aa r X i v : . [ a s t r o - ph ] N ov PASJ:
Publ. Astron. Soc. Japan , 1– ?? , c (cid:13) Relativistic Radiative Flow in a Luminous Disk II
Jun
Fukue and Chizuru
Akizuki
Astronomical Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka [email protected] (Received 0 0; accepted 0 0)
Abstract
Radiatively-driven transfer flow perpendicular to a luminous disk is examined in the relativistic regimeof ( v/c ) , taking into account the gravity of the central object. The flow is assumed to be vertical, andthe gas pressure as well as the magnetic field are ignored. Using a velocity-dependent variable Eddingtonfactor, we can solve the rigorous equations of the relativistic radiative flow accelerated up to the relativistic speed. For sufficiently luminous cases, the flow resembles the case without gravity. For less-luminous orsmall initial radius cases, however, the flow velocity decreases due to gravity. Application to a supercriticalaccretion disk with mass loss is briefly discussed. Key words: accretion, accretion disks — astrophysical jets — radiative transfer — relativity —
1. Introduction
Mass outflow from a luminous disk is a clue to the for-mation mechanism of astrophysical jets and winds in theactive objects. In particular, in a supercritical accretiondisk, the disk local luminosity exceeds the Eddington one,and mass loss from a disk surface driven by radiation pres-sure would take place (see Kato et al. 1998 for a reviewof accretion disks).So far, radiatively driven outflows from a luminousdisk have been extensively studied by many researchers(Bisnovatyi-Kogan, Blinnikov 1977; Katz 1980; Icke 1980;Melia, K¨onigl 1989; Misra, Melia 1993; Tajima, Fukue1996, 1998; Watarai, Fukue 1999; Hirai, Fukue 2001;Fukue et al. 2001; Orihara, Fukue 2003), and by numer-ical simulations (Eggum et al. 1985, 1988; Okuda 2002;Ohsuga et al. 2005; Ohsuga 2006). In almost all of thesestudies, however, the luminous disk was treated as an ex-ternal radiation source, and the radiation transfer in theflow was not solved.Radiation transfer in the disk, on the other hand, wasinvestigated in relation to the structure of a static diskatmosphere and the spectral energy distribution fromthe disk surface (e.g., Meyer, Meyer-Hofmeister 1982;Cannizzo, Wheeler 1984; Shaviv, Wehrse 1986; Adam etal. 1988; Hubeny 1990; Ross et al. 1992; Artemova et al.1996; Hubeny, Hubeny 1997, 1998; Hubeny et al. 2000,2001; Davis et al. 2005; Hui et al. 2005). In these studies,however, the vertical movement and the mass loss werenot considered.Recently, mass outflow as well as radiation transfer havebeen examined for the first time in the subrelativistic(Fukue 2005a, 2006a) and fully relativistic cases (Fukue2005b). In the latter case, it is pointed out some sin-gular behavior inherent in the relativistic radiative flow(e.g., Turolla, Nobili 1988; Turolla et al. 1995; Dullemond1999). When the gaseous flow is radiatively acceleratedup to the relativistic regime, the velocity gradient becomes very large in the direction of the flow. As a result, the ra-diative diffusion may become anisotropic in the comovingframe of the gas. Hence, in a flow that is accelerated fromsubrelativistic to relativistic regimes, the Eddington fac-tor should be different from 1 / velocity-dependent Eddington factor , which de-pends on the flow velocity v : f ( β ) = 1 + 2 β , (1)where β = v/c . In Fukue (2006b) this form (1) wasadopted as the simplest one among various forms, whichsatisfy several necessary conditions to avoid the singular-ity. Physically speaking, this form (1) can be interpretedas follows. In the high velocity regime, where the radiativediffusion may become anisotropic in the comoving frame,the ‘apparent’ optical depth τ would be of the order of1 + τ = 1 β . (2)That is, as the flow is accelerated and approaches to thespeed of light, the optical depth becomes zero (outward peaking ). In this case, the form (1) can be read as f ( τ ) = 3 + τ τ , (3)which recovers a similar form of a usual