New Model For Black Hole Accretion: Beautiful - But Is It Real?

Though a certain minority of physicists-astrophysicists (such as Lawrence Krauss) continues to believe black holes are some kind of myth or abstract confection with no grounding in reality, most of us don't buy that. Indeed, if it were true, we'd never see the frequency of papers on black holes published in reputable journals including the Astrophysical Journal, Nature, and Science.

Over the years the dynamics of the black hole as part of a binary system have been especially well investigated given that such pairing is the only way we can detect their presence. Usually, this is by the x-radiation give off in the process of "accretion" or layers of the companion star being pulled off and into the black hole with the friction unleashing the x-rays.

The Schwarzschild radius  provides the theoretical basis for the formation of most supermassive black holes and is given by:

R(s) = 2GM/ c ²


Where c is the speed of light, M is the gravitating or collapsed mass, and G the Newtonian gravitational constant. Thus, the value R(s) denotes the radius of the putative black hole given the mass M as the source. By way of insight, for the Sun R(s) would be about 3 km, but of course this is purely a theoretical limit given our star is simply not massive enough to collapse down to that size!  Not so for truly massive supergiants in the 10- 20 solar mass range, and further the super black hole at the center of our galaxy with 9.7 billion times the mass of the Sun.


Fig :Showing 3 different numerical  modelings-simulations.

In a recent numerical simulation study published in Science, (Vol. 345, p. 1330), the authors consider a scenario (depicted in Fig. 1) in which a low mass Population III remnant black hole (BH) remains embedded in a nuclear star cluster fed by cold gas flows and under the right conditions has the potential to grow rapidly. The simulation, model is beautiful and self-consistent but the question remains whether it is real, that is, has a correspondent system in physical reality. (I am writing not just about the black hole but the aggregate system).

In the model, the stars and the gas are "virialized" in the cluster potential - see e.g.

http://brane-space.blogspot.com/2010/11/basic-problems-in-astrophysics-4.html


So that basically the binding energy of the star cluster (E(s):

E(s) =  K  + W  = W/2 = -K

Thus, the total energy of the  star cluster E(S) is equal to half the gravitational potential energy (e.g. W/2)

The black hole is initially a "test particle" in equipartition with the stars. Then gas within the accretion (capture) radius of the BH

_{r a}   = [2 c ² /  c' ²  +   v ²  ] _{r g}

is dynamically bound to it. (Note: the gravitational radius  _{r g}  =  2 R(s) the Schwarzschild radius)  Note also that c' is the gas sound speed, i.e. in the cold flow far from the BH - and is a measure of the star cluster's gravitational potential.  Meanwhile, v is the BH velocity relative to the gas.  The authors point out that "prompt accretion requires gas to flow from r a   into the black hole on a specific trajectory with low angular momentum j =  4   _r g c.  and through the innermost stable periapse distance  r p, "    They note it is this angular momentum barrier not the Eddington limit (for which outward gas pressure balances gravity) that is the main obstacle to super-exponential growth.

Other points noted:

- The BH is more massive than a cluster star so that  v ²  <  c' ²  (The accretion flow is quasi-spherical)

- In the idealized case (flow radial and adiabatic) the Bondi solution is assumed such that:

_MB   = [ π  / Ö2 ]  ( r a ² )  r c'

(With adiabatic index   g  = 4/3  assumed)

- The stronger than linear dependence of the accretion rate on the BH mass leads to a solution that diverges supra-exponentially.

Focusing now on Fig. 1, the graphic shows dense cold gas (green) flowing to the center (X) of the stellar cluster (light blue region) of total mass:

_M o  = N_{o   M o}  +  Mg

And radius  R c  which contains  Ns  stars (yellow circles) of mass M  with velocity v, and gas of mass   Mg.

The gas is nearly pressure supported and close to the virial temperature, which from the previous link to my post on the virial theorem would be found from:  E(S) = - 3/2 [ g  - 1] U  where the internal energy U = f(T). A stellar black hole (BH) which is accreting from its capture radius (dark blue circle) is initially in a dissipation equilibrium with the stars and is scattered by them (black dashed line) over the distance: D (red circle).

