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GENERALIZED SHOCK SOLUTIONS FOR HYDRODYNAMIC BLACK HOLE ACCRETION

Tapas K.Das1

Racah Instituteof Physics, Hebrew University, Jerusalem 91045, Israel; and Inter-University Centre for Astronomy and Astrophysics, Post Bag 4,Ganeshkhind, Pune 411 007, India; tapas@iucaa.ernet.in

Received 2001 December 22;accepted 2002 June 3

ABSTRACT

For the first time, all available pseudo-Schwarzschild potentials are exhaustively used to investigate thepossibility of shock formation in hydrodynamic, inviscid, black hole accretion disks. It is shown that a signifi-cant region of parameter space spanned by important accretion parameters allows shock formation for flowin all potentials used in this work. This leads to the conclusion that the standing shocks are essential ingre-dients in accretion disks around nonrotating black holes in general. Using a complete general relativistic

framework, equations governing multitransonic black hole accretion and wind are formulated and solved,and the condition for shock formation in such flows is also derived in the Schwarzschild metric. Shock solu-tions for accretion flow in various pseudopotentials are then compared with such general relativistic solutionsto identify which potential is the best approximation of Schwarzschild spacetime as far as the question ofshock formation in black hole accretion disks is concerned.

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

1. INTRODUCTION

The process by which any gravitating, massive, astro-physical object captures its surrounding fluid is called accre-tion. Depending on the rotational energy content of theinfalling material, accretion flows onto black holes (BHs)may be broadly classified into two different categories, i.e.,nonrotating (spherical) and rotating (accretion disks) accre-tion. If the instantaneous dynamical velocity and localacoustic velocity of the accreting fluid, moving along a spacecurve parameterized by r, are ur and ar, respectively,

then the local Mach number Mr of the fluid can be definedas Mr ur=ar. The flow will be locally subsonic orsupersonic according to Mr < 1 or Mr > 1, i.e., accord-ing to ur < ar or ur > ar. The flow is transonic if atany moment it crosses M 1. This happens when a sub-sonic-to-supersonic or supersonic-to-subsonic transitiontakes place either continuously or discontinuously. Thepoint(s) where such crossing takes place continuously is(are) called the sonic point(s) and where such crossing takesplace discontinuously are called shocks or discontinuities. Itis generally argued that, in order to satisfy the inner boun-dary conditions imposed by the event horizon, accretiononto black holes exhibits transonic properties in general,which further indicates that formation of shock waves is

possible in astrophysical fluid flows onto Galactic andextragalactic black holes. One also expects that shock for-mation in black hole accretion might be a general phenom-ena because shock waves in rotating and nonrotating flowsare convincingly able to provide an important and efficientmechanism for conversion of a significant amount of thegravitational energy (available from deep potential wellscreated by these massive compact accretors) into radiationby randomizing the directed infall motion of the accreting

fluid. Hence, shocks possibly play an important role in gov-erning the overall dynamical and radiative processes takingplace in accreting plasma. Thus, the study of steady, station-ary shock waves produced in black hole accretion hasacquired a very important status in recent years, and it isexpected that shocks may be an important ingredient in anaccreting black hole system in general.

While the possibility of the formation of a standing spher-ical shock around compact objects was first conceived longago (Bisnovatyi-Kogan, Zeldovich, & Sunyaev 1971), mostof the works on shock formation in spherical accretionshare more or less the same philosophy that one shouldincorporate shock formation to increase the efficiency ofdirected radial infall in order to explain the high luminosityof active galactic nuclei (AGNs) and quasi-stellar objectsand to model their broadband spectrum (Jones & Ellison1991). Considerable work has been done in this directionwhere several authors have investigated the formation anddynamics of standing shock in spherical accretion (Mes-zaros & Ostriker 1983; Protheroe & Kazanas 1983; Chang& Ostriker 1985; Kazanas & Ellision 1986; Babul, Ostriker,& Meszaros 1989; Park 1990a, 1990b). Ideas and formal-isms developed in these works have been applied to studyrelated interesting problems such as entropic-acoustic orvarious other instabilities in spherical accretion (Foglizzo &Tagger 2000; Blondin & Ellison 2001; Lai & Goldreich2000; Foglizzo 2001; Kovalenko & Eremin 1998), produc-tion of high-energy cosmic rays from AGNs (Protheroe &Szabo 1992), study of the hadronic model of AGNs (Blon-din & Konigl 1987; Contopoulos & Kazanas 1995), highlyenergetic emission from relativistic particles in our Galacticcenter (Markoff, Melia, & Sarcevic 1999), explanation ofhigh lithium abundances in the late-type, low-mass compan-ions of soft X-ray transients (Guessoum & Kazanas 1999),and post-Newtonian (Das 1999, 2000, 2002) as well as com-plete general relativistic (Das 2001a) study of accretion-powered spherical winds emanating from Galactic andextragalactic black hole environments.

1 Current address: Division of Astronomy, University of California atLos Angeles, Box 951562, Los Angeles, CA 90095-1562; tapas@astro.ucla.edu.

TheAstrophysical Journal, 577:880892, 2002October 1

# 2002.The American AstronomicalSociety. All rightsreserved.Printedin U.S.A.

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With equal (if not more) importance and rigor, the ques-tion of shock formation in accretion disks around Schwarz-child black holes has been addressed by several authors.While the initial works in this direction can be attributed toFukue (1983), Hawley, Wilson, & Smarr (1984), Ferrari etal. (1985), Sawada, Matsuda, & Hachisu (1986), and Spruit(1987), it was Fukue (1987) and Chakrabarti and his collab-orators (Chakrabarti 1989, 1996a and references therein;Abramowicz & Chakrabarti 1990; Chakrabarti & Molteni1993) who were the first to provide the satisfactory semiana-lytical or numerical global shock solution for transonic,inviscid, Keplerian, or sub-Keplerian rotating accretionaround a Schwarzchild black hole. Consequently, theirworks were further supported and improved by severalother independent works (Yang & Kafatos 1995, hereafterYK; Caditz & Tsuruta 1998; Toth, Keppens, & Botchev1998). Because of the inner boundary conditions imposedby the event horizon, shocks form in BH accretion disksonly if the flow has more than one real physical X-type sonicpoint (multitransonic flow). For a particular set of initialboundary conditions, some of the above mentioned worksreport multiplicity in shock location, but such a degeneracycan ultimately be removed by local stability analysis, allow-ing one to assert that only one stable shock location is possi-ble. Hereafter, whenever we use the word shock, it is tobe understood that we in general always refer to only thestable shock location unless otherwise mentioned.

