PASJ: Publ. Astron. Soc. Japan 64, 132, 2012 December 25 c 2012. Astronomical Society of Japan.
Radiative Transfer in Anisotropic Scattering Media Jun F UKUE Astronomical Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582-8582
[email protected] (Received 2012 March 8; accepted 2012 June 27)
Abstract Transfer and moment equations in anisotropic scattering media are formulated under a simple linear model using the Eddington approximation. In the transfer equation, a term including the radiative flux appears, and the first moment equation is also modified, while the zeroth moment equation does not change, since the scattering processes exchange momentum, but not energy. Moment equations and transfer equation are solved for several simple cases to clarify the qualitative properties of radiative transfer in anisotropic scattering media. When forward scattering dominates, the radiative intensity is enhanced in some cases, while the gradient of the mean intensity decreases in other cases, depending on the boundary conditions for the problem. These analytic expressions are helpful in studying and understanding the radiative properties of stars with dusty envelopes, interstellar matter, protoplanetary disks and star forming regions, and the dusty universe. Key words: accretion, accretion disks — ISM: dust, extinction — radiative transfer — scattering — stars: atmosphere
1.
Introduction
The research field of radiative transfer in astrophysics has a century long history, since the 1900s (Schuster 1905; Milne 1921; Eddington 1926; Kosirev 1934; Chandrasekhar 1934), and there are many excellent monographs on radiative transfer and radiation hydrodynamics (Chandrasekhar 1960; Mihalas 1970; Rybicki & Lightman 1979; Mihalas & Mihalas 1984; Shu 1991; Peraiah 2002; Castor 2004). The methods of radiative transfer are quite important tools for treating and understanding all of the luminous phenomena in the universe: gaseous nebulae and interstellar matter, protoplanetary disks, stellar atmospheres and winds, novae and supernovae, hypernovae and gamma-ray bursts, accretion disks and jets, and first stars and the proto universe. Radiative transfer is also applied to and extensively studied in the field of the planetary atmosphere and oceans (Thomas & Stamnes 1999). In this latter case, scattering is remarkably important and thoroughly studied (e.g., Sobolev 1975; Yanovitskij 1997; Kokhanovsky 1999). Light scattering is also important in various scenes in astrophysics; from Thomson scattering in the high energy regime, such as black-hole accretion disks, Rayleigh scattering in the middle-energy regime, such as the solar atmosphere, to Mie scattering in the low energy regime, such as interstellar matter. For example, radiative transfer in accretion disks was examined by, e.g., Fukue and Akizuki (2006), Fukue (2011, 2012), for isotropic scattering cases. If scattering is weakly anisotropic, like Thomson or Rayleigh, the formulation is not changed as long as the radiation field is isotropic. If, however, scattering is highly anisotropic, the usual formulation may be violated. Particularly, anisotropic scattering by, e.g., dust grains is important in the dusty envelope of stars, interstellar clouds, protoplanetary disks and star forming regions, and the early dusty universe. In astrophysics, however, the radiative transfer
problems for such anisotropic scattering media have not been well examine from a theoretical viewpoint, except for numerical simulations. In planetary science, on the other hand, anisotropic scattering by, e.g., aerosols has been widely examined theoretically (e.g., Sobolev 1975; Yanovitskij 1997; Kokhanovsky 1999), although it is not familiar in astrophysics. In addition, these studies in planetary atmospheres are rather complicated in general, and it is difficult to apply each problem in astrophysical contexts. In this paper we thus consider radiative transfer in anisotropic scattering media under a simple assumption that the scattering phase function has only a linear term on the direction cosine. We then formulate the moment equations under the Eddington approximation, which is familiar in astrophysics, and analytically solve the transfer problem. In the next section we assume the analytic expression for the phase function. In section 3 we derive the transfer and moment equations in general and plane-parallel cases. We obtain analytical solutions for stellar atmospheres in section 4, for illuminated interstellar clouds in section 5, and for accretion disks in section 6. The final section is devoted to concluding remarks. 2.