Figure 2 summarizes the results of three different numerical simulations including a Monte Carlo run. Note that the vertical axis gives the angular momentum ratio   j a  / j iso    ie. in terms of gas captured by the BH, as a function (abscissa) of the BH mass and the corresponding time ratio t/ t' for Bondi accretion.   The authors note that the initial stages of BH growth is computed in the "ballistic wind accretion limit:  using an angular momentum capture efficiency of  h = 1/3 (red line).validated against results from a Monte Carlo integration over the exact capture cross section (tiny blue circles along the analytic  h = 1/3 red graph.

Note that  j a  falls to zero at _M o  = 20 solar masses (where the density and velocity gradients cancel each other). The vertical line displayed at _{M eq}  = 25 solar masses marks the transition to a dynamical regime where two -body relaxation can no longer establish equipartition of energy between the BH and stars.

Comments:

Examining the authors' model and their inputs as well as the model parameters (Table,  p. 1331) it appears they have a brilliant simulation for a rapidly growing black hole in a star cluster with particular dynamical properties in relation to it. I also, in 1977, believed I had a brilliant model for Epsilon Aurigae - to account for its binary eclipse phase-  until actual observations revealed I was wrong.  But this is the problem inherent in all numerical models. You carefully design them and they can entertain and inform...only up to the point that actual observations can confirm them.

I have no issues with the authors' modeling and simulations but I would like to see some kind of validation - preferably using a 'real world' system that displays similar properties to what the authors show in their Table.

The Making of a Terminal Bore: Wojciech Langer

The spherical harmonic function for problem (1) which I assigned Langer. He punted on all three cosmological questions that a first year physics student could have solved.


I wasn't remotely aware of the individual named Wojciech Langer until perusing some of the book reviews to do with cosmology on amazon.com. Then I beheld him cropping up like a bad penny each time five stars was awarded in a review for a book challenging the god assumption. The most recent example was Lawrence Krauss' excellent summation of recent cosmology, A Universe Out of Nothing.

In comment after comment going after 5-star reviewers of anti-creationist cosmology books, Langer's irritation - like most would-be godmongers, appeared to be that atheism was infiltrating or being infused into cosmology. Evidently the guy didn't realize or know this has been so for some time!

For example, there was the terrific book, ‘Great Ideas and Theories of Cosmology’ by Jagjit Singh (Dover, 1961).

I reference this book, because Singh’s book was among the first to boldly lay waste the idea of a “creator” being responsible for the cosmos (Chapter XVI, p. 252), and indeed he notes:

"No one… would dream of defending surrealism and cubism by an appeal to the tensor calculus or quantum theory, and it is as illogical to invoke scientific cosmology in support of God.

For the practice of rationalism is an irreversible process. If once one loses the innocence of naïve belief by venturing to stray into rational thought, there can be no honest way of recovering it. When one has cut himself off from God by a first sip of the cup of knowledge, one will not rediscover Him by drinking its dregs, no matter how hard they may be boiled

Even Stephen Hawking revived this tradition in his book, ‘A Brief History of Time’ though many readers may not have carefully parsed his words and only interpreted what they wanted to. For example, he clearly notes on p. 122 - after an audience with the pope, following a scientific conference at the Vatican:

“At the end of the conference, the participants were granted an audience with the pope. He told us that it was all right to study the evolution of the universe after the Big Bang, but we should not inquire into the Big Bang itself because that was the moment of creation and therefore the work of God.

I was glad then that he did not know the subject of the talk I had just given at the conference - the possibility that space-time was finite but had no boundary - which means that it had no beginning, no moment of creation.I had no desire to share the fate of Galileo, with whom I have a strong sense of identity”.

A few pages later in the same chapter, emphasizing the consequences of the quantum, boundary free cosmos, Hawking asks: "What need then for a Creator?"