The above mentioned works deserve attention becausethe shocked flows studied there are expected to explain thespectral properties of BH candidates. However, thus far inthe astrophysical literature, the theoretical study of steady,standing shock formation in accretion disks around nonro-tating BHs has suffered from two general limitations. First,the shock solutions were obtained either on a case-by-casebasis or, even when successful attempts were made to pro-vide a more complete analysis, the boundary of the parame-ter space responsible for shock formation was obtained onlyfor global variation of the total specific energy E (or accre-tion rate _MM) and specific angular momentum of the flowand not for variations of the polytropic constant of theflow; rather, accretion was always considered to be ultrare-lativistic,2 which may not always be a realistic assumption.Since is expected to have great influence on the radiativeproperties of the flow in general, we think that ignoring theexplicit dependence of shock solutions on limits claims ofgenerality. Second, except for YK, all available so-calledglobal shock solutions have been discussed only in the con-text of one particular type of BH potential, namely, the Pac-zynski & Wiita (1980) potential [1r]. Along with the1r, recent studies (Das & Sarkar 2001 and referencestherein) enhance the importance of also considering threeother pseudo-Schwarzschild BH potentials, one [2r] pro-posed by Nowak & Wagoner (1991) and two others [3rand 4r] due to Artemova, Bjornsson, & Novikov (1996,hereafter ABN), in mimicking the complete general relativ-istic spacetime for accretion around a Schwarzschild blackhole. Hence, we believe that being restricted to only one spe-cific pseudo-Schwarzschild BH potential does not guaranteethe claimed global nature of so-called global shock solu-tions present in the literature; rather, one must study the

transonic disk structure as well as shock formation in allavailable BH potentials to firmly assert the ubiquity ofshock formation in a multitransonic accretion disk arounda Schwarzschild BH. In this context, it is to be mentionedhere that YK deserve special importance because the shocksolution due to YK appears to be the only work available inthe literature that provides the complete general relativisticdescription of shock formation exclusively for a nonrotatingBH. Nevertheless, this work deals with isothermal accre-tion, but one understands that global isothermality in BHaccretion is difficult to achieve for realistic flows, and a moregeneral kind of BH accretion is expected to be governed bya polytropic equation of state. Also, YK do not provide theglobal parameter space dependence of shock solutions. Afew authors claim to provide the full general relativisticshock solutions for Schwarzschild BHs as a limiting case oftheir results obtained in Kerr geometry (Chakrabarti 1996b,1996c; Lu et al. 1997). In doing so, a number of assumptionsare made, some of which, however, may not appear to befully convincing. For example, either the disk is supposed tobe in conical equilibrium (Lu et al. 1997), which should notbe the case in reality because the realistic accretion flowshould be in vertical equilibrium (Chakrabarti 1996a andreferences therein), or some results valid for isothermalaccretion are directly applied to study the polytropic accre-tion in an ad hoc manner (Chakrabarti 1996b), or some ofthe Newtonian approximations are not very convincinglycombined with complete general relativistic equations(Chakrabarti 1996c), which does not strengthen their claimfor a full general relativistic treatment of shock formation.Hence, it is fair to say that although literature on general rel-ativistic hydrodynamic BH accretion is well enriched by anumber of important works (the following is an incompletelist of relevant papers on the subject: Novikov & Thorne1973; Bardeen & Petterson 1975; Abramowicz, Jaroszynski,& Sikora 1978; Lu 1985, 1986; Karas & Mucha 1993;Bjornsson 1995; Riffert & Herold 1995; Ipser 1996; Pariev1996; Peitz & Appl 1997; Bao, Wiita, & Hadrava 1998;Gammie & Popham 1998; Gammie 1999), no well-acceptedcomplete general relativistic global shock solution exclu-sively obtained for a hydrodynamic accretion disk around aSchwarzschild BH has yet appeared in the literature.

Motivated by the above mentioned limitations encoun-tered by previous works in this field, the major aim of ourwork presented in this paper is to provide a generalized for-malism that is expected to handle the formation of steady,standing Rankine- Hugoniot shock (RHS) in multitran-sonic hydrodynamic BH accretion flow and to identifywhich region of parameter space (spanned by every impor-tant accretion parameter, namely, E, , and ) will beresponsible for such shock formation for all availablepseudo-Schwarzschild BH potentials. We would also like tocompare the properties of multitransonic accretion in theseBH potentials with complete general relativistic BH accre-tion as far as the issue of shock formation is concerned.

Hereafter, we will define the Schwarzschild radius rg as

rg 2GMBH

c2

(where MBH is the mass of the black hole and G is the uni-versal gravitational constant) so that the marginally boundcircular orbit rb and the last stable circular orbit rs take thevalues 2rg and 3rg, respectively, for a typical Schwarzschild

2 By the terms ultrarelativistic and purely nonrelativistic, we meana flow with 4=3 and 5=3, respectively, according to the terminologyused in Frank,King,& Raine (1992).

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black hole. Also, total mechanical energy per unit mass onrs (sometimes called efficiency, e) may be computed as0.057 for this case. Also, we will use a simplified geometricunit throughout this paper where radial distance r is scaledin units of rg, radial dynamical velocity u and polytropicsound speed a ofthe flow are scaled in units ofc (the velocityof light in vacuum), mass m is scaled in units ofMBH, and allother derived quantities would be scaled accordingly. Also,for simplicity, we will use G c MBH 1. Inthe nextsec-tion, we briefly describe a few important features of the fourdifferent pseudo-Schwarzschild effective BH potentialsused in this work. In x 3, we show how we formulate andsolve the equations governing multitransonic BH accretionin these potentials that may have shocks. In x 4, we studymultitransonic BH accretion using the full general relativis-tic framework and argue which potential is expected to bethe closest approximation of actual general relativistic solu-tions for which regions of parameter space spanned by E, ,and , as long as one concentrates only on shocked flows.Finally, in x 5 we draw our conclusion by highlighting someof the possible important impacts of the study of shock for-mation on related fields.

2. PROPERTIES OF FOUR PSEUDO-SCHWARZSCHILD BLACK

HOLE POTENTIALS

Rigorous investigation of the complete general relativisticmultitransonic BH accretion disk structure is extremelycomplicated. At the same time, it is understood that, sincerelativistic effects play an important role in the regions closeto the accreting black hole (where most of the gravitationalpotential energy is released), a purely Newtonian gravita-tional potential [in the form Nr 1=r in the system ofunits used here] cannot be a realistic choice to describe tran-sonic black hole accretion in general. To compromisebetween the ease of handling of a Newtonian description ofgravity and the realistic situations described by complicatedgeneral relativistic calculations, a series of modified Newtonian potentials have been introduced to describe thegeneral relativistic effects that are most important for accre-tion disk structure around Schwarzschild and Kerr blackholes (see ABN for further discussion). Introduction of suchpotentials allows one to investigate the complicated physicalprocesses taking place in disk accretion in a semi-Newto-nian framework by avoiding pure general relativistic calcu-lations, so that most of the features of spacetime around acompact object are retained and some crucial properties ofthe analogous relativistic solutions of disk structure can bereproduced with high accuracy. Hence, those potentialsmight be designated as pseudo-Kerr or pseudo-Schwarzschild potentials, depending on whether they areused to mimic the spacetime around a rapidly rotating ornonrotating/slowly rotating (Kerr parameter a $ 0) blackhole, respectively. Below we describe four such pseudo-Schwarzschild potentials on which we concentrate in thispaper. It is important to note that as long as one is not inter-ested in astrophysical processes extremely close (within1rg 2rg) to a black hole horizon, one may safely use the fol-lowing BH potentials to study accretion onto a Schwarzs-child black hole with the advantage that the use of thesepotentials would simplify calculations by allowing one touse some basic features of flat geometry (additivity of energyor decoupling of various energy components, i.e., thermal

