aa r X i v : . [ a s t r o - ph ] N ov PASJ:
Publ. Astron. Soc. Japan , 1– ?? , c (cid:13) Radiative Transfer and Limb Darkening of Accretion Disks
Jun
Fukue and Chizuru
Akizuki
Astronomical Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka [email protected] (Received 0 0; accepted 0 0)
Abstract
Transfer equation in a geometrically thin accretion disk is reexamined under the plane-parallel approx-imation with finite optical depth. Emergent intensity is analytically obtained in the cases with or withoutinternal heating. For large or infinite optical depth, the emergent intensity exhibits a usual limb-darkeningeffect, where the intensity linearly changes as a function of the direction cosine. For small optical depth,on the other hand, the angle-dependence of the emergent intensity drastically changes. In the case withoutheating but with uniform incident radiation at the disk equator, the emergent intensity becomes isotropicfor small optical depth. In the case with uniform internal heating, the limb brightening takes place forsmall optical depth. We also emphasize and discuss the limb-darkening effect in an accretion disk forseveral cases.
Key words: accretion, accretion disks — black holes — galaxies: active — radiative transfer —relativity
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,in cataclysmic variables and supersoft X-ray sources, ingalactic X-ray binaries and microquasars, and in activegalaxies and quasars. Accretion-disk models have beenextensively studies during these three decades (see Katoet al. 1998 for a review).Radiative transfer in the accretion 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; Kˇriˇz and Hubeny 1986; Shaviv, Wehrse1986; 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 etal. 2005). In many cases the diffusion approximationor Eddington one was employed; it provides a satisfactorydescription at large optical depth, although the emergentradiation field originates at optical depth of the order ofunity. Furthermore, gray and non-gray models of accre-tion disks were constructed under numerical treatments(Kˇriˇz and Hubeny 1986; Shaviv and Wehrse 1986; Adamet al. 1988; Ross et al. 1992; Shimura and Takahara 1993;Hubeny, Hubeny 1997, 1998; Hubeny et al. 2000, 2001;Davis et al. 2005; Hui et al. 2005) and under analyticalones (Hubeny 1990; Artemova et al. 1996).In these studies, however, the vertical movement andthe mass loss were not considered. Hence, recently, in re-lation to the radiative disk wind, radiative transfer in amoving disk atmosphere was also investigated (e.g., Fukue2005a, b, 2006a, b). In contrast to the static atmosphere,in the moving atmosphere the boundary condition at the surface 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; Turolla et al. 1995; Dullemond1999), and the velocity-dependent variable Eddington fac-tor was proposed (Fukue 2006b).In the usual studies of radiative transfer in the disk,the emergent intensity has not been fully obtained, sincethe attention were usually focused on the disk internalstructure such as a temperature distribution. In addition,the effect of limb darkening has not been well examinedexcept for a few cases relating to cataclysmic variables(e.g., Diaz et al. 1996; Wade, Hubeny 1998; see also Fukue2000; Hui et al. 2005 for high energy cases).In this paper, we thus reexamine radiative transferin the accretion disk with finite optical depth under theplane-parallel approximation, and analytically obtain anemergent intensity for the cases with or without internalheating. Besides cataclysmic variables, we also emphasizeand discuss the limb-darkening effect in an accretion diskfor various cases.In the next section we describe the basic equations. Insection 3, we show analytical solutions. In section 4, wediscuss several cases of accretion disk models, and empha-size the importance of the limb-darkening effect. The finalsection is devoted to concluding remarks.
