Airmass
705635
199207803
2008-03-18T23:04:14Z
JeffConrad
1801965
Formulae → formulas: conform to preferred usage outside of acadamia.
: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
[[extinction (astronomy)#Atmospheric extinction|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]], 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. 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
[[#CITEREFBemporad1904|Bemporad (1904)]], [[#CITEREFAllen1976|Allen (1976)]],<ref>
Allen's airmass table was an abbreviated compilation of values from earlier sources, primarily
[[#CITEREFBemporad1904|Bemporad (1904)]].
</ref>
and [[#CITEREFKastenYoung1989|Kasten and Young (1989)]].
==Calculating airmass==
[[Image:AirmassFormulaePlots.png |thumb|right|Plots of airmass using various formulas.]]
===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 ([[#CITEREFYoung1994|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.<ref>
At very high zenith angles, airmass 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 airmass near the horizon
is all but impossible.</ref>
===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 <math>X</math> then is simply the [[secant]] of the
[[celestial coordinate system|zenith angle]] <math>z</math>:
:<math>X = \sec\, z</math>
At a zenith angle of 60° (i.e., at an [[altitude (astronomy)|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 <math>X = \sec\, z</math>
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
[[#CITEREFYoung and Irvine1967|Young and Irvine (1967)]] included a simple
corrective term:
:<math>X = \sec\,z_\mathrm t \, \left [ 1 - 0.0012 \,(\sec^2 z_\mathrm t - 1) \right ],</math>
where <math>z_\mathrm t</math> 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.
[[#CITEREFHardie1962|Hardie (1962)]] introduced a polynomial in <math>\sec\,z - 1</math>:
:<math>X = \sec\,z \,-\, 0.0018167 \,(\sec\,z \,-\, 1) \,-\, 0.002875 \,(\sec\,z \,-\, 1)^2
\,-\, 0.0008083 \,(\sec\,z \,-\, 1)^3 ,
</math>
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.
[[#CITEREFRozenberg1966|Rozenberg (1966)]] suggested
:<math>X = \left (\cos\,z + 0.025 e^{-11 \cos\, z} \right )^{-1},</math>
which gives reasonable results for high zenith angles, with a horizon airmass of 40.
[[#CITEREFKastenYoung1989|Kasten and Young (1989)]] developed
:<math>X = \frac{1} { \cos\, z + 0.50572 \,(96.07995 - z)^{-1.6364}}\;,</math>
which gives reasonable results for zenith angles of up to 90°, with an
airmass of approximately 38 at the horizon. Here the second <math>z</math>
term is in ''degrees''.
[[#CITEREFYoung1994|Young (1994)]] developed
:<math>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 }\,,
</math>
in terms of the true zenith angle <math>z_\mathrm t</math>, 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 ([[#CITEREFSchoenberg1929|Schoenberg 1929]], 173)
that the path <math>s</math> of a light ray at zenith angle
<math>z</math> through a radially symmetrical atmosphere of height
<math>y_{\mathrm {atm}}</math> is given by
:<math>
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\,,
</math>
or alternatively,
:<math>
s = \sqrt {\left ( R_\mathrm {E} + y_\mathrm{atm} \right )^2
- R_\mathrm {E}^2 \sin^2 z}
- R_\mathrm {E} \cos\, z\, ,
</math>
where <math>R_\mathrm E</math> 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 <math>y_{\mathrm
{atm}}</math>, and the relative airmass is
:<math>
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
</math>
If density is constant, [[hydrostatic]] considerations give the atmospheric height as
:<math>y_\mathrm{atm} = \frac {kT_0} {mg}\,,</math>
where <math>k</math> is [[Boltzmann's constant]], <math>T_0</math> is the
sea-level temperature, <math>m</math> is the molecular mass of air, and
<math>g</math> 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 <math>T_0</math> = 288.15 K,
<math>m</math> = 28.9644×1.6605×<math>10^{-27}</math> kg,
and <math>g</math> = 9.80665 <math>\mathrm{m/s}^2</math>
gives <math>y_\mathrm{atm}</math> ≈ 8435 m. Using
Earth's mean radius of 6371 km, the sea-level airmass at the horizon is
:<math>
X_\mathrm{horiz} = \sqrt {1 + 2 \frac {R_\mathrm {E}} {y_\mathrm{atm}}} \approx 38.87
</math>
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 <math>z</math> = 88°
gives <math>y_\mathrm{atm}</math> ≈ 10,096 m and
<math>X_\mathrm{horiz}</math> ≈ 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.<ref>
Although acknowledging that an isothermal or polytropic
atmosphere would have been more realistic,
[[#CITEREFJaniczek and DeYoung1987|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.