[a2= 1 ], kinetic [u2=2], or gravitational [], etc.; see sub-sequent discussions), which is not possible for calculationsin a purely Schwarzschild metric (see x 4). Also, one canstudy more complex many-body problems, such as accre-tion from an ensemble of donors, or overall efficiency ofaccretion onto an ensemble of black holes in a galaxy, ornumerical hydrodynamic or magnetohydrodynamic accre-tion flows around a black hole, as simply as can be done in aNewtonian framework but with far better accuracy. So webelieve that a comparative study of multitransonic accretionflow as well as shock formation using all these potentialsmight be quite useful in understanding some important fea-tures of various shock-related astrophysical phenomena, atleast until one can have a complete and self-consistenttheory of complete general relativistic shock formationexclusively for a Schwarzschild BH. However, one shouldbe careful in using these potentials because none of thepotentials discussed here are exact in the sense that theyare not directly derivable from the Einstein equations. Thesepotentials could be used only to obtain more accurate cor-rection terms over and above the purely Newtonian results,and any radically new results obtained using thesepotentials should be cross-checked very carefully with theexact general relativistic theory.

Paczynski & Wiita (1980) proposed a pseudo-Schwarzs-child potential of the form

1r 1

2r 1; 1a

which accurately reproduces the positions of rs and rb andgives the value of efficiency to be 0.0625, which is in closestagreement with the value obtained in full general relativisticcalculations. Also, the Keplerian distribution of angularmomentum obtained using this potential is exactly the sameas that obtained in pure Schwarzschild geometry. It is worthmentioning here that this potential was first introduced to

study a thick accretion disk with super-Eddington luminos-ity. Also, it is interesting to note that although it had beenthought of in terms of disk accretion, 1r is sphericallysymmetric with a scale shift ofrg.

To analyze the normal modes of acoustic oscillationswithin a thin accretion disk around a compact object (slowlyrotating black hole or weakly magnetized neutron star),Nowak & Wagoner (1991) approximated some of the domi-nant relativistic effects of the accreting black hole (slowlyrotating or nonrotating) via a modified Newtonian poten-tial of the form

2r 1

2r1

3

2r 12

1

2r

2" #; 1b

2r has the correct form ofrs as in the Schwarzschild met-ric but is unable to reproduce the value ofrb. This potentialhas the correct general relativistic value of the angularvelocity s at rs. Also, it reproduces the radial epicyclic fre-quency (for r > rs) close to its value obtained from generalrelativistic calculations, and among all BH potentials, 2rprovides the best approximation fors and . However, thispotential gives the value of efficiency as 0.064, which islarger than that produced by1r; hence, the disk spectrumcomputed using 2r would be more luminous comparedto a disk structure studied using1r.

Considering the fact that free-fall acceleration plays avery crucial role in Newtonian gravity, ABN proposed two

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different BH potentials to study disk accretion around anonrotating black hole. The first potential proposed bythem produces exactly the same value of the free-fall accel-eration of a test particle at a given value of r as is obtainedfor a test particle at rest with respect to the Schwarzschildreference frame and is given by

3r 1 1 1

r 1=2

: 1c

The second one gives a value of free-fall acceleration that isequal to the value of the covariant component of the threedimensional free-fall acceleration vector of a test particlethat is at rest in the Schwarzschild reference frame and isgiven by

4r 1

2ln 1

1

r

: 1d

Efficiencies produced by 3r and 4r are 0.081 and0.078, respectively. The magnitude of efficiency producedby3r being maximum, calculation of disk structure using3r will give the maximum amount of energy dissipation,

and the corresponding spectrum would be the most lumi-nous one. Hereafter, we will refer to all these four potentialsby ir in general, where i 1; 2; 3; 4f g would corre-spond to1 (eq. [1a]), 2 (eq. [1b]), 3 (eq. [1c]), and4 (eq.[1d]), respectively. One should notice that while all otherir have singularity at r rg, only 2r has a singularityat r 0. It can be shown that for r > 2rg, while 2r is flat-ter compared to purely Newtonian potentialNr, all otherir are steeper toNr.

At any radial distance r measured from the accretor, onecan define the effective potentialeffi r to be the summationof the gravitational potential and the centrifugal potentialfor matter accreting under the influence of the ith pseudopo-tential. The effective potentialeffi r can be expressed as

effi r ir 2r

2r2; 2a

where r is the nonconstant distance dependent specificangular momentum of accreting material. One then easilyshows that r may have an upper limit

upi r r3=2

ffiffiffiffiffiffiffiffiffiffiffi0ir

q; 2b

where 0ir represents the derivative ofir with respect tor. For weakly viscous or inviscid flow, angular momentumcan be taken as a constant parameter (), and equation (2a)can be approximated as

effi r ir 2

2r2: 2c

For general relativistic treatment of accretion, the effectivepotential cannot be decoupled into its gravitational and cen-trifugal components. For a Schwarzschild metric of theform

ds2 gldxldx

1 1

r

dt2 1

1

r

1dr2

r2 d2 sin2 d2

;

the world line of the accreting fluid is timelike, and the four-velocity of the fluid satisfies the normalization condition

ulul 1 ;

where ulul is the contra(co)variant four-velocity of thefluid. The angular velocity r of the fluid can be computedas

r u

ut gtt

g r 1

2r3;

where u=ut is the specific angular momentum that isconserved for fluid dynamics as well as for particle dynamicsfor inviscid flow. The general relativistic effective potentialeffGRr (excluding the rest mass) experienced by the fluidaccreting onto a Schwarzschild BH can be expressed as

effGRr r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffir 1

r3 2 1 r

s 1 : 2d

One can understand that the effective potentials in generalrelativity cannot be obtained by linearly combining its grav-

itational and rotational contributions because various ener-gies in general relativity are combined together to producenonlinearly coupled new terms.

