DISK-OUTFLOW COUPLING: ENERGETICS AROUND SPINNING BLACK HOLES

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10Draft version February 15, 2010Preprint typeset using LATEX style emulateapj v. 03/07/07

DISK-OUTFLOW COUPLING: ENERGETICS AROUND SPINNING BLACK HOLES

Debbijoy Bhattacharya1, Shubhrangshu Ghosh2, Banibrata Mukhopadhyay3

Astronomy and Astrophysics Programme, Department of Physics, Indian Institute of Science, Bangalore-560012, India.Draft version February 15, 2010

ABSTRACT

The mechanism by which outflows and plausible jets are driven from black hole systems, still re-mains observationally elusive. Notwithstanding, several observational evidences and deeper theoreticalinsights reveal that accretion and outflow/jet are strongly correlated. We model an advective disk-outflow coupled dynamics, incorporating explicitly the vertical flux. Inter-connecting dynamics ofoutflow and accretion essentially upholds the conservation laws. We investigate the properties of thedisk-outflow surface and its strong dependence on the rotation parameter of the black hole. Theenergetics of the disk-outflow strongly depend on the mass, accretion rate and spin of the black holes.The model clearly shows that the outflow power extracted from the disk increases strongly with thespin of the black hole, inferring that the power of the observed astrophysical jets has a proportionalcorrespondence with the spin of the central object. In case of blazars (BL Lacs and Flat SpectrumRadio Quasars), most of their emission are believed to be originated from their jets. It is observedthat BL Lacs are relatively low luminous than Flat Spectrum Radio Quasars (FSRQs). The lumi-nosity might be linked to the power of the jet, which in turn reflects that the nuclear regions of theBL Lac objects have a relatively low spinning black hole compared to that in the case of FSRQ. Ifthe extreme gravity is the source to power strong outflows and jets, then the spin of the black hole,perhaps, might be the fundamental parameter to account for the observed astrophysical processes inan accretion powered system.

Subject headings: accretion, accretion disks — black hole physics — hydrodynamics — galaxies: active— galaxies: jets — X-rays: binaries

1. INTRODUCTION

High resolution observations show strong outflows andjets in black hole accreting systems, both in active galac-tic nuclei (AGNs) or quasars (Begelman et al. 1984;Mirabel 2003) and microquasars (SS433, GRS 1915+105)[Margon 1984; Mirabel & Rodriguez 1994, 1998]. Ex-tragalactic radio sources show evidence of strong jets inthe vicinity of spinning black holes (Meier et al. 2001;Meier 2002). Outflows are also observed in neutron starlow mass X-ray binaries (LMXBs) (Fender et al. 2004;Migliari & Fender, 2006) and also in young stellar objects(Mundt 1985). It has been argued (Ghosh & Mukhopad-hyay 2009; Ghosh et al. 2010; hereinafter GM09, G10respectively, and references therein) that outflows andjets are more prone to emanate from strong advectiveaccretion flows; the said paradigm is more susceptiblefor super-Eddington and sub-Eddington accretion flows.However, the exact (global if any) mechanism of forma-tion of the jet, its collimation, acceleration, compositionfrom the accretion powered systems still remain incon-clusive.Extensive works have been pursued on the origin of

outflow/jet, since the pioneering work of Blandford &Payne (1982) (e.g. Pudritz & Norman 1986; Contopou-los 1995; Ostriker 1997), where the authors used a self-similar approach to demonstrate that the poloidal com-ponent of the magnetic field can be seemingly used tolaunch outflowing matter from the disk. The formationof the jet is directly related to the efficacy of extrac-

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

tion of angular momentum and energy from the accre-tion disk. Physical understanding of the hydromagneticoutflows from disks has been developed from magnetohy-drodynamic (MHD) simulations (Shibata & Uchida 1986;Ustyogova et al. 1999; De Villiers et al. 2005; Haw-ley & Krolik 2006) both in non-relativistic as well as inrelativistic regimes, mostly in the Keplerian paradigm.However, the large Lorentz factors as well as Faradayrotation measures suggest that the observed VLBI jetsin quasars and active galaxies are in the Poynting fluxregime (Homan et al. 2001; Lyutikov 2006). On theother front, strong radiation pressure can serve as a dif-ferent mechanism to effuse outflows/jets. This is likelyto occur when the accretion rate is super-Eddingtonor super-critical (Fabrika 2004; Begelman et al. 2006;GM09,G10 and references therein) and the accretion diskis precisely “radiation trapped” as in ultra-luminous X-ray (ULX) sources. It has been confirmed by radiationhydrodynamic simulation at super-critical accretion rate(Okuda et al. 2009) as well to explain luminosity andmass outflow rate of relativistic outflows from SS433.The under-luminous accreting sources, having high sub-critical accretion flow, were explained by an advectiondominated accretion flow (ADAF) model (Narayan & Yi1994). The promising outcome of this model lies in thelarge positive value of the Bernoulli parameter becauseof the small radiative energy loss. It leads to conceivethat the gas in the inflowing disk is susceptible to escape,leading to strong unbounded flows in the form of outflowsand jets. This also signifies that even in the absence ofmagnetic field and radiation pressure, outflows are plau-sible from strongly advective accretion flow if the systemis allowed to perturb. Nevertheless, the definitive under-

2

standing of the origin of outflows/jets is sill unknown.It remains one of the most compelling problems in highenergy astrophysics.Whatever might be the reason for the origin of out-

flow and then jet from the disk, one aspect is howeverdefinite that strong outflows producing relativistic jetsare powered by extreme gravity. Although seems para-doxical, the strength and the length scale of observedastrophysical jets vary directly with the strength of thecentral gravitating potential. Observationally, it is evi-dent that strong outflows and relativistic jets are morepowerful in observed AGNs and quasars, harboring su-permassive black holes, compared to that in black holeX-ray binaries (XRBs). In addition, the jets observed inAGNs and quasars have greater length scale compared tothat seen in stellar mass black hole systems. One of themost important signatures of relativistic gravitation isthe spin of the central object. It is presumably believedthat the spin (practically specific angular momentum) ofthe neutron star is less than that of a black hole, andthus the observed jet from black holes is much strongerand powerful than that of neutron star sources. In early,Blandford & Znajek (1977) demonstrated that if thereis a magnetic field associated with the black hole dueto threading of magnetic field lines from the disk andthe angular momentum of the Kerr black hole is largeenough, then the energy and the angular momentum canbe extracted from the underlying black hole by a purelyelectromagnetic mechanism, which can thus be expectedto power the jet in an AGN. These imply that the spinmight play a significant role in powering jets, both inmicroquasars and in AGNs, rather, it can act as a fun-damental parameter in an accretion powered system.Most of the studies of accretion disk and studies of re-

