Without degeneracy, when would Solar cores collapse?

[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.

 

[PeterDonis]

a device for locating the "go no further" sign is quite different from a physical description of what is actually going on.

I disagree. Forces are just as "actual" as energies, and a description in terms of forces is just as much a description of "what is actually going on" as a description in terms of energies. Both can be true and valid at the same time. One or the other might be preferred for a particular purpose or by a particular person, but that doesn't make the other one any less "actual" or valid.

 

[Ken G]

I disagree. Forces are just as "actual" as energies, and a description in terms of forces is just as much a description of "what is actually going on" as a description in terms of energies. Both can be true and valid at the same time.

Yes, they are just as actual, but that's not the issue at all. It just doesn't have to do with forces. But wait on that, forget the critique of pedagogy, this whole thread is to avoid subjective critiques. We have a real question on the table, which again is, why can you point to a 2 or 3 solar mass iron core that is ideal gas and see that it is just having no problem at all, at this moment, maintaining force balance in a star with a mass of like 30 solar masses, but you will never be able to do that with a highly degenerate core. What's more, if you enter weirdo classical electron world, you can start with a 1 solar mass highly degenerate core, add a bunch of weirdo electrons, and have no problem going past 1.4 solar masses (if you do it fast enough), but you'll never be able to do that with highly degenerate real electrons. Why is that?

 

[PeterDonis]

to understand why degenerate gas can never go above 1.4 solar masses, while ideal gas can

I still don't understand what comparison you are trying to make here or why you think it's so important. Yes, there are a variety of objects that astronomers look at, some of which have degeneracy as a significant factor in their behavior, and some of which don't. But I don't see what particular pair of such objects are "similar" enough to make the kind of comparison you are making in the quote above.

 

[PeterDonis]

why can you point to a 2 or 3 solar mass iron core that is ideal gas and see that it is just having no problem at all, at this moment, maintaining force balance in a star with a mass of like 30 solar masses, but you will never be able to do that with a highly degenerate core. Why is that?

The ideal gas core in this case is nowhere near white dwarf densities, so again I don't see why I would even want to compare it with a degenerate core that is at white dwarf densities in some other star with some other mass.

 

[PeterDonis]

forget the critique of pedagogy

I don't see how I can, since the main problem you are having in this thread is one of pedagogy. It's not a matter of me disagreeing with any actual physics you are expounding (leaving aside the issue I have already pointed out about thought experiments that violate the laws of physics). I just don't get why you are concentrating on the particular things you are concentrating on or why I should think they are important. That's a problem of pedagogy.

 

[Ken G]

The ideal gas core in this case is nowhere near white dwarf densities, so again I don't see why I would even want to compare it with a degenerate core that is at white dwarf densities in some other star with some other mass.

Then consider the situation where the densities are the same, the real electrons and the weirdo electrons. That is the device to understand what degeneracy is actually doing, because the only difference in the two situations is the particle distribution functions. Answer in that situation, why can the weirdo electrons go past 1.4 solar, but the highly degenerate ones cannot? What would actually happen to those populations as you quickly (but in force balance) add mass from 1 to 1.4? (That's not a pedagogical question, it's a physics question, albeit with hypothetical elements.) And by the way, it will be important that both these situations involve not just electrons, but also ions. (And it's not the charges that matter.)

 

[Vanadium 50]

I'm still unclear on what will happen when the magic want is waved.

Electrons become distinguishable, so the emtropy goes away up. The internal energy initially stays the same, because the system is virialized, and so the temperature goes way up. That surely cranks up the fusion rate, so you end up again with very different stars.

 

[PeterDonis]

consider the situation where the densities are the same, the real electrons and the weirdo electrons.

In this situation, as I have already said, the real electrons will stop contracting due to degeneracy, whereas the weirdo electrons won't. The weirdo electron core will keep contracting, so if, say we have both cores at 1 solar mass and a typical white dwarf density for that mass, then by the time both cores have grown to, say, 1.2 solar mass, the weirdo electron core will be significantly more dense than the real electron core. And that, to me, means the chances of endothermic processes like electron capture will be significantly higher in the weirdo electron core (since the rates for such processes become significant at densities right at the high end of the white dwarf density range), meaning that the weirdo electron core will likely start a catastrophic collapse at a mass below the Chandrasekhar limit, whereas the real electron core, whose density will stay lower, will not catastrophically collapse until it exceeds the limit.

why can the weirdo electrons go past 1.4 solar

Can they? Under what circumstances? See above.

 

[Ken G]

I'm still unclear on what will happen when the magic want is waved.

