would be no reason for any significant fraction of that heat to end up in the ions, so there would not be a helium flash. Remarkably, what happens in the limit of T equilibration is that most of the added heat ends up in the ions, and the electrons actually lose kinetic energy. The reason the electrons don't end up receiving much heat is an issue of heat capacity-- whenever you have two substances in thermal contact (i.e., same T), and you add heat, the heat ends up partitioning in proportion to the heat capacity of the substances. Degenerate electrons have a tiny heat capacity-- you need to add very little heat to them to get a big jump in temperature, because adding heat breaks the degeneracy.