2. Basic Equations
We here assume the followings: (i) The disk is steadyand axisymmetric. (ii) It is also geometrically thin andplane parallel. (iii) As a closure relation, we use theEddington approximation. (iv) The gray approxima-tion, where the opacity does not depend on frequency,is adopted. (v) The viscous heating rate is concentrated J. Fukue and C. Akizuki [Vol. ,at the equator or uniform in the vertical direction.The radiative transfer equations are given in severalliteratures (Chandrasekhar 1960; Mihalas 1970; Rybicki,Lightman 1979; Mihalas, Mihalas 1984; Shu 1991; Kato etal. 1998). For the plane-parallel geometry in the verticaldirection ( z ), the frequency-integrated transfer equation,the zeroth moment equation, and the first moment equa-tion become, respectively,cos θ dIdz = ρ (cid:20) j π − ( κ abs + κ sca ) I + κ sca c π E (cid:21) , (1) dFdz = ρ ( j − cκ abs E ) , (2) dPdz = − ρ ( κ abs + κ sca ) c F, (3)where θ is the polar angle, I the frequency-integrated spe-cific intensity, E the radiation energy density, F the verti-cal component of the radiative flux, P the zz -componentof the radiation stress tensor, ρ the gas density, and c thespeed of light. The mass emissivity j and opacity κ abs and κ sca are assumed to be independent of the frequency(gray approximation).For matter, the vertical momentum balance and energyequation are, respectively,0 = − dψdz − ρ dpdz + κ abs + κ sca c F, (4)0 = q +vis − ρ ( j − cκ abs E ) , (5)where ψ is the gravitational potential, p the gas pressure,and q +vis the viscous-heating rate. In this paper, we do notsolve the hydrostatic equilibrium (4). Generally speaking,when the contribution of the radiative flux is small, com-pared with the pressure gradient term, the gas pressuredominates in the atmoshere, and the density distributionwill not be constant. When the radiative flux is strong,on the other hand, the radiation pressure dominates, andthe density may be approximately constant throughoutmuch of the disk. Anyway, we suppose that the densitydistribution would be ajusted so as to hold the hydro-static equilibrium (4) through the main part of the diskatmosphere, under the radiative flux obtained later.Using this energy equation (5) and introducing the op-tical depth, defined by dτ ≡ − ρ ( κ abs + κ sca ) dz, (6)we rewrite the radiative transfer equations: µ dIdτ = I − c π E − π κ abs + κ sca q +vis ρ , (7) dFdτ = − κ abs + κ sca q +vis ρ , (8) c dPdτ = F, (9) cP = 13 cE, (10)where µ ≡ cos θ . Final equation is the usual Eddingtonapproximation. As for the boundary condition at the disk surface of τ = 0, we impose a usual condition:3 cP s = cE s = 2 F s at τ = 0 , (11)where the subscript s denotes the values at the disk sur-face.For the internal heating, we consider two extreme cases:(i) No heating ( q +vis = 0), where the viscous heating isconcentrated at the disk equator and there is no heat-ing source in the atmosphere. (ii) Uniform heating in thesense that q +vis / ( κ abs + κ sca ) ρ =constant. The latter casemeans that the kinematic viscosity ν is constant in thevertical direction, since q +vis /ρ = ν ( rd Ω /dr ) , as long asthe opacities are constant.Finally, the disk total optical depth becomes τ = − Z H ρ ( κ abs + κ sca ) dz, (12)where H is the disk half-thickness.
3. Analytical Solutions
Except for the emergent intensity I , several analyti-cal expressions for moments as well as temperature distri-butions were obtained by several researchers (e.g., Laor,Netzer 1989; Hubeny et al. 2005; Artemova et al. 1996).For the completeness, we recalculate them as well as theintensity I . We first consider the case without heating in the diskatmosphere: q +vis = 0, but with uniform incident intensity I from the disk equator, where the viscous heating isassumed to be concentrated.In this case, the analytical solutions of moment equa-tions are easily given as F = F s = πI , (13)3 cP = cE = 3 F s (cid:18)
23 + τ (cid:19) . (14)This is a familar solution under the Milne-Eddington ap-proximation for a plane-parallel geometry. It should benoted that the vertical radiative flux F is conserved, andequals to πI at the disk equator.Since we obtain the radiation energy density E in theexplicit form, we can now integrate the radiative transferequation (7). After several partial integrations, we obtainboth an outward intensity I ( τ, µ ) ( µ >
0) and an inwardintensity I ( τ, − µ ) as I ( τ, µ ) = 3 F s π (cid:20)
23 + τ + µ − (cid:18)
23 + τ + µ (cid:19) e ( τ − τ ) /µ (cid:21) + I ( τ , µ ) e ( τ − τ ) /µ , (15) I ( τ, − µ ) = 3 F s π (cid:20)
23 + τ − µ − (cid:18) − µ (cid:19) e − τ/µ (cid:21) , (16)where I ( τ , µ ) is the boundary value at the midplane ofthe disk.o. ] Radiative Transfer and Limb Darkening of Accretion Disks 3 m I n t e n s it y t Fig. 1.