Analytic Expressions for the Phase Function
We define the phase function, (cos ), or the scattering redistribution function, (l , l 0 ), by the normalized condition R ( d Ω = 1), where l and l 0 are the direction cosine vectors of the incident and scattered light, respectively, and the angle between both light rays. For isotropic scattering = 1=4, whereas for Thomson scattering 1 3 1 3 (1) 1 + .l l 0 /2 = 1 + cos2 : T = 4 4 4 4 The phase function for Rayleigh scattering has the same form
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as the frequency-dependent cross-section. The phase function for the anisotropically scattering particles cannot be expressed by a simple analytic form, but is expressed by an infinite series, or approximated by analytical functions. For example, in the field of planetary atmospheres the phase function is expanded by the Legendre polynomials to reproduce the experimental values (e.g., Kokhanovsky 1999). On the other hand, often used for dust grain mixtures is the Henyey-Greenstein (1941) phase function, HG =
1 1 g2 ; 2 4 .1 + g 2gcos /3=2
4.0 0.5 3.0 4πφ0 2.0
(2)
where parameter g is the first moment; g = hcosR i. This phase function has the correct first moment, hcos i = HG cos d Ω = g, but of course does not reproduce the real phase function perfectly. For example, HG cannot reproduce Rayleigh scattering in the limit of g ! 0, where HG ! 1=4. In order to express the real phase function more precisely, more complex formulae were proposed by, e.g., Draine (2003), where the proposed phase function has two parameters. Trading off the accuracy, to examine the properties of radiative transfer in anisotropic scattering media, we approximate the anisotropy by a very simple function, as described below. We can suppose several forms expressed by the power of the direction cosine: 1 .1 + a/; (3) 1 = 4 1 1 Œ1 + a.1 + /2 ; (4) 2 = 4 4 1+ a 3 1 1 3 = (5) Œ1 + a.1 + /3 ; 4 1 + 2a where a is a free parameter and = cos . Of these, 2 and 3 are shown in figure 1 with HG . The linear function 1 is too simple to express the anisotropy of the phase function. On the other hand, both the quadratic function, 2 , and cubic one, 3 , can reproduce the trend of anisotropy. As can be seen in figure 1, the cubic case is not much better than the quadratic case. In addition, for the cubic function the third moment appears in the basic equations. We also checked two parameter cases, but the trend does not improve. Hence, in this paper we adopt a quadratic expression, 2 , to examine the qualitative properties of radiative transfer in anisotropically scattering media. 3. Moment Formalism In this section we first write down the basic equations of radiative transfer with the phase function (4) under the moment formalism (see, e.g., Kato et al. 2008). 3.1. Transfer Equation By means of the mass emissivity, j , the mass absorption coefficient, , the mass scattering coefficient, , and the mass density, , the transfer equation for the specific intensity, I (l , t ), is expressed as 1 @I + .l r / I c @t
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0.2
1.0 0 -1.0
0
1.0
μ
Fig. 1. Approximate expressions of the phase function. Thick solid curves are HG with g = 0.2 and 0.5 for weak and strong anisotropy, respectively. Thin solid ones show 2 with a = 0.6 and 2.625. Thin dashed ones mean 3 with a = 0.4 and 1.75.
1 =
j I I + 4
Z
.l ; l 0 /I .l 0 /d Ω0 ; (6)
where (l , l 0 ) is the scattering redistribution function. Inserting the phase function (4) into the third term on the right-hand side, the transfer equation (6) becomes 1 @I + .l r / I c @t 1 =
j I I 4 1 + .1 + a/J + 2a.l H / + a `i `j Kij ; (7) 4 1+ a 3 where J is the mean intensity (= cE =4, E being the radiation energy density), H the Eddington flux vector (H ij = F =4, F the radiative flux vector), K the mean ij ij ij radiation stress tensor (K = cP =4, P the radiation stress tensor). ij Under the Eddington approximation of K = ı ij J =3, the transfer equation for the present anisotropic case is finally expressed as 1 @I + .l r / I c @t =
2
1 6
j I I + 4J + 4
3 2a 7 .l H /5 : 4 1+ a 3 (8)
As can be seen in equation (8), there appears a term including the radiative flux, H , which is absent in the isotropic case as well as the Thomson and Rayleigh cases. It should be noted that in the field of the planetary atmosphere the phase function is expanded by the Legendre polynomials and in some case the
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first linear term is examined (e.g., Yanovitskij 1997), although it is not the case in astrophysics. In the field of astrophysics the Legendre expansion is also shown to fit, e.g., the Milky Way dust model (e.g., Draine 2003). 3.2.