variableEddington factor (see, e.g., Tamazawa et al. 1975).Hence, the applicability and accuracy of the form (1) froma low speed regime to a high speed one would be similarto those of a variable Eddington factor from an opticallythick regime to an optically thin one.Similarly, for a spherically symmetric case, Akizuki andFukue (2006) proposed a variable Eddington factor, whichdepends on the flow velocity β as well as the optical depth τ : J. Fukue and C. Akizuki [Vol. , f ( τ, β ) = γ (1 + β ) + τγ (1 + β ) + 3 τ , (4)where γ (= 1 / p − β ) is the Lorentz factor.In Fukue (2006b), the plane-parallel case was examinedas an example, although the gravity of the central ob-ject was ignored for simplicity. In this paper, we thusconsider the radiatively driven vertical outflow – movingphotosphere – in a luminous flat disk within the frame-work of radiation transfer in the relativistic regime using f ( β ), while taking into account the gravity of the centralobject, although the gas pressure is ignored.In the next section we describe the basic equations inthe vertical direction. In section 3 we show our numericalexamination of the radiative flow. In section 4 we brieflyapply the present model to the case of a supercritial ac-cretion disk. The final section is devoted to concludingremarks.
2. Basic Equations and Boundary Conditions
Let us suppose a luminous flat disk, inside of whichgravitational or nuclear energy is released via viscous heat-ing or other processes. The radiation energy is trans-ported in the vertical direction, and the disk gas, itself,also moves in the vertical direction due to the action ofradiation pressure. For simplicity, in the present paper,the radiation field is considered to be sufficiently intensethat the gas pressure can be ignored: tenuous cold normalplasmas in the super-Eddington disk, cold pair plasmasin the sub-Eddington disk, or dusty plasmas in the sub-Eddington disk. As for the order of the flow velocity v ,we consider the fully relativistic regime, where the termsare retained up to the second order of ( v/c ). Under these assumptions, the radiation hydrodynamicequations for steady vertical ( z ) flows without rotation aredescribed as follows (Kato et al. 1998; Fukue 2006b).The continuity equation is ρcu = J (= const . ) , (5)where ρ is the proper gas density, u the vertical four ve-locity, J the mass-loss rate per unit area, and c the speedof light. The four velocity u is related to the proper threevelocity v by u = γv/c , where γ is the Lorentz factor, γ = √ u = 1 / p − ( v/c ) .The equation of motion is c u dudz = − GM z ( R − r g ) R + κ abs + κ sca c (cid:2) F γ (1 + 2 u ) − ( cE + cP ) γ u (cid:3) , (6)where M is the mass of the central object, R = √ r + z , r being the radius, r g (= 2 GM/c ) is the Schwarzschildradius, κ abs and κ sca are the absorption and scatteringopacities (gray), defined in the comoving frame, E theradiation energy density, F the radiative flux, and P theradiation pressure observed in the inertial frame. The first term in the square bracket on the right-hand side ofequation (6) means the radiatively-driven force, which ismodified to the order of u , whereas the second term is theradiation drag force, which is also modified, but roughlyproportional to the velocity. As for the gravity, we adoptthe pseudo-Newtonian potential (Paczy´nski, Wiita 1980).When the gas pressure is ignored, the advection termsof the energy equation are dropped (cf. Kato et al. 1998),and heating is balanced with the radiative terms,0 = q + − ρ (cid:0) j − κ abs cEγ − κ abs cP u + 2 κ abs F γu (cid:1) , (7)where q + is the internal heating and j is the emissivitydefined in the comoving frame. In this equation (7), thethird and fourth terms on the right-hand side appear inthe relativistic regime.For radiation fields, the zeroth-moment equation be-comes dFdz = ργ (cid:2) j − κ abs cE + κ sca ( cE + cP ) u + κ abs F u/γ − κ sca F (1 + v /c ) γu (cid:3) . (8)The first-moment equation is dPdz = ργc (cid:20) uγ j − κ abs F + κ abs cP uγ − κ sca F (1 + 2 u ) + κ sca ( cE + cP ) γu (cid:3) . (9)In order to close moment equations for radiation fields,we need some closure relation. Instead of the usualEddington approximation, we here adopt a velocity-dependent variable Eddington factor f ( β ), P = f ( β ) E (10)in the comoving frame, where P and E are the quantitiesin the comoving frame. If we adopt this form (10) as theclosure relation in the comoving frame, the transformedclosure relation in the inertial frame is cP (cid:0) u − f u (cid:1) = cE (cid:0) f γ − u (cid:1) + 2 F γu (1 − f ) , (11)or equivalently, cP (cid:0) − f β (cid:1) = cE (cid:0) f − β (cid:1) + 2 F β (1 − f ) . (12)As a form of the function f ( β ) we adopt the simplest one: f ( β ) = 13 + 23 β (13)for a plane-parallel geometry (Fukue 2006b; cf. Akizukiand Fukue 2006 for a spherically symmetric geometry).Eliminating j and E with the help of equations (7) and(11), equations (6), (8), and (9) become c u dudz = − c r g z R − r g ) R + κ abs + κ sca c γ F ( f γ + u ) − cP (1 + f ) γuf γ − u , (14) dFdz = q + γ − ρ ( κ abs + κ sca ) u F ( f γ + u ) − cP (1 + f ) γuf γ − u , (15)o. ] Relativistic Radiative Flow in a Luminous Disk II 3 dPdz = q + uc − ρ κ abs + κ sca c γ F ( f γ + u ) − cP (1 + f ) γuf γ − u . (16)Introducing the optical depth by dτ = − ( κ abs + κ sca ) ρdz, (17)and using continuity equation (5), equations (14)–(17) arerearranged as c J dudτ = cκ abs + κ sca c r g z R − r g ) R − γ F ( f γ + u ) − cP (1 + f ) γuf γ − u , (18) J dzdτ = − cuκ abs + κ sca , (19) dFdτ = − q + ( κ abs + κ sca ) ρ γ + u F ( f γ + u ) − cP (1 + f ) γuf γ − u , (20) c dPdτ = − q + ( κ abs + κ sca ) ρ u + γ F ( f γ + u ) − cP (1 + f ) γuf γ − u . (21)In this paper we assume that the heating takes placedeep inside the disk at the midplane and in the atmosphere q + = 0. However, it is straightfoward to consider moregeneral cases, where, e.g., the heating q + is proportionalto the gas density ρ (cf. Fukue 2005a, b).We solved equations (18)–(21) for appropriate bound-ary conditions at the moving surface with a variableEddington factor (13). As already pointed out in Fukue (2005a), the usualboundary conditions for the static atmosphere cannot beused for the present radiative flow, which moves with ve-locity at the order of the speed of light.When there is no motion in the gas (“static photo-sphere”), the radiation field just above the surface un-der the plane-parallel approximation is easily obtained.Namely, just above the disk with surface intensity I s , theradiation energy density E s , the radiative flux F s , and theradiation pressure P s are (2 /c ) πI s , πI s , and (2 / c ) πI s ,respectively, where the subscript s denotes the values atthe disk surface. However, the radiation field just abovethe surface changes when the gas itself does move upward(“moving photosphere”), since the direciton and inten-sity of radiation change due to relativistic aberration andDoppler effect (cf. Kato et al. 1998; Fukue 2000).Let us suppose a situation that a flat infinite photo-sphere with surface intensity I s in the comoving frame isnot static, but moving upward at a speed v s (= cβ s , andthe corresponding Lorentz factor is γ s ). Then, just abovethe surface, the radiation energy density E s , the radiativeflux F s , and the radiation pressure P s measured in the inertial frame become, respectively, cE s = 2 πI s γ + 3 γ s u s + u , (22) F s = 2 πI s γ + 8 γ s u s + 3 u , (23) cP s = 2 πI s γ + 3 γ s u s + 3 u , (24)where u s (= γ s v s /c ) is the flow four velocity at the surface(Fukue 2005a).Thus, we impose the following boundary conditions: Atthe flow base (deep “inside” the atmosphere) with an ar-bitrary optical depth τ , the flow velocity u is zero, theradiative flux is F (which is a measure of the strength ofradiation field), and the radiation pressure is P (whichconnects with the radiation pressure gradient and relatesto the internal structure), where the subscript 0 denotesthe