aa r X i v : . [ a s t r o - ph ] N ov PASJ:
Publ. Astron. Soc. Japan , 1– ?? , c (cid:13) Radiative Transfer in Accretion Disk Winds
Jun
Fukue
Astronomical Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka [email protected] (Received 0 0; accepted 0 0)
Abstract
Radiative transfer equation in an accretion disk wind is examined analytically and numerically underthe plane-parallel approximation in the subrelativistic regime of ( v/c ) , where v is the wind vertical velocity.Emergent intensity is analytically obtained for the case of a large optical depth, where the flow speed andthe source function are almost constant. The usual limb-darkening effect, which depends on the directioncosine at the zero-optical depth surface, does not appear, since the source function is constant. Becauseof the vertical motion of winds, however, the emergent intensity exhibits the velocity-dependent limb-darkening effect, which comes from the Doppler and aberration effects. Radiative moments and emergentintensity are also numerically obtained. When the flow speed is small ( v ≤ . c ), the radiative structureresembles to that of the static atmosphere, where the source function is proportional to the optical depth,and the usual limb-darkening effect exists. When the flow speed becomes large, on the other hand, theflow speed attains the constant terminal one, and the velocity-dependent limb-darkening effect appears.We thus carefully treat and estimate the wind luminosity and limb-darkening effect, when we observe anaccretion disk wind. Key words: accretion, accretion disks — galaxies: active — radiative transfer — relativity — X-rays:stars
1. Introduction
Accretion disks are now widely believed to be energysources in various active phenomena in the universe:in protoplanetary nebulae around young stellar objects(YSOs), in cataclysmic variables (CVs) and supersoft X-ray sources (SSXSs), in galactic X-ray binaries and mi-croquasars ( µ QSOs), and in active galaxies (ANGs) andquasars (QSOs). Accretion-disk models have been ex-tensively studies during these three decades (see Katoet al. 1998 for a review). Besides the traditional stan-dard disk by Shakura and Sunayev (1973), new type disks,such as advection-dominated accretion flows (ADAF) orradiatively-inefficient accretion flows (RIAF) for the verysmall mass-accretion rate (e.g., Narayan, Yi 1994), andsupercritical accretion disks or so-called slim disks for thevery large mass-accretion rate (e.g., Abramowicz et al.1988).Accretion disk winds have been also extensively exam-ined in relation to astrophysical jets and outflows: in bipo-lar outflows from YSOs, in mass outflows from CVs andSSXSs, in relativistic jets from µ QSOs, AGNs, QSOs, andin gamma-ray bursts (GRBs). In particular, intense radi-ation fields of luminous supercritical accretion disks maybe responsible for relativistic jets from super-Eddingtonsources, such as luminous µ QSOs, GRS 1915+105 andSS 433, luminous QSOs, 3C 273, and energetic GRBs (see,e.g., Fukue 2004 for references).In such circumstances, radiative transfer in accretiondisk winds as well as accretion disks becomes more andmore important. Radiative transfer in the standard disk has been inves-tigated in relation to the structure of a static disk atmo-sphere and the spectral energy distribution from the disksurface (e.g., Meyer, Meyer-Hofmeister 1982; Cannizzo,Wheeler 1984). Furthermore, gray and non-gray mod-els of accretion disks were constructed under numericaltreatments (Kˇriˇz and Hubeny 1986; Shaviv and Wehrse1986; Adam et al. 1988; Mineshige, Wood 1990; Ross etal. 1992; Shimura and Takahara 1993; Hubeny, Hubeny1997, 1998; Hubeny et al. 2000, 2001; Davis et al. 2005;Hui et al. 2005) and under analytical ones (Hubeny 1990;Artemova et al. 1996; Fukue, Akizuki 2006a).Radiative transfer in the accretion disk wind, on theother hand, has not been well considered both in the non-relativistic and relativistic regimes. Recently, radiativetransfer in a moving disk atmosphere was firstly investi-gated in the subrelativistic regime (Fukue 2005a, 2006a),and in the relativistic regime (Fukue 2005b, 2006b; Fukue,Akizuki 2006b). In contrast to the static atmosphere, inthe moving atmosphere the boundary