Normalized emergent intensity as a function of µ forthe case without heating. The numbers attached on eachcurve are values of τ at the disk midplane. The dashedstraight line is for the usual plane-parallel case with infiniteoptical depth. In the geometrically thin disk with finite optical depth τ and uniform incident intensity I from the disk equator,the boundary value I ( τ , µ ) of the outward intensity I consists of two parts: I ( τ , µ ) = I + I ( τ , − µ ) , (17)where I is the uniform incident intensity and I ( τ , − µ ) isthe inward intensity from the backside of the disk beyondthe midplane. Determining I ( τ , − µ ) from equation (16),we finally obtain the outward intensity as I ( τ, µ ) = 3 F s π (cid:20)
23 + τ + µ − µe ( τ − τ ) /µ − (cid:18) − µ (cid:19) e ( τ − τ ) /µ (cid:21) + I e ( τ − τ ) /µ = 3 F s π (cid:20)
23 + τ + µ + (cid:18) − µ (cid:19) e ( τ − τ ) /µ − (cid:18) − µ (cid:19) e ( τ − τ ) /µ (cid:21) , (18)where we have used F s = πI .For sufficiently large optical depth τ , this equation (18)reduces to the usual Milne-Eddington solution: I = 3 F s π (cid:18)
23 + τ + µ (cid:19) . (19)Finally, the emergent intensity I (0 ,µ ) emitted from thedisk surface for the finite optical depth becomes I (0 , µ ) = 3 F s π (cid:20)
23 + µ + (cid:18) − µ (cid:19) e − τ /µ − (cid:18) − µ (cid:19) e − τ /µ (cid:21) . (20)In figure 1, the emergent intensity I (0 ,µ ) normalized bythe isotropic value ¯ I (= F s /π ) is shown for several values of τ as a function of µ .As is easily seen in figure 1, for large optical depth( τ >
10) the angle-dependence of the emergent intensityis very close to the case for a usual plane-parallel case withinfinite optical depth. Therefore, the usual limb-darkeningeffect is seen. Namely, in the case of a semi-infinite diskwith large optical depth, the energy density increases lin-early with the optical depth in the atmosphere, and thetemperature increases accordingly. As a result, an ob-server at a pole-on position of µ = 1 will see deeper in thedisk, where the temperature (and therefore, the sourcefunction) is larger than that observed by an observer atan edge-on position of µ = 0. Thus, the observed intensitywill be higher at µ = 1. This is just a usual limb-darkening.For small optical depth, however, the angle-dependenceis drastically changed. When the optical depth is a few,the vertical intensity ( µ ∼
1) decreases due to the finitenessof the optical depth. That is, we cannot see the ‘deeper’position in the atmosphere, compared with the case of asemi-infinite disk. Furthermore, when the optical depthis less than unity, the intensity in the direction of small µ increases, and the emergent intensity becomes isotropicwith a uniform value I at the disk equator; the limb-darkening effect disappears. Indeed, the limiting case of τ ∼ I (0 , µ ) ∼ F s /π . That is, in this case of very smalloptical depth, the source function is dominated by theisotropic source at the midplane. Now, we consider the case with uniform heating: q +vis / ( κ abs + κ sca ) ρ =constant.Integrating the equation (8) under the following bound-ary conditions: F = 0 at τ = τ ,F = F s at τ = 0 , (21)we obtain F = F s (cid:18) − ττ (cid:19) . (22)The radiative flux F linearly increases from 0 to the sur-face value F s .Substituting equation (22) into equation (9), and inte-grating the resultant equation under the boundary condi-tion (11), we obtain3 cP = cE = 3 F s (cid:18)
23 + τ − τ τ (cid:19) . (23)This expression (23) for finite optical depth is seen in,e.g., Laor and Netzer (1989). A similar but more generalexpression was obtained by Hubeny (1990). In any case,this expression reduces to the Milne-Eddington solutionfor sufficiently large optical depth. In the case of finiteoptical depth, the radiation energy density and pressuredecrease from the midplane to the surface in the quadraticform. It should be noted that at the midplane of the diskof τ = τ , J. Fukue and C. Akizuki [Vol. ,3 cP = cE = 3 F s (cid:18)
23 + τ (cid:19) . (24)As already mentioned by Hubeny (1990), the energy den-sity at the disk midplane is the half of the correspondingstellar atmospheric one. This is explained by the fact thatthe radiation from the disk midplane may escape equallyto both sides of the disk.Since we obtain the radiation energy density E in theexplicit form (23), we can now integrate the radiativetransfer equation (7). After several partial integrations,we obtain both an outward intensity I ( τ, µ ) ( µ >
0) andan inward intensity I ( τ, − µ ) as I ( τ, µ ) = 3 F s π (cid:20)
