"Gravitational Compression in Neutron Stars"

[billj]

What happens to the neutrons in a neutron star as it collapses Into a black hole?

 

[phinds]

What happens to the neutrons in a neutron star as it collapses Into a black hole?

Same thing as happens to ALL matter that gets into a black hole, it disappears into the singularity. Now this is not believed to be physical but it's what the current model shows. Expectations are that if/when loop quantum gravity becomes a solid theory we might understand what's REALLY happening, but for now we don't.

 

[mathman]

We don't really know what happens to anything inside a black hole.

 

[Bernie G]

What happens to the neutrons in a neutron star as it collapses Into a black hole?

We don't know that neutron stars collapse into black holes. Maybe a better question is what happens to neutrons if there is core collapse in a neutron star.

 

[Bernie G]

What happens to the neutrons in a neutron star as it collapses Into a black hole?

Rephrasing: Total neutron collapse would mean star collapse, but does that happen in reality? If some neutrons collapse in a neutron star do all neutrons collapse? Maybe a better question would be: What happens if there is some neutron collapse in a neutron star?

 

[Drakkith]

If some neutrons collapse in a neutron star do all neutrons collapse?

Pretty much, yes, because if the core of the star starts to collapse, the rest of the star suddenly has nothing supporting it and collapses inward as well.

 

[Bernie G]

Collider experiments show that when a nucleus collapses what is produced is from 1% quark type matter and 99% energy to 10% quark type matter and 90% energy. What if a small percentage of core neutrons (<0.01R) collapsed into this instead of nothing? Normally we think of “photons” as weightless but here there would briefly be zillions of tons of photons with a pressure of (rho)(c^2)/3. I think this explosive pressure would temporarily heat and support the neutron star, or blast out of the star if it had a channel. A magnetic solenoid is an easy way out.

 

[sevenperforce]

If the pressure at the center of a neutron star were to exceed the limits of neutron degeneracy pressure, then the neutrons would presumably start to collapse into a black hole. If this black hole were small enough (e.g., on the order of a few thousand tonnes), then the radiation pressure from Hawking radiation could potentially be high enough to arrest further collapse. Don't know whether it would be stable, though.

 

[Bernie G]

If the pressure at the center of a neutron star were to exceed the limits of neutron degeneracy pressure, then the neutrons would presumably start to collapse into a black hole.

That would only be true if the collapsed neutrons had a volume that approached zero.

 

[Jonathan Scott]

That would only be true if the collapsed neutrons had a volume that approached zero.

What makes you think so?

As far as I know, even if neutrons did not collapse, GR says that a large enough neutron star would become a black hole anyway.

If neutrons do collapse, then if they collapse to a sufficiently dense form, it would merely cause a black hole to occur at a lower mass.

On the other hand, if they collapse to a form which is not sufficiently dense to cause an immediate black hole, then what happens beyond that would depend on the nature of that form and in particular the pressure it could support, but that form would also be certain to collapse to a black hole at a smaller mass than if it were able to remain as a neutron star because it would have greater density.

 

[Bernie G]

Let me rephrase the statement:
That would only be true if the collapsed neutrons had significantly less volume.

 

[sevenperforce]

Let me rephrase the statement:
That would only be true if the collapsed neutrons had significantly less volume.

Not necessarily. The density at the center of a neutron star is believed to exceed that of an atomic nucleus: 8e17 kg/m³. Of course, such high gravity is going to warp space pretty significantly, so Euclidean geometry doesn't exactly hold here...but taking the Euclidean approximation, a core which grows to 4.8 solar masses at this density will become a black hole in its own right without needing to collapse at all. If quark-degenerate matter starts to form at the core of a neutron star as neutrons begin to break down, then the density is expected to be around 1.7e18 kg/m³; such a quark-matter core would satisfy the condition for a black hole with just under 3.5 solar masses. A non-Euclidean formation would likely decrease these requirements significantly.

 

[Bernie G]

So far there are about 2000 observed neutron stars all with a maximum mass limit of about 2M☉. If neutron stars collapsed directly into black holes there should be black holes starting at 2M☉ but none have been observed yet. To me it looks like there is some kind of process intrinsic to neutron stars that limits their mass to about 2M☉.

