Abstract

The quasi-random model of band absorption provides a means for the accurate representation of molecular absorption over finite frequency intervals. It does not suffer from the limitations of the Elsasser and statistical models. The spacing between actual spectral lines is not constant nor does it vary in a random manner. The essential features of the actual arrangement of these lines are reproduced in the quasi-random model by the division of the frequency interval of interest into much smaller intervals within which the lines are assumed to be arranged at random. The actual intensity distribution including the important effect of the numerous weak lines is accurately simulated. A new expression for the transmittance related to a finite frequency interval is derived. The absorption from the wings of the spectral lines in other intervals is taken into account. Through these and other features the quasi-random model provides a means of accurately representing molecular absorptance over finite frequency intervals.

© 1962 Optical Society of America

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References

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  1. W. M. Elsasser, Phys. Rev. 54, 126 (1938).
    [Crossref]
  2. H. Mayer, “Methods of Opacity Calculations,” Los Alamos, LA-647 (1947).
  3. R. M. Goody, Quart. J. Roy. Meteorol. Soc. 78, 165 (1952).
    [Crossref]
  4. L. D. Kaplan, Proc. Toronto Meteorol. Conf.,  43 (1953).
  5. G. N. Plass, J. Opt. Soc. Am. 48, 690 (1958).
    [Crossref]
  6. D. Burch and D. Williams, Appl. Opt. 1, 587 (1962).
    [Crossref]
  7. P. J. Wyatt, V. R. Stull, and G. N. Plass, Appl. Opt. (to be published); Aeronutronic Division, Ford Motor Company, Publication No. U-1717 (1962).
  8. V. R. Stull, P. J. Wyatt, and G. N. Plass, Applied Optics (to be published); Aeronutronic Division, Ford Motor Company, Publication U-1718 (1962).
  9. G. N. Plass, J. Opt. Soc. Am. 50, 868 (1960).
    [Crossref]
  10. R. Landenberg and F. Reiche, Ann. Physik 42, 181 (1913).
    [Crossref]
  11. W. S. Benedict (private communication).
  12. C. G. J. Jacobi, J. für Math. 15, 12 (1836).
  13. J. H. Graf and E. Gubler, Einleitung in die Theorie der Besselshen Funktionen (K. J. Wyss, Bern, 1898), Vol. I, p. 86.

1962 (1)

1960 (1)

1958 (1)

1953 (1)

L. D. Kaplan, Proc. Toronto Meteorol. Conf.,  43 (1953).

1952 (1)

R. M. Goody, Quart. J. Roy. Meteorol. Soc. 78, 165 (1952).
[Crossref]

1947 (1)

H. Mayer, “Methods of Opacity Calculations,” Los Alamos, LA-647 (1947).

1938 (1)

W. M. Elsasser, Phys. Rev. 54, 126 (1938).
[Crossref]

1913 (1)

R. Landenberg and F. Reiche, Ann. Physik 42, 181 (1913).
[Crossref]

1836 (1)

C. G. J. Jacobi, J. für Math. 15, 12 (1836).

Benedict, W. S.

W. S. Benedict (private communication).

Burch, D.

Elsasser, W. M.

W. M. Elsasser, Phys. Rev. 54, 126 (1938).
[Crossref]

Goody, R. M.

R. M. Goody, Quart. J. Roy. Meteorol. Soc. 78, 165 (1952).
[Crossref]

Graf, J. H.

J. H. Graf and E. Gubler, Einleitung in die Theorie der Besselshen Funktionen (K. J. Wyss, Bern, 1898), Vol. I, p. 86.

Gubler, E.

J. H. Graf and E. Gubler, Einleitung in die Theorie der Besselshen Funktionen (K. J. Wyss, Bern, 1898), Vol. I, p. 86.

Jacobi, C. G. J.

C. G. J. Jacobi, J. für Math. 15, 12 (1836).

Kaplan, L. D.

L. D. Kaplan, Proc. Toronto Meteorol. Conf.,  43 (1953).

Landenberg, R.

R. Landenberg and F. Reiche, Ann. Physik 42, 181 (1913).
[Crossref]

Mayer, H.

