Transonic flows around black holes and their observational properties
Sandip K. Chakrabarti
S.N. Bose National Centre for Basic Sciences and
Centre for Space Physics

Well known picture of a black hole with an accretion disk:
What is the description of the flow around a black hole?

Binary accretion: usual picture from a distance

• It the flow spherical in nature or disk-like?
• Is the viscosity always sufficient to form a specific angular momentum distribution (e.g.,
Keplerian)?
• Is the flow smooth?
• Is the flow stationary or time-dependent?
• Do the radiative properties of model flows match with observations?

On a super-massive black hole, the accretion could be from a bunch of stars embedded in a gas
(or from a star that is torn apart inside the Roche radius)

Brief history of accretion/wind study around compact objects
• Bondi accretion (1952- )
• Shakura-Sunyaev ‘Standard Keplerian disk’ accretion (1973- )
• Liang & Thomson/Abramowicz & Zurek/Paczynski & Czerny transonic flows (1980-1982)
• Thick accretion disks/Polish doughnuts/ion pressure supported tori (1978-1985)
• Global transonic solutions with shocks and inefficient emissions (late 80s)
• Dissipative shocked flows, theory & simulations (early 90s)
• Advection dominated accretion flows or ADAFs (1994-1999)
• Inflow-outflow solutions; Spectral and timing properties (mid to late 90s)
• Refinements (continuing)

In a Bondi Flow, spherically symmetric matter falls on a gravitating object. Adiabatic matter passes through ONE sonic point
(Bondi 1952).
Radial Momentum equation
Continuity equation
Velocity equation (derived)
Flow derivatives at the sonic point

Two solutions: One for accretion and the other for the wind (Parker)
On a black hole
On a neutron star

•Meanwhile, Quasars are discovered and they are found to be highly luminous.
•It was impossible to explain them by the radiativly inefficient Bondi flow
•It was necessary to slow down the matter by addition of angular motion
•Most natural choice was the Keplerian distribution:
Along came the most well known accretion disk model of
Shakura & Sunyaev (1973)

Essence of the Shakura-Sunyaev (1973) disk
Angular momentum distribution
Accretion rate equation
Viscous stress responsible for angular momentum transport and heating the gas
Flux of radiation

Flux
Various quantities can be written down analytically as a function of the radial distance
Surface density
Vertical height
Density
Temperature
Opacity
Luminosity

Keplerian Disk and its spectrum

• At all? When not and why not?
• Always?
• Viscosity (alpha, beta, or what ever)?
• What are the signatures (as far as spectral properties are concerned) that the disks are different?
• Sunyaev-Titarchuk work (1985) suggested that the power-law component should be due to
Comptonization of thermal photons.
• HMXB vs. LMXB --- do they have any distinction?
• Can timing properties distinguish one disk from another?

• Ideally, never! The disk will always pass through the inner sonic point and will become supersonic to enter into the black hole.
• It can also pass through shocks where supersonic to subsonic transitions occur.
• For neutron stars also, a shock and a boundary layer will form (Chakrabarti 1989).
• In order to become transonic, the flow requires a ‘preparatory zone’ extending r>3r g
, the marginally stable value (where the standard disk is cut-off).

• Since the viscous stress has the same dimension as that of the pressure, SS assumed that they are proportional to each other
W r f
=a
P
Is P the thermal pressure, or gas pressure or total pressure?
Since radial velocity was ignored in the subsonic Keplerian disk, the net pressure may be taken to be P+ r v 2 since v ~ c close to the horizon and the ram pressure is as important as the thermal pressure. Furthermore, P+ r v 2 is continuous across the shock discontinuities. So, in order to have continuity in viscous stress, we must have P+ r v 2 as the pressure to take care of the generalized flow (Chakrabarti &
Molteni, 1995)

• Not absolutely essential for a black hole accretion. We can have flow of constant energy falling into the hole as long as the pressure (radiation or thermal or both) is high enough.
• On the other hand, if the angular momentum of the supplied matter is high, it has to be reduced substantially before it can be accreted at a moderately high pressure! This is the job of viscosity.

Potential barrier to be overcome by matter before entering into a black hole (here we draw potential barriers for different angular momentum)

There is nothing wrong in the ‘standard’ formalism itself, except that the assumptions about the sub-sonic nature should be relaxed and more accurate study must be made, for instance, which included transonic solutions close to the inner edge of the flow.

