Abstract

An exhaustive, theoretical study is made of the various models that have been proposed to represent band absorption. These models are compared with each other; a derivation is given of the regions where they predict the same absorption. The regions of validity for various useful approximations to these models are also given. The statistical model is extended to include the random superposition of a finite number of Elsasser bands. Thus a continuous spectrum of absorption curves is obtained between the results for the Elsasser and the pure statistical models. The absorption predicted by the statistical model when there is a specific number of spectral lines in the frequency interval under consideration is compared with the limit as the number of lines approaches infinity. It is shown that the shape of the absorption curve obtained from the statistical model is independent of the distribution of line intensities in the band for most cases of interest. The absorption from the statistical model including the effects of overlapping is shown to depend only on an average equivalent width for a single line. This result is used to derive the band absorption for the statistical model with Lorentz, Doppler, and other line shapes.

© 1958 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, Oct.31, 1947.
  3. R. M. Goody, Quart. J. Roy. Meteorol. Soc. 78, 165 (1952).
    [Crossref]
  4. R. Landenberg and F. Reiche, Ann. Physik 42, 181 (1913).
    [Crossref]
  5. W. M. Elsasser, Heat Transfer by Infrared Radiation in the Atmosphere, Harvard Meteor. Studies No. 6 (Harvard University Press, Cambridge, Massachusetts, 1942).
  6. G. N. Plass, J. Meteorol.,  11, 163 (1954). Only values of β less than unity were considered in this reference. Some of the limits are generalized in the present paper to include all values of β.
    [Crossref]
  7. O. Struve and C. T. Elvey, Astrophys. J. 79, 409 (1934).
    [Crossref]
  8. G. N. Plass and D. I. Fivel, Astrophys. J. 117, 225 (1953).
    [Crossref]
  9. R. M. Goody, The Physics of the Stratosphere (University Press, Cambridge, England, 1954), p. 163.
  10. Equations essentially identical with Eqs. (26) and (50) have been derived by L. D. Kaplan, Proc. Toronto Meteorol, Conf.1953, 43.
  11. A. Erdélyi and et al., Tables of Integral Transforms (McGraw-Hill Book Company, Inc., New York, 1954), Vol. I., p. 195.
    [Crossref]
  12. G. N. Plass, J. Opt. Soc. Am. 42, 677 (1952).
    [Crossref]
  13. G. N. Plass, Quart. J. Roy. Meteorol. Soc. 82, 30 (1956).
    [Crossref]
  14. G. N. Plass, Quart. J. Roy. Meteorol. Soc. 82, 310 (1956).

1956 (2)

G. N. Plass, Quart. J. Roy. Meteorol. Soc. 82, 30 (1956).
[Crossref]

G. N. Plass, Quart. J. Roy. Meteorol. Soc. 82, 310 (1956).

1954 (1)

G. N. Plass, J. Meteorol.,  11, 163 (1954). Only values of β less than unity were considered in this reference. Some of the limits are generalized in the present paper to include all values of β.
[Crossref]

1953 (1)

G. N. Plass and D. I. Fivel, Astrophys. J. 117, 225 (1953).
[Crossref]

1952 (2)

G. N. Plass, J. Opt. Soc. Am. 42, 677 (1952).
[Crossref]

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

1947 (1)

H. Mayer, “Methods of opacity calculations,” Los Alamos, LA-647, Oct.31, 1947.

1938 (1)

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

1934 (1)

O. Struve and C. T. Elvey, Astrophys. J. 79, 409 (1934).
[Crossref]

1913 (1)

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

Elsasser, W. M.

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

W. M. Elsasser, Heat Transfer by Infrared Radiation in the Atmosphere, Harvard Meteor. Studies No. 6 (Harvard University Press, Cambridge, Massachusetts, 1942).

Elvey, C. T.

O. Struve and C. T. Elvey, Astrophys. J. 79, 409 (1934).
[Crossref]

Erdélyi, A.

A. Erdélyi and et al., Tables of Integral Transforms (McGraw-Hill Book Company, Inc., New York, 1954), Vol. I., p. 195.
[Crossref]

Fivel, D. I.

G. N. Plass and D. I. Fivel, Astrophys. J. 117, 225 (1953).
[Crossref]

Goody, R. M.

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

R. M. Goody, The Physics of the Stratosphere (University Press, Cambridge, England, 1954), p. 163.

Kaplan, L. D.

