STABILITY OF ACCRETION DISKS AROUND ROTATING BLACK HOLES: A

STABILITY OF ACCRETION DISKS AROUND ROTATING BLACK HOLES: APSEUDO–GENERAL-RELATIVISTIC FLUID DYNAMICAL STUDY

BanibrataMukhopadhyay

Inter-University Centre for Astronomy andAstrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, IndiaReceived 2002 September 8; accepted 2002 December 6

ABSTRACT

We discuss the solution of an accretion disk when the black hole is chosen to be rotating. We study how thefluid properties are affected for different rotation parameters of the black hole. We know that no cosmicobject is static in the universe. Here the effect of the rotation of the black hole on spacetime is considered, fol-lowing an earlier work of the author, where the pseudo-Newtonian potential was prescribed for the Kerrgeometry. We show that, with the inclusion of rotation of the black hole, the valid disk parameter region dra-matically changes, and the disk becomes unstable. Also we discuss the possibility of shocks in accretion disksaround rotating black holes. When the black hole is chosen to be rotating, the sonic locations of the accretiondisk get shifted or disappear, making the disk unstable. To bring it into the stable situation, the angularmomentum of the accreting matter has to be reduced/enhanced (for co/counterrotating disk) by means ofsome physical process.

Subject headings: accretion, accretion disks — black hole physics — gravitation — hydrodynamics —shock waves

1. INTRODUCTION

Over the last three decades, fluid dynamical studies ofaccretion disks around black holes have been extensivelyperformed. Shakura & Sunyaev (1973) initiated this discus-sion considering a very simplistic but effective model ofaccretion disks. They chose the Newtonian gravitationalpotential. Novikov & Thorne (1973) and Page & Thorne(1974) studied a few aspects of the accretion disk in a fullyrelativistic treatment. Paczynski & Wiita (1980) proposedone pseudopotential which could approximately describerelativistic properties of the accretion disk around nonrotat-ing black holes. Using that pseudopotential, Abramowicz &Zurek (1981) studied some transonic properties of accretionflows. With time, a number of studies of accretion diskshave been carried out with modified disk structures (e.g.,Abramowicz et al. 1988; Abramowicz & Kato 1989; Chak-rabarti 1990, 1996a; Narayan & Yi 1994; Narayan et al.1997, 1998; Kato, Fukue, & Mineshinge 1998; Mukhopad-hyay 2002a) for nonrotating black holes. Also the studies ofaccretion disks in Kerr geometry have been explored by sev-eral authors (e.g., Sponholz & Molteni 1994; Chakrabarti1996b, 1996c; Abramowicz et al. 1996; Peitz & Appl 1997;Gammie & Popham 1998; Popham & Gammie 1998; Miwaet al. 1998; Lu &Yuan 1998;Manmoto 2000). Those studieshave been made either in a full general relativistic or in apseudo-Newtonian approach.

The study of stability of the accretion disk is one of theimportant criteria in this context. The possibility of steadyand stable disk formation by incoming matter toward ablack hole is allowed only for certain sets of initial parame-ters. Abramowicz & Zurek (1981) studied the effect of angu-lar momentum on the accretion and the correspondingstability of the transonic nature of the infalling matter ontothe black hole. After that, Chakrabarti (1989, 1990) dis-cussed the formation of sonic points and shock waves inaccretion disks. He showed that the formation of stablesonic points and shocks in the disk are possible for a partic-ular range of physical parameters. Recently, Mukhopad-

hyay & Chakrabarti (2001) have analyzed the stability ofaccretion disk in the presence of nucleosynthesis. As thetemperature of the disk is very high (at least of the order of109 K), nuclear burning can take place and generate and/orabsorb energy on a large scale in the disk. Because of thisgeneration/absorption of nuclear energy, fundamentalproperties of the disk, like sonic locations, temperature,Mach number, shock location (if any), etc., may be affected.Thus the stability gets influenced by the nuclear burning. Ifthe rate of generation/absorption of nuclear energy is large,the disk structure may become unstable. In their discussion,Mukhopadhyay & Chakrabarti (2001) have also shown thatthe sonic locations disappear for a particular shell of anaccretion disk where the nuclear energy is significant com-pared to the mechanical energy of the accreting matter.

In all the above mentioned works, when the stability wasanalyzed for the accretion disk, the central black hole waschosen to be nonrotating. As it is well known that no objectis static in the universe, before making any serious conclu-sions about the inner properties of the accretion disk, con-sideration of rotation of a black hole is essential. Though afew works have been done for Kerr black holes, no suchcomparative study has been explored so far with differentKerr parameters. Thus, automatically, two questions arise.Does the rotation of the black hole play an important rolein the accretion properties of a disk? Does the considerationof rotation of the black hole affect the disk structure for aparticular parameter regime that is known to be stable forthe nonrotating case? Here, we would like to address thesequestions one by one.