 

[Bernie G]

On the other hand, if they collapse to a form which is not sufficiently dense to cause an immediate black hole, then what happens beyond that would depend on the nature of that form and in particular the pressure it could support, but that form would also be certain to collapse to a black hole at a smaller mass than if it were able to remain as a neutron star because it would have greater density.

What if that new form was ultra relativistic quark matter? Ultra relativistic matter would either heat or escape the star.

 

[sevenperforce]

So far there are about 2000 observed neutron stars all with a maximum mass limit of about 2M☉. If neutron stars collapsed directly into black holes there should be black holes starting at 2M☉ but none have been observed yet. To me it looks like there is some kind of process intrinsic to neutron stars that limits their mass to about 2M☉.

Indeed.

To begin with, there needs to be an explanation on the formative side of things. Current models suggest strongly that there is a certain mass/metallicity threshold required for the formation of a black hole by a collapsing star which ensures black holes will have at least five solar masses. Below this threshold, the collapse is much more energetic and will accelerate most of the star's material away, leaving no more than two solar masses to collapse into a stellar remnant. However, this fails to explain why a neutron star could not subsequently grow above this threshold. There are a few possibilities for neutron stars which exceed approximately two solar masses (by accretion or by a different kind of collapse):

1. They immediately collapse, with the collapse generating enough strong-interaction-bound and gravitational-potential-bound energy to exceed the relativistic gravitational binding energy of the object, blowing it apart completely.
2. The pressure at the core begins to "burn" neutrons by collapsing them into quarks, and that energy somehow escapes into polar jets.
3. They exceed two solar masses without incident, but this is so rare that we have not yet discovered one. Or, if we have discovered one, it isn't in the right place to have its mass measured so we don't know yet.

If 1 or 2 above are correct, it should be noted that this eliminates the need for an explanation on the formation side of things; a neutron star COULD form with a mass greater than 2 solar masses, but it would blow itself up (or, in the other case, shrink) rapidly. If the answer is 3, then the formative explanation is still needed.

 

[Jonathan Scott]

Collider experiments show that when a nucleus collapses what is produced is from 1% quark type matter and 99% energy to 10% quark type matter and 90% energy.

In the case you quote, the energy comes from the collider.

As far as I know, unless baryon number can be violated (which would be a non-mainstream assumption outside the scope of these forums), the effective rest energy (including internal kinetic energy) of the components of a neutron cannot be less than that of a proton, and quarks cannot be isolated, so very little additional kinetic energy can be obtained by breaking down a neutron into its components.

 

[sevenperforce]

As far as I know, unless baryon number can be violated (which would be a non-mainstream assumption outside the scope of these forums), the effective rest energy (including internal kinetic energy) of the components of a neutron cannot be less than that of a proton, and quarks cannot be isolated, so very little additional kinetic energy can be obtained by breaking down a neutron into its components.

So the binding energy between the quarks in quark-degenerate matter or a quark-gluon plasma is exactly identical to the binding energy between the quarks in a neutron? That doesn't quite make sense; breaking a bunch of neutrons down into quark-degenerate matter ought to release at least some of the strong-interaction-binding energy that kept the quarks in a baryonic configuration. Baryon number wouldn't be violated because you still have the same number of quarks, right?

 

[Bernie G]

... so very little additional kinetic energy can be obtained by breaking down a neutron into its components.

So are you saying when a 1000 MeV neutron disintegrates all we get out of it is some quarks with about 10 MeV rest mass?

 

[Jonathan Scott]

What if that new form was ultra relativistic quark matter? Ultra relativistic matter would either heat or escape the star.

This appears to be a personal theory of yours which you have already posted in some other threads, and I pointed out that you should start a new thread and provide acceptable references if you wished to continue to discuss it.

Your idea that something inside the neutron star could have enough energy to rise to the surface and escape does not make sense from an energy point of view.

As most of the energy per particle is simply derived from gravity, the only way for anything other than electromagnetic radiation and neutrinos to escape from the surface is if there is some effect such as a significant fusion explosion of accumulated matter which generates a huge amount of energy over a very short time. That could then result in a flash of neutron star surface material being ejected into space, as a cloud or shell containing traces of elements such as iron.

