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.

'Interstellar' - A Film Not To Miss! (Warning: Some Spoilers!)

In some early reviews the new science fiction film 'Interstellar' has been compared to Stanley Kubrick's '2001- A Space Odyssey' which is not too far off. As in that 1968 film there are segments of this one that leave the viewer spellbound and often in wonderment (and mystery) at being enfolded in an almost metaphysical realm (especially one sequence where the astronaut 'Coop' finds himself inside a hyper-dimensional tesseract.)

The film, directed by Christopher Nolan, stars Matthew McConaughey, Anne Hathaway, Jessica Chastain, and Michael Caine. It is basically about a team of space travelers who travel through a wormhole in search of a new habitable planet. The opening scenes (roughly 20 minutes) show why. The Earth is now ravaged by monstrous dust storms and all its grain crops are dying out or have already become extinct because of a blight.

The scenes are realistic as we see people covered in dust, which gets into every nook and cranny, and also invades living spaces, smothers tables, linens, and incites chronic coughs. One of the most upsetting scenes occurs during a baseball game when a giant Haboob bears down and all the spectators must run for cover.

We don't know how the planet came to be this way or what caused the blight affecting the crops, but it doesn't take a lot of imagination to think that it has been spawned from radical climate change, at a time when even the seasons have stopped. The film cites the population being "9 billion" which would put the year at about 2040. (David Suzuki's projection for the first "year of no seasons" - marking the beginning of the runaway greenhouse effect).

Ultimately, the has-been astronaut Coop gets a new lease on his astronaut life when he ends up locating (using Morse code decoding of a gravimetric signal in his home) the only existing NASA facility. It is surrounded by barbed wire and kept ultra-secret, because as Prof. Brand (Michael Caine's character) tells him, you can't be spending billions on space ships when the world is starving and the only surviving grain crop is corn - which is also on its way out.

In this hidden enclave, Coop learns that 12 earlier "Lazarus" missions have been sent out - using a wormhole discovered near Saturn. Some have sent back signals indicating planets that might be able to support human life. These planets are all located in another galaxy, so clearly conventional rocket power would never be adequate to reach them. Thus has Coop been recruited to pilot Endurance, an experimental spacecraft, to follow the Lazarus Missions, through the wormhole to survey potential planets' long-term habitability. (By the time of Coop's launch the data from the Lazarus Mission has given NASA three potentially habitable planets: Miller, Edmunds, and Mann, named after the astronauts who carried out the surveys.)

The Endurance trek to the vicinity of Saturn alone takes two years, requiring all the astronauts to enter a hibernation condition to save oxygen. The wormhole itself is really a sphere, and by now the astronauts are awake as Coop enters through it - with incredible cgi graphics images depicting the transition.

The remainder of the film depicts astronauts Coop, Brand (Prof. Brand's daughter, played by Anne Hathaway) and Bromley trying their best to at least get to one more habitable world - after a disaster befalls them on Miller's planet (where one hour equals 7 years because of the gravitational slowing of time) and the craft is upended by a monstrous tidal wave some half- mile high, killing one of their original team.

A key segment, which I won't go into as it gives too much away, is the plan to pilot Endurance to a "gentle" black hole's ergosphere, jettison an AI designed robot into the event horizon and have it send data regarding the singularity within the black hole back to Earth.

What impressed me the most about the film is how realistically it depicted what an interstellar voyage would really be like, even assisted by a convenient wormhole and relativity. Bottom line: No picnic! Unlike most such portrayals of locating an alter-Earth (cf. the TV series 'Earth Two') there are no flora -abundant, hospitable planets, but mostly barren and hostile - with t he possible exception of "Edmund's planet".  We understand from the get- go life will largely be a matter of survival not simply landing in 'paradise' in the midst of avocado and mango trees.

One other point: Some reviews, for example in The Financial Times, have complained about needing "a baccalaureate degree in astrophysics" to understand the scientific interjections "about once every fifteen minutes". But first, the references were not that frequent (more like once every half hour) and second, they were totally within the grasp of a reasonably scientifically literate person - say who has read Hawking's 'Brief History of Time' or similar works.  Nor were any equations actually used in the explanations, we only see them occasionally when the blackboard is shown in Prof. Brand's (the elder's) office. All the equations are based on actual astrophysical theory and were pre-prepared by Kip Thorne of Caltech, one of the pre-eminent researchers of black holes and one of the film advisors.

By all means, if you have the chance to see this film, see it. It offers hope that one day, indeed, we may actually transfer the vision of settling other words to really doing so - as opposed to waiting for a monster, planet-killing asteroid (or something else) to take us out.

Stephen Hawking's Simple Explanation for How the Cosmos Originated...Godlessly!

Last night's Planet Green documentary on God and the Origin of the Universe in their 'Curiosity' series, was excellent, and perhaps Stephen Hawking's most masterful explanation yet for why no extraneous agents are needed to explain the origin of the cosmos. By the end of the program even a person with no mathematical background could grasp Hawking's simple arguments which followed from gravitational time dilation, e.g. the slowing of clocks as experienced near black holes or other gravitating masses, basic logic to do with time....as required for causality-causal nexus, and simple illustrations, examples. Altogether it was bot an entertaining and informative hour.

