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Stellar-mass black hole formation sequence

  1. Jan 14, 2016 #1
    I feel like this could go in quite a few of the Physics subforums (Quantum Physics, Beyond the Standard Model, Special and General Relativity, or High Energy, Nuclear, Particle Physics) instead of Astronomy and Cosmology, but hopefully this will work. This is my first question I've posed here, btw, so please forgive (but feel free to correct) any missteps.

    (Background: B.S. in physics, pretty good grasp of GR and SR, wide familiarity with astrophysics, less robust grasp of QM; this question is personal interest alone.)

    As far as I've been able to tell, the sequence of events in a massive star's collapse aren't very well-understood yet. It's not known whether the iron core collapses first to an unstable neutron star and then into a black hole, or if the collapse goes directly to a black hole. Moreover, there's a gap between the largest theoretically possible neutron stars (~3 M) and the smallest observed stellar-mass black holes (~5 M); presumably, this would be the result of some aspect of the collapse sequence, but that's not really known either.

    There is a great deal of discussion about the theoretical minimum-mass black hole, usually in the context of quantum gravity and the potential for quantized black holes. The entropic argument for black holes (that the event horizon of a black hole must be quantized to the Planck area in order to limit information to discrete bits) doesn't seem to hold up well and has been generally dismissed by the larger physics community, but at the same time it seems quite reasonable to assume that some minimum to the mass of a black hole must exist. If there was no lower limit on the mass of a black hole, then the power output of Hawking radiation would go to infinity, which means either there's some minimum on black hole mass or the Hawking radiation equations break down at quantum levels.

    On the other side of things: all the discussions/models/depictions I've ever seen of stellar-mass black hole formation feature a macroscopic event horizon, with the Schwarzschild radius and the surface radius growing closer (either because the Schwarzschild radius is increasing due to added mass or because the surface radius is shrinking due to increased density) until they finally meet and the black hole is formed.

    However, in the collapse of a massive star where neutron degeneracy pressure is insufficient to arrest collapse and a black hole forms, it doesn't seem that there would be anything preventing the density at the center from increasing without bound. The only pressure would be radiation pressure from the massive amounts of energy received, but due to the gravitational field the incoming radiation would be dramatically blueshifted while outgoing radiation would be dramatically redshifted, so that the pressure gradient goes to infinity as you approach the center. Is it possible, then, that the exponential increase in density at the center will reach the necessary conditions for the formation of a minimum-mass black hole before the conditions for a black hole are met at a larger radius, such black holes originate at the center and propagate outward?

    If so, that might lead to some interesting predictions about the rest of the black hole formation sequence, including a possible explanation for why the gap exists between neutron stars and black holes as well as a more promising basis for black hole quantization.

    Of course, I fully recognize that trying to model a minimum-mass black hole using GR and the characteristics of Hawking radiation may well be comparable to trying to model a thermonuclear explosion using a 2nd-grade chemistry set...but I figure it's worth a shot anyway.
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  3. Jan 14, 2016 #2

    Simon Bridge

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    Not all the mass of a star need form a super-dense object. i.e. neutron star formation.
    You are asking, I guess, if the huge pressure inside a very massive collapsing body could produce such high density inside that a black hole region could form which would not otherwise form by collapse from it's own mass alone? Off the top of my head: sure.
    However, I suspect you are also wondering if this means that maybe such a "light" black hole could remain afterwards ... that would be a no. Once formed, the light hole would quickly gobble more mass until you had a quite conventional black hole. However, this could be seen as a stage in the transition from regular degenerate matter to a black hole... probably very rapid if the transition to a Neutron star in a supernova is anything to go by.

    But I didn't look anything up - there's probably a wrinkle I didn't think of right away. There may be no need, particularly in the quantum regime, for there to be an intermediate stage at all.

    This is a highly speculative area after all.
  4. Jan 15, 2016 #3
    Indeed. In fact, this is one of the things that makes the observed mass gap so odd. Neutron stars can be produced by O+Ne+Mg-core stars in the 8-10 M range, by Fe-core stars in the 10-25 M range, or by larger-mass progenitors if the stellar metallicity is high enough, but in no circumstance does the degenerate remnant exceed 3 M. In contrast, black holes can form from progenitors above 25 solar masses with lower stellar metallicity, sometimes involving the fallback of matter onto an initial neutron star, but in no circumstance is the black hole less than 5 M. There must be some aspect of the collapse process in which collapsing stars which meet the conditions for creating a neutron star remnant "blow away" all but <3 M to escape velocity, but collapsing stars which meet the conditions for creating a black hole remnant always manage to retain >5 M.

