I Without degeneracy, when would Solar cores collapse?

[Ken G]

TL;DR Summary

It is often said that stellar core collapse happens when gravity overcomes the pressure caused by electron degeneracy. Since degeneracy is a consequence of the Pauli exclusion principle, this suggests that the nature of core collapse can test that electrons are indistinguishable on the scale of a stellar core.

I believe I know the answer to this question, but it is still very informative to ask: Would iron stellar cores still collapse when they reached some mass without degeneracy (by which I mean, if electrons were not indistinguishable, so did not obey the Pauli exclusion principle on such large scales as a star), and would that mass be higher or lower than the Chandrasekhar mass? Consider first the answer you come to if you take at face value the standard language that "degeneracy pressure" is a mysterious form of additional pressure that is produced by degeneracy as electrons "get in each other's way" or "cannot be squeezed any closer", and it is finally overcome by gravity when the core reaches the Chandrasekhar mass of about 1.4 solar masses. What answer to the above question does that language lead you to?

 

[JLowe]

I mean, do electrons still repulse each other in your scenario?

 

[Ken G]

Yes, they still have negative charges, but only because we don't want to change anything except the Pauli exclusion principle, nothing else. (It turns out that particle charges play almost no role in the force balance of a star, all those positive and negative charges cancel out to a high degree. You can estimate the characteristic interaction energy between nearest neighbors, for example, and see that it is quite small compared to the overall gravitational binding energy per particle.) So the charges are there, but forget about them, stars are basically made of gravitationally bound gases.

 

[snorkack]

If electrons were distinguishable, or had integer spin, they would behave as an ideal gas. Which means that pV=nRT, and V goes to 0 as T goes to 0. Degeneracy means that V does not go to 0 for a finite P even if T does go to 0.

 

[Ken G]

Yes, that is all true. So, let us address the question. We have the core of a massive star, made of iron. It is rather rapidly compiling more mass as silicon burns in a shell around the iron core, so the iron core is at silicon burning temperature. But there's no PEP. Will the core collapse, and if so, at mass less, greater, or the same as 1.4 solar masses?

 

[snorkack]

The answer is that ideal gas of bosons will collapse at arbitrarily small mass. It cannot occupy any volume unless it has nonzero temperature, and it will be radiating away energy at any nonzero temperature.

 

[Ken G]

You are talking about the fact that there will never be a permanent (eternal) equilibrium for that core, but I'm talking about core collapse, as in a supernova. We have to look at the conditions a bit closer, and to avoid a full simulation, make some simplifying assumptions. This core is inside a shell of burning silicon, so it is not an eternal equilibrium we mean, but rather an equilibrium within the context of an evolution toward core collapse. The burning silicon is quite hot, and its temperature is not self regulated, it just burns at whatever rate its temperature dictates, and the core will not cool below that. So we needn't be concerned with zero temperature (the temperature is not zero in a core collapse), that is an idealization that is more relevant for cooling white dwarfs.

So let us consider the issue in its more appropriate context, the core is not losing heat because it is sitting inside a fusion shell. We can debate the precise assumptions that should be made, but for argument's sake, let's take the core as adiabatic, insulated from losing heat by the fusion shell around it. The main point in core collapse is that new matter is added to the core with less kinetic energy than it would need to be "virialized", that is, the shell was sitting on top of the core and supported by it (so the matter in the shell has kinetic energy that is less than virialized, it has less kinetic energy than it would need to support itself without help from the core). This means ash from the shell creates a gravitational load on the core as it joins it.

This is the normal scenario of core collapse, iron "ash" is added to the core with less kinetic energy than it would need to support itself, so it causes the core to contract a bit, and the question is, can the core contract enough to take on that new load. It is common to hear that in this scenario, core collapse is a dynamical instability that kicks in when this increased gravity overwhelms even the strong additional pressure supplied by electron degeneracy. My question is, what do you expect would happen to such a core if there is no PEP, and no "degeneracy pressure" from it? You are saying you expect collapse at any mass, but you are taking a zero temperature. What do you get if we instead imagine the core is adiabatic, and the mass being added to it increases its gravity but does not carry enough kinetic energy on its own to keep the core virialized without the core contracting some in response? (If you think the adiabatic idealization is too stringent, then consider the situation where the temperature of the core rises as needed to maintain virialization, even if it is not the adiabatic temperature. I think it would be hard to argue the core could cool to lower temperature, since the fusion shell can get very hot.)

 

[PeterDonis]

@Ken G, when you say there is no Pauli Exclusion Principle, do you mean we should treat the particles in the star as bosons, as @snorkack assumed in post #6? Or do you mean we should treat them as "classical" distinguishable particles (i.e., Maxwell-Boltzmann statistics)?

 

[Ken G]

I'm not sure it makes any difference in the physics we are using here. But just in case the spin of the electron ever does come up, which I don't expect, we can keep them as fermions. Again, I can't see that it will matter, probably only their mass is relevant. (@snorkack did not need to assume they were bosons; if they are distinguishable, his statement holds either way.)

 

[Vanadium 50]

I actually don't think this is going to work in either scenario.

If nucleons and electrons are not fermions, there is nothing to prevent the end state to be one big ball of nucleons. Fusion doesn't stop at iron - it just keeps going and going and going. Depending on how BBN goes, you may not get stars at all, and certainly not stars as we know them - especially red giants and SNe.

This is a general problem with counterfactual physics assumptions.

 

[Ken G]

We do want electrons to behave like electrons in terms of things like beta decay and whatnot. But I don't know of any nuclear physics that cares they are indistinguishable until you get to very high energies, at which point the core collapse has in a sense already happened (things like photodisintegration and Urca processes, the core is a goner by the time those endothermic reactions kick in, we can think of this whole question as being about what mass a core will need to accumulate to get to that point). I never said the nucleus is not comprised of fermions, of course we don't want to change the nuclear physics here. BBN nucleosynthesis does not seem relevant at all, but if you are concerned about it, just turn off the PEP after the BB is over.

Counterfactual physics assumptions are a valuable tool for exploring our understanding. Let me give you an example. Let's say you can run a MESA interiors code to decide the hydrodynamic equilibrium in any star you like. Now let's say you can go into the guts of MESA, and just require free electrons to follow a Maxwell Boltzmann distribution, rather than Fermi Dirac. Poof, you have precisely the scenario we are discussing. The question is then, can we use what we know about physics to predict the behavior of that code as the stellar core evolves toward what would normally be core collapse? It's a very straightforward physics question.

