In astronomy, there is a term
“Airmass”. Note that this is not “Air mass” (mass of air); it
refers to the ratio of the atmospheric
optical thickness in the direction of zenith distance z to the atmospheric optical thickness in the
zenith direction. I currently also cannot find what
its Chinese name is; anyway, I feel that if it is translated as
“Atmospheric Mass” it would be very strange, so I will temporarily
translate it as “Atmospheric Thickness Index”. Now it is known as
“Atmospheric Optical Mass”, generally denoted by X. As shown in the figure below, X = BC/AC.
In a relatively small area, the atmosphere and the ground can both be regarded as parallel planes. In this case, there is a good approximation formula: X = \sec z For current middle school textbooks, some readers may not know what \sec is. In fact: \sec z = \frac{1}{\cos z}.
At 60^\circ, the airmass is approximately equal to 2. However, the Earth is not flat. Depending on different accuracy requirements, the peak value of z is 60^\circ–75^\circ. As the zenith angle increases, the accuracy of this formula decreases rapidly; the formula tends toward infinity at the horizon, whereas according to actual curved atmospheric conditions, the airmass usually does not exceed 40.
In fact, if atmospheric refraction is not considered, the Earth is viewed as a sphere, and the atmosphere is also viewed as a hollow sphere with height y, looking only from a geometric perspective, the atmospheric thickness that light emitted from zenith angle z must penetrate is (R_E is the Earth’s radius): s = \sqrt{R_E^2 \cos^2 z + 2R_E \cdot y + y^2} - R_E \cdot \cos z Or written as: s = \sqrt{(R_E + y)^2 - R_E^2 \sin^2 z} - R_E \cdot \cos z
But in practice, factors such as refraction must also be considered, so many interpolation formulas have emerged. For example, Young and Irvine added a correction factor to X = \sec z in 1967: X = \sec z_t [1 - 0.0012(\sec^2 z_t - 1)] Here z_t is the true zenith angle, which is the zenith angle after correcting for atmospheric refraction. After this treatment, the peak value of the zenith angle z can reach 80^\circ. However, the accuracy also decreases rapidly as the zenith angle increases; the formula reaches a maximum value of 11.13 at z = 86.6^\circ and tends toward negative infinity at the horizon.
Hardie used a polynomial of \sec z - 1 for correction in 1962: X = \sec z - 0.0018167(\sec z - 1) - 0.002875(\sec z - 1)^2 - 0.0008083(\sec z - 1)^3 This provides a peak value as high as 85^\circ, but like the previous formula, it reaches a maximum value and then tends toward negative infinity at the horizon.
In 1966, Rozenberg proposed: X = (\cos z + 0.025e^{-11 \cos z})^{-1} This method still yields reasonable values at the horizon (at z = 90^\circ, airmass is approximately 40).
Kasten and Young developed this in 1989 into: X = \frac{1}{\cos z + 0.50572(96.07995 - z)^{-1.6364}} The formula remains quite reasonable when z is close to 90^\circ, with the result at the horizon being approximately 38. Note that z here must be in degrees.
In 1994, Young derived: X = \frac{1.002432 \cos^2 z_t + 0.148386 \cos z_t + 0.0096467}{\cos^3 z_t + 0.149864 \cos^2 z_t + 0.0102963 \cos z_t + 0.000303978} Similarly, z_t is the true zenith angle. This formula can control the error (even for results at the horizon) within 0.0037.
In 2002, Pickering derived the formula: X = \frac{x}{\sin(h + \frac{244}{165 + 47h^{1.1}})} where h = 90^\circ - z.
The curves in the figure below compare the accuracy of different interpolation formulas:
Interpolation formulas are approximate formulas derived from different calculation processes. They only consider the factor z, because in this problem, z is our only variable. To a certain extent, interpolation formulas facilitate our calculations. Although some formulas may look cumbersome, for the computer age, the key to a formula is: effective and accurate.
For more information, please refer to: http://en.wikipedia.org/wiki/Airmass
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