condition at thesurface of zero optical depth should be modified (Fukue2005a, b). Moreover, the usual Eddington approximationviolates in the highly relativistic flow (Fukue 2005b; seealso Turolla, Nobili 1988; Nobili et al. 1991; Turolla etal. 1995; Dullemond 1999), and the velocity-dependentvariable Eddington factor was proposed (Fukue 2006b fora plane-parallel case; Akizuki, Fukue 2007 for a sphericalcase).Radiation hydrodynamical (RHD) simulations werealso performed for radiation-dominated supercritical diskswith winds by several researchers (Eggum et al. 1985, J. Fukue [Vol. ,1988; Okuda et al. 1997, 2005; Okuda, Fujita 2000; Okuda2002; Ohsuga et al. 2005; Ohsuga 2006). In these currentstudies of RHD simulations for disks and winds, they weredone in the subrelativistic regime up to the order of ( v/c ) ,using the moment formalism and the flux-limited diffusion(FLD) approximation (Levermore, Pomraning 1981). Theflux-limited diffusion method provides good approxima-tions to the exact solutions but only if they are derivedfrom transfer equations in which terms of the order of( v/c ) or higher have been retained (Yin, Miller 1995).Radiative transfer problems on accretion disks andwinds are not well understood yet, in particular for therelativistic cases. Hence, in order clarify the physics, inaddition to RHD simulations, we must treat the simplifiedproblem in the analytical way.In this paper, we thus examine radiative transfer inthe accretion disk wind, which is assumed to blow offfrom the luminous disk in the vertical direction (plane-parallel approximation), and analytically and numericallyobtain the flow solutions for the case without internalheating. In the previous studies, radiative flows in thevertically moving atmosphere were solved in the subrela-tivistic regime (Fukue 2005a, 2006a), and in the relativis-tic regime (Fukue 2005b, 2006b; Fukue, Akizuki 2006b),where only radiative moments were obtained. In thepresent paper, we further obtain the radiation intensityas well as radiative moments.In the next section we describe the basic equations. Insection 3, we show analytical solutions, while we presentnumerical solutions in section 4. The final section is de-voted to concluding remarks.
2. Basic Equations
Let us suppose a luminous flat disk, deep inside 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 as a disk wind dueto the action of radiation pressure (i.e., plane-parallel ap-proximation). For simplicity, in the present paper, theradiation field is considered to be sufficiently intense thatboth the gravitational field of, e.g., the central object andthe gas pressure can be ignored. We also assume the grayapproximation, where the opacities do not depend on thefrequency. As for the order of the flow velocity v , we con-sider the subrelativistic regime, where the terms of thefirst order of ( v/c ) are retained, in order to take accountof radiation drag.The radiative transfer equations are given in severalliteratures (Chandrasekhar 1960; Mihalas 1970; Rybicki,Lightman 1979; Mihalas, Mihalas 1984; Shu 1991; Katoet al. 1998). For the plane-parallel geometry in the ver-tical direction ( z ), the radiation hydrodynamic equationsare described as follows (Kato et al. 1998). It should benoted that the basic equations below are the same as thosegiven in Fukue (2005a), except for the transfer equationfor the radiation intensity.For matter, the continuity equation is ρv = J (= const . ) , (1)where ρ is the gas density, v the vertical velocity, and J the mass-loss rate per unit area. The equation of motionis v dvdz = κ abs + κ sca c [ F − ( E + P ) v ] , (2)where κ abs and κ sca are the absorption and scatteringopacities (gray), which are defined in the comoving (fluid)frame, E the radiation energy density, F the radiativeflux, and P the radiation pressure in the vertical direc-tion, which are measured in the fixed (laboratory) frame.In a gas-pressureless approximation, the energy equationis reduced to0 = q + − ρ (cid:18) j − cκ abs E + κ abs F vc (cid:19) , (3)where q + is the heating and j is the emissivity, which ismeasured in the comoving frame.For radiation fields, the frequency-integrated transferequation, the zeroth moment equation, and