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| {{About||air mass in meteorology|air mass|practical application|air mass (solar energy)}}
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| In [[astronomy]], '''air mass''' (or '''airmass''') is the optical path length through [[Earth's atmosphere|Earth’s atmosphere]] for [[light]] from a [[celestial source]]. As it passes through the atmosphere, light is attenuated by [[scattering]] and [[absorption (electromagnetic radiation)|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 law|Beer–Lambert–Bouguer law]].
| |
| | |
| “Air mass” normally indicates ''relative air mass'', the path length relative to that at the [[zenith]] at [[sea level]], so by definition, the sea-level air mass at the zenith is 1. Air mass increases as the angle between the source and the zenith increases, reaching a value of approximately 38 at the horizon. Air mass can be less than one at an elevation greater than sea level; however, most [[closed-form expression]]s for air mass do not include the effects of elevation, so adjustment must usually be accomplished by other means.
| |
| | |
| 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 coefficient#Cases|air mass zero]]” (AM0).
| |
| | |
| Tables of air mass have been published by numerous authors, including [[#CITEREFBemporad1904|Bemporad (1904)]], [[#CITEREFAllen1976|Allen (1976)]],<ref>
| |
| Allen’s air mass table was an abbreviated compilation of values from earlier sources, primarily
| |
| [[#CITEREFBemporad1904|Bemporad (1904)]].
| |
| </ref>
| |
| and [[#CITEREFKastenYoung1989|Kasten and Young (1989)]].
| |
| | |
| ==Calculating air mass== | |
| [[Image:AirmassFormulaePlots.png|thumb|350px|right|Plots of air mass using various formulas.]]
| |
| | |
| ===Zenith angle and altitude===
| |
| {{main|Celestial coordinates}}
| |
| | |
| 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 (astronomy)|altitude]]'', the angle above the geometric horizon; the altitude <math>h</math> and the zenith angle <math>z</math> are thus related by
| |
| | |
| :<math>h = 90^\circ - z \,.</math>
| |
| | |
| ===Atmospheric Refraction===
| |
| [[Atmospheric refraction]] causes light to follow an approximately circular
| |
| path that is slightly longer than the geometric path, and the air mass 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 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.<ref>
| |
| 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.</ref>
| |
| | |
| ===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 <math>X</math> then is simply the [[trigonometric functions|secant]] of the
| |
| zenith angle <math>z</math>:
| |
| | |
| :<math>X = \sec\, z \,.</math>
| |
| | |
| At a zenith angle of 60°, the air mass 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; 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
| |
| [[#CITEREFYoungIrvine1967|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 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.
| |
| | |
| [[#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 air mass 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 air mass of 40.
| |
| | |
| [[#CITEREFKastenYoung1989|Kasten and Young (1989)]] developed<ref>
| |
| The Kasten and Young formula was originally given in terms of ''altitude'' <math>\gamma</math> as
| |
| :<math>X = \frac{1} { \sin\, \gamma + 0.50572 \,(\gamma + 6.07995^\circ )^{-1.6364}}\;;</math>
| |
| in this article, it is given in terms of zenith angle for consistency with the other formulas. | |
| </ref>
| |
| | |
| :<math>X = \frac{1} { \cos\, z + 0.50572 \,(6.07995^\circ + 90 - z)^{-1.6364}} \,,</math> | |
| | |
| which gives reasonable results for zenith angles of up to 90°, with an
| |
| air mass 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 air mass.
| |
| | |
| [[#CITEREFPickering2002|Pickering (2002)]] developed
| |
| :<math>X = \frac{1} { \sin (h + {244}/(165+47 h^{1.1}) ) } \,,</math>
| |
| | |
| where <math>h</math> is apparent altitude <math>(90^\circ - z)</math> in degrees. Pickering claimed his equation to have a tenth the error of [[#CITEREFSchaefer1998|Schaefer (1998)]] near the horizon.<ref>
| |
| [[#CITEREFPickering2002|Pickering (2002)]] uses [[#CITEREFGarfinkel1967|Garfinkel (1967)]] as the reference for accuracy.
