Abstract

Precise calculations of the total Rayleigh-scattering optical depth have been performed at 88 wavelengths ranging from 0.20 to 4.00 µm for the six well-known standard atmosphere models by integrating the volume Rayleigh-scattering coefficient along the vertical atmospheric path from sea level to a 120-km height. The coefficient was determined by use of an improved algorithm based on the Ciddor algorithm [ Appl. Opt. 35, 1566 ( 1996)], extended by us over the 0.20–0.23-µm wavelength range to evaluate the moist air refractive index as a function of wavelength, air pressure, temperature, water-vapor partial pressure, and CO2 volume concentration. The King depolarization factor was also defined taking into account the moisture conditions of air. The results indicate that the influence of water vapor on Rayleigh scattering cannot be neglected at tropospheric altitudes: for standard atmospheric conditions represented in terms of the U.S. Standard Atmosphere (1976) model, the relative variations produced by water vapor in the Rayleigh scattering parameters at a 0.50-µm wavelength turn out to be equal to −0.10% in the moist air refractivity at sea level (where the water-vapor partial pressure is equal to ≈7.8 hPa), −0.04% in the sea-level King factor, −0.24% in the sea-level Rayleigh-scattering cross section, and −0.06% in the Rayleigh-scattering optical depth.

© 2005 Optical Society of America

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2002 (2)

C. E. Sioris, W. F. J. Evans, R. L. Gattinger, I. C. McDade, D. A. Degenstein, E. J. Llewellyn, “Ground-based Ring-effect measurements with the OSIRIS development model,” Can. J. Phys. 80, 483–491 (2002).
[CrossRef]

P. E. Ciddor, “Refractive index of air: 3. The roles of CO2, H2O, and refractivity virials,” Appl. Opt. 41, 2292–2298 (2002).
[CrossRef] [PubMed]

2000 (1)

A. Smirnov, B. N. Holben, O. Dubovik, N. T. O’Neill, L. A. Remer, T. F. Eck, I. Slutsker, D. Savoie, “Measurement of atmospheric optical parameters on U.S. Atlantic coast sites, ships, and Bermuda during TARFOX,” J. Geophys. Res. 105, 9887–9901 (2000).
[CrossRef]

1999 (1)

B. A. Bodhaine, N. B. Wood, E. G. Dutton, J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Oceanic Technol. 16, 1854–1861 (1999).
[CrossRef]

1998 (1)

1996 (1)

1995 (1)

1994 (2)

N. Larsen, B. Knudsen, T. S. Jorgensen, A. di Sarra, D. Fuà, P. Di Girolamo, G. Fiocco, M. Cacciani, J. M. Rosen, N. T. Kjome, “Backscatter measurements of stratospheric aerosols at Thule during January–February 1992,” Geophys. Res. Lett. 21, 1303–1306 (1994).
[CrossRef]

K. P. Birch, M. J. Downs, “Correction to the updated Edlén equation for the refractive index of air,” Metrologia 31, 315–316 (1994).
[CrossRef]

1993 (1)

K. P. Birch, M. J. Downs, “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
[CrossRef]

1992 (4)

J. Beers, T. Doiron, “Verification of revised water vapour correction to the refractive index of air,” Metrologia 29, 315–316 (1992).
[CrossRef]

R. S. Davis, “Equation for the determination of the density of moist air (1981/91),” Metrologia 29, 67–70 (1992).
[CrossRef]

E. G. Dutton, J. R. Christy, “Solar radiative forcing at selected locations and evidence for global lower tropospheric cooling following the eruptions of El Chichón and Pinatubo,” Geophys. Res. Lett. 19, 2313–2316 (1992).
[CrossRef]

K. J. Thome, B. M. Herman, J. A. Reagan, “Determination of precipitable water from solar transmission,” J. Appl. Meteorol. 31, 157–165 (1992).
[CrossRef]

1990 (2)

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 19, 677–715 (1990).
[CrossRef]