Moment Equations
Integrating the transfer equation (8) over a solid angle, we obtain the zeroth moment. Integrating it over a solid angle, after being multiplied by the direction cosine, we obtain the first moment. The zeroth and first moments of equation (8) are, respectively, j @J @Hk =
J + (9) ; c@t @x k 4 2a 1 i @Hi @Kik = . + /Hi + + H : (10) 4 3 c@t @x k 1+ a 3 The right-hand side of the zeroth moment (9), which expresses the energy exchange, is the same as those for the isotropic case, whereas on the right-hand side of the first moment (10), which expresses the momentum exchange, there appears a new term proportional to the radiative flux vector. These are easily understood because the scattering processes exchange momentum, but not energy. On the right-hand side of equation (10), the first term means the decrease of momentum of the radiation field (the first term on the left-hand side); the radiation field gives momentum to gas through the momentum exchange between radiation and gas in the usual isotropic case. On the other hand, the second new term means an increase of momentum of the radiation fields, if forward scattering dominates over back scattering (a > 0); the anisotropically scattering enhances the radiative flux in the same direction. In the extremely anisotropic case of a ! 1, the increase of the flux is limited as Hi =2. Hence, the momentum reduction by the first term always dominates the momentum increase due to the anisotropically scattering effect. 3.3.
Plane–Parallel Forms
In the steady plane–parallel case, the transfer equation (8) and moment equations, (9) and (10), are respectively expressed as 0 13 2
dI B 6 j = 4 . + /I + @J + dz 4
dH j =
J ; dz 4
2a C7 HA5 ; 4 1+ a 3 (11) (12)
2 a 3 H ; (13) 4 1+ a 3 where (= cos ) is the direction cosine, and K the zzcomponent of the radiation stress tensor. Introducing the optical depth, defined by d ( + )dz, we rewrite the radiative transfer equations in the form: dK = . + /H + dz
132-3 1
0
dI j B = I .1 " /@J + d 4. + /
2a C HA ; 4 1+ a 3 (14)
j dH = " J ; d 4. + / 3 2 2 a dK 6 7 = 41 .1 " / 3 H = ı H ; 4 5 d 1+ a 3 where " is the photon destruction probability, " = ; +
(15)
(16)
(17)
and the parameter ı is defined as 2 a : (18) ı 1 .1 " / 3 4 1+ a 3 For the weak anisotropic case of a = 0.6, the factor (2a=3)=(1 + 4a=3) is 2=9, for the strong anisotropic case of a = 3, it is 2=5, and for the extreme case of a = 1, it is 1=2. 4.