| |
| </ref>
| |
| | |
| ===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 radially symmetrical atmosphere====
| |
| [[File:airmass geometry.png|thumb|350px|right|Atmospheric effects on optical transmission can be modelled as if the atmosphere is concentrated in approximately the lower 9 km.]]
| |
| 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 [[Homogeneity (physics)|homogeneous]] (i.e., [[density]] is constant), the
| |
| path at zenith is simply the atmospheric height <math>y_{\mathrm
| |
| {atm}}</math>, and the relative air mass 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|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> = 28.9644×1.6605×10<sup>−27</sup> kg,
| |
| and <math>g</math> = 9.80665 m/s<sup>2</sup>
| |
| gives <math>y_\mathrm{atm}</math> ≈ 8435 m. Using
| |
| Earth’s mean radius of 6371 km, the sea-level air mass 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 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
| |
| | |
| :<math>\frac {R_\mathrm{E}} {y_\mathrm{atm}}
| |
| = \frac {X^2 - 1} {2 \left ( 1 - X \cos z \right )} \,;</math>
| |
| | |
| matching Bemporad’s value of 19.787 at <math>z</math> = 88°
| |
| gives <math>R_\mathrm{E} / y_\mathrm{atm}</math> ≈ 631.01 and
| |
| <math>X_\mathrm{horiz}</math> ≈ 35.54. With the same value for <math>R_\mathrm{E}</math> as above, <math>y_\mathrm{atm}</math> ≈ 10,096 m.
| |
| | |
| 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, including those greater than 90° (''see'' [[#Homogeneous spherical atmosphere with elevated observer|Homogeneous spherical atmosphere with elevated observer]] ''below''). The model
| |
| requires comparatively little computational overhead, and if high accuracy is
| |
| not required, it gives reasonable results.<ref>
| |
| Although acknowledging that an isothermal or polytropic
| |
| atmosphere would have been more realistic,
| |
| [[#CITEREFJaniczekDeYoung1987|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, 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 decreases with elevation above
| |
| [[mean sea level]]. The ''absolute air mass''
| |
| <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 air mass then is
| |
| | |
| :<math>X = \frac \sigma {\sigma_\mathrm{zen}} \,.</math>
| |
| | |
| The absolute air mass 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 air mass 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|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 air mass 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 air mass integral becomes<ref>
| |
| See [[#CITEREFThomasonHermanReagan1983|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 [[Gladstone–Dale relation]]
| |
| | |
| :<math>\frac {n - 1} {n_\mathrm{obs} - 1} = \frac {\rho} {\rho_\mathrm{obs}} \,.</math>
| |
| | |
| Rearrangement and substitution into the absolute air mass 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>
| |
| | |
| ====Homogeneous spherical atmosphere with elevated observer====
| |
| <!-- this section is linked from Homogeneous atmosphere above -->
| |
| [[File:HomogSphElevObsAM.png|thumb|250px|right|Air mass for elevated observer in homogeneous spherical atmosphere]]
| |