 

[Ken G]

On the issue of the core temperature, it is going to be important to decide what assumptions to use there. This is a good thing, as it shifts the physics focus right where it needs to be: on the energy equation, not the force equation. The hydrostatic equilibrium in any gas, whether Maxwell Boltzmann or Fermi Dirac, is pretty simple if you are tracking the kinetic energy: the pressure is 2/3 the internal kinetic energy divided by the volume in any cell in your calculation. So if you know what the energy is doing, you never need to know anything about distinguishability or indistinguishability. I hope you can all see that if you don't know it already, it's an elementary consequence of every equation you find in Ballentine that relates to nonrelativistic gases. (It will matter that the gases become relativistic, but the complication imposed by the particle distribution function is minor, it has to do with the fact that these distributions have somewhat different relativistic tails at any temperature, but those tails are not where the pressure comes from and this is really not going to be the key issue here.)

So once we understand that discussing the fate of a core has to do with the effects of the particle statistics on energy transport, not some direct effect on the force equation, we have already made a big step forward. This is why @snorkack was concerned about the temperature going to zero, that's all about energy transport. But the temperature will not go to zero, the core will get very hot because it is contracting and it is inside a fusing silicon shell. So I suggest we assume the core will stay at whatever temperature is required for it to be virialized. That seems the natural assumption to me, but if someone thinks that will not happen, then that is certainly grounds for discourse.

 

[PeterDonis]

I'm not sure it makes any difference in the physics we are using here. But just in case the spin of the electron ever does come up, which I don't expect, we can keep them as fermions.

This doesn't make sense. If there is no Pauli Exclusion Principle applied to them, then they aren't fermions. We have to have a consistent theoretical model in order to make predictions, and without making predictions we can't possibly answer the question you pose. What model should we use?

 

[Vanadium 50]

But I don't know of any nuclear physics that cares they are indistinguishable until you get to very high energies

The fact that nucleons are fermions matters to nuclear physics exactly as much as the fact that electrons are fermions matters to atomic physics, I gave a very specific example: fusion doesn't stop at iron.

I am reasonably sure that such a universe would have much, much higher metalicity than our own, so stars would be very different. Specifically, you will get stable nuclei with A = 5 and 8, so you don't need triple-alpha to build up big nuclei.

 

[Ken G]

The fact that nucleons are fermions matters to nuclear physics exactly as much as the fact that electrons are fermions matters to atomic physics, I gave a very specific example: fusion doesn't stop at iron.

I know that nucleons being fermions matters to nuclear physics. This has nothing to do with that. (Perhaps I should have specified electron degeneracy in the OP, but the situation is clearly spelled out there.)

I am reasonably sure that such a universe would have much, much higher metalicity than our own, so stars would be very different. Specifically, you will get stable nuclei with A = 5 and 8, so you don't need triple-alpha to build up big nuclei.

Sounds like you are still talking about nucleons not being indistinguishable. Is the indistinguishability of electrons playing any role in what you are talking about? This thread is about if electrons did not obey the PEP, so followed Maxwell Boltzmann statistics not Fermi Dirac. That's all I have in mind here, nothing more.

 

[Ken G]

This doesn't make sense. If there is no Pauli Exclusion Principle applied to them, then they aren't fermions.

I just meant they can still have spin 1/2, and be distinguishable, and they will not obey the PEP. We might not call them fermions in that situation, but it doesn't matter what we call them. In any event, I don't think spin is going to matter at all, it seems like an extraneous issue but if it matters, make it 1/2. Literally what I have in mind is take a simulation code of a stellar interior, and simply force the electrons to be an ideal gas (Maxwellian). That's it, nothing more. The issue is, what does that do to the maximum mass that can be reached before core collapse occurs, where core collapse is defined as a core that reaches, on core evolution timescales, internal energies capable of devastatingly endothermic processes like photodisintegration and runaway neutrino generation.

 

[PeterDonis]

I just meant they can still have spin 1/2, and be distinguishable

Ok, "distinguishable" is the key, that means they obey Maxwell-Boltzmann statistics, not Fermi-Dirac statistics.

We might not call them fermions in that situation

Yes, we wouldn't, because they don't obey Fermi-Dirac statistics. See above.

 

[Ken G]

I mean, do electrons still repulse each other in your scenario?

Yes I understand you mean that, and yes, they still do. They act like electrons in all ways, except kinetic energy is distributed over them following the Maxwell Boltzmann distribution (meaning they are an ideal gas) rather than the Fermi Dirac distribution (invoked for gas that gets called degenerate, for some unknown reason). But honestly I don't think their repulsion is going to matter much, it is generally ignored anyway.

 

[Ken G]

Ok, "distinguishable" is the key, that means they obey Maxwell-Boltzmann statistics, not Fermi-Dirac statistics.

Correct.

Yes, we wouldn't, because they don't obey Fermi-Dirac statistics. See above.

Let's call them classical electrons.

 

[PeterDonis]

Would iron stellar cores still collapse when they reached some mass without degeneracy

Since we have established that "without degeneracy" means "Maxwell-Boltzmann statistics", then Shapiro & Teukolsky, Chapter 3 (the sections I referenced in my Insights article on maximum mass limits) gives a simple answer: both the (positive) kinetic energy, and therefore the pressure, and the (negative) gravitational potential energy of a Maxwell-Boltzmann gas scale like $1 / R$. That means such an object can contract indefinitely by radiating away heat, decreasing $R$ and therefore decreasing its total energy (since, as you have yourself pointed out, the virial theorem says the absolute value of the kinetic energy is half the absolute value of the potential energy). So of course any such object will collapse, regardless of its mass.

In fact the whole point of S&T's Chapter 3 is to show that, for any such object to stop collapsing, there must be some other mechanism that comes into play to change the behavior from that of a Maxwell-Boltzmann gas. The mechanism they analyze, which is the one that our standard theory of compact objects says is the one that actually comes into play in cases like white dwarfs and neutron stars, is Fermi-Dirac statistics becoming important. There are multiple ways one could describe this in ordinary language; one could say it's "degeneracy pressure", or one could say it's the Fermi sea being filled that prevents further heat loss (as you have described it in other discussions). But the fundamental point remains the same: a Maxwell-Boltzmann gas will always collapse. There is no mass threshold for such an object at all. So for there to be any mass threshold, the behavior must stop being that of a Maxwell-Boltzmann gas at some point.