the first mo-ment equation become, respectively, µ dIdz = ρ (cid:20) (1 + 3 βµ ) j π − ( κ abs + κ sca ) (1 − βµ ) I + κ sca π { (1 + 3 βµ ) cE − F β } i , (4) dFdz = ρ ( j − cκ abs E + κ abs F β − κ sca F β ) , (5) dPdz = ρc [ jβ − ( κ abs + κ sca ) F + κ abs cP β + κ sca ( cE + cP ) β ] , (6)where µ is cos θ , θ being the polar angle, and β = v/c . Wefurther adopt the Eddington approximation in the comov-ing frame, which is transformed into P = 13 E + 43 Fc β (7)in the fixed frame (Kato et al. 1998). Here, the transferequation (4) is corrected to the order of ( v/c ) (Kato etal. 1998).Eliminating j with the help of equation (3), and intro-ducing the optical depth by dτ = − ( κ abs + κ sca ) ρdz, (8)we can rearrange the basic equations up to the order of( v/c ) as c J dβdτ = − ( F − cP β ) , (9) µ dIdτ = − π q + ( κ abs + κ sca ) ρ (1 + 3 βµ )+ (1 − βµ ) I − (1 + 3 βµ ) 14 π ( cE − F β ) , (10) dFdτ = − q + ( κ abs + κ sca ) ρ + F β, (11) c dPdτ = − q + ( κ abs + κ sca ) ρ β + F − cP β, (12)o. ] Radiative Transfer in Accretion Disk Winds 3 J dzdτ = − κ abs + κ sca ) cβ. (13)Finally, integrating the sum of equations (9) and (12)gives the momentum flux conservation in the present ap-proximation, c Jβ + cP = cP − Z q + ( κ abs + κ sca ) ρ βdτ = cP , (14)when there is no heating ( q + = 0). In addition, the sub-script 0 means the value at some reference position (i.e.,the wind base). Similarly from equations (9) and (11) wehave the energy flux conservation,12 Jv + F = F − Z q + ( κ abs + κ sca ) ρ dτ = F , (15)when there is no heating ( q + = 0). Here, the first termon the left-hand side is eventually dropped, although weretain it here to clarify the physical meanings.We solve equations (9), (14), (15), and (10) for appro-priate boundary conditions, and we obtaine analytic solu-tions.As for the boundary conditions at the wind base of τ = τ and at the wind top of τ = 0, we impose the followingconditions.At the wind base on the disk surface with an arbitraryoptical depth τ , the flow velocity β is zero, the radiativeflux is F (which is a measure of the strength of radiationfield), and the radiation pressure is P (which connectswith the radiation pressure gradient), where the subscript0 denotes the values at the wind base.At the wind top, on the other hand, as already pointedout in Fukue (2005b), the usual boundary conditions forthe static atmosphere cannot be used for the present ra-diative wind, which moves with velocity at the order of thespeed of light. Namely, the radiation field just above thewind top changes when the gas itself does move upward,since the direciton and intensity of radiation change dueto relativistic aberration and Doppler effect (cf. Kato etal. 1998; Fukue 2000). If a flat infinite plane with surfaceintensity I s in the comoving frame is not static, but mov-ing upward with a speed v s (= cβ s , and the correspondingLorentz factor is γ s ), where the subscript s denotes thevalues at the surface, then, just above the surface, theradiation energy density E s , the radiative flux F s , andthe radiation pressure P s measured in the inertial framebecome, respectively, cE s = 2 πI s γ + 3 γ s u s + u , (16) F s = 2 πI s γ + 8 γ s u s + 3 u , (17) cP s = 2 πI s γ + 3 γ s u s + 3 u , (18)where u s (= γ s v s /c ) is the flow four velocity at the surface(Fukue 2005b). As a result, we have the boundary condi-tion at the wind top within the present approximation, cP s F s = 2 + 6 β s + 6 β β s + 3 β ∼
23 + 29 β s . (19) This is the consistent boundary condition under thepresent subrelativistic regime up to the order of ( v/c ) ,although Fukue (2005a) have approximately used theboundary condition, cP s /F s = 2 /
3, for a static atmosphere.In general, at the wind top of τ = 0, the boundary condi-tion (19) is not satisfied. Hence, for given parameters, weadjust and obtain the mass-loss rate J as an eigen value,so as to satisfy the boundary condition (19).As already stated, for the internal heating in the wind,we assume that there is no heating source ( q + = 0), al-though it is straightfoward to extend the model to thecase with heating. On the other hand, at the wind baseon the luminous disk, there is assumed to be a uniformsource of I .