| In the figure at right, an observer at O is at an elevation <math>y_\mathrm{obs}</math> above sea level in a uniform radially symmetrical atmosphere of height <math>y_\mathrm{atm}</math>. The path length of a light ray at zenith angle <math>z</math> is <math>s</math>; <math>R_\mathrm{E}</math> is the radius of the Earth. Applying the [[law of cosines]] to triangle OAC,
| |
| | |
| :<math>\begin{align}
| |
| {{\left( {{R}_{\text{E}}}+{{y}_{\text{atm}}} \right)}^{2}} & ={{s}^{2}}+{{\left( {{R}_{\text{E}}}+{{y}_{\text{obs}}} \right)}^{2}}-2\left( {{R}_{\text{E}}}+{{y}_{\text{obs}}} \right)s\cos \left( 180{}^\circ -z \right) \\
| |
| & ={{s}^{2}}+{{\left( {{R}_{\text{E}}}+{{y}_{\text{obs}}} \right)}^{2}}+2\left( {{R}_{\text{E}}}+{{y}_{\text{obs}}} \right)s\cos z\text{ ;}
| |
| \end{align}</math>
| |
| | |
| expanding the left- and right-hand sides, eliminating the common terms, and rearranging gives
| |
| | |
| :<math>{{s}^{2}}+2\left( {{R}_{\text{E}}}+{{y}_{\text{obs}}} \right)s\cos z-2{{R}_{\text{E}}}{{y}_{\text{atm}}}-y_{\text{atm}}^{2}+2{{R}_{\text{E}}}{{y}_{\text{obs}}}+y_{\text{obs}}^{2}=0 \,.</math>
| |
| | |
| Solving the quadratic for the path length ''s'', factoring, and rearranging,
| |
| | |
| :<math>s=\pm \sqrt{{{\left( {{R}_{\text{E}}}+{{y}_{\text{obs}}} \right)}^{2}}{{\cos }^{2}}z+2{{R}_{\text{E}}}\left( {{y}_{\text{atm }}}-{{y}_{\text{obs}}} \right)+y_{\text{atm}}^{2}-y_{\text{obs}}^{2}}-({{R}_{\text{E}}}+{{y}_{\text{obs}}})\cos z \,.</math>
| |
| | |
| The negative sign of the radical gives a negative result, which is not physically meaningful. Using the positive sign, dividing by <math>y_\mathrm{atm}</math>, and cancelling common terms and rearranging gives the relative air mass:
| |
| | |
| :<math>X=\sqrt{{{\left( \frac{{{R}_{\text{E}}}+{{y}_{\text{obs}}}}{{{y}_{\text{atm}}}} \right)}^{2}}{{\cos }^{2}}z+\frac{2{{R}_{\text{E}}}}{y_{\text{atm}}^{2}}\left( {{y}_{\text{atm}}}-{{y}_{\text{obs}}} \right)-{{\left( \frac{{{y}_{\text{obs}}}}{{{y}_{\text{atm}}}} \right)}^{2}}+1}-\frac{{{R}_{\text{E}}}+{{y}_{\text{obs}}}}{{{y}_{\text{atm}}}}\cos z \,.</math>
| |
| | |
| With the substitutions <math>\hat{r} = R_\mathrm{E} / y_\mathrm{atm}</math> and <math>\hat{y} = y_\mathrm{obs} / y_\mathrm{atm}</math>, this can be given as
| |
| | |
| :<math>X=\sqrt{{{(\hat{r}+\hat{y})}^{2}}{{\cos }^{2}}z + 2 \hat{r} (1-\hat{y}) - \hat{y}^2 +1} \; - \; (\hat{r}+\hat{y})\cos z \,.</math>
| |
| | |
| When the observer’s elevation is zero, the air mass equation simplifies to
| |
| | |
| :<math>X=\sqrt{{{\left( \frac{{{R}_{\text{E}}}}{{{y}_{\text{atm}}}} \right)}^{2}}{{\cos }^{2}}z+\frac{2{{R}_{\text{E}}}}{{{y}_{\text{atm}}}}+1}-\frac{{{R}_{\text{E}}}}{{{y}_{\text{atm}}}}\cos z \,.</math>
| |
| | |
| <br/>'''Maximum zenith angle'''<br/> | |
| [[File:HomogSphElevObsZmax.png|thumb|250px|right|Maximum zenith angle for elevated observer in homogeneous spherical atmosphere]]
| |
| When the observer is at an elevation greater than that of the horizon, the zenith angle can be greater than 90°. The maximum possible zenith angle occurs when the ray is tangent to Earth’s surface; from triangle OCG in the figure at right,
| |
| | |
| :<math>\cos \gamma =\frac{{{R}_{\text{E}}}+{{y}_{\text{obs}}}-h}{{{R}_{\text{E}}}+{{y}_{\text{obs}}}} \,,</math>
| |
| | |
| where <math>h</math> is the observer’s height above the horizon. The geometrical dip of the horizon <math>\gamma</math> is related to <math>z_\mathrm{max}</math> by
| |
| | |
