## 1. Introduction

*n*

_{s}is the refractive index of air and

*λ*is the wavelength of light in micrometers. This equation is for “standard” air, which is defined as dry air at 760 mm Hg (1013.25 mb), 15°C (288.15 K), and containing 300 ppm CO

_{2}. It is an empirical relationship derived by fitting the best available experimental data and is dependent on the composition of air, particularly CO

_{2}and water vapor. Next, Penndorf (1957) calculated the Rayleigh scattering coefficient for standard air using the classic equation that is presented in many textbooks (e.g., van de Hulst 1957; McCartney 1976):

*σ*is the scattering cross section per molecule;

*N*

_{s}is molecular density; the term (6 + 3

*ρ*)/(6 − 7

*ρ*) is called the depolarization term,

*F*(air), or the King factor;and

*ρ*is the depolarization factor or depolarization ratio, which describes the effect of molecular anisotropy. The

*F*(air) term is the least known for purposes of Rayleigh scattering calculations and is responsible for the most uncertainty. The depolarization term does not depend on temperature and pressure, but does depend on the gas mixture. Also,

*N*

_{s}depends on temperature and pressure, but does not depend on the gas mixture. The resulting value of

*σ,*the scattering cross section per molecule of the gas, calculated from Eq. (2), is independent of temperature and pressure, but does depend on the composition of the gas. Note that

*N*

_{s}depends on Avogadro’s number and the molar volume constant, and is expressed as molecules per cubic centimeter, and that values for

*n*

_{s}and

*N*

_{s}must be expressed at the same temperature and pressure. However, since (

*n*

^{2}

_{s}

*n*

^{2}

_{s}

*N*

_{s}, the resulting expression for

*σ*is independent of temperature and pressure (McCartney 1976; Bucholtz 1995). Note that the usual approximation

*n*

^{2}

_{s}

*n*

_{s}from the 1953 formula was given as only 1.4 × 10

^{−8}. Edlén (1953, 1966) also discussed the variation of refractive index with temperature and pressure, and also with varying concentrations of CO

_{2}and water vapor. In light of the Edlén (1966) revisions, Owens (1967) presented an in-depth treatment of the indexes of refractions of dry CO

_{2}-free air, pure CO

_{2}, and pure water vapor, and provided expressions for dependence on temperature, pressure, and composition. However, Owens’ (1967) main interest was in temperature and pressure variations, and his analysis does not significantly impact our present work because our calculations are performed at the temperature and pressure of “standard” air. Peck and Reeder (1972) further refined the currently available data for the refractive index of air and suggested the formula

_{2}. Also, they repeat Edlén’s (1966) formula, which had clearly defined standard air as having 300 ppm CO

_{2}, but state that it applies to air having 330 ppm CO

_{2}. Here we will use the equation of Peck and Reeder (1972) and assume that it holds for standard air having 300 ppm CO

_{2}.

Possible errors in the depolarization term were considered by Hoyt (1977), Fröhlich and Shaw (1980), and Young (1980, 1981). The correction proposed by Young (1981) had been accepted for modern Rayleigh scattering calculations in atmospheric applications. In brief, Young (1981) suggested that the value *F*(air) = (6 + 3*ρ*)/(6 − 7*ρ*) = 1.0480 be used rather than the value 1.0608 used by Penndorf (1957). This effect alone reduced Rayleigh scattering values by 1.2%; however, it cannot be applied over the entire spectrum because *F*(air) is dependent on wavelength. Furthermore, since the depolarization has been measured for the constituents of air (at least in a relative sense), it is possible in principle to estimate the depolarization of air as a function of composition. Bates (1984) and Bucholtz (1995) discussed the depolarization in detail. It appears that currently the best estimates for (6 + 3*ρ*)/(6 − 7*ρ*) use the equations given by Bates (1984) for the depolarization of N_{2}, O_{2}, Ar, and CO_{2} as a function of wavelength. It is therefore possible to calculate the depolarization of air as a function of CO_{2} concentration.

_{2}as a function of wavelength as

_{2}as

*F*(air) be calculated using Eqs. (5) and (6), assuming that

*F*(Ar) = 1.00,

*F*(CO

_{2}) = 1.15, and ignoring the other constituents of air.