 

[Jonathan Scott]

So the binding energy between the quarks in quark-degenerate matter or a quark-gluon plasma is exactly identical to the binding energy between the quarks in a neutron? That doesn't quite make sense; breaking a bunch of neutrons down into quark-degenerate matter ought to release at least some of the strong-interaction-binding energy that kept the quarks in a baryonic configuration. Baryon number wouldn't be violated because you still have the same number of quarks, right?

So are you saying when a 1000 MeV neutron disintegrates all we get out of it is some quarks with about 10 MeV rest mass?

Baryon conservation and the fact that quarks can't be isolated together mean that per original neutron the internal kinetic energy of the bound systems of quarks plus the rest mass of any components with rest mass cannot add up to less than the mass of a proton. If there is sufficient energy around, then obviously one can create additional matching particle/antiparticle pairs or equivalent, but the quarks and gluons cannot "cool" back to anything less than a proton.

 

[sevenperforce]

Baryon conservation and the fact that quarks can't be isolated together mean that per original neutron the internal kinetic energy of the bound systems of quarks plus the rest mass of any components with rest mass cannot add up to less than the mass of a proton.

Forgive me if this is an elementary or obvious question, but why can't quarks released by the collapsing neutrons be bound in quark-degenerate or strange matter? Would that violate baryon conservation, or would that somehow constitute "quark isolation" and thus be prevented?

 

[Jonathan Scott]

Forgive me if this is an elementary or obvious question, but why can't quarks released by the collapsing neutrons be bound in quark-degenerate or strange matter? Would that violate baryon conservation, or would that somehow constitute "quark isolation" and thus be prevented?

I don't see any reason why alternative forms should be prevented. Baryon number conservation doesn't prevent the quarks being arranged in other ways or being excited to other levels such as strange quarks (with the same baryon number). However, any bound group of quarks and gluons could only be isolated if the total baryon number is a whole number (which implies groups of three plus optional particle / antiparticle pairs).

 

[sevenperforce]

I don't see any reason why alternative forms should be prevented. Baryon number conservation doesn't prevent the quarks being arranged in other ways or being excited to other levels such as strange quarks (with the same baryon number). However, any bound group of quarks and gluons could only be isolated if the total baryon number is a whole number (which implies groups of three plus optional particle / antiparticle pairs).

Naturally.

So what, then, is to prevent a gravitationally-bound collection of neutrons from collapsing into a soup of strong-interaction-bound quark matter with matching baryon number but lower binding energy, for a net exothermic process? I'm assuming that 21 quarks bound together in quark-degenerate plasma is going to have a lower binding energy than 7 neutrons...

 

[Jonathan Scott]

Naturally.

So what, then, is to prevent a gravitationally-bound collection of neutrons from collapsing into a soup of strong-interaction-bound quark matter with matching baryon number but lower binding energy, for a net exothermic process? I'm assuming that 21 quarks bound together in quark-degenerate plasma is going to have a lower binding energy than 7 neutrons...

If that was possible and you took that 21-quark unit out of that environment without adding energy, it couldn't decay back to protons and neutrons without adding energy, so either it or some decay product would be stable but have a mass less than the corresponding number of protons. I don't find that plausible.

I've been assuming that if "quark-degenerate plasma" is a possible result from compressing neutrons then giving way to the compressive forces would allow a little extra energy to be acquired from the environment, but I don't see any mechanism for releasing additional energy.

I should point out that my replies on this subject are not based on any specific familiarity with this area but rather on basic physics principles such as energy and quantum number conservation laws.

 

[sevenperforce]

If that was possible and you took that 21-quark unit out of that environment without adding energy, it couldn't decay back to protons and neutrons without adding energy, so either it or some decay product would be stable but have a mass less than the corresponding number of protons. I don't find that plausible.

I guess it would only be possible if strangelets were stable.

