Abstract

An exact expression for the emissivity of an ideally diffuse, gray, isothermal, spherical cavity is derived directly, making no geometrical approximations, and is shown to agree with the results of DeVos, Gouffé, and Sparrow and Jonsson, as compared by Fecteau. It appears that, even under ideal conditions, the spherical configuration is the only one that will have uniform isotropic emissivity over a wide angle, approaching a full hemisphere, across the entire aperture.

© 1968 Optical Society of America

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References

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  1. M. L. Fecteau, Appl. Opt. 7, 1363 (1968).
    [CrossRef] [PubMed]
  2. J. C. DeVos, Physica 20, 669 (1954).
    [CrossRef]
  3. André Gouffé, Rev Opt. 24, 1 (1945).
  4. E. M. Sparrow, V. K. Jonsson, J. Heat Transfer 84, 188 (1962); also NASA Tech. Note D–1289, National Aeronautics and Space Administration, Washington, D.C. (June1962).
    [CrossRef]
  5. F. E. Nicodemus, Appl. Opt. 4, 767 (1965).
    [CrossRef]
  6. W. L. Wolfe, F. E. Nicodemus, in Handbook of Military Infrared Technology, W. L. Wolfe, Ed. (Office of Naval Research, Washington, D.C., 1965), Chap. 2.
  7. F. E. Nicodemus, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic Press Inc., New York, 1967), Vol. 4, Chap. 8.
  8. F. E. Nicodemus, Am. J. Phys. 31, 368 (1963).
    [CrossRef]
  9. W. E. Sumpner, Proc. Phys. Soc. (London) 12, 10 (1892); also Phil. Mag. 35, 81 (1893).
    [CrossRef]

1968 (1)

1965 (1)

1963 (1)

F. E. Nicodemus, Am. J. Phys. 31, 368 (1963).
[CrossRef]

1962 (1)

E. M. Sparrow, V. K. Jonsson, J. Heat Transfer 84, 188 (1962); also NASA Tech. Note D–1289, National Aeronautics and Space Administration, Washington, D.C. (June1962).
[CrossRef]

1954 (1)

J. C. DeVos, Physica 20, 669 (1954).
[CrossRef]

1945 (1)

André Gouffé, Rev Opt. 24, 1 (1945).

1892 (1)

W. E. Sumpner, Proc. Phys. Soc. (London) 12, 10 (1892); also Phil. Mag. 35, 81 (1893).
[CrossRef]

DeVos, J. C.

J. C. DeVos, Physica 20, 669 (1954).
[CrossRef]

Fecteau, M. L.

Gouffé, André

André Gouffé, Rev Opt. 24, 1 (1945).

Jonsson, V. K.

E. M. Sparrow, V. K. Jonsson, J. Heat Transfer 84, 188 (1962); also NASA Tech. Note D–1289, National Aeronautics and Space Administration, Washington, D.C. (June1962).
[CrossRef]

Nicodemus, F. E.

F. E. Nicodemus, Appl. Opt. 4, 767 (1965).
[CrossRef]

F. E. Nicodemus, Am. J. Phys. 31, 368 (1963).
[CrossRef]

W. L. Wolfe, F. E. Nicodemus, in Handbook of Military Infrared Technology, W. L. Wolfe, Ed. (Office of Naval Research, Washington, D.C., 1965), Chap. 2.

F. E. Nicodemus, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic Press Inc., New York, 1967), Vol. 4, Chap. 8.

Sparrow, E. M.

E. M. Sparrow, V. K. Jonsson, J. Heat Transfer 84, 188 (1962); also NASA Tech. Note D–1289, National Aeronautics and Space Administration, Washington, D.C. (June1962).
[CrossRef]

Sumpner, W. E.

W. E. Sumpner, Proc. Phys. Soc. (London) 12, 10 (1892); also Phil. Mag. 35, 81 (1893).
[CrossRef]

Wolfe, W. L.

W. L. Wolfe, F. E. Nicodemus, in Handbook of Military Infrared Technology, W. L. Wolfe, Ed. (Office of Naval Research, Washington, D.C., 1965), Chap. 2.