## 2. Optical depth

*I*

*λ*

*I*

_{0}

*λ*

*τ*

*λ*

*θ*

*I*

_{0}(

*λ*) is the extraterrestrial flux at wavelength

*λ, I*(

*λ*) is the flux reaching the ground,

*θ*is the solar zenith angle, and

*τ*(

*λ*) is the optical depth. Clear-sky measurements of

*I*(

*λ*) as a function of

*θ,*and plotted as ln

*I*(

*λ*) versus sec

*θ,*should yield a straight line with slope −

*τ*(

*λ*) and intercept

*I*

_{0}(extrapolated back to sec

*θ*= 0). An excellent example, along with a discussion of this process, is shown by Stephens (1994) in his Fig. 6.1. An important point is that

*τ*(

*λ*), the total optical depth, may be composed of several components given by

*τ*

*λ*

*τ*

_{R}

*λ*

*τ*

_{a}

*λ*

*τ*

_{g}

*λ*

*τ*

_{R}(

*λ*) is the Rayleigh optical depth,

*τ*

_{a}(

*λ*) is aerosol optical depth, and

*τ*

_{g}(

*λ*) is the optical depth due to absorption by gases such as O

_{3}, NO

_{2}, and H

_{2}O. In principle it is possible to measure

*τ*(

*λ*) and then derive aerosol optical depth by subtracting estimates of

*τ*

_{R}(

*λ*) and

*τ*

_{g}(

*λ*). In practice, however, arriving at reasonable estimates of these quantities can be difficult, particularly during fairly clean atmospheric conditions such as those found at Mauna Loa, Hawaii. At this point it should be apparent that in order to isolate the individual components of optical depth it is necessary to provide accurate estimates of Rayleigh optical depth.

*P*is the pressure,

*A*is Avogadro’s number,

*m*

_{a}is the mean molecular weight of the air, and

*g*is the acceleration of gravity. Note that

*m*

_{a}depends on the composition of the air, whereas

*A*and

*g*are constants of nature.

*g*may be considered a constant of nature, it does vary significantly with height and location on the earth’s surface and may be calculated according to the formula (List 1968)

*g*

^{−2}

*g*

_{0}

^{−4}

^{−7}

*ϕ*

*z*

^{−11}

^{−13}

*ϕ*

*z*

^{2}

^{−17}

^{−20}

*ϕ*

*z*

^{3}

*ϕ*is the latitude,

*z*is the height above sea level in meters, and

*g*

_{0}is the sea level acceleration of gravity given by

*g*

_{0}

*ϕ*

^{2}

*ϕ*

## 3. Approximations for Rayleigh optical depth

*τ*

*λ*

*Aλ*

^{−B}

*A*and

*B*are constants to be determined from a power-law fit and the equation is normalized to 1013.25-mb pressure. An example was given by Dutton et al. (1994), who performed such a fit over the visible range and provided the equation

*p*is the site pressure,

*p*

_{0}is 1013.25 mb, and

*λ*is in micrometers. Clearly, one problem with this approximation is that it cannot be extrapolated to other parts of the spectrum, particularly the UV, where the power-law exponent is significantly different. To account for the fact that the exponent changes, some authors (e.g., Fröhlich and Shaw 1980; Nicolet 1984) used equations of the form

^{−2}is calculated from the surface pressure, as explained above. This equation is likely to be more accurate over a greater range of the spectrum. A slightly different approach was taken by Hansen and Travis (1974), who suggested the equation

*τ*

_{R}

*λ*

*λ*

^{−4}

*λ*

^{−2}

*λ*

^{−4}

*τ*

_{R}(

*λ*) is normalized to 1013.25 mb. As a final example Stephens (1994) suggested the equation

*τ*

_{R}

*λ*

*λ*

^{(−4.15+0.2λ)}

*e*

^{(−0.1188z−0.00116z2)}

## 4. Suggested method to calculate Rayleigh optical depth of air

Here we suggest a method for calculation of Rayleigh optical depth that goes back to first principles as suggested by Penndorf (1957) rather than using curve-fitting techniques, although it is true that the refractive index of air is still derived from a curve fit to experimental data. We suggest using all of the latest values of the physical constants of nature, and we suggest including the variability in refractive index, and also the mean molecular weight of air, due to CO_{2} even though these effects are in the range of 0.1%–0.01%. It should be noted that aerosol optical depths are often as low as 0.01 at Mauna Loa. Since Rayleigh optical depth is of the order of 1 at 300 nm, it is seen that a 0.1% error in Rayleigh optical depth translates into a 10% error in aerosol optical depth. Furthermore, it simply makes sense to perform the calculations as accurately as possible.

We should note that the effects of high concentrations of water vapor on the refractive index of air may be of the same order as CO_{2} (Edlén 1953, 1966). However, for practical atmospheric situations the total water vapor in the vertical column is small and does not significantly affect the above calculations. Furthermore, the water vapor in the atmosphere is usually confined to a thin layer near the surface, which significantly complicates the calculation, whereas CO_{2} is generally well mixed throughout the atmosphere.