values at the flow base. At the flow top (“surface”of the atmosphere) where the optical depth is zero, theradiation field should satisfy the values above a movingphotosphere given by equations (22)–(24): i.e., cP s F s = 2 + 6 β s + 6 β β s + 3 β , (25)where β s is a final speed at the disk surface.Physically speaking, in the radiative flow starting fromthe flow base with an arbitrary optical depth τ , for initialvalues of F and P at the flow base, the final value ofthe flow velocity v s at the flow top can be obtained bysolving basic equations. Furthermore, the mass-loss rate J is determined as an eigenvalue so as to satisfy the bondarycondition (23) at the flow top (cf. Fukue 2005a in thesubrelativistic regime).However, the permitted region for J is very tight, and itis difficult to search the value of J . Hence, in this paper,as a mathematically equivalent way, we fix the value of J ,and search the value of P so as to satisfy the boundarycondition (23).
3. Relativistic Radiative Flow under Gravity
In this section we show the relativistic radiative verticalflow in the luminous disk under the influence of gravityof the central object. In order to obtain the solution,as already stated, we numerically solve equations (18)–(21), starting from τ = τ at z = 0 with appropriate initialconditions for v , F , and P , down to τ = 0 so as to satisfythe boundary conditions (25) there. The parameters arethe initial radius r on the disk, the initial optical depth τ , which relates to the disk surface density, the initialflux F , which is the measure of the strength of radiationfield to gravity, and the initial radiation pressure P atthe disk base, which connects with the radiation pressuregradient in the vertical direction and relates to the diskinternal structure. The mass-loss rate J is determined asan eigenvalue of the boundary condition at the flow top.Several examples of numerical calculations are shown infigures 1–3. Physical quantities are normalized in terms of J. Fukue and C. Akizuki [Vol. , t v FP z Fig. 1.
Flow velocity v (thick solid curve), flow height z (chain-dotted one), radiative flux F (dashed one), and radia-tion pressure P (dotted one), as a function of the optical depth τ . The parameters are r = 3, τ = 1, F = 1, and P = 1 . J = 1. The quantities are normalized inunits of c , r g , and L E / (4 πr ). the speed of light c , the Schwarzschild radius r g , and theEddington luminosity L E [= 4 πcGM/ ( κ abs + κ sca )]. Theunits of F , cP , and c J is L E / (4 πr ). It should be notedthat the solutions can be obtained for arbitrary opticaldepths τ at the flow base (see Fukue 2006b, and section4). We here show, however, the cases of τ = 1, where thevelocity change is remarkable.In figure 1 we show the flow velocity v (solid curve),the flow height z (chain-dotted one), the radiative flux F (dashed one), and the radiation pressure P (dotted one),as a function of the optical depth τ for r = 3, τ = 1, F = 1,and P = 1 .
23 (i.e., J = 1).As the optical depth decreases from the flow base atthe disk equator to the flow top at the disk surface, theradiative flux slightly decreases while the flow velocity in-creases; the radiative energy is converted to the flow bulkmotion in the vertical direction. As usually known, ina static plane-parallel atomosphere, under the radiativeequilibrium with gray approximation, the vertical flux F is conserved without heating source. In the present rela-tivistically moving atmosphere , on the contrary, the radia-tive flux F decreases via F u term, which acts to drive thegas toward the vertical direction. In the case of figure 1,the initial flux ( F = 1) is nearly the local Eddington one,and therefore, the final flow velocity is mildly relativisticdue to the effect of gravity of the central object. Otherparameter dependences are shown in figures 2 and 3.In figure 2 the flow velocity v are shown for severalparameter set: A thick solid curve is for the typical caseof r = 3, τ = 1, F = 1, and P = 1 .