lated outflow/jet have been evolved separately, assumingthese two apparently to be dissimilar objects. However,several observational inferences (for details see GM09,G10 and references therein) and improved understand-ing of accretion flow and outflow reveal that accretionand outflow/jet are strongly correlated. The unifyingscheme of disk and outflow is essentially governed by con-servation laws; conservations of matter, energy and mo-mentum. Hence, in modeling the accretion and outflowsimultaneously in any accretion powered system, follow-ing aspects should be taken into account: (1) the effectof relativistic central gravitational potential including itsspin, (2) proper mechanism of origin of outflow/jet fromthe disk, (3) appropriate hydrodynamic equations (con-sidering that the accreting gas be treated as a continuumfluid), capturing the information about the intrinsic cou-pling between inflow and outflow which are governed bythe conservation laws in a strongly advective paradigm.Recently, GM09, G10 made an endeavor to explore the

dynamics of the accretion-induced outflow around blackholes/compact objects in a 2.5-dimensional paradigm.The authors formulated the disk-outflow coupled modelin a more self-consistent way by solving a complete set ofcoupled partial differential hydrodynamic equations in ageneral advective regime through a self-similar approachin an axisymmetric, cylindrical coordinate system. Theyexplicitly incorporated the information of the verticalflux in their model. However, they restricted their studyto Newtonian regime, thus negating the requisite effect ofgeneral-relativity, especially the effect of spin of the cen-

tral object. Based on the model, the authors computedthe mass outflow rate and the power extracted by theoutflow from the disk self-consistently, without propos-ing any prior relation between the inflow and outflow.In the present paper, we propose a new model for

the accretion-induced outflow by extending the work ofGM09, G10, by incorporating the general relativistic ef-fect of the central potential without limiting ourselvesto a self-similar regime. As the definitive mechanism oflaunching of outflows/jets is still evasive, we do not em-brace any specific mechanism of outflow like inclusion ofthe magnetic field into our model equations. Neverthe-less, the importance of the magnetic field can not be,in principle, discarded to explain the launching and col-limation of jets off the accretion disk. Here, owing toour inability to solve coupled partial differential MHDequations in a 2.5-dimensional advective regime, we ne-glect the influence of the magnetic field in our modelequations. However, the implicit coupling between theinflow and outflow, dictated by the conservation equa-tions, have been taken into account appropriately. Wearrange our paper as follows. In the next section, wepresent the formulation of our model. The section 3 de-scribes the computational procedure to solve the modelequations of the accretion-induced outflow. In sections 4and 5, we study the dynamics and the energetics of theflow respectively. Finally, we end up in section 6 with asummary and discussion.

2. MODELING THE CORRELATED DISK-OUTFLOWSYSTEM

We formulate the disk-outflow coupled model by con-sidering a geometrically thick accretion disk, which isstrongly advective as strong outflows/jets are more likelyto eject from a thick/puffed up region of the accretionflow (GM09,G10). The vertical flow is explicitly includedin the system. The basic features of the model are similarto that in GM09. We adopt the cylindrical coordinatesystem to describe a steady, axisymmetric accretion flow.The dynamical flow parameters, namely, radial velocity(vr), specific angular momentum (λ), vertical velocity oroutflow velocity (vz), adiabatic sound speed (cs), massdensity (ρ) and pressure (P ) depend both on radial andvertical coordinates. We have already highlighted theimportance of the spin of black hole to power the out-flow/jet and its presumed relative effect on the obser-vation of various AGN classes. Thus we have includedthe effect of the spin in our model. As the spin of theblack hole is a signature of pure general relativity, i.e.,Einstein’s gravitation, its effect on the accretion flow,especially in the inner region of the disk, is mimicked ap-proximately with the use of pseudo-general-relativisticor pseudo-Newtonian potential (PNP). Because of thedisk-outflow system to be geometrically thick and theflow to be 2.5-dimensional (not confined to the equato-rial plane), we use the PNP of Ghosh & Mukhopadhyay(2007), which is a pseudo-Newtonian vector potential, tocapture the inner disk properties of the accretion flowaround a Kerr black hole approximately.At the first instant we neglect the effect of viscosity in

our system. One of the reasons behind it is the unavail-ability of the effective computational technique to solvecoupled partial differential viscous hydrodynamic conser-vation equations for the compressible flow. To make the

3

inviscid assumption more arguable, it can also be notedthat the angular momentum transport in the accretionflow, for which the necessity of turbulent viscosity is in-voked, can alternatively take place purely by outflow. Atthis extreme end, the outflow extracts angular momen-tum from the disk allowing the matter to get accretedtowards the black hole and hence the inviscid assump-tion can be adopted. This is in essence similar to theBlandford & Payne (1982) mechanism to extract energyand angular momentum from the magnetized disk, wherethe extraction of the angular momentum and energy isessentially done by the outflow, and not due to the vis-cous dissipation. The outflow originates from just abovethe equatorial plane of the disk and this lower boundaryis maintained at z = 0, when vz = 0, unlike other workwhere the outflow is hypothesized to effuse out from thedisk surface (e.g. Xie & Yuan 2008). Our model is ef-fectively valid only in the region where disk and outfloware coupled, i.e. the region from where essentially theoutflow is emanated from the accretion flow. Hence ourstudy will remain confined within this predefined region.Further, we neglect the contribution of the magnetic fieldas argued in §1.We circumvent the idea of vertical integration of the

flow equations. The validity and the reliability of theheight integrated equations is normally gratifying in thegeometrically thin limit. In that circumstances, the flowvelocities are likely to be more or less independent of thedisk scale-height, which is not the case of the presentparadigm of interest. We further consider that the diskto be non-self-gravitating, assuming that the mass of thedisk to be much less than that of the black hole. Theradial and vertical coordinates are expressed in units ofGM/c2, flow velocities in c, time in GM/c3 and the spe-cific angular momentum in the unit of GM/c. Here G,M and c are gravitational constant, mass of the blackhole and speed of light respectively. The steady state,axisymmetric disk-outflow coupled equations in cylindri-cal geometry in the inviscid limit are then as follows.(a) Mass transfer:

