Airmass

: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 isattenuated by scattering and absorption; the more atmospherethrough 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 pathlength relative to that at the zenith, so by definition, theairmass at the zenith is 1. Airmass increases as the angle between thesource and the zenith increases, reaching a value of approximately 38at the horizon. Airmass can also be less than one, for example, by increasing altitude from the reference level. The solar intensity above the atmophere is referred to as the “Air Mass Zero” (or AM0) spectrum.

Tables of airmass have been published by numerous authors, including
Bemporad (1904), Allen (1976), [Allen's airmass table was an abbreviated compilation of values from earlier sources, primarily
Bemporad (1904).] and Kasten and Young (1989).

Calculating airmass

Atmospheric Refraction

Atmospheric refraction causes light to follow an approximately circularpath that is slightly longer than the geometric path, and the airmass musttake into account the longer path (Young 1994).Additionally, refraction causes a celestial body to appear higher above thehorizon than it actually is; at the horizon, the difference between thetrue zenith angle and the apparent zenith angle is approximately 34 minutesof arc. Most airmass formulas are based on the apparent zenith angle, butsome are based on the true zenith angle, so it is important to ensure thatthe correct value is used, especially near the horizon. [At very high zenith angles, airmass is strongly dependent on local atmosphericconditions, 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. Manyauthors have cautioned that accurate calculation of airmass near the horizonis all but impossible.]

Plane-parallel atmosphere

When the zenith angle (or zenith distance) is small to moderate, agood approximation is given by assuming a homogeneous plane-parallelatmosphere (i.e., one in which density is constant and Earth's curvature isignored). The airmass $X$ then is simply the secant of the
zenith angle $z$:

:$X\; =\; sec,\; z$

At a zenith angle of 60° (i.e., at an altitudeof 90° − zenith angle = 30°) 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 $X\; =\; sec,\; z$becoming infinite atthe 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 simplecorrective term:

:$X\; =\; sec,z\_mathrm\; t\; ,\; left\; [\; 1\; -\; 0.0012\; ,(sec^2\; z\_mathrm\; t\; -\; 1)\; ight\; ]\; ,$

where $z\_mathrm\; t$ is the true zenith angle. This gives usableresults up to approximately 80°, but the accuracy degrades rapidly atgreater zenith angles. The calculated airmass reaches a maximum of 11.13at 86.6°, becomes zero at 88°, and approaches negative infinity atthe horizon. The plot of this formula on the accompanying graph includes acorrection for atmospheric refraction so that the calculated airmass is forapparent rather than true zenith angle.

Hardie (1962) introduced a polynomial in $sec,z\; -\; 1$:

:$X\; =\; sec,z\; ,-,\; 0.0018167\; ,(sec,z\; ,-,\; 1)\; ,-,\; 0.002875\; ,(sec,z\; ,-,\; 1)^2\; ,-,\; 0.0008083\; ,(sec,z\; ,-,\; 1)^3\; ,$

which gives usable results for zenith angles of up to perhaps 85°. Aswith the previous formula, the calculated airmass reaches a maximum, andthen approaches negative infinity at the horizon.

Rozenberg (1966) suggested

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

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

Kasten and Young (1989) developed

:$X\; =\; frac\{1\}\; \{\; cos,\; z\; +\; 0.50572\; ,(96.07995\; -\; z)^\{-1.6364;,$

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

Young (1994) developed

:$X\; =\; frac\{\; 1.002432,\; cos^2\; z\_mathrm\; t\; +\; 0.148386\; ,\; cos,\; z\_mathrm\; t\; +\; 0.0096467\; \}\{\; cos^3\; z\_mathrm\; t\; +\; 0.149864,\; cos^2\; z\_mathrm\; t\; +\; 0.0102963\; ,\; cos,\; z\_mathrm\; t\; +\; 0.000303978\; \},,$

in terms of the true zenith angle $z\_mathrm\; t$, for which heclaimed a maximum error (at the horizon) of 0.0037 airmass.

Atmospheric models

Interpolative formulas attempt to provide a good fit to tabular values ofairmass using minimal computational overhead. The tabularvalues, however, must be determined from measurements or atmosphericmodels that derive from geometrical and physical considerations of Earth andits atmosphere.

