MNRAS 436, 2550 2559 (2013) Advance Access publication 2013 October 22 doi:10.1093/mnras/stt1751 Radiative transfer in a floating stratus irradiated by a luminous source I. Basics a plane flat source J. Fukue Astronomical Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582-8582, Japan Accepted 2013 September 15. Received 2013 September 14; in original form 2013 March 6 ABSTRACT Radiative transfer in a geometrically thin stratus (i.e. a sheet-like gaseous cloud with finite optical depth), floating above a luminous source such as an accretion disc, is examined under the non-grey frequency-dependent treatment. Emergent intensity other radiative quantities are analytically obtained under the Eddington approximation for two typical cases: purely scattering local thermodynamic equilibrium (LTE). In the purely scattering case without emission, the mean intensity is a linear function of the optical depth, the flux is constant, the emergent intensity has angle dependency as a result of scattering. In this case, the emergent spectra have the same profile as those of the central source. In the LTE case, however, there appears to be a reprocessed thermal component in addition to a scattered one. As a result, the behaviour of the mean intensity flux depends on the fraction of the thermal component. The emergent intensity shows the usual limb-darkening or limb-brightening effect, as well as the scattering effect. In this case, the emergent spectra have thermal scattering components, show complicated profiles. We also show illustrative examples of a stratus floating over a plane flat source: observational images modified spectra due to reprocessing. Key words: accretion, accretion discs black hole physics radiative transfer galaxies: active. 1 INTRODUCTION In the Universe, there are many luminous objects surrounded by translucent envelopes, including massive hot stars, novae supernovae, accretion discs. For example, accretion discs have been operating as the central engines in various active phenomena in the Universe (i.e. protostars protoplanetary nebulae, cataclysmic variables supersoft X-ray sources, such as white dwarfs, microquasars galactic X-ray binaries, including neutron stars black holes, ultraluminous X-ray sources, quasars active galaxies). Accretion-disc models have been extensively studied during the past four decades (for a review, see Kato, Fukue & Mineshige 2008). However, surrounding such accretion discs, there can exist various gaseous components, such as inflowing gas, rotating debris, outflowing wind, high-speed jets floating cloud. These gaseous materials have been observed in, for example, active galactic nuclei (e.g. Arav et al. 2001; Alonso-Herrero et al. 2011; King, Miller & Raymond 2012), have been discussed widely (e.g. Elvis 2000, 2012; Czerny & Hryniewicz 2012). Such gaseous envelopes or environments around accretion discs have been examined by several researchers. For example, dense clouds in the central region of active galactic nuclei have been E-mail: fukue@cc.osaka-kyoiku.ac.jp discussed by several researchers (Rees 1987; Sivron & Tsuruta 1993; Na & George 1994). These gas clouds reprocess reflect the non-thermal radiation (Guilbert & Rees 1988; Lightman & White 1988). Reprocessed iron-line profiles have been examined by a quasi-spherical distribution of clouds (Karas et al. 2000), or by dense clouds embedded in a hot gas (Hartnoll & Blackman 2001). Floating clouds above accretion discs have also been discussed in the literature (Fukue 1996; Kitabatake & Fukue 2003b). Gaseous clouds above accretion discs also reprocess disc radiation (Kitabatake & Fukue 2003a,b). In these current studies on the gaseous environments around accretion discs or the central engine of active galactic nuclei, the researchers have discussed the dynamics or reprocessed effects of clouds. Neither radiative transfer effects nor the observational properties of gaseous clouds/layers surrounding a luminous central source have been examined. Thus, in this paper, we start the investigation of radiative transfer in gaseous clouds (sheet-like layers; strati) around a central luminous source irradiated by a source, such as an accretion disc. Here, we first consider the basic properties of radiative transfer in a floating stratus irradiated by an external luminous source, including both scattering reprocessed lights. We analytically obtain the emergent intensity as well as other radiative quantities under the Eddington approximation. C 2013 The Author Published by Oxford University Press on behalf of the Royal Astronomical Society