1

r

∂

∂r(rρvr) +

∂

∂z(ρvz) = 0, (1)

where the first term is the signature of accretion and thesecond term attributes to outflow. As the outflow startsfrom just above z = 0 surface, within the inflow regionitself, we make a reasonable hypothesis that within theprescribed disk region the variation of the dynamical flowparameters with z is much less than that with r, allowingus to choose ∂A/∂z ≈ O(A/z); for any parameter A.Strictly speaking, the weak variation with z ensures thatthe outflow originating from the mid-plane of the diskdoes not disrupt the disk structure, and thus allowing asmooth accretion flow towards the black hole. Thus Eqn.(1) then reduces to

1

r

∂

∂r(rρvr) +

ρvzz

= 0. (2)

(b) Radial momentum balance:

vr∂vr∂r

+ vz∂vr∂z

− λ2

r3+ FGr +

1

ρ

∂P

∂r= 0, (3)

where FGr is radial component of the gravitational force.As discussed earlier, here FGr is the radial force corre-sponding to the PNP in cylindrical coordinate system

given by Ghosh & Mukhopadhyay (2007) containing theinformation of spin of the black hole. With the similarargument as above the equation then reduces to

vr∂vr∂r

+ vzvr − vr0

z− λ2

r3+ FGr +

1

ρ

∂P

∂r= 0, (4)

where vr0 is the radial velocity at the mid-plane of thedisk.(c) Azimuthal momentum balance:

1

r2∂

∂r

(

r2ρvrvφ)

+∂

∂z(ρvφvz) = 0, (5)

where vφ is the azimuthal velocity of the flow. The firstterm of this equation signifies the radial transport of theangular momentum, while the second term describes theextraction of angular momentum due to mass loss in theoutflow. If we consider that the net angular momentumextracted by the outflow be λj , and the remaining angu-lar momentum retained by the disk λd, then total angu-lar momentum λ = λj + λd can be assumed to remainconstant throughout the flow within our predefined disk-outflow region by the virtue of an inviscid flow. There-fore,

λ = constant. (6)

(d) Vertical momentum balance:

vr∂vz∂r

+ vz∂vz∂z

+ FGz +1

ρ

∂P

∂z= 0, (7)

where FGz is the vertical component of the gravitationalforce, described by Ghosh & Mukhopadhyay (2007). Fol-lowing previous arguments this reduces to

vr∂vz∂r

+v2zz

+ FGz +1

ρ

P − P0

z= 0, (8)

where P0 is the pressure of the flow at the mid-plane ofthe disk. If there is no outflow, then vz = 0, and Eqn. (8)reduces to the well known hydrostatic equilibrium condi-tion in the disk, and the hydrostatic disk scale-height canbe obtained. Similarly, one can customarily extend thevertical momentum equation to compute the disk scale-height when there is an outflow coupled with the disk.Thus we will use Eqn. (8) to obtain the scale-heightof the disk-outflow coupled system. We can reasonablyassume that at height h (i.e. at z=h), the pressure ofthe disk is much less compared to that at the equatorialplane to prevent any disruption of the disk, permittinga steady structure of accretion flow. The variation ofdensity along z direction is approximately kept constant(ρ0 ∼ ρ) which in turn means that the disk-outflow cou-pled region is weakly stratified. Considering the abovefacts, Eqn. (8) is simplified to obtain the scale-height as

vr|h∂vz∂r

∣

∣

∣

∣

h

+v2z |hh

+ FGz|h − P0

hρ= 0, (9)

where h is thus the the root of the above transcendentalequation. Thus Eqns. (2), (4), (6) & (9) will simultane-ously have to be solved to obtain the dynamics and theenergetics of the accretion-induced outflow.

4

3. SOLUTION PROCEDURE AND THE DISK-OUTFLOWSURFACE

We assume that the accretion flow follows an adiabaticequation of state P = Kρ1+1/n where n = 1/(γ − 1), nand γ are the polytropic and the adiabatic indices re-spectively. Note that constant K carries the informa-tion of entropy (e.g. Mukhopadhyay 2003) of the flow.Thus for an adiabatic flow, cs = (γ P

ρ )1/2. In an ear-

lier work (GM09), a reasonable assumption was madethat h ∼ r/2, which can be approximately accepted for ageometrically thick, 2.5-dimensional disk structure. Pre-suming that the outflow velocity is not likely to exceedthe sound speed at the disk-outflow surface (the outerboundary of the accretion flow in the z direction), wepropose a simplified relation between cs and vz as

vz <∼ 2(

z

r) cs. (10)

Hence, at z = h ∼ r/2, we obtain

vz <∼ cs. (11)

This implies that the outflow can ideally come out off thedisk surface which can appropriately be termed as thesonic surface in the vertical direction. With this notionin mind let us generalize this particular scaling betweenvz and cs, pertaining to our formalism as

vz = ı (z

r)µ cs, (12)

where ı and µ are the constants which will be deter-mined by substituting Eqn. (12) in our model conser-vation equations described in §2. The index µ measuresthe degree of subsonic nature of the vertical flow withinthe disk-outflow coupled region (i.e. in between the mid-plane and the surface of the accretion flow).Generalizing the procedure adopted in earlier works

for 1.5-dimensional disks (Chakrabarti 1996; Mukhopad-hyay 2003; Mukhopadhyay & Ghosh 2003), we solveEqns. (2), (4), (6) & (9). Using Eqn. (12) along withEqn. (6) and combining Eqns. (2) & (4), we obtain

∂vr∂r

=λ2

r3 − FGr +c2sr + ı

z (zr )

µcs(vr0 − vr +c2svr)

vr − c2svr

=N

D.

