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.

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.

Atmospheric physicists also need airmass values when inferring the amount of molecular absorption from telluric lines in the solar spectrum. 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.

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 Andrew T. Young

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