Air mass (astronomy)#Homogeneous atmosphere

The absolute air mass is defined as:

$$\backslash sigma\; =\; \backslash int\; \backslash rho\; \backslash ,\; \backslash mathrm\; ds\; \backslash ,.$$

where $\backslash rho$ is volumetric density of air. Thus $\backslash sigma$ is a type of oblique column density.

In the vertical direction, the absolute air mass at zenith is:

$$\backslash sigma\_\backslash mathrm\{zen\}\; =\; \backslash int\; \backslash rho\; \backslash ,\; \backslash mathrm\; dz$$

So $\backslash sigma\_\backslash mathrm\{zen\}$ is a type of vertical column density.

Finally, the relative air mass is:

$$X\; =\; \backslash frac\; \backslash sigma\; \{\backslash sigma\_\backslash mathrm\{zen\}\}$$

Assuming air density to be uniform allows removing it from the integrals. The absolute air mass then simplifies to a product:

$$\backslash sigma\; =\; \backslash bar\backslash rho\; s$$

where $\backslash bar\backslash rho\; =\; \backslash mathrm\{const.\}$ is the average density and the arc length $s$ of the oblique and zenith light paths are:

$$\backslash begin\{align\}$$

s &= \int \, \mathrm ds \\

s_\mathrm{zen} &= \int \, \mathrm dz

\end{align}

In the corresponding simplified relative air mass, the average density cancels out in the fraction, leading to the ratio of path lengths:

$$X\; =\; \backslash frac\; s\; \{s\_\backslash mathrm\{zen\}\}\; \backslash ,.$$

Further simplifications are often made, assuming straight-line propagation (neglecting ray bending), as discussed below.

Image:Viewing_angle_and_air_masses.svg

The angle of a celestial body with the zenith is the zenith angle (in astronomy, commonly referred to as the zenith distance). A body's angular position can also be given in terms of altitude, the angle above the geometric horizon; the altitude $h$ and the zenith angle $z$ are thus related by

$$h\; =\; 90^\backslash circ\; -\; z\; \backslash ,.$$

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 {{harv|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 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.At very high zenith angles, air mass is strongly dependent on local atmospheric conditions, including temperature, pressure, and especially the temperature gradient near the ground. In addition low-altitude extinction is strongly affected by the aerosol concentration and its vertical distribution. Many authors have cautioned that accurate calculation of air mass near the horizon is all but impossible.

When the zenith angle 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 air mass $X$ then is simply the secant of the zenith angle $z$:

$$X\; =\; \backslash sec\backslash ,\; z\; \backslash ,.$$

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 $X\; =\; \backslash sec\backslash ,\; z$ becoming infinite at the horizon; the horizon air mass in the more realistic spherical atmosphere is usually less than 40.

Many formulas have been developed to fit tabular values of air mass; one by {{harvtxt|Young|Irvine|1967}} included a simple corrective term:

$$X\; =\; \backslash sec\backslash ,z\_\backslash mathrm\; t\; \backslash ,\; \backslash left\; [\; 1\; -\; 0.0012\; \backslash ,(\backslash sec^2\; z\_\backslash mathrm\; t\; -\; 1)\; \backslash right\; ]\; \backslash ,,$$

where $z\_\backslash mathrm\; t$ 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.

{{harvtxt|Hardie|1962}} introduced a polynomial in $\backslash sec\backslash ,z\; -\; 1$:

$$X\; =\; \backslash sec\backslash ,z\; \backslash ,-\backslash ,\; 0.0018167\; \backslash ,(\backslash sec\backslash ,z\; \backslash ,-\backslash ,\; 1)\; \backslash ,-\backslash ,\; 0.002875\; \backslash ,(\backslash sec\backslash ,z\; \backslash ,-\backslash ,\; 1)^2$$

\,-\, 0.0008083 \,(\sec\,z \,-\, 1)^3

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.

