Topics in Particle and Dispersion Science

Solving RTE in the small-angle scattering approximation for a medium illuminated at normal incidence

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The RTE expressed in a simplfied form in Eq. 3 of Radiative transfer equation (RTE) in the small-angle approximation .. can be solved by expanding both the azimuthally-averaged phase function, p_a (defined in Eq. 2 of that topic), and the radiance, L, into a series of Legendre polynomials:

pa(κ', κ ) = ∑ j = 0 to ∞ hj Pj(κ) Pj(κ' )

(1)

where κ and κ' are the cosines of the elevation angles of the observation and incidence directions respectively, (see also Fig. 1), h_j are the expansion coefficients of the phase function, p(θ), with θ being the scattering angle [a shortcut notation to replace the (κ', κ ) pair]:

p(θ ) = ∑ j = 0 to ∞ hj Pj(cos θ)

(2)

and:

L(κ, τ ) = ∑ j = 0 to ∞ vj(τ) Pj(κ)

(3)

and substituting the expansions into Eq. 3 of Radiative transfer equation (RTE) in the small-angle approximation .. . With these substitutions, one readily obtains:

dvj / dτ = - vj + ω0 hj / (2j + 1)

(4)

with the use of the orthogonality condition for the Legendre polynomials:

∫-11 Pi(κ) Pj(κ) dκ = 2 δij / (2j + 1)

(5)

where δ_ij = 1 for i = j and equals zero otherwise. By integrating Eq. 4 one obtains:

vj(τ) = Aj exp( -cj τ )

(6)

where A_j and

cj = 1 - ω0 hj / (2j + 1)

(7)

are constants to be determined from the boundary conditions. In particular, by assuming that

L(κ, 0) = E0δ(1 - κ),

(8)

where E₀ is the irradiance at τ = 0, and δ(x) is the delta function, and expanding the latter in terms of Legendre polynomials:

δ(1 - κ) = [1 / (4π)] ∑ j = 0 to ∞ (2j + 1) Pj(κ)

(9)

one obtains:

Aj = [E0 / (4π)] (2j + 1)

(10)

Note that the use of irradiance in Eq. 8 is required by the dimension of the delta function, as applicable here, i.e. sr ^-1. Thus, one obtains an analytical solution for the transmitted radiance as a function of the optical thickness of the turbid medium and the observation angle, γ (Fig. 1), as represented by its cosine, κ:

L(κ, τ) = [E0 / (4π)] ∑ j = 0 to ∞ (2j + 1) exp( -cj τ ) Pj(κ)

(11)

I discuss a sample application of the small-angle approximation in Transmission function in the small-angle approximation for a medium illuminated at normal incidence.

CITATION: Kokhanovsky A. A. 2007. Solving RTE in the small-angle scattering approximation for a medium illuminated at normal incidence (www.tpdsci.com/Tpc/RTESAANmlSol.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 14-Feb-2007 Modified: 05-Feb-2007 Peer-reviewed: PENDING

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