Relativistic Radiative Flow in a Luminous Disk II

Transcription

1 PASJ: Publ. Astron. So. Japan 58, , 2006 Deember Astronomial Soiety of Japan. Relativisti Radiative Flow in a Luminous Disk II Jun FUKUE and Chizuru AKIZUKI Astronomial Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka fukue@.osaka-kyoiku.a.jp (Reeived 2006 July 15; aepted 2006 Otober 8) Abstrat Radiatively driven transfer flow perpendiular to a luminous disk is examined in the relativisti regime of (v/) 2, while taking into aount the gravity of the entral objet. The flow is assumed to be vertial, and the gas pressure as well as the magneti field are ignored. Using a veloity-dependent variable Eddington fator, we an solve the rigorous equations of the relativisti radiative flow aelerated up to relativisti speeds. For suffiiently luminous ases, the flow resembles the ase without gravity. For less-luminous or small initial radius ases, however, the flow veloity dereases due to gravity. Appliation to a superritial aretion disk with mass loss is briefly disussed. Key words: aretion, aretion disks astrophysial jets radiative transfer relativity 1. Introdution Mass outflow from a luminous disk is a lue to the formation mehanism of astrophysial jets and winds in the ative objets. In partiular, in a superritial aretion disk, the disk loal luminosity exeeds the Eddington one, and mass loss from a disk surfae driven by radiation pressure would take plae (see Kato et al for a review of aretion disks). So far, radiatively driven outflows from a luminous disk have been extensively studied by many researhers (Bisnovatyi- Kogan, Blinnikov 1977; Katz 1980; Ike 1980; Melia, Königl 1989; Misra, Melia 199; Tajima, Fukue 1996, 1998; Watarai, Fukue 1999; Hirai, Fukue 2001; Fukue et al. 2001; Orihara, Fukue 200), and by numerial simulations (Eggum et al. 1985, 1988; Okuda 2002; Ohsuga et al. 2005; Ohsuga 2006). In almost all of these studies, however, the luminous disk was treated as an external radiation soure, and radiation transfer in the flow was not solved. Radiation transfer in the disk, on the other hand, was investigated in relation to the struture of a stati disk atmosphere and the spetral energy distribution from the disk surfae (e.g., Meyer, Meyer-Hofmeister 1982; Cannizzo, Wheeler 1984; Shaviv, Wehrse 1986; Adam et al. 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 vertial movement and the mass loss were not onsidered. Reently, mass outflow as well as radiation transfer has been examined for the first time in the subrelativisti (Fukue 2005a, 2006a) and fully relativisti ases (Fukue 2005b). In the latter ase, it is pointed out some singular behavior inherent in relativisti radiative flow (e.g., Turolla, Nobili 1988; Turolla et al. 1995; Dullemond 1999). When the gaseous flow is radiatively aelerated up to the relativisti regime, the veloity gradient beomes very large in the diretion of the flow. As a result, the radiative diffusion may beome anisotropi in the omoving frame of the gas. Hene, in a flow that is aelerated from subrelativisti to relativisti regimes, the Eddington fator should be different from 1/, even in the optially thik diffusion limit. In order to avoid the singular behavior of suh a relativisti regime, for a plane-parallel ase, Fukue (2006b) proposed a veloity-dependent Eddington fator, whih depends on the flow veloity v, f (β)= 1+2β, (1) where β = v/. In Fukue (2006b) this form (1) was adopted as the simplest one among various forms, whih satisfy several neessary onditions to avoid a singularity. Physially speaking, this form (1) an be interpreted as follows. In the high-veloity regime, where the radiative diffusion may beome anisotropi in a omoving frame, the apparent optial depth, τ, would be on the order of 1+τ = 1 β. (2) That is, as the flow is aelerated and approahes the speed of light, the optial depth beomes zero (outward peaking). In this ase, form (1) an be read as f (τ)= +τ +τ, () whih reovers a similar form of a usual variable Eddington fator (see, e.g., Tamazawa et al. 1975). Hene, the appliability and auray of form (1) from a low-speed regime to a high-speed one would be similar to those of a variable Eddington fator from an optially thik regime to an optially thin one. Similarly, for a spherially symmetri ase, Akizuki and Fukue (2006) proposed a variable Eddington fator, that depends on the flow veloity, β,as wellasthe optial depth,τ: γ (1 + β) +τ f (τ,β)= γ (1 + β) +τ, (4) where γ (= 1/ 1 β 2 ) is the Lorentz fator. In Fukue (2006b), the plane-parallel ase was examined as an example, although the gravity of the entral objet was