    For two collapsing progenitors of equal mass but different metallicity, such that the first will leave a neutron star remnant and the latter will leave a black hole remnant, the former's collapse sequence must necessarily be markedly more energetic than the latter. There do not seem to be any cases in which a neutron star grows to its maximum mass and then collapses into a black hole of 3-5 M thereafter.

    Well, yes, but in a less general case. Specifically, can the density gradient go up rapidly enough that a black hole of arbitrarily low mass forms?

    The density/mass threshold for black holes looks like this (log plot):
    If the tremendous pressure inside the center of a collapsing body causes the central density to increase more rapidly than the inverse-squared relationship between density and mass, then the black hole will start at the center at an arbitrarily low mass. I'm just trying to figure out if that's possible. It seems possible; after all, in a collapse scenario where degeneracy pressure is no longer able to function, radiation pressure would be the only remaining force other than gravitational force, and the strong wavelength shifting of the gravitational field would presumably make inward radiation pressure exponentially greater than outward radiation pressure.

    That's not quite what I was wondering, but close. If the black hole forms at an arbitrarily low mass, then presumably this would be the minimum possible mass of a black hole. Since this would likely be near the Planck scale, the decay lifetime of such a black hole would be almost infinitely brief (if, in fact, Hawking's predictions hold for this scale), too brief for it to "gobble up" any more mass. Which could perhaps lead to some interesting predictions. But I know that I'll first need to figure out whether the initial concept (exponential density gradient causing an initial black hole to form with arbitrarily low mass) is even a possibility.
  5. Jan 16, 2016 #4
    Isn't the maximum observed neutron star mass only about 2.01 M☉? To me this indicates neutron stars greater than this are unstable and eject mass. Just my opinion. Instead of a collapsing star blowing away all but <3 M☉maybe neutron stars >2.01 M☉ blow away mass by themselves. Its a good idea but maybe you are incorrect about there being an exponential increase in density at the center during collapse so as to form a mini black hole. In a stable gravitational structure the core density is typically only 2.5 times that of the average star density. I think you are raising important issues that should be discussed more. Q: What happens to an existing accreting neutron star when its mass exceeds 2.01 (or 3 if you prefer) M☉?
  6. Jan 17, 2016 #5

    Simon Bridge

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    Except the hawking radiation stuff assumes there is vacuum outside the even't horizon - in this case the miniature hole is surrounded by other not quite dense enough material, so this reasoning won't hold.
    You would basically have to do the maths. It may be something more like the supermassive black holes in a galactic core but on a very small scale.
  7. Jan 18, 2016 #6


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    An obvious problem with current theories for stellar mass black holes is the mass gap between neutron stars and stellar mass black holes. The former has an upper observational limit of around 2 solar, and the latter has a lower observational limit of about 5 solar. This is very interesting. While the mass of both objects are difficult to quantify save in binary systems, it clearly suggests physics beyond our current understanding.
  8. Jan 18, 2016 #7
    AFAIK, there are a few different/competing explanations for how Hawking radiation is produced. One possibility is that virtual particles arising from the quantum vacuum are boosted into existence by the gravitational field of the black hole and escape; another is that mass-energy inside the black hole can tunnel through the event horizon and escape in that way. The latter explanation would seem slightly more promising, as quantum tunneling tends to fit more neatly with the whole concept of the black hole becoming more energetic as its event horizon shrinks, but then again that's just a conceptual/qualitative intuition.

    Applying conceptual intuition to these things is rarely fruitful, but it's worth a short.

    Let's suppose, for the sake of argument, that it is possible for a strongly self-gravitating object to satisfy the condition RS(r1)/r1 > RS(r2)/r2 with r1 < r2. In other words, regardless of how space is being curved (because that's going to mess with volume/density/etc.), the mass contained within r1 is "closer" to satisfying the conditions for a black hole than the mass contained within r2 even though r2 > r1. Let's further suppose that this condition is satisfied for all r1 < r2, so that RS(r)/r increases monotonically as r decreases. If that is the case, then during a collapse a black hole will form at the center with the currently-unknown minimum possible mass for a black hole.

    What might the maths look like? Well, such a black hole may be totally outside of classical behavior and bounds, but if it does fit, then we can put some constraints on it. Its decay lifetime and Schwarzschild radius need to be greater than the Planck time and Planck length, respectively, and ideally its mass would be lower than the Planck mass.

    If Hawking's equations are applicable, then the power output of a black hole is going to go to infinity as the mass decreases, so at some point the remaining mass-energy is going to become too low to sustain the required power output within its remaining lifetime. That's where I think we can tentatively put the minimum mass of a black hole. As the black hole shrinks, its temperature will also go to infinity, requiring that the peak wavelength of its blackbody spectrum goes to zero. Since the energy of a particle is defined by its wavelength, this offers a possible way to relate the energy of the discrete Hawking radiation to the total mass-energy of the black hole.