 

[Ken G]

Since we have established that "without degeneracy" means "Maxwell-Boltzmann statistics", then Shapiro & Teukolsky, Chapter 3 (the sections I referenced in my Insights article on maximum mass limits) gives a simple answer: both the (positive) kinetic energy, and therefore the pressure, and the (negative) gravitational potential energy of a Maxwell-Boltzmann gas scale like $1 / R$. That means such an object can contract indefinitely by radiating away heat, decreasing $R$ and therefore decreasing its total energy (since, as you have yourself pointed out, the virial theorem says the absolute value of the kinetic energy is half the absolute value of the potential energy). So of course any such object will collapse, regardless of its mass.

As I mentioned to @snorkack, we are not talking about the eventual contraction of the core, which will depend on the ability of the core to lose heat. That's what you are describing as well, one has to take into account that this core is inside a silicon fusing shell, and can easily surpass the Chandrasekhar mass while it is waiting to be able to lose enough heat to follow that inevitable eventual path of contraction that you are describing. In core collapse, there is not just heat loss, there is also a rise in mass, as the core is adding iron "ash" from the silicon fusion shell around it. When the core is degenerate, its ability to lose heat is inhibited, so the adding of mass becomes the central issue that causes core collapse, and that's why it is a mass limit (the Chandra mass) where it happens. If the core were not obeying the PEP, then we'd have a new question, central to this thread, which is whether the (potentially gradual) loss of heat from the core is the key evolutionary driver, or if it is the addition of mass. I am focusing on the latter, asking if there is a mass limit, like the Chandra mass, where the addition of mass puts it over the edge. If the issue is heat loss instead, then the question becomes, how big is the mass when the core collapse occurs? That is an issue you have not covered, because the Chapter 3 you discuss assumes fixed mass, and that's not what is happening in core collapse. But I can agree it connects to the central issue here, which is timescale for heat loss, versus timescale for mass addition. It may well be that the final answer requires knowing those two things, so let us then consider the situation where the addition of mass is relatively rapid. (In real stars, silicon fusion can add significantly to the core mass in just a week or two!)

Put differently, everything you are describing, and what @snorkack was talking about, is happening under the aegis of continuous force balance (as you mention the virial theorem) with a constant total mass. Core collapse is sometimes described as a loss of force balance due to a rise in mass, so that's something quite different. Personally, I don't think loss of force balance is necessary, the key point is that the energy scale must reach the devastating levels of endothermic processes like neutron loss and iron photodisintegration. But if we allow the core to stay virialized (i.e., stay in force balance, as in the Chapter 3 you describe) the entire time, then we can easily determine is temperature from its mass. The question then becomes, at what mass does the core reach the temperature of those core collapsing endothermic mechanisms? And we can compare that to the Chandra mass, to answer the query in the OP.

In fact the whole point of S&T's Chapter 3 is to show that, for any such object to stop collapsing, there must be some other mechanism that comes into play to change the behavior from that of a Maxwell-Boltzmann gas.

Yes, that is correct, this is the right way to think about what degeneracy is doing. It is the "other mechanism" that is needed to inhibit heat loss and prevent the gradual contraction you are describing. I believe I have made that point in the past, now you can see what I meant. But for core collapse, it is essential to bring in two additional aspects that have so far not been included: the rapidly rising core mass, and the central importance of relativity.

But the fundamental point remains the same: a Maxwell-Boltzmann gas will always collapse. There is no mass threshold for such an object at all. So for there to be any mass threshold, the behavior must stop being that of a Maxwell-Boltzmann gas at some point.

The mass threshold you describe is not the relevant one here (though it is in your excellent Insights article). You are talking about the fact that below the Chandra mass, Fermi-Dirac electrons will cease to be able to lose heat, so cannot continue the path that Maxwell electrons will take. That is certainly an important point in any constant mass situation that is evolving over long timescales, but core collapse is very different because we have rising mass, and the evolutionary timescale is quite short. The rising mass is crucial because a degenerate gas cannot go past about 1.4 solar without reaching core collapse energies, but what is so different about an ideal gas, that has not been penetrated to yet in this thread, makes it able to go way past that limit before collapsing. So we still have two things to understand:
1) why is degenerate gas so different as 1.4 solar masses is reached, and
2) what is the right way to think about an ideal gas core that is adding iron ash.
In my next post I'll start in on #2, and #1 is something of an "aha" that can come later.

 

[PeterDonis]

If the core were not obeying the PEP, then we'd have a new question, central to this thread, which is whether the (potentially gradual) loss of heat from the core is the key evolutionary driver, or if it is the addition of mass. I am focusing on the latter, asking if there is a mass limit, like the Chandra mass, where the addition of mass puts it over the edge.

The answer to that is easy: if you're assuming the core is a Maxwell-Boltzmann gas (at least as far as the electrons are concerned), then no, there can't be any mass threshold for collapse. There is no additional mechanism that would provide any such threshold: the only mechanisms in play, if I'm understanding your hypothetical correctly, are mass accretion and radiative heat loss. Both of those are continuous for a Maxwell-Boltzmann gas, with no thresholds anywhere.

what is so different about an ideal gas, that has not been penetrated to yet in this thread, makes it able to go way past that limit before collapsing

You haven't shown that this will actually be the case. In fact my initial take is that adding mass to a Maxwell-Boltzmann gas will make it radiate more and thus lose heat and contract faster. Adding mass increases the absolute value of the (negative) gravitational potential energy, and by the virial theorem, that also increases the (positive) kinetic energy, and hence the temperature, and hence the radiation rate.

This also brings up the fact that, if the core is a Maxwell-Boltzmann gas, it won't even be stable to start with, at any mass. It's not even a question of having a stable object that gets "pushed over the edge" by adding enough mass to it, and asking whether the "edge" is different than for the actual case in the real world, where electrons obey Fermi-Dirac statistics. There is no stable equilibrium in the given mass and size range to start with. The object will always be contracting.

 

[PeterDonis]

If the issue is heat loss instead, then the question becomes, how big is the mass when the core collapse occurs?

There is no such thing as "core collapse" in the sense you are using the term if the core electrons are a Maxwell-Boltzmann gas at all densities. There is no mass threshold at all, as I have already said. The core will always be contracting; it will never be in a stable equilibrium at all. So the whole structure of the object, as well as its evolution over time, will be different than in the actual case where Fermi-Dirac statistics are significant.

