Are there widespread misconceptions about degeneracy pressure?

[Ken G]

Two statements that are often made about degeneracy pressure are:
1) It is a new or special kind of pressure that requires quantum mechanics, in contrast with ideal gas pressure, which in effect involves somewhat mysterious forces that emerge from the Pauli exclusion principle, and
2) it behaves in such a way that degenerate gases do not expand like ideal gases when heat is added to them, which allows the heat to build up and fusion to run away (which causes helium flashes and type Ia supernovae).

For example, these claims can be found in many textbooks, and in Wiki:
http://en.wikipedia.org/wiki/Electro...eracy_pressure
"The Pauli exclusion principle disallows two half integer spin particles (fermions) from simultaneously occupying the same quantum state. The resulting emergent repulsive force is manifested as a pressure against compression of matter into smaller volumes of space."
So that certainly sounds like a "T" for (1). Then we have:
http://en.wikipedia.org/wiki/Type_Ia_supernova
"A main sequence star supported by thermal pressure would expand and cool in order to counterbalance an increase in thermal energy. However, degeneracy pressure is independent of temperature; the white dwarf is unable to regulate the burning process in the manner of normal stars, and is vulnerable to a runaway fusion reaction."
If the point being made here seems unclear, it is often explained further in the quite similar conditions that appear in a helium flash:
http://en.wikipedia.org/wiki/Helium_flash
"A helium flash occurs in these situations because the helium is degenerate, meaning it is supported against gravity by quantum mechanical pressure rather than thermal pressure. Thus an increase in the temperature in the material undergoing fusion does not act to expand the material and by doing so cool it, and there is no regulation of the rate of fusion. "

Certainly there are always idealizations and generalizations needed to simplify complex physics, but do the above two statements about degeneracy pressure really encapsulate the essence of the phenomena encountered, or are they pretty much false myths that are propagated simply because they are not subjected to critical scrutiny? What do people think? The poll possibilities are TT, which is both statements are mostly true, or TF, so statement (1) is mostly true but statement (2) is basically a myth, or FT if the opposite, or FF if both statements are mostly myths that do more to foster misconceptions than bring insights.

 

[Ken G]

No opinions on the matter? Are the people on this forum not very familiar with degeneracy pressure? You could read the cited Wiki article, and see if you agree with the two statements in the OP. The physics is elementary quantum mechanics, but there are interesting subtleties when it comes to the interpretation, hence the need to gauge the value of those two claims.

 

[alialice]

The two statements are both true.
The second one, related to helium flash, can be explained with the concept of thermal stability. If a degenerate system goes under compression, it cools down (it becomes "more degenerate" because density increases). Thermal stability is achieved only by classical system, for which:

sign(d T) = sign (d rho)

d =: "variation of"
T=central temperature
rho=central density

 

[Ken G]

So you would then vote "TT" in the poll. You are certainly not alone, many textbooks make similar claims. But if we look at them more carefully, are they actually correct? How many forum members see those statements as true, or do some find falsehood there? If you analyze more carefully, I wager you will see the situation, at best, is not so cut and dried, and at worst, is hopelessly confused by those two claims. (Let me interject that I agree with your statement that degenerate gases are thermally unstable and ideal gases are thermally stable, if a global force balance is maintained the whole time, but the issue is--- why?)

 

[Ken G]

Perhaps degeneracy pressure is not of great interest in this forum, it is a primarily astrophysical topic. All the same, both statements made about it in the OP are essentially completely false, even though they are propagated widely. Degeneracy is an entirely thermodynamic effect, it has no mechanical consequences that distinguish it in any way from an ideal gas. That is, it is a constraint on the ratio of temperature to energy per particle, but it has no effect on pressure whatsoever in any situation where the energy per particle is already specified. As such, it is a perfectly garden variety pressure, if the processes that set the energy are already being tracked. Where degeneracy is important is in the thermodynamics, that is, when we want to track the heat transport via knowledge of the temperature. That will of course affect the kinetic energy and the pressure of the gas, but that's the only place where it connects with pressure.

What's more, it is completely false that degenerate gases do not expand when heat is added to them, they expand exactly the same as ideal gases do. This is an elementary result, derivable from the very same equations you will find in the textbooks that say degenerate gas doesn't expand when it is heated. To see this, what is required is more precise thermodynamic usage, where "heat" and "temperature" are not treated as interchangeable concepts. The thermal instability of a degenerate gas has nothing at all to do with the gas not expanding, and indeed it is patently false that the gas does not expand. I would be happy to expound on these points if people are curious, the treatment is all undergraduate level physics.