Dusty Stellar Atmosphere
In the following three sections, we examine several astrophysical applications. In this section we first consider the basic case of the Eddington model for the stellar atmosphere. If we assume a gray atmosphere, the basic equations (14)– (16) are expressed by the frequency-integrated form: 1 0
j dI B =I .1 "/ @J + d 4. + /
2a C H A ; 4 1+ a 3
(19) j dH = "J ; (20) d 4. + / dK = ıH; (21) d where the quantities are frequency-integrated one, and " and ı are respectively defined as "= ; (22) + 2 a : (23) ı = 1 .1 "/ 3 4 1+ a 3 In the radiative equilibrium case, and under the Eddington approximation, the basic equations (19)–(21) can be written as
dI = I J .1 "/ d
dH = 0; d
2a H; 4 1+ a 3
(24)
(25)
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dJ = 3ıH: d Hence, the flux is constant: H. / = H.0/;
(27)
J. / = H.0/.2 + 3ı /:
(28)
This means that when the radiative flux is constant throughout the atmosphere, the gradient of the mean intensity can become shallow due to forward scattering. Using these solutions, we can easily integrate the transfer equation (24) under the semi-inifite boundary condition. The outward and inward intensities become, respectively: (29) I + . ; / = H.0/.2 + 3 + 3ı /; = I . ; / = I .0/ + H.0/Œ.2 + 3/.1 e / + 3ı : (30) In addition, the emergent intensity is I + .0; / = H.0/.2 + 3/:
(31)
Surprisingly, under the radiative equilibrium and Eddington approximation, the emergent intensity has the same form with the usual Eddington model, although the internal intensity is slightly changed. This unexpected situation is caused because the decrease of the mean intensity is cancelled out by the increase of the flux term in the transfer equation (24). If we assume LTE, this is not true, as shown in section 6. 5. Illuminated Interstellar Cloud Next, we consider a sheet-like interstellar cloud illuminated by a nearby luminous source of uniform intensity, I . For simplicity, we consider no emission (j = 0) and no absorption ( = 0), but only scattering. In this case the basic equations (14)–(16) are expressed as: 1 0 dI B = I @J + d
1.0
(26)
and, for the Lambertian boundary condition, which assumes the surface radiation is isotropic, of J (0) = 2H (0), equation (26) is integrated to yield
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2a C H A ; 4 1+ a 3
(32)
dH = 0; (33) d dK = ıH ; (34) d where 2 1+ a 3 : (35) ı= 4 1+ a 3 When the cloud is illuminated from the top-side, the boundary conditions for the moments are J .0/ = 2H .0/ + .1 A /W I ; J . b / = 2H . b /;
(36) (37)
where A is the surface albedo, W the dilution factor of the source, and b the optical thickness of the cloud. It should
0.4
Jν
f
1
0
1 0.4
Hν -1.0 0
0.2
0.4
0.6 τν
0.8
1.0
Fig. 2. Radiative quantities normalized by (1 A )W I as a function of the optical depth. Thin curves are J and H for the isotropic scattering case of ı = 1 (a = 0), while thick ones are for the anisotropic case of ı = 0.4 (a = 3). The other parameter is b = 1.
be noted that equation (36) for the top-side is the Lambertian boundary condition with external illumination, while equation (37) for the back-side is the usual Lambertian boundary condition without illumination. Under these boundary conditions and the Eddington approximation, the solutions of the moment equations (33) and (34) become, respectively: 2 + 3ı (38) .1 A /W I ; J = 1 4 + 3ı b 1 H = .1 A /W I : (39) 4 + 3ı b Examples of radiative quantities are shown in figure 2. As can be seen in figure 2, in this case the mean intensity linearly decreases from the illuminated top-side to the unilluminated bottom-side. Comparing the isotropic scattering case (a = 0 and ı = 1; thin curves) with the anisotropic scattering one (a > 0 and ı < 1; thick ones), the gradient of the mean intensity becomes shallower due to forward scattering. Similarly, the radiative flux toward the bottom-side in the anisotropically scattering case becomes larger due to forward scattering. Inserting the solutions of the moment equations into the transfer equation (32), we can integrate the transfer equation (32) under the appropriate boundary condition. The upward and downward intensities become, respectively: I+ . b ; / . b /= I+ . ; / = e .1 A /W I .1 A /W I 2 + 3ı b 3 + 1 e . b /= 4 + 3ı b 3ı b e . b /= ; 4 + 3ı b
(40)
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Radiative Transfer in Anisotropic Scattering Media
downward intensity, I (solid curves), shows the usual limbdarkening. Conversely, seen from the top-side, the mean intensity, J , decreases toward the inward (downward) direction, as shown in figure 2. Hence, seen from the top-side, the upward intensity, I+ (dashed curves), shows anti-limbdarkening, or limb-brightening.