Recently, we have proposed a new pseudo-Newtonianpotential (Mukhopadhyay 2002b, hereafter Paper I) todescribe the accretion disk around Kerr black holes. Fol-lowing Paper I, we will describe our accretion disk in apseudo-Newtonian manner. The potential that we are usingcan describe approximately all the essential general relativ-istic properties of the accretion disk. It can reproduce thelocations of marginally stable and bound orbits exactly oralmost exactly for different Kerr parameters (see Paper I),

The Astrophysical Journal, 586:1268–1279, 2003 April 1

# 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.

1268

which are the pure general relativistic properties. Therefore,we call this study a pseudo–general-relativistic approach.As our main interest is to study the inner region of the accre-tion disk, we will concentrate upon the sub-Keplerian flow,where the effect of rotation of the black hole is significant.In Paper I, we already raised several questions, which areimportant to address in the physics of accretion disk. Herewe will show how useful the pseudopotential described inPaper I is in studying the global behavior of the accretiondisk. In full general relativity, the basic equations of theaccretion disk are very complicated and tedious to handle.In early works, there were no suitably good pseudopoten-tials for a rotating black hole. It seems that Paper I has comeup with the answer, and one of our aims in this paper is toshow its applicability. As the location of horizon, margin-ally bound and marginally stable orbits change for differentKerr parameters, any inner property of the disk is verymuch related to the rotation of black hole. We will followMukhopadhyay & Chakrabarti (2001) to analyze the stabil-ity of the disk. We will study whether or not the sonic loca-tion(s), shock formation(s) (if any), etc., get affected byapplying rotation to the black hole. In an accretion disk, ifthe sonic point loses entropy or disappears by the rotationaleffect of black hole, we will understand that the sonic pointas well as the disk is unstable because a stable sonic point isnecessary for accretion onto the black hole. In the next sec-tion, we will present the basic equations needed to describethe accretion disk. In x 3, we will discuss about the parame-ter space of accretion disk and how the rotation affects it.Subsequently, in x 4, we will describe the fluid dynamicalresults. Finally, in x 5, we will sum up all the results andmake the overall conclusions.

2. BASIC EQUATIONS

Throughout in our calculations, we express the radialcoordinate in units of GM=c2, where M is the mass of theblack hole, G is the gravitational constant, and c is the speedof light. We also express the velocity in units of the speed oflight and the angular momentum in units of GM=c. Theequations to be solved are given below as

d

dxðx�vÞ ¼ 0 ; ð1Þ

vdv

dxþ 1

�

dP

dx� �2

x3þ FðxÞ ¼ 0 ; ð2Þ

where � is the vertically integrated density and FðxÞ is thegravitational pseudo-Newtonian force given in Paper I as

FðxÞ ¼ ðx2 � 2affiffiffix

pþ a2Þ2

x3ffiffiffix

pðx� 2Þ þ a½ �2

; ð3Þ

where a indicates the specific angular momentum of theblack hole (the Kerr parameter). We do not consider anykind of energy dissipation in the accretion disk, as we wantto check how the black hole rotation solely can affect thedisk properties. Thus the angular momentum of the accret-ing fluid (�) remains constant throughout a particular flow.Following Matsumoto et al. (1984), we can calculate thevertically integrated density as

� ¼ In�ehðxÞ ; ð4Þ

where �e is the density at equatorial plane, hðxÞ is the half-thickness of the disk, and In ¼ ð2nn!Þ2= ð2nþ 1Þ!½ �, where n isthe polytropic index. From the vertical equilibrium assump-tion, the half-thickness can be written as

hðxÞ ¼ csx1=2F�1=2 ; ð5Þ

where cs is the speed of sound. We also consider the equa-tion of state to be �P=� ¼ c2s , where � is the gas constant.

Now, combining equations (1) and (2) we get

dv

dx¼ �2

x3� FðxÞ þ c2s

� þ 1

3

x� 1

F

dF

dx

� �� v� 2c2s

ð� þ 1Þv

� �:

ð6Þ

Far away from the black hole, v < cs, and close to it, v > cs,thus there is an intermediate location where the denomina-tor of equation (6) must vanish. Therefore, to have a smoothsolution at that location, the numerator has to be zero. Thislocation is called the sonic point or critical point (xc). Theexistence of this sonic location plays an important role inthe accretion phenomena. For an accretion disk around ablack hole, a sonic point must exist. From the global analy-sis of a sonic point, one can understand the stability of thephysical parameter region that we will discuss in the nextsection.

As it is described above that at x ¼ xc, dv=dx ¼ 0=0, usingl’Hopital’s rule and after some algebra, we can get the veloc-ity gradient of accreting matter at the sonic location as

dv

dx

��c¼ �

BþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 � 4AC

q2A

; ð7Þ

where

A ¼ 1þ 2c2sc

ð� þ 1Þv2cþ 4c2scð� � 1Þ

v2cð� þ 1Þ2;

B ¼ 4c2scð� � 1Þvcð� þ 1Þ2

3

xc� 1

Fc

dF

dx

��c

� �;

C ¼ c2sc� þ 1

1

Fc

dF

dx

��c

� �2

� 1

Fc

d2F

dx2

��c� 3

x2c

" #� 3�2

x4c� dF

dx

��c

� c2scð� � 1Þð� þ 1Þ2

3

xc� 1

Fc

dF

dx

��c

� �2: ð8Þ

Thus, we have to integrate equations (6) and (7) with anappropriate boundary condition to get the fluid propertiesin the accretion disk. From equation (6), we can easily findout theMach number at the sonic point as