 

[alialice]

What's more, it is completely false that degenerate gases do not expand when heat is added to them, they expand exactly the same as ideal gases do. This is an elementary result, derivable from the very same equations you will find in the textbooks that say degenerate gas doesn't expand when it is heated. To see this, what is required is more precise thermodynamic usage, where "heat" and "temperature" are not treated as interchangeable concepts. The thermal instability of a degenerate gas has nothing at all to do with the gas not expanding, and indeed it is patently false that the gas does not expand. I would be happy to expound on these points if people are curious, the treatment is all undergraduate level physics.

Why do you say that? So how do you explain helium flash? After the nuclear explosion the gas is no more degenerate and it expand regularly.

 

[alialice]

That is, it is a constraint on the ratio of temperature to energy per particle, but it has no effect on pressure whatsoever in any situation where the energy per particle is already specified. As such, it is a perfectly garden variety pressure, if the processes that set the energy are already being tracked. Where degeneracy is important is in the thermodynamics, that is, when we want to track the heat transport via knowledge of the temperature. That will of course affect the kinetic energy and the pressure of the gas, but that's the only place where it connects with pressure.

I surely refer to a thermodynamic interpretation in what I'm goin to write. I don't agree on the point where you say that degeneracy has no effect on pressure, or that it has only in connection with heat transport. White dwarfs and neutron stars are maintained by degeneracy pressure because they the correct density and temperature relation to obtain this condition. And the degenerate component in most case doesn't correspond with the responsible of heat transfer.

 

[cristo]

NB. Polls are not permitted in the science forums, since they do not give any added benefit to the thread.

 

[Ken G]

Why do you say that? So how do you explain helium flash? After the nuclear explosion the gas is no more degenerate and it expand regularly.

It expands the same when heat is added, it makes no difference if it is ideal or degenerate. The helium flash has nothing to do with presence or absence of expansion, and the way to see that is to ask, would there be a helium flash if the helium was as degenerate as the electrons? The answer is no-- yet nothing you will typically find in the erroneous explanations for the helium flash can account for this uncontroversial fact.

 

[Ken G]

I surely refer to a thermodynamic interpretation in what I'm goin to write. I don't agree on the point where you say that degeneracy has no effect on pressure, or that it has only in connection with heat transport. White dwarfs and neutron stars are maintained by degeneracy pressure because they the correct density and temperature relation to obtain this condition. And the degenerate component in most case doesn't correspond with the responsible of heat transfer.

I would argue that it is very misleading to claim that degeneracy causes the pressure in white dwarfs. Degeneracy is not the reason a white dwarf is small, that is simply due to its history of losing heat. Degeneracy is also not the reason that the particles have lots of kinetic energy, that is due to the virial theorem, indeed it is a trivial example of the virial theorem. The only reason that degeneracy matters in a white dwarf is that it shuts off the thermodynamic heat transfer from the degenerate gas to its environment, which prevents further contraction. That is not a cause of pressure.

 

[zonde]

There certainly are widespread misconceptions about degeneracy pressure.

Two statements that are often made about degeneracy pressure are:
1) It is a new or special kind of pressure that requires quantum mechanics, in contrast with ideal gas pressure, which in effect involves somewhat mysterious forces that emerge from the Pauli exclusion principle, and
2) it behaves in such a way that degenerate gases do not expand like ideal gases when heat is added to them, which allows the heat to build up and fusion to run away (which causes helium flashes and type Ia supernovae).

First statement is certainly false but it is a bit more complicated with second. The idea that you can add a lot of energy without getting much pressure in response is because particles become relativistic i.e. energy is going up but speed of particles is bound to be no more than c. I believe the argument had some more steps before you arrived at contraction. Something about increasing gravity.

 

[Ken G]

You are right that things change as the particles go relativistic, but note that the helium flash is often the place where these kinds of arguments about degeneracy pressure appear, and that stays pretty nonrelativistic, though it's only an approximation so one needs to choose how accurate one wants to be.

 

[zonde]

Wikipedia page about helium flash does not seem very clear.
It speaks about degeneracy pressure when talking about core helium flash. But it says that "The [core] helium flash is not directly observable on the surface by electromagnetic radiation." So it's hypothetical phenomena.
In case of observable helium flashes wikipedia page does not give much of the explanation.