Nonrefracting radially symmetrical atmosphere

If refraction is ignored, it can be shown from simple geometricalconsiderations (Schoenberg 1929, 173)that the path $s$ of a light ray at zenith angle$z$ through a radially symmetrical atmosphere of height$y\_\{mathrm\; \{atm$ 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,

:$s\; =\; sqrt\; \{left\; (\; R\_mathrm\; \{E\}\; +\; y\_mathrm\{atm\}\; ight\; )^2\; -\; R\_mathrm\; \{E\}^2\; sin^2\; z\}\; -\; R\_mathrm\; \{E\}\; cos,\; z,\; ,$

where $R\_mathrm\; E$ is the radius of the Earth.

Homogeneous atmosphere

If the atmosphere is homogeneous (i.e., density is constant), thepath at zenith is simply the atmospheric height $y\_\{mathrm\{atm$, and the relative airmass is

:$X\; =\; frac\; s\; \{y\_mathrm\{atm\; =\; frac\; \{R\_mathrm\; \{E\; \{y\_mathrm\{atm\; sqrt\; \{cos^2\; z\; +\; 2\; frac\; \{y\_mathrm\{atm\; \{R\_mathrm\; \{E\; +\; left\; (\; frac\; \{y\_mathrm\{atm\; \{R\_mathrm\; \{E\; ight\; )^2\; \}\; -\; frac\; \{R\_mathrm\; \{E\; \{y\_mathrm\{atm\; cos,\; z$

If density is constant, hydrostatic considerations give the atmospheric height as

:$y\_mathrm\{atm\}\; =\; frac\; \{kT\_0\}\; \{mg\},,$

where $k$ is Boltzmann's constant, $T\_0$ is thesea-level temperature, $m$ is the molecular mass of air, and$g$ is the acceleration due to gravity. Although this is thesame as the pressure scale height of an isothermal atmosphere, theimplication is slightly different. In an isothermal atmosphere, 37% of theatmosphere is above the pressure scale height; in a homogeneous atmosphere,there is no atmosphere above the atmospheric height.

Taking $T\_0$ = 288.15 K,$m$ = 28.9644×1.6605×$10^\{-27\}$ kg,and $g$ = 9.80665 $mathrm\{m/s\}^2$gives $y\_mathrm\{atm\}$ ≈ 8435 m. UsingEarth's mean radius of 6371 km, the sea-level airmass at the horizon is

:$X\_mathrm\{horiz\}\; =\; sqrt\; \{1\; +\; 2\; frac\; \{R\_mathrm\; \{E\; \{y\_mathrm\{atm\}\; approx\; 38.87$

The homogeneous spherical model slightlyunderestimates the increase in airmass very close to the horizon; a reasonable overallfit to values determined from more rigorous models can be had by setting theairmass to match a value at a zenith angle less than 90°.For example, matching Bemporad's value of 19.787 at $z$ = 88°gives $y\_mathrm\{atm\}$ ≈ 10,096 m and$X\_mathrm\{horiz\}$ ≈ 35.54.

While a homogeneous atmosphere isn't a physically realistic model, the approximation is reasonableas 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, andrequires comparatively little computational overhead; if high accuracy isnot required, it gives reasonable results. [Although acknowledging that an isothermal or polytropicatmosphere would have been more realistic,
Janiczek and DeYoung (1987) used thehomogeneous spherical model in calculating illumination from the Sun andMoon, with the implication that the slightly reduced accuracy was more thanoffset by the considerable reduction in computational overhead.] However, a better fit to accepted values of airmass can be had with severalof the interpolative formulas.

Variable-density atmosphere

In a real atmosphere, density decreases with elevation above
mean sea level. The "absolute airmass"$sigma$ then is

:$sigma\; =\; int\; ho\; ,\; mathrm\; d\; s$

For the geometrical light path discussed above, this becomes, for a sea-level observer,

:$sigma\; =\; int\_0^\{y\_mathrm\{atm\; frac\; \{\; ho\; ,\; left\; (\; R\_mathrm\; \{E\}\; +\; y\; ight\; )\; mathrm\; d\; y\}\; \{sqrt\; \{R\_mathrm\; \{E\}^2\; cos^2\; z\; +\; 2\; R\_mathrm\; \{E\}\; y\; +\; y^2$

The relative airmass then is

:$X\; =\; frac\; sigma\; \{sigma\_mathrm\{zen$

The absolute airmass at zenith $sigma\_mathrm\{zen\}$ is also known asthe "column density".

