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UDC 523.035.2
ASTRONOMY
Corresponding Member of the USSR Academy of Sciences V. V. SOBOLEV
DIFFUSION OF RADIATION IN A MEDIUM OF LARGE OPTICAL THICKNESS WITH ANISOTROPIC SCATTERING
In the study of planetary atmospheres and water basins one encounters the problem of the diffusion of radiation in a plane layer of large optical thickness illuminated by parallel rays. For the case of a spherical scattering indicatrix this problem was solved earlier (¹). Now, by the same method, it is solved for the case of anisotropic scattering of light. In the first part of the present note the radiation field in the deep layers of a semi-infinite medium is determined; in the second, asymptotic formulas are derived for the intensity of radiation emerging from a plane layer of large optical thickness.
- Let parallel rays, making an angle \(\arccos \zeta\) with the normal, fall on a semi-infinite medium and produce an illumination of the area perpendicular to them equal to \(\pi S\). Denote by \(I(\tau,\eta,\zeta)\) the azimuth-averaged intensity of the diffuse radiation traveling at optical depth \(\tau\) at an angle \(\arccos \eta\) to the normal, and by \(B(\tau,\eta,\zeta)\) the azimuth-averaged “source function.” As is known, these quantities are related by the equations:
\[ \eta\, dI(\tau,\eta,\zeta)/d\tau=-I(\tau,\eta,\zeta)+B(\tau,\eta,\zeta), \tag{1} \]
\[ B(\tau,\eta,\zeta)=\frac{\lambda}{2}\int_{-1}^{+1}p(\eta,\eta')I(\tau,\eta',\zeta)\,d\eta' +\frac{\lambda}{4}S p(\eta,\zeta)e^{-\tau/\zeta}; \tag{2} \]
where \(\lambda\) is the photon survival probability in an elementary act of scattering,
\[ p(\eta,\eta')=\frac{1}{2\pi}\int_{0}^{2\pi}\chi\!\left(\eta\eta' +\sqrt{(1-\eta^2)(1-\eta'^2)}\cos\varphi\right)d\varphi, \tag{3} \]
\(\chi(\cos\gamma)\) is the scattering indicatrix.
From (1) and (2), with the boundary condition \(I(0,\eta,\zeta)=0\) for \(\eta>0\), we obtain an integral equation for determining the source function:
\[ \begin{aligned} B(\tau,\eta,\zeta) ={}&\frac{\lambda}{2}\int_{0}^{1}p(\eta,\eta')\,d\eta' \int_{0}^{\tau} B(\tau',\eta',\zeta)e^{-(\tau-\tau')/\eta'}\frac{d\tau'}{\eta'} \\ &+\frac{\lambda}{2}\int_{0}^{1}p(\eta,-\eta')\,d\eta' \int_{\tau}^{\infty} B(\tau',-\eta',\zeta)e^{-(\tau'-\tau)/\eta'}\frac{d\tau'}{\eta'} \\ &+\frac{\lambda S}{4}p(\eta,\zeta)e^{-\tau/\zeta}. \end{aligned} \tag{4} \]
For the deep layers of a semi-infinite medium we have
\[ B(\tau,\eta,\zeta)=Sc(\zeta)b(\eta)e^{-k\tau}, \tag{5} \]
\[ I(\tau,\eta,\zeta)=Sc(\zeta)i(\eta)e^{-k\tau}, \tag{6} \]
where
\[ i(\eta)=b(\eta)/(1-k\eta). \tag{7} \]
In this case equation (4) can be written in the form
\[ B(\tau,\eta,\xi)=\frac{\lambda}{2}\int_{0}^{1}p(\eta,\eta')\,d\eta' \int_{-\infty}^{\tau}B(\tau',\eta',\xi)e^{-(\tau-\tau')/\eta'}\frac{d\tau'}{\eta'} + \frac{\lambda}{2}\int_{0}^{1}p(\eta,-\eta')\,d\eta' \int_{\tau}^{\infty}B(\tau',-\eta',\xi)e^{-(\tau'-\tau)/\eta'}\frac{d\tau'}{\eta'} . \tag{8} \]
Substituting (5) into (8), we arrive at an equation for determining the function \(b(\eta)\)
\[ b(\eta)=\frac{\lambda}{2}\int_{-1}^{+1}p(\eta,\eta')\, \frac{b(\eta')}{1-k\eta'}\,d\eta' . \tag{9} \]
The quantity \(k\) is found from the solvability condition for this equation. For normalizing \(b(\eta)\) one may use the condition
\[ \frac{1}{2}\int_{-1}^{+1}b(\eta)\,d\eta=1 . \tag{10} \]
Equation (9) was first obtained by V. A. Ambartsumian \((^{2})\). It was subsequently solved in works by the author \((^{3})\), Kuper \((^{4})\), and others. However, as far as we know, the function \(c(\xi)\) has not yet been determined. To determine it, let us differentiate both sides of equation (4) with respect to \(\tau\) and compare the result obtained with equation (4). Doing this, we find