23 + τ + µ + 1 τ (cid:18) − τ − µτ − µ (cid:19) − (cid:18)
23 + τ τ − µ τ (cid:19) e ( τ − τ ) /µ (cid:21) + I ( τ , µ ) e ( τ − τ ) /µ , (25) I ( τ, − µ ) = 3 F s π (cid:20)
23 + τ − µ + 1 τ (cid:18) − τ µτ − µ (cid:19) − (cid:18) − µ + 13 τ − µ τ (cid:19) e − τ/µ (cid:21) , (26)where I ( τ , µ ) is the boundary value at the midplane ofthe disk.In the case with uniform heating and without the inci-dent intensity, the boundary value I ( τ ,µ ) of the outwardintensity I is I ( τ , µ ) = I ( τ , − µ ) , (27)and we finally obtain the outward intensity as I ( τ, µ ) = 3 F s π (cid:20)
23 + τ + µ + 1 τ (cid:18) − τ − µτ − µ (cid:19) − (cid:18) − µ + 13 τ − µ τ (cid:19) e ( τ − τ ) /µ . (cid:21) (28)For sufficiently large optical depth τ , this equation (28)also reduces to the usual Milne-Eddington solution (19).Finally, the emergent intensity I (0 ,µ ) emitted from thedisk surface for the finite optical depth becomes I (0 , µ ) = 3 F s π (cid:20)
23 + µ + 1 τ (cid:18) − µ (cid:19) − (cid:18) − µ + 13 τ − µ τ (cid:19) e − τ /µ (cid:21) . (29)In figure 2, the emergent intensity I (0 ,µ ) normalized bythe isotropic value ¯ I (= F s /π ) is shown for several valuesof τ as a function of µ .As is easily seen in figure 2, for large optical depth( τ >
10) the angle-dependence of the emergent intensityis very close to the case with a usual plane-parallel casewith infinite optical depth. Therefore, the usual limb-darkening effect is seen, as already stated at the end ofsection 3.1.For small optical depth, however, the angle-dependence m I n t e n s it y t Fig. 2.
Normalized emergent intensity as a function of µ for the case with uniform heating. The numbers attached oneach curve are values of τ at the disk midplane. The dashedstraight line is for the usual plane-parallel case with infiniteoptical depth. is drastically changed similar to the case without heat-ing. In the vertical direction of µ ∼
1, the emergent in-tensity decreases as the optical depth decreases. This isdue to the finiteness of the optical depth. That is, wecannot see the ‘deeper’ position in the atmosphere, com-pared with the case of a semi-infinite disk. In the inclineddirection of small µ , on the other hand, the emergent in-tensity becomes larger than that in the case of the infiniteoptical depth. Moreover, when the optical depth is lessthan unity, the emergent intensity for small µ is greaterthan unity: the limb brightening takes place. Indeed, inthe limiting case of τ ∼ I (0 , µ ) ∼ ( F s /π ) / (2 µ ). This isbecause that the path length is longer for such a case ofsmall µ . That is, in this case for low optical depth, thesource function is very uniform. This, coupled with theabsence of an isotropic source at the midplane, is why thegeometric effect (longer path length) is dominant and onefinds limb ‘brightening’. In this subsection, we briefly discuss the validity of theclosure relation in the present treatment. In this paper, wehave adopted the usual Eddington approximation, wherethe ratio of the radiation pressure to the energy densityis fixed as 1/3, to close the moment equations. As is wellknown, this approximation is correct in the limit of anisotropic radiation field. Hence, in the problem of limb-darkening, where the deviation from the isotropy is es-sential, this approximation is only approximately correct,although it is used in the usual Milne-Eddington approx-imation.For example, let us suppose the case of a semi-infinitedisk with an infinite optical depth τ , In this case, weeasily calculate the energy density as well as the radia-tion pressure from the derived intensity (19) of the Milne-Eddington solution. At the deeper position in the atmo-o. ] Radiative Transfer and Limb Darkening of Accretion Disks 5sphere, where the integration is done in all directions, there-calculated variables satisfy the condition of P/E = 1 / τ = 0, we integrate the emergent intensity toyield: cE = 3 F s Z (cid:18)
23 + µ (cid:19) dµ = 3 F s , (30) cP = 3 F s Z (cid:18)
23 + µ (cid:19) µ dµ = 13 3 F s , (31)or P/E = (1 / / anti-peaking effect. As is seen in figure 2, the limb-brightening becomes stronger and stronger for small op-tical depth. Hence, for such a case of very small opticaldepth, the Eddington approximation would not be good,although the qualitative properties would not be changed.In order to obtain the intensity distribution more pre-cisely, we, for example, introduce a variable Eddingtonfactor, that is beyond the scope of the present paper.