Isothermal atmosphere

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

:$ho\; =\; ho\_0\; e^\{-y\; /\; H\},,$

where $ho\_0$ is the sea-level density and $H$ isthe pressure scale height. When the limits of integration are zero andinfinity, and some high-order terms are dropped, this model yields(Young 1974, 147),

:$X\; approx\; sqrt\; \{\; frac\; \{pi\; R\}\; \{2\; H\; exp\; \{left\; (\; frac\; \{R\; cos^2\; z\}\; \{2\; H\}\; ight\; )\}\; ,\; mathrm\; \{erfc\}\; left\; (\; sqrt\; \{frac\; \{R\; cos^2\; z\}\; \{2\; H\; ight\; )$

An approximate correction for refraction can be made by taking(Young 1974, 147)

:$R\; =\; 7/6\; ,\; R\_mathrm\; E,,$

where $R\_mathrm\; E$ is the physical radius of the Earth. At thehorizon, the approximate equation becomes

:$X\_mathrm\{horiz\}\; approx\; sqrt\; \{\; frac\; \{pi\; R\}\; \{2\; H$

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

:$X\_mathrm\{horiz\}\; approx\; 37.20$

Polytropic atmosphere

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

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

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

:$ho\; =\; ho\_0\; left\; (\; 1\; -\; frac\; alpha\; T\_0\; y\; ight\; )^\{1\; /\; (kappa\; -\; 1)\},,$

where $kappa$ 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, sothe integration usually is performed numerically.

Compound atmosphere

Earth's atmosphere consists of multiple layers with differenttemperature and density characteristics; common atmospheric modelsinclude the International Standard Atmosphere and the
US Standard Atmosphere. A good approximation for many purposes is apolytropic troposphere of 11 km height with a lapse rate of6.5 K/km and an isothermal stratosphere of infinite height(Garfinkel 1967), which corresponds very closelyto the first two layers of the International Standard Atmosphere. Morelayers can be used if greater accuracy is required. [ The notes for ReedMeyer's [http://reed.gigacorp.net/vitdownld.html#airmass airmass calculator] describe an atmospheric model using eight layers and using polynomialsrather than simple linear relations for temperature lapse rates.]

Refracting radially symmetrical atmosphere

When atmospheric refraction is considered, the absolute airmass integral becomes [See Thomason, Herman, and Reagan (1983) fora derivation of the integral for a refracting atmosphere.]

:$sigma\; =\; int\_\{r\_mathrm\{obs^\{r\_mathrm\{atm\; frac\; \{\; ho,\; mathrm\; d\; r\}\; \{sqrt\; \{\; 1\; -\; left\; (\; frac\; \{n\_mathrm\{obs\; n\; frac\; \{r\_mathrm\{obs\; r\; ight\; )^2\; sin^2\; z,,$

where $n\_mathrm\{obs\}$ is the index of refraction of air at theobserver's elevation $y\_mathrm\{obs\}$ above sea level,$n$ is the index of refraction at elevation$y$ above sea level, $r\_mathrm\{obs\}\; =\; R\_mathrm\{E\}\; +\; y\_mathrm\{obs\}$,$r\; =\; R\_mathrm\{E\}\; +\; y$ is the distance from the center ofthe Earth to a point at elevation $y$, and $r\_mathrm\{atm\}=\; R\_mathrm\{E\}\; +\; y\_mathrm\{atm\}$ is distance to the upper limit ofthe atmosphere at elevation $y\_mathrm\{atm\}$. The index ofrefraction in terms of density is usually given to sufficient accuracy(Garfinkel 1967) by the Dale-Gladstonerelation

:$frac\; \{n\; -\; 1\}\; \{n\_mathrm\{obs\}\; -\; 1\}\; =\; frac\; \{\; ho\}\; \{\; ho\_mathrm\{obs$

Rearrangement and substitution into the absolute airmass integralgives

:$sigma\; =\; int\_\{r\_mathrm\{obs^\{r\_mathrm\{atm\; frac\; \{\; ho,\; mathrm\; d\; r\}\; \{sqrt\; \{\; 1\; -\; left\; (\; frac\; \{n\_mathrm\{obs\; \{1\; +\; (\; n\_mathrm\{obs\}\; -\; 1\; )\; ho/\; ho\_mathrm\{obs\; ight\; )^2\; left\; (\; frac\; \{r\_mathrm\{obs\; r\; ight\; )^2\; sin^2\; z$