In Figure 1, we plot effi r (obtained from eq. [2c]) andeffGRr as a function ofr in logarithmic scale. The value ofistaken tobe 2 inunitsof 2GM=c. The effr curves for dif-ferent ir are marked exclusively in the figure, and thecurve marked by GR represents the variation ofeffGRr withr. One can observe thateff1 r is in excellent agreement witheffGRr; only for a very small value of r (r ! rg),

eff1 r

starts deviating from effGRr, and this deviation keepsincreasing as matter approaches closer and closer to theevent horizon. All other effi r approach

effGRr at a radial

distance (measured from the BH) considerably larger com-

pared to the case for eff1 r. If one defines Deffi r to be themeasure of the deviation ofeffi r with

effGRr at any point

r,

Deffi r

effi r

effGRr ;

Fig. 1.Effective BH potentials for general relativistic [effGRr] as wellas for pseudogeneral relativistic [effi r] accretion disks as a function ofthe distance (measured from the event horizon in units ofrg) plotted in log-arithmic scale. The specific angular momentumis chosento be 2 in geomet-ricunits. Seetext fordetails.

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one observes that Deffi r is always negative for eff1 r, but

for other effi r, it normally remains positive for low valuesof but may become negative for a very high value of . IfjDeffi rj be the modulus or the absolute value ofD

effi r, one

can also see that, although only for a very small range ofradial distance very close to the event horizon, Deff3 r ismaximum for the whole range of the distance scale while1r is the best approximation of general relativistic space-time, 2 r is the worst approximation, and 4r and 3rare the second- and the third-best approximations as faras the total effective potential experienced by the accretingfluid is concerned. It can be shown that jDeffi rj nonlinearlyanticorrelates with . The reason behind this is understand-able. As decreases, rotational mass as well as its couplingterm with gravitational mass decreases for general relativis-tic accretion material, while for accretion in any ir, cen-trifugal force becomes weak and gravity dominates; hence,deviation from the general relativistic case will be moreprominent because general relativity is basically a manifes-tation of strong gravity close to the compact objects.

From the figure it is clear that for effGRr as well as for alleffi r, a peak appears close to the horizon. The height ofthese peaks may roughly be considered as the measure ofthe strength of the centrifugal barrier encountered by theaccreting material for respective cases. The deliberate use ofthe word roughly instead of exactly is due to the factthat here we are dealing with fluid accretion, and unlike par-ticle dynamics, the distance at which the strength of the cen-trifugal barrier is maximum is located further away fromthe peak of the effective potential because here the totalpressure contains the contribution due to fluid or ram pressure also. Naturally, the peak height for effGRr as wellas for effi r increases with increase of , and the locationof this barrier moves away from the BH with higher valuesof angular momentum. If the specific angular momentum ofaccreting material lies between the marginally bound andmarginally stable value, an accretion disk is formed. Forinviscid or weakly viscous flow, the higher the value of ,the higher the strength of the centrifugal barrier and themore the amount of radial velocity or the thermal energythat the accreting material must have to begin with so that itcan be made to accrete onto the BH. In this connection, it isimportant to observe from the figure that accretion under1r will encounter a centrifugal barrier farthest away fromthe BH compared to other ir. For accretion under allir except 1r, the strength of the centrifugal barrier ata particular distance will be more compared to its value forfull general relativistic accretion.

3. MULTITRANSONIC FLOW IN VARIOUS BLACK

HOLE POTENTIALS AND SHOCK FORMATION

Following the standard literature, we consider a thin,rotating, axisymmetric, inviscid steady flow in hydrostaticequilibrium in the transverse direction. The assumption ofhydrostatic equilibrium is justified for a thin flow becausefor such flows the infall timescale is expected to exceed thelocal sound crossing timescale in the direction transverse tothe flow. The flow is also assumed to possess a considerablylarge radial velocity that makes the flow advective. Thecomplete solutions of such a system require the dimension-less equations for conserved specific energy E and angularmomentum of the accreting material, the mass conserva-tion equations supplied by the transonic conditions at the

sonic points, and the Rankine-Hugoniot conditions at theshock. The local half-thickness, hir, of the disk for anyir can be obtained by balancing the gravitational forceby the pressure gradient and can be expressed as

hir affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r= 0i q

;

where 0i dir=dr. For a nonviscous flow obeying thepolytropic equation of state p K (Kis a measure of thespecific entropy of the flow), integration of the radialmomentum equation

udu

dr

1

dP

dr

d

dreffi r

0

leads to the following energy conservation equation insteady state:

E 1

2u2e

a2e

1

2

2r2 ir 0 ; 3a

and the continuity equation,

d

dr urhir 0 ;

can be integrated to obtain the barion number conservationequation:

_MMin

ffiffiffi1

sueaeer

3=2 0i 1=2

: 3b

Following Chakrabarti (1989), one can define the entropyaccretion rate _MM _MMinK1= 1 1= 1 , which undergoes adiscontinuous transition at the shock location rsh, wherelocal turbulence generates entropy to increase _MM for post-shock flows. For our purpose, the explicit expression for _MMcan be obtained as

_MM

ffiffiffi1

suea

1 = 1 e r

3=2 0i 1=2

: 3c

In equations (3a)(3c), the subscript e indicates the valuesmeasured on the equatorial plane of the disk; however, wewill drop e hereafter if no confusion arises in doing so. Onecan simultaneously solve equations (3a)(3c) for any partic-ular ir and for a particular set of values of E; ; f g.Hereafter, we will use the notation Pi for a set of values ofE; ; f g for any particular i.

For a particular value of Pi , it is now quite straightfor-ward to derive the space gradient of dynamical flow velocity

du=dr i for flow in any particular ith BH potentialir:du

dr

i

2=r3 0ir

a2= 1 3=r 00i r=0ir

u 2a2=u 1

;

4a

where 00i represents the derivative of0i. Since the flow is

assumed to be smooth everywhere, if the denominator ofequation (4a) vanishes at any radial distance r, the numera-tor must also vanish there to maintain the continuity of theflow. One therefore arrives at the so-called sonic-point

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(alternately, the critical-point ) conditions by simultane-ously making the numerator and denominator of equation(4a) equal zero. The sonic-point conditions can be expressedas

ais

ffiffiffiffiffiffiffiffiffiffiffi1

2

ruis

0ir 0ir

r22 r30ir

30ir r00i r

!s

;

4b

where the subscript s indicates that the quantities are to bemeasured at the sonic point(s). For a fixed Pi , one cansolve the following polynomial of r to obtain the sonicpoint(s) of the flow:

E2

2r2 ir

!s

2

2 1

0ir 0ir

r22 r30ir

30ir r00i r

!s

0 : 4c

Similarly, the value of du=di i at its corresponding sonicpoint(s) can be obtained by solving the following equation:

4

1

du

dr

2s; i

2us 1

1

3

r00i r

0ir

!s

du

dr

s; i

a2s

(000i r

0ir

2

1 200i r

0ir

!2

6 1

1 200i r

0ir

6 2 1

2 1 2

)s

00i js 32

r4s 0 ; 4d

where the subscript s; i indicates that the correspondingquantities for any ith potential are being measured at its cor-responding sonic point(s), and000i r d