P. M. Teillet, “Rayleigh optical depth comparisons from various sources,” Appl. Opt. 29, 1897–1900 (1990).
[CrossRef] [PubMed]

1988 (1)

R. Muijlwijk, “Update of the Edlen formulae for the refractive index of air,” Metrologia 25, 189 (1988).
[CrossRef]

1987 (1)

S. Solomon, A. Schmeltekopf, R. W. Sanders, “On the interpretation of zenith sky absorption measurements,” J. Geophys. Res. 92, 8311–8319 (1987).
[CrossRef]

1986 (4)

1984 (2)

C. Tomasi, “Vertical distribution features of atmospheric water vapor in the Mediterranean, Red Sea, and Indian Ocean,” J. Geophys. Res. 89, 2563–2566 (1984).
[CrossRef]

D. R. Bates, “Rayleigh scattering by air,” Planet. Space Sci. 32, 785–790 (1984).
[CrossRef]

1982 (2)

G. E. Shaw, “Atmospheric turbidity in the polar regions,” J. Appl. Meteorol. 21, 1080–1088 (1982).
[CrossRef]

P. Giacomo, “Equation for the determination of the density of moist air (1981),” Metrologia 18, 33–40 (1982).
[CrossRef]

1981 (3)

F. E. Jones, “The refractivity of air,” J. Res. Natl. Bur. Stand. 86, 27–32 (1981).
[CrossRef]

A. T. Young, “On the Rayleigh-scattering optical depth of the atmosphere,” J. Appl. Meteorol. 20, 328–330 (1981).
[CrossRef]

A. T. Young, “Rayleigh scattering,” Appl. Opt. 20, 533–535 (1981).
[CrossRef] [PubMed]

1980 (4)

1978 (1)

A. C. Simmons, “The refractive index and Lorentz-Lorenz functions of propane, nitrogen and carbon-dioxide in the spectral range 15803–22002 cm−1 and at 944 cm−1,” Opt. Commun. 25, 211–214 (1978).
[CrossRef]

1977 (2)

W. F. Murphy, “The Rayleigh depolarization ratio and rotational Raman spectrum of water vapor and the polarizability components for the water molecule,” J. Chem. Phys. 67, 5877–5882 (1977).
[CrossRef]

D. V. Hoyt, “A redetermination of the Rayleigh optical depth and its application to selected solar radiation problems,” J. Appl. Meteorol. 16, 432–436 (1977).
[CrossRef]

1973 (1)

A. Bideau-Mehu, Y. Guern, R. Abjean, A. Johannin-Gilles, “Interferometric determination of the refractive index of carbon dioxide in the ultraviolet region,” Opt. Commun. 9, 432–434 (1973).
[CrossRef]

1972 (1)

1971 (1)

1968 (1)

F. Kasten, “Rayleigh-Cabannes Streuung in trockener Luft unter Berücksichtigung neuerer Depolarisations-Messungen,” Optik (Stuttgart) 27, 155–166 (1968).

1967 (2)

1966 (1)

B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

1962 (1)

1960 (1)

A. Dalgarno, A. E. Kingston, “Refractive indices and Verdet constants of the inert gases,” Proc. R. Soc. London Ser. A 259, 424–429 (1960).
[CrossRef]

1957 (1)

1953 (1)

1939 (1)

H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
[CrossRef]

1935 (1)

V. H. Volkmann, “Messungen des Depolarisationsgrades bei der molekularen Lichtzerstreuung,” Ann. Phys. (Leipzig) 24, 457–484 (1935).
[CrossRef]

1923 (1)

L. V. King, “On the complex anisotropic molecule in relation to the dispersion and scattering of light,” Proc. R. Soc. London Ser. A 104, 333–357 (1923).
[CrossRef]

1921 (1)

J. Cabannes, “Sur la diffusion de la lumière par les molécules des gaz transparents,” Ann. Phys. (Paris) 15, 5–150 (1921).

Abjean, R.