Mc ¼vccsc

¼

ffiffiffiffiffiffiffiffiffiffiffi2

� þ 1

sð9Þ

and the corresponding sound speed as

csc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið� þ 1Þ �2

x3c� Fc

� �1

Fc

dF

dx

��c� 3

xc

� ��1s

: ð10Þ

Now integrating equations (1) and (2), we can write downthe energy and entropy of the flow at a sonic point as

Ec ¼2�

ð� � 1Þ�2

x3c� Fc

� ��1

Fc

dF

dx

��c

� 3

xc

� �� þ Vc þ

�2

2x2c;

ð11Þ

ACCRETION DISK STABILITY AROUND BLACK HOLES 1269

and

_MMc ¼ ð�KÞn _MM ¼ x3=2c F

�1=2c ð� þ 1Þq=2

� �2

x3c� Fc

� ��1

Fc

dF

dx

��c

� 3

xc

� �� = ��1ð Þ; ð12Þ

where K is basically the entropy of the system, Vc ¼ðRFdxÞjc and q ¼ ð� þ 1Þ= 2ð� � 1Þ½ �. Actually, _MM itself is

not entropy but carries the information about it. For anondissipative system, _MM is conserved for a particular _MM,unless the shock forms there. We have to supply the sonicenergy Ec as the boundary condition for a particular flow.Then from equation (11), we can find out the sonic locationxc. Therefore, knowing xc, one can easily find out the fluidvelocity and sound speed at the sonic point from equations(9) and (10) for a particular accretion flow. These have to besupplied as further boundary conditions of the flow.

Another important issue is the formation of the shock inthe discussion of the accretion disk. Here, whenever wemention the shock, will mean the Rankine-Hugoniot shock(Landau & Lifshitz 1987). If we generalize the conditions toform a shock in accretion disk given by Chakrabarti (1989)to the case of a rotating black hole, we get

12M

2þc

2sþ þ nc2sþ ¼ 1

2M2�c

2s� þ nc2s� ; ð13Þ

c�sþ_MMþ

2�

3� � 1þ �M2

þ

� �¼ c�s�

_MM�

2�

3� � 1þ �M2

�

� �; ð14Þ

_MMþ > _MM� ; ð15Þ

where

_MM ¼ Mc2ðnþ1Þs

x3=2sffiffiffiffiffiffiffiffiffiffiffiffiFðxsÞ

p : ð16Þ

Here, the subscripts ‘‘�’’ and ‘‘ + ’’ indicate the quantitiesjust before and after the shock, respectively, and xs indicatesthe shock location.M denotes theMach number of the mat-ter and � ¼ 3� � 1ð Þ= � � 1ð Þ. From equations (13)–(16), itis very clear that except the entropy expression, all remainunchanged with respect to the case of a nonrotating blackhole. Also from equations (14) and (16), we get the shockinvariant quantity as

C ¼ 2=Mþ þ ð3� � 1ÞMþ½ �2

M2þð� � 1Þ þ 2

¼ 2=M� þ ð3� � 1ÞM�½ �2

M2�ð� � 1Þ þ 2;

ð17Þ

which remains unchanged with respect to the nonrotatingcase. If all the conditions (13)–(15) and (17) are simultane-ously satisfied by the matter, the shock will form in anaccretion disk.

3. ANALYSIS OF THE PARAMETER SPACE

One of our aims is to check how the rotation affects thedisk parameter region known for a Schwarzschild blackhole. Therefore, we have to analyze globally the accretiondisk properties around rotating black holes. We also haveto check, how does the rotation of the black hole affect thesonic location, structure as well as the stability of disk. InFigure 1a, we show the variation of disk entropy as a func-

tion of sonic location. The intersections of all the curves bythe horizontal line (which is a constant entropy line) indi-cate the sonic points of the accretion disk for that particularentropy and rotation of the black hole. It is clearly seen that,at a particular entropy, if the rotation of the black holeincreases, marginally bound (xb) and stable (xs) orbits aswell as sonic points shift to a more inner region and the pos-sibility to have all four sonic points in the disk outside thehorizon increases. As an example, for a ¼ 0:998, the inneredge of the accretion disk enlarges in such a manner that the

Fig. 1.—Variation of (a) entropy and (b) energy as a function of soniclocation for various values of Kerr parameters a. Central solid line indicatesnonrotating case (a ¼ 0), while the results in regions of either side of it indi-cated by ‘‘+ ’’ and ‘‘�’’ are for prograde and retrograde orbits, respec-tively. Different curves from the central solid line downward are fora ¼ 0:1; 0:5; 0:998 and upward for a ¼ �0:1;�0:5;�0:998. The horizontalline indicates the curve of (a) constant entropy of 5� 10�5 and (b) constantenergy of 0.0065. � ¼ 3:3, � ¼ 4=3 for all the curves.