</ref>
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''
<math>\sigma</math> then is
:<math>\sigma = \int \rho \, \mathrm d s</math>
For the geometrical light path discussed above, this becomes, for a sea-level observer,
:<math>
\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}}
</math>
The relative airmass then is
:<math>X = \frac \sigma {\sigma_\mathrm{zen}}</math>
The absolute airmass at zenith <math>\sigma_\mathrm{zen}</math> 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
:<math>\rho = \rho_0 e^{-y / H}\,,</math>
where <math>\rho_0</math> is the sea-level density and <math>H</math> is
the pressure [[scale height]]. When the limits of integration are zero and
infinity, and some high-order terms are dropped, this model yields
([[#CITEREFYoung1974|Young 1974]], 147),
:<math>
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 )
</math>
An approximate correction for refraction can be made by taking
([[#CITEREFYoung1974|Young 1974]], 147)
:<math>R = 7/6 \, R_\mathrm E\,,</math>
where <math>R_\mathrm E</math> is the physical radius of the Earth. At the
horizon, the approximate equation becomes
:<math>X_\mathrm{horiz} \approx \sqrt { \frac {\pi R} {2 H}}</math>
Using a scale height of 8435 m, Earth's mean radius of 6371 km,
and including the correction for refraction,
:<math>X_\mathrm{horiz} \approx 37.20</math>
====Polytropic atmosphere====
The assumption of constant temperature is simplistic; a more realistic
model is the [[polytropic]] atmosphere, for which
:<math>T = T_0 - \alpha y\,,</math>
where <math>T_0</math> is the sea-level temperature and <math>\alpha</math>
is the temperature [[lapse rate]]. The density as a function of elevation
is
:<math>\rho = \rho_0 \left ( 1 - \frac \alpha T_0 y \right )^{1 / (\kappa - 1)}\,,</math>
where <math>\kappa</math> is the polytropic exponent (or polytropic index).
The airmass integral for the polytropic model does not lend itself to a
[[closed-form expression|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
([[#CITEREFGarfinkel1967|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.<ref> The notes for Reed
Meyer's
[http://reed.gigacorp.net/vitdownld.html#airmass airmass calculator]
describe an atmospheric model using eight layers and using polynomials
rather than simple linear relations for temperature lapse rates.</ref>
====Refracting radially symmetrical atmosphere====
When atmospheric refraction is considered, the absolute airmass integral becomes<ref>
See [[#CITEREFThomason et al1983|Thomason, Herman, and Reagan (1983)]] for
a derivation of the integral for a refracting atmosphere.