Radiative transfer in a floating stratus I 2551 In the next section, we describe the basic equations. We examine the two cases in Sections 3 4, respectively. The final section is devoted to concluding remarks. 2 BASIC EQUATIONS In the present work, we assume the following situations for a stratus floating in an active galactic nucleus or another environment (Fig. 1). (i) The stratus is made of ionized hydrogen. (ii) Its shape is sheet-like is treated as a plane-parallel layer. (iii) The optical depth of a stratus in the vertical direction is finite. (iv) The emissivity other quantities do not depend on the optical depth. (v) The radiation field is isotropic, we adopt the Eddington approximation. (vi) There is no internal heating in the stratus. (vii) The stratus is irradiated by the central engine from its base, while at its top there is no illuminating source. Under these situations, we examine the radiative transfer problem in such a floating stratus (see Fukue & Akizuki 2006; Fukue 2011, 2012a,b). It should be noted that, in a textbook situation (e.g. Rybicki & Lightman 1979; Rutten 2003), assumptions (i) (vi) are usually used for a scattering isothermal atmosphere in order to explain the effect of scattering. However, as far as we know, there are no relevant calculations including assumption (vii) in the context of the present situation, where a floating stratus exists above a luminous source, such as an accretion disc. The radiative transfer equations have previously been presented by several authors (Chrasekhar 1960; Mihalas 1970; Rybicki & Lightman 1979; Mihalas & Mihalas 1984; Kato et al. 2008; see also Fukue 2011). For the plane-parallel geometry in the vertical direction (z), the frequency-dependent transfer equation, the zeroth moment equation the first moment equation become μ di [ ] ν dz = ρ jν 4π (κ ν + σ ν ) I ν + σ ν J ν, (1) dh ν dz ( ) = ρ jν 4π κ νj ν dk ν dz = ρ(κ ν + σ ν )H ν, (3) Figure 1. Schematic diagram of a stratus floating above a luminous flat source. The optical depth of the stratus is finite, the source emits isotropic radiation. The observer is located far above the stratus in the vertical direction. (2) respectively. Here, μ is the direction cosine (= cos θ), I ν is the specific intensity, J ν is the mean intensity (= ce ν /4π, wheree ν is the radiation energy density), H ν is the Eddington flux (H ν = F ν /4π, wheref ν is the vertical component of the radiative flux), K ν is the mean radiation stress (K ν = cp ν /4π, wherep ν is the zzcomponent of the radiation stress tensor), ρ is the gas density c is the speed of light. The mass emissivity j ν, the absorption opacity κ ν the scattering opacity σ ν generally depend on the frequency. The Eddington approximation is K ν = J ν /3. It should be noted that equation (1) is usually derived for the isotropic scattering case, does not generally hold. However, for the anisotropic case, such as electron scattering, equation (1) holds as long as the radiation field is isotropic, the Eddington approximation can be used. Introducing the optical depth, defined by dτ ν ρ(κ ν + σ ν )dz, we rewrite the radiative transfer equations in the following forms: μ di ν 1 j ν = I ν dτ ν κ ν + σ ν 4π (1 ε ν)j ν, (4) dh ν dτ ν = ε ν J ν 1 κ ν + σ ν j ν 4π, (5) dk ν = 1 dj ν = H ν. (6) dτ ν 3 dτ ν Here, ε ν is the photon destruction probability: ε ν = κ ν. (7) κ ν + σ ν In the present problem, a floating stratus, a sheet-like cloud, has finite optical depth τ νb = 0 H ρ(κ ν + σ ν )dz, (8) where H is the geometrical thickness of the stratus. This sheet-like cloud is assumed to be irradiated from its bottom side of τ ν = τ νb by uniform intensity Iν, we impose the following boundary conditions. For intensities, the boundary conditions at the top side (τ ν = 0) at the bottom side (τ ν = τ νb )are Iν (0,μ) = 0, (9) I ν + (τ νb,μ) = Iν, (10) respectively. Hence, for radiative moments, the boundary condition at the top side becomes J ν (0) 2H ν (0) = 0, (11) where we assume an isotropically radiating Lambertian surface. However, the boundary condition at the illuminated bottom side is J ν (τ νb ) + 2H ν (τ νb ). = WIν, (12) Here, W = W J + 2W H is the total dilution factor, where W J W H are the dilution factors for the mean intensity flux, respectively. For sources with uniform intensity, these dilution factors can be calculated geometrically as W J = 1 d, (13) 4π W H = 1 cos θ d, (14) 4π
2552 J. Fukue while they should be calculated with source intensities for sources with non-uniform intensity (e.g. Fukue 2012a). For an infinitely extending uniform source, the dilution factors are W J = 1/2, W H = 1/4 W = 1. It should be noted that the dilution factors do appear in the boundary conditions for moment quantities, but they do not appear in the boundary conditions for intensities; this is because, as is well known, the intensity is conserved along the light ray, while the radiation field is generally diluted far from the source. 3 SCATTERING CASE First, we consider the purely scattering case: no emission (j ν = 0) no absorption (κ ν = 0), but only scattering. In this case, the basic equations (4) (6) are expressed as μ di ν dτ ν = I ν J ν, (15) dh ν dτ ν = 0 (16) dj ν dτ ν = 3H ν. (17) 3.1 Radiative moments Under the boundary conditions (11) (12), the solutions of the moment equations (16) (17), respectively, become (Fukue 2012a) J ν = 2 + 3τ ν WIν 4 + 3τ, (18) νb 1 H ν = WIν 4 + 3τ. (19) νb In the optically thick limit of τ νb, both the mean intensity flux become very small at the top side of the stratus (τ ν = 0). In the optically thin limit of τ νb 0, however, these become J ν J ν (0) 1 2 WI ν, (20) H ν H ν (0) 1 4 WI ν. (21) These are just the properties of the usual optically thin radiation field. Examples of radiative quantities are shown in Fig. 2. As is seen in Fig. 2, the mean intensity linearly decreases from the illuminated bottom side to the unilluminated top side, whereas the radiative flux is constant. In addition, from solution (18), as the total optical depth becomes small, the gradient of the mean intensity is steeper, vice versa. 