2 1074 J. Fukue and C. Akizuki [Vol. 58, ignored for simpliity. In this paper, we thus onsider the radiatively driven vertial outflow moving photosphere in a luminous flat disk within the framework of radiation transfer in the relativisti regime using f (β), while taking into aount the gravity of the entral objet, although the gas pressure is ignored. In the next setion we desribe the basi equations in the vertial diretion. In setion we show our numerial examination of the radiative flow. In setion 4 we briefly apply the present model to the ase of a superritial aretion disk. The final setion is devoted to onluding remarks. 2. Basi Equations and Boundary Conditions Let us suppose a luminous flat disk, inside of whih gravitational or nulear energy is released via visous heating or other proesses. The radiation energy is transported in the vertial diretion, and the disk gas, itself, also moves in the vertial diretion due to the ation of radiation pressure. For simpliity, in the present paper, the radiation field is onsidered to be suffiiently intense that the gas pressure an be ignored: tenuous old normal plasmas in the super-eddington disk, old pair plasmas in the sub-eddington disk, or dusty plasmas in the sub-eddington disk. As for the order of the flow veloity v, we onsider the fully relativisti regime, where the terms are retained up to the seond order of (v/) Basi Equations Under these assumptions, the radiation hydrodynami equations for steady vertial (z) flows without rotation are desribed as follows (Kato et al. 1998; Fukue 2006b). The ontinuity equation is ρu = J (= onst.), (5) where ρ is the proper gas density, u the vertial four veloity, J the mass-loss rate per unit area, and the speed of light. The four-veloity u is related to the proper three-veloity v by u = γv/,whereγ is the Lorentz fator, γ = 1+u 2 = 1/ 1 (v/) 2. The equation of motion is 2 u du dz = GMz (R r g ) 2 R + κ abs + κ sa [ Fγ(1 + 2u 2 ) (E + P )γ 2 u ], (6) where M is the mass of the entral objet, R = r 2 + z 2, r being the radius, r g (= 2GM/ 2 ) is the Shwarzshild radius, κ abs and κ sa are the absorption and sattering opaities (gray), defined in the omoving frame, E the radiation energy density, F the radiative flux, and P the radiation pressure observed in the inertial frame. The first term in the square brakets on the right-hand side of equation (6) means the radiatively driven fore, whih is modified to the order of u 2, whereas the seond term is the radiation drag fore, whih is also modified, but roughly proportional to the veloity. As for the gravity, we adopt the pseudo-newtonian potential (Pazyński, Wiita 1980). When the gas pressure is ignored, the advetion terms of the energy equation are dropped (f. Kato et al. 1998), and heating is balaned with the radiative terms, 0=q + ρ ( j κ abs Eγ 2 κ abs P u 2 +2κ abs Fγu ), (7) where q + is the internal heating and j is the emissivity defined in the omoving frame. In this equation (7), the third and fourth terms on the right-hand side appear in the relativisti regime. For radiation fields, the zeroth-moment equation beomes df dz = ργ [ j κ abs E + κ sa (E + P )u 2 + κ abs Fu/γ κ sa F (1 + v 2 / 2 )γu ]. (8) The first-moment equation is dp dz = ργ [ u γ j κ absf + κ abs P u γ ] κ sa F (1 + 2u 2 )+κ sa (E + P )γu. (9) In order to lose moment equations for radiation fields, we need some losure relation. Instead of the usual Eddington approximation, we here adopt a veloity-dependent variable Eddington fator, f (β), P 0 = f (β)e 0, (10) in the omoving frame, where P 0 and E 0 are the quantities in the omoving frame. If we adopt this form (10) as the losure relation in the omoving frame, the transformed losure relation in the inertial frame is P (1 + u 2 fu 2 )=E (fγ 2 u 2 )+2Fγu(1 f ), (11) or equivalently, P (1 fβ 2 )=E (f β 2 )+2Fβ(1 f ). (12) As a form of the funtion f (β) we adopt the simplest one, f (β)= β, (1) for a plane-parallel geometry (Fukue 2006b; f. Akizuki and Fukue 2006 for a spherially symmetri geometry). Eliminating j and E with the help of equations (7) and (11), equations (6), (8), and (9) beome 2 u du dz = 2 r g z 2(R r g ) 2 R df dz = q + γ + κ abs + κ sa γ F (fγ2 + u 2 ) P (1 + f )γu fγ 2 u 2, (14) ρ(κ abs + κ sa )u F (fγ2 + u 2 ) P (1 + f )γu fγ 2 u 2, (15) dp dz = q + u ρ κ abs + κ sa γ F (fγ2 + u 2 ) P (1 + f )γu fγ 2 u 2. (16)

