The relative airmass is the ratio of the amount of air in the line of sight (at some zenith distance z) to the amount in the zenith. So it's exactly unity at the zenith; at the horizon, it's about 38, at sea level. The term “air mass” was coined by Bouguer, whose monograph on photometry contains the world's first table of air masses.

Traditionally, the airmass function is denoted by M(z), though other notations are in use. Bemporad, in his classic work on the subject, used F(z); Hardie, in a widely-cited article, used X for the airmass, based on the incorrect notion that the magnitude of a star would be a linear function of airmass. (Hardie also gave an approximation formula for airmass with a misleading number of significant figures.)

Airmass is closely linked to refraction by Laplace's extinction theorem. Like refraction, airmass near the horizon depends on the structure of the atmosphere. Unlike refraction, even airmass at modest zenith distances still depends on atmospheric structure. Fortunately, however, the dependence at the zenith distances where photometry and radiometry are feasible is quite weak; so it is usually ignored.

As an aside, note that the relative airmass function would be just the secant of the zenith distance, if the Earth were flat. The Earth isn't  flat, but people often use sec z as an approximation to M(z) near the zenith.

Who needs airmass?

Airmass values, obtained either from standard tables of the airmass function, or from approximation formulae, are used by astronomical photometrists in correcting observed stellar magnitudes for atmospheric extinction. They are also used in the field of solar energy to estimate the amount of solar power transmitted by the Earth's atmosphere at different times during the day.

Both these groups need airmass as a function of the geometric zenith distance of the Sun or some other star. However, airmass tables (such as Bemporad's widely-reproduced one) and formulae (such as those published by Kasten) nearly always have the apparent  (i.e., refracted) zenith distance as argument. This means that the geometric Z.D. calculated from the time and the positions of the observer and the observed object MUST be converted to apparent (refracted) Z.D. before using the tables or formulae.

Using the wrong argument can produce serious errors!   This is a very common mistake.

Atmospheric physicists also need airmass values when inferring the amount of molecular absorption from telluric lines in the solar spectrum. But note that the usual airmass factor applies only  to the well-mixed gases (oxygen [O2], nitrogen, and perhaps carbon dioxide, and some very stable pollutants like fluorocarbons) — but certainly not  water vapor, reactive species such as OH, oxides of chlorine, nitrogen, and sulfur, or photochemically active species like ozone [O3]. Here again, the airmass is usually calculated from times and positions, not from observed zenith distances; so the precaution just mentioned must be carefully observed.

The usual airmass is not enough

Another common problem is to assume that “the” airmass function is all you need to deal with atmospheric extinction. But it only tells you how much extinction is due to the well-mixed gases in the atmosphere: the part of the extinction due to Rayleigh scattering. A different  relative airmass factor must be used to deal with aerosol scattering and water vapor, because these sources of extinction have a different vertical structure (they are more concentrated toward the Earth's surface).

These additional sources of extinction become relatively much more important near the horizon. They are extremely variable, so any attempt to describe their effective “airmass functions” is necessarily approximate. The airmass factor for water vapor can be several times larger at the horizon than the molecular-atmosphere value of about 38. The horizontal extinction due to aerosols can be hundreds of times larger than the zenith value, particularly if a low-level inversion is present.

Some of the green-flash simulations illustrate the effects of variable aerosol extinction near the horizon. (See also the discussion of GF colors.)

In addition, a significant amount of the visible extinction is due to ozone, which is concentrated in a layer near 25 km height, in the stratosphere. The relative ozone airmass near the horizon is considerably less  than that of the mixed molecular atmosphere. This effect has also been included in making the realistic simulations of green flashes.


Copyright © 2007 – 2008, 2012, 2020 Andrew T. Young

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