Which is precisely the point! Science and modern scientific theories - bearing predictions which can be verified - make religious or supernaturalist intrusion redundant. They add nada to the quality of predictions, nor do they afford a new variant of suggested observations based on existing ones. In other words, all they offer is some childish security blanket for the timid or mentally un-tough, but not much more.

I used all this as a basis to dismiss all Langer's comments to the effect that this mindset "has no place in a cosmology text". YES, it does!

As for Wojciech Langer - the "industrial chemist"(from his amazon profile) - I suspected he didn't know diddly squat about any cosmology so that all his criticisms were hollow. They merely needed to be exposed, as he did. So I put him to the test by giving him three simple cosmological questions, of which I'd have been satisfied to get even one back correct. They were:

1) The data obtained from balloon-borne microwave telescopes, e.g. Maxima and Boomerang (cf. Physics Today: ‘Balloon Measurements of the Cosmic Microwave Background Strongly Favor Flat Cosmos’, July 2000, p 17)enabled a power spectrum of spatial temperature fluctuations to be assembled (see graphic) using a spherical harmonic function fit.

The multi-humped graph (ibid) plotted mean square temp. fluctuation (in micro-Kelvin on vertical axis) vs. multipole order l on the abscissa. This multipole order also concides with the spherical harmonic order index l.

The temperature differences dT_i appear at angular separations of π/ l and display a non-uniformity on angular scales of about 1 deg. If the first peak occurs at l= 200 and the 2nd peak is asymmetrically spaced relative to it, predict the next 2 minima displayed (i.e. their spherical harmonic order indices) and show working.

2) If the plasma is treated as an ion acoustic plasma what is the most direct (and simplest interpretation) of the “humps” in the function? According to the article (ibid.) the propagation speed of sound in such a plasma would be expected to be v(s) = c/ [3]^ ½

The acoustic properties of the plasma therefore create “standing waves”.

Assuming v(s) to be the “ion sound speed” how would one use it estimate the temperature of the plasma?

3) Consider 3 galaxies: A, B and C, i.e. as represented below:

(0.7c) <--------(B)----(A)-----(C)-------->(0.7c)

An observer in A measures the velocities of B and C and finds they are moving in opposite directions - each with a speed of 0.7c relative to him.
What is the speed of A observed by someone in B? What is the speed of C observed by someone in B?

And what was Langer's reply? (Even after I let him know that calculus wasn't needed for any of the problems).I produce it below:

"No, I am not able and not willing ..-I read POPULAR books (in case you did not managed to verify it), and inability to deal with high level of math (I do practical work not theoretical science) does not disqualify me from reviewing and commenting on them Same relates to majority of reviewers for this and other pop-cosmo books, therefore you show disrespect towards them as well.

But of course, he had no problems showing disrespect to another commentator who also questioned his criticisms on cosmology issues, especially when his profile showed "industrial chemist". His reply to her? Her comment was "stupid". In fact, it was spot -on and she just didn't have the background to expose him for the fake he is, and why his comments - at least to do with cosmology texts, have no merit. If a profuse, relentless critic of cosmology books and reviews can't even solve one simple cosmological problem he definitely doesn't deserve to have his opinions or comments taken with more than a grain of salt!

Lawrence Krauss' Black Hole Problem

Brought to my attention recently by a good friend is an article about Lawrence Krauss evidently rejecting the concept of black holes, e.g.

http://news.sciencemag.org/sciencenow/2007/06/21-01.html#.Tt50MKy3IYI

The article notes of the phenomenon of "Hawking radiation", whereby quantum-scale black holes can indeed "evaporate", "so that anything that enters is eventually released over billions or even trillions of years". The question is then raised: "How can something be both airtight and leaky"? But no one ever claimed quantum holes were both, only stellar black holes - the collapsed cores of massive stars, qualified on both those counts.

The threshold for making this cut is given by the well known Schwarzschild radius or:

R(s) = 2GM/c^2

where G is the Newtonian gravitational constant, c is the speed of light in vacuo, and M is the gravitating mass. Once a stellar remnant collapses within this radius, light cannot escape and the object is no longer visible, hence effectively "air tight" to use the words of the article. It is a characteristic radius associated with every mass of macroscopic scale. (As I noted, Hawking radiation preents quantum black holes from qualifying for the effective shutting off of radiant energy, because of their quantum properties which allow for quantum tunneling, superposition of states-interference effects etc. not to mentionbeing subject to the energy -time uncertainty principle).

