Air mass: Analytical and empirical treatment; an improved formula for air mass

In this paper, relative air mass is defined analytically and treated empirically, and a new formula is proposed. This formula takes into account not only the zenith angle, but also the Earth's curvature, altitude and temperature. The new formula and the old one (m = 1cos θz) are in good agreement for zenith angles up to 70°, but for bigger angles the new formula gives smaller values of air mass. In particular, it gives a finite value for air mass when the zenith angle is 90°. It is shown that the old formula is a special case of the new one. The values of relative air mass calculated with the new formula depend on the effective thickness of the atmosphere. Experimentally, it is found that the effective thickness of the atmosphere is linearly related to the average thickness of the atmosphere, which depends on latitude and meteorological conditions. It is concluded that for values of θz up to 70°, with little error, the old formula can be used, but that for larger zenith angles (especially for times near sunrise or sunset), only the new formula can give proper and meaningful results. The influence of the zenith angle, the altitude and the temperature on the relative air mass is analyzed.