Equations essentially identical with Eqs. (26) and (50) have been derived by L. D. Kaplan, Proc. Toronto Meteorol, Conf.1953, 43.

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, Oct.31, 1947.

Plass, G. N.

G. N. Plass, Quart. J. Roy. Meteorol. Soc. 82, 30 (1956).
[Crossref]

G. N. Plass, Quart. J. Roy. Meteorol. Soc. 82, 310 (1956).

G. N. Plass, J. Meteorol.,  11, 163 (1954). Only values of β less than unity were considered in this reference. Some of the limits are generalized in the present paper to include all values of β.
[Crossref]

G. N. Plass and D. I. Fivel, Astrophys. J. 117, 225 (1953).
[Crossref]

G. N. Plass, J. Opt. Soc. Am. 42, 677 (1952).
[Crossref]

Reiche, F.

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

Struve, O.

O. Struve and C. T. Elvey, Astrophys. J. 79, 409 (1934).
[Crossref]

Ann. Physik (1)

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

Astrophys. J. (2)

O. Struve and C. T. Elvey, Astrophys. J. 79, 409 (1934).
[Crossref]

G. N. Plass and D. I. Fivel, Astrophys. J. 117, 225 (1953).
[Crossref]

J. Meteorol. (1)

G. N. Plass, J. Meteorol.,  11, 163 (1954). Only values of β less than unity were considered in this reference. Some of the limits are generalized in the present paper to include all values of β.
[Crossref]

J. Opt. Soc. Am. (1)

Los Alamos, LA-647 (1)

H. Mayer, “Methods of opacity calculations,” Los Alamos, LA-647, Oct.31, 1947.

Phys. Rev. (1)

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

Quart. J. Roy. Meteorol. Soc. (3)

G. N. Plass, Quart. J. Roy. Meteorol. Soc. 82, 30 (1956).
[Crossref]

G. N. Plass, Quart. J. Roy. Meteorol. Soc. 82, 310 (1956).

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

Other (4)

W. M. Elsasser, Heat Transfer by Infrared Radiation in the Atmosphere, Harvard Meteor. Studies No. 6 (Harvard University Press, Cambridge, Massachusetts, 1942).

R. M. Goody, The Physics of the Stratosphere (University Press, Cambridge, England, 1954), p. 163.

Equations essentially identical with Eqs. (26) and (50) have been derived by L. D. Kaplan, Proc. Toronto Meteorol, Conf.1953, 43.

A. Erdélyi and et al., Tables of Integral Transforms (McGraw-Hill Book Company, Inc., New York, 1954), Vol. I., p. 195.
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Absorption for an Elsasser band as a function of β2x=2παSu/d2. The linear and square-root approximations [Eqs. (6) and (8)] are also indicated to show their regions of validity. Curves are shown for β=0.01, 0.1, 1.0, 10, where β=2πα/d.

Fig. 2
Fig. 2

Transmission for the statistical model of a band as a function of β2x=2παSu/d2. It is assumed that all spectral lines are equally intense and that the Lorentz line shape is valid. Curves are shown for two, ten, and an infinite number of lines with random spacing, but with the same mean spacing, d, in each case. The frequency interval is d in each case.

Fig. 3
Fig. 3

Absorption for the statistical model of a band as a function of β2x. It is assumed that all of the spectral lines are equally intense, that the Lorentz line shape is valid, and that the absorption is from a large number of spectral lines with an average spacing d. The absorption calculated from the linear and square-root approximations, Eqs. (6) and (8), respectively, is also shown.

Fig. 4
Fig. 4

Absorption for the statistical and Elsasser models of a band. In each case it is assumed that all of the spectral lines are equally intense and that the Lorentz line shape is valid.

Fig. 5
Fig. 5

Absorption as a function of βx=Su/d for the statistical model with an exponential distribution of line intensities [P(S)=S0−1 exp(−S/S0)] and with all lines equally intense [P(S)=δ(SS0)]. For comparison purposes, S0 for the exponential distribution has been multiplied by 1 4 π in order to compare with the case where all the lines are equally intense (see Sec. VIII). For comparison the absorption curves for an Elsasser band are also shown.

Fig. 6
Fig. 6

Absorption as a function of β D 2 x D = π 1 2 ( ln 2 ) 1 2 Δ ν D S u d - 2 for the Doppler, Lorentz, and square line shapes. In each case the statistical model of the band has been used with all spectral lines assumed to have the same intensity. For comparison purposes β D and x D were set equal to β and x for the Lorentz line shape and to ( π 1 2 / 2 ) ( δ / d ) and ( 2 / π 1 2 ) ( u S 0 / d ) for the square line shape respectively.