3.2 Albedo for scattering case Using the results of the transfer problem, we can calculate determine the albedo of the layer. For an infinitely extending uniform source, where W = 1, we can calculate the scattering albedo A ν,as follows. Figure 2. Radiative quantities normalized by WIν as a function of the optical depth. The solid dashed curves are J ν H ν, respectively. The total optical depth is τ νb = 1. For a infinitely extending uniform source of uniform intensity I ν, the incident flux H ν is H ν = 1 4 I ν, (22) while the flux in the stratus at τ ν = τ νb is 1 H ν = Iν 4 + 3τ. (23) νb Hence, the scattering albedo is determined as A ν (1/4)I ν [1/(4 + 3τ νb)]i ν (1/4)I ν = 3τ νb 4 + 3τ νb (24) (see Fig. 8). In the optically thick limit of τ νb, the scattering albedo becomes unity. In the optically thin limit of τ νb 0, however, it approaches 0, as expected. 3.3 Emergent intensity By inserting the solutions of moment equations into the transfer equation (15), we can integrate the transfer equation (15) under the boundary conditions (9) (10). The upward I ν + downward intensities become I ν I + ν (τ ν,μ) WI ν Iν (τ ν,μ) WIν = I + ν (τ νb,μ) WI ν e (τν τ νb)/μ + 2 + 3μ 4 + 3τ νb [ 1 e (τ ν τ νb )/μ ] + 3 4 + 3τ νb [ τν τ νb e (τν τ νb)/μ ], (25) = I ν (0,μ) WI ν τν /μ e + 2 + 3μ ( ) 1 e τ ν /μ 3 + τ ν, (26) 4 + 3τ νb 4 + 3τ νb respectively, where I + ν (τ νb,μ) = I ν (27)
Radiative transfer in a floating stratus I 2553 Here, A is the frequency-integrated albedo, W H is the dilution factor for the flux I (= Iν dν) is the total irradiated intensity (see Fukue 2012a). It should be noted that the frequency-integrated albedo A should also be calculated as a result of the transfer problem, similar to the scattering case. The albedo is determined in Section 4.2. In this case, by eliminating H ν from equations (5) (6), using the isothermal assumption, we obtain the second-order form of the transport equation: 1 d 2 (J 3 dτν 2 ν B ν ) = ε ν (J ν B ν ). (33) Here, ε ν is the photon destruction probability defined by equation (7). Figure 3. Emergent intensities normalized by WIν as a function of μ for various values of τ νb W. The solid curves are the case of the uniform infinite source (W = 1/2), while the dashed curves denote the no-dilution case (W = 1). The values of τ νb are given for each solid curve: 0.1, 0.5, 1 10. I ν (0,μ) = 0. (28) Hence, the emergent intensity from the top side becomes I + ν (0,μ) WI ν = 1 W e τ νb/μ + 2 + 3μ 4 + 3τ νb ( 1 e τ νb /μ ) 3τ νb e τνb/μ. (29) 4 + 3τ νb In the optically thick limit of τ νb, the emergent intensity becomes very small. In the optically thin limit of τ νb 0, however, it becomes I + ν (0,μ) I ν. (30) This is just the incident intensity. The emergent intensities normalized by WIν are shown in Fig. 3 for various values of τ νb W. We can see several prominent properties in Fig. 3. For example, there is a limb-darkening-like effect. Seen from the top side, the mean intensity J ν increases inwards, as seen in Fig. 2. Hence, seen from the top side, the upward intensity I ν + shows the pseudo-limb-darkening effect, which is not equivalent to the usual limb-darkening because, in the present case, there is only scattering. This is the diffusion effect via scattering. In addition, for small optical depth τ νb, the light ray can almost pass through the layer, the emergent intensity approaches the incident uniform intensity Iν, except for small μ. 4 LOCAL THERMODYNAMIC EQUILIBRIUM CASE Next, we consider the local thermodynamic equilibrium (LTE) case: j ν /4π = κ ν B ν. In addition, we assume that the cloud layer is isothermal in the vertical direction B ν = B ν (0), (31) with temperature T eff determined by the irradiation heating: 1 π σt4 eff = (1 A)W H I. (32) 4.1 Radiative moments If the photon destruction probability ε ν has the same value at all depths, the linear differential equation (33) is easily integrated to yield the general solution: J ν = B ν + C 1 e 3ε ν τ ν + C 2 e 3ε ν τ ν. (34) Here, C 1 C 2 are integral constants, H ν = 1 dj ν 3εν = (C 1 e 3ε ν τ ν C 2 e ) 3ε ν τ ν. (35) 3 dτ ν 3 Under the boundary conditions (11) (12), the integral constants are determined as C 1 = 1 (ε /ε + )e 3εν τ νb B ε + (ε /ε 2 + )e 2 3ε ν τ ν νb C 2 = (ε /ε + )e 3εν τ νb WI ε + (ε /ε 2 + )e 2 3ε ν τ νb ν, (36) [1 (ε /ε + )e 3ε ν τ νb ] ε + (ε 2 /ε + )e 2 3ε ν τ νb e 3ε ν τ νb e 3ε ν τ νb B ν + WI ε + (ε /ε 2 + )e 2 3ε ν τ νb ν, (37) where ε + = 1 + 2 3 3εν (38) ε = 1 2 3 3εν. (39) In the optically thick limit of τ νb, these become B ν (0) J ν B ν (0) 1 + (2/3) e 3εν τ ν (40) 3ε ν 3εν B ν (0) H ν 3 1 + (2/3) e 3εν τ ν, (41) 3ε ν or at the top side (τ ν = 0) J ν (0) (2/3) 3ε ν B ν (0) 1 + (2/3) (42) 3ε ν 3εν B ν (0) H ν (0) 3 1 + (2/3). (43) 3ε ν