4. Discussion
As was derived in the previous section, the emergentintensity I of the accretion disk depends on the disk totaloptical depth τ as well as the direction cosine µ . Thelimb-darkening effect is considerably modified for small τ ,compared with the usual case for infinite optical depth.Even for the case with sufficiently large optical depth,limb darkening in the luminous accretion disk must beimportant, and should be examined more carefully.In this section, we discuss several cases in turn, andcall the attention to the importance of the limb-darkeningeffect. The optical depth at the midplane in the inner regionof a geometrically thin standard disk (Shakura, Sunyaev1973; see also Kato et al. 1998) is expressed as τ = 12 κ Σ = 20 α − ˙ m − ˆ r / − r r ! − , (32)where κ is the electron scattering optical depth, α theviscous parameter, ˙ m the mass accretion rate normalizedby the critical rate ˙ M (= L E /c ), L E being the Eddingtonluminosity of the central object, ˆ r the radius normalizedby the Schwarzschild radius r g (= 2 GM/c ). This optical depth becomes small for large ˙ m and/orsmall ˆ r . For a slightly large accretion rate, inside somecritical radius r cr = 2 ˙ m, (33)the disk shifts to a supercritical regime, whereas the diskis a standard regime outside r cr (Fukue 2004). At thiscritical radius, the optical depth becomes τ cr ∼ α − ˙ m / . (34)Hence, in a usual situation the optical depth of the innerregion of the standard disk is greater than several tens.In such a situation, however, due to a usual limb-darkening effect for semi-infinite medium, the emergentradiation toward the pole-on direction is enchanced by20 percent, while the emergent radiation seen from theedge-on direction diminishes by 50 percent. Thus, in cal-culating the flux and spectrum of the standard disk, wecarefully consider the limb-darkening effect.In the region inside the inner edge at 3 r g , the disk gasfreely falls toward the central black hole, and the surfacedensity (i.e., the disk optical depth) quickly drops. Hence,the emergent spectrum from the innermost region insidethe inner edge would be greatly modified from the opti-cally thick case and the optically thin one. In the supercritical accretion disk, where the mass ac-cretion rate exceeds the critical rate, the expression for thedisk optical depth is changed. For example, in the self-similar model without mass loss (Fukue 2000), the opticaldepth at the midplane of the disk is τ = κ π ˙ Mc α √ GM r = ˙ m √ c α r , (35)where c is a coefficient of the order of unity. At thecritical radius, the disk optical depth is τ cr ∼ c α ˙ m / . (36)In the critical accretion disk (Fukue 2004), where themass accretion rate exceeds the critical one, but the excessmass is expelled by the wind mass loss, the optical depthat the midplane of the disk is τ = 16 √ α ˆ r / = 39 . α − ˆ r / . (37)At the critical radius, the disk optical depth is τ cr ∼ α − ˙ m / . (38)Hence, in a usual situation the optical depth of the su-percritical/critical disk is also greater than several tens.In such a situation, however, the usual limb-darkeningeffect for semi-infinite medium is also important (seeFukue 2000). In addition to the limb-darkening effect,the geometrical effects, such as a projection effect and a self-occultation , should be considered in calculating theflux and spectrum of the supercritical disk (e.g., Fukue2000; Watarai et al. 2005; Kawata et al. 2006). J. Fukue and C. Akizuki [Vol. , If a luminous disk is sandwiched by a disk corona, thesituation is similar to the case without heating, but withthe incident intensity from the midplane. Hence, whenthe optical depth of the corona is sufficiently smaller thanunity, the emergent intensity is just that of the disk, ex-cept for a very small direction cosine. However, when theoptical depth of the corona is a few, limb darkening takesplace, and the emergent intensity toward the edge-on di-rection remarkably reduces.On the other hand, if the accretion rate is quite small,and the inner region of the disk becomes an optically-thin advection dominated state (ADAF), the situation issimilar to the case with internal heating, although, rig-orously speaking, the plane-parallel approximation maybe invalid. When the optical depth of an ADAF region issufficiently smaller than unity, the limb brightening wouldtake place.In both cases with hot gas, however, the effect of theCompton scattering may be important. Hence, the angledependence of the intensity would be modified by the an-gle dependence of the Compton scattering, and the trans-fer problem should be treated more carefully.In addition, in the latter case of ADAF, the accretionflow is supposed to be conical or spherical. Hence, whenthe opening angle is small, the limb brightening wouldqualitatively take place. When the opening angle is large,on the other hand, the extension of the emitting region islarge, and the angle dependence of the emergent intensitywould become much more complicated than the presentsimple case.