{{harvtxt|Rozenberg|1966}} suggested

$$X\; =\; \backslash left\; (\backslash cos\backslash ,z\; +\; 0.025\; e^\{-11\; \backslash cos\backslash ,\; z\}\; \backslash right\; )^\{-1\}\; \backslash ,,$$

which gives reasonable results for high zenith angles, with a horizon air mass of 40.

{{harvtxt|Kasten|Young|1989}} developedThe Kasten and Young formula was originally given in terms of altitude $\backslash gamma$ as

$$X\; =\; \backslash frac\{1\}\; \{\; \backslash sin\backslash ,\; \backslash gamma\; +\; 0.50572\; \backslash ,(\backslash gamma\; +\; 6.07995^\backslash circ\; )^\{-1.6364\}\}\backslash ;;$$ in this article, it is given in terms of zenith angle for consistency with the other formulas.

$$X\; =\; \backslash frac\{1\}\; \{\; \backslash cos\backslash ,\; z\; +\; 0.50572\; \backslash ,(6.07995^\backslash circ\; +\; 90^\backslash circ\; -\; z)^\{-1.6364\}\}\; \backslash ,,$$

which gives reasonable results for zenith angles of up to 90°, with an air mass of approximately 38 at the horizon. Here the second $z$ term is in degrees.

{{harvtxt|Young|1994}} developed

$$X\; =\; \backslash frac$$

{ 1.002432\, \cos^2 z_\mathrm t + 0.148386 \, \cos\, z_\mathrm t + 0.0096467 }

{ \cos^3 z_\mathrm t + 0.149864\, \cos^2 z_\mathrm t + 0.0102963 \, \cos\, z_\mathrm t + 0.000303978 } \,

in terms of the true zenith angle $z\_\backslash mathrm\; t$, for which he claimed a maximum error (at the horizon) of 0.0037 air mass.

{{harvtxt|Pickering|2002}} developed

$$X\; =\; \backslash frac\{1\}\; \{\; \backslash sin\; (h\; +\; \{244\}/(165+47\; h^\{1.1\})\; )\; \}\; \backslash ,,$$

where $h$ is apparent altitude $(90^\backslash circ\; -\; z)$ in degrees. Pickering claimed his equation to have a tenth the error of {{harvtxt|Schaefer|1998}} near the horizon.{{harvtxt|Pickering|2002}} uses {{harvtxt|Garfinkel|1967}} as the reference for accuracy. (p. 22, continuation of footnote 39 beginning on p. 21)

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.

File:airmass geometry.png

If atmospheric refraction is ignored, it can be shown from simple geometrical considerations (Schoenberg 1929, 173) that the path $s$ of a light ray at zenith angle

$z$ through a radially symmetrical atmosphere of height $y\_\{\backslash mathrm\; \{atm\}\}$ above the Earth is given by

s = \sqrt {R_\mathrm {E}^2 \cos^2 z + 2 R_\mathrm {E} y_\mathrm{atm}

+ y_\mathrm{atm}^2}

- R_\mathrm {E} \cos\, z \,

or alternatively,

s = \sqrt {\left ( R_\mathrm {E} + y_\mathrm{atm} \right )^2

- R_\mathrm {E}^2 \sin^2 z}

- R_\mathrm {E} \cos\, z \,

where $R\_\backslash mathrm\; E$ is the radius of the Earth.

The relative air mass is then:

X = \frac s {y_\mathrm{atm}}

= \frac {R_\mathrm {E}} {y_\mathrm{atm}} \sqrt {\cos^2 z

+ 2 \frac {y_\mathrm{atm}} {R_\mathrm {E}}

+ \left ( \frac {y_\mathrm{atm}} {R_\mathrm {E}} \right )^2 }

- \frac {R_\mathrm {E}} {y_\mathrm{atm}} \cos\, z \,.