We start with two identical cores of highly degenerate electrons (and ideal iron ions). Both are at 1 solar mass, so they look a lot like white dwarfs. We wave the wand over one of them, and strip them of their PEP, that is, we label each electron so they are no longer indistinguishable, but all this does is release them from the PEP, nothing else.

Electrons become distinguishable, so the emtropy goes away up. The internal energy initially stays the same, because the system is virialized, and so the temperature goes way up. That surely cranks up the fusion rate, so you end up again with very different stars.

You are correct in all those statements, except the fusion. This is iron.

 

[Ken G]

In this situation, as I have already said, the real electrons will stop contracting due to degeneracy, whereas the weirdo electrons won't.

Yes, I said that also, from the very start of the thread. But remember, the degenerate electrons won't actually stop contracting, because mass is being added. We completely agree that the weirdo electrons will have a very high T and may start losing heat prodigiously, even if that is somewhat limited by that fusing silicon shell around them. But there is a timescale involved there, and it should be rather long because energy transport timescales are generally much longer than momentum transport (i.e., free fall) timescales, until you get into the core collapse phase.

The weirdo electron core will keep contracting, so if, say we have both cores at 1 solar mass and a typical white dwarf density for that mass, then by the time both cores have grown to, say, 1.2 solar mass, the weirdo electron core will be significantly more dense than the real electron core.

Yes, but the key point is this takes time, and we have a silicon burning shell adding mass quite fast. It takes only a few weeks to significantly enhance the core mass. This is the key player in all core collapse scenarios, the core mass is rising rapidly, and the new material is generally coming in "undervirialized", meaning it is not pulling its own weight in terms of the kinetic energy it shows up with. The core sags under the new weight, until it recovers virialization. However, the weirdo electrons are also losing heat at the same time, and this is where the timescale competition comes in that I mentioned early in the thread. The key is that the mass has to be added faster than the heat loss is happening to the ideal core, so the main contraction is from the rising mass and not the contraction from heat loss. That is what happens, for example, in a regular core collapse in a real star, the core collapse is ushered in more by adding mass than by waiting for the core to lose heat. I grant you that we may have to add the mass pretty quickly, but so be it, that's the scenario under consideration. We are asking why adding mass to a degenerate core makes it collapse, when it does not cause collapse of an ideal core with all the same properties except the particle distribution functions, i.e., it has the same density and energy but the kinetic energy is distributed differently. In the latter case, you can point to it and say, look, that same core we had a bit ago is now less contracted than the degenerate one, even though the degenerate one is supposed to have all that extra quantum mechanical pressure force! The remaining question is, why does this happen? There is a very specific reason. You are still wondering if it happens, I'm saying, add mass fast enough so that it does.

And that, to me, means the chances of endothermic processes like electron capture will be significantly higher in the weirdo electron core (since the rates for such processes become significant at densities right at the high end of the white dwarf density range), meaning that the weirdo electron core will likely start a catastrophic collapse at a mass below the Chandrasekhar limit, whereas the real electron core, whose density will stay lower, will not catastrophically collapse until it exceeds the limit.

That would certainly be true if the ideal core had enough time to lose heat and contract more than the degenerate core does. But if the mass is added quickly, the degenerate core will contract much faster, because it must reach zero radius by the time the Chandra mass is reached, if it didn't have all those endothermic processes kick in first. (The other timescale of interest is the force balance timescale, but that's the free fall time and is very fast indeed, so we will always assume force balance until collapse kicks in.)

Can they? Under what circumstances? See above.

If you don't give the heat loss enough time to act. Heat transport is generally slow, though I admit we have a lot of processes like neutrino escape and ultimately neutronization and photodisintegration. But those latter really kick in when you have core collapse, and we are analyzing the time before that, when energy transport timescales are still slow.

 

[PeterDonis]

If you don't give the heat loss enough time to act.

When I say the ideal core will keep contracting while the degenerate one will stop, I'm not just talking about heat loss at a fixed mass (or more precisely a fixed number of baryons and electrons). I am talking about the effect of adding more mass. As mass is added, the ideal core will compress more than the degenerate core.

Here's why I think that: for an ideal Maxwell-Boltzmann gas, the kinetic energy, and hence the pressure, goes like $1 / R$ always. So it compresses the same way in all regimes.

But for a non-relativistic Fermi gas, the kinetic energy, and hence the pressure, goes like $1 / R^2$. That means that the pressure increases faster as the gas is compressed.

Once the Fermi gas becomes relativistic, the kinetic energy, and hence, the pressure, goes like $1 / R$, as with the ideal gas, so in that regime the gas will compress the same as an ideal gas.