1270 MUKHOPADHYAY Vol. 586

fourth sonic point in the disk appears outside the horizon.On the other hand, for retrograde orbits (counterrotatingcases), xb and xs move to greater radii, and all the sonicpoints come close together for a particular Kerr parameter.Similar features are reflected in Figure 1b, where the sonicenergy is plotted as a function of sonic location. Here also,the intersections of the horizontal line (which indicates aconstant energy line) with all the curves indicate the sonicpoints of the accretion disk for that particular energy androtation of the black hole. For both Figures 1a and 1b, sonicpoints with negative slope of the curve indicate the locationsof ‘‘ saddle-type ’’ sonic point and positive slopes indicate

the ‘‘ center-type ’’ sonic point. Thus the rotation of blackhole plays an important role in the formation and locationof sonic points which are related to the structure of accre-tion disk.

In Figure 2, with the consideration of rotation of blackhole, we show how the disk region having three sonic pointsis affected. Figures 2a, 2b, and 2c show the variation of sonicentropy as a function of sonic point, when the angularmomentum of accreting matter is chosen as the parameterfor three different rotations of the black hole asa ¼ 0:5;�0:5; 0. If we join the extrema of the curves, we getthe bounded parameter region of the disk where the three

Fig. 2.—(a–c) Variation of entropy as a function of sonic location for a set of disk angular momentum (�), when (a) a ¼ 0:5, (b) a ¼ �0:5, and (c) a ¼ 0.Solid curve indicates the flow with critical angular momentum (�c) for which the flow has just a single critical point which is (a) �c ¼ 2:94, (b) �c ¼ 3:04, and(c) �c ¼ 2:982. From the top to bottom curve, � ¼ 2:8; �c; 3:2; 3:3; 3:4; 3:5; 3:6. Any flow of � > �c has three and � < �c has one critical point. (d ) Variation ofextremum values of entropy for curves (a), (b), and (c) as a function of �2 for a ¼ 0 (solid curve), a ¼ 0:5 (dotted curve), and a ¼ �0:5 (dashed curve). � ¼ 4=3for all the curves.

No. 2, 2003 ACCRETION DISK STABILITY AROUND BLACK HOLES 1271

sonic points exist. As the angular momentum of accretingmatter decreases, the possibility of forming three sonicpoints decreases. This is very well understood physically; asthe angular momentum of accreting matter decreases, thedisk tends to a Bondi-Flow like structure, which has singlesaddle-type sonic point. The solid curves of all three figuresindicate the locus of sonic entropy with a single extremumpoint. Only the flows of angular momentum greater thanthat of the solid curve have three sonic points, otherwise fora particular flow, we have only one sonic point. In Figure2d, we show the projection of all the curves in Figure 2a, 2b,and 2c to the _MMcm-�

2 plane, where _MMcm indicates theextremum entropy at the sonic points. Figure 2d indicatesthe valid physical parameter region of the accretion diskthat has three sonic points in the flow. It should be noted

that, for a particular Kerr parameter, the accreting flow canhave three sonic locations for different pairs of (�; _MMcm).However, for a particular �, if _MMcm decreases, the stabilityof the sonic points as well as that of the disk diminishes too.Thus, in a section of the diagram for particular a, all pointsare not an equally probable set of physical parameters in theformation of three sonic points. It is clear that, as the coro-tation of black hole increases, the parameter region en-larges, so that three sonic points may exist. Therefore, thedisk with three sonic points becomes more probable at highcorotation. On the other hand, this region decreases for thecounterrotating flow. Similar features are noticed from Fig-ure 3, where the variation of sonic energies are depicted. Ina similar manner, Figure 3d indicates that the parameterspace with three sonic points enlarges if the corotation of

Fig. 3.—Same as Fig. 2 except the energy is plotted in place of entropy


the black hole increases, where Ecm indicates the extremumenergy of the different curves in Figures 3a–3c. The physicalreasons are the following. Higher corotation results in ashift of xb and xs to an edge of the accretion disk farther inand an enlargement of the stable disk region (as we knowthat the stable circular orbit is possible in the disk up to theradius xs). On the other hand, higher counterrotation resultsin an outward shift of xb and xs, and the size of the stabledisk region decreases.

Figure 4 indicates the variation of energy with entropy atsonic locations for different Kerr parameters. ‘‘ I ’’ indicatesthe branch of inner ‘‘ X-type ’’ sonic point that is differentfor different Kerr parameters, while ‘‘ O ’’ indicates the outer

‘‘ X-type ’’ sonic point branch. Along the O branch , all thecurves (solid, dotted, and dashed ) lie on top of each other.Therefore it is clear that the rotation of the black hole doesnot affect the outer ‘‘ X-type ’’ sonic point. Figures 4a and4b show that with the increase of Kerr parameter (corota-tion), inner sonic points of the disk shift toward the lower-entropy region, and the disk becomes unstable. But theouter sonic point branch remains unchanged for all valuesof rotation. We know that, if there is a possibility of matterin the outer sonic point branch of lower entropy jumping tothe inner sonic point branch of higher entropy, a shock canform in the accretion disk. As, with the increase of Kerrparameter, the inner sonic point branch shifts to the lower