3 No. 6] Relativisti Radiative Flow in a Luminous Disk II 1075 Introduing the optial depth by dτ = (κ abs + κ sa )ρdz, (17) and using ontinuity equation (5), equations (14) (17) are rearranged as 2 J du dτ = κ abs + κ sa 2 r g z 2(R r g ) 2 R γ F (fγ2 + u 2 ) P (1 + f )γu fγ 2 u 2, (18) J dz dτ = u, (19) κ abs + κ sa df dτ = q + (κ abs + κ sa )ρ + u F (fγ2 + u 2 ) P (1 + f )γu fγ 2 u 2, (20) dp dτ = q + J (κ abs + κ sa )ρ + γ F (fγ2 + u 2 ) P (1 + f )γu fγ 2 u 2. (21) In this paper we assume that heating takes plae deep inside the disk at the midplane and in the atmosphere where q + =0. However, it is straightforward to onsider more general ases, where, e.g., the heating q + is proportional to the gas density, ρ (f. Fukue 2005a, b). We solved equations (18) (21) for appropriate boundary onditions at the moving surfae with a variable Eddington fator (1) Boundary Conditions As already pointed out in Fukue (2005a), the usual boundary onditions for the stati atmosphere annot be used for the present radiative flow, whih moves with a veloity on the order of the speed of light. When there is no motion in the gas ( stati photosphere ), the radiation field just above the surfae under the planeparallel approximation is easily obtained. Namely, just above the disk with surfae intensity I s, the radiation energy density, E s, the radiative flux, F s, and the radiation pressure, P s are (2/) πi s, πi s, and (2/) πi s, respetively, where the subsript s denotes the values at the disk surfae. However, the radiation field just above the surfae hanges when the gas itself does move upward ( moving photosphere ), sine the diretion and intensity of radiation hange due to relativisti aberration and Doppler effet (f. Kato et al. 1998; Fukue 2000). Let us suppose a situation where a flat infinite photosphere with surfae intensity I s in the omoving frame is not stati, but moving upward at a speed v s (= β s, and the orresponding Lorentz fator is γ s ). Then, just above the surfae, the radiation energy density, E s, the radiative flux, F s, and the radiation pressure, P s, measured in the inertial frame beome, respetively, γs 2 E s =2πI s γs 2 F s =2πI s +γ s u s + u 2 s, (22) +8γ s u s +u 2 s, (2) 6 γs 2 +γ s u s +u 2 s P s =2πI s, (24) where u s (= γ s v s /) is the flow four veloity at the surfae (Fukue 2005a). Thus, we impose the following boundary onditions: At the flow base (deep inside the atmosphere) with an arbitrary optial depth, τ 0, the flow veloity u is zero, the radiative flux is F 0 (whih is a measure of the strength of radiation field), and the radiation pressure is P 0 (whih onnets with the radiation pressure gradient and relates to the internal struture), where the subsript 0 denotes the values at the flow base. At the flow top ( surfae of the atmosphere) where the optial depth is zero, the radiation field should satisfy the values above a moving photosphere given by equations (22) (24): i.e., P s F s = 2+6β s +6βs 2 +8β s +βs 2, (25) where β s is a final speed at the disk surfae. Physially speaking, in the radiative flow starting from the flow base with an arbitrary optial depth, τ 0, for initial values of F 0 and P 0 at the flow base, the final value of the flow veloity, v s at the flow top an be obtained by solving basi equations. Furthermore, the mass-loss rate J is determined as an eigenvalue so as to satisfy the boundary ondition (2) at the flow top (f. Fukue 2005a in the subrelativisti regime). However, the permitted region for J is very tight, and it is diffiult to searh the value of J. Hene, in this paper, as a mathematially equivalent way, we fix the value of J,and searh for the value of P 0 so as to satisfy the boundary ondition (2).. Relativisti Radiative Flow under Gravity In this setion we show the relativisti radiative vertial flow in the luminous disk under the influene of gravity of the entral objet. In order to obtain the solution, as already stated, we numerially solved equations (18) (21), starting from τ = τ 0 at z = 0 with appropriate initial onditions for v, F,andP, down to τ = 0 so as to satisfy the boundary onditions (25) there. The parameters were the initial radius, r, on the disk, the initial optial depth, τ 0, whih relates to the disk surfae density, the initial flux F 0, whih is the measure of the strength of radiation field to gravity, and the initial radiation pressure P 0 at the disk base, whih onnets with the radiation pressure gradient in the vertial diretion and relates to the disk internal struture. The mass-loss rate J was determined as an eigenvalue of the boundary ondition at the flow top. Several examples of numerial alulations are shown in figures 1. The physial quantities were normalized in terms of the speed of light,, the Shwarzshild radius, r g,andthe Eddington luminosity, L E [= 4πGM/(κ abs + κ sa )]. The units of F, P,and 2 J are L E /(4πr 2 g). It should be noted that the solutions an be obtained for arbitrary optial depths, τ 0,

4 1076 J. Fukue and C. Akizuki [Vol. 58, Fig. 1. Flowveloity v (thik solid urve), flow height z (hain-dotted one), radiative flux F (dashed one), and radiation pressure P (dotted one), as a funtion of the optial depth τ. The parameters are r =, τ 0 =1,F 0 =1,andP 0 =1.2. The mass-loss rate is J = 1. The quantities are normalized in units of, r g,andl E /(4πrg 2 ). Fig. 2. Flow veloity v as a funtion of the optial depth τ. A thik solid urve is for the typial ase of r =,τ 0 =1,F 0 =1,andP 0 =1.2 (i.e., J = 1). The solid urve is for the ase of r =,τ 0 =1,F 0 =10, and P 0 =10.6(i.e.,J = 1), while the dashed one is for the ase of r =, τ 0 =1,F 0 =1,andP 0 =1.041 (i.e., J =0.005). The quantities are normalized in units of, r g,andl E /(4πrg 2 ). at the flow base (see Fukue 2006b, and setion 4). We here show, however, the ases of τ 0 = 1, where the veloity hange is remarkable. In figure 1 we show the flow veloity v (solid urve), the flow height z (hain-dotted one), the radiative flux F (dashed one), and the radiation pressure P (dotted one), as a funtion of the optial depth τ for r =,τ 0 =1,F 0 =1,andP 0 =1.2 (i.e., J =1). As the optial depth dereases from the flow base at the disk equator to the flow top at the disk surfae, the radiative flux slightly dereases while the flow veloity inreases; the radiative energy is onverted to the flow bulk motion in the vertial diretion. As is usually known, in a stati planeparallel atmosphere, under the radiative equilibrium with the gray approximation, the vertial flux, F, is onserved without any heating soure. In the present relativistially moving atmosphere, on the ontrary, the radiative flux, F, dereases via the Futerm, whih ats to drive the gas toward the vertial diretion. In the ase of figure 1, the initial flux (F 0 =1)is nearly the loal Eddington one, and therefore, the final flow veloity is mildly relativisti due to the effet of gravity of the entral objet. Other parameter dependenes are shown in figures 2 and. In figure 2 the flow veloity v is shown for several parameter set: The thik solid urve is for the typial ase of r =,τ 0 =1, F 0 =1,andP 0 =1.2 (i.e., J = 1). The solid urve is for the ase of r =,τ 0 =1,F 0 = 10, and P 0 =10.6 (i.e.,j = 1), while the dashed one is for the ase of r =,τ 0 =1,F 0 =1,andP 0 =1.041 (i.e., J =0.005). As is easily expeted, the flow veloity remarkably inreases when the initial flux, F 0, is large (a solid urve). When the initial flux is small, on the other hand, the flow veloity beomes small. Even for the same initial flux, when the mass-loss rate is small, the flow veloity remarkably inreases (a dashed urve). This is beause for a small mass-loss rate the density dereases, and therefore the vertial height, z, beomes large, ompared with the ase for a large mass-loss rate with the same optial depth. As a result, the gas is aelerated along the long distane to reah the highly relativisti regime. In figure the dependene on the initial radius r is shown: r = (a thik solid urve), r = 2 (a solid one), and r =1.5 (a dashed one). Other parameters are τ 0 =1,F 0 =1,andP 0 = 1.2 (i.e., J =1). For the same initial onditions, the flow veloity dereases as the initial radius dereases. This is just the effet of gravity of the entral objet. In the ase of r =1.5, the veloity is slightly deelerated toward the surfae due to gravity. 4. Relativisti Radiative Flow in the Critial Disk In this setion we apply the present model to the mass outflow in the luminous superritial aretion disks (f. Fukue 2006a for a subrelativisti ase). When the mass-aretion rate, Ṁ, in the disk around a entral objet of mass M highly exeeds the ritial rate, Ṁ rit,definedbyṁ rit L E / 2, the disk is believed to be in the superritial regime, and the disk luminosity exeeds the Eddington one. Suh a superritial aretion disk, a soalled slim disk, has been extensively studied, both numerially