3. Analytical Solutions
In Fukue (2005a), analytical solutions for radiative mo-ments were already derived. For the completeness, inthis section, we first recalculate them under the consistentboundary condition. Using the analytical expressions formoments, we then calculate the radiative intensity, whichwe wish to know in the present paper.
Under the present subrelativistic regime, equation (15)means that the radiative flux F is conserved: F = F = F s . (20)Using the boundary conditions at the wind base ( v = 0),equation (14) is expressed as Jv + P = P . (21)Hence, equation (9) becomes cJ dvdτ = − ( F s − P v ) , (22)that can be analytically solved to yield v = F s P " − e P cJ ( τ − τ ) . (23)Thus, the radiative flow from the luminous disk withoutheating is expressed in terms of the boundary values andthe mass-loss rate. In addition, the flow velocity v s at thewind top ( τ = 0) is v s = F s P − e − P cJ τ ! . (24)Using the boundary condition at the wind top, wefurther impose a condition on the values at boundaries.Inserting boundary values (19) into momentum equation(21), using equation (23), we have the following relation: cP F s = 23 + (cid:18) cJ P + F s cP (cid:19) − e − P cJ τ ! . (25)That is, for given τ and P at the wind base, the mass-loss rate J is determined in units of F s /c , as an eigen J. Fukue [Vol. ,value. Compared with Fukue (2005a), the second term inthe first parentheses on the right-hand side is an additionalone, which appears due to the present corrected boundarycondition (19).As already stated in Fukue (2005a), the mass-loss rateincreases as the initial radiation pressure increases, whilethe flow terminal speed increases as the initial radiationpressure and the loaded mass decrease.Moreover, as easily shown from equation (25), in orderfor the flow to exist, the radiation pressure P at the flowbase is restricted in some range,2 + √ < cP F s <
23 + τ , (26)which is slightly modified from that given in Fukue(2005a), due to the corrected boundary condition. At theupper limit of cP /F s = 2 / τ , the loaded mass divergesand the flow terminal speed becomes zero. On the otherhand, at the lower limit of cP /F s = (2 + √ /
6, the pres-sure gradient vanishes, the loaded mass becomes zero andthe terminal speed approaches the saturation speed of β ∞ = − √ ∼ . , (27)where the radiative flux F is balanced by the radiationdrag 4 P v with the boundary condition (19).In figure 1 we show analytical solutions, the flow veloc-ity v (solid curves) in units of c , the radiation pressure P (dashed ones) in units of F s /c , and the source function S (chain-dotted ones) in units of F s as a function of theoptical depth τ for several values of P at the wind basein a few cases of τ .When the initial radiation pressure P at the wind baseis large, the pressure gradient between the wind base andthe wind top is also large. As a result, the loaded mass J also becomes large, but the flow final speed v s is small dueto momentum conservation (14). When the initial radia-tion pressure P is small, on the other hand, the pressuregradient becomes small, and the loaded mass is also small,but the flow final speed becomes large. In the latter case,the source function S becomes almost constant.Within the present approximation of ( v/c ) with theboundary condition (19), the flow final speed saturates at0 . c . Now, we turn to the transfer equation (10), or µ dIdτ = (1 − βµ ) I − (1 + 3 βµ ) 14 π ( cE − F β ) , (28)when there is no heating.Under the present approximation up to the order of( v/c ) , F is constant, but E ( P ) and β are generally func-tions of τ . If, however, the radiation field is sufficientlyintense, the wind flow quickly saturates; the radiativeflux, F , is balanced by the radiation drag force, ( E + P ) v ,and the wind speed reaches the saturation terminal one, F/ ( E + P ). In such a case, the wind speed and the radi-ation quantities are almost constant. In other words, the t (a) v v P S t (b) v v P S Fig. 1.