^{23}molecules mol

^{−1}), and molar volume at 273.15 K and 1013.25 mb (22.4141 L mol

^{−1}) were taken from Cohen and Taylor (1995). In order to calculate the mean molecular weight of dry air with various concentrations of CO

_{2}, the percent by volume of the constituent gases in air were taken from Seinfeld and Pandis (1998), and the molecular weights of those gases were taken from the

*Handbook of Physics and Chemistry*(CRC 1997). These results are shown in Table 1. The mean molecular weights (

*m*

_{a}) for dry air were calculated from the formula

*m*

_{a}and CO

_{2}concentration,

*m*

_{a}may be estimated from the equation

*m*

_{a}= 15.0556(CO

_{2}) + 28.9595 gm mol

^{−1}, where CO

_{2}concentration is expressed as parts per volume (use 0.000 36 for 360 ppm).

_{2}concentration:

_{2}concentration using the formula

_{2}concentration is expressed as parts per volume (Edlén 1966). Thus the refractive index for dry air with zero ppm CO

_{2}is

_{2}is

*λ*in units of micrometers.

^{2}molecule

^{−1}) of air be calculated from the equation

*n*is the refractive index of air at the desired CO

_{2}concentration,

*λ*is expressed in units of centimeters,

*N*

_{s}= 2.546 899 × 10

^{19}molecules cm

^{−3}at 288.15 K and 1013.25 mb, and the depolarization ratio

*ρ*is calculated as follows. Using the values for depolarization of the gases O

_{2}, N

_{2}, Ar, and CO

_{2}provided by Bates (1984), we recommend that the depolarization of dry air be calculated using Eqs. (5)–(6) and the following equation to take into account the composition of air:

*C*

_{CO2}

_{2}expressed in parts per volume by percent (e.g., use 0.036 for 360 ppm). The results of Eq. (23) for standard air (300 ppm CO

_{2}) are shown in Fig. 1. Note that the value of

*N*

_{s}in Eq. (22) was calculated from Avogadro’s number and the molar volume, and then scaled to 288.15 K according to the formula

*P*is the surface pressure of the measurement site (dyn cm

^{−2}),

*A*is Avogadro’s number, and

*m*

_{a}is the mean molecular weight of dry air calculated from the formula

*m*

_{a}= 15.0556(CO

_{2}) + 28.9595, as in Eq. (17). The value for

*g*needs to be representative of the mass-weighted column of air molecules above the site, and should be calculated from Eqs. (10)–(11), modified by using a value of

*z*

_{c}determined from the

*U.S. Standard Atmosphere,*as provided by List (1968). To determine

*z*

_{c}we used List’s (1968, p. 267) table of the density of air as a function of altitude and calculated a mass-weighted mean above each altitude value, using an average altitude and average density for each layer listed in the table. Next a least squares straight line was passed through the resulting

*z*

_{c}values up to 10 500 m, giving the following equation:

*z*

_{c}

*z*

*z*is the altitude of the observing site and

*z*

_{c}is the effective mass-weighted altitude of the column. For example, an altitude of

*z*= 0 m yields an effective mass-weighted column altitude of

*z*

_{c}= 5517.56 m to use in the calculation of

*g.*The resulting values or

*τ*

_{R}should be considered the best currently available values for the most accurate estimates of optical depth.

## 5. Optical depths of the constituents of air

_{2}to the Rayleigh optical depth of air may be estimated as a function of wavelength by using the above formulas expressed for CO

_{2}. Owens (1967) gives the refractive index of CO

_{2}at 15°C and 1013.25 mb as

*λ*is expressed in units of micrometers as before. Next the scattering cross section of a CO

_{2}molecule can be calculated from Eq. (22), where

*N*

_{s}= 2.546 899 × 10

^{19}molecules cm

^{−3}at 288.15 K and 1013.25 mb as before, and the King factor

*F*(CO

_{2}) taken to be 1.15, as suggested by Bates (1984). Finally

*τ*(CO

_{2},

*λ*) can be calculated using Eq. (25), where

*m*= 44.01 (the molecular weight of CO

_{2}), and multiplying by 0.000 36 (for a CO

_{2}concentration of 360 ppm) to estimate the number of CO

_{2}molecules. Note that

*τ*(H

_{2}O,

*λ*) for 44 kg m

^{−2}column water vapor was calculated in a similar manner using the refractive index of H

_{2}O given by Harvey et al. (1998) and a depolarization ratio of 0.17 for H

_{2}O given by Marshall and Smith (1990). The results of these calculations are shown in Table 2, where the change of optical depth Δ

*τ*(H

_{2}O) is given for the case where dry air molecules are replaced by H

_{2}O molecules for 44 kg m

^{−2}column water vapor. Thus for a Rayleigh optical depth of 1.224 for air at 300 nm, a CO

_{2}concentration of 360 ppm would contribute an optical depth about 0.001.