5 No. 6] Relativisti Radiative Flow in a Luminous Disk II 1077 H r = = 1 ( 4 ln 1+ ṁ ), (28) 20 where is some numerial oeffiient determined by the similar proedure in Narayan and Yi (1994) for optiallythin advetion-dominated disks. The seond equality of this relation omes from a numerial alulation (Watarai et al. 2000). Although the mass loss was not onsidered in Watarai et al. (2000), we adopted this relation as some measure: when the normalized aretion rate ṁ is 1000, the oeffiient beomes Furthermore, in Fukue (2004), several alternatives are disussed, and some of them gives the physial quantities of the ritial aretion disk with mass loss as τ 0 = 16 6 r, (29) α r g F 0 = 12 6 α L E 4πr 2 r r g, (0) Fig.. Flow veloity, v, as a funtion of the optial depth, τ, for several initial radii r: r = (a thik solid urve), r = 2 (a solid one), and r =1.5 (a dashed one). The other parameters are τ 0 =1,F 0 =1,and P 0 =1.2 (i.e., J = 1). The quantities are normalized in units of, r g, and L E /(4πrg 2 ). and analytially (Abramowiz et al. 1988; Eggum et al. 1988; Szuszkiewiz et al. 1996; Beloborodov 1998; Watarai, Fukue 1999; Watarai et al. 2000; Mineshige et al. 2000; Fukue 2000; Kitabatake et al. 2002; Okuda 2002; Ohsuga et al. 2002, 200, 2005; Watarai, Mineshige 200; Fukue 2004; Ohsuga 2006). It was found that the optially-thik superritial disk is roughly expressed by a self-similar model (e.g., Watarai, Fukue 1999; Fukue 2000; Kitabatake et al. 2002; Fukue 2004). Exept for the ase of Fukue (2004), however, many of these analytial models did not onsider the mass outflow from the disk surfae. Hene, in this paper we adopt a model developed by Fukue (2004), as a bakground superritial disk model. In the ritial aretion disk onstruted by Fukue (2004), the mass-aretion rate is assumed to be regulated just at the ritial rate with the help of wind mass-loss. Outside some ritial radius, the disk is in a radiation-pressure dominated standard state, while inside the ritial radius the disk is in a ritial state, where the exess mass is expelled by wind and the aretion rate is kept to be just at the ritial rate at any radius. Here, the ritial radius is derived as r r = 9 σ T 16πm p Ṁinput 1.95ṁr g, (26) where Ṁ input is the aretion rate at the outer edge of the disk, and ṁ = Ṁ input /Ṁ rit. Outside r r, the aretion rate is onstant, while inside r r the aretion rate would vary as Ṁ(r)= 16πm p 9 r r = Ṁ input. (27) σ T r r In suh a ritial aretion disk, the disk thikness, H,is onial as P 0 = GM 1 τ 0 κ r 2 = 16 6GM ακ 1 r r 2, (1) r g where α is the visous parameter. In the present non-dimensional unit in terms of, r g,and the Eddington luminosity L E, these physial quantities are expressed as H = r, (2) τ 0 = 16 6 r, () α F 0 = 12 6 α 1, (4) r/2 P 0 = , (5) α r/2 where the symbol hat (say, ˆr) is dropped. Using these relations, we an solve the basi equations, and obtain numerial solutions at eah radius r. The example in the ase of ṁ = 1000 and α = 1 is shown in figures 4 and 5. In figure 4 we show the flow veloity v as a funtion of optial depth, τ, for several values of r. The quantities are normalized in units of and r g. The parameters of the ritial disk are ṁ = 1000 and α =1. As an be seen in figure 4, the flow veloity, v, varies selfsimilarly for different values of the radii. This may be refleted the initial self-similar models. As a result, in eah radius with different optial depth, the flow final speed is almost the same. In figure 5 we show several quantities for eah radius r: The disk height, H (dashed urve), and the optial depth, τ 0 (haindotted one), are from the ritial aretion disk model, while the height, z s, and veloity, v s (solid urves), at the flow top and the mass-loss rate, J (dashed one), are the results of numerial alulations. The quantities are normalized in units of and r g. The parameters of the ritial disk is ṁ = 1000 and α =1. As an be seen in figure 5, the flow height, z s, is enormously large. Hene, rigorously speaking, the present plane-parallel