Fig. 7
Fig. 7

Absorption as a function of βx=Su/d for β=0.1. The absorption is shown for a single Elsasser band (N=1); the random superposition of two Elsasser bands (N=2) where the intensities are equal (S1=S2) and where one band is ten times as intense as the other (S1=10S2); the random superosition of five Elsasser bands (N=5) where the intensities in each band are equal (S1=S2=⋯=S5); the statistical model where all the intensities are equal.

Tables (1)

Tables Icon

Table I Regions of validity of various absorption models and approximations for Lorentz line shape. a

Equations (71)

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W s l = Δ ν ( 1 - e - u S b ) d ν ,
0 b ( ν ) d ν = 1.
b ( ν ) = α / { π [ ( ν - ν 0 ) 2 + α 2 ] } ,
W s l = 2 π α x e - x [ I 0 ( x ) + I 1 ( x ) ] ,
x = S u / 2 π α ,
W s l = 2 π α x = d β x ,             x 0.02 q ,
β = 2 π α / d ,
W s l = 2 α ( 2 π x ) 1 2 = d ( 2 β 2 x / π ) 1 2 ,             x 12.5 q - 1 .
W s l , D = 2 π - 1 d β x tan - 1 ( n π β - 1 ) ;
W s l , D = ( 2 β 2 x / π n 2 ) 1 2 [ 1 - ϕ ( z ) ] + 1 - exp ( - z 2 ) ,
z 2 = 2 β 2 x / π 2 n 2 , ϕ ( z ) = 2 π - 1 2 0 z exp ( - z 2 ) d z ,
S b ( ν ) = k 0 exp { - ( ν - ν 0 ) 2 ln 2 / Δ ν D 2 } ,
k 0 = ( S / Δ ν D ) ( ln 2 / π ) 1 2 ,
W s l = d β D x D n = 0 ( - 1 ) n x D n ( n + 1 ) ! ( n + 1 ) 1 2 ,
W s l = 2 π - 1 2 d β D ( ln x D ) 1 2 [ 1 + ( C / 2 ln x D ) - ] ,
x D = [ ( ln 2 ) / π ] 1 2 ( S u / Δ ν D ) ,
β D = ( π / ln 2 ) 1 2 ( Δ ν D / d ) .
W s l = d ( 2 β 2 x / π ) 1 2 { 1 - ( 1 - 3 2 a - 2 ) ( 8 x ) - 1 + } ,
a = ( α / Δ ν D ) ( ln 2 ) 1 2 ,
A = 1 - ( 2 π ) - 1 - π π × exp [ - β x sinh β / ( cosh β - cos z ) ] d z ,
A = ϕ [ ( 1 2 β 2 x ) 1 2 ] ,
A = 1 - e - β x ,
A = 1 - - 1 2 D 1 2 D - 1 2 D 1 2 D N ( ν 1 , , ν n ) d ν 1 d ν n 0 0 i P ( S i ) exp [ - u S i b ( ν i ) ] d S i - 1 2 D 1 2 D - 1 2 D 1 2 D N ( ν 1 , , ν n ) d ν 1 d ν n 0 0 i P ( S i ) d S i ,
0 P ( S i ) d S i = 1.
A = 1 - D - n - 1 2 D 1 2 D - 1 2 D 1 2 D d ν 1 d ν n 0 × 0 i P ( S i ) exp [ - u S i b ( ν i ) ] d S i .
A = 1 - { D - 1 - 1 2 D 1 2 D d ν 0 P ( S ) e - u S b ( ν ) d S } n .
A = 1 - { 1 - D - 1 - 1 2 D 1 2 D d ν × 0 P ( S ) [ 1 - e - u S b ( ν ) ] d S } n .
A = 1 - { 1 - D - 1 0 P ( S ) d S - 1 2 D 1 2 D ( 1 - e - u S b ) d ν } n .
A = 1 - { 1 - D - 1 0 W s l , D ( S , α ) P ( S ) d S } n .