| :<math>\gamma ={{z}_{\text{max}}}-90{}^\circ \,,</math>
| |
| | |
| so that
| |
| | |
| :<math>\cos \gamma =\cos \left( {{z}_{\text{max}}}-90{}^\circ \right)=\sin {{z}_{\text{max}}} \,.</math>
| |
| | |
| Then
| |
| | |
| :<math>\sin {{z}_{\text{max}}}=\frac{{{R}_{\text{E}}}+{{y}_{\text{obs}}}-h}{{{R}_{\text{E}}}+{{y}_{\text{obs}}}} \,.</math>
| |
| | |
| For a non-negative height <math>h</math>, the angle <math>z_\mathrm{max}</math> is always ≥ 90°; however, the inverse sine functions provided by most calculators and programming languages return values in the range ±90°. The value can be placed in the proper quadrant by
| |
| | |
| :<math>{{z}_{\text{max}}}=180{}^\circ -{{\sin }^{-1}}\frac{{{R}_{\text{E}}}+{{y}_{\text{obs}}}-h}{{{R}_{\text{E}}}+{{y}_{\text{obs}}}} \,.</math>
| |
| | |
| If the horizon is at sea level, <math>y_\mathrm{obs} = h</math>, and this simplifies to
| |
| :<math>{{z}_{\text{max}}}=180{}^\circ -{{\sin }^{-1}}\frac{{{R}_{\text{E}}}}{{{R}_{\text{E}}}+h}\,.</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 [[refractive index#Dispersion and absorption|extinction coefficient]] depends on elevation, it
| |
| must be determined as part of the air mass integral, as described by
| |
| [[#CITEREFThomasonHermanReagan1983|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 air mass formulas and separately
| |
| determining extinction coefficients for each of the attenuating species
| |
| ([[#CITEREFGreen1992|Green 1992]], [[#CITEREFPickering2002|Pickering 2002]]).
| |
| | |
| == Air mass and astronomy ==
| |
| [[File:Atmospheric electromagnetic transmittance or opacity.jpg|thumb|Atmospheric transmittance across the [[electromagnetic spectrum]].]]
| |
| | |
| {{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 ''[[Astronomical seeing|seeing]]''.<ref>[http://www.astro.uni-bonn.de/~mischa/obstips/airmass.html Observing tips: air mass and differential refraction] retrieved 15 May 2011.</ref> On bigger telescopes, such as the [[William Herschel Telescope|WHT]] ([[#CITEREFWynneWorsick1988|Wynne and Warsick 1988]]) and [[Very Large Telescope|VLT]] ([[#CITEREFAvilaRupprechtBeckers1997|Avila, Rupprecht, and Becker 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
| |
| 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 ([[#CITEREFVanDerTolVanDerVeen2007|van der Tol and van der Veen 2007]]; [[#CITEREFDeVosGunstNijboer2009|de Vos, Gunst, and Nijboer 2009]]), but on the other hand can also study the ionosphere by instead measuring these distortions ([[#CITEREFThide2007|Thidé 2007]]).
| |
| | |
| == Air mass and solar energy ==
| |
| [[File:Solar Spectrum.png|thumb|Solar irradiance spectrum above atmosphere and at surface]]
| |
| | |
| {{Main|Air mass (solar energy)}}
| |
| | |
| 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 [[Sunlight#Composition|spectral irradiance]]. [[Photovoltaic module]]s are commonly rated using spectral irradiance for an air mass of 1.5 (AM1.5); tables of these standard spectra are given in [[#CITEREFASTM_G173|ASTM G 173-03]]. The extraterrestrial spectral irradiance (i.e., that for AM0) is given in [[#CITEREFASTM_E490|ASTM E 490-00a]].<ref>
| |
| ASTM E 490-00a was reapproved without change in 2006.
| |
| </ref>
| |
| | |
| 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 the section [[#Plane-parallel atmosphere|Plane-parallel atmosphere]].
