Air mass (astronomy)


In astronomy, air mass or airmass is the "amount of air that one is looking through" when seeing a star or other celestial source from below Earth's atmosphere. It is formulated as the integral of air density along the light ray.
As it penetrates the atmosphere, light is attenuated by scattering and absorption; the thicker atmosphere through which it passes, the greater the attenuation. Consequently, celestial bodies when nearer the horizon appear less bright than when nearer the zenith. This attenuation, known as atmospheric extinction, is described quantitatively by the Beer–Lambert law.
"Air mass" normally indicates relative air mass, the ratio of absolute air masses at oblique incidence relative to that at zenith. So, by definition, the relative air mass at the zenith is 1. Air mass increases as the angle between the source and the zenith increases, reaching a value of approximately 38 at the horizon. Air mass can be less than one at an elevation greater than sea level; however, most closed-form expressions for air mass do not include the effects of the observer's elevation, so adjustment must usually be accomplished by other means.
Tables of air mass have been published by numerous authors, including Bemporad, Allen,
and Kasten and Young.

Definition

The absolute air mass is defined as:
where is volumetric density of air.
Thus is a type of oblique column density.
In the vertical direction, the absolute air mass at zenith is:
So is a type of vertical column density.
Finally, the relative air mass is:
Assuming air density is uniform allows removing it out of the integrals.
The absolute air mass then simplifies to a product:
where is the average density
and the arc length of the oblique and zenith light paths are:
In the corresponding simplified relative air mass, the average density cancels out in the fraction, leading to the ratio of path lengths:
Further simplifications are often made, assuming straight-line propagation, as discussed below.

Calculation

Background

The angle of a celestial body with the zenith is the zenith angle. A body’s angular position can also be given in terms of altitude, the angle above the geometric horizon; the altitude and the zenith angle are thus related by
Atmospheric refraction causes light entering the atmosphere to follow an approximately circular
path that is slightly longer than the geometric path. Air mass must
take into account the longer path.
Additionally, refraction causes a celestial body to appear higher above the
horizon than it actually is; at the horizon, the difference between the
true zenith angle and the apparent zenith angle is approximately 34 minutes
of arc. Most air mass formulas are based on the apparent zenith angle, but
some are based on the true zenith angle, so it is important to ensure that
the correct value is used, especially near the horizon.

[|Plane-parallel atmosphere]

When the zenith angle is small to moderate, a
good approximation is given by assuming a homogeneous plane-parallel
atmosphere. The air mass then is simply the secant of the
zenith angle :
At a zenith angle of 60°, the air mass is approximately 2.
However, because the Earth is not flat,
this formula is only usable for zenith angles up to about 60° to 75°, depending on accuracy requirements.
At greater zenith angles, the accuracy degrades rapidly, with
becoming infinite at
the horizon; the horizon air mass in the more-realistic spherical atmosphere is usually less than 40.

Interpolative formulas

Many formulas have been developed to fit tabular values of air mass; one by
Young and Irvine included a simple
corrective term:
where is the true zenith angle. This gives usable
results up to approximately 80°, but the accuracy degrades rapidly at
greater zenith angles. The calculated air mass reaches a maximum of 11.13
at 86.6°, becomes zero at 88°, and approaches negative infinity at
the horizon. The plot of this formula on the accompanying graph includes a
correction for atmospheric refraction so that the calculated air mass is for
apparent rather than true zenith angle.
Hardie introduced a polynomial in :
which gives usable results for zenith angles of up to perhaps 85°. As
with the previous formula, the calculated air mass reaches a maximum, and
then approaches negative infinity at the horizon.
Rozenberg suggested
which gives reasonable results for high zenith angles, with a horizon air mass of 40.
Kasten and Young developed
which gives reasonable results for zenith angles of up to 90°, with an
air mass of approximately 38 at the horizon. Here the second
term is in degrees.
Young developed
in terms of the true zenith angle, for which he
claimed a maximum error of 0.0037 air mass.
Pickering developed
where is apparent altitude in degrees. Pickering claimed his equation to have a tenth the error of Schaefer near the horizon.