But from the above, we can see that a real electron gas will have a "bump" in its pressure behavior, while it is in the non-relativistic degenerate regime, whereas the weirdo electron gas will not. And while the real core is in the "bump" regime, it will compress less than the weirdo core does. This is true regardless of what is causing the compression, mass being added or heat loss. So once the "bump" regime is entered, the real core will be less compressed than the weirdo core, for the rest of the "experiment".

 

[Ken G]

When I say the ideal core will keep contracting while the degenerate one will stop, I'm not just talking about heat loss at a fixed mass (or more precisely a fixed number of baryons and electrons). I am talking about the effect of adding more mass. As mass is added, the ideal core will compress more than the degenerate core.

This is indeed the crucial issue, let us see where it leads.

Here's why I think that: for an ideal Maxwell-Boltzmann gas, the kinetic energy, and hence the pressure, goes like $1 / R$ always. So it compresses the same way in all regimes.

But for a non-relativistic Fermi gas, the kinetic energy, and hence the pressure, goes like $1 / R^2$. That means that the pressure increases faster as the gas is compressed.

This is a very important point. If indeed the degenerate gas stayed nonrelativistic, which we would normally assume for simplicity, then it is true that the degenerate gas would not contract as much. But the reason there is this 1.4 solar mass limit is that the degenerate gas goes relativistic. So that's why the radius actually goes to zero, in force balance, for completely degenerate gas. All you have to do is add mass to get to 1.4, maintain force balance and complete degeneracy, and the degenerate core is gone, it has fallen into core collapse.

To expound a bit, the reason highly relativistic gases contract to zero radius is that their virial theorem has a qualitatively different character. If you add undervirialized mass, a nonrelativistic gas can contract, release gravitational energy, and it only takes half that released energy to maintain the previous level of virialization, the remainder is available to improve the degree of virialization (if that excess kinetic energy is not lost as heat, so this is why we have to add the mass fast). But highly relativistic gas has no such margin for revirializing itself, it needs all the released gravitational energy just to maintain whatever level of virialization it had before. Hence adding undervirialized gas is catastrophic for highly relativistic gas, and this is a crucial ingredient of core collapse.

But the ideal gas core does not suffer that fate. This is the punchline. Why doesn't the ideal gas core also go relativistic, and also contract to zero radius? Why can it sail past 1.4 solar masses and not be relativistic, even these weirdo electrons that started out at the same energy as their degenerate cousins in the scenario where they were both at the same total energy and density when they had 1 solar mass? (And here is where the ions matter.)

Once the Fermi gas becomes relativistic, the kinetic energy, and hence, the pressure, goes like $1 / R$, as with the ideal gas, so in that regime the gas will compress the same as an ideal gas.

Ah yes, I see you have already realized the key point. But something is missing.

But from the above, we can see that a real electron gas will have a "bump" in its pressure behavior, while it is in the non-relativistic degenerate regime, whereas the weirdo electron gas will not. And while the real core is in the "bump" regime, it will compress less than the weirdo core does. This is true regardless of what is causing the compression, mass being added or heat loss.

Everything you are saying is correct, but forgetting one critical point: the ions!

 

[PeterDonis]

the reason highly relativistic gases contract to zero radius is that their virial theorem has a qualitatively different character

Yes, because the ratio of kinetic energy to (absolute value of) potential energy goes from $1/2$ in the non-relativistic limit to $1$ in the relativistic limit. But this will be true whether the relativistic gas is an ideal gas or a Fermi gas. So it will affect both cases (real electrons and weirdo electrons) the same, as soon as each case reaches the density where it becomes relativistic. And, as already noted, the weirdo electrons will be compressed more so they will reach that density sooner.

the ions!

The ions are an ideal gas in both cases (they don't become degenerate until neutron star densities are reached), so I don't see any difference in behavior there between the two cases.

 

[Vanadium 50]

Ah yes, I see you have already realized the key point. But something is missing.

I really, really, really hope you are not playing Twenty Questions here.

 

[Vanadium 50]

You are correct in all those statements, except the fusion. This is iron.

It will induce fusion in the envelope, and it will increase the rate of fusion before you get to iron. But no matter - let's start the clock with an iron core where electrons suddenly behave classically.

As we agreed, the temperature goes way, way up. So the radiation emitted also goes way, way up. As energy is lost, the temperature goes up (the specific heat is negative) and the radiation increases more. My copy of S&T is in a box at the moment, but I think this is similar to their argument. This in turn causes the core to contract more. This slows the process, but cannot stop it - it doesn;t stop until the neutron degeneracy kicks in.