 

[Ken G]

Yes, the "flash" is too deep in the star to observe, it is just a theoretical expectation. Still, the transition from a red giant to a "horizontal branch" star is observed, and that is supposed to be the change that the helium flash brings about, but still it is just a theoretical step whose details are probably not well known. I'm referring more to our pedagogical understanding of what should be happening, more so than any details that can be observed!

 

[Drakkith]

I don't quite understand your questions, Ken. Per wiki: http://en.wikipedia.org/wiki/Helium_flash

The explosive nature of the helium flash arises from its taking place in degenerate matter. Once the temperature reaches 100 million–200 million kelvins and helium fusion begins using the triple-alpha process, the temperature rapidly increases, further raising the helium fusion rate and, because degenerate matter is a good conductor of heat, widening the reaction region.
However, since degeneracy pressure (which is purely a function of density) is dominating thermal pressure (proportional to the product of density and temperature), the total pressure is only weakly dependent on temperature. Thus, the dramatic increase in temperature only causes a slight increase in pressure, so there is no stabilizing cooling expansion of the core.
This runaway reaction quickly climbs to about 100 billion times the star's normal energy production (for a few seconds) until the temperature increases to the point that thermal pressure again becomes dominant, eliminating the degeneracy. The core can then expand and cool down and a stable burning of helium will continue.[1]

This seems pretty clear that the addition of thermal energy only adds a small amount of pressure at first, so the temperature increases many many times what it was, leading to an "explosive" burning of helium as the temperature skyrockets, lasting until the gas can expand and cool off once more.

Are you saying this is wrong?

 

[Ken G]

Yes, I'm saying that the text in bold is pretty close to completely wrong. The only reason it isn't 100% wrong is that it tends to focus its attention on temperature, and indeed the temperature response of a degenerate gas is a bit bizarre. However, it fails to describe what is happening with energy, and of course following the energy is always a crucially important thing to do in physics. To expose the flaws in the bold text, simply ask this question: does the helium flash happen if helium is just as degenerate as the electrons? It is easy to see that the answer is no, yet what part of that Wiki explanation would suggest even in the least bit that no helium flash occurs if helium is also degenerate?

What is actually happening in a helium flash is infinitely more interesting than that description. The first thing to get is that the pressure in any nonrelativistic gas, be it ideal or degenerate, is 2/3 the kinetic energy density. This is an elementary result, I'm sure you can derive it in ten seconds. So if you follow the energy, it becomes much more clear that adding heat to a degenerate gas creates exactly the same expansion as for an ideal gas. There is no need to wait for the gas to become ideal, expansion comes with energy deposition by fusion, period. Now, it is true that the temperature spikes much more rapidly in a degenerate gas, but here's the interesting part-- the temperature rises even though the internal energy per particle drops (due to expansion work). The latter is elementary, the T rise is what is subtle and relies on degeneracy.

But now here's the kicker-- if the helium is also degenerate, then the T rise would not correspond to raising the energy of the helium, so would not cause more fusion. The gas would be thermally stable, and for exactly the same reason that hydrogen fusion in the Sun is stable-- adding heat would cause expansion, which would do work, remove internal energy, and shut off the helium burning. Where is that accounted for in the essentially incorrect bolded text above?

(ETA: the point is, the bottom line is that the helium flash has to do with the strange temperature behavior of degenerate gas, which the Wiki quote does allude to, but it has nothing whatsoever to do with anything going on with pressure. What is happening with pressure is completely mundane, and is the same as for an ideal gas. The issue is all thermodynamic, pressure and expansion are red herrings.)

 

[Drakkith]

I'm sorry, Ken, I don't know enough to really understand you.

To expose the flaws in the bold text, simply ask this question: does the helium flash happen if helium is just as degenerate as the electrons?

I have no idea. Are you asking if a helium flash happens if the ions are degenerate like the electrons are? If so, why does that even matter in this situation?

The first thing to get is that the pressure in any nonrelativistic gas, be it ideal or degenerate, is 2/3 the kinetic energy density.

Is degenerate gas in the core of a star a nonrelativistic gas?

So if you follow the energy, it becomes much more clear that adding heat to a degenerate gas creates exactly the same expansion as for an ideal gas.