Flow velocity v (solid curves) in units of c , radiationpressure P (dashed ones) in units of F s /c , and source function S in units of F s (chain-dotted ones) as a function of the opticaldepth τ for several values of P at the wind base in a few casesof τ : (a) τ = 1 and (b) τ = 10. From top to bottom of v and from bottom to top of P and S , the values of P are 0.75,1, 1.2, 1.4 in (a), and 0.75, 1, 2, 5 in (b). source function, the second term on the right-hand side ofequation (28), is almost constant (see figure 1), since inthe present case the source function S is expressed as S = cE − F β π = 3 cP − F β π . (29)Here, we thus assume that the wind velocity is constant,and analytically integrate the transfer equation (28).When the wind velocity becomes a constant terminalone, using equations (23), (21), and (7), we have β = β s = F s cP , (30) cP = cP − c Jβ = 23 F s + 118 F cP , (31) cE = 3 cP − F β = 2 F s − F cP . (32)Hence, cE − F β = 2 F s − F cP = 2 F s − F s β. (33)o. ] Radiative Transfer in Accretion Disk Winds 5That is, the source function measured in the fixed frameslightly decreases due to the effect of the relativistic mo-tion.Under the above situations, we can now integrate theradiative transfer equation (28), similar to Fukue andAkizuki (2006). After several partial integrations, we ob-tain both an outward intensity I ( τ, µ ) ( µ >
0) and an in-ward intensity I ( τ, − µ ) as I ( τ, µ ) = cE − F β π βµ − βµ h − e − βµµ ( τ − τ ) i + I ( τ , µ ) e − βµµ ( τ − τ ) , (34) I ( τ, − µ ) = cE − F β π βµ − βµ h − e − − βµµ τ i , (35)where I ( τ , µ ) is the boundary value at the wind base onthe luminous disk.In general case with finite optical depth τ and uniformincident intensity I from the disk, the boundary value I ( τ , µ ) of the outward intensity I consists of two parts: I ( τ , µ ) = I + I ( τ , − µ ) , (36)where I (= F s /π ) is the uniform incident intensity and I ( τ , − µ ) is the inward intensity from the backside of thedisk beyond the midplane. Determining I ( τ , − µ ) fromequation (35), we finally obtain the outward intensity as I ( τ, µ ) = cE − F β π βµ − βµ h − e − βµµ ( τ − τ ) i + I e − βµµ ( τ − τ ) , ∼ F s π (cid:26)(cid:18) − β + 83 βµ (cid:19) h − e − βµµ ( τ − τ ) i + 43 e − βµµ ( τ − τ ) (cid:27) , (37)where we have used F s = πI .Finally, the emergent intensity I (0 ,µ ) emitted from thewind top becomes I (0 , µ ) = 3 F s π (cid:20)(cid:18) − β + 83 βµ (cid:19) (cid:16) − e − − βµµ τ (cid:17) + 43 e − − βµµ τ (cid:21) , ∼ F s π (cid:18) − β + 83 βµ (cid:19) for large τ . (38)Under the present approximation, where the sourcefunction is constant, the usual limb darkening does notappear: e.g., (3 F s / π )(2 / τ ) for the Milne-Eddingtonsolution. Due to the Doppler and aberration effects orig-inating from the vertical motion of winds, however, theemergent intensity (38) depends on the wind velocity aswell as the direction cosine. This is the velocity-dependent limb-darkening effect.In figure 2, the emergent intensity I (0 ,µ ) normalized bythe isotropic value ¯ I (= F s /π ) is shown for several values of β as a function of µ . Although the present approximationmay be valid for β ≤ .