Atmospheric models

Interpolative formulas attempt to provide a good fit to tabular values of
air mass using minimal computational overhead. The tabular
values, however, must be determined from measurements or atmospheric
models that derive from geometrical and physical considerations of Earth and
its atmosphere.

Nonrefracting spherical atmosphere

If atmospheric refraction is ignored, it can be shown from simple geometrical
considerations
that the path of a light ray at zenith angle
through a radially symmetrical atmosphere of height
above the Earth is given by
or alternatively,
where is the radius of the Earth.
The relative air mass is then:

Homogeneous atmosphere

If the atmosphere is homogeneous, the atmospheric height follows from hydrostatic considerations as:
where is Boltzmann’s constant, is the
sea-level temperature, is the molecular mass of air, and
is the acceleration due to gravity. Although this is the
same as the pressure scale height of an isothermal atmosphere, the
implication is slightly different. In an isothermal atmosphere, 37% of the
atmosphere is above the pressure scale height; in a homogeneous atmosphere,
there is no atmosphere above the atmospheric height.
Taking = 288.15 K,
= 28.9644×1.6605×10−27 kg,
and = 9.80665 m/s2
gives ≈ 8435 m. Using
Earth’s mean radius of 6371 km, the sea-level air mass at the horizon is
The homogeneous spherical model slightly underestimates the rate of increase in air mass near the horizon; a reasonable overall
fit to values determined from more rigorous models can be had by setting the
air mass to match a value at a zenith angle less than 90°. The air mass equation can be rearranged to give
matching Bemporad’s value of 19.787 at = 88°
gives ≈ 631.01 and
≈ 35.54. With the same value for as above, ≈ 10,096 m.
While a homogeneous atmosphere isn’t a physically realistic model, the approximation is reasonable
as long as the scale height of the atmosphere is small compared to the radius of the planet.
The model is usable at all zenith angles, including those greater than 90°. The model
requires comparatively little computational overhead, and if high accuracy is
not required, it gives reasonable results.
However, for zenith angles less than 90°, a better fit to accepted values of air mass can be had with several
of the interpolative formulas.

Variable-density atmosphere

In a real atmosphere, density is not constant (it decreases with elevation above mean sea level.
The absolute air mass for the geometrical light path discussed above, becomes, for a sea-level observer,

Isothermal atmosphere

Several basic models for density variation with elevation are commonly used. The simplest, an
isothermal atmosphere, gives
where is the sea-level density and is
the pressure scale height. When the limits of integration are zero and
infinity, and some high-order terms are dropped, this model yields
,
An approximate correction for refraction can be made by taking
where is the physical radius of the Earth. At the
horizon, the approximate equation becomes
Using a scale height of 8435 m, Earth’s mean radius of 6371 km,
and including the correction for refraction,

Polytropic atmosphere

The assumption of constant temperature is simplistic; a more realistic
model is the polytropic atmosphere, for which
where is the sea-level temperature and
is the temperature lapse rate. The density as a function of elevation
is
where is the polytropic exponent.
The air mass integral for the polytropic model does not lend itself to a
closed-form solution except at the zenith, so
the integration usually is performed numerically.

Layered atmosphere

consists of multiple layers with different
temperature and density characteristics; common atmospheric models
include the International Standard Atmosphere and the
US Standard Atmosphere. A good approximation for many purposes is a
polytropic troposphere of 11 km height with a lapse rate of
6.5 K/km and an isothermal stratosphere of infinite height
, which corresponds very closely
to the first two layers of the International Standard Atmosphere. More
layers can be used if greater accuracy is required.

