Abstract

The transmission of a randomly spaced array of Lorentz lines whose intensities are exponentially distributed may be simply corrected for doppler effects. Evaluation of the correction exponent requires an integration over the Voigt line profile, which may be done once and for all. The integration is described and results presented in a table. From these results, the mean curve of growth of a single Voigt line in such an array may easily be calculated. Curtis’ approximation, applied to such a band, gives equivalent widths several tens of percent too large in many situations.

© 1968 Optical Society of America

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References

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  1. R. M. Goody, Quart. J. Roy. Meteorol. Soc. 78, 165 (1952).
    [CrossRef]
  2. C. D. Rodgers, C. D. Walshaw, Quart. J. Roy. Meteorol. Soc. 92, 67 (1966).
    [CrossRef]
  3. J. C. Gille, Bull. Am. Meteorol. Soc. 48, 202 (1967).
  4. D. K. Edwards, W. A. Menard, Appl. Opt. 3, 847 (1964).
    [CrossRef]
  5. J. C. Bates, D. S. Hanson, F. B. House, R. O’B Carpenter, J. C. Gille, “The Synthesis of 15-μInfrared Horizon Radiance Profiles from Meteorological Data Inputs,” NASA Contractor Rept. CR—724 (1967).
  6. R. M. Goody, Atmospheric Radiation, I; Theoretical Basis (Clarendon Press, Oxford, 1964), pp. 99, 133.
  7. R. M. Goody, M. J. S. Belton, Planetary Space Sci. 15, 247 (1967).
    [CrossRef]
  8. C. Young, J. Quant. Spectry. Radiative Transfer 5, 549 (1965).
    [CrossRef]
  9. C. D. Walshaw, Proc. Phys. Soc. London A68, 530 (1955).
  10. S. R. Drayson, “Atmospheric Slant Path Transmission in the 15 μ, CO2Band,” University of Michigan Tech. Rept. 05863–6–T (1964).
  11. C. Young, “Tables for Calculating the Voigt Profile”, University of Michigan Tech. Rept. 05863–7–T (1965).
  12. G. N. Plass, J. Opt. Soc. Am. 48, 690 (1958).
    [CrossRef]
  13. P. A. Jansson, C. L. Korb, to appear in J. Quantum Spectry. Radiative Transfer (1968).

1967 (2)

J. C. Gille, Bull. Am. Meteorol. Soc. 48, 202 (1967).

R. M. Goody, M. J. S. Belton, Planetary Space Sci. 15, 247 (1967).
[CrossRef]

1966 (1)

C. D. Rodgers, C. D. Walshaw, Quart. J. Roy. Meteorol. Soc. 92, 67 (1966).
[CrossRef]

1965 (1)

C. Young, J. Quant. Spectry. Radiative Transfer 5, 549 (1965).
[CrossRef]

1964 (1)

1958 (1)

1955 (1)

C. D. Walshaw, Proc. Phys. Soc. London A68, 530 (1955).

1952 (1)

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

Bates, J. C.

J. C. Bates, D. S. Hanson, F. B. House, R. O’B Carpenter, J. C. Gille, “The Synthesis of 15-μInfrared Horizon Radiance Profiles from Meteorological Data Inputs,” NASA Contractor Rept. CR—724 (1967).

Belton, M. J. S.

R. M. Goody, M. J. S. Belton, Planetary Space Sci. 15, 247 (1967).
[CrossRef]

Drayson, S. R.

S. R. Drayson, “Atmospheric Slant Path Transmission in the 15 μ, CO2Band,” University of Michigan Tech. Rept. 05863–6–T (1964).

Edwards, D. K.

Gille, J. C.

J. C. Gille, Bull. Am. Meteorol. Soc. 48, 202 (1967).

J. C. Bates, D. S. Hanson, F. B. House, R. O’B Carpenter, J. C. Gille, “The Synthesis of 15-μInfrared Horizon Radiance Profiles from Meteorological Data Inputs,” NASA Contractor Rept. CR—724 (1967).

Goody, R. M.

R. M. Goody, M. J. S. Belton, Planetary Space Sci. 15, 247 (1967).
[CrossRef]

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

R. M. Goody, Atmospheric Radiation, I; Theoretical Basis (Clarendon Press, Oxford, 1964), pp. 99, 133.

Hanson, D. S.

J. C. Bates, D. S. Hanson, F. B. House, R. O’B Carpenter, J. C. Gille, “The Synthesis of 15-μInfrared Horizon Radiance Profiles from Meteorological Data Inputs,” NASA Contractor Rept. CR—724 (1967).

House, F. B.

J. C. Bates, D. S. Hanson, F. B. House, R. O’B Carpenter, J. C. Gille, “The Synthesis of 15-μInfrared Horizon Radiance Profiles from Meteorological Data Inputs,” NASA Contractor Rept. CR—724 (1967).

Jansson, P. A.

P. A. Jansson, C. L. Korb, to appear in J. Quantum Spectry. Radiative Transfer (1968).

Korb, C. L.

P. A. Jansson, C. L. Korb, to appear in J. Quantum Spectry. Radiative Transfer (1968).

Menard, W. A.

O’B Carpenter, R.