1, we show the cases for β ≤ . m I n t e n s it y b Fig. 2.
Normalized emergent intensity as a function of µ for the case without heating. The numbers attached on eachcurve are values of β of wind velocity. The dashed straight lineis for the usual Milne-Eddinton solution for the plane-parallelcase. As is easily seen in figure 2, as the velocity becomeslarge, the limb-darkening effect becomes prominant. Thatis, the emergent intensity increases in the poleward direc-tion, while it decreases in the edgeward direction. As aresult, a wind luminosity would be overestimated by apole-on observer and underestimated by an edge-on ob-server, when we observe an optically-thick accretion diskwind.It is interesting that the intensity does not change inthe direction of µ = 2 /
4. Numerical Solutions
In this section, we numerically solve equations (9) and(12) [or (14)] with constant F , and further solve equation(10) [or (28)] for several cases, and compare the resultswith that of analytical solutions to check the accuracyand limitations of analytical solutions. In figure 3 we show several numerical solutions, the flowvelocity v (thick solid curves) in units of c and the radia-tion pressure P (thick dashed curves) in units of F s /c asa function of the optical depth τ for several values of P at the wind base in a few cases of τ . Corresponding an-alytical solutions are shown by thin curves, although themass-loss rates are slightly different so as to satisfy theupper boundary condition (19).As is seen in figure 3, in both analytical and numericalcases, when the initial pressure P and the loaded massare large, the flow speed is small ( v ≤ . c ). In such a case,the radiative structure resembles to that of the static at-mosphere, where the source function is proportional to theoptical depth. When the initial pressure and the loadedmass decrease, on the other hand, the flow speed becomeslarge to attain the saturation one. In such a case, thesource function is almost constant.In addition, for the same parameters the analytical J. Fukue [Vol. , t (a) v vP t (b) v vP Fig. 3.
Flow velocity v (thick solid curves) in units of c andradiation pressure P (thick dashed curves) in units of F s /c as a function of the optical depth τ for several values of P at the wind base in a few cases of τ : (a) τ = 1 and (b) τ = 10. From top to bottom of v and from bottom to top of P , the values of P are 0.75 and 1.4 in (a), and 1 and 5 in (b).Corresponding analytical solutions are shown by thin curves. and numerical solutions are slightly different, although wehave solved the same basic equations. This is understoodas follows. Our basic equations are up to the order of( v/c ) , and therefore have the accuracy of the same or-der. In deriving the analytical solutions, we have droppedthe terms of order of ( v/c ) , while we have retained thoseterms in solving the numerical solutions. Hence, the ana-lytical and numerical solutions should be same within theaccuracy of ( v/c ) . After obtaining the numerical solutions for v and P (i.e., E ), we can further numerically integrate the transfer equa-tion (28) for many angles µ as a function of τ under ap-propriate boundary conditions and given initial conditionsof τ and P . As boundary conditions, we set a uniformsource of I at the wind base, while there is no incidentintensity at the wind top: I + ( τ ,µ ) = I + I − ( τ , − µ ) and I − ( τ , − µ ) = 0. We use meshes of 100 for µ and 500 for τ .Finally, the emergent intensity I (0 , µ ) emitted from thewind top is numerically obtained as a function of angle m I n t e n s it y b Fig. 4.