(13)

Equation (13) shows that to guarantee a smooth solutionat the “critical point”, N = D = 0. From the criticalpoint condition we obtain vrc = csc, at r = rc. Heresubscript c is referred to critical point. The radius rc isalso called the “sonic radius” since no disturbance cre-ated within this radius can cross the radius (also knownas sound horizon) and escape to infinity. Conditions atcritical/sonic radius give

vrc = csc = − ı

2z(z

rc)µcs0crc +

[{

ı

2z

(

z

rc

)µ

cs0crc

}2

+rcFGrc −λ2c

r2c

]1/2

,(14)

where cs0c = vr0c = (rcFGr0c − λ2c/r

2c )

1/2, is the soundspeed or the radial velocity of the flow at the sonic ra-dius in the disk mid-plane (z = 0). FGrc is the radial

gravitational force at the sonic radius and FGr0c is thecorresponding value in the disk mid-plane.Now at sonic location ∂vr

∂r = 0/0. Hence by applyingl’Hospital’s rule, Eqn. (13) reduces to

∂vr∂r

∣

∣

∣

∣

c

= −[

2nvrccsc

(−B +√B2 − 4AC2A

)

+vrcrc

+ı

z

(

z

rc

)µ

csc

]

, (15)

where A = F1(r, z)|c, B = F2(r, z)|c and C = F3(r, z)|c,are complicated functions of r and z at sonic location;F1, F2 & F3 have been explicitly given in the appendix.Equations (13) and (15) are then solved with an appro-priate boundary condition to obtain vr and cs as func-tions of r. The value of specific angular momentum inour flow always remains constant and is same as λc, thevalue at sonic radius, by virtue of Eqn. (6).Until now, we have emphasized on velocity profiles of

the accretion-induced outflow at any arbitrary z. Wehave, however, mentioned before that the intrinsic cou-pling of inflow and outflow is confined within a speci-fied region, from where the outflow emanates. The disk-outflow inter-correlated region is bounded vertically fromthe mid-plane (z = 0) to an upper surface above whichany inflow of matter ceased to exist. We aim at pre-cisely investigating the nature and dynamics of the flowat different layers within this disk-outflow coupled region.Therefore, we first need to calculate the scale-height ofthe accretion flow, based on our proposed model, andobtain the disk-outflow surface (upper boundary). It isto be noted that the transcendental Eqn. (9) cannot beindependently used to calculate the scale-height h or itsgeneral variance with r, for unknown variables vr and cs.Nevertheless, we can easily obtain the scale-height h atrc, say hc, from Eqn. (9) as radial velocity and soundspeed at the sonic location are known. Thus at rc, Eqn.(9) simplifies to

ı

(

hc

rc

)µ

csc|h[

ı

(

hc

rc

)µcsc|hhc

− µcsc|hrc

+

(

∂cs∂r

)

c

∣

∣

∣

∣

h

]

+FGzc|h − c2s0cγhc

= 0, (16)

where ∂cs/∂r|ch is ∂cs/∂r at the sonic location in theplane with scale-height h. Then hc obtained from Eqn.(16) can be inserted into Eqn. (15) to obtain ∂vr/∂r|cat scale-height h.The disk scale-height h without any outflow is seen to

be linearly increasing with r (approximately by dimen-sional analysis). For a 2.5-dimensional disk-outflow sys-tem, it seems that h and r can also be linearly connected,based on the order of magnitude analysis (GM09). How-ever, unlike the thin disk, strong gas pressure gradient inthe disk-outflow coupled system leads to a geometricallythick 2.5-dimensional disk. Therefore, for the presentpurpose we make a generalized scaling of h with r as

h ∼ δ r, (17)

where δ be a dimensionless arbitrary constant or anynumerical variable (discussed in detail later). The im-pressionable choice about the factor δ would be that itshould contain precisely the information of the nature of

5

the flow. The only physical information we can extractrelated to the scale-height from our model equations ishc. Thus we rationally demand an approximate expres-sion for dimensionless parameter δ as

δ ∼ hc

rc. (18)

It is found that the value of δ circumvents around 1/2,for the entire range of spin parameter of the black hole,which here acts as a normalization constant. Thus even-tually, Eqns. (13), (15), (17) & (18) are combined to-gether to obtain vr and cs along the scale-height h for anaccretion-induced outflow. It also appears that under allthese circumstances, and for a physically acceptable flow,the constant ı and the index µ of Eqn. (12) connectingvz and cs yield to be ∼ 1 and ∼ 3/2 respectively.

3.1. Construction of disk-outflow surface

After establishing the scale-height h, we in princi-ple can obtain the profiles of the dynamical parametersat different layers of the disk, i.e. at z = ℓh, where0 ≤ ℓ ≤ 1, within our prescribed disk-outflow region. Wefind that the radial velocity profile vr along all layers ofdisk for any arbitrary spin parameter exhibits some un-usual, yet very interesting behavior. For any specific zand spin parameter a, vr attains a negative value at aradius greater than a certain distance r = rb. The mag-nitude of rb decreases with the increase of ℓ, which indi-cates that the positive trait of vr is greater for lower z.The negative value of vr does not necessarily reflect anyunphysical behavior. If the outflow originates from anyparticular layer ℓh at a radius r ≥ rb, it then appears thatthe coupling between the disk and the outflow ceased toexist. It infers that along the layer ℓh, both accretionand outflow simultaneously occur up to the radial dis-tance rb, and beyond which the solution (with negativevr) reflects the truncation of the disk (along the specifiedlayer). Let us consider different layers of z. As we as-cend to layers from a lower to a higher z, the truncationof the disk-outflow region along these layers occurs at aradius smaller than that of the lower z. As accretion isthe source of the outflow, we can ostensibly conclude thatthe region from where the outflowing matter originates inthe disk shrinks as we go to higher latitudes. Note that rbcorresponds to zero vr. Therefore, we restrict our studyup to the boundary rb. With this insight, we simply com-pute rb along various layers of the disk and construct asurface in the r − z plane connecting all rb for differentlayers. The enclosed region bounded by this surface isdefined by the positivity of vr, where both the accretionand outflow simultaneously persist and are intrinsicallycoupled to each other. We attribute the outer surface ofthe region as disk-outflow surface (hsurf ) and above thislayer no inflow takes place. The arrows in the diagramshown in Fig. 1 reveal the direction of flow. The arrowexactly at rb points vertically upwards, which indicatesthat just at rb the flow pattern exclusively correspondsto vertical motion and any accretion flow ceased to ex-ist. However, more realistic disk-outflow surface couldbe visualized with a thick-solid line shown in Fig. 1.As in our system we have neglected the viscosity and

any radiative loss, and thus the flow is considered to bestrongly advective, presumably it is gas pressure dom-inated. Notwithstanding, various scattering processes

Fig. 1.— Nature and geometry of the disk-outflow coupledregion and the outflow surface. Discussed in detail in §3.1.

