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Airmass
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- For air mass in meteorology, see air mass.
In astronomy, airmass is the optical path length through Earth's atmosphere for light from a celestial source. As it passes through the atmosphere, light is attenuated by scattering and absorption; the more atmosphere through which it passes, the greater the attenuation. Consequently, celestial bodies at the horizon appear less bright than when at the zenith. The attenuation, known as atmospheric extinction, is described quantitatively by the Beer-Lambert-Bouguer law.
“Airmass” normally indicates relative airmass, the path length relative to that at the zenith at sea level, so by definition, the sea-level airmass at the zenith is 1.
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- For air mass in meteorology, see air mass.
In astronomy, airmass is the optical path length through Earth's atmosphere for light from a celestial source. As it passes through the atmosphere, light is attenuated by scattering and absorption; the more atmosphere through which it passes, the greater the attenuation. Consequently, celestial bodies at the horizon appear less bright than when at the zenith. The attenuation, known as atmospheric extinction, is described quantitatively by the Beer-Lambert-Bouguer law.
“Airmass” normally indicates relative airmass, the path length relative to that at the zenith at sea level, so by definition, the sea-level airmass at the zenith is 1. Airmass increases as the angle between the source and the zenith increases, reaching a value of approximately 38 at the horizon. Airmass can be less than one at an elevation greater than sea level; however, most closed-form expressions for airmass do not include the effects of elevation, so adjustment must usually be accomplished by other means.
In some fields, such as solar energy, airmass is indicated by the acronym AM; additionally, the value of the airmass is often given by appending its value to AM, so that AM1 indicates an airmass of 1, AM2 indicates an airmass of 2, and so on. The region above Earth’s atmoshphere, where there is no atmospheric attenuation of solar radiation, is considered to have
“air mass zero” (AM0).
Tables of airmass have been published by numerous authors, including Bemporad (1904), Allen (1976),
and Kasten and Young (1989).
Calculating airmass
Atmospheric Refraction
Atmospheric refraction causes light to follow an approximately circular
path that is slightly longer than the geometric path, and the airmass must
take into account the longer path (Young 1994).
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 airmass 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 (or zenith distance) is small to moderate, a
good approximation is given by assuming a homogeneous plane-parallel
atmosphere (i.e., one in which density is constant and Earth's curvature is
ignored). The airmass then is simply the secant of the
zenith angle :
At a zenith angle of 60° (i.e., 90° − altitude angle = zenith angle) the airmass is approximately 2.
The Earth is not flat, however, and, depending on accuracy requirements,
this formula is usable for zenith angles up to about 60° to 75°.
At greater zenith angles, the accuracy degrades rapidly, with
becoming infinite at
the horizon, while the horizontal airmass in the curved atmosphere is usually less than 40.
Interpolative formulas
Many formulas have been developed to fit tabular values of airmass; one by
Young and Irvine (1967) 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 airmass 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 airmass is for
apparent rather than true zenith angle.
Hardie (1962) introduced a polynomial in :
which gives usable results for zenith angles of up to perhaps 85°. As
with the previous formula, the calculated airmass reaches a maximum, and
then approaches negative infinity at the horizon.
Rozenberg (1966) suggested
which gives reasonable results for high zenith angles, with a horizon airmass of 40.
Kasten and Young (1989) developed
which gives reasonable results for zenith angles of up to 90°, with an
airmass of approximately 38 at the horizon. Here the second
term is in degrees.
Young (1994) developed
in terms of the true zenith angle , for which he
claimed a maximum error (at the horizon) of 0.0037 airmass.
Atmospheric models
Interpolative formulas attempt to provide a good fit to tabular values of
airmass 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 radially symmetrical atmosphere
If refraction is ignored, it can be shown from simple geometrical
considerations (Schoenberg 1929, 173)
that the path of a light ray at zenith angle
through a radially symmetrical atmosphere of height
is given by
or alternatively,
where is the radius of the Earth.
Homogeneous atmosphere
If the atmosphere is homogeneous (i.e., density is constant), the
path at zenith is simply the atmospheric height , and the relative airmass is
If density is constant, hydrostatic considerations give the atmospheric height 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× kg,
and = 9.80665
gives ˜ 8435 m. Using
Earth's mean radius of 6371 km, the sea-level airmass at the horizon is
The homogeneous spherical model slightly
underestimates the increase in airmass very close to the horizon; a reasonable overall
fit to values determined from more rigorous models can be had by setting the
airmass to match a value at a zenith angle less than 90°.
For example, matching Bemporad's value of 19.787 at = 88°
gives ˜ 10,096 m and
˜ 35.54.
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 (i.e., it does not diverge or go to zero) at all zenith angles, and
requires comparatively little computational overhead; if high accuracy is
not required, it gives reasonable results.
However, a better fit to accepted values of airmass can be had with several
of the interpolative formulas.
Variable-density atmosphere
In a real atmosphere, density decreases with elevation above
mean sea level. The absolute airmass
then is
For the geometrical light path discussed above, this becomes, for a sea-level observer,
The relative airmass then is
The absolute airmass at zenith is also known as
the column density.
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
(Young 1974, 147),
An approximate correction for refraction can be made by taking
(Young 1974, 147)
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 (or polytropic index).
The airmass 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.
Compound atmosphere
Earth's 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
(Garfinkel 1967), 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, the absolute airmass 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
(Garfinkel 1967) by the Dale-Gladstone
relation
Rearrangement and substitution into the absolute airmass integral
gives
The quantity is quite small; expanding the
first term in parentheses, rearranging several times, and ignoring terms in
after each rearrangement, gives
(Kasten and Young 1989)
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 (Hayes and Latham 1975):
Rayleigh scattering by air molecules, Mie scattering by
aerosols, and molecular absorption (primarily by
ozone). 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 airmass integral, as described by
Thomason, Herman, and Reagan (1983). A
compromise approach often is possible, however. Methods for separately
calculating the extinction from each species using
closed-form expressions are described in
Schaefer (1993) and
Schaefer (1998). 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 airmass formulas and separately
determining extinction coefficients for each of the attenuating species
(Green 1992).
Airmass and solar energy
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 airmass of 1.5 (AM1.5); tables of these standard spectra are given in ASTM G 173-03. The extraterrestrial spectral irradiance (i.e., that for AM0) is given in ASTM E 490-00a.
For most solar energy applications, airmass is determined to sufficient accuracy with the simple secant formula described in the section Plane-parallel atmosphere.
See also
External links
- An via the AAVSO
- Reed Meyer's (notes in the source code describe the theory in detail)
- A source for electronic copies of some of the references.
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