</ref>
:<math>
\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}}\,,
</math>
where <math>n_\mathrm{obs}</math> is the index of refraction of air at the
observer's elevation <math>y_\mathrm{obs}</math> above sea level,
<math>n</math> is the index of refraction at elevation
<math>y</math> above sea level, <math>r_\mathrm{obs} = R_\mathrm{E} + y_\mathrm{obs}</math>,
<math>r = R_\mathrm{E} + y</math> is the distance from the center of
the Earth to a point at elevation <math>y</math>, and <math>r_\mathrm{atm}
= R_\mathrm{E} + y_\mathrm{atm}</math> is distance to the upper limit of
the atmosphere at elevation <math>y_\mathrm{atm}</math>. The index of
refraction in terms of density is usually given to sufficient accuracy
([[#CITEREFGarfinkel1967|Garfinkel 1967]]) by the [[Dale-Gladstone]]
relation
:<math>\frac {n - 1} {n_\mathrm{obs} - 1} = \frac {\rho} {\rho_\mathrm{obs}}</math>
Rearrangement and substitution into the absolute airmass integral
gives
:<math>
\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}}
</math>
The quantity <math>n_\mathrm{obs} - 1</math> is quite small; expanding the
first term in parentheses, rearranging several times, and ignoring terms in
<math>(n_\mathrm{obs} - 1)^2</math> after each rearrangement, gives
([[#CITEREFKastenYoung1989|Kasten and Young 1989]])
:<math>
\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}}
</math>
===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 ([[#CITEREFHayesLatham1975|Hayes and Latham 1975]]):
[[Rayleigh scattering]] by air molecules, [[Mie scattering]] by
[[Particulate|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
[[#CITEREFThomason et al1983|Thomason, Herman, and Reagan (1983)]]. A
compromise approach often is possible, however. Methods for separately
calculating the extinction from each species using
[[closed-form expression]]s are described in
[[#CITEREFSchaefer1993|Schaefer (1993)]] and
[[#CITEREFSchaefer1998|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
([[#CITEREFGreen1992|Green 1992]]).
==Notes==
{{reflist}}
==References==
* <span id="CITEREFAllen1976">Allen, C. W. 1976. ''Astrophysical Quantities'', 3rd ed. 1973, reprinted with corrections, 1976. London: Athlone, 125.</span> ISBN 0-485-11150-0
* <span id="CITEREFBemporad1904">Bemporad, A. 1904. Zur Theorie der Extinktion des Lichtes in der Erdatmosphäre. ''Mitteilungen der Großherzoglichen Sternwarte zu Heidelberg'' Nr. 4, 1–78. </span>
* <span id="CITEREFGarfinkel1967">Garfinkel, B. 1967. Astronomical Refraction in a Polytropic Atmosphere. ''Astronomical Journal'' 72:235–254.</span>
* <span id="CITEREFGreen1992">Green, Daniel W. E. 1992. Magnitude Corrections for Atmospheric Extinction. ''International Comet Quarterly'' 14, July 1992, 55–59.</span>
* <span id="CITEREFHardie1962">Hardie, R. H. 1962. In ''Astronomical Techniques''. Hiltner, W. A., ed. Chicago: University of Chicago Press, 184–.</span> LCCN 62009113
* <span id="CITEREFHayesLatham1975">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.</span>
* <span id="CITEREFJaniczek_and_DeYoung1987">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.</span>
* <span id="CITEREFKastenYoung1989">Kasten, F., and A. T. Young. 1989. Revised optical air mass tables and approximation formula. ''Applied Optics'' 28:4735–4738.
* <span id="CITEREFRozenberg1966">Rozenberg, G. V. 1966. ''Twilight: A Study in Atmospheric Optics''. New York: Plenum Press, 160.</span> Translated from the Russian by R. B. Rodman. LCCN 65011345
* <span id="CITEREFSchaefer1993">Schaefer, B. E. 1993. Astronomy and the Limits of Vision. ''Vistas in Astronomy'' 36:311–361.</span>
* <span id="CITEREFSchaefer1998">———. 1998. To the Visual Limits. ''Sky & Telescope'', May 1998, 57–60.</span>
* <span id="CITEREFSchoenberg1929">Schoenberg, E. 1929. Theoretische Photometrie, g) Über die Extinktion des Lichtes in der Erdatmosphäre. In ''Handbuch der Astrophysik''. Band II, erste Hälfte. Berlin: Springer.</span>
* <span id="CITEREFThomason_et_al1983">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.</span>
* <span id="CITEREFYoung1974">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.</span> ISBN 0-12-474912-1
* <span id="CITEREFYoung1994">Young, A. T. 1994. Air mass and refraction. ''Applied Optics''. 33:1108–1110.</span>
* <span id="CITEREFYoung_and_Irvine1967">Young, A. T., and W. M. Irvine. 1967. Multicolor photoelectric photometry of the brighter planets. I. Program and procedure. ''Astronomical Journal'' 72:945–950.</span>
==See also==
* [[Extinction (astronomy)#Atmospheric extinction|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.
[[Category:Observational astronomy]]
[[de:Air_Mass]]