 

[Bernie G]

As most of the energy per particle is simply derived from gravity, the only way for anything other than electromagnetic radiation and neutrinos to escape from the surface is if there is some effect such as a significant fusion explosion of accumulated matter which generates a huge amount of energy over a very short time. That could then result in a flash of neutron star surface material being ejected into space, as a cloud or shell containing traces of elements such as iron.

Fusion reactions do not produce enough velocity for nuclei to escape a neutron star's surface.

 

[Jonathan Scott]

Fusion reactions do not produce enough velocity for nuclei to escape a neutron star's surface.

Continuous or frequent fusion would not produce enough energy per particle, but if material builds up for a while before a fusion chain reaction, then the resulting shock wave might well propel a small amount of material to escape velocity.

 

[PeterDonis]

Moderator's note: I have deleted a number of off topic posts, and added several posts below to try to refocus the discussion. Please keep things on topic and bear in mind the rules on speculative posts.

 

[PeterDonis]

It might be helpful to take a step back and look at the starting premise of this thread:

What happens to the neutrons in a neutron star as it collapses Into a black hole?

First you need to ask the question: why is the neutron star collapsing?

If a neutron star is below the maximum mass limit for neutron stars (analogous to Chandrasekhar's limit for white dwarfs--our current best estimate is that the limit for neutron stars is somewhere between 1.5 and 3 solar masses), it will never collapse; the neutron star will remain stable indefinitely.

If an object is above the maximum mass limit for neutron stars (say the collapsing remnant of a massive star's core after it has undergone a supernova), then it will never form a neutron star in the first place; it will collapse straight to a black hole. (Note that this conclusion assumes that there is no other stable state of matter that the neutron star could collapse to. See further comments on that below.)

So in order to even make sense of the question quoted above, we have to find a plausible scenario for a neutron star collapsing into a black hole. One such scenario would be a neutron star that is below the maximum mass limit, but not by much, accreting enough mass onto it to push it over the limit (for example, the neutron star could be in a binary system with a massive companion and material from the companion could fall onto the neutron star). If that scenario seems ok to everyone, then further discussion can be based on it. But it's meaningless to try to discuss the question without any scenario in mind at all.

 

[PeterDonis]

If the pressure at the center of a neutron star were to exceed the limits of neutron degeneracy pressure, then the neutrons would presumably start to collapse into a black hole. If this black hole were small enough (e.g., on the order of a few thousand tonnes), then the radiation pressure from Hawking radiation could potentially be high enough to arrest further collapse.

No, this won't work. It is true that, if we look at the event horizon in a spacetime where an object like a neutron star (or an ordinary star) is collapsing to a black hole, the horizon forms at the center, ##r = 0##, and moves outward until it reaches the Schwarzschild radius associated with the total mass of the object. But that does not mean the mass of the black hole starts at zero and slowly grows; what it means is that, until all of the matter in the object has collapsed below the event horizon, there is no clean way to separate the "black hole" from "the rest of the object".

Another way to put this is: when the event horizon forms at ##r = 0## and starts moving outward, it won't be producing Hawking radiation (at least according to our best understanding of Hawking radiation), for at least two reasons. First, the horizon is not in vacuum--it is embedded in the collapsing matter. The derivation of Hawking radiation being emitted from a horizon assumes vacuum. Second, the horizon is not a trapped surface--in other words, its area is not constant. The area of the horizon grows until all the collapsing matter has fallen inside it. The derivation of Hawking radiation, if you look at the details, assumes that the horizon is a trapped surface--that its area is not growing.

So this proposed mechanism for stopping a black hole from forming, at least if we use the current understanding of Hawking radiation, won't work. However, it should be noted that our current understanding of Hawking radiation and how it is produced might not be correct. There are speculations that quantum gravity effects might change things, even to the point where they prevent black holes (i.e., event horizons) from ever forming at all. But there are other speculations that say that quantum gravity effects only become important when spacetime curvature is large enough--the usual rule of thumb is that the energy density must be of the order of one Planck energy per Planck volume. This won't happen until well after the collapsing object, whatever it is, has formed an event horizon and all of the matter has fallen inside it. So on this view, while quantum gravity might prevent a singularity from forming inside a black hole, it won't prevent the black hole itself from forming. We won't know for sure which viewpoint is right until we have figured out the correct theory of quantum gravity.