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

r

hsu

rf

Fig. 2.— Variation of disk-outflow surface with radial coordinatefor different spin of the black hole. Solid, dashed, dot-dashed anddotted curves are for a = 0, 0.5, 0.9, 0.998 respectively. γ = 1.5.

will indeed produce radiation in the system, whose con-tribution could be insignificant. Hence, γ ∼ 3/2 is rea-sonably an appropriate choice in our model 4. Note thatthis is in essence similar to the choice of the earlier au-thors who modeled gas dominated low mass advectiondominated accretion flows (e.g. Narayan & Yi 1994).We would also like to clarify that γ = 5/3 correspondsonly to the case of zero angular momentum. Figure 2shows the profiles of disk-outflow surface for various spinparameters a of the black hole. All the input parameterscorresponding to each value of a are listed in Table 1. Itis seen that the disk-outflow region and the peak of thesurface namely, Rjs, shifts to the vicinity of the blackhole for a higher spin.The marginally stable orbit in a Keplerian accretion

flow defines the inner edge of the disk (with zero torquecondition). However the orbit is more or less an arti-fact of the flow. For a transonic advective disk (as is

4 The relationship between γ and β, the ratio of gas pressure to

the total pressure, is given by β = 6γ−83(γ−1)

(GM09).

6

Table 1

Input parameters for various a

a rc λ

0.0 6.15 3.3

0.3 5.7 2.9

0.5 5.5 2.6

0.7 5.3 2.3

0.9 5.0 1.9

0.95 4.4 1.6

0.97 4.3 1.5

0.998 4.2 1.3

the present case), an apriori definition of inner edge issomewhat fuzzy, due to which we can pretend to choosethat the disk extends to the black hole horizon. In pres-ence of an outflow intrinsically coupled to the disk, thereis supposed to be an inner boundary beyond which anyoutflow and then jet will ceased to exist. We define thisinner boundary of the disk-outflow coupled region in ex-plaining the energetics of the flow. Also, with the in-crease of the spin of the black hole, it is seen that thedisk-outflow coupled region shrinks considerably, attain-ing a steeper nature. Literally speaking, the spin of theblack hole directly influences the nature and region of theoutflow. The tendency of the outflow region to get con-tracted to the inner radius suggests that the outflow ismore likely to exuberantly emanate from inner region ofthe disk for rapidly spinning black holes which have moregravitating power. Therefore, with the increase in spin,the disk-outflow region becomes more dense and moresusceptible to eject the matter with a greater power.

4. DYNAMICS AND NATURE OF THE FLOW

As the system is gas pressure dominated and stronglyadvective, it is more receptive to strong outflows. Figures3 and 4 describe the variation of flow parameters as func-tions of radial coordinate r along the disk-outflow surfacefor various spin of the black hole. With the increase ofthe spin of the black hole, vr increases at the inner re-gion of the disk. It is seen that along the disk-outflowsurface, vr becomes zero beyond a certain distance Rjs,which is also termed as zero vr surface illustrated in §3.1,which is explained it detail in the next section in orderto understand the outflow power. The zero vr surfaceextends upto more inner region of the disk with the in-crease of spin of the black hole, indicating the fact thatoutflows and then jets originate from more inner regionof the disk for rapidly spinning black holes. The increaseof cs with spin (Fig. 4) indicates that the temperatureof the disk-induced outflow is higher for rapidly spinningblack holes. It is found from Fig. 4 that the maximumtemperature of the accretion-induced outflow varies from∼ 1011 to 1012K corresponding to zero to maximal spinof the black hole. The truncation of the curves in theinner region symbolizes the inner boundary of the disk-outflow region as mentioned in §3.1.

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

r

v r

Fig. 3.— Variation of radial velocity with radial coordi-nate. Solid, dashed, dot-dashed and dotted curves are for a =0, 0.5, 0.9, 0.998 respectively. Other parameter γ = 1.5.

0 5 10 15 20 25 30 35 40 45 500.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

r

c s

Fig. 4.— Same as that of Fig. 3 but variation of sound speed.

7

5. ENERGETICS OF THE FLOW

Accretion by a black hole is the primary source of en-ergy of the mass outflow from the inner disk region. Theenergetics of the accretion-induced outflow are mainlyattributed to the mass outflow rate and the power ex-tracted by the outflow from the disk. The derivation ofthe mass outflow rate had been elaborated in G10. Wefollow the same procedure here, however including thespin information of the black hole. The mass outflowrate is given by

Mj(r) = −∫

4πrρ(hsurf )vz(hsurf ) dr + cj, (19)

where the constant cj is determined by an appropriateboundary condition. From the adiabatic conditions, ρcan be written in terms of cs which is already deter-mined as a function of r. The corresponding proportion-ality constant is determined at a radius, outside whichthe contribution to the mass outflow rate is negligible(vz = 0). However, the total mass accretion rate M(which is sum of the inflow rate and the outflow rate)can be obtained by integrating the continuity equationalong the radial and vertical directions. Hence, the con-stant is computed easily by supplying the values of vrand cs at that radius in the expression of M . As we havediscarded any possible small amount of the outflow atthat radius, the actual value of the constant, and hencethe outflow power, could be slightly over estimated.Mj(r) in Eqn. (19) refers to the rate at which the

vertical mass flux ejects from the disk-outflow surface.Figure 5 shows the variation of Mj profiles with the spinof the black hole. It is seen that with the increase ofspin of the black hole, Mj increases. All the profileshave been shown considering a supermassive black holeof mass ∼ 108M⊙ with a mass accretion rate at infinity,

M ∼ 10−2MEdd, where MEdd is the Eddington mass ac-cretion rate ∼ 1.44×1025gm/s. Low mass accretion rate(sub Eddington accretion flow) is in conformity with ourgas pressure dominated advective disk paradigm. It isfound that Mj increases with the increase of mass of theblack hole and mass accretion rate as well. The trunca-tion of the curves at an inner radius indicates the innerboundary of the disk-outflow coupled region, explainedin detail in the next paragraph.In computing the power extracted by the outflow from

the disk, we follow the same procedure as in G10. Thusthe power of the outflow is given by

Pj(r) =

∫

4πr

[(

v2

2+

γ

γ − 1

P

ρ+ φG

)

ρvz

]∣

∣

∣

∣

hsurf

dr,

(20)

which is the total power removed from the disk by theoutflow along the disk-outflow surface. In measuring thenet power of any astrophysical jet, it appears that thecomputed power Pj will then be the initial power of thejet. Figure 6 depicts the variation of the power Pj withr for various spin parameters of the black hole. If wemeticulously investigate the nature of the power profiles,we observe that when the radial distance is less thana certain value, Pj begins to decrease. The decreasingtrend of Pj infers that the characteristic flow is bounded(integrand of Eqn. (20) carrying the information of thevertical energy flux becomes negative).