While this model was an immediate success in explaining quasar luminosity, for stellar mass compact objects the disk model immediately required modification. This was sensed by Eardley et al (1976) and Sunyaev & Trümper (1978). The spectrum required a COMPTON Cloud.

Came along thick accretion disks/Polish doughnuts
Radiation pressure was added ….
resulting in non-
Keplerian angular momentum ….
and a thick disk is formed

• Paczyński and his group (~1978-1982) started to ‘repair’ inner edge of the disk by adding transonic flows.
• They also came up with the so-called thick accretion disk in which the angular momentum distribution is either conserved or different from Keplerian. This disk puffs up due to radiation/thermal pressure.
Radial velocity was ignored.

Equipotential Surfaces in a Polish Doughnut

This gave rise to the Ion pressure supported tori (British doughnuts)

This cannot be the whole story since the important ingredient, namely, the radial velocity is still missing from the equations.
Fortunately, Liang & Thomson (1980) have been looking into this problem with full set of equations and found that the flow can have two saddle type Sonic points:
But they did not extrapolate the conclusion further.

Meanwhile, Solar wind people were already way ahead, they found multiple sonic points, shocks etc. in the winds. (1960-1970s)
Fukue (1987) showed that such solutions should exist in black hole accretion also
(though he had multiple shocks).

The main premise: Whatever be the mass of the black hole, the inner boundary condition is the same, namely, matter enters through the horizon with a velocity of light.
How do we study such a mess when we have all types of black holes in all types of surroundings?
We use mathematical properties of black hole horizons!

Generic properties of a black hole accretion:
A black hole accretion is necessarily supersonic on the horizon
The flow must pass through at least one sonic point
In presence of rotation it can have
TWO saddle type sonic points and thus,
It can have standing, oscillating or propagating shock waves

These are super-sonic to sub-sonic transitions of matter at around tens of Schwarzschild radii where matter piles up. They are primarily centrifugally supported.
The shocks are centrifugal pressure supported since l
2
/ r
3 
1 / r
2 close to a black hole

Let’s think of a more generalized steady flow around a compact object:

What should be the topologies of this flow?
Specific energy of matter of an isothermal flow:
Very close to the black hole or very far away from the black hole:
At intermediate distance (x ~ λ )
Net topology
+ +
+

All possible flow topologies
Classification of solutions for a Kerr black hole
Chakrabarti 1996
Same as in
Schwarzschild black hole: SKC 1989

All possible solutions of an inviscid flow on a
Schwarzschild black hole
Chakrabarti, 1989

(a) Chakrabarti and Molteni (1993) showed that shocks indeed form. This was done using Molteni’s SPH code
(b) Molteni, Lanzafame and Chakrabarti
(1994) published papers on shock formation in two dimension flow. Again the shocks were found to be formed without any problem.

• The post-shock flow is as close to Polish doughnuts as one can get (Molteni et al. 1994)
• It is the boundary layer of the black hole (Chakrabarti et al. 1996)
• It is the Compton cloud as far as the radiative properties are concerned (Chakrabarti & Titarchuk, 1995)

• Viscosity can re-distribute matter and make Keplerian disks out of a low angular momentum flow
(Chakrabarti, 1990; 1996; Chakrabarti & Molteni, 1995)
• The topologies change of the flow change and the shocks form in a smaller region of the parameter space.
Variation of Topologies with viscosity in an accretion flow

For higher viscosity, the disk becomes Keplerian when the shock propagates to a large distance (Chakrabarti and Molteni 1995).

An angular momentum distribution could be created by some viscosity:
When viscosity is low, the entire disk may not become Keplerian, but when viscosity is high enough, the Keplerian disk is formed, first by forming a shock which propagates outward. For low viscosity flows the shock stays at a given place or just oscillates back and forth.
Chakrabarti + Molteni 1995

• There exists a critical viscosity above which the flow cannot have shocks (Chakrabarti
1990). So we can think of a disk with a variation of viscosity in the vertical direction, high viscosity in the equatorial plane 
Keplerian disk and low viscosity away from the equatorial plane  sub-Keplerian disk above and below.
• This is the genesis of the two component model (Chakrabarti & Titarchuk, 1995)

In a binary system, especially in LMXBs, it is OK to consider large Keplerian disks forming out of
Roche lobe overflow. For HMXBs like Cyg X-1, winds are accreted and sub-Keplerian flows are expected. For super-massive black holes, most of the matter could have very little angular momentum (Sub-Keplerian), especially when the matter comes from of stellar winds.