 

[Ken G]

Fortunately for us, there is already a well known situation in stellar evolution that deals with an ideal gas core that is having mass added to it: and it is called the "Schonberg-Chandrasekhar limit." (https://en.wikipedia.org/wiki/Schönberg–Chandrasekhar_limit) This happens in stars of masses around 2 to 8 solar masses, when they have an ideal gas helium core that is having mass added to it as a shell of hydrogen burns. This would be entirely analogous to an iron core surrounded by silicon burning, and the key point is that the ideal gas core is isothermal because it comes to the temperature of the fusing shell. There is no obvious difficulty in the force balance here, so you might think you could add mass to the core indefinitely. (Gravity never "overcomes" the pressure in the ideal gas, even though it has no "additional degeneracy pressure", because the ideal gas just contracts however much it needs to in order to stay at the shell temperature and have the pressure it needs, something it can always do based on a trivial consequence of the nonrelativistic virial theorem). This is also not a contradiction to the Chapter 3 results we heard about, because those do not keep the star at the temperature of its fusion shell (and they are not intended to handle a rising mass either.) It's just not the same physical situation of a stellar core that is at a particular evolutionary stage and only concerned about its immediate prospects!

However, there is a fly in the ointment: the core has a higher mass per particle than what surrounds it, and this leads to a strange predicament that is the "Schonberg-Chandrasekhar limit," which relates to higher mass particles being susceptible to collapse under their self gravity when surrounded by lower mass ones (it's a subtle effect, vaguely similar to the Jeans criterion if you have seen that). When the core gets to about 1/10 or maybe 1/7 of the total mass of the star, the core's own self gravity becomes too strong to also hold up the weight of the rest of the star, and the core collapses (causing what gets called the "Hertsprung gap" in the HR diagram by astronomers).

Notice this is a totally different cause of core collapse (and in the case of the Hertsprung gap, it only boosts the core to having a temperature gradient, it doesn't get it to the violent levels of a core collapse supernova). A degenerate core collapses for reasons that relate to certain interesting and underappreciated things that the Fermi Dirac distribution is doing (hint: it has to do with specific heat of degenerate gas, and of course, relativity, that's the stuff in #1 above). An ideal gas core at the same temperature as its fusion shell would never collapse at all, if that fusion temperature wasn't as high as the endothermic processes that cause the collapse, except if the core reached that 1/10 or 1/7 of the mass of the whole star. For a 20 solar mass star, the core could go to 2 or 3 solar masses! Well past the Chandra limit, without any "additional degeneracy pressure" for gravity to "overcome." Not quite what we expected!

 

[Ken G]

So what this means is, if you put the two 20 solar mass stars side by side, and wait until they get an iron core about about 1 solar mass, this is what you would see. The core obeying the PEP will have a tiny degenerate core with very high energy electrons and ions, while the core not obeying the PEP will have a much larger ideal gas core at similar temperature but much lower energy per particle (because it's not degenerate), both in good force balance. Both are adding iron ash, that is producing a load on the core that is causing it to contract. The PEP one has all this "degeneracy pressure", which gets "overcome by gravity" when it reaches 1.4 solar masses (for reasons that are quite interesting), while the ideal one has just plain old ideal gas pressure, but it sails well past 1.4 with no problem.

 

[PeterDonis]

This would be entirely analogous to an iron core surrounded by silicon burning

Would it? The iron core will be about 14 times as dense as a helium core, so given the same mass (1 solar mass in your hypothetical), it will be smaller and hence its temperature will be higher by about the same factor. That seems like a significant change to the conditions of the analysis.

 

[Ken G]

Would it? The iron core will be about 14 times as dense as a helium core, so given the same mass (1 solar mass in your hypothetical), it will be smaller and hence its temperature will be higher by about the same factor. That seems like a significant change to the conditions of the analysis.

It will be hotter, yes. But then, the physics of the SC effect is not temperature sensitive, it all scales. I'm sure S&T has a chapter on it, you can see if anything in there is going to care if the temperature is hydrogen burning (10 million Kelvin) or silicon burning (3 billion Kelvin). The issue is what other processes can happen, there will certainly be a lot more in the way of neutrino losses at that temperature. But we know the degenerate core is also going to be at temperatures similar to the silicon burning, and the degenerate electron kinetic energies are higher still, so those processes are already happening. All the SC effect needs is an isothermal core for an ideal gas.

What's more, I believe we are seeing exactly the effect I'm talking about if you look at figure 1 in https://www.nature.com/articles/s41586-020-03059-w. That is a plot of the density as a function of mass coordinate, and you can see that for the larger mass stars, above 20 solar, the core mass is already well past the Chandra mass even before they begin their core collapse simulations. I think that's a clear sign that these cores are actually quite close to ideal gases, if they were highly degenerate they'd be past the Chandra limit. If the stars went up to 40 solar masses or more, I think this would be even more clear, you'd have very ideal gas cores that are below the SC limit and having no issue with force balance, though if they were degenerate they would have collapsed.

The reason for this, the thing that makes degeneracy susceptible to creating collapse, is the most interesting thing of all that degeneracy does, hardly ever even mentioned!

 

[Ken G]

The answer to that is easy: if you're assuming the core is a Maxwell-Boltzmann gas (at least as far as the electrons are concerned), then no, there can't be any mass threshold for collapse. There is no additional mechanism that would provide any such threshold: the only mechanisms in play, if I'm understanding your hypothetical correctly, are mass accretion and radiative heat loss. Both of those are continuous for a Maxwell-Boltzmann gas, with no thresholds anywhere.

Core collapse basically happens any time the electron energies reach the levels of the key endothermic process of electron capture, although there is also the endothermic process of photodisintegration of the iron, and there is neutrino escape going on the whole time. So these endothermic processes are the problem, because the core always needs kinetic energy to avoid collapse, and these processes rob it. So we can take as a working hypothesis that core collapse is equivalent to reaching the energies needed for these processes to run away. So that would happen to a Maxwellian gas, but there is a very interesting reason why it is much harder for a Maxwellian electron gas to get to this point than a Fermi Dirac electron gas. It's quite a remarkable process that involves degeneracy, should always be mentioned in any context involving core collapse.