such high gravity is going to warp space pretty significantly, so Euclidean geometry doesn't exactly hold here

Not only that, you are assuming a static system; a black hole is not a static system. What's more, you can't even have a static system with a radius just a little bit larger than the Schwarzschild radius associated with its mass. There is a theorem called Buchdahl's theorem which says that the minimum radius that any static system can have is 9/8 of the Schwarzschild radius associated with its mass. That means there is a finite "gap" between an object being stable in a static configuration and an object being a black hole; there is no continuous sequence of static configurations with gradually increasing mass that suddenly turns into black holes without any collapse in between.

 

[PeterDonis]

breaking a bunch of neutrons down into quark-degenerate matter ought to release at least some of the strong-interaction-binding energy that kept the quarks in a baryonic configuration

This reasoning would be valid if quark-degenerate matter were a possible state of matter at zero temperature, as baryonic configurations are. But it isn't; it can only exist to begin with at very high temperatures. Which means that the transition from baryonic configurations to quark-degenerate matter requires an input of energy; it is not a transition that will release energy, whether it's "binding energy" or anything else.

Note also that you can't use the kinetic energy gained in the collapse as the input of energy, at least not permanently, because that energy can be radiated away. In other words, if we took a bunch of neutrons and forced them to collapse somehow, and were able to control the process so that the kinetic energy of the collapse converted everything into quark-degenerate matter, the quark-degenerate matter could still emit radiation and convert right back into neutrons, since neutrons are a lower energy configuration.

why can't quarks released by the collapsing neutrons be bound in quark-degenerate or strange matter?

Because those configurations are higher energy configurations than neutrons. See above.

I'm assuming that 21 quarks bound together in quark-degenerate plasma is going to have a lower binding energy than 7 neutrons

And that is the incorrect assumption that is leading you astray. See above.

 

[PeterDonis]

Maybe core neutrons collapse into 1% quark matter with 99% energy that result in super intense X-rays which could result in intense positron/electron production.

As Jonathan Scott says, this would violate baryon number conservation. But we can look at it even without reference to baryons. Neutrons are composed of three quarks--one up quark and two down quarks. There are no antiquarks. So there is no way to convert 99% of the neutron's mass into energy; that would require matter-antimatter annihilation, i.e., quark-antiquark annihilation.

You could reduce a neutron's mass by a percent or two via weak interactions, converting it into a proton, if the neutron were in free space and not bound into a neutron star. But in neutron stars, neutrons are stable; weak interactions to convert them into protons would require an input of energy, and the resulting protons would just turn right back into neutrons, because the neutrons are a lower energy configuration. (As my response to sevenperforce should make clear, under the conditions inside a neutron star, neutrons are the lowest energy configuration possible, as far as I know.)

I have deleted all posts relative to ultra-relativistic jets; they are an interesting topic but not relevant to this discussion. Bernie G, please start a separate thread if you want to discuss the jets further, and please refrain from speculation.

 

[sevenperforce]

It might be helpful to take a step back and look at the starting premise of this thread:

We have to find a plausible scenario for a neutron star collapsing into a black hole. One such scenario would be a neutron star that is below the maximum mass limit, but not by much, accreting enough mass onto it to push it over the limit (for example, the neutron star could be in a binary system with a massive companion and material from the companion could fall onto the neutron star). If that scenario seems ok to everyone, then further discussion can be based on it.

Sounds good to me.

This reasoning would be valid if quark-degenerate matter were a possible state of matter at zero temperature, as baryonic configurations are. But it isn't; it can only exist to begin with at very high temperatures. Which means that the transition from baryonic configurations to quark-degenerate matter requires an input of energy; it is not a transition that will release energy, whether it's "binding energy" or anything else.

Ah. Quark-degenerate matter is a higher-energy state than baryonic matter. Got it!

No, this won't work. It is true that, if we look at the event horizon in a spacetime where an object like a neutron star (or an ordinary star) is collapsing to a black hole, the horizon forms at the center, ##r = 0##, and moves outward until it reaches the Schwarzschild radius associated with the total mass of the object. But that does not mean the mass of the black hole starts at zero and slowly grows; what it means is that, until all of the matter in the object has collapsed below the event horizon, there is no clean way to separate the "black hole" from "the rest of the object".