If the atmosphere is homogeneous (i.e., density is constant), the atmospheric height $y\_\{\backslash mathrm\; \{atm\}\}$ follows from hydrostatic considerations as:{{citation needed|date=May 2020}}

$$y\_\backslash mathrm\{atm\}\; =\; \backslash frac\; \{kT\_0\}\; \{mg\}\; \backslash ,,$$

where $k$ is the Boltzmann constant, $T\_0$ is the sea-level temperature, $m$ is the molecular mass of air, and $g$ 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% (1/e) of the atmosphere is above the pressure scale height; in a homogeneous atmosphere, there is no atmosphere above the atmospheric height.

Taking $T\_0\; =\; \backslash mathrm\{288.15~K\}$, $m\; =\; \backslash mathrm\{\; 28.9644\; \backslash times\; 1.6605\; \backslash times\; 10^\{-27\}\; ~\; kg\}$, and $g\; =\; \backslash mathrm\{9.80665\; ~\; m/s^2\}$ gives $y\_\backslash mathrm\{atm\}\; \backslash approx\; \backslash mathrm\{8435\; ~\; m\}$. Using Earth's mean radius of 6371 km, the sea-level air mass at the horizon is

X_\mathrm{horiz} = \sqrt {1 + 2 \frac {R_\mathrm {E}} {y_\mathrm{atm}}} \approx 38.87 \,.

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

$$\backslash frac\; \{R\_\backslash mathrm\{E\}\}\; \{y\_\backslash mathrm\{atm\}\}$$

= \frac {X^2 - 1} {2 \left ( 1 - X \cos z \right )} \,;

matching Bemporad's value of 19.787 at $z$ = 88°

gives $R\_\backslash mathrm\{E\}\; /\; y\_\backslash mathrm\{atm\}$ ≈ 631.01 and

$X\_\backslash mathrm\{horiz\}$ ≈ 35.54. With the same value for $R\_\backslash mathrm\{E\}$ as above, $y\_\backslash mathrm\{atm\}$ ≈ 10,096 m.

While a homogeneous atmosphere is not 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, including those greater than 90° (see {{slink||Homogeneous spherical atmosphere with elevated observer}}). The model requires comparatively little computational overhead, and if high accuracy is not required, it gives reasonable results.Although acknowledging that an isothermal or polytropic atmosphere would have been more realistic, {{harvtxt|Janiczek|DeYoung|1987}} used the homogeneous spherical model in calculating illumination from the Sun and Moon, with the implication that the slightly reduced accuracy was more than offset by the considerable reduction in computational overhead.

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.

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,

\sigma = \int_0^{y_\mathrm{atm}}

\frac {\rho \, \left ( R_\mathrm {E} + y \right ) \mathrm d y}

{\sqrt {R_\mathrm {E}^2 \cos^2 z + 2 R_\mathrm {E} y + y^2}} \,.

Several basic models for density variation with elevation are commonly used. The simplest, an isothermal atmosphere, gives

$$\backslash rho\; =\; \backslash rho\_0\; e^\{-y\; /\; H\}\; \backslash ,,$$

where $\backslash rho\_0$ is the sea-level density and $H$ is the density scale height. When the limits of integration are zero and infinity, the result is known as Chapman function. An approximate result is obtained if some high-order terms are dropped, yielding {{harv|Young|1974|p=147}},

X \approx \sqrt { \frac {\pi R} {2 H}}

\exp {\left ( \frac {R \cos^2 z} {2 H} \right )} \,

\mathrm {erfc} \left ( \sqrt {\frac {R \cos^2 z} {2 H}} \right ) \,.