1.0 0.4
Intensity
0.8 0.6
1 1 δ
0.4
6.
0.2 0.4 0 0
0.2
0.4
μ
0.6
0.8
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1.0
Fig. 3. Emergent intensities normalized by (1 A )W I as a function of for various values of ı. Solid curves are the downward intensities I , while dashed ones are the upward ones I+ . The values of ı are from 1 (isotropic) to 0.4 (anisotropic) in steps of 0.1. The other parameters are b = 1 and W = 1.
I .0; / I . ; / = e = .1 A /W I .1 A /W I 2 + 3ı b 3 + 1 e = 4 + 3ı b 3ı : 4 + 3ı b The boundary conditions are I+ . b ; / = 0; I .0; / = .1 A /I :
Accretion Disk
We now examine a geometrically thin accretion disk irradiated by the central star. We here consider a passive isothermal disk (see, e.g., Fukue 2012), but the case of an active disk with internal heating is straightforward. If we assume LTE (strictly speaking, we assume that j =4 = B ), the basic equations (14)–(16) become the following forms: dI 2a H ; = I S .1 " / (46) 4 d 1+ a 3 dH = J S ; (47) d dK = ı H ; (48) d where S is the source function, 1 j S = J = " B + .1 " /J : + + 4 +
(41)
Eliminating H from equations (47) and (48), and using the Eddington approximation, we have the second order form of the transport equation, 1 d 2 J = ı " .J B /: 3 d 2
(42) (43)
Hence, the emergent intensities become
(50)
For a vertically isothermal disks irradiated by the central object, the Planck function B ( ), the thermal component of the reprocessed radiation, is constant in the vertical direction, B = B .0/;
I+ .0; / 2 3 3ı b = ; 1 e b = + .1 A /W I 4 + 3ı b 4 + 3ı b
(49)
(51)
with temperature Teff determined by the condition (44)
I .0; / 1 b = 2 + 3 = + 1 e b = e .1 A /W I W 4 + 3ı b 3ı b b = e ; (45) 4 + 3ı b where we change ! for I (0, ). The emergent intensities normalized by (1 A )W I are shown in figure 3 for various values of ı. We can see several prominent properties in figure 3. First, as expected, the anisotropic scattering enhances the forward intensity, the downward intensity I (solid curves) in the present case, where the top-side is illuminated. Conversely, the backward/upward intensity I+ (dashed curves) decreases due to anisotropic scattering. The limb-darkening effect is also noted. Seen from the bottom-side, the mean intensity, J , increases toward the inward (upward) direction, as can be seen in figure 2. Hence, seen from the bottom-side, the
4 = .1 A/WH T4 ; Teff
(52)
where A is the albedo, WH the dilution factor for the flux (see Fukue 2012), and T is the surface temperature of the central object, which is assumed to radiate uniform intensity I = T4 = and specific intensity, I = B (Teff ). In this case, for transfer in the vertical direction, equation (50) can be expressed as 1 d2 .J B / = ı " .J B /: 3 d 2
(53)
If the photon destruction probability, " , has the same value at all depths, this linear differential equation (53) is easily integrated to yield the general solution, J B = C1 e
p 3ı "
where C1 and C2 = (1=3ı )dJ =d .
+ C2 e
p 3ı "
;
are integral constants,
(54) and H
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2.0
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6.0
(a)
(b)
4.0
Jν
f 1.0
2.0
Bν
f Jν
2.0
4.0
6.0 τν
Hν
-2.0
10Hν 0 0
0 Bν
8.0 10.0
-4.0 0
2.0
4.0
6.0
τν
8.0 10.0
Fig. 4. Radiative quantities normalized by B (0) as a function of the optical depth: (a) the small irradiation case of (1 A )W I =B (0) = 0.1, and (b) the large irradiation case of (1 A )W I =B (0) = 10. Solid curves represent J , dashed ones H , and the Planck function B is constant. Thin curves are for the isotropic scattering case (a = 0 and ı = 1), and thick ones for the anisotropic case (a = 3). The optical depth is b = 1, and the values of the parameter " are 1, 0.5, and 0.1 from top to bottom for J , while from bottom to top for H in (a), and vice versa in (b).