2554 J. Fukue These show the usual ε ν -law. Namely, the thermal component B ν becomes small because of scattering as ε ν B ν (see Rybicki & Lightman 1979; Rutten 2003). In the optically thin limit of τ νb 0, however, these become J ν J ν (0) 1 [ ] Bν (0) + WIν, (44) 2 H ν H ν (0) 1 4 WI ν. (45) These are again just the properties of the usual optically thin radiation field, although there exists a thermal component. For example, in equation (44), the terms on the right-h side are expressed as a thermal component B ν (0), its scattered reduction (1/2)B ν (0) the irradiated component from the bottom side (1/2)WIν. These analytical solutions (34) (35), normalized by the incident irradiation value WIν,areshowninFigs4 6forseveral values of the thermal component [B ν (0)/W Iν = 1, 10, 0.1], of the total optical depth (τ νb = 10 1) of the photon destruction probability (ε ν = 1, 0.5, 0.1, 0.01). In Fig. 4, radiative quantities for the medium thermal value [B ν (0)/W Iν = 1] are shown as a function of the optical depth τ ν for the large small optical depths of τ νb = 10 1. The values of ε ν are 1, 0.5, 0.1 0.01 from top to bottom for J ν. In Fig. 4(a), where the total optical depth is 10, the solutions are quite similar to the usual isothermal atmosphere. When there is no scattering (ε ν = 1), the mean intensity J ν at τ ν = 0ishalfofthe thermal value, because there is no photon from the top side. As the photon destruction probability ε ν decreases, both the mean intensity flux decrease; this is just the scattering effect or the so-called εν -law. In the limit of scattering only (ε ν = 0), these solutions approach those for the scattering case. In Fig. 4(b), where the disc optical depth is unity, the mean intensity decreases, while the flux increases, compared with Fig. 4(a). This is the translucent effect resulting from the small optical depth. Because of scattering, the mean intensity further decreases, while the flux becomes flat. In Fig. 5, radiative quantities for the large thermal value [B ν (0)/W Iν = 10] are shown as a function of the optical depth τ ν for the large small optical depths of τ νb = 10 1. The values of ε ν are 1, 0.5, 0.1 0.01 from top to bottom for J ν. This thermal-component-dominated case is impressive. That is, the mean intensity drops on both sides of the layer because of the lack of photons from the top bottom sides. Moreover, in the case of the large optical depth (Fig. 5a), it decreases because of scattering, whereas it decreases because of the scattering translucent effects in the case of the small optical depth (Fig. 5b). However, the behaviour of the flux is similar to that of the medium thermal value in Fig. 4. It is noted, however, that in the case of the small optical depth (Fig. 5b), the flux can become negative for large ε ν, because thermal radiation flows downwards for such a case, In Fig. 6, radiative quantities for the small thermal value [B ν (0)/W Iν = 0.1] are shown as a function of the optical depth τ ν for the large small optical depths of τ νb = 10 1. The values of ε ν are 1, 0.5, 0.1 0.01 from bottom to top for J ν.in this case, the situation is again different from both the medium large thermal values of Figs 4 5. In Fig. 6(a), where the total optical depth is 10, the values of ε ν are 1, 0.5, 0.1 0.01 from bottom to top for J ν. When there is no scattering (ε ν = 1), the mean intensity J ν at the top side of τ ν = 0 is half of the thermal value because of the lack of an external radiation field, whereas at the bottom side of τ νb, it is half of the value of irradiation. The flux is large at the bottom side. As the photon destruction probability ε ν Figure 4. Radiative quantities normalized by WIν as a function of the optical depth for the value of the thermal component of B ν (0)/W Iν = 1: (a) the large optical depth of τ νb = 10; (b) the small optical depth of τ νb = 1. Solid dotted curves represent J ν H ν, respectively, the thick solid lines denote B ν. The values of the parameter ε ν are 1, 0.5, 0.1 0.01 from top to bottom for J ν from inclined to flat for H ν. decreases, the mean intensity increases, whereas the flux becomes flat; this is also the scattering effect. In the limit of scattering only (ε ν = 0), these solutions approach those for the scattering case. In Fig. 6(b), where the disc optical depth is unity, the mean intensity decreases, while the flux increases, compared with Fig. 6(a). This is also the translucent effect resulting from the small optical depth, similar to other cases. Because of scattering, the mean intensity further decreases, while the flux becomes flat. 4.2 Albedo for local thermodynamic equilibrium case In the LTE case, the calculation of the albedo is difficult, compared with the scattering case. One reason for this is that the solutions in the LTE case are more complicated than those in the scattering case. However, the main reason is that the frequency-integrated albedo A is included in the thermal component, as in equation (32). Hence, here we use the grey approximation frequency-integrated equations, assume that the frequency-dependent albedo A ν is equal to the frequency-integrated albedo A.