I cannot follow the energy, and what you're saying goes against everything I've read so far. Can you show some math or something?

 

[Ken G]

Are you asking if a helium flash happens if the ions are degenerate like the electrons are? If so, why does that even matter in this situation?

It's not so much that the words in the Wiki quote are literally incorrect, it is that they do not convey the real reason there is a helium flash. Pressure plays no role at all, neither does expansion. The key issue is that adding heat causes the temperature of a degenerate gas to rise, because it softens the degeneracy. But the energy per particle of the degenerate gas falls, because of expansion-- the expansion that is not supposed to be happening. However, that doesn't matter, because if the temperature rises, the energy of the ideal helium nuclei goes up. It's all about breaking degeneracy, expansion doesn't matter a whit.

Is degenerate gas in the core of a star a nonrelativistic gas?

To a decent approximation, yes, at least for the helium flash.

I cannot follow the energy, and what you're saying goes against everything I've read so far. Can you show some math or something?

Let's agree that pressure is 2/3 the kinetic energy density, ideal or degenerate. Are we good there?

 

[Drakkith]

It's not so much that the words in the Wiki quote are literally incorrect, it is that they do not convey the real reason there is a helium flash. Pressure plays no role at all, neither does expansion. The key issue is that adding heat causes the temperature of a degenerate gas to rise, because it softens the degeneracy. But the energy per particle of the degenerate gas falls, because of expansion-- the expansion that is not supposed to be happening. However, that doesn't matter, because if the temperature rises, the energy of the ideal helium nuclei goes up. It's all about breaking degeneracy, expansion doesn't matter a whit.

What I'm getting from this:

Electron energy per particle falls because it is expanding and becoming non-degenerate.
Nuclei energy increases because the temperature is increasing.

Is that correct?

To a decent approximation, yes, at least for the helium flash.Let's agree that pressure is 2/3 the kinetic energy density, ideal or degenerate. Are we good there?

Uhh, sure?

 

[Ken G]

What I'm getting from this:

Electron energy per particle falls because it is expanding and becoming non-degenerate.
Nuclei energy increases because the temperature is increasing.

Is that correct?

Yes, exactly.

Uhh, sure?

The point there is that if degeneracy doesn't affect the pressure once we specify the energy density, then it also doesn't affect the pressure once we say how much heat is being added by fusion. So pressure is a complete red herring, it has nothing to do with the interesting things that degeneracy is doing in the helium flash. The clear misconception fostered by the Wiki quote, and a million other places, is that degenerate gas doesn't expand when heat is added, so the heat piles up. That's what is wrong (although that particular Wiki quote doesn't actually say that, I'll give it credit for that). The heat does not pile up, but the temperature does rise. That's what we need to understand about degeneracy-- how it let's the temperature rise even as the average energy per electron is dropping. Nothing in the Wiki quote conjures that crucial state of affairs, so it really misses the boat on what is subtle and interesting about degeneracy. It just has nothing to do with pressure, and neither does the helium flash.

 

[Drakkith]

That's what is wrong (although that particular Wiki quote doesn't actually say that, I'll give it credit for that). The heat does not pile up, but the temperature does rise. That's what we need to understand about degeneracy-- how it let's the temperature rise even as the average energy per electron is dropping. Nothing in the Wiki quote conjures that crucial state of affairs, so it really misses the boat on what is subtle and interesting about degeneracy. It just has nothing to do with pressure, and neither does the helium flash.

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

Also, perhaps it is confusing because the electrons are degenerate while the ions are not. How does temperature work in this case where there are two clearly different states; the degenerate electrons and the non-degenerate ions.

 

[Ken G]

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. This is what we need to understand about degenerate gas, its rather unusual thermodynamics. Focusing on pressure is exactly the wrong place to look, nothing is happening with pressure that matters at all.

Also, perhaps it is confusing because the electrons are degenerate while the ions are not. How does temperature work in this case where there are two clearly different states; the degenerate electrons and the non-degenerate ions.

Each has its own way of arriving at a temperature, and this is the crucial issue in the helium flash. It's thermodynamics, not mechanics.

 

[Drakkith]

Tell me if the following is correct:

The pressure in a degenerate gas depends only on the speed of the degenerate particles NOT the temperature of the gas. But to change the speed of degenerate particles requires A LOT of energy because they are locked into place against each other. Adding heat only causes the non-degenerate particles to move faster, but the degenerate ones supplying the pressure are unaffected.