You haven't shown that this will actually be the case. In fact my initial take is that adding mass to a Maxwell-Boltzmann gas will make it radiate more and thus lose heat and contract faster. Adding mass increases the absolute value of the (negative) gravitational potential energy, and by the virial theorem, that also increases the (positive) kinetic energy, and hence the temperature, and hence the radiation rate.

You are imagining a gas in free space, these cores are inside fusion shells. They're not radiating, they are getting radiated on! (They only lose heat as mass is added to them and they contract, which does cause them to get hotter and that can be radiated away, but only down to the level of the shell around them.) But at least we are focusing on the energy transport, that is always the key issue in core collapse (not the force equation, which is trival, it's just the virial theorem as you say).

Look at figure 1 in https://www.nature.com/articles/s41586-020-03059-w . Stars with higher initial mass start out farther from degeneracy, so higher mass stars have less degenerate cores. Notice what is happening there, the cores don't have a sudden transition to degeneracy. I'm pretty sure those smooth density distributions are much less degenerate, possibly almost ideal gases.

This also brings up the fact that, if the core is a Maxwell-Boltzmann gas, it won't even be stable to start with, at any mass. It's not even a question of having a stable object that gets "pushed over the edge" by adding enough mass to it, and asking whether the "edge" is different than for the actual case in the real world, where electrons obey Fermi-Dirac statistics. There is no stable equilibrium in the given mass and size range to start with. The object will always be contracting.

I'm not sure what you mean here that it will not be stable. Any nonrelativistic gas is dynamically stable, that's what the virial theorem says. Whether it is thermally stable depends on a lot of things, but there's no reason to think it wouldn't be in this case. Can you elaborate your thinking? I don't think "stability" is what you are thinking it is.

 

[PeterDonis]

the core not obeying the PEP will have a much larger ideal gas core at similar temperature but much lower energy per particle

If this configuration is actually possible, then it should exist in our actual world. In other words, even with electrons that obey the PEP, it should be possible in our actual world for an isothermal iron core to exist in thermal equilibrium with a surrounding silicon burning shell at a size large enough that the density is well below the density at which the PEP becomes significant.

Is that in fact the case? That is, do massive stars exist in our actual world that have such iron cores? If they don't, then I question whether your claimed configuration is actually possible, since the presence or absence of the PEP should be insignificant for such a configuration.

OTOH, if massive stars with large iron ideal gas cores do exist in our actual world, then it seems to me that the other case you describe, a massive star with an electron degenerate iron core at a much smaller size, should not exist in our actual world--because there should be no way to ever get to such a configuration. The ideal gas core should, according to you, remain in equilibrium until its mass is well above the Chandrasekhar limit, and when the ideal gas core finally does collapse, it will be too massive to reach an equilibrium at white dwarf size.

So what does the actual world tell us?

 

[Ken G]

There is no such thing as "core collapse" in the sense you are using the term if the core electrons are a Maxwell-Boltzmann gas at all densities.

Again, yes there is. It is the same thing that always causes core collapse: runaway endothermic processes. Let's focus on an important one: electron capture. When electrons get to high enough energies, they can enter nuclei and essentially bond with the protons, turning them into neutrons. The neutrons then leak out of the nuclei, this is "neutronization", and it causes the loss of kinetic energy and removes pressure support as a result. That causes even more electron capture, so too much of that process and the core just falls in on itself, it's an endothermic runaway. That will happen in an ideal gas just as much as in a degenerate one, it's core collapse.

There is no mass threshold at all, as I have already said. The core will always be contracting; it will never be in a stable equilibrium at all.

Always contracting is not at all the same thing as a "stable equilibrium." Things can be, and usually are, always contracting, even when very very close to force balance. The Earth is, for example! But stability is very different, it has to do with response to perturbations. Something that is dynamically unstable, if you kick it, falls into itself in a free fall time. Core collapse is more like a thermal instability, but the principle is similar, it's more or less the reverse instability of a thermonuclear explosion like a type Ia supernova (which is exothermic rather than endothermic).

So the whole structure of the object, as well as its evolution over time, will be different than in the actual case where Fermi-Dirac statistics are significant.

I don't know what you mean the the "structure of the object", the key equations in both cases are hydrostatic equilibrium (on free fall timescales only, as usual), and energy transport. Particle statistics only affect the latter, if you are satisfying the virial theorem for the former.

 

[PeterDonis]

I'm sure S&T has a chapter on it

No, they don't. S&T is mainly concerned with "cold" configurations of matter, i.e., configurations in which thermal energy is not significant. It talks some about processes of contraction that lead to such configurations, but it does not give a general discussion of the structure of objects like ordinary stars that are not cold.

 

[PeterDonis]

You are imagining a gas in free space, these cores are inside fusion shells. They're not radiating, they are getting radiated on!

Well, in thermal equilibrium the net radiation heat transfer is zero, but yes, I see what you mean. But one still has to ask what kinds of configurations of this sort are possible. That's why I posed the questions I did in post #29.

 

[PeterDonis]

It is the same thing that always causes core collapse: runaway endothermic processes.

Obviously we can always make the star do whatever we want if we allow arbitrary endothermic or exothermic processes. But I'm still struggling to see how this gets us to any kind of clean, simple comparison of the sort your OP made me think you were looking for.

 

[PeterDonis]

Always contracting is not at all the same thing as a "stable equilibrium."

I know, that's what I said.

Things can be, and usually are, always contracting, even when very very close to force balance. The Earth is, for example!

Um, what? Are you saying the Earth is always contracting? Where are you getting that from?

 

[PeterDonis]

I don't know what you mean the the "structure of the object"

The questions I asked in post #29 do a better job of getting at the point I had in mind.

 

[Ken G]

If this configuration is actually possible, then it should exist in our actual world. In other words, even with electrons that obey the PEP, it should be possible in our actual world for an isothermal iron core to exist in thermal equilibrium with a surrounding silicon burning shell at a size large enough that the density is well below the density at which the PEP becomes significant.

That is correct, it exists in every massive star whose iron core is not yet highly degenerate. For very high mass stars, that is quite common indeed. I suspect it holds all the way up to core collapse energies, if the mass is high enough, but at very high mass there are other instabilities also, like pair creation. So core collapse looks a bit different for the stars that just fall right into black holes, but I suspect they are ideal gases the entire time.

Is that in fact the case? That is, do massive stars exist in our actual world that have such iron cores?