An approximate correction for refraction can be made by taking {{harv|Young|1974|p=147}}

$$R\; =\; 7/6\; \backslash ,\; R\_\backslash mathrm\; E\; \backslash ,,$$

where $R\_\backslash mathrm\; E$ is the physical radius of the Earth. At the horizon, the approximate equation becomes

$$X\_\backslash mathrm\{horiz\}\; \backslash approx\; \backslash sqrt\; \{\; \backslash frac\; \{\backslash pi\; R\}\; \{2\; H\}\}\; \backslash ,.$$

Using a scale height of 8435 m, Earth's mean radius of 6371 km, and including the correction for refraction,

$$X\_\backslash mathrm\{horiz\}\; \backslash approx\; 37.20\; \backslash ,.$$

The assumption of constant temperature is simplistic; a more realistic model is the polytropic atmosphere, for which

$$T\; =\; T\_0\; -\; \backslash alpha\; y\; \backslash ,,$$

where $T\_0$ is the sea-level temperature and $\backslash alpha$ is the temperature lapse rate. The density as a function of elevation is

$$\backslash rho\; =\; \backslash rho\_0\; \backslash left\; (\; 1\; -\; \backslash frac\; \backslash alpha\; T\_0\; y\; \backslash right\; )^\{1\; /\; (\backslash kappa\; -\; 1)\}\; \backslash ,,$$

where $\backslash kappa$ is the polytropic exponent (or polytropic index). 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.

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 {{harv|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.The notes for Reed Meyer's [http://reed.gigacorp.net/vitdownld.html#airmass air mass calculator] describe an atmospheric model using eight layers and using polynomials rather than simple linear relations for temperature lapse rates.

When atmospheric refraction is considered, ray tracing becomes necessary {{harv|Kivalov|2007}}, and the absolute air mass integral becomesSee {{harvtxt|Thomason|Herman|Reagan|1983}} for a derivation of the integral for a refracting atmosphere.

\sigma = \int_{r_\mathrm{obs}}^{r_\mathrm{atm}} \frac {\rho\, \mathrm d r}

{\sqrt { 1 - \left ( \frac {n_\mathrm{obs}} n \frac {r_\mathrm{obs}} r \right )^2 \sin^2 z}} \,

where $n\_\backslash mathrm\{obs\}$ is the index of refraction of air at the observer's elevation $y\_\backslash mathrm\{obs\}$ above sea level, $n$ is the index of refraction at elevation $y$ above sea level, $r\_\backslash mathrm\{obs\}\; =\; R\_\backslash mathrm\{E\}\; +\; y\_\backslash mathrm\{obs\}$,

$r\; =\; R\_\backslash mathrm\{E\}\; +\; y$ is the distance from the center of the Earth to a point at elevation $y$, and $r\_\backslash mathrm\{atm\}\; =\; R\_\backslash mathrm\{E\}\; +\; y\_\backslash mathrm\{atm\}$ is distance to the upper limit of the atmosphere at elevation $y\_\backslash mathrm\{atm\}$. The index of refraction in terms of density is usually given to sufficient accuracy (Garfinkel 1967) by the Gladstone–Dale relation

$$\backslash frac\; \{n\; -\; 1\}\; \{n\_\backslash mathrm\{obs\}\; -\; 1\}\; =\; \backslash frac\; \{\backslash rho\}\; \{\backslash rho\_\backslash mathrm\{obs\}\}\; \backslash ,.$$

Rearrangement and substitution into the absolute air mass integral gives

\sigma = \int_{r_\mathrm{obs}}^{r_\mathrm{atm}} \frac {\rho\, \mathrm d r}

{\sqrt { 1 - \left ( \frac {n_\mathrm{obs}} {1 + ( n_\mathrm{obs} - 1 ) \rho/\rho_\mathrm{obs}} \right )^2 \left ( \frac {r_\mathrm{obs}} r \right )^2 \sin^2 z}} \,.