Radiative transfer in a floating stratus I 2555 Figure 5. Radiative quantities normalized by WIν as a function of the optical depth for the value of the thermal component of B ν (0)/W Iν = 10: (a) the large optical depth of τ νb = 10; (b) the small optical depth of τ νb = 1. Solid dotted curves represent J ν H ν, respectively, the thick solid lines denote B ν. The values of the parameter ε ν are 1, 0.5, 0.1 0.01 from top to bottom for J ν from inclined to flat for H ν. In this case, for a infinitely extending uniform source of uniform intensity I, the incident flux H is H = 1 4 I, (46) while the frequency-integrated flux in the stratus at τ = τ b is 3ε H (τ b ) = (C 1 e 3ετ b C 2 e ) 3ετ b, (47) 3 where C 1 C 2 are given by equations (36) (37), respectively, under the grey approximation frequency-integrated forms, ε is the frequency-integrated photon destruction probability. In addition, the thermal component (32) is expressed as B = 1 π σt4 eff = (1 A)W H I = (1 A) 1 4 I. (48) Figure 6. Radiative quantities normalized by WIν as a function of the optical depth for the value of the thermal component of B ν (0)/W Iν = 0.1: (a) the large optical depth of τ νb = 10; (b) the small optical depth of τ νb = 1. Solid dotted curves represent J ν H ν, respectively, the thick solid lines denote B ν. The values of the parameter ε ν are 1, 0.5, 0.1 0.01 from bottom to top for J ν from inclined to flat for H ν. In this frequency-integrated case, the frequency-integrated albedo in the LTE case is determined by A H H (τ b ) = (1/4)I H (τ b ). (49) H (1/4)I We further rewrite this expression as (1 A) 1 4 I = H (τ b ). (50) Inserting equation (47) with equation (48) into equation (50), after several manipulations, we obtain the frequency-integrated
2556 J. Fukue depth: I ν + (τ ν,μ) = I ν + (τ νb,μ)e (τν τνb)/μ + B ν (0)[1 e (τν τνb)/μ ] + (1 ε ν)c 1 [e 1 + 3ε ν τ ν e (τν τ νb)/μ ] 3ε ν τ νb 3ε ν μ + (1 ε ν)c 2 [e 1 3ε ν τ ν e (τν τ νb)/μ+ ] 3ε ν τ νb. (54) 3ε ν μ Figure 7. Frequency-integrated albedo in the LTE case as a function of the total optical depth τ b for several values of ε. The values of ε are0,0.1,0.5 1, from top to bottom for the solid curves, while the thick dashed curve denotes the scattering albedo. albedo in the explicit form: [ 3ε ( ) ][ 1 A = 4 ε + + ε e 2 3ετ b ε+ 2 3 ε2 e 2 3ετ b where 3ε ) + (ε + ε e 3ετ b )(1 ] 1 e 3ετ b, (51) 3 ε ± = 1 ± 2 3 3ε. (52) The frequency-integrated albedo (51) in the LTE case is shown in Fig. 7 as a function of the total optical depth τ b for several values of ε. For given values of ε τ b, the frequency-integrated albedo is determined uniquely, as in equation (51) in Fig. 7. When ε = 0 (purely scattering case), the albedo approaches unity as τ b becomes large. The thick dashed curve, which follows the curve of ε = 0, is the frequency-integrated equation (24). When ε 0, however, as the value of ε becomes large, the albedo decreases. It should be noted that in the optically thick limit of τ b, the frequency-integrated albedo becomes A = 1 ( 3ε/3) 1 +, (53) 3ε which is less than unity, except for ε = 0,asshowninFig.7.Thisis because a part of the incident flux is consumed to the thermalization of the stratus. In the optically thin limit of τ b 0, however, it approaches 0, as expected. 4.3 Emergent intensity Using the solutions for radiative moments from Section 4.1, we can also solve the transfer equation (4) to obtain the intensity I ν (τ ν ). After some manipulations, we obtain the upward I ν + downward intensities for the present isothermal case with finite optical I ν Iν (τ ν,μ) = Iν (0,μ)eτν /μ + B ν (0)(1 e τν /μ ) + (1 ε ) ν)c 1 (e 1 + 3ε ν τ ν e τν /μ 3ε ν μ + (1 ε ν)c 2 1 3ε ν μ (e 3ε ν τ ν e τν /μ ). (55) Under the boundary conditions (9) (10), we finally obtain the emergent intensity I ν (0, μ) emitted from the top side of a stratus irradiated from the bottom side (see Fukue 2012a): I ν + (0,μ) = I ν e τ νb/μ + B ν (0)(1 e τνb/μ ) + (1 ε [ ] ν)c 1 1 + 1 e (1+ 3ε ν μ)τ νb /μ 3ε ν μ + (1 ε ν)c 2 1 3ε ν μ [ ] 1 e (1 3ε ν μ)τ νb /μ. (56) The terms on the right-h side of this emergent intensity (56) have the following meanings. The first term is the extinguished intensity of irradiation incoming from the bottom side of the stratus, the second term is the thermal radiation of the reprocessed light of irradiation the third fourth terms are the scattering radiation of thermal irradiated components, respectively. In the optically thick limit of τ νb, equation (56) reduces to the usual solution: I ν + (0,μ) B 1 ε ν ν(0) (1 + 3ε ν μ)[1 + (2/3) 3ε ν ] B ν(0). (57) This is just the usual scattered radiation obeying the ε ν -law in the stellar atmosphere with a semi-infinite