From: http://www.astronomynotes.com/evolutn/s10.htm (Step 2)

I assume they are mostly unaffected, not completely unaffected?

 

[Ken G]

Tell me if the following is correct:

The pressure in a degenerate gas depends only on the speed of the degenerate particles NOT the temperature of the gas. But to change the speed of degenerate particles requires A LOT of energy because they are locked into place against each other. Adding heat only causes the non-degenerate particles to move faster, but the degenerate ones supplying the pressure are unaffected.

I would say that quote misses the mark even worse than the Wiki quote. It is just loaded with the potential for fostering misconceptions, I really don't see much of value in it to be quite honest. This is why I started with the point that the pressure is 2/3 the kinetic energy density, whether the gas is ideal or degenerate. Once you recognize that, you really have to wonder why this quote is trying to suggest that somehow adding heat to a degenerate gas doesn't raise the pressure like it would an ideal gas. It's all just a question of how much heat you are adding compared to how much internal energy is already there, none of that has anything directly to do with degeneracy, nor tells us anything interesting about degeneracy. Yes a white dwarf has a lot of internal energy, the virial theorem tells you that. It would have the same internal energy if it were an ideal gas of bosons of the same mass and radius. This is just missing the issue of what degeneracy does-- it makes the temperature very low compared to an ideal gas of bosons, all else being equal. There just are no pressure implications, that's a complete red herring.

 

[Drakkith]

I think I'm just more confused now than before I got into the thread.

 

[Ken G]

That's because you have been told a lot of hooey about degeneracy pressure that you first have to unlearn!

Let me give you a concrete example. We have a nonrelativistic gas with pressure P and density n. We then know its average energy per particle, that is U = 3/2 P/n. We say the gas is self-gravitating and obeys the virial theorem, which means that changes in its internal energy will always obey dU/U = -dR/R = 1/3 dn/n. This is saying the self-gravitating gas is staying in a homologous force balance as n changes, which is a standard assumption used to analyze all the things we have been talking about.

Now say we add heat per particle dQ. The first law of thermodynamics says dU = dQ + P dn/n² = dQ + 2/3 U dn/n = dQ + 2 dU. Hence we derive dU = -dQ, the standard outcome of the virial theorem. It says that if you add heat to a self-gravitating non-relativistic gas, you reduce the internal energy by the amount you think you should be increasing it by. This reduction is due to expansion work.

Now here comes the kicker: where did I need to say if this gas was ideal or degenerate? I didn't, I never had to say which it is-- because the whole issue of degenerate vs. ideal behavior has nothing to do with how the gas responds to having heat added, unless I need to talk about the temperature. Since I didn't need to talk about temperature to derive dU = -dQ, it has nothing to do with degeneracy. Neither does dn/n = 3 dU/U = -3 dQ/U. So this is what I mean, degeneracy has nothing to do with the way pressure responds to heat being added, and nothing to do with how the gas expands when heat is added. Degeneracy is about a rather bizarre type of temperature behavior, and a lot of what is written about it completely obfuscates this important point.

The Wiki quote, and those course notes, are classic examples of the mire you get into when you mix the things that don't care about degeneracy with the things that do. Degeneracy has no pressure implications in any situation where you have already specified the energy density and particle density, it's just that some combinations of those parameters might make it impossible to find a physically possible temperature if the gas is fermionic, so those energy densities are impossible for degenerate gases of given density. But if you have a possible energy density and particle density, and you don't care what the temperature is doing (as for solving the expansion and the internal energy change, as above), then degeneracy plays no role whatsoever. I could have just said: degeneracy plays no role at all in the virial theorem.

 

[zonde]

the pressure is 2/3 the kinetic energy density, whether the gas is ideal or degenerate.

It's wrong.
Look, kinetic theory explains pressure as result of collisions between particles or collisions of particles with container. But collision change quantum states of colliding particles. So outcome of collision is restricted by available quantum states. And if there are no accessible quantum states there are no collisions and no pressure. Degenerate particles just move on their way no matter how much kinetic energy they have.

 

[Ken G]

But you are simply demonstrating that I am correct in the title of this thread. I don't blame you for holding that false impression, there are so many places that say degeneracy pressure is some strange kind of quantum mechanical force that appears when you have a Pauli exclusion principle. But it's all baloney-- degeneracy pressure is gas pressure, and it is very definitely 2/3 the kinetic energy density in any nonrelativistic gas. The same sites that make the spurious claims about it will usually invoke this fact at some point.