The quantity $n\_\backslash mathrm\{obs\}\; -\; 1$ is quite small; expanding the first term in parentheses, rearranging several times, and ignoring terms in $(n\_\backslash mathrm\{obs\}\; -\; 1)^2$ after each rearrangement, gives {{harv|Kasten|Young|1989}}

\sigma = \int_{r_\mathrm{obs}}^{r_\mathrm{atm}} \frac {\rho\, \mathrm d r}

{\sqrt { 1 - \left [ 1 + 2 ( n_\mathrm{obs} - 1 )(1 - \frac \rho {\rho_\mathrm{obs}} ) \right ]

\left ( \frac {r_\mathrm{obs}} r \right )^2 \sin^2 z}} \,.

{{original research section|date=February 2019}}

File:HomogSphElevObsAM.png

In the figure at right, an observer at O is at an elevation $y\_\backslash text\{obs\}$ above sea level in a uniform radially symmetrical atmosphere of height $y\_\backslash mathrm\{atm\}$. The path length of a light ray at zenith angle $z$ is $s$; $R\_\backslash mathrm\{E\}$ is the radius of the Earth. Applying the law of cosines to triangle OAC,

$$\backslash begin\{align\}$$

\left(R_{E}+y_\text{atm}\right)^2 & =s^2+\left(R_{E}+y_\text{obs}\right)^2-2\left(R_{E}+y_\text{obs}\right)s \cos\left(180^{\circ}-z\right)\\

& =s^2+\left(R_{E}+y_\text{obs}\right)^2+2\left(R_{E}+y_\text{obs}\right)s\cos z\end{align}

expanding the left- and right-hand sides, eliminating the common terms, and rearranging gives

$$\{\{s\}^2\}+2\backslash left(\; \{R\_\{\backslash text\{E\}\}\}+\{y\_\{\backslash text\{obs\}\}\}\; \backslash right)s\backslash cos\; z-2\{R\_\{\backslash text\{E\}\}\}\{y\_\{\backslash text\{atm\}\}\}-y\_\{\backslash text\{atm\}\}^2+2\{R\_\{\backslash text\{E\}\}\}\{y\_\{\backslash text\{obs\}\}\}+y\_\{\backslash text\{obs\}\}^2=0\; \backslash ,.$$

Solving the quadratic for the path length s, factoring, and rearranging,

$$s=\backslash pm\; \backslash sqrt\{\{\{\backslash left(\; \{R\_\{\backslash text\{E\}\}\}+\{y\_\{\backslash text\{obs\}\}\}\; \backslash right)\}^2\}\{\{\backslash cos\; \}^2\}z+2\{R\_\{\backslash text\{E\}\}\}\backslash left(\; \{y\_\{\backslash text\{atm\; \}\}\}-\{y\_\{\backslash text\{obs\}\}\}\; \backslash right)+y\_\{\backslash text\{atm\}\}^2-y\_\{\backslash text\{obs\}\}^2\}-(\{R\_\{\backslash text\{E\}\}\}+\{y\_\{\backslash text\{obs\}\}\})\backslash cos\; z\; \backslash ,.$$

The negative sign of the radical gives a negative result, which is not physically meaningful. Using the positive sign, dividing by $y\_\backslash mathrm\{atm\}$, and cancelling common terms and rearranging gives the relative air mass:

$$X=\backslash sqrt\{\{\{\backslash left(\; \backslash frac\{\{R\_\{\backslash text\{E\}\}\}+\{y\_\{\backslash text\{obs\}\}\}\}\{\{y\_\{\backslash text\{atm\}\}\}\}\; \backslash right)\}^2\}\{\{\backslash cos\; \}^2\}z+\backslash frac\{2\{R\_\{\backslash text\{E\}\}\}\}\{y\_\{\backslash text\{atm\}\}^2\}\backslash left(\; \{y\_\{\backslash text\{atm\}\}\}-\{y\_\{\backslash text\{obs\}\}\}\; \backslash right)-\{\{\backslash left(\; \backslash frac\{\{y\_\{\backslash text\{obs\}\}\}\}\{\{y\_\{\backslash text\{atm\}\}\}\}\; \backslash right)\}^2\}+1\}-\backslash frac\{\{R\_\{\backslash text\{E\}\}\}+\{y\_\{\backslash text\{obs\}\}\}\}\{\{y\_\{\backslash text\{atm\}\}\}\}\backslash cos\; z\; \backslash ,.$$