A. Bideau-Mehu, Y. Guern, R. Abjean, A. Johannin-Gilles, “Interferometric determination of the refractive index of carbon dioxide in the ultraviolet region,” Opt. Commun. 9, 432–434 (1973).
[CrossRef]

Anderson, G. P.

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km)),” , AFGL-TR-86-0110 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

Barrell, H.

H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
[CrossRef]

Barrett, J. J.

Bates, D. R.

D. R. Bates, “Rayleigh scattering by air,” Planet. Space Sci. 32, 785–790 (1984).
[CrossRef]

Beers, J.

J. Beers, T. Doiron, “Verification of revised water vapour correction to the refractive index of air,” Metrologia 29, 315–316 (1992).
[CrossRef]

Bideau-Mehu, A.

A. Bideau-Mehu, Y. Guern, R. Abjean, A. Johannin-Gilles, “Interferometric determination of the refractive index of carbon dioxide in the ultraviolet region,” Opt. Commun. 9, 432–434 (1973).
[CrossRef]

Birch, K. P.

K. P. Birch, M. J. Downs, “Correction to the updated Edlén equation for the refractive index of air,” Metrologia 31, 315–316 (1994).
[CrossRef]

K. P. Birch, M. J. Downs, “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
[CrossRef]

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, J. Spronck, “Measurements of the refractive index of air using interference refractometers,” Metrologia 22, 279–287 (1986).
[CrossRef]

K. P. Birch, M. J. Downs, “The precise determination of the refractive index of air,” (National Physical Laboratory, Teddington, U.K., 1988).

Bodhaine, B. A.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Oceanic Technol. 16, 1854–1861 (1999).
[CrossRef]

Bréon, F.-M.

Bucholtz, A.

Byrne, D. M.

M. D. King, D. M. Byrne, J. A. Reagan, B. M. Herman, “Spectral variations of optical depth at Tucson, Arizona between August 1975 and December 1977,” J. Appl. Meteorol. 19, 723–732 (1980).
[CrossRef]

Cabannes, J.

J. Cabannes, “Sur la diffusion de la lumière par les molécules des gaz transparents,” Ann. Phys. (Paris) 15, 5–150 (1921).

Cacciani, M.

N. Larsen, B. Knudsen, T. S. Jorgensen, A. di Sarra, D. Fuà, P. Di Girolamo, G. Fiocco, M. Cacciani, J. M. Rosen, N. T. Kjome, “Backscatter measurements of stratospheric aerosols at Thule during January–February 1992,” Geophys. Res. Lett. 21, 1303–1306 (1994).
[CrossRef]

Cacciari, A.

A. Cacciari, C. Tomasi, A. Lupi, V. Vitale, S. Marani, “Radiative forcing effects by aerosol particles in Antarctica,” in Eighth Workshop Italian Research on Antarctic Atmosphere, M. Colacino, G. Giovanelli, eds. (Societa Italiana di Fisica, Bologna, Italy, 2000), Vol. 69, pp. 455–467.

Cheesman, L. E.

Chetwynd, J. H.

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km)),” , AFGL-TR-86-0110 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

Christy, J. R.

E. G. Dutton, J. R. Christy, “Solar radiative forcing at selected locations and evidence for global lower tropospheric cooling following the eruptions of El Chichón and Pinatubo,” Geophys. Res. Lett. 19, 2313–2316 (1992).
[CrossRef]

Chubachi, S.

S. Chubachi, “Annual variation of total ozone at Syowa station, Antarctica,” in Atmospheric Ozone, Vol. I of the Proceedings of the XVIII Quadriennial Ozone Symposium, R. D. Bojkov, G. Visconti, eds. (L’Aquila, Italy, 1998), pp. 193–196.

Ciddor, P. E.

Clough, S. A.

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km)),” , AFGL-TR-86-0110 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

Dalgarno, A.

A. Dalgarno, A. E. Kingston, “Refractive indices and Verdet constants of the inert gases,” Proc. R. Soc. London Ser. A 259, 424–429 (1960).
[CrossRef]

Davis, R. S.