0 10 20 30 40 500

1

2

3

x 1022

r

Mas

s ou

tflow

rat

e (g

m s

−1)

(a)

0 10 20 30 40 500

2

4

6

x 1022

r

Mas

s ou

tflow

rat

e (g

m s

−1)

(b)

0 10 20 30 40 500

2

4

6

8

x 1022

r

Mas

s ou

tflow

rat

e (g

m s

−1)

(c)

0 10 20 30 40 500

2

4

6

8

10

x 1022

r

Mas

s ou

tflow

rat

e (g

m s

−1)

(d)

Fig. 5.— Variation of mass outflow rate as a function of radialcoordinate, when (a) a = 0, (b) a = 0.5, (c) a = 0.9, (d) a = 0.998.Other parameter γ = 1.5.

0 10 20 30 40 500

1

2

3

x 1041

r

Pj (

erg

s−1

)

(a)

0 10 20 30 40 500

0.5

1

1.5

2

x 1042

r

Pj (

erg

s−1

)

(b)

0 10 20 30 40 500

2

4

6

x 1042

r

Pj (

erg

s−1

)

(c)

0 10 20 30 40 500

5

10

15

x 1042

r

Pj (

erg

s−1

)

(d)

Fig. 6.— Variation of outflow power as a function of radialcoordinate, when (a) a = 0, (b) a = 0.5, (c) a = 0.9, (d) a = 0.998.Other parameter γ = 1.5.

This occurs owing to the fact that in the extreme in-ner region of disk, due to its strong gravitating power thestarved black hole sucks all of the matter in its sphereof influence, even if there is any outflow emanating fromthe disk. We identify this inner transition radius as Rjt,beyond which no outflow occurs. We attribute Rjt as theinner boundary of the disk-outflow region. In all of theprevious profiles, the said truncation of the curves is as-cribed to Rjt. We describe the dynamical variables in ourstudy within the region between an outer boundary andRjt, and within this prescribed region strong outflows aremost plausible to originate, intrinsically coupled to thedisk. The power profile shows that with the increase in a,Pj increases as well as Rjt gradually shifts to the vicinityof the black hole, indicating the fact that outflow regionmoves further inward. We show the variation of total Pj

extracted from the disk-outflow region with the spin ofthe black hole in Fig. 7a. Considering a black hole ofmass ∼ 108M⊙, as seen in AGNs and quasars, accreting

with M ∼ 10−2MEdd, it is seen that Pj ∼ 1041 erg/s fora = 0. For a maximally spinning black hole (a = 0.998)with same parameters, the computed Pj ∼ 1043 erg/s.

8

0 0.2 0.4 0.6 0.8 10.01

0.5

1

1.5

2

a

Pj (

erg

s−1

)/10

43

(a)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

a

Rjt

(b)

0 0.2 0.4 0.6 0.8 110

15

20

25

30

a

Rjs

(c)

Fig. 7.— Variations of (a) net outflow power in the units of1043 erg s−1, (b) inner transition radius of the outflow, (c) peak ofdisk-outflow surface, as functions of spin of the black hole.

Thus, there is an increase of two orders of magnitudeof Pj with the increase of spin of the black hole from 0to 0.998. In an earlier work, Donea & Biermann (1996)showed that the power extracted by the outflow/jet fromthe disk increases with a. However, they did not com-pute the power extracted from the disk explicitly. Thenumerical simulations by De Villiers et al. (2005), ina different accretion paradigm including magnetic field,also concluded that the jet efficiency is possible to in-crease with the increase of spin of the black hole from 0to 0.998. Figure 7b shows the variation of Rjt with a. Indescribing the disk-outflow surface in §3.1, we have ar-ticulated the impact of spin on the disk-outflow coupledregion. The geometry of the surface for a particular a isidentified with a parameter Rjs (see Fig. 2). We showexclusively the variation of Rjs with the spin of the blackhole in Fig. 7c. which too reveals that the fast rotatingblack hole retracts the outflow region towards it; a signof a pure relativistic gravitation.

6. DISCUSSION

Blandford-Znajek process (Blandford & Znajek 1977)is still one of the most promising mechanisms to drivepowerful jets in AGNs and XRBs. Although the exactmechanism of formation of the jet in the vicinity of theblack holes is still elusive, and whatever might be thereason for the origin of jet, the said work is significantmostly due to two underlying reasons: (1) extreme grav-ity is indispensable to effuse strong unbounded flows inthe vertical direction from the inner region of the accre-tion disk, (2) the spin of the black holes, which is purelya relativistic effect, is essential to power strong outflowsand jets.In the present study, we have neither laid importance

to the nature of the outflow nor invoked any aspect forthe origin of outflows or jets. Indeed, the distinctive ordefinitive understanding of the formation of strong out-flows or jets is till unknown. Notwithstanding, the mostdistinguishable and obvious picture in this case is thatoutflows and jets observed in AGNs and XRBs can onlyoriginate in an accretion powered system, where the ac-cretion of matter around relativistic gravitating objectslike black holes acts as a source, and outflow and then jet

takes the form of one of the possible sinks (the other sinkis the central nucleus). The dynamics of the outflowingmatter should then be intrinsically coupled to the accre-tion dynamics macroscopically through the fundamentallaws of conservation (of matter, momentum and energy).The outflow is unbounded and the total energy just atthe base of the outflow should be positive. The accretionflow should be bounded in the vicinity of the central ob-ject as the central potential is attractive. However, thestrength of the unbounded flows in the form of jets isdistinctly proportional to the attractiveness of the cen-tral gravitational potential field. This paradox is wellmanifested in the observed universe, as relativistic jetsare more populated around extreme gravitating objectslike black holes. Also noticeably, length scale of jets in-creases from microquasars to quasars, which are suppos-edly harboring stellar mass and supermassive black holesrespectively.Thus in any theoretical modeling of the accretion and