Now, according to the article:

"Physicist Lawrence Krauss and Case Western Reserve colleagues think they have found the answer to the paradox. In a paper accepted for publication in Physical Review D, they have constructed a lengthy mathematical formula that shows, in effect, black holes can't form at all. The key involves the relativistic effect of time, .....n effect, Krauss says, time effectively stops at that point, meaning time is infinite for black holes. If black holes radiate away their mass over time, as Hawking showed, then they should evaporate before they even form, Krauss says. It would be like pouring water into a glass that has no bottom. In essence, physicists have been arguing over a trick question for 40 years"

But not really! The fact is that Krauss' formula doesn't apply to macro-scale black holes, of the type that form from collapsed stellar cores, but only to quantal scale ones that have the potential to emit Hawking radiation. Thus, Krauss' error lies in extrapolating his results from the quantum-scale black holes to the macro-scale ones (of which some new findings report holes of several million solar masses at the center of our galaxy.

Krauss, in the article argues:

"'How do you know they're black holes? No one has actually seen a black hole', he says, and anything with a tremendous amount of gravity--such as the supermassive remnants of stars--could exert effects similar to those researchers have blamed on black holes. "

But this is being too clever by half. The fact is that given the preceding theoretical threshold for the Schwarzschild radius we have an excellent default scale point for macro-emergence, and we are also able to empirically validate the limiting parameters once the hole is associated with a known stellar binary system. Thus, in the case of macro-scale black holes, we observe them indirectly as a members of binary (double) star systems, in order to infer their presence from x-rays given off when the companion star’s gaseous layers are sucked into it.As the extraneous matter is severely compressed by the black hole's gravity, it gives off the characteristic x-rays. We can say this with very high probability, irrespective of Krauss' formula.

In the case of supermassive galactic black holes, such as the pair recently discovered within clusters of ellptical galaxies more than 300 million light years distant, we have one "record breaker" at 9.7 BILLION time the mass of the Sun. For such a monster, the Schwarzschild radius would work out to 2.89 x 10^13 m, or about 193 astronomical units (AU). This is about 4.5 times the diameter of the solar system. The astronomers who effectively detected it used the Hubble Space Telescope and supercomputers to crunch the velocities observed for the stars in the vicinity of the objects, which would have been moving considerably faster than otherwise expected. (Recall the vis viva equation in obtaining the velocity of a body within a mutually orbiting system).

Can Krauss refute their findings, which were published Monday in the journal Nature? Then let him go for it!

Further, as astrophysicist Kimberly Weaver of NASA's Goddard Space Flight Center in Greenbelt, Maryland notes in the article, we haven't actually seen or detected Hawking radiation - the putative signature for Krauss' equations - either. So what's with that?

To make a long story short: the bottom line here is that the Schwarzschild radius provides an excellent theoretical threshold for setting the limit for emergence of actual macro-scale super-massive black holes. Unless Krauss can disprove it or attack its physical assumptions or basis, most of us will stick to the paradigm that black holes can and do exist, until such time

The Space Shuttle Was No “Dud”!

For a fairly bright guy, Lawrence Krauss (Director of the Origins Project at Arizona State University) can get some things mighty wrong! Among these is his take on the U.S. space program and specifically the Space Shuttle, appearing in an op-ed in yesterday’s Wall Street Journal (‘The Shuttle was a Dud But Space is Our Destiny’, p. A13). Krauss claims that the Shuttle was a “Failed program” and “a colossal waste of resources, time and creative energy”.

He also claims:

“The Space Shuttle program failed to live up to its primary goal of providing relatively cheap and efficient human space travel”

However, Krauss’ error – like so many others – is not looking at the program within the fiscal context of the time. Detached from this context, of course the Shuttle appears as a disappointment when so many expected lunar bases (the natural extrapolated outcome of the Apollo missions) by 1980. Then, we ought to have had a Mars base by 1990! But Krauss forgets or never processes a little impediment that intruded: the $269 b eventual cost of the wasteful (in both blood and treasure) Vietnam War.

