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 Lambert-Beer law.

“Airmass” normally indicates relative airmass, the path length relative to that at the zenith, so by definition, the 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. Tables of airmass have been published by numerous authors, including Bemporad (1904), Allen (1976), and Kasten and Young (1989).

1 Calculating airmass
2 Notes
3 References
4 See also
5 External links

[edit] Calculating airmass

Plots of airmass using various formulae.

[edit] 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 formulae 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 at the horizon.^[1]

[edit] 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 $X$ then is simply the secant of the zenith angle $z$ :

$X = \sec\, z$

At a zenith angle of 60° (i.e., at an altitude of 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 airmass becoming infinite at the horizon.

[edit] Interpolative formulae

Many formulae have been developed to fit tabular values of airmass; one by Young and Irvine (1967) included a simple corrective term:

$X = \sec\,z_\mathrm t \, \left [ 1 - 0.0012 \,(\sec^2 z_\mathrm t - 1) \right ],$

where $z t$ is the true zenith angle. This gives usable results up to approximately 80°, but the accuracy degrades rapidly at greater zenith angles. The calculated 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 $\sec\,z - 1$ :

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

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

Rozenberg (1966) suggested

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

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 an airmass 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 $z t$ , for which he claimed a maximum error (at the horizon) of 0.0037 airmass.

[edit] Atmospheric models

Interpolative formulae 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.

[edit] Nonrefracting radially symmetrical atmosphere

If 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 $y atm$ 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,

$s = \sqrt {\left ( R_\mathrm {E} + y_\mathrm{atm} \right )^2 - R_\mathrm {E}^2 \sin^2 z} - R_\mathrm {E} \cos\, z\, ,$

where $R E$ is the radius of the Earth.

[edit] Homogeneous atmosphere

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

$X = \frac s {y_\mathrm{atm}} = \frac {R_\mathrm {E}} {y_\mathrm{atm}} \sqrt {\cos^2 z + 2 \frac {y_\mathrm{atm}} {R_\mathrm {E}} + \left ( \frac {y_\mathrm{atm}} {R_\mathrm {E}} \right )^2 } - \frac {R_\mathrm {E}} {y_\mathrm{atm}} \cos\, z$

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 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% of the atmosphere 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 $m / s 2$ gives $y atm$ ≈ 8435 m. Using Earth'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 atmosphere isn't a very realistic model, so the atmospheric height determined above has little physical significance. The 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 $z$ = 88° gives $y atm$ ≈ 10,096 m and $X horiz$ ≈ 35.54.

The homogeneous spherical model is usable at all zenith angles, and requires comparatively little computational overhead; if high accuracy is not required, it gives reasonable results.^[2] However, a better fit to accepted values of airmass can be had with several of the interpolative formulae.

[edit] Variable-density atmosphere

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

$\sigma = \int \rho \, \mathrm d s$

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

$\sigma = \int_0^{y_\mathrm{atm}} \frac {\rho \, \left ( R_\mathrm {E} + y \right ) \mathrm d y} {\sqrt {R_\mathrm {E}^2 \cos^2 z + 2 R_\mathrm {E} y + y^2}}$

The relative airmass then is

$X = \frac \sigma {\sigma_\mathrm{zen}}$

The absolute airmass at zenith $σ zen$ is also known as the column density.

[edit] Isothermal atmosphere

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

$\rho = \rho_0 e^{-y / H}\,,$

where $ρ 0$ is the sea-level density and $H$ 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),

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

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

$R = 7/6 \, R_\mathrm E\,,$

where $R E$ is the physical radius of the Earth. At the horizon, 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,

$X_\mathrm{horiz} \approx 37.20$

[edit] Polytropic atmosphere

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

$T = T_0 - \alpha y\,,$

where $T 0$ is the sea-level temperature and $α$ is the temperature lapse rate. The density as a function of elevation is

$\rho = \rho_0 \left ( 1 - \frac \alpha T_0 y \right )^{1 / (\kappa - 1)}\,,$

where $κ$ 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.

[edit] 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 (Garfinkle 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.^[3]

[edit] Refracting radially symmetrical atmosphere

When atmospheric refraction is considered, the absolute airmass integral becomes^[4]

$\sigma = \int_{r_\mathrm{obs}}^{r_\mathrm{atm}} \frac {\rho\, \mathrm d r} {\sqrt { 1 - \left ( \frac {n_\mathrm{obs}} n \frac {r_\mathrm{obs}} r \right )^2 \sin^2 z}}\,,$

where $n obs$ is the index of refraction of air at the observer's elevation $y obs$ above sea level, $n$ is the index of refraction at elevation $y$ above sea level, $r obs = R E + y obs$ , $r = R E + y$ is the distance from the center of the Earth to a point at elevation $y$ , and $r atm = R E + y atm$ is distance to the upper limit of the atmosphere at elevation $y atm$ . The index of refraction in terms of density is usually given to sufficient accuracy (Garfinkle 1967) by the Dale-Gladstone relation

$\frac {n - 1} {n_\mathrm{obs} - 1} = \frac {\rho} {\rho_\mathrm{obs}}$

Rearrangement and substitution into the absolute airmass 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 $n obs - 1$ is quite small; expanding the first term in parentheses, rearranging several times, and ignoring terms in $(n obs - 1) 2$ after each rearrangement, gives (Kasten and Young 1989)

$\sigma = \int_{r_\mathrm{obs}}^{r_\mathrm{atm}} \frac {\rho\, \mathrm d r} {\sqrt { 1 - \left [ 1 + 2 ( n_\mathrm{obs} - 1 )(1 - \frac \rho {\rho_\mathrm{obs}} ) \right ] \left ( \frac {r_\mathrm{obs}} r \right )^2 \sin^2 z}}$

[edit] 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 formulae and separately determining extinction coefficients for each of the attenuating species (Green 1992).

[edit] Notes

^ At very high zenith angles, airmass is strongly dependent on local atmospheric conditions, including temperature, pressure, and aerosol concentration. Many authors have cautioned that accurate calculation of airmass near the horizon is all but impossible.
^ Although acknowledging that an isothermal or polytropic atmosphere would have been more realistic, Janiczek and 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.
^ The notes for Reed Meyer's airmass calculator describe an atmospheric model using eight layers and using polynomials rather than simple linear relations for temperature lapse rates.
^ See Thomason, Herman, and Reagan (1983) for a derivation of the integral for a refracting atmosphere.

[edit] References

Allen, C. W. 1976. Astrophysical Quantities, 3rd ed. 1973, reprinted with corrections, 1976. London: Athlone, 125. ISBN 0485111500
Bemporad, A. 1904. Zur Theorie der Extinktion des Lichtes in der Erdatmosphäre. Mitteilungen der Großherzoglichen Sternwarte zu Heidelberg Nr. 4, 1–78.
Garfinkle, 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 0124749121
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.

[edit] See also

[edit] External links

An online airmass and scintillation calculator via the AAVSO
Reed Meyer's downloadable airmass calculator, written in C (notes in the source code describe the theory in detail)
NASA Astrophysics Data System A source for electronic copies of some of the references.

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Category: Spherical astronomy