23 (i.e., J = 1). Asolid curve is for the case of r = 3, τ = 1, F = 10, and P = 10 . J = 1), while a dashed one is for the caseof r = 3, τ = 1, F = 1, and P = 1 .
041 (i.e., J = 0 . F is large (a solid curve).When the initial flux is small, on the other hand, the flowvelocity becomes small.Even for the same initial flux, when the mass-loss rate t v Fig. 2.
Flow velocity v as a function of the optical depth τ . A thick solid curve is for the typical case of r = 3, τ = 1, F = 1, and P = 1 .
23 (i.e., J = 1). A solid curve is for thecase of r = 3, τ = 1, F = 10, and P = 10 . J = 1),while a dashed one is for the case of r = 3, τ = 1, F = 1, and P = 1 .
041 (i.e., J = 0 . c , r g , and L E / (4 πr ). t v Fig. 3.
Flow velocity v as a function of the optical depth τ for several initial radii r : r = 3 (a thick solid curve), r = 2 (asolid one), and r = 1 . τ = 1, F = 1, and P = 1 .
23 (i.e., J = 1). The quantities arenormalized in units of c , r g , and L E / (4 πr ). is small, the flow velocity remarkably increases (a dashedcurve). This is because for a small mass-loss rate thedensity decreases, and therefore, the vertical height z be-comes large, compared with the case for a large mass-lossrate with the same optical depth. As a result, the gasis accelerated along the long distance to reach the highlyrelativistic regime.In figure 3 the dependence on the initial radius r isshown: r = 3 (a thick solid curve), r = 2 (a solid one),and r = 1 . τ = 1, F = 1, and P = 1 .
23 (i.e., J = 1).For the same initial conditions, the flow velocity de-creases as the initial radius decreases. This is just the ef-fect of gravity of the central object. In the case of r = 1 . decelarated toward the surface dueto gravity.o. ] Relativistic Radiative Flow in a Luminous Disk II 5
4. Relativistic Radiative Flow in the CriticalDisk
In this section we apply the present model to the massoutflow in the luminous supercritical accretion disks (cf.Fukue 2006a for a subrelativistic case).When the mass-accretion rate ˙ M in the disk arounda central object of mass M highly exceeds the criticalrate ˙ M crit , defined by ˙ M crit ≡ L E /c , the disk is believedto be in the supercritial regime, and the disk luminos-ity exceeds the Eddington one. Such a supercritical ac-cretion disk, a so-called slim disk, has been extensivelystudied, both numerically and analytically (Abramowiczet al. 1988; Eggum et al. 1988; Szuszkiewicz et al. 1996;Beloborodov 1998; Watarai, Fukue 1999; Watarai et al.2000; Mineshige et al. 2000; Fukue 2000; Kitabatake etal. 2002; Okuda 2002; Ohsuga et al. 2002, 2003, 2005;Watarai, Mineshige 2003; Fukue 2004; Ohsuga 2006). Itwas found that the optically-thick supercritical disk isroughly expressed by a self-similar model (e.g., Watarai,Fukue 1999; Fukue 2000; Kitabatake et al. 2002; Fukue2004). Except for the case of Fukue (2004), however, manyof these analytical models did not consider the mass out-flow from the disk surface. Hence, in this paper we adoptthe model developted by Fukue (2004), as a backgroundsupercritical disk model.In the critical accretion disk constructed by Fukue(2004), the mass-accretion rate is assumed to be regulatedjust at the critical rate with the help of wind mass-loss.Outside some critical radius, the disk is in a radiation-pressure dominated standard state, while inside the criti-cal radius the disk is in a critical state, where the excessmass is expelled by wind and the accretion rate is kept tobe just at the critical rate at any radius. Here, the criticalradius is derived as r cr = 9 √ σ T πm p c ˙ M input ∼ .