So if you turn off electron degeneracy, you skip the whit dwarf phase (and the core of a red giant is white dwarf-like) and go straight to the neutron star phase. As expected.

 

[Ken G]

Yes, because the ratio of kinetic energy to (absolute value of) potential energy goes from $1/2$ in the non-relativistic limit to $1$ in the relativistic limit. But this will be true whether the relativistic gas is an ideal gas or a Fermi gas. So it will affect both cases (real electrons and weirdo electrons) the same, as soon as each case reaches the density where it becomes relativistic. And, as already noted, the weirdo electrons will be compressed more so they will reach that density sooner.

Right, another way to say the same thing is that when you contract undervirialized nonrelativistic particles, half the gravitational energy released goes into maintaining the previous level of virialization, and half goes into making up the previous shortfall, so in that way contraction brings the gas closer to being virialized. But for highly relativistic particles, essentially all of the liberated gravitational energy goes into simply maintaining the previous degree of virialization, and no "catching up" can occur, so at that point adding any more mass causes complete contraction to zero. When you mix highly nonrelativistic particles with highly relativistic particles, you can derive that the half of the released energy that would have gone into catching up gets reduced to just the fraction of the kinetic energy that is in the nonrelativistic population.

The weirdo electrons do lose more heat, so would contract faster if losing heat was the only issue. But if revirialization after mass increase is the only issue, then both groups contract similarly. So you correctly point out that when we take that into account and tack on the extra heat loss, the loser would be weirdo electrons, but only if they had the same degree of relativity as the degenerate case.

The ions are an ideal gas in both cases (they don't become degenerate until neutron star densities are reached), so I don't see any difference in behavior there between the two cases.

The ions don't go relativistic. I might be overstressing the importance of this, given that there is only one iron ion for twenty six electrons (it's more important for carbon in type Ia supernovae, where you have one ion for six electrons), but a very important thing that happens when you mix ions and degenerate electrons is that the degenerate electrons steal almost all of the kinetic energy of the ions (for very high degeneracy). This is the thing that degeneracy does that rarely gets mentioned, it is the reason that the valence electrons in a metal fork have way more kinetic energy than the ions in there (but they don't burn us because they are degenerate). So this is the way that degeneracy actually assists collapse: the electrons get all the kinetic energy, and when they go relativistic, there is no nonrelativistic fraction of the kinetic energy to produce any "catching up" of virialization when mass is added. But if the gas is ideal, then the ions get their fair share of the kinetic energy, and when that stays nonrelativistic, it allows for some protection against contraction via revirialization. Given that in core collapse, we have only one such ion for twenty six electrons, it might not be enough to induce less contraction in the ideal gas, so the ideal gas might not win after all in an actual simulation of this scenario rather than this idealized one.

Still, we can use this to understand what actually does happen in core collapse. We have a core where two things are happening at the same time, it is contracting for two reasons at once. One is that it is losing heat, because it is probably above the SC limit so must have a temperature gradient (it is not yet fully degenerate so shares some ideal gas behavior). The other is that undervirialized mass is being added to it, producing the need for "catching up" the virialization. All this contraction is causing the energy scales to rise. But because it is losing heat, its degree of degeneracy is rising, so there is one more thing happening: the degenerate electrons are stealing kinetic energy from the highly nonrelativistic ions, making everything more relativistic, and providing less ability to "catch up" whenever the virialization falls behind due to the gravitational load of the added mass. So degeneracy is not always helping prevent contraction, it only helps when interdicting heat loss is more important than increasing degree of relativity.

There is nothing like that in any explanation that says gravity is overcoming some added type of pressure in the core collapse process. But if you simply track energy, and say where the energy is going, the picture is crystal clear.

 

[PeterDonis]

the degenerate electrons steal almost all of the kinetic energy of the ions

Or, to put this another way, the ion temperature goes way down (because it is trying to equilibrate with the electron temperature, which is way down because the electrons are degenerate--their kinetic energy is not thermal). So another difference between the "real electron" core and the "weirdo electron" core is that the "real electron" core will be much colder than the surrounding shell in which fusion is taking place, while the "weirdo electron" core will be at least as hot and likely hotter.

This means that, while the "weirdo electron" core will be losing heat, the "real electron" core will not--it will be gaining heat (as well as mass in the form of iron ash) from the surrounding shell. This seems like yet another effect that would cause the "real electron" core to contract more slowly than the "weirdo" electron core, and therefore to take longer to become relativistic. Remember that the only regime where there is a difference between the pressure behavior of the two cores is the regime in which the electrons are non-relativistically degenerate. And in that regime, the real electron core contracts more slowly.