3ir=dr3.For alli, we find a significant region of parameter space

spanned by Pi that allows the multiplicity of sonic pointsfor accretion as well as for wind where two real physicalinner and outer (with respect to the BH location) X-typesonic points rin and rout encompass one O-type unphysicalmiddle sonic point rmid in between. For a particular ir, ifAiPi denotes the universal set representing the entireparameter space covering all values ofPi and ifBiPi rep-resents one particular subset ofAiPi that contains onlythe particular values of Pi for which the above mentionedthree sonic points are obtained, then BiPi can further bedecomposed into two subsetsCiPi andDiPi such that

CiPi BiPi only for _MM rin > _MM rout ;

DiPi BiPi only for _MM rin < _MM rout ;

then for Pi 2 CiPi we get multitransonic accretion, andfor Pi 2 DiPi one obtains multitransonic wind. In Figure2 we plot

Ei; i 2 Pi 2 CiPi BiPi ;

Ei; i 2 Pi 2 DiPi BiPi

for all ir (marked in the figure) when 4=3. While thespecific energy E is plotted along the Y-axis, the specificangular momentum is plotted along the X-axis. For1r,

the shaded region PQR represents the parameter spacespanned byE and for which three sonic points will form inaccretion (PQR C1 P1 ), while the wedge-shapedunshaded region PSR represents the parameter space forwhich three sonic points are formed in wind(PSR D1 P1 ). A similar kind of parameter-space divi-sion is shown for other ir as well. A careful analysis ofFigure 2 reveals the fact that, at least for ultrarelativisticflow, no region of parameter space common to all ir isfound for which Pi 2 CiPi or Pi 2 DiPi . However, asignificant region of parameter space is obtained for whichPi 2 CiPi or Pi 2 DiPi for 2r and 3r, and a

very small region of such a common zone in the parameterspace is obtained (only for extremely low values of the

energy and angular momentum of the accreting matter) for2r, 3r, and 4r. As the flow approaches its purelynonrelativistic limit, i.e., as we make ! 5=3, the tendencyfor such a mutual overlap of parameter space for 2r,3r, and 4r increases. Nevertheless, 1r still remains untouchable by2r and 3r; only a particular regionof parameter space (fairly low energy accretion with inter-mediate value of angular momentum) is commonly sharedby4r and1r.

One also observes that ifEmaxi and maxi are the maximum

available energy and angular momentum of the flow for anyir for which Pi 2 CiPi or Pi 2 DiPi , one can write

Emax3 > E

max4 > E

max1 > E

max2 ;

max1 > max4 >

max2 >

max3 :

The above trend remains unaltered as ! 5=3, and weobserve that both Emaxi and

maxi nonlinearly anticorrelate

with .If shock forms in accretion (in this work we do not study

shock formation in wind), then Pi responsible for shockformation must be somewhere from the region for whichPi 2 CiPi , although not all Pi 2 CiPi will allow shock

transition. Using equations (3a)(3c), we combine the threestandard Rankine-Hugoniot conditions (Landau & Lifshitz1959) for vertically integrated pressure and density (seeMatsumoto et al. 1984) to derive the following relation,

Fig. 2.Parameter space division for multitransonic, ultrarelativisticaccretion and wind in four different pseudo-Schwarzschild BH potentials.Seetext fordetails.

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which is valid only at the shock location:

1 _MM

_MM

log1E kith

1 Rcomp 1

1 1 0 ; 5

where E kith is the total specific thermal plus mechanicalenergy of the accreting fluid: E kith fE2=2r2 i g; Rcomp and are the density compression

and entropy enhancement ratios, respectively, defined asRcomp = and _MM=

_MM; 1 1 and

Rcomp; and + and refer to the postshock andpreshock quantities. The shock strengthSi (ratio of the pre-shock to postshock Mach number of the flow) can be calcu-lated as

Si Rcomp 1 : 6

Equations (5) and (6) cannot be solved analytically becausethey are nonlinearly coupled. However, we have been ableto simultaneously solve equations (3a)(6) using iterativenumerical techniques. We have developed an efficient

numerical code that takes Pi and i as its input and cancalculate rsh along with any sonic or shock quantity as afunction ofPi . It is to be noted that like the references citedin x 1, we also obtain multiplicity in the shock location. Weperform the local stability analysis and find that only one rshthat forms in between rout and rmid is stable for allir.

Let Pi 2 FiPi CiPi represent the region of param-eter space for which a multitransonic supersonic flow isexpected to encounter an RHS at rsh. At the shock, thesupersonic flow becomes hotter, shock compressed, andsubsonic. This shocked subsonic flow will become super-sonic again after passing through rin and will ultimatelycross the event horizon. For multitransonic flows with ini-tial boundary conditions different from those discussed

above, one can define Pi 2 GiPi , which is complement ofFiPi related toCiPi , so that for

GiPi j Pi 2 CiPi ; Pi 62 FiPi f g ;

the shock location becomes imaginary in GiPi ; hence, nostable RHS forms in that region; rather, the shock keepsoscillating back and forth. We anticipate that GiPi is alsoan important zone that might be responsible for the quasi-periodic oscillation (QPO) of the BH candidates (see x 5).

Figure 3 demonstrates few typical flow topologies of theintegral curves of motion for ultrarelativistic ( 4=3)shocked flows in various i (indicated in the figure). Whilethe distance from the event horizon of the central BH(scaled in units ofr

gand plotted in logarithmic scale) is plot-

ted along the X-axis, the local Mach number of the flow isplotted along the Y-axis. One can easily obtain such a set offigures for any (and Pi ) that allows shock formation. Forall figures, ABCD represents the transonic accretion passingthrough the outer sonic point rout (marked as B) if a shockwould not form. However, since _MM of the flow is higher atthe inner sonic point rin compared to _MM at rout, the flowmust encounter a shock at C (the vertical line CE marked byan arrowhead represents the shock transition), becomessubsonic, and jumps on the branch EF, which ultimatelyhits the event horizon supersonically after it passes throughthe inner sonic point rin, which is marked on EFby the smallcircle with a dot at the center. An asterisk in the figure indi-