95 ˙ mr g , (26)where ˙ M input is the accretion rate at the outer edge of thedisk, and ˙ m = ˙ M input / ˙ M crit . Outside r cr , the accretionrate is constant, while, inside r cr the accretion rate wouldvary as˙ M ( r ) = 16 πcm p √ σ T r = ˙ M input rr cr . (27)In such a critical accretion disk, the disk thickness H isconical as Hr = √ c = 14 ln (cid:18) m (cid:19) , (28)where c is some numerical coefficient determined by thesimilar procedure in Narayan and Yi (1994) for optically-thin advection-dominated disks. The second equalityof this relation comes from the numerical calculation(Watarai et al. 2000). Although the mass loss was notconsidered in Watarai et al. (2000), we adopted this re-lation as some measure: when the normalized accretionrate ˙ m is 1000, the coefficient √ c becomes 0.983. t v Fig. 4.
Flow velocity v as a function of τ for several valuesof r . The values of r are, from left to right, 3, 4, 5, 6, and7. The quantities are normalized in units of c and r g . Theparameters of the critical disk is ˙ m = 1000 and α = 1. Furthermore, in Fukue (2004), several alternatives arediscussed, and some of them gives the physical quantitiesof the critical accretion disk with mass loss as τ = 16 √ α r rr g , (29) F = 12 √ α √ c L E πr r rr g , (30) cP = cGMκ √ c τ r = 16 √ cGMακ √ c r r rr g , (31)where α is the viscous parameter.In the present non-dimensional unit in terms of c , r g ,and the Eddington luminosity L E , these physical quanti-ties are expressed as H = √ c r (32) τ = 16 √ α √ r, (33) F = 12 √ α √ c r / , (34) P = 16 √ α √ c r / , (35)where the symbol “hat” (say, ˆ r ) is dropped.Using these relations, we can solve the basic equations,and obtain numerical solutions at each radius r . The ex-ample in the case of ˙ m = 1000 and α = 1 is shown in figures4 and 5.In figure 4 we show the flow velocity v as a function ofoptical depth τ for several values of r . The quantities arenormalized in units of c and r g . The parameters of thecritical disk is ˙ m = 1000 and α = 1.As can be seen in figure 4, the flow velocity v varies self-similarly for different values of radii. This may be reflectedthe initial self-similar models. As a result, in each radiuswith different optical depth, the flow final speed is almostsame.In figure 5 we show several quantities for each radius r :The disk height H (dashed curve) and the optical depth J. Fukue and C. Akizuki [Vol. , rv H z t /10 J /10 s s Fig. 5.
Several quantities for each radius r : The disk height H (dashed curve) and the optical depth τ (chain-dotted one)are from the critical model, while the height z s and velocity v s (solid curves) at the flow top and the mass-loss rate J (dashedone) are the results of the present numerical calculations. Thequantities are normalized in units of c and r g . The parametersof the critical disk is ˙ m = 1000 and α = 1. τ (chain-dotted one) are from the critical accretion diskmodel, while the height z s and velocity v s (solid curves)at the flow top and the mass-loss rate J (dashed one) arethe results of numerical calculations. The quantities arenormalized in units of c and r g . The parameters of thecritical disk is ˙ m = 1000 and α = 1.As can be seen in figure 5, the flow height z s is enor-mously large. Hence, rigorously speaking, the presentplane-parallel approximation violates in this applicationof ˙ m = 1000, and two-dimensional numerical simulationshould be needed in such a case. However, we can seeseveral insights from the present case.First, the final speed of the flow accelerated in the lu-minous critical disk becomes sufficiently relativistic. Inother words, relativistic jets can form in such a luminousaccretion disk. In addition, this final speed does not de-pend on the initial radius so much due maybe to the initialself-similarity. Second, on contrary to the final speed, themass-loss rate per unit area increases as the initial radiusdecreases; it is roughly approximated by J ∼ /r . On theother hand, the model mass-loss rate (Fukue 2004, 2006a)becomes J = 1 /r , that is qualitatively same, but quantita-tively different from the present numerical values. Hence,the mass loss from the critical disk may be concentrated inthe inner region, although the true mass-loss rate cannotbe determined at the present simple state. This nature,however, is also convenient for centrally concentrated jets.