Fig. 4.—Variation of energy with entropy at sonic points when disk angular momentum � ¼ 3:3 for (a) a ¼ 0 (solid curve), a ¼ 0:1 (dotted curve), anda ¼ 0:5 (dashed curve); (b) a ¼ 0 (solid curve) and a ¼ 0:998 (dotted curve); (c) a ¼ 0 (solid curve), a ¼ �0:1 (dotted curve), and a ¼ �0:5 (dashed curve); and (d )a ¼ 0 (solid curve) and a ¼ �0:998 (dotted curve). O and I indicate the branch of outer and inner sonic points, respectively, for a ¼ 0. � ¼ 4=3 for all the curves.


entropy region, the stability of both the shock and the diskis at stake. For example, in Figure 4a, if we join the pointsA, B and A, D, we will find two sections of the curve,namely, ABC (solid curve) and ADE (dashed curve) whichare the parameter regions where the shock may form in theaccretion disk for a ¼ 0 and a ¼ 0:5, respectively. If wedraw the lines of constant energy (which are horizontal inFig. 4a), connecting the outer sonic point branch (AC fora ¼ 0 and AE for a ¼ 0:5), to the inner one (BC for a ¼ 0and DE for a ¼ 0:5), section ABC will contain more lineswith respect to section ADE (as the section ABC is biggerthan ADE). On the other hand, a constant energy line indi-cates the possibility of a shock (as it shows the possibility ofan increment of entropy in the flow, keeping the energy con-stant). Clearly, for a corotating black hole, the possibility offormation of a shock decreases, as well as the shock itselfbecomes unstable. The physical reason is that, with theincrease of Kerr parameter, the angular momentum of thesystem increases, which helps the disk to maintain a highazimuthal speed of matter up to a very close proximity tothe black hole, and as a result the radial matter speed canovercome the corresponding sound speed at an inner radiusonly. On the other hand, the inner edge of the accretion diskis comparatively less stable as the entropy decreases at lowerradii. As the inner sonic points form at an inner radii, withthe increase of a, the disk tends to an unstable situation thatdecreases the value of the entropy. Figure 4b shows that thepossibility of a shock disappears completely when a ¼0:998, as there is no transition possible from the outer toinner sonic branch that can increase the entropy of the sys-tem. Here, the inner edge is extremely unstable in such amanner that its entropy is even lower than that of the outersonic point branch. As the inner edge of the disk spreads outfor a ¼ 0:998, there are two branches of the center-typesonic point.

In Figure 4c, the situation is opposite. As the counterrota-tion of the black hole increases in magnitude, the inner sonicpoint branch shifts toward a higher-entropy region. Thus,the disk, as well as the shock, becomes more stable. Still, theouter sonic point branch is not affected due to the rotationof black hole and the branches for different Kerr parametersare superimposed. In Figure 4c, the sections ABC (solidcurve) and ADE (dashed curve) for a ¼ 0 and a ¼ �0:5,respectively, show the regions where the shock may form inthe disk. The physical reason to have a more stable innersonic point branch for the counterrotating case is as follows.A higher counterrotation implies that the net angularmomentum of the system decreases; as a result, the matterfalls faster at larger radii, and it becomes supersonic at acomparatively outer edge of the disk that is more stable. Ina similar manner, as explained for Figure 4a, the greater sizeof the section ADE than that of ABC implies a greater pos-sibility of a shock. However, for a very high counterrotationof the black hole (see Fig. 4d), the entire system shifts to theouter side that is more stable, and the inner sonic pointbranch itself tends to merge with the branch of the outersonic point. Since the net angular momentum of the systembecomes very small for higher counterrotation, the corre-sponding centrifugal pressure onto the matter becomesinsignificant; as a result, the possibility of shock formationdiminishes again. Therefore, we can conclude that, if theangular momentum of the black hole increases or decreasessignificantly, the shock wave in the accretion disk becomesunstable and the disk itself loses stability.

4. FLUID PROPERTIES OF ACCRETION DISK

We will discuss now the effect of rotation of the black holeon the disk fluid. In the previous section, we have alreadyconfirmed that the rotation significantly affects the globalparameter space of the accretion disk. Our intention is toincorporate the rotation of the black hole and to see thechange of fluid properties known for the nonrotating case.