 

[Ken G]

It will induce fusion in the envelope, and it will increase the rate of fusion before you get to iron. But no matter - let's start the clock with an iron core where electrons suddenly behave classically.

Yes, you have a point, rising the core temperature will likely bump up the shell fusion a bit, and I'm not including that. So yes, there is always the issue of how far down the road you want to track the counterfactuality!

As we agreed, the temperature goes way, way up. So the radiation emitted also goes way, way up.

Probably neutrino emission would be the biggest deal, given that it doesn't have to wait like radiative diffusion.

As energy is lost, the temperature goes up (the specific heat is negative) and the radiation increases more. My copy of S&T is in a box at the moment, but I think this is similar to their argument. This in turn causes the core to contract more. This slows the process, but cannot stop it - it doesn;t stop until the neutron degeneracy kicks in.

That is the endothermic runaway that causes core collapse, so when thermal timescales start to compete on the free fall time, that's core collapse. You are saying that might be ushered in by the weirdo electron case, which is also what @PeterDonis is worried about. You're not wrong. But the energy losses are not just a function of temperature, many of the endothermic processes care more about the electron energy than the temperature. For those, like electron capture, what matters is the gravitational energy scale, so those are the same for real and weirdo electrons as long as they are at the same stage of contraction. I grant you that the ones that depend on temperature will be more active in the weirdo case, this all falls under the heading of heat transport. Remember we are considering the situation where mass is being added faster than heat transport timescales, which are generally pretty slow compared to virialization timescales (the latter being the free fall time). But I'm not disputing that these timescales are not being carefully tracked in this hypothetical scenario, it's not clear that the mass can be added fast enough to override the effects you and @PeterDonis are talking about.

So if you turn off electron degeneracy, you skip the whit dwarf phase (and the core of a red giant is white dwarf-like) and go straight to the neutron star phase. As expected.

It was never disputed that the ultimate endpoint will be a neutron star or black hole, the issue was always if there will come a point where you can point at, say, a 2 solar mass core that is still in pretty good force balance in the case of the weirdo electrons. That is what you know for certain you will never be able to do for the highly degenerate real electrons. @PeterDonis pointed out that the weirdo electrons are always more apt to contract as heat is lost, so that 2 solar mass situation should never occur if it doesn't for the degenerate electrons starting out from the same state of contraction. I pointed out that the weirdos have one ace in the hole: their kinetic energy is shared (somewhat) by nonrelativistic gas, making them better at revirializing when mass is added and potentially allowing them to sail through the 1.4 solar mass limit. But yeah, the timescales are not completely clear, and there are not a lot of iron nuclei, so this might not happen.

Still, in summary, the whole enterprise was just a device to bring us into a conversation that is focusing where it needs to be: on the energy, and its transport, and how that energy is shared between low temperature, high temperature, relativistic, and nonrelativistic particles. Not forces, or weird quantum mechanical additions to them. I am very pleased that in all of the well posed challenges you both have placed on the scenario I was advocating, neither of you ever found any use in citing any kind of modification to a force! I'm not surprised by that, because degeneracy is a thermodynamic effect, not a mechanical one.

 

[PeterDonis]

the weirdos have one ace in the hole: their kinetic energy is shared (somewhat) by nonrelativistic gas

The ions being non-relativistic only matters if the electrons are relativistic--but we have already agreed that degeneracy doesn't make a difference if the electrons are relativistic, the kinetic energy/pressure goes like $1 / R$ in both cases. It is only if the electrons are non-relativistic that degeneracy changes the kinetic energy/pressure behavior to $1 / R^2$, which slows the contraction as mass is added.

And since we are considering cores with the same mass (adding mass to both at the same rate since they are both surrounded by the same kind of shell in which fusion is taking place), the electrons will become relativistic at the same size range for both, so the only question is which one reaches that size range first. So far, both differences between the cores--degeneracy changing the pressure behavior, and the degenerate core being colder--look to me to cause the contraction to be slower in the degenerate case. That means the "weirdo electron" core should reach the size range at which the electrons become relativistic first.

 

[PeterDonis]

I am very pleased that in all of the well posed challenges you both have placed on the scenario I was advocating, neither of you ever found any use in citing any kind of modification to a force!

Not quite. I cited a difference in the pressure behavior in the regime where the electrons are non-relativistically degenerate, a $1 / R^2$ dependence instead of $1 / R$. You can of course equally well describe this as a difference in the kinetic energy behavior. As I have already commented, both descriptions, the force/pressure description and the energy description, are equally valid. You prefer the energy description so that has been the description we have been mainly using in this discussion. But that doesn't make the force/pressure description invalid or wrong. Both things will coexist in any scenario.