5. Concluding Remarks
In this paper we have examined the relativistic radia-tive transfer flow in a luminous disk in the relativisticregime of ( v/c ) , taking account of gravity of the cen-tral object. In such a relativistic regime, we adopt thevelocity-dependent variable Eddington factor. The flow isassumed to be vertical, and the gas pressure is ignored forsimplicity. The basic equations are numerically solved asa function of the optical depth τ , and the flow velocity v , the height z , the radiative flux F , and the radiation pres-sure P are obtained for a given radius r , the initial opticaldepth τ , and the initial coditions at the flow base (disk“inside”), whereas the mass-loss rate J is determined as aneigenvalue of the boundary condition at the flow top (disk“surface”). For sufficiently luminous cases, the flow re-sembles the case without gravity. For less-luminous cases,however, the flow velocity decreases.Application to the critical accretion disk was also exam-ined. If the disk thickness becomes so large, the presentplane-parallel approximation violates and other treat-ment, such as numerical simulations, should be needed.The radiative flow investigaed in the present paper mustbe a quite fundamental problem for accretion-disk physicsand astrophysical jet formation, although there are manysimplifications at the present stage. For example, we haveignored the gas pressure. In general cases, where the gaspressure is considered, there usually appears sonic points,and the flow is accelerated from subsonic to supersonic.In this paper we consider a purely vertical flow, and thecross section of the flow is constant. If the cross section ofthe flow increases along the flow, the flow properties suchas a transonic nature would be influenced. Moreover, wedo not consider the rotation of the gas. In accretion disksaround a black hole, the gas usually rotates around thehole at the relativistic speed. In the vicinity of the equa-tor, the vertical flow approximation safely holds, since theradial gravity is balanced with the centrifugal force. Whenthe flow is accelerated to be lift up the large height, thestreamline would be curved outward, since the centrifu-gal force dominates. This would violate the vetical flowapproximation.There remain many problems to be solved.The authors thank an anonymous referee for useful com-ments, which greatly improved the original manuscript.This work has been supported in part by a Grant-in-Aidfor the Scientific Research (18540240 J.F.) of the Ministryof Education, Culture, Sports, Science and Technology. References
Abramowicz, M. A., Czerny, B., Lasota, J. P., & Szuszkiewicz,E. 1988, ApJ, 332, 646Adam, J., St¨orzer, H., Shaviv, G., & Wehrse, R. 1988, A&A,193, L1Akizuki, C., & Fukue, J. 2006, PASJ submittedArtemova, I. V., Bisnovatyi-Kogan, G. S., Bj¨ornsson, G., &Novikov, I. D. 1996, ApJ, 456, 119Beloborodov, A. M. 1998, MNRAS, 297, 739Bisnovatyi-Kogan, G. S., & Blinnikov, S. I. 1977, A&A, 59,111Cannizzo, J. K., & Wheeler, J. C. 1984, ApJS, 55, 367Davis, S. W., Blaes, O. M., Hubeny, I., & Turner, N. J. 2005,ApJ, 621, 372Dullemond, C. P. 1999, A&A, 343, 1030Eggum, G. E., Coroniti, F. V., & Katz, J. I. 1985, ApJ, 298,L41Eggum, G. E., Coroniti, F. V., & Katz, J. I. 1988, ApJ, 330,142 o. ] Relativistic Radiative Flow in a Luminous Disk II 7o. ] Relativistic Radiative Flow in a Luminous Disk II 7