Actually, it's kind of an interesting question as to why you just said is wrong. One might imagine that collisions would be changed by the availability of states, and indeed degenerate gas is highly conducting of heat and can even act as a superfluid (though I don't know many details about that). But collisions between particles has nothing to do with gas pressure, collisionless gases have the same pressure as collisional ones do. It's all about kinetic energy density, that's an elementary result of the definition of pressure.

 

[Drakkith]

Are you saying degeneracy pressure is gas pressure because it is the result of the kinetic energy of the electrons, like normal gas pressure is the result of the kinetic energy of the gas particles?

 

[Ken G]

Yes.

 

[Drakkith]

Yes.

Oh good, I was hoping I'd put everything together correctly.

Any chance you can answer the following?If we have a degenerate stellar core at around 100 million k, and we increase the temperature to 200 million k, what exactly happens?

• How much might the core expand?

• What would the thermal pressure of the ions be compared to the pressure of the degenerate electrons?

• How would the increase in energy be distributed between the ions and electrons?

 

[zonde]

But you are simply demonstrating that I am correct in the title of this thread.

If what I say is wrong then they are my more or less private misconceptions. Certainly not widespread.

I don't blame you for holding that false impression, there are so many places that say degeneracy pressure is some strange kind of quantum mechanical force that appears when you have a Pauli exclusion principle. But it's all baloney-- degeneracy pressure is gas pressure, and it is very definitely 2/3 the kinetic energy density in any nonrelativistic gas. The same sites that make the spurious claims about it will usually invoke this fact at some point.

Actually, it's kind of an interesting question as to why you just said is wrong. One might imagine that collisions would be changed by the availability of states, and indeed degenerate gas is highly conducting of heat and can even act as a superfluid (though I don't know many details about that). But collisions between particles has nothing to do with gas pressure, collisionless gases have the same pressure as collisional ones do. It's all about kinetic energy density, that's an elementary result of the definition of pressure.

Definition of pressure is force per unit area applied in a direction perpendicular to the surface of an object.
Without collisions how is force applied to anything? How is energy transferred from the gas to the object that is measuring force?

 

[Ken G]

If we have a degenerate stellar core at around 100 million k, and we increase the temperature to 200 million k, what exactly happens?

Temperature is very complicated with degenerate gas, that's the issue. It is much easier to know what happens to the internal energy if you add heat (dU = -dQ). To solve for the temperature requires solving for the degeneracy parameter, it's a bit tedious but is in standard references on the degeneracy parameter. It acts a lot like a thermodynamic chemical potential. But I'll give you basic non-quantitative answers:

• How much might the core expand?

This depends on just how degenerate the gas is originally. 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. Note this is what I have been saying-- degeneracy is a temperature effect, so changing the temperature is about changing the degeneracy. Adding heat will change the degeneracy, which will change the temperature. Expansion is irrelevant, and it is false to say the gas doesn't expand when you add heat. If you track the heat added, the expansion is identical to an ideal gas. If you track the change in temperature, the degenerate situation is completely different, because degeneracy is all about temperature.

• What would the thermal pressure of the ions be compared to the pressure of the degenerate electrons?

The ion pressure will double if this happens with little expansion, but will still be way less than the electron pressure, since we are assuming the degeneracy is staying high.

• How would the increase in energy be distributed between the ions and electrons?

This is the crux of the helium flash-- if you add a unit of heat, the total internal energy goes down by 1 unit (as required by the virial theorem), but the electron energy goes down by 2 units and the ion energy goes up by 1 unit. 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.

 

[Ken G]

If what I say is wrong then they are my more or less private misconceptions. Certainly not widespread.

Well, your take on the issue is not necessarily in the textbooks, but it is already more deeply related to the actual physics of degeneracy than what you normally find there. In all honesty, I have as yet not spoken with a single person who really understands degeneracy, and neither did I just a few short months ago. The misconceptions are very widespread.

Definition of pressure is force per unit area applied in a direction perpendicular to the surface of an object.
Without collisions how is force applied to anything? How is energy transferred from the gas to the object that is measuring force?

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. That's another widespread myth about pressure. 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. 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.

 

[Drakkith]

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

 

[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?