Refracting radially symmetrical atmosphere

When atmospheric refraction is considered, ray tracing becomes necessary, and the absolute air mass integral becomes
where is the index of refraction of air at the
observer’s elevation above sea level,
is the index of refraction at elevation
above sea level,,
is the distance from the center of
the Earth to a point at elevation, and is distance to the upper limit of
the atmosphere at elevation. The index of
refraction in terms of density is usually given to sufficient accuracy
by the Gladstone–Dale relation
Rearrangement and substitution into the absolute air mass integral
gives
The quantity is quite small; expanding the
first term in parentheses, rearranging several times, and ignoring terms in
after each rearrangement, gives

Homogeneous spherical atmosphere with elevated observer

In the figure at right, an observer at O is at an elevation above sea level in a uniform radially symmetrical atmosphere of height. The path length of a light ray at zenith angle is ; is the radius of the Earth. Applying the law of cosines to triangle OAC,
expanding the left- and right-hand sides, eliminating the common terms, and rearranging gives
Solving the quadratic for the path length s, factoring, and rearranging,
The negative sign of the radical gives a negative result, which is not physically meaningful. Using the positive sign, dividing by, and cancelling common terms and rearranging gives the relative air mass:
With the substitutions and, this can be given as
When the observer’s elevation is zero, the air mass equation simplifies to
In the limit of grazing incidence, the absolute airmass equals the distance to the horizon.
Furthermore, if the observer is elevated, the horizon zenith angle can be greater than 90°.

Nonuniform distribution of attenuating species

Atmospheric models that derive from hydrostatic considerations
assume an atmosphere of constant composition and a single mechanism
of extinction, which isn’t quite correct. There are three main sources of
attenuation :
Rayleigh scattering by air molecules, Mie scattering by
aerosols, and molecular absorption. The relative contribution of each source varies with elevation
above sea level, and the concentrations of aerosols and ozone cannot be
derived simply from hydrostatic considerations.
Rigorously, when the extinction coefficient depends on elevation, it
must be determined as part of the air mass integral, as described by
Thomason, Herman, and Reagan. A
compromise approach often is possible, however. Methods for separately
calculating the extinction from each species using
closed-form expressions are described in
Schaefer and
Schaefer. The latter reference includes
source code for a BASIC program to perform the calculations.
Reasonably accurate calculation of extinction can sometimes
be done by using one of the simple air mass formulas and separately
determining extinction coefficients for each of the attenuating species
.

Implications

Air mass and astronomy

In optical astronomy, the air mass provides an indication of the deterioration of the observed image, not only as regards direct effects of spectral absorption, scattering and reduced brightness, but also an aggregation of visual aberrations, e.g. resulting from atmospheric turbulence, collectively referred to as the quality of the "seeing". On bigger telescopes, such as the WHT and VLT, the atmospheric dispersion can be so severe that it affects the pointing of the telescope to the target. In such cases an atmospheric dispersion compensator is used, which usually consists of two prisms.
The Greenwood frequency and Fried parameter, both relevant for adaptive optics, depend on the air mass above them.
In radio astronomy the air mass is not relevant. The lower layers of the atmosphere, modeled by the air mass, do not significantly impede radio waves, which are of much lower frequency than optical waves. Instead, some radio waves are affected by the ionosphere in the upper atmosphere. Newer aperture synthesis radio telescopes are especially affected by this as they “see” a much larger portion of the sky and thus the ionosphere. In fact, LOFAR needs to explicitly calibrate for these distorting effects, but on the other hand can also study the ionosphere by instead measuring these distortions.

Air mass and solar energy

In some fields, such as solar energy and photovoltaics, air mass is indicated by the acronym AM; additionally, the value of the air mass is often given by appending its value to AM, so that AM1 indicates an air mass of 1, AM2 indicates an air mass of 2, and so on. The region above Earth's atmosphere, where there is no atmospheric attenuation of solar radiation, is considered to have "air mass zero".
Atmospheric attenuation of solar radiation is not the same for all wavelengths; consequently, passage through the atmosphere not only reduces intensity but also alters the spectral irradiance. Photovoltaic modules are commonly rated using spectral irradiance for an air mass of 1.5 ; tables of these standard spectra are given in ASTM G 173-03. The extraterrestrial spectral irradiance is given in ASTM E 490-00a.
For many solar energy applications when high accuracy near the horizon is not required, air mass is commonly determined using the simple secant formula described in the section Plane-parallel atmosphere.