Air mass (astronomy)
In
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 (as defined above) at oblique incidence relative to that at
Tables of air mass have been published by numerous authors, including Bemporad (1904), Allen (1973),[1] and Kasten & Young (1989).
Definition
The absolute air mass is defined as:
In the
So is a type of
Finally, the relative air mass is:
Assuming air density to be uniform allows removing it from the integrals. The absolute air mass then simplifies to a product:
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 (neglecting ray bending), as discussed below.
Calculation
Background
The angle of a celestial body with the zenith is the
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 (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.[2]
Plane-parallel atmosphere
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 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 & Irvine (1967) included a simple corrective term:
Hardie (1962) introduced a polynomial in :
Rozenberg (1966) suggested
Kasten & Young (1989) developed[3]
Young (1994) developed
Pickering (2002) developed
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 (Schoenberg 1929, 173) that the path of a light ray at zenith angle through a radially symmetrical atmosphere of height above the Earth is given by
The relative air mass is then:
Homogeneous atmosphere
If the atmosphere is homogeneous (i.e., density is constant), the atmospheric height follows from
Taking , , and gives . 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
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 § Homogeneous spherical atmosphere with elevated observer). The model requires comparatively little computational overhead, and if high accuracy is not required, it gives reasonable results.[5] 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
Isothermal atmosphere
Several basic models for density variation with elevation are commonly used. The simplest, an
An approximate correction for refraction can be made by taking (Young 1974, p. 147)
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
Layered atmosphere
Refracting radially symmetrical atmosphere
When atmospheric refraction is considered, ray tracing becomes necessary (Kivalov 2007), and the absolute air mass integral becomes[7]
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 (Kasten & Young 1989)
Homogeneous spherical atmosphere with elevated observer
This section possibly contains original research. (February 2019) |
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,
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
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 (
Rigorously, when the extinction coefficient depends on elevation, it must be determined as part of the air mass integral, as described by 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 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 air mass formulas and separately determining extinction coefficients for each of the attenuating species (Green 1992, Pickering 2002).
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".[8] On bigger telescopes, such as the WHT (Wynne & Worswick 1988) and VLT (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
In
Air mass and solar energy
In some fields, such as
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
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 § Plane-parallel atmosphere.
See also
- 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
Notes
- ^ Allen's air mass table was an abbreviated compilation of values from earlier sources, primarily Bemporad (1904).
- ^ 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.
- ^ The Kasten and Young formula was originally given in terms of altitude as
in this article, it is given in terms of zenith angle for consistency with the other formulas.
- ^ Pickering (2002) uses Garfinkel (1967) as the reference for accuracy.
- ^ Although acknowledging that an isothermal or polytropic atmosphere would have been more realistic, 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.
- ^ The notes for Reed Meyer's air mass calculator describe an atmospheric model using eight layers and using polynomials rather than simple linear relations for temperature lapse rates.
- ^ See Thomason, Herman & Reagan (1983) for a derivation of the integral for a refracting atmosphere.
- ^ Observing tips: air mass and differential refraction retrieved 15 May 2011.
- ^ ASTM E 490-00a was reapproved without change in 2006.
References
- Allen, C. W. (1973). Astrophysical Quantities (3rd ed.). London: Athlone, 125.: Athlone Press. )
- ASTM E 490-00a (R2006). 2000. Standard Solar Constant and Zero Air Mass Solar Spectral Irradiance Tables. West Conshohocken, PA: ASTM. Available for purchase from ASTM.Optical Telescopes of Today and Tomorrow
- ASTM G 173-03. 2003. Standard Tables for Reference Solar Spectral Irradiances: Direct Normal and Hemispherical on 37° Tilted Surface. West Conshohocken, PA: ASTM. Available for purchase from ASTM.
- Avila, Gerardo; Rupprecht, Gero; Beckers, J. M. (1997). Arne L. Ardeberg (ed.). "Atmospheric dispersion correction for the FORS Focal Reducers at the ESO VLT". Optical Telescopes of Today and Tomorrow. Proceedings of SPIE. 2871 Optical Telescopes of Today and Tomorrow: 1135–1143. S2CID 120965966.
- Bemporad, A. 1904. Zur Theorie der Extinktion des Lichtes in der Erdatmosphäre. Mitteilungen der Grossh. Sternwarte zu Heidelberg Nr. 4, 1–78.
- Garfinkel, Boris (1967). "Astronomical Refraction in a Polytropic Atmosphere". The Astronomical Journal. 72: 235–254. doi:10.1086/110225.
- 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. Bibcode 1962aste.book.....H.
- Kivalov, Sergey N. (2007). "Improved ray tracing air mass numbers model". Applied Optics. 46 (29): 7091–8. PMID 17932515.
- Hayes, D. S.; Latham, D. W. (1975). "A Rediscussion of the Atmospheric Extinction and the Absolute Spectral-Energy Distribution of Vega". The Astrophysical Journal. 197: 593–601. ISSN 0004-637X.
- 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. Bibcode 1987USNOC.171.....J.
- Kasten, F.; Young, A. T. (1989). "Revised optical air mass tables and approximation formula". Applied Optics. 28 (22): 4735–4738. PMID 20555942.
- Pickering, K. A. (2002). "The Southern Limits of the Ancient Star Catalog" (PDF). DIO. 12 (1): 20–39.
- Rozenberg, Grzegorz V. (1966). Twilight: A Study in Atmospheric Optics. New York: Plenum Press. OCLC 1066196615.
- Schaefer, Bradley E. (1993). "Astronomy and the Limits of Vision". Vistas in Astronomy. 36 (4): 311–361. .
- Schaefer, B. E. 1998. To the Visual Limits: How deep can you see?. Sky & Telescope, May 1998, 57–60.
- Schoenberg, E. 1929. Theoretische Photometrie, Über die Extinktion des Lichtes in der Erdatmosphäre. In Handbuch der Astrophysik. Band II, erste Hälfte. Berlin: Springer.
- Thidé, Bo (2007-12-01). "Nonlinear physics of the ionosphere and LOIS/LOFAR". Plasma Physics and Controlled Fusion. 49 (12B): B103–B107. S2CID 18502182.
- Thomason, L. W.; Herman, B. M.; Reagan, J. A. (1983-07-01). "The Effect of Atmospheric Attenuators with Structured Vertical Distributions on Air Mass Determinations and Langley Plot Analyses". Journal of the Atmospheric Sciences. 40 (7): 1851–1854. ISSN 0022-4928.
- van der Tol, Sebastiaan; van der Veen, Alle-Jan (2007). "Ionospheric Calibration for the LOFAR Radio Telescope". 2007 International Symposium on Signals, Circuits and Systems. Vol. 2. pp. 1–4. ISBN 978-1-4244-0968-6.
- de Vos, M.; Gunst, A. W.; Nijboer, R. (2009). "The LOFAR Telescope: System Architecture and Signal Processing" (PDF). Proceedings of the IEEE. 97 (8): 1431–1437. S2CID 4411160.
- Wynne, C. G.; Worswick, S. P. (1988-02-01). "Atmospheric dispersion at prime focus". Monthly Notices of the Royal Astronomical Society. 230 (3): 457–471. ISSN 0035-8711.
- 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, Andrew T. (1994-02-20). "Air Mass and Refraction". Applied Optics. 33 (6): 1108–1110. PMID 20862124.
- Young, Andrew T.; Irvine, William M. (1967). "Multicolor photoelectric photometry of the brighter planets. I. Program and Procedure". The Astronomical Journal. 72: 945–950. doi:10.1086/110366.
External links
- 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.