Normalized emergent intensity as a function of µ for the case without heating. The numbers attached on eachcurve are values of β s of wind terminal velocity. The ini-tial optical depth τ is 1 and the initial pressure P is 1.4( β s = 0 . β s = 0 . for given τ and P . Examples of the results are shown infigure 4.In figure 4, the emergent intensity obtained numericallyis shown for the initial optical depth of τ = 1, and forseveral values of β s , that is attached on each curve. Whenthe terminal speed is small, as already stated, the windstructure resembles to that of the static atmosphere. Inthe present case, however, the optical depth of wind isfinite. As a result, at around µ ∼ I (i.e., the peakingeffect diminishes), while the usual limb-darkening effectrecovers for small µ , where the line-of-sight length is long.When the terminal speed is large, on the other hand, thevelocity-dependent limb-darkening effect appears againfor the numerical solutions. That is to say, as stated infigure 2, due to the Doppler and aberration effects orig-inating from the vertical motion of winds, the emergentintensity depends on the wind velocity as well as the di-rection cosine. In particular, the emergent intensity isenhanced and increases as µ increases.By the way, in figure 4 the emergent intensity entirelydecreases for all µ as the velocity increases. Apparently,this seems to be curious, because the relativistic boostswork in the regime with large velocity. The reason is alsothe relativistic effect. The present subrelativistic flow upto the order of ( v/c ) , the momentum and energy of ra-diation fields accelerate the flow. Although the flux F isconserved within the present order of ( v/c ) , the radia-tion intensity diminishes due to the interaction betweenthe field and the flow [the 2 F β term in the second termon the right-hand side of equation (28)]. In other words,the source term effectively decreases. As a result, theemergent intensity entirely decreases, compared with thenon-relativistic case.We here briefly check the consistency of the numericalresults. In order to obtain the radiation intensity I fromequation (28), we use the radiation energy density E mom and the radiative flux F mom (= F s ) obtained from the mo-o. ] Radiative Transfer in Accretion Disk Winds 7 t Fig. 5.
Ratios between the quantities obtained from the mo-ment equations and those obtained from the intensity. Solidcurves and dashed ones denote the energy density and theflux, respectively. In the small velocity case of β s = 0 . β s = 0 .
337 (thin curves), on the otherhand, the consistency becomes worse. In addition, chain–dotted curves mean the three times Eddington factor for thequantities obtained from the intensity. ment equations (14). Once the radiation intensity I ( τ, µ )is obtained, we can calculate the radiation energy den-sity E sim and the radiative flux F sim by the definition. Ifthe quantities obtained from the moment equations andthose obtained from the intensity coincide each other, thesolutions are consistent, and vice versa.In figure 5 we plot the ratios E sim /E mom (solid curves)and F sim /F mom (dashed ones) for the small velocity caseof β s = 0 .
150 (thick curves) and for the large velocity caseof β s = 0 .
337 (thin curves). In addition, we also plotthe quantities ( E sim + 4 F sim v ) / (3 P sim ), the three timesEddington factor, by chain-dotted curves.As is seen from figure 5, in the case of small veloc-ity, where the structure resembles to that of the staticcase, the ratios are almost unity and the solutions areconsistent. In the small velocity case, furthermore, theEddington factor is almost 1 /
3. In the case of large veloc-ity, on the other hand, the ratios differ from unity. Themain reason of this discrepancy is the relativistic effect.As already stated, the radiation intensity diminishes dueto the interaction between the field and the flow. As aresult, the intensity I ( τ, µ ) as well as the emergent inten-sity I (0 ,µ ) entirely decreases. Thus, F sim becomes smallerthan F mom . Although the reason that the energy densityratio increases is less clear, it seems to be the relativis-tic peaking effect (Doppler boost and relativistic abbera-tion). Indeed, the Eddington factor in the large velocitycase is somewhat larger than unity. This would be alsothe relativistic peaking effect. Anyway, as the velocitybecomes large, the consistency becomes worse, and thepresent treatment under the order of ( v/c ) would be in-valid.
5. Concluding Remarks