Indeed, as the red ink bled, both from the war and LBJ’s “great Society” programs, it was evident by 1971 that: a) the Apollo missions would have to be truncated, and b) the manned space program – if kept – would devolve and diminish to a low Earth orbit substitute of earlier aspirations. When then President Richard Nixon (just before his Watergate crisis) confronted NASA’s administrators in 1972, they were basically informed of the writing on the wall: Either come up with a much tailored down program for manned space exploration, or have nothing at all.

Given the choice between something and nothing, NASA chose the first – which meant pursuing the less costly Space Shuttle program, already on the drawing boards. Alas, since the Shuttle was really designed for re-supply of existing space stations and none existed yet, this meant it would be constrained mainly to ‘show and tell’ events in low Earth orbit. There were no other choices, it was either this, leading on to the eventual substantive Shuttle missions 20 years later, or no manned flights.

Once Reagan assumed office in January, 1981, the constrictions on spending for space grew much more formidable, thanks to his massive tax cuts (going from a maximal marginal rate near 70% to 28%) and the $2.7 trillion defense spending spree (which combination was in fact responsible for converting us into a global debtor). Naturally, in this anemic and hostile (to space spending) fiscal environment the screws tightened even more on NASA and they were forced to lowball cost estimates and cut corners for future flights. The latter included ramping up an already overly ambitious launch schedule. If they didn’t do this, they’d be totally left out of the manned space budget. In these circumstances, the Challenger disaster must be understood and referenced- as the ultimate result of excessive cutting corners, and rushing launch schedules (the Challenger never should have been launched under those frigid conditions – and the O-ring risk had been noted by engineers at Morton Thiokol)

Krauss also lampoons the space station, which he describes as a “$100 billion boondoggle orbiting no farther from Earth than New York is from Washington”

But again, Krauss is naively measuring his standard for success against that for an ideal fiscal world, particularly one devoid of budget-busting wars! We did not have such a world, and from Reagan’s era deficits have only compounded making it clearly impossible (especially with so many occupations of choice) we ever will. You only have so much money to spend on so many different things – and the huge bank bailouts didn’t help! Krauss processes none of this background. Meanwhile, factoring in such background, I see the ISS as a stupendous feat of human engineering which will not only provide the basis to acclimatize to living and working in space, but also do experiments not possible from Earth – as well as analysis. And we have already seen manufacturing benefits from micro-gravity processes.

The space station has also afforded two former combatant nations the opportunity to cooperate in an extended space endeavor, detached from military agendas. It has thereby forged bonds, which will place us in good stead for future cooperative missions. Far from being a “boondoggle” the space station is easily the best first stepping stone on the path to Mars and more distant destinations.

While Krauss, like Steven Weinberg before him, does make the solid case that the “real science” has mainly been achieved by NASA’s robot probes, this doesn’t mean that it would have been smart to allow the manned effort to hibernate for 30 years! Sure, we may have had more probes to the outer planets, even to Mars or the asteroids, but with the manned space effort we now have a leg up on mastering the ability to LIVE and WORK in space over extended times, certain prerequisites for being able to survive 8-10 month journeys to Mars! Thus, the ISS and Shuttle have certainly not been for naught.

I do agree that in spending hundreds of billions one needs a rational plan “that can excite the imagination of the next generation”, as Krauss maintains. But let us agree that such a rational plan can only emerge in a fiscal environment which nurtures such expectations. Most of the cost overruns of the shuttle, meanwhile, were due to crimping its budget from the outset forcing NASA to lowball estimates to unrealistic levels.

Krauss ends by citing Richard Feynman’s famous quotation following the Challenger investigation:

“Reality must take precedence over public relations, for nature cannot be fooled”


To which I would only add:

“And sufficient money must be in place before reality can be manifested from dreams or imagination!”