    At the extreme, the micro-black-hole emits two identical Hawking radiation particles in opposite directions (so as to satisfy conservation of momentum), each containing half its mass-energy. To satisfy the blackbody spectrum, these must be emitted at the peak wavelength λ = b/TH, where b is Wien's displacement constant and TH is the Hawking temperature. However, λ = b/TH corresponds to the wavelength as seen by an observer at infinity, so we have to account for redshift. Each particle is effectively being emitted "from" the event horizon of the black hole but is leaving behind a mass distribution half the size of the black hole, and since the Schwarzschild radius is proportional to mass, the Schwarzschild metric dictates that the Hawking radiation at infinity will have a wavelength λf = sqrt(2)*λi, where λi is the emitted wavelength. Thus the wavelength of our particles is λ = b/sqrt(2)*TH, and using the equation for Hawking radiation temperature (TH = ħc3/8πGMkB) allows us to put this in terms of the mass:

    λ = 8πGMkBb/sqrt(2)ħc3

    If each of these particles contains half the mass-energy of the black hole (which we can find quite easily with E = Mc2), then we can use E = hc/λ to find that λ = 2h/Mc, and we can combine this with our former equation then solve for M:

    M = ħc*21/4/sqrt(2*GbkB)

    A little math, and this evaluates to 1.627e-8 kg. A black hole with mass 1.627e-8 kg, about 3/4 of the Planck mass, would have a Schwarzschild radius of 1.5 Planck lengths and an evaporation lifetime of around 6700 Planck times. 6700 Planck times is probably not enough time for this micro black hole to "suck up" anything so it is going to evaporate into those two ultra-high-energy particles before it can grow. Each of those particles has enough energy that anything they come into contact with will probably collapse into a micro black hole as well, repeating the process in a chain reaction.

    I don't want to speculate too far, but if this chain reaction were to progress until the entire 5+ stellar masses had been "consumed", then perhaps collective gravitation of the whole micro-black-hole cloud would contain them, time-dilate them, and redshift their Hawking radiation so as to match the classical characteristics of a typical stellar-mass black hole.
  9. Jan 18, 2016 #8
    That's an intriguing possibility, but I'm not quite sure what mechanism could exist to allow a neutron star to eject mass. Perhaps the surface radiation or magnetic field of a heavy neutron star is energetic enough that any infalling matter above that threshold gets blasted into unstable particles which rapidly decay and escape? One problem would be that unlike black holes, neutron stars can have a wide range of temperatures and magnetic fields depending on their age, so this wouldn't necessarily hold true for a single mass limit. Also, this would suggest that infalling matter would produce spectacularly brilliant flashes of light which would probably be visible if you knew where to look for them.

    That's a really good question. It should be noted that a black hole is actually the minimum potential energy configuration for a given amount of mass; a neutron star has dramatically more potential energy than a black hole of equivalent mass. That potential energy is bound up in the strong force interaction between the quarks that make up the neutron star's baryons as well as the gravitational potential energy of the object. It may be that once a neutron star becomes so heavy that the pressure at the center exceeds the limits of neutron degeneracy pressure, the release of potential energy exceeds the gravitational binding energy of the body and it simply blows itself apart. Such an event might be a candidate for the creation of certain gamma ray bursts, according to at least one source I found.

    Neutrons are composed of two down quarks and one up quark, which have bare masses of 4.8 MeV/c2 and 2.3 MeV/c2, respectively. Neutrons, however, have a mass of about 940 MeV/c2, meaning that fully 99.2% of the neutron's mass is strong-interaction-bound. Lattimer's speculative neutron star relativistic equations of state suggest that the relativistic gravitational binding energy fraction (ratio of binding energy to total mass-energy) of a neutron star can be found as 0.6β/(1 - β/2), where β = RS/2R = GM/Rc2. So for a neutron star of 2.01 M and a radius of roughly 11 km, β = 0.27 and so the star's gravitational binding energy is about 18.7% of its total mass-energy. Thus, if the initial collapse rapidly reduced at least 19% of the neutron star's neutrons to quark-degenerate matter, the release of strong-interaction-bound energy would exceed the binding energy of the neutron star and the whole thing would blow itself to bits.