For the present problem, the boundary conditions at the disk surface ( = 0) and equator ( = b ) become, respectively, J .0/ 2H .0/ = .1 A /W I ; H . b / = 0;
(55) (56)
where A is the frequency-dependent albedo, and W the total dilution factor defined by W = WJ + 2WH , WJ and WH being the dilution factors for the mean intensity and flux, respectively [see Fukue (2012)]. In the present case the concrete expressions of dilution factors are not necessary. Under these boundary conditions, the integral constants are determined as C1 = C ; C2 = C e
p 2 3ı " b
(57) ;
(58)
where C is defined as C
B .0/ .1 A /W I p : 2 p 2 p 1+ 3ı " + 1 3ı " e 2 3ı " b 3ı 3ı (59)
Hence, the analytical solutions for the radiative moments become p p p J = B .0/ C e 3ı " + e 3ı " 2 3ı " b ; (60) p p p 3ı " p3ı " H = C e e 3ı " 2 3ı " b : (61) 3ı In the optically thick limit of b ! 1, these become J B .0/
B .0/ .1 A /W I p3ı " e ; 2 p 1+ 3ı " 3ı
(62)
p 3ı " B .0/ .1 A /W I p3ı " e ; (63) 2 p 3ı 1+ 3ı " 3ı or at the disk surface ( = 0) 2 p 3ı " B .0/ + .1 A /W I 3ı J .0/ ; (64) 2 p 1+ 3ı " 3ı p 3ı " B .0/ .1 A /W I : (65) H .0/ 2 p 3ı 1+ 3ı " 3ı In the numerator of equation (64), the first term is the thermal component of the disk, which is preprocessed radiation from the p proportional to " =ı , instead of the usual " -law, while the second term represents the irradiated component of the diluted radiation from the central star. These analytical solutions, (60) and (61), normalized by the thermal value, B (0), are shown in figure 4 for small and large irradiation cases. In figure 4a radiative quantities for the small irradiation case are shown as a function of the optical depth, , for the isotropic case (a = 0, thin curves), and for the anisotropic case (a = 3, thick ones). The disk optical depth is infinite, and the values of the parameter, " , are 1, 0.5, and 0.1 from top to bottom for J , while from bottom to top for H . Comparing the isotropic scattering case (thin curves), the anisotropic solutions (thick ones) show a slight decrease in the mean intensity, J , while increasing the flux H , and the difference becomes large as " becomes large. This is just the anisotropic scattering effect. In general, the scattering changes the effective optical depth. As can be seen in the solution (62) for J , anisotropic scattering enhances this effect; the effective H
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Radiative Transfer in Anisotropic Scattering Media
optical depth becomes p p 3" ! 3ı " :
(66)
In other words, the thermalization length Λ becomes longer in the anisotropic case as 1 1 !p : p 3" 3ı "
(67)
This is the reason of the changes in J and partly in H . In addition, the mean flux (63) increases by a factor of 1=ı due to forward scattering. This is the another reason for the change in H . In figure 4b, radiative quantities for the large irradiation case are shown as a function of the optical depth, , for the isotropic case (a = 0, thin curves), and for the anisotropic case (a = 3, thick ones). The disk optical depth is infinite, and the values of the parameter " are 1, 0.5, and 0.1 from bottom to top for J , while from top to bottom for H . In this case, the behavior of the solutions is remarkably different from the small-irradiation case of figure 4a. The mean intensity, J , increases as the optical depth decreases, while the flux, H , is negative. This is just the irradiation effect (Fukue 2012). In addition, comparing the isotropic scattering case (thin curves), the anisotropic solutions (thick ones) further increase or decrease as " becomes large. This is understood as being due to the change of the thermalization length (J and H ), and due to the forward scattering effect (H ). Using the above solutions for radiative moments, we can also solve the transfer equation (46) to obtain the intensity, I ( ). After some manipulations, we obtain the outward intensity, I+ , and the inward one, I , for the present isothermal case with finite optical depth: I+ . ; / = I+ . b ; /e . b /= + B .0/Œ1 e . b /= .1 " /.C C0 / p 1 + 3ı " i h p p e 3ı " e =.1+ 3ı " / b = .1 " /.C + C0 / p 1 3ı " i h p p p e 3ı " 2 3ı " b e =.1+ 3ı " / b = :