\[ B'(\tau,\eta,\xi)=-\frac{1}{\xi}B(\tau,\eta,\xi) +\frac{2}{S}\int_{0}^{1}B(\tau,\eta,\eta')B(0,\eta',\xi)\frac{d\eta'}{\eta'} . \tag{11} \]
Substitution of (5) into (11) gives
\[ c(\xi)=\frac{\xi}{1-k\xi}\,\frac{2}{S}\int_{0}^{1}c(\eta)B(0,\eta,\xi)\frac{d\eta}{\eta}. \tag{12} \]
The quantity \(B(0,\eta,\xi)\) entering (12) can be found from equation (4). Setting \(\tau=0\) in it, we obtain
\[ B(0,\eta,\xi)=\frac{\lambda}{4}S\left[ 2\xi\int_{0}^{1}p(\eta,-\eta')\rho(\eta',\xi)\,d\eta' +p(\eta,\xi) \right], \tag{13} \]
where \(\rho(\eta,\xi)\) denotes the brightness coefficient of a semi-infinite medium, defined by the formula
\[ \int_{0}^{\infty}B(\tau,-\eta,\xi)e^{-\tau/\eta}\frac{d\tau}{\eta} =S\rho(\eta,\xi)\xi . \tag{14} \]
Comparing equations (12) and (13) with the equations obtained in the work of V. A. Ambartsumian \((^{5})\), we see that the function \(c(\xi)\) can be represented in the form
\[ c(\xi)=u(\xi)\xi, \tag{15} \]
where \(u(\eta)\) is the intensity of radiation diffusely transmitted by a semi-infinite medium at an angle \(\arccos\eta\) to the normal. Previously the function \(u(\eta)\) was determined only up to a constant factor. We shall now find this factor for the case of deep layers of a semi-infinite medium. To do this, we apply the following device. Substitute in (8), instead of
Replace, in \(B(\tau,\eta,\xi)\), the function by \(b(\eta)e^{-\kappa\tau}\) and write the identity obtained in the form
\[ \begin{aligned} b(\eta)e^{-\kappa\tau}={}& \frac{\lambda}{2}\int_0^1 p(\eta,\eta')\,d\eta' \int_0^\tau b(\eta')e^{-\kappa\tau'-(\tau-\tau')/\eta'}\frac{d\tau'}{\eta'} \\ &+\frac{\lambda}{2}\int_0^1 p(\eta,-\eta')\,d\eta' \int_\tau^\infty b(-\eta')e^{-\kappa\tau'-(\tau'-\tau)/\eta'}\frac{d\tau'}{\eta'} \\ &+\frac{\lambda}{2}\int_0^1 p(\eta,\eta')e^{-\tau/\eta'}i(\eta')\,d\eta' . \end{aligned} \tag{16} \]
Comparing (16) with (4), we have
\[ b(\eta)e^{-\kappa\tau}=\frac{2}{S}\int_0^1 B(\tau,\eta,\eta')i(\eta')\,d\eta' . \tag{17} \]
Relation (17) is valid for arbitrary \(\tau\). Substituting expression (5) into (17) and using formula (15), we obtain
\[ 2\int_0^1 u(\eta)i(\eta)\eta\,d\eta=1 . \tag{18} \]
Formula (18) also makes it possible to find the unknown constant factor in the function \(u(\eta)\). Thus the problem of determining the radiation field in deep layers of a semi-infinite medium is completely solved.
- Let us now consider the diffusion of radiation in a plane layer of large optical thickness \(\tau_0\). As before, suppose that the layer is illuminated by parallel rays.
We shall be interested in the intensities of the radiation emerging from the layer, or in the corresponding brightness coefficients \(\rho(\eta,\xi,\tau_0)\) and \(\sigma(\eta,\xi,\tau_0)\), related to the intensities by the relations
\[ I(0,-\eta,\xi,\tau_0)=S\rho(\eta,\xi,\tau_0)\xi,\qquad I(\tau_0,\eta,\xi,\tau_0)=S\sigma(\eta,\xi,\tau_0)\xi . \tag{19} \]
In the case under consideration (for \(\tau_0\gg 1\)), the results obtained above for deep layers of a semi-infinite medium can be used to determine the brightness coefficients. Let us make, mentally, a cut in the semi-infinite medium at a large optical depth \(\tau_0\). Then it may be assumed that a plane layer of optical thickness \(\tau_0\) is illuminated from above by parallel rays, and from below by diffuse radiation, whose intensity is determined by formula (6) at \(\tau=\tau_0\) and \(\eta<0\).
