Abstract

The net photon current is obtained from the exact transport solution for photon density in an infinite isotropic atmosphere. An analytical-numerical solution for the photon current is developed and presented graphically together with the diffusion theory current versus nondimensional distance.

© 1958 Optical Society of America

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References

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  1. Case, deHoffman, and Placzek, Introduction to the Theory of Neutron Diffusion (U. S. Government Printing Office, 1953).
  2. S. Chandrasekhar, Radiative Transfer (Oxford University Press, New York, 1950).
  3. P. I. Richards, “Light Scattering in Clouds,” , Technical Operations, Inc., Burlington, Massachusetts (1955).
  4. J. Le Caine, “A Table of Integrals involving the Functions En(X),” Chalk River, Ontario NRC, No. 1553.

Case,

Case, deHoffman, and Placzek, Introduction to the Theory of Neutron Diffusion (U. S. Government Printing Office, 1953).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford University Press, New York, 1950).

deHoffman,

Case, deHoffman, and Placzek, Introduction to the Theory of Neutron Diffusion (U. S. Government Printing Office, 1953).

Le Caine, J.

J. Le Caine, “A Table of Integrals involving the Functions En(X),” Chalk River, Ontario NRC, No. 1553.

Placzek,

Case, deHoffman, and Placzek, Introduction to the Theory of Neutron Diffusion (U. S. Government Printing Office, 1953).

Richards, P. I.

P. I. Richards, “Light Scattering in Clouds,” , Technical Operations, Inc., Burlington, Massachusetts (1955).

Other (4)

Case, deHoffman, and Placzek, Introduction to the Theory of Neutron Diffusion (U. S. Government Printing Office, 1953).

S. Chandrasekhar, Radiative Transfer (Oxford University Press, New York, 1950).

P. I. Richards, “Light Scattering in Clouds,” , Technical Operations, Inc., Burlington, Massachusetts (1955).

J. Le Caine, “A Table of Integrals involving the Functions En(X),” Chalk River, Ontario NRC, No. 1553.

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Figures (3)

Fig. 1
Fig. 1

General geometry of the problem.

Fig. 2
Fig. 2

Photon current relative to vacuum for a point source in an infinite medium—isotropic scattering at zero absorption.

Fig. 3
Fig. 3

Photon density relative to vacuum for point source in infinite medium—isotropic scattering with zero absorption.

Equations (20)

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n ( r ¯ ) = n d ( r ¯ ) + c - n ( r ¯ ) e - r - r / λ 4 π ( r ¯ - r ¯ ) 2 λ d 3 r ¯ ,
n ( r ) = 1 4 π r [ k 0 c 2 e - α 0 r + 0 1 g ( c , μ ) e - r / μ d μ μ 2 ] ,
g ( c , μ ) = 1 ( 1 - c μ tan h - 1 μ ) 2 + [ ( π / 2 ) c μ ] 2 ,
1 - c tan - 1 k 0 k 0 = 0 ,
c = α 0 tan h - 1 α 0 ,
J + = j 0 + + j 1 + ,
c d A cos ϕ n ( R ) e - r d 3 R 4 π r 2 .
j 1 + = c n ( R ) e - r d 3 R cos ϕ 4 π r 2 .
r 2 = R 2 + D 2 - 2 D R cos ψ ,
cos ϕ = D - R cos ψ ( R 2 + D 2 - 2 D R cos ψ ) 1 2 .
j 1 + = c θ = 0 2 π ψ = 0 π R = 0 D n ( R ) exp [ - ( R 2 + D 2 - 2 D R cos ψ ) 1 2 ] 4 π ( R 2 + D 2 - 2 D R cos ψ ) { D - R cos ψ ( R 2 + D 2 - 2 D R cos ψ ) 1 2 } sin ψ R 2 d R d ψ d θ + c θ = 0 2 π ψ = cos - 1 ( D / R ) π R = D n ( R ) exp [ - ( R 2 + D 2 - 2 D R cos ψ ) 1 2 ] 4 π ( R 2 + D 2 - 2 D R cos ψ ) { D - R cos ψ ( R 2 + D 2 D R cos ψ ) 1 2 } sin 2 ψ R 2 d R d ψ d θ .
a = ( R 2 + D 2 - 2 D R cos ψ ) ,
a d a R D = sin ψ d ψ .
j 1 = c 4 D 2 a = D - R a = D + R R = 0 R = D n ( R ) e - a a 2 { D 2 - R 2 + A 2 } d a d R + c 4 D 2 a = ( R 2 - D 2 ) 1 2 R + D R = D n ( R ) e - a a 2 { D 2 - R 2 + a 2 } d a d R ,
j 1 + = c 4 D 2 R = 0 D n ( R ) R { ( D + R ) E 2 ( D - R ) - ( D - R ) E 2 ( D + R ) - e - ( D + R ) + e - ( D - R ) } d R + c 4 D 2 R = D n ( R ) R { - ( R 2 - D 2 ) 1 2 E 2 ( R 2 - D 2 ) + ( R - D ) E 2 ( R + D ) - e - ( D + R ) + exp [ - ( R 2 - D 2 ) 1 2 ] } d R ,
E n ( X ) = 1 e - x u u n d u
j 1 + = R = 0 ξ ( R ) d R + R = D ξ ( R ) d R + R = D ξ ( R ) d R ,
R = 0 ξ ( R ) d R = c 4 D 2 R = 0 n ( R ) R { ( D + R ) E 2 ( D - R ) - ( D - R ) E 2 ( D + R ) - e - ( D + R ) + e - ( D - R ) } d R c 16 π D 2 R = 0 { E 2 ( D ) ( D R + 1 ) - E 2 ( D ) ( D R - 1 ) - e - D [ e - R R - e R ] } d r c 8 π D 2 [ E 2 ( D ) + e - D ] R = 0 d R σ 8 π D 2 σ 1 [ E 2 ( D ) + e - D ] .
J + ( D ) = j 0 + + j 1 + ,
= 0.1 D , R = 0.1 D .