(68)
I . ; / = I .0; /e = + B .0/Œ1 e = i .1 " /.C C0 / h p3ı " e = e p 1 + 3ı " .1 " /.C + C0 / p 1 3ı " i hp p p e 3ı " 2 3ı " b e =2 3ı " b ; (69) where C0 = C
2a 4 1+ a 3
p 3ı " : 3ı
(70)
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In a geometrically thin disk with a finite optical depth, b , with an external irradiation of intensity I , the boundary conditions at the disk equator ( = b ) and the disk surface ( = 0) respectively become I+ . b ; / = I . b ; /; I .0; / = .1 A /I .0; /;
(71) (72)
where I ( b , ) is the inward intensity from the backside of the disk beyond the midplane. After some manipulations, we finally obtain the outward intensity: I+ . ; / = .1 A /I e . 2 b /= + B .0/Œ1 e . 2 b /= .1 " /.C C0 / p 1 + 3ı " h p i p e 3ı " e . 2 b /=2 3ı " / b .1 " /.C + C0 / p 1 3ı " h p i p e 3ı " 2 3ı " b e . 2 b /= :
(73)
Finally, the emergent intensity, I (0, ), emitted from the disk surface for the finite optical depth becomes I+ .0; / = .1 A /I e 2 b = + B .0/Œ1 e 2 b = i p .1 " /.C C0 / h 1 e .2+2 3ı " / b = p 1 + 3ı " i .1 " /.C + C0 / h 2p3ı " b e 2 b = : e p 1 3ı "
(74)
The emergent intensities normalized by the thermal value, B (0), are shown in figure 5 for the small and large irradiation cases. In figure 5a the emergent intensity for the small irradiation case is shown as a function of for the isotropic case (a = 0, thin curves), and for the anisotropic case (a = 3, thick ones). The disk optical depth is infinite, and the values of the parameter " are 1, 0.5, 0.1, and 0.01 from top to bottom. In the present isothermal case, there is no limb-darkening effect in its usual meaning. As the photon destruction probintensity decreases. This ability, " , decreases, the emergent p is the scattering effect (the " -law). In addition, the emergent intensity under scattering decreases as decreases. This is not the usual limb-darkening effect, but also the scattering effect, as already stated in a previous study [Fukue (2012)]. Furthermore, in the anisotropic case (thick curves), the emergent intensity is larger than that for the isotropic case (thin ones). This is an effect of forward scattering. In figure 5b the emergent intensity for the large irradiation case is shown as a function of for the isotropic case (a = 0, thin curves), and for the anisotropic case (a = 3, thick ones). The disk optical depth is infinite, and the values of the parameter " are 1, 0.5, 0.1, and 0.01 from bottom to top. In this case the emergent intensity increases due to scattering. In addition, the anisotropic scattering decreases the
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2.0
10.0
(a) εν 1
1.0
(b) εν 0.01
8.0 Intensity
Intensity
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0.5 0.1
6.0
0.1
4.0 0.5
2.0
0.01 0 0
0.2
0.4
μ
0.6
0.8
1.0
1
0 0
0.2
0.4
μ
0.6
0.8
1.0
Fig. 5. Emergent intensities normalized by B (0) as a function of : (a) small irradiation case of [(1 A )W I =B (0) = 0.1], and large irradiation case of [(1 A )W I =B (0) = 10]. Thin curves are for the isotropic scattering case (a = 0 and ı = 1), and thick ones for the anisotropic case (a = 3). The optical depth is b = 1, and the values of the parameter, " , are 1, 0.5, 0.1, and 0.01 from top to bottom in (a), and from bottom to top in (b).