cates the location of the middle sonic point rmid. The corre-sponding values ofrin, rmid, rout, the shock location rsh, andthe shock strength Si are indicated at the top of each figure,while the corresponding values of the total specific energy Eand angular momentum for which the solutions areobtained are indicated inside each figure. GBH representsthe self-wind of the flow, which, in the course of itsmotion away from the BH to infinity, becomes supersonicafter passing through rout at B. Collectively, ABCEF repre-sents the real physical shocked accretion that connects infin-ity with the event horizon. The overall scheme for obtainingthe above mentioned integral curves is as follows: First wecompute rin, rmid, and rout by solving equation (4c). Then weobtain the dynamical velocity gradient of the flow at sonicpoints by solving equation (4d). For a chosen _MMin (scaled inthe units of the Eddington rate _MMEdd), we then compute thelocal dynamical flow velocity ur, the local polytropicsound speed ar, the local radial Mach number Mr, thelocal fluid density r, and any other related dynamical orthermodynamic quantities by solving equations (4a)(4d)from the outer sonic point using the fourth-order Runge-Kutta method. We start integrating from rout in two differ-ent directions. Along BH, we solve only for ur, ar, andMr because shock does not form in subsonic flows. How-ever, integration along BCD involves a different procedure.Along BCD, we not only compute ur, ar, and Mr butalso, at every integration step (with as small a step size aspossible), we keep checking whether equation (5) is beingsatisfied at that point. To do so, at each and every point, westart with a suitable initial guess value ofRcomp andSi andperforms millions of iterations to check whether for any setof Rcomp;Si

equation (5) is satisfied at that point andwhether for such Rcomp;Si

, the value ofobtained from

equation (5) becomes exactly equal to _MMrin= _MMrout, inother words, whether the entropy generated at that point (ifany) becomes exactly equal to the difference between theentropies at the inner and the outer sonic points. If suchconditions are satisfied at some particular point (point C inthe figure), we argue that the shock forms at that point, andwe can calculate any preshock and the postshock dynamicaland thermodynamic quantities at the shock location rsh(i.e., at C). Once a shock is formed, the flow jumps from itssupersonic branch BCD to its subsonic branch EG. Weagain start calculating ur, ar, and Mr and any otherrelated flow quantities by solving equation (4a) using thefourth-order Runge-Kutta method (with the help of eqs.[3a][3c] and eq. [4d]), but this time from the inner sonicpoint rin of the flow.

In Figure 4, we present theFiPi for all four i: 1 (Fig.4a), 2 (Fig. 4b), 3 (Fig. 4c), and 4 (Fig. 4d). The specificenergy E, specific angular momentum , and the polytropicindex of the flow are plotted along the Z-, Y-, and X-axes,respectively. Each surface for a particular ir is drawn fora particular value of. While the first surfaces (which havethe maximum surface areas) on the E- plane representultrarelativistic accretion ( 4=3), successive surfaces arealso shown for higher values of , taking a regular intervalofD 0:025. It is observed that as the flow approaches itspurely nonrelativistic limit, the area of the E- surfacesresponsible for shock formation starts shrinking. We findthat the shock location correlates with . This is obviousbecause the higher the flow angular momentum, the greaterthe rotational energy content of the flow and the higher thestrength of the centrifugal barrier (which is responsible for

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breaking the incoming flow by forming a shock) as well asthe further the location of such barrier from the event hori-zon. However, rsh anticorrelates with E and , which meansthat for the sameE and , shock in the purely nonrelativisticflow will form closer to the event horizon compared to theultrarelativistic flow. We also observe that the shockstrengthSi nonlinearly anticorrelates with the shock loca-tion rsh, which indicates that the closer the shock forms tothe BH, the higher is the strength Si and the entropyenhancement ratio . The ultrarelativistic flows are sup-posed to produce the strongest shocks. The reason behindthis is also easy to understand: the closer the shock forms tothe event horizon, the higher the available gravitationalpotential energy to be released and the higher the radialadvective velocity required to have a more vigorous shock

jump. Compared to2 and3, 1 and4 allow wider spansofas well as for shock formation. IfEmax, max, and max

represent the maximum values of the corresponding param-eters for which shock formation is possible, we obtainEmax 3 > Emax 4 > Emax 1 > Emax 2 , max 1 >

max

4 >

max

3 >

max

2 , and

max

4 >

max 1 > max 3 > max 2 , respectively. Also, weobserve that as the flow approaches its purely nonrelativisticlimit more and more, shock may form for less and less angu-lar momentum. For some ir, even a very small amountof angular momentum ( < 1) allows shock formation,which indicates that for purely nonrelativistic accretion,shock formation may take place even for quasi-sphericalflow.

4. GENERAL RELATIVISTIC MULTITRANSONICACCRETION

Following the arguments provided by Novikov & Thorne(1973) and Chakrabarti (1996b), we derive the expressions

Fig. 3.Solution topologies for multitransonic, ultrarelativistic ( 4=3) shocked flows in different BH potentials as indicated in the figure. See text fordetails.

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for the conserved total specific energy E0 (which includes the

rest mass energy) and the entropy accretion rate _MM as

E0

1

1 a2

ffiffiffiffiffiffiffiffiffiffiffiffiffir 1

1 u2

rr3 2 1 r 1=2

; 7a

_MM 5:657ur1:25ffiffiffiffiffiffiffiffiffiffiffiffiffi

r 1

1 u2

r

a2 1

1 a2

! 1 =2 1 r3 2 1 0:25

: 7b

One can see from equation (7a) that the total specific energy,in this case, cannot be decoupled into various linearly addi-tive contributions of separate physical origin (i.e., kinetic,

thermal, rotational, or gravitational) as could be done for

flows in any pseudopotential.Following the procedure outlined in previous section, one

can derive the dynamical flow velocity gradient for generalrelativistic accretion flow as

du

dr

1

2r

2r3 2

r3 2 1

!

2r 1

2r r 1

2a2

1

5 7r

4r r 1

2 3r2

4 r3 2 1 r

& '

2a2

u u2 1 1

u

1 u2

!1; 8a

Fig. 4.Region of parameter space responsible for shock formation (FiPi ), forfour different BH potentials:1 (a), 2 (b),3 (c), and4 (d).See textfordetails.

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from which the sonic point conditions comes out to be

us

ffiffiffiffiffiffiffiffiffiffiffi2

1

sas

" 1

2

1

2r

2r3 2

r3 2 1

!

2r 1

2r r 1

& '

5 7r4r r 1

2

3r2

4 r3 2 1 r

& '1s

#1=2: 8b

The sonic point(s) could be computed by solving the follow-ing equation:

E02 r3s

2 1 rs

rs 1

1 W rs;

1

rs;

!2 0 ; 8c

where

rs; 1 1

2W rs; ;

W rs; 12rs

2r3s

2

r3s 2 1

! 2rs 1

2rs rs 1

& '

5 7rs

4rs rs 1

2 3r2s4 r3s

2 1 rs

( )1:

The dynamical flow velocity gradient at the sonic point(s)can be obtained by solving the following equation:

2 2 3a2s

1 u2s 1 2 dudr

2s

4 rs; 1 a2s

u2s 1

!du

dr

s

2

1

a2s rs; "2 rs; 1 a2s

1

2rs 1

rs rs 1

3r2s 2

r3s 2 1 rs

40r3s 24r

2s

2 16rs 13

10r 4s 8r3s

2 8r2s 13rs 5

# 0 ; 8d

where

rs; 5 7rs

4rs rs 1

2 3r2s4 r3s

2 1 rs :