The program began with a look at Norsemen and how they viewed solar eclipses as terrifying actions by a particular unfriendly god in trying to eat the Sun and starve it of light. To these primitives, without any understanding or appreciation of astronomy or the movements of the the Moon, Earth in relation to the Sun, it had to be a "god" that took away the sunlight. Today, with our knowledge of celestial dynamics, we know this is nonsense. The Sun's light vanishes for a brief interval simply because the Moon passes directly in front of the Sun in our line of sight and the time interval elapsed for darkness (totality) is defined by the proximity of the Moon to the Earth at the time. The closer it is, the larger the angular diameter and the longer the time the Sun's light is blocked. No mystery! No gods needed!

Hawking argues the origin of the cosmos, and especially the Big Bang arose in a similar fashion of being shrouded in mystery, and superstition. He even recounts how Pope John Paul II warned against trying to investigate the "moment of creation" at one cosmology conference Hawking attended at the Vatican. Now after 3,000 years we no longer require artifacts - divine or other to grasp it. Still, the very concept of an uncaused universe beggars many people's imaginations and intellects to the point of madness, or at least strenous objection...via appeals to "common sense" (despite the fact most of modern physics is beyond common sense!)

Understanding Hawking's arguments even at a basic level, requires a few preliminaries. First, at a basic logic level, we agree and understand that to have causal agents or causality we require time. If there is no time evident or existent, then there can be no nexus or connection from event A to event B. It is time- the chronometric passage of some interval- which allows one to say, e.g.

(Event A) -> (Event B)

If on the other hand, the two events occurred with no time passage, one wouldn't be able to infer causality, say that Event A "caused" B.

Second, we need to grasp at least at a qualitative and rough quantitative level that time slows down in the vicinty of gravitating masses. The way this is approached is usually by using what's called a "metric tensor" such as shown in the bottom half of the lower diagram There are 16 components in this tensor, each of which can be measured by means of clocks...atomic or other. For the form of the metric tensor shown, we have elements conforming to a spherically symmetric distribution with r the radius or distance from the center.

To operate in a typically curved space-time (x,y,z,t) one will usually use what's called the "interval" such that:

ds^2 = {1 - 2GM/r}dt^2 - {1 + 2GM/r}(dx^2 + dy^2 + dz^2)

The key component used in gravitational time dilation measurements is the one at the very upper left of the matrix which we denote by g_oo. All experiments on gravitational time dilation can be regarded as direct measurements of the g_oo component of the metric tensor. For all comparisons with actual experiments, it's convenient to express chronometers at two different distances, say:

dτ2/ dτ1 = [g_oo(2)]^½/ [g_oo(1)]^½

it's convenient to use fractional deviations, e.g.

delta τ/ τ = (dτ2 - dτ1)/ dt ~ GM {1/r1 - 1/r2}

The clock rate can also be related to the frequency f. If a clock registers a large elapsed time it must be ticking fast, and if a small elapsed time, ticking slow. Thus the frequency difference, delta f, must be proportional to delta τ, or:

delta f/f = (f2 - f1)/f = delta τ/ τ ~ GM{1/r1 - 1/r2}

Cutting to the chase and sparing readers a lot of algebra, one ends up with:

delta τ/ τ = g [delta r/ c^2]

As a terrestrial application, say delta r = 10 km (the typical height attained by an aircraft) the time dilation is:

delta τ/ τ = 10^-12

Now switch to the Hawking's black hole example, and one finds as one approaches the center of the hole, at which r = 0, there is no time, period. This is also the point of maximal gravitational acceleration, call it g = oo, or infinite. Thus:

delta τ/τ = oo[ 0/c^2]= delta f/f = 0

In the above we have assumed the reference clock is right at the black hole center, and the difference is now 0. Then the time elapsed is zero. This is confirmed - well, the tendency of clocks to slow in g-fields is confirmed by hydrogen -maser clocks and others. (See, e.g. 'Gravitation and Spacetime', by Ohanian and Ruffini, p. 183. In effect, one simply extrapolates to the most extreme g-field, present in black holes, to infer no time passage that can be reckoned by any chronometer.

If there is no time passage, there can be NO cause! There can be nothing which causes the universe in the first place, hence...no God. Since we already agreed that for causality to operate time must exist, then conversely, if we find time doesn't exist then there can be no cause or causal agent.

Hawking's final argument, illustrated by the top illustration in the lower diagram, is simply that the cosmos (originally with less diameter than a proton) simply "popped" into existence much like we currently observe particles in pair production doing the same via spontaneous quantum fluctuations using the energy-time uncertainty principle. The basic principle underlying these pair production processes is the energy-time uncertainty principle: dE dt > h/2π where h is the Planck constant of action. . Thus, a brief fluctuation occurring over time dt can produce a change in energy dE, which may be sufficient to produce particles. The energy has an estimated magnitude given by the uncertainty principle.

Hawking's other point is that the manifestation of mass in the cosmos always has a complementary action in forming negative energy. He used the analogy of a person forming a large mound or hill by digging into the ground. The larger the hill formed, the bigger the hole from which the hill was excavated. All that negative energy is now scattered across space, as bound systems via gravitational potentials. These potentials represent negative energy reservoirs. For example, gravitational potential energy, say associated with a planet's g-field might be expressed: V(r) = - GMm/r.

If all the positive and negative energy contributions are tallied, the total comes to zero. This zero energy indicates that no agents acted to create the cosmos. We didn't need a "hole digger" to get positive mass from "digging a negative energy hole" because it already inhered in the subatomic cosmic hyperdense state. Moreover, we're still seeing -observing the effects of the initial repulsion of the unstable primeval "atom" as all parts of the cosmos continue to accelerate during the expansion of space.