R. S. Davis, “Equation for the determination of the density of moist air (1981/91),” Metrologia 29, 67–70 (1992).
[CrossRef]

Degenstein, D. A.

C. E. Sioris, W. F. J. Evans, R. L. Gattinger, I. C. McDade, D. A. Degenstein, E. J. Llewellyn, “Ground-based Ring-effect measurements with the OSIRIS development model,” Can. J. Phys. 80, 483–491 (2002).
[CrossRef]

Di Girolamo, P.

N. Larsen, B. Knudsen, T. S. Jorgensen, A. di Sarra, D. Fuà, P. Di Girolamo, G. Fiocco, M. Cacciani, J. M. Rosen, N. T. Kjome, “Backscatter measurements of stratospheric aerosols at Thule during January–February 1992,” Geophys. Res. Lett. 21, 1303–1306 (1994).
[CrossRef]

di Sarra, A.

N. Larsen, B. Knudsen, T. S. Jorgensen, A. di Sarra, D. Fuà, P. Di Girolamo, G. Fiocco, M. Cacciani, J. M. Rosen, N. T. Kjome, “Backscatter measurements of stratospheric aerosols at Thule during January–February 1992,” Geophys. Res. Lett. 21, 1303–1306 (1994).
[CrossRef]

Doiron, T.

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Figures (7)

Fig. 1
Fig. 1

Upper panel: comparison between the spectral curves of standard air refractivity defined by the difference ns(ps, Ts, λ) − 1 within the 0.185–0.250-µm wavelength range obtained with the Edlén37 formula in Eq. (6), the first Peck and Reeder40 formula in Eq. (8), and the second Peck and Reeder40 formula in Eq. (9). Lower panel: spectral curves within the same wavelength range of the spectral ratio φ1(λ) between the standard air refractivity from Eq. (8) and the one from Eq. (9) (solid curve) and the spectral ratio φ2(λ) between the standard air refractivity from Eq. (8) and the one from Eq. (6) (dashed curve).

Fig. 2
Fig. 2

Spectral curves of the moist air refractivity in the 0.2–4.0-µm wavelength range, determined as the difference between n(p, T, e, λ, C) and unity with the Ciddor56 model defined by Eqs. (10)(19) for CO2 volume concentration C = 385 ppmv. The three curves were calculated for the triplets of parameters p, T, and e relative to the three following levels of Model 625: (a) z = 0 km, with p = 1013.00 hPa, T = 288.2 K, and e = 7.85075 hPa; (b) z = 10 km, with p = 265.00 hPa, T = 223.3 K, and e = 1.855 × 10−2 hPa; (c) z = 20 km, with p = 55.29 hPa, T = 216.7 K, and e = 2.16 × 10−4 hPa.

Fig. 3
Fig. 3

Upper panel: spectral curves of the moist air King factor F(λ), obtained in terms of Eq. (22) within the 0.2–4.0-µm wavelength range, for CO2 concentration C = 385 ppmv and the triplets of parameters p, T, and e relative to the three following levels of Model 625: (a) z = 0 km, with p = 1013.00 hPa, T = 288.2 K, and e = 7.85075 hPa; (b) z = 10 km, with p = 265.00 hPa, T = 223.3 K, and e = 1.855 × 10−2 hPa; (c) z = 20 km, with p = 55.29 hPa, T = 216.7 K, and e = 2.16 × 10−4 hPa. Lower panel: spectral curves of the difference 1 − κ(λ), where κ(λ) is the spectral ratio between the moist air King factor given by Eq. (22) and the dry air King factor, as derived from Eq. (22), for CO2 concentration C = 385 ppmv and the three triplets of parameters p, T, and e relative to the same levels considered in the upper panel.