We solve equation (8c) and find that like flows in variousir here also a significant region of parameter spaceallows the multiplicity of sonic points for accretion as wellas for wind where one O-type unphysical middle sonic pointis flanked in between two X-type real physical sonic pointsrin and rout. In Figure 5 we show the regions of parameterspace for which multitransonic flow is obtained for bothaccretion and wind. The dimensionless conserved total spe-cific energy E (excluding the rest-mass energy) is plottedalong the Y-axis, whereas the specific angular momentum is plotted along the X-axis. In the region bounded by PQRand marked by A, three sonic points are formed in accre-tion, and in the region bounded by PRS and marked by W,three sonic points are formed in wind. While the figure isdrawn for ultrarelativistic flows, the corresponding regions

of parameter space can be obtained for any . IfEmax be themaximum value of the energy and if max and min be themaximum and minimum values of the angular momentum,

respectively, for which three sonic points are formed inaccretion for any particular , we observe thatEmax; max; min nonlinearly anticorrelates with . In other

words, as the flow approaches its purely nonrelativisticlimit, the area of the region involved in formation of multi-transonic accretion decreases to a lower value.

In Figure 6, we show the integral curves of motion forgeneral relativistic accretion of ultrarelativistic polytropicfluid. For a particular set of E; ; shown in the figure,ABCD represents the accretion passing through the outersonic point rout (marked in the figure by B), the location ofwhich can be found by solving equation (8c). EBI representsthe self-wind. Flow along EFGH passes through the innersonic point rin (marked in the figure by F) and encompasses

a middle sonic point rmid, the location of which is shown inthe figure using an asterisk. As in Figure 3, here also weobtain the complete solution topology by integrating equa-tion (8a) (with the help of eqs. [7a], [7b], and [8c]) using thefourth-order Runge-Kutta method.

Fig. 5.Parameter space division for ultrarelativistic, multitransonicaccretion and wind in general relativity.

------------------------------------------------------------------------------------------------

Fig. 6.Integral curves of motion for ultrarelativistic, multitransonic,black hole accretion and corresponding self-wind in a Schwarzschild met-ric. Seetext fordetails.

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If and P be the shock compression and the entropyenhancement ratio (at the shock location) for this case( M=M, _MM=

_MM), one can show that the fol-lowing equation will be satisfied when shock forms:

1= 1 T

T

= 1 1 u21 u2

1=4 3 = 1 1 ; 9

where T= and u= are the preshock/postshock

temperature and dynamical velocities of the flow, respec-tively. However, it is our limitation in this paper that wehave not been able to formulate or solve any equation thatcan be used to calculate the shock location in general relativ-istic accretion onto Schwarzschild BHs. Nevertheless, ifshock forms in such flow (which is, of course, expected), it isobvious that the set of E; responsible for shock forma-tion must belong to the region PQR ( PGR 2 CGR PGR ;see x 3) of Figure 5 because shock will form only in multi-transonic accretion. The above argument is useful for com-paring accretion flows in various ir with generalrelativistic accretion (at least as far as the question of shockformation in multitransonic flow is concerned) in the fol-lowing way. Suppose that for ultrarelativistic flows, we takethe region of parameter space Pi 2 FiPi for any irused in this paper (see Fig. 4) and then superpose that regionwith PQR of Figure 5 and study which ir provides themaximum overlap between Pi 2 FiPi and PGR 2CGR PGR . That particular BH potential is then consideredto be the most efficient pseudopotential in approximatingthe general relativistic, multitransonic, shocked BH accre-tion. However, such an efficiency test is not entirelyunambiguous. We have yet to figure out the exactPGR 2FGR PGR . Hence, there may be some possibility

that for any ir, although Pi 2FiPi will overlap withPGR 2 CGR PGR , but instead of falling onto PGR 2FGR PGR , it will rather overlap with PGR 2 GGR PGR because the exact boundary between PGR 2 FGR PGR and PGR 2 GGR PGR could not be explored in our work.Nevertheless, we believe that still our arguments for the effi-ciency test are of some use, at least until one can find out theexact shock formation zone for general relativistic flow.

In Figure 7, we superpose Figure 5 on Pi 2 FiPi forall different ir (marked in the figure) used in our work.Unlike other Pi 2FiPi , P3 2 F3 P3 is drawn usinglong-dashed lines to show its overlap with P2 2F2 P2 .The figure is drawn for ultrarelativistic flow but can also bedrawn for other values of as well. We observe that whileP1 2 F1 P1 has excellent overlap (except at very high

energy) with PGR 2 CGR PGR , no other Pi 2 FiPi have any overlap with it. This leads to the conclusion that atleast for ultrarelativistic flow, 1r is not only a very goodapproximation; rather, it is the only BH potential to approx-imate the general relativistic multitransonic shocked flow.However, as the flow approaches its purely nonrelativisticlimit, we observe that the area of the overlapping zone for1r decreases with higher and P1 2 F1 P1 is pushedback to overlap rather with PGR 2 DGR PGR ; hence,unlike ultrarelativistic accretion, 1r may not be consid-ered such a good approximation for purely nonrelativisticflows. Also, we find that a region of low-energy, highangu-lar momentum P4 2 F4 P4 starts overlapping withPGR 2 CGR PGR . So for high- flows, along with 1r,4r may also be considered as a plausible approximationfor general relativistic accretion. Shocked flows in2r and

3r never show any overlap with PGR 2 CGR PGR for

any value of ; hence, these potentials may not be consid-ered to mimic the general relativistic multitransonic accre-tion flows.

5. CONCLUDING REMARKS

In this paper, we provide a generalized formalism thatcan formulate and solve the equations governing the advec-tive, multitransonic, hydrodynamic BH accretion in allavailable pseudo-Schwarzschild potentials, which may con-tain steady, standing, Rankine-Hugoniot kinds of shocks.We have also formulated and solved the equations govern-ing multitransonic, complete general relativistic BH accre-tion and wind in a Schwarzschild metric and compared our

pseudo-Schwarzschild solutions with the general relativisticone. The main conclusions of this paper are the following:

1. We observe that a significant region of parameterspace (spanned by the conserved total specific energy E, thespecific angular momentum , and the polytropic index ofthe flow) allows shock formation for all potentials, whichleads to the strong conclusion that stable, standing RHSsare inevitable ingredients in multitransonic accretion disksaround nonrotating BHs. The same kind of conclusion wasdrawn by previous works in this field (see x 1) only for ultra-relativistic accretion in the Paczynski & Wiita (1980) poten-tial, whereas we make this conclusion more general byincorporating allavailable BH potentials to study BH accre-tion for allpossible values of.

2. Since the shock forms at a particular radial distance, itis clear that self-similar solutions should not be invokedwhile studying real physical BH accretion and relatedphenomena.