## 6. Some example calculations

Using the above equations we now present example calculations to show new values for the scattering cross section (as a function of wavelength) of dry air containing 360 ppm CO_{2}, similar to the presentations of Penndorf (1957) and Bucholtz (1995). In addition we present new values for Rayleigh optical depth for dry air containing 360 ppm CO_{2} at sea level, 1013.25 mb, and a latitude of 45°; and at Mauna Loa Observatory (MLO) (altitude 3400 m, pressure 680 mb, and a latitude of 19.533°). The results of these calculations are shown in Table 3.

*PA/m*

_{a}

*g*terms (molecules in the column) given in Eq. (25) for the two cases (taking into account the 10

^{−28}factor that was removed from the scattering cross section data to facilitate curve fitting calculations).

## 7. Conclusions

We have presented the latest values of the physical constants necessary for the calculation of Rayleigh optical depth. For the most accurate calculation of this quantity it is recommended that users go directly to first principles and that Peck and Reeder’s (1972) formula be used to estimate the refractive index of standard air. Next, we recommend that Penndorf’s (1957) method be used to calculate the scattering cross section per molecule of air, taking into account the concentration of CO_{2}. In most cases the effects of water vapor may be neglected. The recommendations of Bates (1984) were used for the depolarization of air as a function of wavelength. Next the Rayleigh optical depth should be calculated using the atmospheric pressure at the site of interest. Note the importance of taking into account variations of *g.* We do not necessarily recommend the use of curve-fitting techniques to generate an equation for estimating Rayleigh optical depth because the inaccuracies that arise can equal or even exceed other quantities being estimated, such as aerosol optical depth. Furthermore, all of the above calculations are simple enough to be done in a spreadsheet if desired, or can easily be programmed in virtually any computer language. However, for those who wish to use a simple equation and are satisfied with less accuracy, the techniques used to produce Eqs. (29)–(31) may be of interest. As more accurate estimates of the various parameters discussed above become available, the equations of interest may easily be modified.

In some calculations of optical depth it may be desired to take into account the vertical distribution of the composition of air, particularly CO_{2}. In this case a layer-by-layer calculation may be done using the estimated composition for each layer, and then the total optical depth may be estimated by summing the optical depths for all of the layers.

## Acknowledgments

We thank Gail Anderson for her helpful comments concerning curve-fitting techniques.

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## APPENDIX

### Summary of Constants

Values for the constants of nature that have been used in this paper are listed below.

Avogadro’s number = 6.022 136 7 × 10

^{23}molecules mol^{−1}Molar volume at 273.15 K and 1013.25 mb = 22.4141 L mol

^{−1}Molecular density of a gas at 288.15 K and 1013.25 mb = 2.546 899 × 10

^{19}molecules cm^{−3}Mean molecular weight of dry air (zero CO

_{2}) = 28.9595 gm mol^{−1}Mean molecular weight of dry air (360 ppm CO

_{2}) = 28.9649 gm mol^{−1}Acceleration of gravity (sea level and 45° latitude)

*g*_{0}(45°) = 980.6160 cm s^{−2}Mass-weighted air column altitude

*z*_{c}= 0.737 37*z*+ 5517.56

Percent error for Eq. (29) fit to the scattering cross-section data in Table 3.

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1854:ORODC>2.0.CO;2

Percent error for Eq. (29) fit to the scattering cross-section data in Table 3.

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1854:ORODC>2.0.CO;2

Percent error for Eq. (29) fit to the scattering cross-section data in Table 3.

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1854:ORODC>2.0.CO;2

Constituents and mean molecular weight of dry air.

Optical depths of the constituents of air (standard pressure 1013.25 mb and altitude 0 m).

Scattering cross section (per molecule) and Rayleigh optical depth (*τ _{R}*) for dry air containing 360 ppm CO

_{2}. Rayleigh optical depths are given for a location at sea level, 1013.25 mb, 45° latitude, and at MLO at altitude 3400 m, pressure 680 mb, and latitude 19.533°

(*Continued*)

(*Continued*)