Fig. 4
Fig. 4

Spectral curves of ratios ψ(λ) (solid curve), ϕ(λ) (dotted curve), and χ(λ) (dashed curve) within the 0.2–1.0-µm wavelength range. The values of ψ(λ) were obtained by dividing the present values of the King factor F(λ) (see Table 2) by the original ones defined by Bucholtz4 (see Table 2, Ref. 4) at the same wavelengths and for standard air conditions at sea level. The values of ϕ(λ) were determined by dividing our values of F(λ) given in Table 2 by the original ones defined by Bodhaine et al.5 (see Table 2, Ref. 5) at the same wavelengths and for standard air conditions at sea level. The values of spectral ratio χ(λ) were calculated by dividing the present estimates of the depolarization ratio ρn(λ) defined in Table 2 by the corresponding values of ρn(λ) proposed by Bucholtz4 (see Table 3, Ref. 4) at the same wavelengths and for standard air conditions at sea level.

Fig. 5
Fig. 5

Vertical profile of Rayleigh-scattering cross section per molecule σ(0.50 µm) (solid curve in the left panel) and vertical profile of volume Rayleigh-scattering coefficient β(0.50 µm) (solid curve in the right panel), both determined for Model 6.25 The other curves represent the vertical profiles of parameters σ(0.50 µm) (left panel) and β(0.50 µm) (right panel) calculated for the other five atmosphere models25 (labeled My with y = 1, 2, 3, 4, and 5 as given in Table 3) and normalized to those calculated for Model 6,25 their scales defined at the top of the figure: open squares refer to Model 1, open triangles (up) to Model 2, open diamonds to Model 3, open circles to Model 4, and inverted open triangles to Model 5.

Fig. 6
Fig. 6

Spectral curves of the ratios ζ(λ) (solid curve) and ξ(λ) (dashed curve) within the 0.2–4.0-µm wavelength range. The ratio ζ(λ) yields the present values of the Rayleigh-scattering cross section per molecule σ(λ) (given in Table 4 for the sea-level atmospheric conditions of Model 625) divided by those determined by Bucholtz4 at the same wavelengths for standard air conditions. The ratio ξ(λ) was determined by dividing the values of volume Rayleigh-scattering coefficient β0(λ) (given in Table 4 for the sea-level atmospheric conditions of Model 6) by those determined by Bucholtz at the same wavelengths for standard air conditions.

Fig. 7
Fig. 7

Spectral curves of the ratio η1(λ) between optical depth δ(λ) calculated with the present method for the six U.S. Standard Atmosphere models25 and optical depth δ(λ) calculated by Bu-choltz4 for the 1962 U.S. Standard Atmosphere model23 and the five supplementary models26 and normalized by us to the sea-level air pressure conditions given in Table 3 for the six atmosphere models: Model 1 (filled squares), Model 2 (open squares), Model 3 (filled circles), Model 4 (open circles), Model 5 (filled triangles), and Model 6 (open triangles). The filled diamonds represent the ratio η2(λ) between optical depth δ(λ) calculated with the present method for Model 6 and optical depth δ(λ) obtained for the same atmospheric model calculated with the method proposed by Bodhaine et al.5

Tables (6)

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Table 1 Mean Values of the Depolarization Ratio ρn of Dry Air and the King Factor F Proposed in the Literaturea

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Table 2 Values of the King Factor F(λ) Calculated with Eq. (22) and Depolarization Ratio ρn(λ)a

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Table 3 Values of Sea-Level Meteorological and Rayleigh-Scattering Parameters at Sea Level in the Six Standard Atmospheresa

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Table 4 Values of the Rayleigh-Scattering Cross Section σ0(λ) and Volume Rayleigh-Scattering Coefficient β0(λ) Calculated at the 88 Wavelengths Selected from 0.20 to 4.00 µm with the Present Rayleigh-Scattering Model for Sea-Level Atmospheric Conditions Defined in Model 6a

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Table 5 Values of the Rayleigh-Scattering Optical Depth δ(λ) Obtained for the Six Standard Atmosphere Modelsa

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Table 6 WDCGG Stations, Their Latitude, Longitude, and Height and the Yearly Average and Monthly Mean Ground-Level Values of CO2 Volume Concentration C Measured in 2001

Equations (25)

