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.