Figure 5a shows the variation in Mach number for a par-ticular set of physical parameters. If the black hole is chosento be nonrotating, a shock forms in the disk. When rotationis considered, keeping other parameters unchanged, the sol-ution changes and the shock disappears. If the black holecorotates, the inner edge of the disk becomes unstable, andthe matter does not find any physical path to fall steadilyinto the black hole. On the other hand, if the black holecounterrotates, there is a single sonic point in the disk, andthe matter attains supersonic speed only close to the blackhole and falls into it. Thus, for the nonrotating and counter-rotating cases, there are smooth solutions of matter passingthrough the inner sonic point, while for the corotating cases,it is not so. The physical reasons behind these phenomenacan be explained as follows. When the black hole corotates,the angular momentum of the system increases, and theradial speed of the matter may not be able to overcome thecentrifugal barrier to fall into the black hole for that param-eter set. If the black hole counterrotates, the angularmomentum of the system decreases, the centrifugal barriersmears out, and matter falls steadily into the black hole. Itcan be mentioned here that, for the other choice of physicalparameters (e.g., reducing/increasing the angular momen-tum of accreting matter for co/counterrotation, changingthe sonic locations, etc.), the shock may appear again, evenfor the rotating black holes (a few such examples aredepicted in Fig. 6). Figure 5b shows the corresponding den-sity profiles in units of c6=G3M2 for a particular accretionrate. As the matter slows down abruptly, the density of thedisk fluid jumps up at the shock location for a nonrotatingcase. For the counterrotating black holes, the angularmomentum of the system is less; thus, to overcome the cen-trifugal barrier, matter does not need to attain a high radialspeed away from the black hole. In this situation, the radialmatter speed is less and the corresponding density of accret-ing fluid is high compared to the nonrotating case. We alsoshow the density profiles for corotating, unstable cases. InFigure 5c, we compare the corresponding temperature pro-files in units of 109 K (T9). The variation of temperature issimilar to that of density away from the black hole but isopposite when close to the horizon (as the density variesinversely with the temperature at a small x but not so at alarge x; see eq. [1]). As the higher counterrotation Machnumber decreases, the temperature of the accretion diskbecomes higher.

The variation of the shock invariant quantity,C, is shownin Figure 5d for the nonrotating case. The intersectionsbetween the solution of the inner and outer sonic pointbranches indicate the shock locations. Since the inner shocklocation is unstable, the only shock forms at an outer radius,x ¼ 37:77.

Figure 6 gives examples where the shock forms in anaccretion disk around a rotating black hole. The variationof Mach numbers for different Kerr parameters are shownin Figure 6a. It reflects that, for a small rotation of the blackhole (a ¼ 0:1), matter adjusts the sonic locations in such a


manner that the outer (xo) and inner (xi) sonic locationsshift to a more outer and more inner region, respectively,with respect to a nonrotating case, and the shock formsin the disk at x ¼ 17:18. For the counterrotating case(a ¼ �0:1), xo may remain unchanged, but xi has to beshifted outside to form a shock at x ¼ 17:25 (that keeps theshock location [almost] unchanged). As for the counterro-tating cases, marginally stable (xs) and marginally bound(xb) orbits shift away from the black hole with respect to acorotating one; xi also shifts outside to stabilize the disk. Ifthe corotation of the black hole is higher, say a ¼ 0:5, to

form a stable shock, not only the inner sonic location has tobe smaller, but the disk angular momentum (�) should alsobe reduced. High rotation of a black hole results in the loca-tion of horizon (xþ) as well as xb and xs shifting inward, andthe matter attains a supersonic speed at a point furtherinside of the inner edge of the disk to fall into the black hole.Thus all the phenomena, e.g., sonic transitions, shock for-mations, etc., shift inside. Moreover, for the stable solution,the angular momentum of the system cannot be very high,otherwise matter will be unable to overcome the centrifugalbarrier. Thus, for the high rotation of a black hole, like

Fig. 5.—Variation of (a) Mach number, (b) density, and (c) temperature in units of 109 K as a function of radial coordinate for a ¼ 0 (solid curve), a ¼ 0:1(dotted curve), a ¼ 0:5 (dashed curve), a ¼ �0:1 (long-dashed curve), and a ¼ �0:5 (dot-dashed curve). (d ) Variation of shock invariant quantity (C ) as a func-tion of radial coordinate for inner sonic (solid curve) and outer sonic (dotted curve) point branch when a ¼ 0; the intersection points of two curves indicate theshock locations, which are xs ¼ 33:77; 6:94. Other parameters are M ¼ 10 M�, _MM ¼ 1 Eddington rate, � ¼ 3:3, xo ¼ 70 (solid, dotted, and dashed curves),xi ¼ 5:6 (solid, long-dashed, and dot-dashed curves), � ¼ 4=3.


a ¼ 0:5, � has to be reduced to form a shock (at x ¼ 9:51)and stabilize the disk. However, for a ¼ �0:5 and less, thereis no sub-Keplerian flow for which a shock may form. Forthe higher counterrotation of a black hole, the angularmomentum of the system decreases. Since a higher centrifu-gal barrier is essential to form a shock, at the higher coun-terrotation, the shock completely disappears from theaccretion disk. Figures 6b and 6c show the correspondingvariations of density and temperature, respectively. In Fig-ures 6d, 6e, and 6f, we show the corresponding variation ofshock invariant quantity C for the inner and outer sonicpoint branches, when a ¼ 0:1; 0:5;�0:1, respectively. Againit becomes clear that, for the faster rotation of a black hole,the shock location shifts inside the disk.