    I'm tempted to try and figure out how large the core would need to be in order to be 19% of the neutron star's mass, to see what it predicts for the overall size of the neutron star, but Euclidean geometry breaks down in the highly-curved space of strongly self-gravitating objects.
  10. Jan 19, 2016 #9
    If neutron stars are limited by some process to about 2 M☉its probably due an internal mechanism in the star instead of an external mechanism that prevents further accretion.
  11. Jan 19, 2016 #10
    Suggestion: Some neutrons collapse. As you indicated a neutron contains a few quarks with relatively small mass. 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 collapsed into this stuff: some quark type matter and mostly radiation. 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. 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. I'm suggesting that instead of forming outside of the star, the relativistic jets from neutron stars are radiation with a little quark matter blasting directly out of the core via the magnetic poles.
  12. Jan 19, 2016 #11


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    No, that is not correct, even though it is what is always presented in pop-science. Hawking himself said that the whole "virtual particle" explanation is not at all what's really happening, it's just the only way he could think of to describe to laymen in English what really can only be described in math.
  13. Jan 19, 2016 #12
    We already watch neutron stars pretty closely because most of them are pulsars, producing nice bright beams from their poles. If large neutron stars were occasionally flinging mass out of their poles we would have definitely noticed.

    Ah, I see; thanks. I know that the math behind Hawking radiation basically arises from the necessary temperature of the space just outside of the event horizon as measured by an infalling observer; I wasn't sure whether the virtual particle explanation was viable or just a convenient fiction. Does a notion of quantum tunneling fit more closely with what Hawking has said recently about super translations and the chaotic apparent horizon?

    If the apparent horizon is actually a sparsely-populated spherical of black hole quanta in a probability distribution across the surface of the would-be horizon, then this would seem promising.
  14. Jan 19, 2016 #13
    I'm suggesting the continuous nice bright beams and jets are the mechanism for mass ejection.
  15. Jan 19, 2016 #14
    Mechanism for or result of?

    The polar jets from pulsars are categorized by three energy sources: loss of angular momentum, collapse of accretion disk, and magnetic field decay. It's possible that core neutrons breaking down into quarks+energy could offer a fourth energy source, perhaps to explain certain gamma-ray pulsars. If this is the case, then we would predict that all gamma-ray pulsars are at or near the same limiting mass. So far, we have identified about a hundred gamma-ray pulsars out of 1800 total known pulsars; I don't know if any of those are in binaries or offer any other way of determining their mass.

    I'm still a little skeptical about a mechanism for getting something out of the core of a neutron star, solenoid or no solenoid. Existing models for the creation of pulsar jets all have to do with energy generation at the poles themselves, rather than deep within the most compact macroscopic objects in the universe.
  16. Jan 19, 2016 #15
    No problem. My best guess is: The basic process is matter collapsing into radiation at the core, and the jets are like a cylindrical flashlight beam exiting the rotation axis. If the magnetic field is not aligned it messes things up. A hard neutron crust or whirl pools are not a problem, a hard crust gets blown aside.
  17. Jan 19, 2016 #16
    Rephrasing: I think core neutrons collapse into radiation and quark matter, and big jets are this stuff exiting the star at the rotation axis and/or magnetic poles.
  18. Jan 22, 2016 #17
    I don't see the problem.
    Core collapse happens in a quite small volume of the star - the core is something like 10000 km across. Core collapse creates a "cavity", a region of low density, when it collapses to a tiny object some 10 km across.

    If core collapse creates a NS, almost all remaining matter gets expelled in the supernova: newborn NS is emitting insane amounts of gamma rays, something like 10 billion solar luminosities from each square meter of its surface. Total power is on the order of 10^40 watts. Matter would have very hard time falling onto such object, and one which does will likely immediately undergo thermonuclear fusion.

    If core collapse creates a BH, the situation is very different. It emits no such "wind". It gladly "eats" any infalling matter and radiation. It may well be the case that in typical core collapses to BHs, several solar masses worth of matter fall into the hole immediately after its creation.
  19. Jan 22, 2016 #18
    Not really, quarks inside hadrons move at relativistic velocities, thus a lot of this energy is their kinetic energy.

    "Blowing itself to bits" would require quark-degenerate matter to revert back to neutrons, which takes the same energy. We have a problem with this scenario...
  20. Jan 23, 2016 #19
    Chronos said: https://www.physicsforums.com/threads/stellar-mass-black-hole-formation-sequence.852285/goto/post?id=5347932#post-5347932 [Broken]
    An obvious problem with current theories for stellar mass black holes is the mass gap between neutron stars and stellar mass black holes. The former has an upper observational limit of around 2 solar, and the latter has a lower observational limit of about 5 solar.

    It could well be an unanswered problem. If a large number of neutron stars have been around and accreting for billions of years why don't we see any greater than about 2☉?
    Last edited by a moderator: May 7, 2017
  21. Jan 23, 2016 #20
    Beyond that mass a neutron star can collapse further to become a black hole, (or the hypothetical quark star as an intermediate).
    Neutron stars are small in size but observable, but an even smaller denser object may not be until it acquires additional mass through accretion,
    At some point (around 5☉), it could become detectable again through indirect observation such as gravitational lensing.
    So objects in the 2☉ to 5☉ may well exist, just we have no way to see them with presently understood physics and technology
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