H. Mayer, “Methods of Opacity Calculations,” Los Alamos, LA-647 (1947).

Plass, G. N.

G. N. Plass, J. Opt. Soc. Am. 50, 868 (1960).
[Crossref]

G. N. Plass, J. Opt. Soc. Am. 48, 690 (1958).
[Crossref]

P. J. Wyatt, V. R. Stull, and G. N. Plass, Appl. Opt. (to be published); Aeronutronic Division, Ford Motor Company, Publication No. U-1717 (1962).

V. R. Stull, P. J. Wyatt, and G. N. Plass, Applied Optics (to be published); Aeronutronic Division, Ford Motor Company, Publication U-1718 (1962).

Reiche, F.

R. Landenberg and F. Reiche, Ann. Physik 42, 181 (1913).
[Crossref]

Stull, V. R.

P. J. Wyatt, V. R. Stull, and G. N. Plass, Appl. Opt. (to be published); Aeronutronic Division, Ford Motor Company, Publication No. U-1717 (1962).

V. R. Stull, P. J. Wyatt, and G. N. Plass, Applied Optics (to be published); Aeronutronic Division, Ford Motor Company, Publication U-1718 (1962).

Williams, D.

Wyatt, P. J.

V. R. Stull, P. J. Wyatt, and G. N. Plass, Applied Optics (to be published); Aeronutronic Division, Ford Motor Company, Publication U-1718 (1962).

P. J. Wyatt, V. R. Stull, and G. N. Plass, Appl. Opt. (to be published); Aeronutronic Division, Ford Motor Company, Publication No. U-1717 (1962).

Ann. Physik (1)

R. Landenberg and F. Reiche, Ann. Physik 42, 181 (1913).
[Crossref]

Appl. Opt. (1)

J. für Math. (1)

C. G. J. Jacobi, J. für Math. 15, 12 (1836).

J. Opt. Soc. Am. (2)

Los Alamos, LA-647 (1)

H. Mayer, “Methods of Opacity Calculations,” Los Alamos, LA-647 (1947).

Phys. Rev. (1)

W. M. Elsasser, Phys. Rev. 54, 126 (1938).
[Crossref]

Proc. Toronto Meteorol. Conf. (1)

L. D. Kaplan, Proc. Toronto Meteorol. Conf.,  43 (1953).

Quart. J. Roy. Meteorol. Soc. (1)

R. M. Goody, Quart. J. Roy. Meteorol. Soc. 78, 165 (1952).
[Crossref]

Other (4)

P. J. Wyatt, V. R. Stull, and G. N. Plass, Appl. Opt. (to be published); Aeronutronic Division, Ford Motor Company, Publication No. U-1717 (1962).

V. R. Stull, P. J. Wyatt, and G. N. Plass, Applied Optics (to be published); Aeronutronic Division, Ford Motor Company, Publication U-1718 (1962).

J. H. Graf and E. Gubler, Einleitung in die Theorie der Besselshen Funktionen (K. J. Wyss, Bern, 1898), Vol. I, p. 86.

W. S. Benedict (private communication).

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Figures (3)

Fig. 1
Fig. 1

The frequency interval which contains these six spectral lines is divided into smaller intervals by two different meshes A and B. With mesh A there are three lines in region I and three in region II. With mesh B there are only two lines in region I′, while there are four lines in region II′.

Fig. 2
Fig. 2

The absorptance of a single spectral line as a function of ξζ2 (proportional to pu) for various values of ζ (proportional to p). The absorptance for a finite frequency interval δ is compared with the absorptance for an infinite interval calculated by dividing the equivalent width of the line by δ.

Fig. 3
Fig. 3

The percentage error for the absorptance of a single spectral line calculated over an infinite interval compared to the absorptance for a finite interval.

Tables (1)

Tables Icon

Table I Complementary omega function, 1−Ω%(ξ,ζ). (Absorptance of a single spectral line over a finite frequency interval.)