In Figure 7, we show the fluid properties for a ¼ �0:998.At a very high rotation of black hole, the possibility to forma shock in the accretion disk totally disappears. For thecorotating case, the disk cannot be stable unless the angularmomentum of accreting matter is very low (as we havealready mentioned that the high angular momentum of the

system makes the disk unstable). Therefore, there is noscope to form a shock, as to its formation, matter needs asignificant centrifugal barrier as well as a high angularmomentum of the accreting fluid. Here we choose � ¼ 1:8,and the matter falls almost freely. For the counterrotatingcase, the matter angular momentum need not be low to sta-bilize the disk. However, in the sub-Keplerian accretiondisk, � cannot attain a high value to increase the angularmomentum as well as the centrifugal barrier of a system toform a shock. Thus, the shock also disappears for a highcounterrotating case. Figure 7a shows that xb and xs shiftoutward for the retrograde orbit, and the matter attains asupersonic speed at a radius farther out than in the case of adirect orbit of the disk. However, far away from black hole,the radial velocity is less (as it needs to overcome a smallercentrifugal barrier) than that of the corotating case. Figure7b shows the variation of corresponding density. Lowervelocity implies a high rate of piling up of matter at a partic-ular radius that produces a high density in the disk and thus,the density behaves in an opposite manner toMach number.

Fig. 6.—Variation of (a) Mach number, (b) density, and (c) temperature in unit of 109 K as a function of radial coordinate when the sets of physical parame-ters are (i) a ¼ 0:1, xo ¼ 94, xi ¼ 5:4, � ¼ 3:3 (solid curve); (ii) a ¼ 0:5, xo ¼ 43, xi ¼ 4:2, � ¼ 2:8 (dotted curve), and (iii) a ¼ �0:1, xo ¼ 70, xi ¼ 6:324,� ¼ 3:3 (dashed curve). Variation of shock invariant quantity, C, as a function of radial coordinate for the set of (d ) parameter (i) when xs ¼ 17:18; 9:2; (e)parameter (ii) when xs ¼ 9:51; 8:35; and ( f ) parameter (iii) when xs ¼ 17:25; 13:73. Solid and dotted curves are for inner and outer sonic branches, respec-tively. Other parameters areM ¼ 10M�, _MM ¼ 1 Eddington rate, and � ¼ 4=3.


Figure 7c shows the variation of virial temperature in theaccretion disk. The temperature is low for corotation andhigh for counterrotation. The physical reason behind it is asfollows. For corotation, the net speed as well as the kineticenergy of matter is lower, producing a lower temperature.On the other hand, for a counterrotating case, the situationis opposite. Although we show the virial temperature ofthe disk, in reality several cooling processes may take placethat can reduce the disk temperature by up to 2 orders ofmagnitude.

5. DISCUSSION AND CONCLUSION

Here we have studied the stability of accretion disksaround rotating black holes in a pseudo-Newtonianapproach. Following Paper I, we have incorporated the gen-eral relativistic effects (according to Kerr geometry) on theaccretion disk in such a manner that all the essential relativ-istic properties, e.g., the locations of the marginally boundand stable orbits, black hole horizon and mechanical energyof the accreting matter are reproduced in the disk for vari-ous values of Kerr parameters. Thus we can say our methodis ‘‘ pseudo-general-relativistic.’’ As our interest is to checkhow the rotation of a black hole solely affects the disk struc-ture, we have chosen inviscid fluid in the accretion disk.Most of the earlier studies on structure and stability analysis

of the accretion disk have been done around a nonrotatingblack hole (Schwarzschild geometry). As no cosmic object isstatic, it is very important to consider the effect of rotationof the black hole, particularly for the discussion of an inneredge of an accretion disk. We have found that when therotation of a black hole is incorporated, the valid, knowndisk parameter region of a nonrotating black hole is dra-matically affected. As a result, the location of various sonicpoints and the shock (if any) get shifted severely and influ-ence the disk structure. Therefore, we can conclude that inthe study of an accretion disk, the rotation should be con-sidered. Any related conclusion depends on the rotationparameter of a black hole. So, we can say that the knownphysical parameter regime (Chakrabarti 1989, 1990, 1996a)to form a shock, sonic locations, etc., in the disk around anonrotating black hole is not global. The mentioned diskproperties and phenomena not only get modified and/orshifted in location at the disk but also sometimes completelydisappear. In our study, we have chosen the Kerr parameterfrom a ¼ 0 to the case of an extremely high rotation asa ¼ 0:998, and thus, we now have a clear global picture ofthe accretion disk.

Although, in the literature, there are a few works on theaccretion disk around Kerr black holes (e.g. Chakrabarti1996b, 1996c; Gammie & Popham 1998; Popham & Gam-mie 1998), those do not concentrate on the comparative

Fig. 7.—Variation of (a) Mach number, (b) density, and (c) temperature in units of 109 K as a function of radial coordinate for the sets of parameters (i)a ¼ 0:998, xi ¼ 5:6, � ¼ 1:8 (solid curve) and (ii) a ¼ �0:998, xi ¼ 5:6, � ¼ 3:3 (dotted curve). Other parameters areM ¼ 10M�, _MM ¼ 1 Eddington rate, and� ¼ 4=3.