Am. J. Phys. (1)

F. E. Nicodemus, Am. J. Phys. 31, 368 (1963).
[CrossRef]

Appl. Opt. (2)

J. Heat Transfer (1)

E. M. Sparrow, V. K. Jonsson, J. Heat Transfer 84, 188 (1962); also NASA Tech. Note D–1289, National Aeronautics and Space Administration, Washington, D.C. (June1962).
[CrossRef]

Physica (1)

J. C. DeVos, Physica 20, 669 (1954).
[CrossRef]

Proc. Phys. Soc. (London) (1)

W. E. Sumpner, Proc. Phys. Soc. (London) 12, 10 (1892); also Phil. Mag. 35, 81 (1893).
[CrossRef]

Rev Opt. (1)

André Gouffé, Rev Opt. 24, 1 (1945).

Other (2)

W. L. Wolfe, F. E. Nicodemus, in Handbook of Military Infrared Technology, W. L. Wolfe, Ed. (Office of Naval Research, Washington, D.C., 1965), Chap. 2.

F. E. Nicodemus, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic Press Inc., New York, 1967), Vol. 4, Chap. 8.

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Figures (2)

Fig. 1
Fig. 1

Spherical cavity configuration. A plane section through the center O of a spherical cavity AP0PB with an aperture ADB (perpendicular to this plane section); ACB is the spherical cap that would just cover the aperture to complete the sphere. A ray of radiance Nw is shown emitted by an element dA of the internal wall at P. If Na denotes its radiance as it passes through the aperture and there is no attenuation in the empty cavity, Na = Nw.

Fig. 2
Fig. 2

Solid angle subtended by surface element dA of a spherical surface at P, a point on the same spherical surface.

Equations (27)

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= 1 - ρ d = 1 - π ρ [ dimensionless ] .
a = N a ( T ) / N B ( T ) [ dimensionless ] ,
N w = N a = N e + N r 1 + N r 2 + + N r n + [ W · cm - 2 · sr - 1 ] ,
H i = N e Ω ' w [ W · cm - 2 ] ,
N r 1 = ρ H i = ρ N e Ω w [ W · cm - 2 · sr - 1 ]
N r 2 = ρ N r 1 Ω w = N e ( ρ Ω w ) 2 [ W · cm - 2 · sr - 1 ] .
N r n = ρ N r ( n - 1 ) Ω w = N e ( ρ Ω w ) n [ W · cm - 2 · sr - 1 ] .
N a = N e [ 1 + ( ρ Ω w ) + ( ρ Ω w ) 2 + + ( ρ Ω w ) n + ] = N e / ( 1 - ρ Ω w ) = N B ( T ) / [ 1 - ( 1 - ) Ω w / π ] [ W · cm - 2 · sr - 1 ] .
Ω c / π = s / S [ dimensionless ] ,
Ω w / π = ( π - Ω c ) / π = ( 1 - s / S ) [ dimensionless ] .
a = N a / N B = / { 1 - ( 1 - ) [ 1 - ( s / S ) ] } = / { ( s / S ) + [ 1 - ( s / S ) ] } [ dimensionless ] .
a ( λ ) = ( λ ) / { ( s / S ) + ( λ ) [ 1 - ( s / S ) ] } [ dimensionless ] .
a ( T ) = [ 1 / N B ( T ) ] 0 a ( λ ) N λ , B ( λ , T ) d λ = [ 1 / N B ( T ) ] 0 ( λ ) N λ , B ( λ , T ) d λ ( s / S ) + ( λ ) [ 1 - ( s / S ) ] [ dimensionless ] .
( T ) = [ 1 / N B ( T ) ] 0 ( λ ) N λ , B ( λ , T ) d λ [ dimensionless ] .
P = A Ω N ( x , y , θ , φ ) cos θ d Ω d A [ W ] ,
N 2 P / cos θ A Ω [ W · cm - 2 · sr - 1 ] ,
P = N A Ω cos θ d Ω d A = N J [ W ] ,
J = A Ω cos θ d Ω d A [ cm 2 · sr ]
J = A Ω d A [ cm 2 · sr ] ,
Ω = Ω d Ω = Ω cos θ d Ω = Ω cos θ sin θ d θ d φ [ sr ]
J = Ω A d A = A Ω [ cm 2 · sr ] .
N = P / J [ W · cm - 2 · sr - 1 ] .
d Ω = d A cos θ / ( A P ¯ ) 2 = d A cos θ / ( 2 R cos θ ) 2 [ sr ] = d A / ( 4 R 2 cos θ ) [ sr ] ,
d Ω = cos θ d Ω = d A / 4 R 2 [ sr ] .
Ω = d Ω = ( 4 R 2 ) - 1 d A = A / 4 R 2 [ sr ] ,
Ω h = 4 π R 2 / 4 R 2 = π [ sr ] .
Ω a / π = s / S [ dimensionless ] .

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