J. C. Bates, D. S. Hanson, F. B. House, R. O’B Carpenter, J. C. Gille, “The Synthesis of 15-μInfrared Horizon Radiance Profiles from Meteorological Data Inputs,” NASA Contractor Rept. CR—724 (1967).

Plass, G. N.

Rodgers, C. D.

C. D. Rodgers, C. D. Walshaw, Quart. J. Roy. Meteorol. Soc. 92, 67 (1966).
[CrossRef]

Walshaw, C. D.

C. D. Rodgers, C. D. Walshaw, Quart. J. Roy. Meteorol. Soc. 92, 67 (1966).
[CrossRef]

C. D. Walshaw, Proc. Phys. Soc. London A68, 530 (1955).

Young, C.

C. Young, J. Quant. Spectry. Radiative Transfer 5, 549 (1965).
[CrossRef]

C. Young, “Tables for Calculating the Voigt Profile”, University of Michigan Tech. Rept. 05863–7–T (1965).

Appl. Opt. (1)

Bull. Am. Meteorol. Soc. (1)

J. C. Gille, Bull. Am. Meteorol. Soc. 48, 202 (1967).

J. Opt. Soc. Am. (1)

J. Quant. Spectry. Radiative Transfer (1)

C. Young, J. Quant. Spectry. Radiative Transfer 5, 549 (1965).
[CrossRef]

Planetary Space Sci. (1)

R. M. Goody, M. J. S. Belton, Planetary Space Sci. 15, 247 (1967).
[CrossRef]

Proc. Phys. Soc. London (1)

C. D. Walshaw, Proc. Phys. Soc. London A68, 530 (1955).

Quart. J. Roy. Meteorol. Soc. (2)

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

C. D. Rodgers, C. D. Walshaw, Quart. J. Roy. Meteorol. Soc. 92, 67 (1966).
[CrossRef]

Other (5)

J. C. Bates, D. S. Hanson, F. B. House, R. O’B Carpenter, J. C. Gille, “The Synthesis of 15-μInfrared Horizon Radiance Profiles from Meteorological Data Inputs,” NASA Contractor Rept. CR—724 (1967).

R. M. Goody, Atmospheric Radiation, I; Theoretical Basis (Clarendon Press, Oxford, 1964), pp. 99, 133.

S. R. Drayson, “Atmospheric Slant Path Transmission in the 15 μ, CO2Band,” University of Michigan Tech. Rept. 05863–6–T (1964).

C. Young, “Tables for Calculating the Voigt Profile”, University of Michigan Tech. Rept. 05863–7–T (1965).

P. A. Jansson, C. L. Korb, to appear in J. Quantum Spectry. Radiative Transfer (1968).

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Figures (3)

Fig. 1
Fig. 1

Mean curve of growth for a line in a random exponential array of Voigt lines, for several ratios of Lorentz-to-doppler half width. Curves are marked with values of d = 2αL/αD. Pure Lorentz lines correspond to the line d = ∞. On this scale, the line d = 1 cannot be distinguished from d = ∞.

Fig. 2
Fig. 2

C(R,d) = ĀV/ĀL as a function of u for several ratios of Lorentz-to-doppler half widths. Curves are marked with various values of d = 2αL/αD. The x axis (C = 1) corresponds to pure Lorentz lines. Crosses show values calculated according to the modified Curtis approximation [Eq. (11)] for d = 10−3, 10−2, 10−1, and 1.

Fig. 3
Fig. 3

Ratio of mean absorption from modified Curtis approximation [Eq. (11)] to Voigt absorption, as a function of absorber amount for several values of d.

Tables (1)

Tables Icon

Table I Values of C(R,d) = ĀV/ĀL, Which May be Used to Correct Random Exponential Band Transmissions Based on the Lorentz Line Shape for Doppler Effects, According to Eq. (2)

Equations (13)

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T ¯ = e - A ¯ ,
T ¯ V = e - A ¯ V = exp [ - A ¯ L ( A ¯ V / A ¯ L ) ] = T ¯ L C ,
A ¯ = 1 δ - a σ f ν d ν 1 + a σ f ν ,
f L ν = α L π - 1 [ ( ν - ν 0 ) 2 + α L 2 ] - 1 ;
A ¯ L = 2 π y u ( 2 u + 1 ) - 1 2 ,
f V ν = d 2 α D π 3 2 - e - t 2 d t ( x - t ) 2 + ( d / 2 ) 2 = 1 α D ( π ) 1 2 K ( x , d / 2 ) ,
A ¯ V d 2 y = - R K ( x , d / 2 ) 1 + R K ( x , d / 2 ) d x = I ( R , d ) ,
C ( R , d ) = [ I ( R , d ) ( 2 u + 1 ) 1 2 ] / π u d .
x = η ( 1 - ζ ) / ( 1 + ζ ) ,
I = 4 η - 1 1 R K ( ζ , d ) 1 + R K ( ζ , d ) d ζ ( 1 + ζ ) 2 ,
W C = W L + W D - W L ( W D / W W )
A ¯ C / y = A ¯ L / y + A ¯ D / y - A ¯ L A ¯ D / 2 π u y 2 .
W ¯ ( a ) = 0 W ( S , a ) σ - 1 e - S / σ d S .

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