Certainly. Massive stars have convective cores, which means they burn a whole bunch of silicon completely to iron, creating an iron core all at once. At that moment, the core has just finished nuclear burning, so it is a very good ideal gas. It takes time to lose enough heat to go degenerate. The highest mass stars should undergo core collapse before their cores ever go degenerate, just like we are talking about. But those core masses are way past the Chandra limit (which has no importance for them) from the outset. The case I'm focusing on is when the cores start out below the Chandra limit, and gain mass from the silicon burning shell. Those get rather degenerate before they reach the Chandra limit, but would stay ideal in our hypothetical situation here.

OTOH, if massive stars with large iron ideal gas cores do exist in our actual world, then it seems to me that the other case you describe, a massive star with an electron degenerate iron core at a much smaller size, should not exist in our actual world--because there should be no way to ever get to such a configuration.

They get to that configuration as iron ash is added to them, which causes them to lose heat as they contract under that load. The standard current picture is you have a silicon core that is above the Chandra mass for the most massive stars, and less than the Chandra mass for the middle mass stars. It is only the latter group that have any chance of getting degenerate iron cores. They do so because the core loses heat as mass is added to it. You pointed out in another post that these stars will probably already be past the SC limit when they first create their iron cores, and that's a good point. So their cores will already have undergone a kind of mini collapse, which creates a temperature gradient but might not be enough to get to the full core collapse energy level. Then the temperature gradient will initiate heat loss, and the core will go degenerate. Look at the core of the 10 solar mass star in figure 1 of https://www.nature.com/articles/s41586-020-03059-w . That is already a very degenerate core, there's a very clear transition there. It is also somewhat close to the SC limit, so close that it's not clear if it is above it or not. Maybe it was above it when it first formed, and the resulting T gradient that the SC limit creates is the reason it is so degenerate now. Much more massive stars don't show that sudden transition, those cores will be much less degenerate, maybe closer to ideal. So I believe that graph is showing us both cases you are mentioning, the case of ideal iron cores and degenerate iron cores. Unfortunately it is very hard to find discussions of the structure of the stars prior to the supernova simulations, all the attention goes to what happens in the supernova itself!

The ideal gas core should, according to you, remain in equilibrium until its mass is well above the Chandrasekhar limit, and when the ideal gas core finally does collapse, it will be too massive to reach an equilibrium at white dwarf size.

Yes, correct. It can also be too massive to reach equilibrium at neutron star size, and make a black hole instead. If all cores collapsed when they got to the Chandra mass, we would not see mergers of 10 solar mass black holes.

 

[PeterDonis]

When electrons get to high enough energies

At what density does this happen in an ideal gas?

 

[Ken G]

No, they don't. S&T is mainly concerned with "cold" configurations of matter, i.e., configurations in which thermal energy is not significant. It talks some about processes of contraction that lead to such configurations, but it does not give a general discussion of the structure of objects like ordinary stars that are not cold.

Oh, OK, it's a fairly general extension of the Jeans criterion. The Jeans criterion says that when a gas contained by external pressure starts to also get its own self gravity, at some point you lose a force balance and it all falls in, and that's how stars form (ignoring a lot of other things that also happen!). The SC limit is just like that, where the external pressure comes from the weight of the envelope, and as the core grows in size, it reaches a point where its self gravity causes it to lose force balance. This would happen when the core is like 1/3 of the mass of the whole star, except for the fact that the core has a different composition, it is made of fused nuclei that are higher mass than the envelope, exacerbating this problem so it happens somewhere in the range of 1/10 to 1/7 of the total star mass. But as you rightly point out, the stars that go supernova have cores that are already past this point, so the cores probably never are isothermal. I stand corrected on that, but I don't think it really matters much, as the SC initiated collapse might not have been enough to get full core collapse anyway. Ultimately it is the heat loss that causes the energy scale to rise, and that is true for either degenerate gas or ideal gas, so heat loss is always what causes core collapse (and its absence is always what prevents it, as we have discussed in the past). I agree that an ideal gas core is more susceptible to heat loss, which could in some cases make it undergo core collapse faster, but I think we have also seen cases where it makes the core stave off core collapse because the core mass can go way past the Chandra mass along the way.

What is still not answered is why this is. What is it about degeneracy that disallows a core more massive than 1.4 solar masses, yet that limitation is not suffered by an ideal gas core? Isn't degeneracy supposed to add some kind of mysterious pressure support? The thing that degeneracy is doing that compromises the core's ability to avoid collapse is the most interesting part of the puzzle, almost never mentioned anywhere. Again, it has to do with the specific heat of degenerate gas, and the fact that the low mass of electrons make them more susceptible to both going degenerate, and going relativistic.

 

[PeterDonis]

They get to that configuration as iron ash is added to them, which causes them to lose heat as they contract under that load.

Ok, but that means the relevant comparison between our actual world, where electrons obey the PEP, and your hypothetical world, where they remain a Maxwell-Boltzmann gas indefinitely, is not the one you proposed. The relevant comparison is what happens in the two cases (PEP vs. Ideal Gas) as this contraction process proceeds due to iron ash being added.

In the PEP case, the contraction stops at white dwarf size, provided the core mass at that point is less than the Chandrasekhar limit (you were assuming a 1 solar mass core so this would be the case). The core's size then remains constant until the Chandra limit is exceeded, at which point it collapses.

In the Ideal case, the core keeps contracting, even when its mass is well under the Chandra limit. The core's mass might exceed the Chandra limit during the process, but that won't change anything significant. In this hypothetical world, the core contraction continues until some other effect takes over (if we let the baryons obey the PEP, this would presumably be when neutron star size is reached and the neutronization processes you refer to have taken place and changed the object's chemical composition).

The qualitative behavior in these two cases is indeed quite different, but I'm not sure how it shows any problem with the general viewpoint that "degeneracy pressure stops contraction until gravity overwhelms it".

 

[PeterDonis]

What is it about degeneracy that disallows a core more massive than 1.4 solar masses

Degeneracy doesn't "disallow" a core more massive than 1.4 suns, it just says such a core must be contracting.

Basically the comparison is:

With Degeneracy: Some cores (those below 1.4 suns) will stop contracting at white dwarf size; the contraction will only resume when they gain enough mass to go over the limit.