medium. However, in the optically thin limit of τ νb 0, this solution (56) becomes I ν (0,μ) I ν. (58) This is just the incident irradiation from the bottom side. The emergent intensities normalized by WIν are shown in Figs 8 10 for several values of B ν (0)/W Iν (=1, 10 0.1), for several values of τ νb ε ν. In Fig. 8, the emergent intensities for the medium thermal value [B ν (0)/W Iν = 1] are shown. When the total optical depth is large (τ νb = 10, dashed curves), the emergent intensity shows the limbdarkening-like effect; it decreases as μ decreases. This is the scattering effect, not the usual limb-darkening effect, because there is no limb-darkening in the present isothermal atmosphere. It further decreases as the photon destruction probability ε ν decreases. This is also the scattering effect (Fukue 2012a). Furthermore, when the total optical depth is small (τ νb = 1, solid curves), the emergent intensity becomes large because of the translucent effect. Fig. 9 shows the emergent intensities for the large thermal value [B ν (0)/W Iν = 10]. When the total optical depth is large (τ νb = 10, dashed curves), the emergent intensity shows the limb-darkeninglike effect, resulting from scattering as in Fig. 8. However, when the total optical depth is small (τ νb = 1, solid curves), the emergent
Radiative transfer in a floating stratus I 2557 Figure 8. Normalized emergent intensity as a function of μ for the medium value of the thermal component of B ν (0)/W Iν = 1. The values of the total optical depth τ νb are 10 (dashed curves) 1 (solid curves). The values of the parameter ε ν are 1, 0.5, 0.1 0.01 from top to bottom for each case. Figure 10. Normalized emergent intensity as a function of μ for the small value of the thermal component of B ν (0)/WIν = 0.1. The values of the total optical depth τ νb are 10 (dashed curves) 1 (solid curves). The values of the parameter ε ν are 1, 0.5, 0.1 0.01 from bottom to top for each case. 5 ILLUSTRATIVE EXAMPLES In this section, we show several illustrative examples for observed properties of strati irradiated from the bottom side. We demonstrate masked images deformed spectra due to reprocessing. Figure 9. Normalized emergent intensity as a function of μ for the large value of the thermal component of B ν (0)/W Iν = 10. The values of the total optical depth τ νb are 10 (dashed curves) 1 (solid curves). The values of the parameter ε ν are 1, 0.5, 0.1 0.01 from top to bottom for each case. intensity decreases as μ increases (i.e. the intensity towards the poleward direction is small). This is the translucent effect. That is, the intensity towards the edgeward direction becomes larger than that towards the poleward direction, because the optical depth of the layer towards the edgeward direction might be larger than unity in this case, although the scattering effect does exist. Fig. 10 shows the emergent intensities for the small thermal value [B ν (0)/W Iν = 0.1]. When the total optical depth is large (τ νb = 10, dashed curves), for ε ν = 1, the emergent intensity is almost 0.1 does not depend on μ, we see only the thermal component. For ε ν < 1, the emergent intensity depends on μ, because the irradiation light from the bottom side can escape from the top side because of scattering. When the total optical depth is small (τ νb = 1, solid curves), even for ε ν = 1, the emergent intensity becomes large towards the poleward direction because of the translucent effect. It further becomes large for ε ν < 1 because of the translucent scattering effects. 5.1 Masked images The usual scattering processes are electron scattering in a hightemperature regime, Mie scattering in a low-temperature regime with dust Rayleigh scattering for very small particles. In around protoplanetary discs, the latter two scattering processes operate, while electron scattering is important in around highenergy accretion discs. If the scattering opacity does not depend on the frequency, scattering does not change the source spectrum. In Fig. 11, we illustratively show the masked images of a luminous source, using the emergent intensity (29) in the scattering case (similar results are obtain using the LTE case with ε ν 0). In Fig. 11, we set W = 1/2 μ = 1 (pole-on view), while we artificially change the stratus optical depth τ νb or the source intensity Iν. In Fig. 11(a), the optical depth of the stratus is fluctuated as τ νb = 1 + (1/2) sin(2πx/2) sin(2πy), but the source intensity is constant. In addition to the fluctuation in the uniform stratus, such a case would be realized when many small clouds (such as broad-line regions in active galactic nuclei) cover the central engine. As the optical