study of a rotating and nonrotating black hole and the insta-bility of disk induced by the rotation of black hole. Chakra-barti (1996b, 1996c) mainly concentrated on fast-rotatingblack holes and showed solution topology in full Kerrgeometry. He does not show how the solution varies withthe change of rotation of the black hole, keeping unchangedother physical parameters. Thus, it is not clear from thoseworks how the Kerr parameter affects the disk solutionsolely. On the other hand, Gammie & Popham (1998) andPopham & Gammie (1998) do not discuss the effect of rota-tion on the sonic points globally, which is shown here in Fig-ures 1–4. However, the fluid dynamical results achieved bythem agree with the present work. For example, they (Gam-mie & Popham 1998; Popham&Gammie 1998) showed thatthe effect of nonzero Kerr parameter is dominant close tothe black hole. That is also reflected in Figure 4 of thepresent work, which shows that the outer sonic pointbranches are unchanged, while the inner ones are shifted.They also showed that the (inner) sonic points move tosmaller (larger) radii as corotation (counterrotation)increases in magnitude. In our work, similar features comeout from Figures 1, 2, 3, and 6. Thus, we can conclude onceagain that the pseudopotential prescribed in Paper I canreproduce the general relativistic properties in accretiondisk. Unlike the works by Gammie & Popham (1998) andPopham & Gammie (1998), here we have chosen inviscidflow to understand the sole effect of rotation of the blackhole on the fluid and corresponding disk structure. Finally,we can conclude that the effect is significant.

Here we have carried out the stability analysis of theaccretion disk by choosing the black hole to be rotating.Prasanna & Mukhopadhyay (2003) have worked on thestability of accretion disks around rotating compact objects(but not for black holes) in a perturbative approach. Theyincorporated the rotational effect of central object in a diskindirectly, by the inclusion of Coriolis acceleration term.Here, the compact object is a black hole and its rotationaleffect is brought from the Kerr metric only. As we have avery self-consistent pseudopotential (Paper I), it has beeneasy to study the accretion phenomena around a rotatingblack hole.

From the global analysis of sonic points, we have estab-lished that the disk becomes unstable for a higher corotationat a particular angular momentum of the accreting matter.On the other hand, to form a shock, the accretion disk musthave a stable inner sonic point. Thus, for higher a (Kerrparameter), the shock is unstable, as the inner region of diskitself is not stable. In that regime, the shock may disappearby any disturbance created on the accreting matter, and thusthe matter may not get a steady inner sonic point to form astable inner disk. For the counterrotating cases, the angularmomentum of the system decreases, and the matter falls tothe black hole more rapidly. As there is no significant centri-fugal barrier to slow down the matter, the possibility ofshock decreases again. For a very high counterrotation,

shock disappears completely. Thus we can conclude that theparameter region where the shock is expected to form(Chakrabarti 1989, 1990, 1996a) for nonrotating black holesgets affected for the rotating ones. For a positive a, theshock might have been formed for a lesser value of the angu-lar momentum of accreting matter. On the other hand, for anegative a, to form a shock, the angular momentum ofaccreting matter should have a larger value than that ofnonrotating cases. As we are studying the sub-Keplerianaccretion flow, the angular momentum of matter cannot beunlimitedly high (e.g., for a nonrotating black hole,�d3:6742 at last stable orbit). Thus, to form a shock, mat-ter cannot attain the angular momentum beyond a certainlimit. Also for a highly corotating black hole, to stabilize thedisk, � has to be reduced by some physical process, other-wise the radial speed could not overcome the centrifugalpressure to form a steady inner edge of the disk. Therefore,the disk would not be transonic and could be unstable in thissituation. Because of a reduced �, centrifugal force as wellas shock strength decreases; as a result, the possibility of ashock is also reduced for both the corotating and counterro-tating black holes. However, at an intermediate corotation,the parameter region of the accretion disk enlarges, havingthree sonic points, and there is a possibility of a stable shockformation. On the other hand, for a counterrotation, thevalid parameter region with three sonic points at the diskshortens, and thus, the possibility of shock reduces.

Now the question may arise, how would a flow behave inthe region where sonic points disappear because of thehigher rotation of black hole? It may be that matter wouldtry to pass through same sonic point as of nonrotating (orslowly rotating) case. Because of the absence of sonic points,the searching procedure for the true sonic location shouldgive rise to an unstable solution. Only the time-dependentsimulation of accretion disk can tell whether this idea is cor-rect or not. Overall we can say that the rotation of a blackhole affects the sonic locations of an accretion disk that aredirectly related to the formation of stable disk structure.Therefore, the stability of an accretion disk is stronglyrelated to the rotation of the central black hole.

It has also been confirmed that the potential given inPaper I is very well applicable for the global solution andcorresponding fluid dynamical study of the accretion diskaround a rotating black hole. The set of full general relativ-istic disk equations is very complicated for a Kerr blackhole, particularly when the accreting fluid is chosen to beviscous. Thus, in the study of a relativistic accretion disk,this pseudopotential is very effective to avoid the complex-ities of full general relativistic fluid equations. The next stepshould be the study of a viscous accretion disk using thisgeneral pseudopotential.

I am grateful to my friend Subharthi Ray (of IF-UFF) forcarefully reading the manuscript and amending the gram-matical errors.

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