Without Degeneracy: All cores, once they start contracting (for example, as you have said, a core of iron will contract as iron ash is deposited on it), will keep contracting right through white dwarf size, regardless of their mass.

So degeneracy does add a "support" that isn't there without it: the "support" that stops cores below 1.4 suns from contracting to smaller than white dwarf size.

If you're talking about ideal gas cores much larger than white dwarf size, which are not contracting because they are isothermal and in equilibrium with a layer of fusion surrounding them, then there is no relevant comparison with degeneracy to be made at all; degeneracy is irrelevant at such sizes and densities. So to say that such an ideal gas core "can" go above 1.4 suns, while a degenerate core (which won't even exist at such sizes) "can't", does not seem to me to be a useful statement.

 

[Ken G]

At what density does this happen in an ideal gas?

When all the particles are ideal, they all basically have the same kinetic energy, so the virial theorem tells us that energy is of order the gravitational energy per particle, GMm/R where m is the mean mass per particle. So it's about the M/R ratio, which is like the cube root of the density, times M to the 2/3 power. So at higher M, the density is lower where the collapse happens, and might not be degenerate at all, consistent with the assumption of being ideal (so it's not hypothetical then). What is so different if the gas has gone degenerate by then is, highly degenerate gas has a way higher energy per particle than kT, so the energy per electron is still GMm/R, but that is way more than kT. So the energy of the electrons doesn't care about the fusion temperature of the shell, it can be whatever it needs to be to support the contracting core. Its only limit comes when the core cannot lose any more heat, so cannot contract any more. That's all that's happening with "degeneracy pressure." The sense to which the electrons are "in each other's way" is simply that they can't receive enough energy from the contraction to get them to fit into the necessary deBroglie wavelengths required by that contraction, so energy loss is interdicted by wave interference. They are only getting in each other's way when they try to lose heat, it's not a force, the force is from the kinetic energy that was already there (and that itself is of course only a fictitious force on the gas particles but we're used to it).

 

[PeterDonis]

@Ken G, I'm not sure you understood the question I asked that you responded to in post #41. I was asking at what density electron capture becomes a significant process. Nothing in your post addresses that question. I'm looking for something along the lines of how the electron capture reaction rate depends on the kinetic energy per electron. I already understand how the kinetic energy per electron depends on density via the virial theorem. (The "in an ideal gas" part might not even be relevant because the electron capture rate might not care whether the electrons are degenerate or not; it might just depend on the kinetic energy per electron, no matter where that energy comes from.)

 

[Ken G]

Degeneracy doesn't "disallow" a core more massive than 1.4 suns, it just says such a core must be contracting.

No, it disallows it, because it contracts too fast to get to that mass. I keep stressing the difference between "must be contracting", which is a very weak constraint (most things you can name in astronomy "must be contracting"), versus "would have already contracted in the attempt to create it in the first place", which is what holds for highly degenerate gas above 1.4 solar masses.

Basically the comparison is:

With Degeneracy: Some cores (those below 1.4 suns) will stop contracting at white dwarf size; the contraction will only resume when they gain enough mass to go over the limit.

Yes, only they never get there, the act of trying to create them already causes their collapse.

Without Degeneracy: All cores, once they start contracting (for example, as you have said, a core of iron will contract as iron ash is deposited on it), will keep contracting right through white dwarf size, regardless of their mass.

Yes, but for ideal gases, it might take a long time, long enough to easily reach that mass, and even to go well past it, before that ultimate full contraction even happens. That's what I mean by "passing the mass limit."

So degeneracy does add a "support" that isn't there without it: the "support" that stops cores below 1.4 suns from contracting to smaller than white dwarf size.

Indeed, as I have always said (look back at any of our earlier correspondences). But notice the important nuance of meaning: by "support", you now mean it is a process that interdicts further contraction, not in the force equation (which is not near any kind of limit), but in the energy equation (the system's process of heat loss and contraction has reached an endpoint). I have put it this way: degeneracy is like a signpost that reads "go no further", it is not some new kind of force that is balancing gravity that wouldn't be there without degeneracy.

If you're talking about ideal gas cores much larger than white dwarf size, which are not contracting because they are isothermal and in equilibrium with a layer of fusion surrounding them, then there is no relevant comparison with degeneracy to be made at all; degeneracy is irrelevant at such sizes and densities.

That is true. But there is still a good way to make the comparison, it merely requires a tolerance for magic wands. What you do is take a degenerate gas that is, say, 1 solar masses, maybe a white dwarf. Then you wave a magic wand and make all the electrons distinguishable, so suddenly the PEP no longer applies. Then you ask a simple question: what happens next? The answer is: not much, at least right away. You might think there would be some catastrophic loss of this "extra force that emerges from the PEP" (I choke out the words), but actually nothing happens to the force at all. Instead, the particles start to redistribute into the Maxwellian distribution, and still very little happens, until they notice that they are no longer interdicted from losing heat. This also means their temperature rises dramatically, without any change in their kinetic energy content or pressure. But the higher temperature could cause heat loss, which will cause the removal of the "go no further" signpost. That's it, that's all that happens, that's the comparison between the two that can be made.

Now the interesting question I keep alluding to is, what is also happening that would allow this newfound ideal gas, if mass were being added to it, to have its mass go above 1.4 solar masses, with no special consequences at all, when that would have been disastrous had the gas stayed indistinguishable?

 

[Ken G]

@Ken G, I'm not sure you understood the question I asked that you responded to in post #41. I was asking at what density electron capture becomes a significant process. Nothing in your post addresses that question. I'm looking for something along the lines of how the electron capture reaction rate depends on the kinetic energy per electron. I already understand how the kinetic energy per electron depends on density via the virial theorem. (The "in an ideal gas" part might not even be relevant because the electron capture rate might not care whether the electrons are degenerate or not; it might just depend on the kinetic energy per electron, no matter where that energy comes from.)

I believe a typical scale for electron capture is about 1 MeV, so the electron is starting to get rather relativistic, in keeping with what allows the core collapse (as we will soon discuss when we get there). So if we take as typical a 2 solar mass core for our consideration, and use the expression from the virial theorem, the characteristic density is about 10 million g/cc. That density does indeed apply for both degenerate and ideal gases. Note this falls nicely below the densities found in the initial states of the core collapse simulations, as we'd expect for sims that want to see the onset and not have it already be there.