depth of the stratus becomes large, the masked image dims out. However, when the optical depth is sufficiently smaller than unity, the luminous source itself can be seen. In Fig. 11(b), the optical depth of the stratus is unity, but the source intensity is fluctuated as Iν = 1 + (1/2) cos(2πx/5) cos(2πy/5). In this case, we see the image of the source. As the optical depth of the stratus becomes large, the image dims out, vice versa. In Fig. 11(c), the fluctuations in both the optical depth the source intensity are considered. In this case, the fluctuated source intensity is further changed by the fluctuated stratus. However, from an observational point of view, we cannot distinguish between the
2558 J. Fukue placed above the luminous emitting surface, the emitting surface would warm up from the backscatter, reach a steady state after a long time, with the same flux but at a higher temperature. Hence, in a steady state after a long time, the scattering cloud would shine as brightly as the emitting surface. In order to examine such a steady state, however, we must solve both the scattered layer emitting surface, simultaneously, but this is beyond the scope of this paper. Instead, the situation considered in this paper is not such a steady state, but the temporary state just after the emitting surface is covered with the scattering layer, similar to when the shining Sun is covered with clouds. Figure 11. Illustrated examples of a stratus irradiated by a luminous flat source from the bottom side. The optical depth τ νb the source intensity I ν are artificially fluctuated: (a) τ νb = 1 + (1/2) sin(2πx/2) sin(2πy); (b) I ν = 1 + (1/2) cos(2πx/5) cos(2πy/5); (c) both. Other parameters are W = 1/2 μ = 1 (pole-on view). source the stratus, or both, unless we know further information about the source or the stratus. It should be noted that dimming out is some artificial effect under the present instantaneous condition. If obscuring clouds were 5.2 Modified spectra due to reprocessing On the one h, a scattering process such as electron scattering does not change the profile of the source spectrum; on the other h, the reprocessing of the incident irradiation in the stratus changes the spectrum of the incident irradiation. In such a case, there are generally several components in the spectra: the source radiation, the thermal (reprocessed) radiation from the stratus the scattered light of thermal radiation diluted irradiation. On the right-h side of the emergent intensity (56), the first term is the extinguished source radiation, the second term is the extinguished thermal radiation, the third fourth terms are the scattered light. We illustratively show these in Fig. 12. Fig. 12 shows illustrative spectra of the emergent intensity (56) in the LTE case. In Fig. 12, we set T = 10 5 K(πI = σt 4), W = 10 4 μ = 1 (pole-on view). In Fig. 12(a), ε ν = 0.1 τ νb = 1(A = A ν = 0.362), in Fig. 12(b), ε ν = 0.1 τ νb = 5 (A = A ν = 0.522). In Fig. 12(a) (τ νb = 1), as already noted, we show the source radiation, thermal radiation scattered radiation. The thin solid curves (the lower curve is overlapped by the thick solid curve) denote the source spectrum Iν the extinguished source spectrum Iν e τνb/μ, which is the first term on the right-h side of equation (56). The thin dashed curves represent the thermal spectrum B ν (0) the extinguished thermal spectrum B ν (0)(1 e τνb/μ ), which is the second term on the right-h side of equation (56). A dotdashed curve denotes the scattered spectra, which are the third fourth terms on the right-h side of equation (56). Finally, a thick solid curve is the emergent spectrum I ν + (0,μ). In this case, the source radiation is sufficiently intense, compared with the thermal scattered light. In Fig. 12(b) (τ νb = 5), the optical depth of the stratus becomes large. In this case, the source radiation is extinguished by the stratus (the lower thin curve). The extinguished thermal radiation is almost the same as the thermal radiation (the thin dashed curves). Because the extinguished source radiation is smaller than in Fig. 10(a), the scattered light also becomes small. As a result, the emergent spectrum has two peaks: the source spectrum the thermal spectrum. It should be noted that the emergent spectrum is slightly smaller than the thermal spectrum. This is the manifestation of the scattering effect ( ε ν -law). 6 CONCLUDING REMARKS In this paper, we have analytically solved the radiative transfer problem in a geometrically thin stratus, floating above a luminous source such as an accretion disc. We have obtained analytical solutions of emergent intensities for scattering LTE cases under the Eddington approximation. In the purely scattering case without emission,
