Transonic flows around black holes and their observational properties

Sandip K. ChakrabartiS.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

Binary accretion: up close

• 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)

Angular momentum distribution

Accretion rate equation

Viscous stress responsible for angular momentum transport and heating the gas

Flux of radiation

Essence of the Shakura-Sunyaev (1973) disk

Various quantities can be written down analytically as a function of the radial distance

Flux

Surface density

Vertical height

Density

Temperature

Opacity

Luminosity

Keplerian Disk and its spectrum

Is any accretion disk ‘Standard’ Shakura-Sunyaev (1973) type?

• 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?

Standard disk Always? At all? When not and why not?

• 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>3rg, the marginally stable value (where the standard disk is cut-off).

Shakura-Sunyaev Viscosity parameter, a measure of viscosity?

• Since the viscous stress has the same dimension as that of the pressure, SS assumed that they are proportional to each other

Wr=- PIs 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+ v2 since v ~ c close to the horizon and the ram pressure is as important as the thermal pressure. Furthermore, P+ v2 is continuous across the shock discontinuities. So, in order to have continuity in viscous stress, we must have P+ v2 as the pressure to take care of the generalized flow (Chakrabarti & Molteni, 1995)

Viscosity, do we really need it?

• 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

Polish doughnuts!

• 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

Thick disk

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 advective disks or Transonic flow solutions came to rescue!

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

What are the shocks?

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

close to a black hole

232 /1/ rrl

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

Very close to the black hole or very far away from the black hole:

At intermediate distance (x ~ λ )

Specific energy of matter of an isothermal flow:

Net topology++ +

What should be the topologies of this flow?

Classification of solutions for a Kerr black holeChakrabarti 1996

Same as in Schwarzschild black hole: SKC 1989

All possible flow topologies

All possible solutions of an inviscid flow

on a Schwarzschild

black hole

Chakrabarti, 1989

First confrontation with theory

(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.

Shocks in multi-dimensional flow • 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)

Role of viscosity• 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

Re-distribution of matter in two components is possible

• 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)

The spectral and timing properties would depend on the type of matter

that is accreted

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.

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

So the paradigm picture of accretion on a black hole is:-

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.

So are the jets really form from the post shock region?

• 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 M87Junor, 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

From Quasars to Nano- Quasars, hydrodynamics is

the same, only radiative processes vary!

Quasar

Micro-Quasar

Milli-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

Quasi-Periodic Oscillations –vibrations or oscillations of the disk?

• 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

Chakrabarti, Acharyya and Molteni, 2004

Three snap-shots showing radial and vertical oscillations

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

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.

Chakrabarti and Manickam 2000

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?

Variation of Light curves of GRS 1915+105

(Belloni, 2000; Chakrabarti and Molteni, 2000)

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.

How long does it take to change from one type to another? Not Long! IXAE results!!!

Matter of a few hours! The flow responsible for such rapid changes must be sub-Keplerian, i.e., almost freely falling.

Chakrabarti et al. 2005

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

Another evidence of class transition in a matter of hours.

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

Supports of two components (Keplerian and sub-Keplerian flows) came from all quarters. It appears that it is truly how the disk is accreting.

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

Conclusions

• 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.