6 1078 J. Fukue and C. Akizuki [Vol. 58, Fig. 4. Flow veloity, v, as a funtion of τ for several values of r. The values of r are, from left to right,, 4, 5, 6, and 7. The quantities are normalized in units of and r g. The parameters of the ritial disk is ṁ = 1000 and α =1. approximation violates in this appliation of ṁ = 1000, and two-dimensional numerial simulation should be needed in suh a ase. However, we an see several insights from the present ase. First, the final speed of the flow aelerated in the luminous ritial disk beomes suffiiently relativisti. In other words, relativisti jets an form in suh a luminous aretion disk. In addition, this final speed does not depend on the initial radius very muh, perhaps due to the initial self-similarity. Seond, on the ontrary to the final speed, the mass-loss rate per unit area inreases as the initial radius dereases; it is roughly approximated by J 6/r. On the other hand, the model mass-loss rate (Fukue 2004, 2006a) beomes J =1/r, that is qualitatively same, but quantitatively different from the present numerial values. Hene, the mass loss from the ritial disk may be onentrated in the inner region, although the true mass-loss rate annot be determined at the present simple state. This nature, however, is also onvenient for entrally onentrated jets. 5. Conluding Remarks In this paper we have examined the relativisti radiative transfer flow in a luminous disk in the relativisti regime of (v/) 2, while taking aount of the gravity of the entral objet. In suh a relativisti regime, we adopted the veloitydependent variable Eddington fator. The flow was assumed to be vertial, and the gas pressure was ignored for simpliity. The basi equations were numerially solved as a funtion of the optial depth, τ, and the flow veloity, v, the height, z, the radiative flux, F, and the radiation pressure, P, were obtained for a given radius, r, the initial optial depth, τ 0, and the initial onditions at the flow base (disk inside ), whereas the massloss rate, J, was determined as an eigenvalue of the boundary ondition at the flow top (disk surfae ). For suffiiently Fig. 5. Several quantities for eah radius r: The disk height, H (dashed urve), and the optial depth, τ, (hain-dotted one) are from the ritial model, while the height, z s, and veloity, v s (solid urves), at the flow top and the mass-loss rate, J (dashed one), are the results of the present numerial alulations. The quantities are normalized in units of and r g. The parameters of the ritial disk is ṁ = 1000 and α =1. luminous ases, the flow resembles the ase without gravity. For less-luminous ases, however, the flow veloity dereases. Appliation to the ritial aretion disk was also examined. If the disk thikness beomes so large, the present planeparallel approximation violates and other treatment, suh as numerial simulations, should be needed. The radiative flow investigated in the present paper must be a quite fundamental problem for aretion-disk physis and astrophysial jet formation, although there are many simplifiations at the present stage. For example, we have ignored the gas pressure. In general ases, where the gas pressure is onsidered, there usually appears soni points, and the flow is aelerated from subsoni to supersoni. In this paper we onsidered a purely vertial flow, and the ross setion of the flow is onstant. If the ross setion of the flow inreases along the flow, the flow properties, suh as the transoni nature, would be influened. Moreover, we did not onsider the rotation of the gas. In aretion disks around a blak hole, the gas usually rotates around the hole at relativisti speeds. In the viinity of the equator, the vertial flow approximation safely holds, sine the radial gravity is balaned with the entrifugal fore. When the flow is aelerated to be lifted up to a large height, the streamline would be urved outward, sine the entrifugal fore dominates. This would violate the vertial flow approximation. There remain many problems to be solved. The authors thank an anonymous referee for useful omments, whih greatly improved the original manusript. This work has been supported in part by a Grant-in-Aid for the Sientifi Researh ( J.F.) of the Ministry of Eduation, Culture, Sports, Siene and Tehnology.

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