What's more, you can't even have a static system with a radius just a little bit larger than the Schwarzschild radius associated with its mass. There is a theorem called Buchdahl's theorem which says that the minimum radius that any static system can have is 9/8 of the Schwarzschild radius associated with its mass. That means there is a finite "gap" between an object being stable in a static configuration and an object being a black hole; there is no continuous sequence of static configurations with gradually increasing mass that suddenly turns into black holes without any collapse in between.

Well, in this case, we'd obviously have a collapse happening, so it's definitely not a static system.

I'd also note that the Schwarzschild radius won't "form" at the center and grow outward, because a neutron star already has a pretty significant Schwarzschild radius. Heck, R_S for the Earth is already about half an inch; a two-solar-mass neutron star has a Schwarzschild radius at a little over half of its own radius. When such a neutron star collapses, the Schwarzschild radius will remain constant while the outer layers fall into it; once they've all fallen in, it's a black hole.

However, the core is a good deal denser than the average density of the neutron star, so in some instances the Schwarzschild radius of the core alone may be a greater proportion of the core radius than for the neutron star as a whole. Thus, if the core begins to collapse first (as would be expected), it's possible that the core will fall within its own Schwarzschild radius before the entire neutron star falls within its total Schwarzschild radius. At that point, you would have a core-mass black hole surrounded by a collapsing shell of neutron-star mantle-and-crust. That's the case where you would have a black hole which is "separate" from the rest of the object.

If the collapse of the core causes the density gradient to increase rapidly enough, then you could have the case where the inner core falls within its own Schwarzschild radius before the outer core does. Depending on the shape of that density gradient, the initial black hole could have an arbitrarily low mass, which leads to my suggestion.

I've also been discussing this in this thread that I actually started myself; I'd love to get your input over there.

When the event horizon forms at ##r = 0## and starts moving outward, it won't be producing Hawking radiation (at least according to our best understanding of Hawking radiation), for at least two reasons. First, the horizon is not in vacuum--it is embedded in the collapsing matter. The derivation of Hawking radiation being emitted from a horizon assumes vacuum. Second, the horizon is not a trapped surface--in other words, its area is not constant. The area of the horizon grows until all the collapsing matter has fallen inside it. The derivation of Hawking radiation, if you look at the details, assumes that the horizon is a trapped surface--that its area is not growing.

So this proposed mechanism for stopping a black hole from forming, at least if we use the current understanding of Hawking radiation, won't work. However, it should be noted that our current understanding of Hawking radiation and how it is produced might not be correct.

Indeed. I know that Hawking's equations were "set" using the model of a stable black hole in a vacuum, but there is no vacuum (since the CMBR is always causing SOMETHING to fall into the black hole) and there are no stable black holes (because, Hawking radiation).

Moreover, while the observed match between the spectrum of thermally-radiating objects and the predictions of Planck's Law are always statistical in nature, Hawking radiation is unique in that it is predicts an exact match to Planck's Law. The reason that Planck's Law matched statistical observations, of course, was due to the quantization of energy levels in thermal emission. So even though Hawking's predictions cannot be derived with the sort of micro-black-hole that we're talking about here, I'm still interested in whether applying Hawking's predictions to a Planck-mass model could yield a statistical match to Hawking's predictions for macroscopic black holes, and thus serve as a starting point for a model of quantum gravity, or at least black hole quantization.

 

[PeterDonis]

I'd also note that the Schwarzschild radius won't "form" at the center and grow outward

I didn't say "Schwarzschild radius". I said "event horizon". They're not the same. See below.

a neutron star already has a pretty significant Schwarzschild radius

But it doesn't have an event horizon at all.

When such a neutron star collapses, the Schwarzschild radius will remain constant while the outer layers fall into it

The Schwarzschild radius remaining constant is just another way of saying the externally measured mass remains constant. (This assumes that no radiation is emitted during the collapse process, which is highly unlikely in the real world, but we can assume it for this thought experiment.) It does not mean that there is an event horizon sitting there waiting for things to fall in.