Equations (41)

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T k ( ν ) = 1 δ n k i = 1 n k δ k e - S i u b ( ν , ν i ) d ν i ,
T k ( ν ) = i = 1 5 { 1 δ δ k e - S ¯ i u b ( ν , ν i ) d ν i } n i ,
n k = i = 1 5 n i .
T ( ν ) = j = 1 T j ( ν ) ,
T ¯ ( ν n ) = 1 3 [ T ( ν n - 1 ) + T ( ν n ) + T ( ν n + 1 ) ] ,
b ( ν , ν i ) = α / π [ ( ν - ν i ) 2 + α 2 ]
ξ k = S k u / π α , ρ = 2 α / δ , η = 2 y / δ , = 2 z / δ ,
y = ν k - ν 0 - 1 2 δ , z = ν - ν 0 - 1 2 δ .
T k = 1 δ δ k exp [ - S k u b ( ν , ν k ) ] d ν k = 1 2 - 1 1 exp [ - ρ 2 ξ k ( - η ) 2 + ρ 2 ] d η .
ξ k = ξ k 0 u / p = S k p 0 u / π α 0 p ,
T k = 1 2 - 1 1 exp [ - ρ 2 ξ k ( - η ) 2 ] d η = 1 2 { ( + 1 ) exp [ - A ( + 1 ) 2 ] - ( - 1 ) exp [ - A ( - 1 ) 2 ] } - 1 2 π 1 2 A 1 2 [ erf ( A 1 2 - 1 ) - erf ( A 1 2 + 1 ) ] ,
Ω ( ξ k , ζ ) = 0 1 exp [ - ζ 2 ξ k y 2 + ζ 2 ] d y .
T k = 1 2 - 1 1 exp [ - ρ 2 ξ k ( - η ) 2 + ρ 2 ] d η = 1 2 0 1 - exp [ - ρ 2 ξ k z 2 + ρ 2 ] d z + 1 2 0 1 + exp [ - ρ 2 ξ k z 2 + ρ 2 ] d z = 1 2 ( 1 - ) Ω ( ξ k , ζ ) + 1 2 ( 1 + ) Ω ( ξ k , ζ ) ,
ζ = ρ / ( 1 - ) , ζ = ρ / ( 1 + ) .
Ω ( ξ , ζ ) = exp [ - ξ ζ 2 / ( 1 + ζ 2 ) ] - 1 2 ζ ξ exp ( - 1 2 ξ ) [ I 0 ( 1 2 ξ ) + I 1 ( 1 2 ξ ) ] ( π - ψ ) + 2 ζ exp ( - 1 2 ξ ) n = 1 I n ( 1 2 ξ ) sin n ψ + ζ ξ exp ( - 1 2 ξ ) n = 1 [ I n ( 1 2 ξ ) + I n + 1 ( 1 2 ξ ) ] sin n ψ n ,
Ω ( ξ , ζ ) = 1 - ζ n = 0 ( - 1 ) n ξ n + 1 ( n + 1 ) ! I n ,
I n = [ ζ 2 n - 1 / 2 n ( 1 + ζ 2 ) n ] + ( 2 n - 1 / 2 n ) I n - 1 I 0 = tan - 1 ( 1 / ζ ) .
Ω ( ξ , ζ ) = exp [ - ξ ζ 2 / ( 1 + ζ 2 ) ] - 1 2 π ζ ξ exp ( - 1 2 ξ ) [ I 0 ( 1 2 ξ ) + I 1 ( 1 2 ξ ) ] + 2 ζ ξ m = 0 p - 1 ( - 1 2 ) m F m m ! + 2 ζ ξ R p ,
F m = - ( 2 m - 1 / 2 ξ ) exp [ - ζ 2 ξ / ( 1 + ζ 2 ) ] + ( 2 m - 1 / 2 ξ ) F m - 1 , F 0 = 1 2 ( π ξ ) 1 2 erf ( ξ 1 2 ) , = ζ / ( 1 + ζ 2 ) ,
R p = ( - 1 2 ) p ( p - 1 ) ! 0 y 2 p exp ( - ξ y 2 ) d y 0 1 ( 1 - t ) p - 1 ( 1 - y 2 t ) p - 1 2 d t