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δ ( λ ) = z 0 z β ( λ , z ) d z ,
β ( λ , z ) = N ( z ) σ ( λ , z ) ,
N ( z ) = N s p ( z ) p s T s T ( z ) ,
σ ( λ , z ) = 24 π 3 [ n ( λ , z ) 2 1 ] 2 λ 4 N ( z ) 2 [ n ( λ , z ) 2 + 2 ] 2 F ( λ , z ) ,
F ( λ , z ) = 6 + 3 ρ n ( λ , z ) 6 7 ρ n ( λ , z ) ,
[ n s ( p s , T s , λ ) 1 ] 10 8 = 8342.13 + 2 , 406 , 030 130 λ 2 + 15 , 997 38.9 λ 2 ,
[ n ( p , T , e , λ ) 1 ] 10 8 = ( 2371.34 + 683 , 939.7 130 λ 2 + 4547.3 38.9 λ 2 ) D s ( p e , T ) + ( 6487.31 + 58.058 λ 2 0.71150 λ 4 + 0.08851 λ 6 ) D w ( e , T ) ,
[ n s ( p s , T s , λ ) 1 ] 10 8 = 5 , 791 , 817 238.0185 λ 2 + 167 , 909 57.362 λ 2 ,
[ n s ( p s , T s , λ ) 1 ] 10 8 = 8060.51 + 2 , 480 , 990 132.274 λ 2 + 17 , 455.7 39.32957 λ 2 ,
Φ 1 ( λ ) = 1.000057551 × 1.0011 11.6795 λ + 34.2542 λ 2 1 11.6702 λ + 34.2347 λ 2 ,
n ( p , T , e , λ , C ) 1 = ( ρ a / ρ a x s ) [ n a x s ( ρ s , T s , λ , C ) 1 ] + ( ρ w / ρ w s ) [ n w s ( e * , T * , λ ) 1 ] ,
ρ = ( p M a / Z R T ) [ 1 X w ( 1 M w / M a ) ] ,
M a = 10 3 28.9635 + 12.011 × 10 6 ( C C 1 ) ,
f ( p , T ) = 1.00062 + 3.14 × 10 8 p + 5.6 × 10 7 ( T 273.15 ) 2 ,
E ( T ) = exp ( 1.2378847 × 10 5 T 2 1.9121316 × 10 2 T + 33.93711047 6343.1645 / T ) ;
Z ( p , T , X w ) = 1 ( p / T ) [ a 0 + a 1 ( T 273.15 ) + a 2 ( T 273.15 ) 2 + b 0 X w + b 1 X w ( T 273.15 ) + c 0 X w 2 c 1 X w 2 ( T 273.15 ) + ( p / T ) 2 ( d 0 + d 1 X w 2 ) ,
[ n a s ( p s , T s , λ , C 2 ) 1 ] 10 8 = 5 , 792 , 105 238.0185 λ 2 + 167 , 917 57.362 λ 2 ,
n a x s ( p s , T s , λ , C ) 1 n a s ( p s , T s , λ , C 2 ) 1 = 1 + 0.534 × 10 6 ( C C 2 ) ,
[ n w s ( e * , T * , λ ) 1 ] 10 8 = 1.022 × ( 295.235 + 2.6422 λ 2 0.032380 λ 4 + 0.004028 λ 6 ) ,
F 1 ( λ ) = 1.034 + 3.17 × 10 4 λ 2 ( for the N 2 molecules ) ,
F 2 ( λ ) = 1.096 + 1.385 × 10 3 λ 2 + 1.448 × 10 4 λ 4 ( for the O 2 molecules ) ,
F 3 = 1.00 ( for Ar ) ,
F 4 = 1.15 ( for CO 2 ) .
F ( λ ) = j c j F j ( λ ) j c j ,
F ( λ , z ) = 0.78084 F 1 ( λ ) + 0.20946 F 2 ( λ ) + 0.00934 F 3 + 10 6 C F 4 + [ e ( z ) / p ( z ) ] F 5 0.999640 + 10 6 C + e ( z ) / p ( z ) ,

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