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

:$sigma\; =\; int\_\{r\_mathrm\{obs^\{r\_mathrm\{atm\; frac\; \{\; ho,\; mathrm\; d\; r\}\; \{sqrt\; \{\; 1\; -\; left\; [\; 1\; +\; 2\; (\; n\_mathrm\{obs\}\; -\; 1\; )(1\; -\; frac\; ho\; \{\; ho\_mathrm\{obs\; )\; ight\; ]\; left\; (\; frac\; \{r\_mathrm\{obs\; r\; ight\; )^2\; sin^2\; z$

Nonuniform distribution of attenuating species

Atmospheric models that derive from hydrostatic considerationsassume an atmosphere of constant composition and a single mechanismof extinction, which isn't quite correct. There are three main sources ofattenuation (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 elevationabove sea level, and the concentrations of aerosols and ozone cannot bederived simply from hydrostatic considerations.

Rigorously, when the extinction coefficient depends on elevation, itmust be determined as part of the airmass integral, as described by
Thomason, Herman, and Reagan (1983). Acompromise approach often is possible, however. Methods for separatelycalculating 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 sometimesbe done by using one of the simple airmass formulas and separatelydetermining extinction coefficients for each of the attenuating species(Green 1992).

Notes

References

* Allen, C. W. 1976. "Astrophysical Quantities", 3rd ed. 1973, reprinted with corrections, 1976. London: Athlone, 125. ISBN 0-485-11150-0
* Bemporad, A. 1904. Zur Theorie der Extinktion des Lichtes in der Erdatmosphäre. "Mitteilungen der Großherzoglichen Sternwarte zu Heidelberg" Nr. 4, 1–78.
* Garfinkel, B. 1967. Astronomical Refraction in a Polytropic Atmosphere. "Astronomical Journal" 72:235–254.
* Green, Daniel W. E. 1992. Magnitude Corrections for Atmospheric Extinction. "International Comet Quarterly" 14, July 1992, 55–59.
* Hardie, R. H. 1962. In "Astronomical Techniques". Hiltner, W. A., ed. Chicago: University of Chicago Press, 184–. LCCN 62009113
* Hayes, D. S., and D. W. Latham. 1975. A Rediscussion of the Atmospheric Extinction and the Absolute Spectral-Energy Distribution of Vega. "Astrophysical Journal" 197:593–601.
* Janiczek, P. M., and J. A. DeYoung. 1987. "Computer Programs for Sun and Moon Illuminance with Contingent Tables and Diagrams", United States Naval Observatory Circular No. 171. Washington, D.C.: United States Naval Observatory.
* Kasten, F., and A. T. Young. 1989. Revised optical air mass tables and approximation formula. "Applied Optics" 28:4735–4738.
* Rozenberg, G. V. 1966. "Twilight: A Study in Atmospheric Optics". New York: Plenum Press, 160. Translated from the Russian by R. B. Rodman. LCCN 65011345
* Schaefer, B. E. 1993. Astronomy and the Limits of Vision. "Vistas in Astronomy" 36:311–361.
* ———. 1998. To the Visual Limits. "Sky & Telescope", May 1998, 57–60.
* Schoenberg, E. 1929. Theoretische Photometrie, g) Über die Extinktion des Lichtes in der Erdatmosphäre. In "Handbuch der Astrophysik". Band II, erste Hälfte. Berlin: Springer.
* Thomason, L. W., B. M. Herman, and J. A. Reagan. 1983. The effect of atmospheric attenuators with structured vertical distributions on air mass determination and Langley plot analyses. "Journal of the Atmospheric Sciences" 40:1851–1854.
* Young, A. T. 1974. Atmospheric Extinction. Ch. 3.1 in "Methods of Experimental Physics", Vol. 12 "Astrophysics", Part A: "Optical and Infrared". ed. N. Carleton. New York: Academic Press. ISBN 0-12-474912-1
* Young, A. T. 1994. Air mass and refraction. "Applied Optics". 33:1108–1110.
* Young, A. T., and W. M. Irvine. 1967. Multicolor photoelectric photometry of the brighter planets. I. Program and procedure. "Astronomical Journal" 72:945–950.

ee also

* Atmospheric extinction
* Extinction coefficient
* International Standard Atmosphere
* Beer-Lambert-Bouguer law
* Law of atmospheres

External links

* An [http://www.aavso.org/observing/programs/ccd/airmass.shtml online airmass and scintillation calculator] via the AAVSO
* 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.