( α ) n = α ( α + 1 ) ( α + n - 1 ) , ( α ) 0 = 1.
T ( ν ) = T k ( ν ) i k T i ( ν ) ,
b ( ν , ν i ) = α π [ ( ν - ν i ) 2 + α 2 ] , ν - ν i < d , = A α exp [ - a ν - ν i b ] π [ ( ν - ν i ) 2 + α 2 ] , ν - ν i d ,
T k ( ν ) = 1 δ δ k exp { - A S k u α exp ( - a ν - ν k b ) π [ ( ν - ν k ) 2 + α 2 ] } d ν k 1 2 - 1 1 exp { - A ξ k ρ 2 exp [ - a ( 1 2 δ ) b - η b ] ( - η ) 2 } d η ,
T k ( ν ) = exp { - A ξ k ρ 2 exp [ - a ( 1 2 δ ) b b ] / 2 } .
Ω ( ξ , ζ ) = 0 1 exp ( - δ 2 ξ y 2 + δ 2 ) d y
Ω ( ξ , ζ ) = ζ ζ ( 1 - ζ 2 ) - 1 2 1 exp ( - ξ z 2 ) z 2 ( 1 - z 2 ) 1 2 d z .
Ω ( ξ , ζ ) = exp [ - ξ ζ 2 / ( 1 + ζ 2 ) ] - 2 ζ ξ ζ ( 1 - ζ 2 ) - 1 2 1 ( 1 - z 2 ) 1 2 exp ( - ξ z 2 ) d z .
G 1 = ζ ( 1 + ζ 2 ) - 1 2 1 ( 1 - z 2 ) 1 2 exp ( - ξ z 2 ) d z = 0 1 ( 1 - z 2 ) 1 2 exp ( - ξ z 2 ) d z - 0 ζ ( 1 + ζ 2 ) 1 2 ( 1 - z 2 ) 1 2 exp ( - ξ z 2 ) d z = G 2 - G 3 .
G 2 = 0 1 2 π cos 2 θ exp ( - ξ sin 2 θ ) d θ = 1 4 π exp ( - 1 2 ξ ) [ I 0 ( 1 2 ξ ) + I 1 ( 1 2 ξ ) ] .
= sin - 1 [ ζ / ( 1 + ζ 2 ) ] = tan - 1 ζ ,
G 3 = 0 cos 2 θ exp ( - ξ sin 2 θ ) d θ .
G 3 = 1 4 e - 1 2 ξ [ G 4 + 2 ( G 4 / ξ ) ] ,
G 4 = 0 2 exp ( 1 2 ξ cos ϕ ) d ϕ .
G 4 = 0 2 exp ( - i z cos ϕ ) d ϕ = 0 2 cos ( z cos ϕ ) d ϕ - i 0 2 sin ( z cos ϕ ) d ϕ .
cos ( z cos ϕ ) = J 0 ( z ) + 2 n = 1 ( - 1 ) n J 2 n ( z ) cos 2 n ϕ , sin ( z cos ϕ ) = 2 n = 0 ( - 1 ) n J 2 n + 1 ( z ) cos ( 2 n + 1 ) ϕ .
G 4 = I 0 ( 1 2 ξ ) ψ + 2 n = 1 I n ( 1 2 ξ ) sin n ψ n ,
ψ = 2 = 2 tan - 1 ζ .
G 3 = 1 4 exp ( - 1 2 ξ ) [ I ( 0 1 2 ξ ) + I 1 ( 1 2 ξ ) ] ψ + exp ( - 1 2 ξ ) ξ n = 1 I n ( 1 2 ξ ) sin n ψ + 1 2 exp ( - 1 2 ξ ) n = 1 [ I n ( 1 2 ξ ) + I n + 1 ( 1 2 ξ ) ] sin n ψ n .
Ω ( ξ , ζ ) = exp [ - ξ ζ 2 / ( 1 + ζ 2 ) ] - 1 2 ξ ζ exp ( - 1 2 ξ ) [ I 0 ( 1 2 ξ ) + I 1 ( 1 2 ξ ) ] + 2 ζ ξ 0 ζ ( 1 + ζ 2 ) - 1 2 ( 1 - z 2 ) 1 2 exp ( - ξ z 2 ) d z .
0 ζ ( 1 + ζ 2 ) - 1 2 ( 1 - z 2 ) 1 2 exp ( - ξ z 2 ) d z = m = 0 p - 1 ( - 1 2 ) m F m m ! + R p ,