outflow, it can be presumably argued that the mathe-matical equations governing the dynamics of the inflowand outflow should inherently be correlated and evolvedself-consistently without any ad hoc proposition, as ac-cretion and outflow should not be treated as dissimilarobjects. Second, the relativistic gravitational effect ofthe black hole should be incorporated, as the nature ofgravity is the cornerstone to both the accretion and theunbounded outflow. To capture this essential physics, inour present study to understand the connection betweendisk and outflow, we have incorporated the general rela-tivistic effect of the spinning black hole through a pseudo-Newtonian approach. Although pseudo-Newtonian for-malism is an approximate method to mimic the space-time geometry of the Kerr black hole, yet it captures theimportant salient features of the corresponding metric,and thus can be used to examine the nature of outflowsfrom the inner region of the disk.In §2 we have described the general disk-outflow cou-

pled hydrodynamic equations in the inviscid limit follow-ing GM09. The necessity to simplify our model equationsfor an inviscid flow is discussed in §2. Although GM09explored the 2.5-dimensional accretion-induced outflowfor a fully viscous system, they used a self-similar ap-proach in order to solve the necessary partial coupleddifferential equations. In addition, they neglected themost indispensable effect of relativistic gravitation or,precisely, the effect of spin of the black hole. In thepresent paper, we have solved the disk-outflow modelequations in a more general 2.5-dimensional paradigm,while trying to limit our assumptions to the least possi-ble extent theoretically. One of the important premiseswe have made is the relationship between vz and cs,which we have established empirically. We have not usedthe height integrated equations which are mostly valid inthe circumstances where the dynamical fluid parametersare likely to be independent of z. Without presumingthe fact that the outflow originates from the surface ofthe disk, as most of the authors do, we have logicallyconstructed a disk-outflow surface with properly definedboundary conditions. One of the most important com-putations we have done is to evaluate the mass outflowrate and the power of the outflow extracted from thedisk self-consistently in the inviscid limit, unlike the pre-vious works (e.g. Donea & Biermann 1996; Blandford

9

& Begelman 1999; Xie & Yuan 2008). We have foundthat the spin of the black hole plays a crucial role in de-termining the structure, dynamics and the energetics ofthe outflow coupled to the disk. With the increase ofthe spin, the outflow region extends further inward andthen the disk-outflow region shrinks and compresses (seeFig. 2). As a result, the outflow and then any plausiblejet is likely to eject out from an extreme inner regionof the flow around a rapidly spinning black hole with agreater efficiency. Note that the higher spin results inthe system to get more compressed with a greater out-flow power. Therefore, the efficiency of outflow and jetis directly related to the disk scale-height and hence thedisk-outflow surface. Previously, in a different context,Mckinney & Gammie (2004), while examining the elec-tromagnetic luminosity of a Kerr black hole, assumed theratio of the disk height to the radius (h/r) constant, ir-respective of the black hole spin. However, as seen in thepresent work, the constant h/r for different spin of theblack hole may not be an obvious choice.The power extracted by the outflow from the disk not

only depends directly on the mass of the black hole andthe initial mass accretion rate of the flow, but also onthe spin of the black hole. With our model, keeping theblack hole mass and the accretion rate the same, thepower of the outflow increases with the spin of the blackhole and the computed power differs in two orders ofmagnitude between non-rotating and maximally rotatingblack holes. We have restricted our study vertically upto the region where the inflow and the outflow are leastcoupled, i.e. the disk-outflow surface. Above this surfaceaccretion ceased to exist and probably the outflow getsdecoupled from the disk, accelerates and eventually formsrelativistic jet. The modeling of the astrophysical jet isaltogether a different issue and is beyond the scope of thepresent work. Nevertheless, it can be effectively arguedthat in modeling the dynamics of the jet, the computedoutflow power may serve as an initial power fed to thejet. Thus, the dynamics and the energetics of the jet willeventually be related to the black hole spin.According to the unification scenario, it is possible

to device a single astrophysical scenario, which canbroadly explain the observed multitude of different typesof AGNs (see e.g. Antonucci 1993; Urry & Padovani1995). Flat Spectrum Radio Quasars (FSRQs) andBL Lacs are probably the two most active types ofAGNs which are collectively referred to blazar. It is ob-served that BL Lacs are relatively low luminous than FS-RQs. Although both of these galaxies show high energyemissions, their spectral properties are different (Bhat-tacharya et al. 2009) which indicate that they are differ-ent source classes. According to the unification scheme,for FSRQs the line-of-sight is almost parallel to the jetand, hence, strong relativistic Doppler beaming of the jetemission produces highly variable and continuum domi-nated emission. As one moves away from the jet axis, thecentral continuum emission falls and the nucleus lookslike a FR-II galaxy. Similarly, BL Lacs are considered tobe a sub class of FR-I galaxies whose line-of-sight is al-most parallel to the jet axis. The present work suggeststhat the total mechanical power of an outflow proportion-ately increases with the spin of the central supermassiveblack hole; higher the spin, stronger is the outflow. It isreasonable to consider that strong outflow can lead to a

strong jet, and hence, one can expect to observe higherluminosity. Therefore, the work suggests that BL Lacsare slow rotators than FSRQs.For the theoretical formulation of the disk-outflow cou-

pling even with approximations, one needs to be verythoughtful for the proper foundation of the model whichessentially needs to solve hydrodynamic or magnetohy-drodynamic conservation equations in presence of stronggravity. The flow parameters vary in 2.5-dimension andare coupled to each other. Limited observational inputsput irremediable constraint on the boundary conditionsas well as the fundamental scaling parameters, governingthe coupled dynamics of the accretion and outflow. Theinadequacy of an effective mathematical tool to handlepartial coupled differential hydrodynamic equations forcompressible flow motivates us to invoke approximationsand assumptions. Despite of this fact, one needs to ex-plore the possibilities to examine the accretion-inducedoutflow. The question then arises: what are the valid as-sumptions and to which extent they can be considered?In the present work, we have addressed this question inthe following way.(1) As the extreme gravity is the most important aspectto effuse jet from the disk, we have incorporated its effectthrough a pseudo-general-relativistic potential.(2) The disk and outflow should not be treated as dissim-ilar objects, and hence their correlated-dynamics shouldbe essentially governed by the conservation laws. Theenergetics of the accretion-induced outflow would thenbe evaluated self-consistently as is shown in our work.(3) Any unbounded flow in the form of outflow is moreplausible to emanate from a hot, puffed up region ofthe accretion flow (may be low/hard state of the blackhole). Hence we have formulated our model in a 2.5-dimensional, strongly advective paradigm, surpassing thesimplicity of height integration.(4) We have approximated our system to an inviscidlimit, whose reason has been argued in §2. Nevertheless,in any future work, viscosity should be incorporated intothe flow to make it more realistic.(5) We do not account for the mechanism of formation ofstrong jets, like due to magnetic field or strong radiationpressure, as the definitive mechanism of the formationof jet is still unknown. Indeed, it is beyond the scopeto solve magnetohydrodynamic equations in the presentscenario. However, assuming that the outflow resides,and is coupled to the disk, the governing conservationequations of matter, momentum and energy should betreated accordingly.(6) Power of the outflow and jet is expected to directlydepend on the spin of the black hole. The spin, which isthe signature of general relativity, can make an impacton the nature of the observed AGN classes, and thus thespin of the black has been incorporated in our study. Asspin of the black hole has a direct impact on the flow pa-rameters, we obtain different outflow power for differentspin.What can be the most defined parameter for the