Spectral Properties of such a disk:-
Assumptions:
Inflow has BOTH Keplerian and sub-Keplerian matter
Magnetic field is stochastic and is proportional to the local equi-partition value
Soft photons are supplied by the Keplerian disk and the synchrotron radiation
Electrons in the CENBOL are mixture of thermal and non-thermal distribution.
Non-thermal electrons are produced by shock acceleration in the accretion disk.

The CENBOL is responsible to drive/supply matter from the disk. If CENBOL is present, then only jets are produced.
VERY strong statement, but it is true. In stellar mass black hole cases.
(Chakrabarti 1999)

Parameter space of Keplerian disks in our diagram
(the region with negative energy)

Two component advective flow solution proposed by Chakrabarti and
Titarchuk (1995)

Note the presence of
Bulk motion
Comptonization in Soft State

Keplerian disk centric flow models (pre-
CT95 era)
These models are not found to be useful any more.

So the paradigm picture of accretion on a black hole is:-
The Centrifugal force and Gravity do tug-of-war and produce shocks – the post-shock region behaves as the Compton cloud and influences the spectrum. It is also responsible for the initial ‘drive’ of outflows and winds
We call the post-shock region as
CENBOL: the centrifugal barrier supported boundary layer

Formation of jet depends on how accretion takes place. Jets can be turned on/off depending on the relative ratios of
Keplerian and sub-
Keplerian matter.
Chakrabarti and
Nandi, 2000

Outflow/inflow ratio as a function of the shock strength (Chakrabarti, 1998/1999)
First theoretical proof that the spectral states and jet formations are related.

• Yes! All the cases which have been studied with high resolution, there are enough evidences that the jets are formed from the same region (CENBOL) which is also the Comptonizing the soft photons to produce hard X-rays.
• Quasi-periodic oscillations also due to the oscillations of this region. Thus, spectral states, jets, shocks, QPOs etc. all become interlinked.

A great example is M87
Junor, Biretta and Livio 1999

Cartoon diagram of physical processes which take place in the shape of the spectrum (SKC+Mandal
2006) emitted from the accretion flow

Typical spectrum of a stellar mass black hole. In this case we chose the parameters in a way that the bulk motion Comptonization is also important.

Fitting of the spectra of galactic black hole Cygnus X-1 using our solution in both the hard and the soft states
Chakrabarti+Mandal, ApJL, 2006

Quasar
Milli-Quasar
Micro-Quasar
Nano-Quasar
Spectral properties do not scale with mass

What happens to the spectrum if winds are active (mass loss or mass gain from/by the CENBOL) ?

Effects of winds on spectra (Theory)
Chakrabarti 2000

Effects of winds on spectra (Observations). Thus the pivotal position is shifted towards higher energy during urst-on and burst-off states.
Chakrabarti et al. 2003

• The RMS is a few percent and thus a significant region of the disk actually has to oscillate.
• The region which is Comptonized also seems to oscillate.
• Post-shock region or CENBOL has to be responsible.
• Various length scales correspond to various time periods that are observed

Quasi-periodic oscillations of X-rays: Are they just some perturbation or something more exciting?
First we discussed in the context of massive black holes
Molteni, Sponholz and Chakrabarti, 1996

Situation when the
Rankine-
Hugoniot conditions are not satisfied
Ryu, Chakrabarti and Molteni, 1997

Vertical and Radial oscillations of shocks and simultaneous propagation of shocks in winds.
Chakrabarti, Acharyya and Molteni, 2004

Three snap-shots showing radial and vertical oscillations
Chakrabarti, Acharyya and Molteni, 2004

Our numerical simulations show that QPOs are formed due to shock oscillations around galactic black holes.
Chakrabarti, Acharyya & Molteni 2004 A&A

MHD Shocks occurring close to black holes (~3-4
Schwarzschild radii) will cause high frequency QPOS
Chakrabarti 1990, Das & Chakrabarti 2006

Energy dependence of QPO properties: more juice out of the shocking flow

Chakrabarti and Manickam 2000
Higher energy photons participate in
QPOs!
This is a direct proof that only
Comptonized photon intensity is oscillating due to changing intercepted soft photons as the shock moves back and forth.