W ¯ s l , D ( S 0 , α ) = 0 W s l , D ( S , α ) P ( S ) d S ,
A = 1 - { 1 - ( W ¯ s l , D / n d ) } n .
A = 1 - exp ( - W ¯ s l / d ) ,
A = W ¯ s l , D d - ( 1 - 1 n ) 2 ! ( W ¯ s l , D d ) 2 + ( 1 - 1 n ) ( 1 - 2 n ) 3 ! ( W ¯ s l , D d ) 3 - .
W ¯ s l , D ( S 0 , α ) = W s l , D ( S 0 , α ) ,
P ( S ) = S 0 - 1 exp ( - S / S 0 ) .
W ¯ s l = 4 π 2 α 2 S 0 - 1 u - 1 0 x [ I 0 ( x ) + I 1 ( x ) ] × exp { - [ 1 + ( 2 π α / u S 0 ) ] x } d x .
A = 1 - exp [ - β x 0 / ( 1 + 2 x 0 ) 1 2 ] ,
b ( ν ) = { δ - 1 ν 0 < ν < ν 0 + δ 0 ν < ν 0 ; ν > ν 0 + δ .
W s l = δ { 1 - exp [ - u S / δ ] } .
A = 1 - exp { - δ d - 1 [ 1 - exp ( - u S 0 / δ ) ] } .
W ¯ s l = δ [ 1 + ( δ / u S 0 ) ] - 1 .
A = 1 - exp { - ( δ / d ) [ 1 + ( δ / u S 0 ) ] - 1 } .
W s l = d β D x D n = 0 ( - 1 ) n x D n ( n + 1 ) ! ( n + 1 ) 1 2 ,
W s l = 2 π - 1 2 d β D ( ln x D ) 1 2 ( 1 + C / ln x D 2 - ) ,
β D = π 1 2 ( ln 2 ) - 1 2 ( Δ ν D / d ) ,
x D = ( ln 2 ) 1 2 π - 1 2 ( S u / Δ ν D ) ,
A = 1 - exp ( - β D x D 0 ) ,             x D 0 < 0.28 ,
A = 1 - exp [ - 2 π - 1 2 β D ( ln x D 0 ) 1 2 ] ,             x D 0 > 18 ,
W s l = S u = d β x .
W s l = d ( 2 β 2 x π ) 1 2 [ 1 - 1 8 x ( 1 - 3 2 a 2 ) + ] ,
a = ( ln 2 ) 1 2 ( α / Δ ν D ) ,
δ = ( i = 1 N 1 Δ i ) - 1 .
W E , i ( x i , β Δ i ) = Δ i A E , i ( x i , β Δ i ) ,
A = 1 - ( i = 1 N Δ i ) - 1 - 1 2 Δ 1 1 2 Δ 1 - 1 2 Δ N 1 2 Δ N d ν 1 d ν N 0 × 0 i = 1 N P E ( S i ) exp [ - u S i b i ( ν i ) ] d S i ,
A = 1 - i = 1 N 0 [ 1 - Δ i - 1 W E , i ( x i , β i ) ] P E ( S i ) d S i .
A = 1 - i = 1 N ( 1 - W ¯ E , i Δ i ) .
A = 1 - [ 1 - ( W ¯ E / N δ ) ] N ,
W ¯ s l / d < 0.02 q .
x < 0.000628 q 2 β - 2 , if β < 0.0314 q , x < 0.02 q β - 1 , if β > 0.0314 q .
x < 0.02 q , if β < 1 , x < 0.02 q β - 1 , if β > 1.
12.5 q - 1 < x < 0.000628 q 2 β - 2 ,             β < 7.09 × 10 - 3 q 3 2 .
12.5 q - 1 < x < 0.06 q β - 2 .
x > 12.5 q - 1 .
x < 0.0157 β - 2 , β < 0.157 , x < 0.1 β - 1 , β > 0.157.
x > 8.30 β - 2 , β < 3.6 , x > 2.30 β - 1 , β > 3.6.
x > 2.70 β - 2 , β < 1.17 , x > 2.30 β - 1 , β > 1.17.
x < 0.000628 q 2 β - 2 , if β < 0.0314 q , x < 0.02 q β - 1 , if β > 0.0314 q .
W s l ( S ˜ , α ) = W ¯ s l ( S 0 , α ) .
S ˜ ( 2 ) exp ( - S ˜ ( 2 ) u / 2 π α ) [ I 0 ( S ˜ ( 2 ) u / 2 π α ) + I 1 ( S ˜ ( 2 ) u / 2 π α ) ] = S 0 ( 2 ) [ 1 + ( S 0 ( 2 ) u / π α ) ] - 1 2 .
S ˜ ( 2 ) = 1 4 π S 0 ( 2 ) .
x > 12.5 q - 1 ,