Radiative Transfer in Spherically and Cylindrically Symmetric Media

Abstract

The theory of light scattering and radiative transfer in various media is successfully used to solve scientific and practical problems of astrophysics, geophysics, oceanology, and other fields of science that study the interaction of the electro-magnetic radiation with the matter. Methods of this theory are also used to study the propagation of particles in a substance, in particular, in the study of the diffusion of neutrons. These methods are also applied in studying heat transfer processes.

Appendices

Appendix A: The Results of Calculation by Formula (33)

The radiation intensity I(τ, μ) in an infinite homogeneous dusty nebula surrounding a star, was calculated according to the formula (39) with values of λ = 0.9 and λ = 1 for three phase functions: A − x(γ) = 1, B − x(γ) = 3/4(1 + cos² γ), and C—x(γ) = 1 + cosγ + P ₂(cosγ). The calculations have been based that $\frac{{L\alpha^{2} }}{{8\pi^{2} }} = 1$. The calculation results are shown in Table 1 (the case λ = 0.9) and in Table 2 (the case λ = 1) for different values of the arguments τ and μ. The values of I(τ, µ) if τ > 10 in case B (the Rayleigh phase function) in Table 2 is not shown as within the accepted accuracy they coincide with the corresponding values for isotropic scattering.

Table 1 Values of I(τ, μ) for λ = 0.9

Table 2 Values of I(τ, μ) for λ = 1

When λ < 1, the radiation intensity decreases rapidly with increasing τ. This is because when τ ≫1 the values of I(τ, μ) is proportional to $\frac{{{\text{e}}^{ - k\tau } }}{\tau }$. Differences in the rate of reduction of I(τ, μ) with increasing τ for different scattering phase functions caused mainly by the difference in the values of k (if λ = 0.9 for phase functions A, B, C the values of k are, respectively, 0.5254; 0.5232; 0.4361). When λ = 1, k = 0, the decrease of the values of I(τ, μ) with increasing τ is much slower.

The values I(τ, μ) increase which increasing μ. It is especially noticeable when λ ≈ 1. The radiation intensity becomes infinite in the direction of radiation entered a given point of medium directly from the star (when μ = 1).

Appendix B: The Exactness of Approximate Expressions for the Mean Radiation Intensity and the Radiation Flux

The nonstationary radiation field in a one-dimensional homogeneous infinite medium with a point energy source is calculated in the cases A (t ₁ ≫ t ₂) and B (t ₂ ≫ t ₁). Obtained exact values for the mean radiation intensity $J_{A} (\tau ,u)$, $J_{B} (\tau ,u)$ and the radiation flux $H_{A} (\tau ,u)$, $H_{B} (\tau ,u)$ are compared with corresponding values $J_{D} (\tau ,u)$ and $H_{D} (\tau ,u)$ founded in the diffusion approximation. Results of the comparison are shown in Tables 3, 4, and 5.

Table 3 Ratios of J _A /J _D and H _A /H _D for λ = 1

Table 4 Values of $J_{A} (\tau ,u)$,$J_{D} (\tau ,u)$, $J_{B} (\tau ,u)$, for λ = 0.5

Table 5 Values of $H_{A} (\tau ,u)$, $H_{D} (\tau ,u)$, $H_{B} (\tau ,u)$, for λ = 0.5

Consider first the case A. In the case of pure scattering (i.e., λ = 1), asymptotic expression (197) and (198) when и ≫ 1 just coincide with formulas (199) and (200) for $J_{D} (\tau ,u)$ and $H_{D} (\tau ,u)$. Exact and approximate values of J and H are quite close to each other (see Table 3). So we can assume that in the case A when λ = 1, the application of diffusion approximation will not lead to large errors.

Another situation that occurs is in the presence of true absorption, i.e., when λ < 1. The exact values of $J_{A} (\tau ,u)$ and $H_{A} (\tau ,u)$ may differ significantly from their asymptotic values. Asymptotic expressions (197) and (198) also differ from expressions (199) and (200). The radio J _A /J _D may differ significantly from unity.

Addressing the case B, we first of all note that in this case always и ≫ τ, e.g. $J_{B} (\tau ,u) = 0$ and $H_{B} (\tau ,u) = 0$ when и < τ. But the diffusion equation does not take into account the finite speed of light. Taking into account only и > τ, approximately the same results, as in case A, can be obtained by exact and approximate expressions. When λ = 1, an asymptotic expressions for $J_{B} (\tau ,u)$ and $H_{B} (\tau ,u)$ coincides with expressions (199) and (200) for $J_{D} (\tau ,u)$ and $H_{D} (\tau ,u)$.

The ratio J _B /J _D and H _B /H _D greatly depends on λ (see Tables 4 and 5).

From the above it is concluded that the replacement of the radiation transfer equation to the diffusion equation should give satisfactory results when λ ≈ 1 but may lead to significant errors when λ < 1.