emergent intensity now. This is because in this irradiation dominated case the forward anisotropic scattering enhances the inward/forward intensity, but reduces the outward/backward intensity shown in figure 5b. It should be noted that in the pure scattering case (" = 0) with semi-infinite optical depth ( b = 1), from equation (74), I+ .0; / = .1 A /W I ;
2.0
1 0.5 0.1
εν
(75)
as also shown in figure 5.
3f 1.0 0.1 0.5 1
7. Concluding Remarks In this paper we have examined radiative transfer in anisotropically scattering media from the analytical viewpoint. Transfer and moment equations in anisotropically scattering media are formulated under a simple linear model using the Eddington approximation. In the transfer equation the term including the radiative flux appears, and the first moment equation has some reduction factor due to anisotropic scattering, while the zeroth moment equation does not change, since the scattering processes exchange momentum, but not energy. When forward scattering dominates, the radiative intensity is enhanced in some cases, while the gradient of the mean intensity decreases in other cases, depending on the boundary conditions for the problem. In any case, anisotropic scattering makes the effective optical depth shorter, and therefore the thermalization length longer. These analytic properties are helpful to study and understand the radiative properties in various situations in astrophysics related with anisotropic scattering. In this paper we suppose that the scattering process is anisotropic, but assume that the radiation field is isotropic (Eddington approximation). Since we obtain the radiation intensity, we can recalculate the Eddington factor and check the accuracy of the Eddington approximation.
0 0
2
4
τ
6
8
10
Fig. 6. Recalculated Eddington factor multiplied 3 as a function of the optical depth. The dashed curves are for the small irradiation case of [(1 A )W I =B (0) = 0.1], whereas the solid ones are for the large irradiation case of [(1 A )W I =B (0) = 10]. The disk optical depth is b = 1, and the values of the parameter " are 1, 0.5, 0.1 from bottom to top for the small irradiation case, while from top to bottom for the large irradiation case. The isotropic case (a = 0) and anisotropic one (a = 3) are almost overlap.
In the case of the stellar atmosphere under RE, discussed in section 4, the Eddington factor is slightly larger than 1=3 in the optically thin region of 1 by a factor 10 percent or so, as in the usual stellar case. In the case of an illuminated cloud with finite optical depth, discussed in section 5, the Eddington factor
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Radiative Transfer in Anisotropic Scattering Media
is slightly smaller than 1=3 by a factor ten percent, or less. In the case of the irradiated accretion disk with finite optical depth under LTE discussed in section 6, the recalculated Eddington factor multiplied by 3 is shown in figure 6. In the small irradiation case (dashed curves), the Eddington factor is slightly smaller than 1=3 in the optically thin region. In the large irradiation case (solid curves), on the other hand, the Eddington factor becomes larger than 1=3 up to about 30 percent at around 1 for the scattering-free case (" 1). For the scattering-dominated case (" 0), however, the Eddington factor approaches 1=3, even in the optically thin region. It should be noted that the isotropic case (a = 0) and the anisotropic one (a = 3) almost overlap. That is, the anisotropy
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of scattering does not affect so much the anisotropy of the radiation field. The anisotropy of scattering considered in this paper is not so extreme, but rather mild, in the sense that it is expressed in the quadratic form of the direction cosine. In the extremely anisotropic case, another formulation may be necessary. The author would like to thank an anonymous referee for his valuable comments, which improved the original manuscript. This work has been supported in part by a Grant-in-Aid for Scientific Research (C) of the Ministry of Education, Culture, Sports, Science and Technology (22540251).
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