Some interesting properties of QPOs: The QPO frequency depends on the duration of QPO!
Chakrabarti and Manickam 2000

Puzzling light curves of black holes: who is responsible for the fast changes?

The light curve of GRS 1915+105 changes very rapidly. In a matter of days it goes from one class to another. In a matter of minutes the count rate can change by the factor of a few.
Variation of Light curves of GRS 1915+105
(Belloni, 2000; Chakrabarti and Molteni, 2000)

How long does it take to change from one type to another? Not Long! IXAE results!!!
Chakrabarti et al. 2005
Matter of a few hours! The flow responsible for such rapid changes must be sub-Keplerian, i.e., almost freely falling.

Another evidence of class transition in a matter of hours.
Note noisy spectral slope before a class transition: Indication of turbulence due to change in accretion rates? (Chakrabarti et al. 2005)

Supports of two components (Keplerian and sub-Keplerian flows) came from all quarters. It appears that it is truly how the disk is accreting.
NASA/RXTE All Sky Monitor (ASM) data for the last six years verify CT95 solution for six black holes!
Distinct changes in power-law photon index and flux relation for HMXB and LMXB were reported and were reported to be due to the presence of
Keplerian and sub-Keplerian components

Examples of the solutions of NASA/Berkeley group
(Smith, Heindle, Markwardt and Swank, 2002, ApJ)

Recently another paper was published using
RXTE/INTEGRAL data (Pottshmidt et al. 2006
NASA/ESA collaboration)
Same conclusion!

The history of how the disk model is modified from theoretical consideration.
(a) Chakrabarti 1990a;
Abramowicz &
Chakrabarti (1990);
(b) Chakrabarti &
Titarchuk, 1995; (c)
Ebisawa, Titarchuk &
Chakrabarti, 1996; (d)
Chakrabarti \& Sahu,
1997; (e) Chakrabarti,
1997; (f) Chakrabarti,
1998ab; (g) Chakrabarti \&
Nandi 2000; (h) Nandi et al. 2001; (i)Chakrabarti et al. 2002.

The history of how the disk model is modified from observational considerations
(a) Fender et al.
1999; (b)
Zdziarski, 2000;
(c) Nowak,
2003;
(d) Zdziarski \&
Gierlinksi 2004;
(e) Markoff \&
Nowak 2004;
(f) Titarchuk \&
Shaposhnikov,
2005; (g)
Ferinelli,
Titarchuk \&
Frontera,2007.

General references
• Chakrabarti, S.K., 1989, ApJ, 347, 365
• Chakrabarti, S.K., 1990, Theory of Transonic Astrophysical Flows,
WSC, Singapore
• Chakrabarti, S.K. 1990, MNRAS, 243, 610
• Chakrabarti, S.K. and Molteni, D., 1993, ApJ, 417, 671
• Molteni,D., Lanzafame, G. and Chakrabarti, S.K., 1994, ApJ,
425,161
• Molteni, D., Sponholz, H. and Chakrabarti, S.K., 1996, 457, 805
• Chakrabarti, S.K. and Titarchuk, L.G., 1995, ApJ, 455, 623
• Chakrabarti, S.K., 1996, ApJ, 464, 664
• Ryu, D., Chakrabarti, S.K. & Molteni, D., 1997, ApJ, 474, 378
• Chakrabarti, S.K. & Manickam, S.G., 2000, ApJ, 531, L41
• Chakrabarti, S.K. & Nandi, A, 2000, IJP, 75(B),1, (astro-ph/0012526)
• Chakrabarti. S.K., Acharyya, K. and Molteni, D., 2004, A&A, 421,1
• Chakrabarti, S.K., & Mandal, S., 2006, ApJ, 642, L49

• Both the theoretical and observational considerations suggest that the disks are not just
Keplerian flow with a corona. They have two components.
• For all practical purposes, black holes have a centrifugal pressure supported boundary layer
(CENBOL).
• CENBOL is responsible for ALL Astrophysically interesting observables including spectral state transitions, jets/outflows, Quasi-periodic oscillations.