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 2: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.
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 = No 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.
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.