With the substitutions $\backslash hat\{r\}\; =\; R\_\backslash mathrm\{E\}\; /\; y\_\backslash mathrm\{atm\}$ and $\backslash hat\{y\}\; =\; y\_\backslash mathrm\{obs\}\; /\; y\_\backslash mathrm\{atm\}$, this can be given as

$$X\; =\; \backslash sqrt\{\{(\backslash hat\{r\}+\backslash hat\{y\})\}^2\; \backslash cos^2\; z\; +\; 2\; \backslash hat\{r\}\; (1-\backslash hat\{y\})\; -\; \backslash hat\{y\}^2\; +1\}\; \backslash ;\; -\; \backslash ;\; (\backslash hat\{r\}+\backslash hat\{y\})\; \backslash cos\; z\; \backslash ,.$$

When the observer's elevation is zero, the air mass equation simplifies to

$$X\; =\; \backslash sqrt\{\{\{\backslash left(\; \backslash frac\{R\_\backslash text\{E\}\}\{y\_\backslash text\{atm\}\}\; \backslash right)\}^2\}\; \backslash cos^2\; z\; +\; \backslash frac\{2\; R\_\backslash text\{E\}\}\{y\_\backslash text\{atm\}\}\; +\; 1\}\; -\; \backslash frac\{R\_\backslash text\{E\}\}\{y\_\backslash text\{atm\}\}\; \backslash cos\; z\; \backslash ,.$$

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°.

File:HomogSphElevObsZmax.png

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 {{harv|Hayes|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 air mass integral, as described by {{harvtxt|Thomason|Herman|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 {{harvtxt|Schaefer|1993}} and {{harvtxt|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 air mass formulas and separately determining extinction coefficients for each of the attenuating species ({{harvnb|Green|1992}}, {{harvnb|Pickering|2002}}).

File:Atmospheric electromagnetic transmittance or opacity.jpg]]

{{See also|Telescope#Atmospheric electromagnetic opacity}}

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".[http://www.astro.uni-bonn.de/~mischa/obstips/airmass.html Observing tips: air mass and differential refraction] retrieved 15 May 2011. On bigger telescopes, such as the WHT {{harv|Wynne|Worswick|1988}} and VLT {{harv|Avila|Rupprecht|Beckers|1997}}, 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 (or more specifically, on the zenith angle).

In radio astronomy the air mass (which influences the optical path length) 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 ({{harvnb|van der Tol|van der Veen|2007}}; {{harvnb|de Vos|Gunst|Nijboer|2009}}), but on the other hand can also study the ionosphere by instead measuring these distortions {{harv|Thidé|2007}}.

File:Solar Spectrum.png

{{Main|Air mass (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" (AM0).

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 (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.ASTM E 490-00a was reapproved without change in 2006.

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 {{slink||Plane-parallel atmosphere}}.

{{Div col|colwidth=22em}}

• Air mass (solar energy)
• Atmospheric extinction
• Beer–Lambert–Bouguer law
• Chapman function
• Computation of radiowave attenuation in the atmosphere
• Diffuse sky radiation
• Extinction coefficient
• Illuminance
• International Standard Atmosphere
• Irradiance
• Law of atmospheres
• Light diffusion
• Mie scattering
• Path loss
• Photovoltaic module
• Rayleigh scattering
• Solar irradiation

{{div col end}}

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• Reed Meyer's [http://reed.gigacorp.net/vitdownld.html#airmass downloadable airmass calculator, written in C] (notes in the source code describe the theory in detail)
• [http://adswww.harvard.edu/index.html NASA Astrophysics Data System] A source for electronic copies of some of the references.

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