On the basis of what has been said, we have the following relations:
\[ \rho(\eta,\xi)=\rho(\eta,\xi,\tau_0)+2u(\xi)e^{-\kappa\tau_0} \int_0^1 \sigma(\eta,\eta',\tau_0)i(-\eta')\eta'\,d\eta', \tag{20} \]
\[ u(\xi)i(\eta)e^{-\kappa\tau_0} =\sigma(\eta,\xi,\tau_0)+2u(\xi)e^{-\kappa\tau_0} \int_0^1 \rho(\eta,\eta',\tau_0)i(-\eta')\eta'\,d\eta' . \tag{21} \]
Here, as above, \(\rho(\eta,\xi)\) denotes the brightness coefficient of the semi-infinite medium, assumed known.
From equations (20) and (21) we easily obtain
\[ \rho(\eta,\xi,\tau_0)=\rho(\eta,\xi)- \frac{u(\eta)u(\xi)}{1-N^2e^{-2\kappa\tau_0}}MNe^{-2\kappa\tau_0}, \tag{22} \]
\[ \sigma(\eta,\xi,\tau_0)= \frac{u(\eta)u(\xi)}{1-N^2e^{-2\kappa\tau_0}}Me^{-\kappa\tau_0}, \tag{23} \]
where
\[ Mu(\eta)=i(\eta)-2\int_0^1 \rho(\eta,\eta')i(-\eta')\eta'\,d\eta', \tag{24} \]
\[ N = 2 \int_0^1 u(\eta)i(-\eta)\eta\,d\eta . \tag{25} \]
To determine the constant \(M\), we use formula (18), as well as the relation
\[ i(-\eta)=2\int_0^1 \rho(\eta,\eta')i(\eta')\eta'\,d\eta', \tag{26} \]
which follows from (17) and (14). Multiplying (24) by \(i(\eta)\eta\) and integrating with respect to \(\eta\) from 0 to 1, with the aid of the indicated formulas we find
\[ M=2\int_{-1}^1 i^2(\eta)\eta\,d\eta . \tag{27} \]
Thus, for the desired brightness coefficients \(\rho(\eta,\xi,\tau_0)\) and \(\sigma(\eta,\xi,\tau_0)\) we have obtained the asymptotic formulas (22) and (23), in which the constants \(M\) and \(N\) are determined by formulas (25) and (27), and the function \(u(\eta)\) is normalized according to (18).
To determine the brightness coefficients in the case of pure scattering (i.e., for \(\lambda=1\)), in formulas (22) and (23) one must set \(k\to0\). For small \(k\) we have
\[ i(\eta)=1+\frac{3}{3-x_1}k\eta, \tag{28} \]
where \(x_1\) is the first coefficient in the expansion of the scattering indicatrix in Legendre polynomials. Substituting (28) into (27), we obtain
\[ M=8k/(3-x_1), \tag{29} \]
and from formulas (25) and (18) it follows that
\[ N=1-\frac{12k}{3-x_1}\int_0^1 u(\eta)\eta^2\,d\eta, \tag{30} \]
where the function \(u(\eta)\) now refers to the case \(\lambda=1\).
Substitution of (29) and (30) into (22) and (23) for \(k=0\) gives
\[ \rho(\eta,\xi,\tau_0)=\rho(\eta,\xi)-{}^{4}/_{3}u(\eta)u(\xi)/[(1-x_1/3)\tau_0+\delta], \tag{31} \]
\[ \sigma(\eta,\xi,\tau_0)={}^{4}/_{3}u(\eta)u(\xi)/[(1-x_1/3)\tau_0+\delta], \tag{32} \]
where
\[ \delta=2\int_0^1 u(\eta)\eta^2\,d\eta\bigg/\int_0^1 u(\eta)\eta\,d\eta . \tag{33} \]
Recently, in an article by van de Hulst and Grossman (6), formulas analogous to (22) and (23) were presented, but obtained by another method. They are a generalization of formulas previously obtained by the author (1, 7) for the case of isotropic scattering.
Leningrad State University
named after A. A. Zhdanov
Received
31 X 1967
CITED LITERATURE
- V. V. Sobolev, Transfer of Radiant Energy in the Atmospheres of Stars and Planets, Moscow, 1956.
- V. A. Ambartsumian, Izv. AN SSSR, ser. geogr. i geofiz., 3, 97 (1942).
- V. V. Sobolev, Izv. AN SSSR, ser. geogr. i geofiz., 8, 273 (1944).
- I. Kuscer, J. Math. and Phys., 34, 256 (1956).
- V. A. Ambartsumian, DAN, 43, 106 (1944).
- H. C. van de Hulst, K. Grossman, Proc. of Conference on the Atmospheres of Mars and Venus, Tucson, 1967.
- V. V. Sobolev, DAN, 155, 316 (1964).