[math-fun] black hole size according to Andrew King. Atrocious.
http://arxiv.org/abs/1511.08502 The reason too-large black holes cannot accrete via an accretion disk allegedly is said disks become unstable to self-gravity-caused "clumping" perturbations. Then the gas either forms stars or is expelled by such stars, instead of spiraling into the hole. I'm finding this all extremely unconvincing. King is just multiplying a few letters like a monkey, and the reasons it all is valid (if it is) are very deep and due to other people not King; just tosses all that off in like 1 sentence, intermixing magic numbers like "0.001" and 3*10^16 that come out of the woodwork, as convenient. If this is correct you will be unable to tell it is correct by reading King; you'd also have to read a LOT of other stuff which is way deeper than King. Also, a lot of the letters he multiplies like monkey, he does not even define, he just says stuff like "alpha is the standard viscosity parameter" and eta is "the standard accretion efficiency" whatever those meant. So not only is it impossible to convince yourself King is correct by just reading King -- you cannot even understand what it is saying by just reading King. And although this ought to be possible to express in terms of fundamental constants, that is not possible by just reading King alone, and he did not try. In my opinion this author behavior is unacceptable. But since his paper was already accepted, evidently the journal does not share my notion of what should be "acceptable."
WDS notes on http://arxiv.org/abs/1511.08502 How Big Can a Black Hole Grow? By Andrew King. King's alleged mass upper bound for supermassive black hole forming from accretion disk: Mmax = 5 * 10^10 * Msun * alpha01^(7/3) * eta01^(4/13) * (L / Ledd)^(-4/13) * f5^(-27/26) where f5=1 approximately, L is the accretion-disk luminosity, Ledd is the Eddington luminosity, 0.1*alpha01 is the standard viscosity parameter, and 0.1*eta01 is the standard accretion efficiency (concepts King makes no attempt whatever to define). Apparently King intends that alpha01 and eta01 be dimensionless (?) and thinks they have order 1 (?) although he never actually states what units they might be in. He says this mass limit implies a luminosity limit Lmax = 6.5 * 10^48 * alpha01^(7/13) * eta01^(4/13) * f5^(-27/26) erg/sec Of course the key numbers here are 5*10^10 Msun and 6.5*10^48 erg/sec both of which King basically pulls out of thin air, with no attempt to express them in terms of anything such as Planck constant or whatever. Further, the whole argument expressed as poetry rather than mathematics (since King never actually does any mathematics) is that too-large black holes cannot accrete via an accretion disk allegedly since said disks become unstable to self-gravity-caused "clumping" perturbations. Then the gas hopefully either forms stars or is expelled by radiation from such stars, instead of spiraling into the hole. However, just last month I saw a lecture by an alleged expert on supermassive black holes who claimed nobody understood accretion disks and their luminosity, for example at least some such supermassive black holes expel relativistic jets out of their north and south poles and the reason is not known, but my guess would be it cannot be explained with just fluid dynamics and gravity – it also requires magneto-hydro-dynamics of plasmas to understand this. And certainly the flow is turbulent. King never even mentions those two things. So given this state of non-understanding I can hardly place much confidence in whatever King “derives,” especially given the utterly atrocious way he does so. It ought to be unacceptable for any author to refuse to define the variables he uses in his formulas. At the very least. And he pulls numerous giant magic numbers out of his ass like 0.001, 7.7*10^5 Msun, and 3*10^16 cm, inserting them into his formulae as he goes. And where are the disks of stars orbiting supermassive black holes which according to King’s claims should be created via his clumping mechanism? Observationally, such disks do not exist. So while I would like for King’s claims to be correct, and I would like to know what this reasoning actually yields as a formula in terms of fundamental constants… it seems to me that (1) what he did should have been unacceptable and cannot be taken seriously at least until it is completely redone in an acceptable manner, (2) it may be quite difficult for anybody other than King to express his formulae in terms of such constants. King assumes with no justification stated (indeed not even explicitly stating he is assuming this) the disk has constant thickness and constant surface density. He says the stability condition is (Disk surface density) / (disk thickness) = (disk mass density) = rho < M/(2*pi*R^3) where M is the black hole mass and R the local disk radius, and most critically would be that R is the outermost disk radius. Equivalently (he then claims) (Accretion disk mass) < 0.001 * (Black hole mass). He then magically says the “full disk equations” (what they are, he does not say) magically imply (he does not give any clue how) his EQ4, involving the magic number 3*10^16 cm, giving the disk’s outer radius. He then says obviously this disk outer radius must exceed its inner radius, which is the min-radius stable orbit about the hole, which is 5*f5*G*M/c^2 where G=gravitational constant, c=lightspeed, and allegedly f5 is about 1. Hence King derives his upper bound on M, the mass of the black hole. Here are some of King’s key references now rewritten in actual English, for the benefit of whoever might want to read them. I'm afraid I have to give up. And of course King never gives even a single equation number from any cited reference, as part of his intentional effort to make it harder for a reader to confirm, deny, or recompute his garbage: S.Collin-Souffrin, AM Dumont: Line and continuum emission from the outer regions of accretion discs in active galactic nuclei. II - Radial structure of the disc, Astron & Astrophys 229,2 (1990) 292-328 Juhan Frank, Andrew King, Derek J Raine: Accretion Power in Astrophysics, 3rd edn., Cambridge Univ. Press, Cambridge 2002. QB466.A25 F73 R.Genzel et al: The Stellar Cusp around the Supermassive Black Hole in the Galactic Center, Astrophys J 594,2 (2003) 812-832 AR King, JE Pringle, JA Hofmann: The Evolution of Black Hole Mass and Spin in Active Galactic Nuclei, MNRAS 385 (2008) 621- http://arxiv.org/abs/0801.1564 JE Pringle: Accretion Discs in Astrophysics, Annual Review of Astronomy and Astrophysics 19 (1981) 137-160 Alar Toomre: On the gravitational stability of a disk of stars, Astrophysical Journal 139 (1964) 1217–1238. https://en.wikipedia.org/wiki/Toomre%27s_Stability_Criterion
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Warren D Smith