 

[PeterDonis]

it disallows it, because it contracts too fast to get to that mass

Yes, only they never get there, the act of trying to create them already causes their collapse.

for ideal gases, it might take a long time, long enough to easily reach that mass, and even to go well past it, before that ultimate full contraction even happens

I don't understand what you are comparing here. If a core is large enough to still be ideal in our actual world, as I have already said, degeneracy is irrelevant, so there is nothing to compare. To make any comparison involving degeneracy, we have to consider cores that are small enough to be degenerate in the actual world (as opposed to still being ideal gases in your hypothetical world where electrons always obey Maxwell-Boltzmann statistics). But that doesn't seem to be what you're comparing here.

 

[PeterDonis]

by "support", you now mean it is a process that interdicts further contraction, not in the force equation (which is not near any kind of limit), but in the energy equation (the system's process of heat loss and contraction has reached an endpoint)

Again I don't understand the distinction you're drawing here. What would "reaching a limit" in the force equation even mean?

In any case, you can indeed analyze this case using force (or more precisely pressure) instead of energy, the same way force analysis can be used for any kind of equilibrium: you look at whether a restoring force is created if the system is perturbed away from the equilibrium point. The fact that you don't like such analyses does not mean they don't exist.

 

[Ken G]

I don't understand what you are comparing here. If a core is large enough to still be ideal in our actual world, as I have already said, degeneracy is irrelevant, so there is nothing to compare. To make any comparison involving degeneracy, we have to consider cores that are small enough to be degenerate in the actual world (as opposed to still being ideal gases in your hypothetical world where electrons always obey Maxwell-Boltzmann statistics). But that doesn't seem to be what you're comparing here.

There are two comparisons happening, one is possible in the real world, which looks at actual stars that have cores that might stay fairly ideal for the entire core collapse leading to supernova (pretty massive stars, more than 40 solar masses perhaps, but to some extent 30 also), versus stars that don't undergo core collapse until they are already degenerate those are the 10 to 20 solar mass types most typically talked about). The second is in hypothetical world, where we can put weirdo classical electrons right next to degenerate electrons and notice the differences in their behavior. The second comparison is a device for understanding the behavior of the first, and for testing our understanding of how degeneracy works.

So let's start with the second one. Here, we can have the exact same volume and pressure in both degenerate electrons and weirdo classical electrons, and ask what is happening differently. That's where we see that ideal gas weirdo electrons can easily surpass 1.4 solar masses with no problem, as long as we add the mass quickly enough that it doesn't matter so much they have a much higher T so are losing heat much faster than the real degenerate versions. Then we see what degeneracy is actually doing, and why ideal gases don't suffer the same disastrous contraction (we're not quite at that punchline yet). The purpose is to take that knowledge over to the first situation, the real one, and notice that the weirdo electrons work just like real ones as long as they real ones stay ideal the whole time (as happens for the more massive stars).

 

[PeterDonis]

take a degenerate gas that is, say, 1 solar masses, maybe a white dwarf. Then you wave a magic wand and make all the electrons distinguishable, so suddenly the PEP no longer applies. Then you ask a simple question: what happens next? The answer is: not much, at least right away. You might think there would be some catastrophic loss of this "extra force that emerges from the PEP" (I choke out the words), but actually nothing happens to the force at all. Instead, the particles start to redistribute into the Maxwellian distribution, and still very little happens, until they notice that they are no longer interdicted from losing heat. This also means their temperature rises dramatically, without any change in their kinetic energy content or pressure.

When you violate the laws of physics in a thought experiment, you can't reliably conclude anything. In particular, you can't conclude that the pressure of a bunch of suddenly distinguishable electrons that are still in a Fermi-Dirac distribution will be identical to the pressure of the same electrons with the same total energy once they have redistributed themselves into a Maxwell-Boltzmann distribution. The equation that we actually use for the pressure of an electron gas assumes that the electrons obey the PEP; you can't assume the same relationship will hold if you wave your magic wand and make them distinguishable.

This is why we frown on such "magic wand" thought experiments here at PF: it is impossible to resolve any questions or disputes that they create.

 

[PeterDonis]

let's start with the second one. Here, we can have the exact same volume and pressure in both degenerate electrons and weirdo classical electrons, and ask what is happening differently. That's where we see that ideal gas weirdo electrons can easily surpass 1.4 solar masses with no problem

I still don't see the point. To make any comparison with degeneracy relevant, we have to be looking at white dwarf densities. At those densities the key difference I see is still the one I said earlier: degeneracy means objects below a certain mass stop contracting, whereas hypothetical always ideal gas objects don't stop contracting regardless of their mass. The "surpass 1.4 solar masses" case is above that limit, so we would not expect degeneracy to stop the contraction anyway--and ideal gas objects don't stop contracting at those densities either.

 

[Ken G]

Again I don't understand the distinction you're drawing here. What would "reaching a limit" in the force equation even mean?

Good question, that's why I don't use that language. Others do though, they talk about degeneracy pressure reaching a limit where it can no longer be as strong as gravity, as if this was a force limit (since both those things are in the language of forces).

In any case, you can indeed analyze this case using force (or more precisely pressure) instead of energy, the same way force analysis can be used for any kind of equilibrium: you look at whether a restoring force is created if the system is perturbed away from the equilibrium point. The fact that you don't like such analyses does not mean they don't exist.

What you can do in the force equation is solve for a limiting case. That's exactly what is normally done, one starts with the assertion that the gas is fully degenerate, and asks what the force (per area) would be at given density. Then you compare that to the force of gravity (again per area) at that same density, and look for a problem. Or you can vary the radius and look for a problem, under those same assumptions, it's the same thing. But a device for locating the "go no further" sign is quite different from a physical description of what is actually going on. However, let's not get into that, because it is pedagogy, and that is always somewhat subjective. The purpose of this thread is actually to understand what physically happens with core collapse, either in degenerate or ideal gas, and to understand why degenerate gas can never go above 1.4 solar masses, while ideal gas can. We don't need to critique other pedagogies, we have a test: find the pedagogy that actually lets you get the answer right. So far, that pedagogy has not been unearthed, because so far we have that ideal gas can't avoid core collapse at any mass, but that's wrong, because core collapse is not about the absence of some final equilibrium, it is about whether you can literally look at a core that has not yet collapsed and say "hey, that core contains 2 solar masses!" Can't do it with fully degenerate gas, but you can with ideal gas. Why is that? This is the actual question, and the answer is quite fascinating.