Radiative transfer in a floating stratus I 2559 In astrophysical situations, the luminous sources might not be plane flat sources, but could be hot luminous stars, luminous discs (a distant approximation of accretion discs) or accretion discs. Moreover, the configuration of strati would be a plane-parallel flat slab, a spherical shell, or some other configuration. The applications of the present results to these various cases are examined in a separate paper (Fukue 2013). In the present study, we have assumed that the strati are floating above a luminous source. In an active surrounding environment (e.g. luminous accretion discs rotating around a central black hole), many strati would move in the outflowing wind, inflowing flow or rotational streams. Radiative transfer in such a moving stratus is a problem to be addressed in the future. ACKNOWLEDGEMENTS The author would like to thank an anonymous referee for useful comments, including the albedo problem. This work has been supported in part by the Grant-in-Aid for Scientific Research (C) of the Ministry of Education, Culture, Sports, Science, Technology (22540251 JF). REFERENCES Figure 12. Illustrative spectra of a stratus irradiated by a luminous source from the bottom side. The thin solid curves are the source extinguished source spectra. The thin dashed curves are the thermal extinguished thermal spectra. The dot-dashed curve is the scattered spectra. The thick solid curve is the emergent spectra. The photon destruction probability, total optical depth albedo are, respectively: (a) ε ν = 0.1, τ νb = 1 A = A ν = 0.362; (b) ε ν = 0.1, τ νb = 5A = A ν = 0.522. Other parameters are T = 10 5 K, W = 10 4 μ = 1 (pole-on view). the emergent spectra have the same profiles as those of the central source. In the LTE case, however, the emergent spectra have thermal scattering components, show complicated profiles. In addition, the emergent intensity shows the usual limb-darkening effect or limb-brightening effect, as well as the scattering effect. We have assumed that the scattering is isotropic. For the weakly anisotropic case, such as electron scattering, the present basic equations hold, as long as the radiation field is isotropic. However, for the general case of anisotropic scattering, such as Mie scattering, the formalism should be reconsidered (Fukue 2012b). In addition, we have assumed that the scattering process does not depend on the frequency, bearing in mind electron scattering. There are frequencydependent scattering processes, such as the Rayleigh scattering. We have shown illustrative examples of a stratus floating over a plane flat source: observational images modified spectra due to reprocessing. In realistic situations, the optical depth of the stratus would be not constant, but there are irregularities (inhomogeneity). Hence, we have assumed sinusoidal irregularities in the illustrative examples. Alonso-Herrero A. et al., 2011, ApJ, 736, 82 Arav N. et al., 2001, ApJ, 561, 118 Chrasekhar S., 1960, Radiative Transfer. Dover Press, New York Czerny B., Hryniewicz K., 2012, J. Phys.: Conf. Ser., 372, 012013 Elvis M., 2000, ApJ, 545, 63 Elvis M., 2012, in Chartas G., Hamann F., Leighly K. M., eds, ASP Conf. Ser. Vol. 460, AGN Winds in Charleston. Astron. Soc. Pac., San Francisco, p. 186 Fukue J., 1996, PASJ, 48, 89 Fukue J., 2011, PASJ, 63, 1273 Fukue J., 2012a, PASJ, 64, 106 Fukue J., 2012b, PASJ, 64, 132 Fukue J., 2013, MNRAS, 436, 2560 (Paper II) Fukue J., Akizuki C., 2006, PASJ, 58, 1039 Guilbert P. W., Rees M. J., 1988, MNRAS, 233, 475 Hartnoll S. A., Blackman E. G., 2001, MNRAS, 324, 257 Karas V., Czerny B., Abrassart A., Abramowicz M. A., 2000, MNRAS, 318, 547 Kato S., Fukue J., Mineshige S., 2008, Black-Hole Accretion Discs Toward a New Paradigm. Kyoto Univ. Press, Kyoto King A. L., Miller J. M., Raymond J., 2012, ApJ, 746, 2 Kitabatake E., Fukue J., 2003a, PASJ, 55, 267 Kitabatake E., Fukue J., 2003b, PASJ, 55, 1115 Lightman A. P., White T. R., 1988, ApJ, 335, 57 Mihalas D., 1970, Stellar Atmospheres. W. H. Freeman, San Francisco Mihalas D., Mihalas B. W., 1984, Foundations of Radiation Hydrodynamics. Oxford Univ. Press, Oxford Na K., George I. M., 1994, MNRAS, 267, 974 Rees M. J., 1987, MNRAS, 228, 47p Rutten R. J., 2003, Radiative Transfer in Stellar Atmospheres, Utrecht University lecture notes, 8th edn. Available at: http://www.nhn.ou.edu/ baron/sa/afy.pdf Rybicki G. B., Lightman A. P., 1979, Radiative Processes in Astrophysics. Wiley, New York Sivron R., Tsuruta S., 1993, ApJ, 402, 420 This paper has been typeset from a TEX/LATEX file prepared by the author.