What actually happens, as I said before, is that, as the star collapses, an event horizon forms at the very center, at ##r = 0##, and moves outward (increases in radius) as the collapse continues. At the point where the surface of the star is just passing through the horizon, the horizon is just at the Schwarzschild radius corresponding to the mass of the system--and once there, it stays there.

Actually, though, this way of putting things can be misleading. A better way to put it starts with recognizing the definition of the event horizon: it is the boundary of the region of spacetime from which light signals cannot escape. So what is actually happening is that there is a particular event at ##r = 0## such that, if an outgoing light signal is emitted from that event, it will intersect the surface of the collapsing star just at the Schwarzschild radius corresponding to the mass of the system, and will then be trapped there forever, unable to move any further outward. The event horizon is just the set of all possible outgoing light rays emitted from that event at ##r = 0##, in all possible directions; they will all intersect the star's surface at the same radius (because we are assuming spherically symmetric collapse), and will all be trapped there, forming a 2-sphere of outgoing light rays that stays at that same radius forever.

I would advise rethinking the rest of your proposed scenarios (with varying densities of core vs. outer parts, etc.) in the light of the above.

I'd love to get your input over there.

I'll take a look.

there is no vacuum (since the CMBR is always causing SOMETHING to fall into the black hole)

Yes. This is usually taken to mean that no black hole can actually emit Hawking radiation until the temperature of the CMBR falls below its Hawking temperature, which will take something like ##10^{67}## years for a one solar mass black hole (and far longer for the much larger holes at the centers of galaxies and quasars). But nobody has actually done a rigorous analysis of this, as far as I know; it's just the obvious heuristic guess based on what we currently know.

applying Hawking's predictions to a Planck-mass model

I'm not sure what you are thinking of here. Hawking's prediction for a Planck mass black hole is that it will evaporate immediately, with no time lapse--i.e., that such a hole can't really exist since it will evaporate as soon as it is formed. This is not something that can be usefully analyzed statistically, as far as I can see.

 

[sevenperforce]

The Schwarzschild radius remaining constant is just another way of saying the externally measured mass remains constant. (This assumes that no radiation is emitted during the collapse process, which is highly unlikely in the real world, but we can assume it for this thought experiment.) It does not mean that there is an event horizon sitting there waiting for things to fall in.

Sure, I get that.

Actually, though, this way of putting things can be misleading. A better way to put it starts with recognizing the definition of the event horizon: it is the boundary of the region of spacetime from which light signals cannot escape. So what is actually happening is that there is a particular event at ##r = 0## such that, if an outgoing light signal is emitted from that event, it will intersect the surface of the collapsing star just at the Schwarzschild radius corresponding to the mass of the system, and will then be trapped there forever, unable to move any further outward.

I would advise rethinking the rest of your proposed scenarios (with varying densities of core vs. outer parts, etc.) in the light of the above.

I'm not quite sure how this would change the scenario. If there is a particular event at ##r = 0## such that an outgoing light signal emitted from that event would intersect the surface of the collapsing core just at the Schwarzschild radius corresponding to the mass of the core, then you have a core-mass black hole already inside the collapsing neutron star. Similarly, if there is a particular event at ##r = 0## such that an outgoing light signal emitted from that event would intersect the inner-core/outer-core boundary just at the Schwarzschild radius corresponding to the mass of the inner core, then you have an inner-core-mass black hole at the center of the collapsing core.

Hawking's prediction for a Planck mass black hole is that it will evaporate immediately, with no time lapse--i.e., that such a hole can't really exist since it will evaporate as soon as it is formed. This is not something that can be usefully analyzed statistically, as far as I can see.

Well, any analysis might be completely pointless if Hawking radiation predictions break down at a larger scale, but if they don't, then there might be a useful statistical analysis of what would happen as the Planck scale is approached, even if we're not dealing specifically with the Planck mass. For instance, trying to derive the minimum mass by looking at where the math would no longer make sense, like when the peak wavelength of emitted radiation would correspond to a particle energy exceeding half the energy of the object.