It is relevant to remember that when calculating air mass, we are actually calculating the relative air mass. The term "relative air mass" refers to the path length relative to that of when the light source (such as the sun) is at the zenith at sea level. The value of sea level air mass at the zenith is therefore taken as 1, and the relative air mass is measures according to it.
Before we start the calculations of air mass, we must first understand the concept of attenuation of celestial light. The attenuation of light refers to the phenomenon of scattering and absorption of light as it passes through the atmosphere. It is due to attenuation that the sun or the moon appears less bright in the horizon than at the zenith.
The reason behind this is that light at the zenith has to travel less distance to reach Earth’s surface than when at the horizon. The attenuation of light is also thereby referred to as atmospheric extinction. The Beer Lambert Bouguer law quantitatively measures the atmospheric extinction rate of light. Due to the attenuation of light the exact air mass is difficult to calculate, however, we can still get precise calculations of air mass.
The mathematical expression for calculation of air mass is AM=A1/Az. Here, AM is the air mass index. A1 is the value of the actual air mass and Az the air mass at the zenith. A1 and Az are calculated by the same methods of calculations. The value of air mass at the zenith is 1, which is the reason we say it is the relative air mass. The air mass index is a ratio of two values with similar units.
Take the air mass at the zenith length to be "x" and the air mass at the actual position length to be "y." Take the angle between the zenith (i.e at 90 degrees) and the actual position of the air mass to be "z." Therefore, we get a right-angled triangle with y as hypotenuse and x as the perpendicular, with z the angle between them. From basic trigonometry, we can derive that sec z = y/x. We already know that y/x is actually the ratio of A1 and Az, which is the air mass index, or AM.