In the case of the relativistic standard disk (e.g.,Novikov, Thorne 1973; Page, Thorne 1974), the situa-tion is similar to the non-relativistic case. However, thedirection cosine of the local emergent intensity in the diskto the observer is changed by two additional reasons: (i)the light trajectory is bent by the space-time curvature,and (ii) the emission from the gas rotating around a blackhole suffers from the special relativistic aberration. Limb-darkening effect in the comoving frame on the spectralenergy distribution (SED) from the relativistic accretiondisk was considered by, e.g., Fu and Taam (1990) andGierli´nski et al. (2001).Recently, due to submilliarcsecond astrometry, imagingof a black-hole silhouette is expected to become possible inthe near future at infrared and submillimetre wavelengths.The “photographs” of relativistic accretion disks arounda black hole have been obtained by many researchers(Luminet 1979; Fukue, Yokoyama 1988; Karas et al. 1992;Jaroszy´nski et al. 1992; Fanton et al. 1997; Fukue 2003;Takahashi 2004, 2005). In these studies, however, theusual limb darkening was not considered.Examples of silhouettes of a dressed black hole areshown in figures 3 and 4. In figure 3 an edge-on viewwith inclination angle of 80 ◦ is shown, while a pole-onview is expressed in figure 4. In both figures, the left pan- Fig. 3.
Edge-on view of a dressed black hole without (leftpanel) and with (right panel) limb darkening. The inclinationangle is 80 ◦ . Fig. 4.
Pole-on view of a dressed black hole without (leftpanel) and with (right panel) limb darkening. The inclinationangle is 0 ◦ . els are for the case without limb darkening, whereas theright panels are for the case with limb darkening.In the edge-on view (figure 3), the image of a limb-darkening disk darkens as expected. This is due mainly tothe usual limb-darkening effect for small direction cosine.Surprisingly, on the other hand, in the pole-on view (figure4), the image of a limb-darkeing disk also darkens! This isbecause that the local direction cosine becomes small dueto the light aberration associated with the disk rotation.Thus, in calculating spectra and observed fluxed of rel-ativistic disks as well as in taking black hole silhouettes,we should carefully consider limb darkening. For example,besides Fu and Taam (1990) and Gierli´nski et al. (2001),Hui et al. (2005) discussed limb darkening in some de-tails in their non-LTE calculation of accretion disk spectraaround intermediate-mass black holes.
5. Concluding Remarks
In this paper we analytically solve the radiative trans-fer problem of a geometrically thin accretion disk withfinite optical depth, and obtain analytical expressions foremergent intensity from the surface of the disk. For smalloptical depth, the angle-dependence of the emergent in-tensity drastically changes from the case with large opticaldepth. In the case without heating but with uniform inci-dent radiation at the disk equator, the emergent intensityo. ] Radiative Transfer and Limb Darkening of Accretion Disks 7becomes isotropic for small optical depth. In the case withuniform internal heating, the limb brightening takes placefor small optical depth. We also emphasize the impor-tance of limb darkening in the accretion disk study, anddiscuss several cases, including relativistic silhouettes.In the calculation of spectra from, e.g., cataclysmic vari-ables, the effect of limb darkening was considered (e.g.,Diaz et al. 1996; Wade, Hubeny 1998). In the cases ofhigh energy and relativistic regimes, we should also con-sider limb darkening (cf. Hui et al. 2005).In addition, if there exist intense radiation sources, suchas neutron stars or radiating jets, irradiation takes placeand the outer boundary condition changes (e.g., Hubeny1990). In such cases under strong incident radiation, limbbrightening may occur (cf. Stibbs 1971), even for largeoptical depth.Finally, if there exists mass loss from the disk surface,the radiative transfer problem becomes much more com-plicated. In order to examine spectra, fluxes, eclipsinglight curves, we must calculate the emergent intensity an-alytically or numerically.This work has been supported in part by a Grant-in-Aidfor Scientific Research (18540240 J.F.) of the Ministry ofEducation, Culture, Sports, Science and Technology.