accretion powered system — mass accretion rate, massof the central star/black hole or the spin of the centralstar/black hole? Our analysis has shown that the powerextracted by the outflow from the disk proportionatelyincreases with the spin of the black hole. It infers thatif extreme gravity is essential to power the jet, then the

10

strength and the length-scale of the observed astrophys-ical jets directly depend on the spin of the black hole.If it is believed that the distant quasars harbor massiveblack holes of same mass scale and accrete matter withsimilar rate, perhaps, spin be the guiding parameter forthe different observed AGN classes. We can end withthe specific question: are the observed AGN classesastrophysical laboratories to measure the spin of thesupermassive black holes?

This work is supported by a project, Grant No.SR/S2HEP12/2007, funded by DST, India. The authorswould like to thank the referee for making importantcomments which helped to prepare the final version ofthe paper.

APPENDIX

Equation (15) consists of complicated terms, whereA,B and C are given by following equations

A = 4n+ 2, (A1)

B =

(

FGrc −λ2c

r3c

)(

1

2n+ 1

)

1

vrc+

vrcrc

(

1− 1

2n

)

− ı

z

(

z

rc

)µ

vr0c

(

1

n+ 1

)

+ 2vrcrc

+ 2ı

z

(

z

rc

)µ

vrc, (A2)

and

C =

(

FGrc−λ2c

r3c

)

1

2nrc+

v2rc2nr2c

− ı

2nz

(

z

rc

)µ

vr0cvrc

[

1

rc+

ı

z

(

z

rc

)µ]

+1

2n

(

∂

∂rFGr

∣

∣

∣

∣

c

+3λ2c

r4c

)

+v2rc2nr2c

+µı

2nz

zµ

rµ+1c

vr0cvrc

+ı

2nz

(

z

rc

)µ

vrc

[

2nvr0ccs0c

(−B0 +√

B20 − 4A0C0

2A0

)

+vr0crc

]

,

(A3)where,

A0 = 2 + 4n, (A4)

B0 =(

FGr0c−λ2c

r3c

)( 1

2n+1

) 1

vr0c+

vr0crc

(

1− 1

2n

)

+2vr0crc

,

(A5)

and

C0 =(

FGr0c −λ2c

r3c

) 1

2nrc+

v2r0cnr2c

+1

2n

( ∂

∂rFGr0

∣

∣

c+ 3

λ2c

r4c

)

.

(A6)

REFERENCES

Antonucci, R. 1993, ARA&A, 31, 473.Begelman, M.C., Blandford, R. D., & Rees, M.J. 1984, RvMP, 56,

255.Begelman, M.C., King, A.R., & Pringle, J.E. 2006, MNRAS, 370,

399.Bhattacharya, D., Sreekumar, P., & Mukherjee, R., 2009, RAA, 9,

85.Blandford, R. D., & Begelman, M.C. 1999, MNRAS, 303, 1.Blandford, R. D., & Payne, D.G. 1982, MNRAS, 199, 883.Blandford R. D., & Znajek R. L. 1977, MNRAS, 179, 433.Chakrabarti, S. K. 1996, ApJ, 464, 664.Contopoulos, J. 1995, ApJ, 450, 616.De Villiers, J., Jean, P., Hawley, J. F., Krolik, J. H., & Hirose, S.

2005, ApJ, 620. 878.Donea, A. C., & Biermann, P. L. 1996, A&A, 316, 43.Fabrika, S. 2004, ASPRv, 12, 1.Fender, R. P., Belloni, T. M., & Gallo, E. 2004, MNRAS, 355, 110.Ghosh, S., & Mukhopadhyay, B. 2007, ApJ, 667, 367.Ghosh, S., & Mukhopadhyay, B. 2009, RAA, 9, 157; GM09.Ghosh, S., Mukhopadhyay, B., Krishan, V., & Khan, M. 2010, New

Astron, 15, 83; G10.Hawley, J. F., & Krolik, J. H. 2006, ApJ, 641, 103.Homan, D. C. 2001, ApJ, 549, 840.Kording, E., Falcke, H., & Markoff, S. 2002, A&A, 382, 13.

Lyutikov, M. 2006, MNRAS, 369, 5.Margon, B., 1984, ARA&A, 22, 507.Mckinney, J. C., & Gammie, C. F. 2004, ApJ, 611, 977.Meier, D. L. 2002, NewAR, 46, 247.Meier, D. L., Shinji, K., & Uchida, Y. 2001, Sci, 291, 84.Migliari, S., & Fender, R. P. 2006, MNRAS, 366, 79.Mirabel, I. F. 2003, New Ast. Rev., 47, 471.Mirabel, I. F., Rodriguez, L. F. 1994, Nature, 371, 46.Mirabel, I. F., Rodriguez, L. F. 1998, Nature, 392, 673.Mukhopadhyay, B. 2003, ApJ, 586, 1268.Mukhopadhyay, B., & Ghosh, S. 2003, MNRAS, 342, 274.Mundt, R. 1985, Obs, 105, 224.Narayan, R., & Yi, I. 1994, ApJ, 428, 13.Okuda, T., Lipunova, G. V., & Molteni, D. 2009, MNRAS, 398,

1668.Ostriker E. C. 1997, ApJ, 486, 291.Shibata, K., & Uchida, Y. 1986, PASJ, 38, 631.Urry, P., & Padovani, P. 1995, PASP, 107, 803.Ustyugova G. V., Koldoba A. V., Romanova M. M., Chechetkin V.

M., & Lovelace R. V. E. 1999, ApJ, 516, 221.

Xie, F-G., & Yuan, F. 2008, ApJ, 681, 499.