3. It is sometimes argued that a nonstanding oscillatingshock may modulate the disk spectrum in order to explainthe dwarf novae outburst (Mauche, Raymond, & Mattei1995) or QPO (Hua, Kazanas, & Titarchuk 1997). In thiscontext, the region of parameter space, for which three sonicpoints are formed in accretion but still no steady, standingshock is found (see x 3), can be considered as quite an impor-tant zone because Pi 2 GiPi may provide the relevantparameters responsible for such physical processes.

Fig. 7.Comparison of parameter space producing shocked multitran-sonic accretion in various BH potentials with parameter space representingmultitransonic black hole accretion and wind in general relativity. The fig-ureis drawn forthe ultrarelativistic flow. Seetext fordetails.

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4. As far as the shock formation in ultrarelativistic blackhole accretion is concerned, the Paczynski & Wiita (1980)potential 1r is the only pseudopotential that can mimicthe solutions of general relativistic accretion disks aroundnonrotating BHs in a very efficient way. However, in thepurely nonrelativistic limit ( ! 5=3), along with 1r,another BH potential, 4r, proposed by ABN is alsoobserved to mimic the general relativistic solutions, at leastfor low-energy, highangular momentum flows. However,it is interesting to note one important feature of the Paczyn-ski & Wiita potential 1r: like spherically symmetricaccretion (see Das & Sarkar 2001), for an accretion diskalso, 1r is observed to be in excellent agreement with sol-utions for ultrarelativistic flow in a pure Schwarzschild met-ric; however, it starts loosing (albeit very slowly) itsefficiency in mimicking a full general relativistic solutionwith higher values of , i.e., as the flow reaches its purelynonrelativistic limits, although the exact reason behind thisis not quite clear to us.

Hot, dense, and exoentropic postshock regions in advec-tive accretion disks are used as a powerful tool in under-standing the spectral properties of BH candidates (Shrader

& Titarchuk 1998 and references therein) and in theoreti-cally explaining a number of diverse phenomena, includingmillisecond variability in the X-ray emission from low-massX-ray binaries and the generation mechanism for high-fre-quency QPOs in general (Titarchuk, Lapidus, & Muslimov1998 and references therein), high-energy emission fromcentral engines of AGNs (Sivron, Caditz, & Tsuruta 1996),formation of heavier elements in BH accretion disks vianonexplosive nucleosynthesis (Mukhopadhyay & Chakra-barti 2000), formation and dynamics of accretion-poweredGalactic and extragalactic jets, quiescent states of X-raynovae systems, and the outflow-induced low luminosity ofour Galactic center (Das 1998, 2001b; Das & Chakrabarti1999). A number of observational evidences are also present

that are in close agreement with the theoretical predictionsobtained from shocked accretion models (Rutledge et al.1999; Muno, Morgan, & Remillard 1999; Webb & Malkan2000; Rao, Yadav, & Paul 2000; Smith, Heindl, & Swank2002). Thus, we believe that our present work may have farreaching consequences because of the following reasons:

1. Our generalized formalism assures that our model isnot just an artifact of a particular type of potential only,and inclusion of every BH potential allows a substantiallyextended zone of parameter space allowing for the possibil-ity of shock formation.

2. Of course, there is the possibility that in future some-one may come up with a pseudo-Schwarzschild potentialbetter than

1r, which will be the best approximation for

complete general relativistic investigation of multitransonicshocked flow. In such a case, if one already formulates ageneralized model for a multitransonic shocked accretiondisk for any arbitrary r, exactly what we have done inthis paper, then that generalized model will be able to read-ily accommodate that new r without having any signifi-cant change in the fundamental structure of the formulationand solution scheme of the model, and we need not have toworry about providing any new scheme exclusively validonly for that new potential, if any.

3. Even if someone can provide a completely satisfactorymodel for shock formation in full general relativistic(Schwarzschild) BH accretion, still the utility of this work

may not be completely irrelevant. Rigorous investigation ofsome of the shock-related phenomena is extremely difficult(if not completely impossible) to study using a full generalrelativistic framework. Hence, one is expected to always relyon these pseudopotentials because of the ease of handlingthem. For example, it was shown that (see x 4) the totalenergy of the general relativistic accretion flow cannot bedecoupled into its constituent contributions, whereas forany kind of pseudopotential (see x 2), all individual energycomponents are linear under addition. This providesenough freedom and ease to simply add any extra compo-nent in the expression for energy to introduce any newphysics in the system (radiative forces or magnetic fields, forexample), which is certainly not possible while dealing withfull general relativistic astrophysical flows around nonrotat-ing BHs.

Thus, for the above mentioned reasons, we believe thatcompared to all previous works based solely on ultrarelativ-istic accretion in 1, our model is better equipped for han-dling various shock-related phenomena.

It is noteworthy that the idea of shock formation inadvective BH accretion is contested by some authors (Nar-

ayan, Kato, & Honma 1997 and references therein). How-ever, the fact that their claim against shock formation is,perhaps, inappropriate for many reasons has been shown(Molteni, Gerardi, & Valenza 2001) from energy considera-tions. One can understand that the problem of not findingshocks lies in the fact that nonshock advection-dominatedaccretion flow models are, perhaps, unable to produce mul-titransonic flows because only one inner sonic point close tothe BH is explored by such works.

One can observe that flows characterized by Pi 2FiPi in our work may contain low intrinsic angularmomentum for some cases (especially for purely nonrelativ-istic flow in some of the BH potentials). However, suchweakly rotating flows are expected to be allowed by nature

for various real physical situations like detached binary sys-tems fed by accretion from OB stellar winds (Illarionov &Sunyaev 1975; Liang & Nolan 1984), semidetached, low-mass, nonmagnetic binaries (Bisikalo et al. 1998), andsupermassive BHs fed by accretion from slowly rotatingcentral stellar clusters (Illarionov 1987; Ho 1999 and refer-ences therein).

Even 28 years after the discovery of standard accretiondisk theory (Shakura & Sunyaev 1973), exact modeling ofviscous multitransonic BH accretion, including proper heat-ing and cooling mechanisms, is still quite an arduous task,and we have not yet fully attempted this. However, our pre-liminary calculations show that the introduction of viscosityvia a radius-dependent power-law distribution for angular

momentum pushes the shock location closer to the BH;details of this work will be discussed elsewhere.

The author acknowledges discussions with Igor D. Novi-kov, A. R. Rao, and Sandip K. Chakrabarti. His very spe-cial thanks go to Paul J. Wiita for reading the manuscriptextremely carefully and for providing a number of usefulsuggestions. He is also thankful to Aveek Sarkar for check-ing some of the algebra. Finally, the hospitality provided bythe Racah Institute of Physics, The Hebrew University ofJerusalem, Israel, where a part of this paper was written, isacknowledged.

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