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| ==See also==
| |
| {{Div col|cols=3}}
| |
| * [[Air mass (solar energy)]]
| |
| * [[Extinction (astronomy)#Atmospheric extinction|Atmospheric extinction]]
| |
| * [[Beer–Lambert law|Beer–Lambert–Bouguer law]]
| |
| * [[Diffuse sky radiation]]
| |
| * [[refractive index#Dispersion and absorption|Extinction coefficient]]
| |
| * [[Illuminance]]
| |
| * [[International Standard Atmosphere]]
| |
| * [[Irradiance]]
| |
| * [[Law of atmospheres]]
| |
| * [[Light diffusion]]
| |
| * [[Mie scattering]]
| |
| * [[Photovoltaic module]]
| |
| * [[Rayleigh scattering]]
| |
| * [[Solar irradiation]]
| |
| {{Div col end}}
| |
| | |
| ==References==
| |
| * {{wikicite |ref="CITEREFAllen1976" |reference=Allen, C. W. 1976. ''Astrophysical Quantities'', 3rd ed. 1973, reprinted with corrections, 1976. London: Athlone, 125. ISBN 0-485-11150-0.}}
| |
| * {{wikicite |ref="CITEREFASTM_E490" |reference=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 [http://www.astm.org/Standards/E490.htm ASTM].}}
| |
| * {{wikicite |ref="CITEREFASTM_G173" |reference=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 [http://www.astm.org/Standards/G173.htm ASTM].}}
| |
| * {{wikicite |ref="CITEREFAvilaRupprechtBeckers1997"|reference=Avila, Gerardo, Gero Rupprecht, and J. M. Beckers. 1997. Atmospheric dispersion correction for the FORS Focal Reducers at the ESO VLT. ''Proceedings of SPIE''. 2871(March):1135–1143. Optical Telescopes of Today and Tomorrow. Ed. Arne L. Ardeberg. [[Digital object identifier|doi]]: [http://dx.doi.org/10.1117/12.269000 10.1117/12.269000]. [[Bibcode]] [http://adsabs.harvard.edu/abs/1997SPIE.2871.1135A 1997SPIE.2871.1135A]. Available as [http://www.hq.eso.org/paranal/instruments/fors1/inst/Papers/Spie_96/ladc_paper.ps.gz PS] from [http://www.hq.eso.org/ ESO.org].}}
| |
| * {{wikicite |ref="CITEREFBemporad1904" |reference=Bemporad, A. 1904. Zur Theorie der Extinktion des Lichtes in der Erdatmosphäre. ''Mitteilungen der Grossh. Sternwarte zu Heidelberg'' Nr. 4, 1–78.}}
| |
| * {{cite any |Garfinkel|1967|reference=Garfinkel, B. 1967. Astronomical Refraction in a Polytropic Atmosphere. ''Astronomical Journal'' 72:235–254. [[Digital object identifier|doi]]: [http://dx.doi.org/10.1086/110225 10.1086/110225]. [[Bibcode]] [http://adsabs.harvard.edu/abs/1967AJ.....72..235G 1967AJ.....72..235G].}}
| |
| * {{wikicite |ref="CITEREFGreen1992" |reference=Green, Daniel W. E. 1992. [http://www.icq.eps.harvard.edu/ICQExtinct.html Magnitude Corrections for Atmospheric Extinction]. ''International Comet Quarterly'' 14, July 1992, 55–59.}}
| |
| * {{wikicite |ref="CITEREFHardie1962" |reference=Hardie, R. H. 1962. In ''Astronomical Techniques''. Hiltner, W. A., ed. Chicago: University of Chicago Press, 184–. LCCN 62009113. [[Bibcode]] [http://adsabs.harvard.edu/abs/1962aste.book.....H 1962aste.book.....H].}}
| |
| * {{wikicite |ref="CITEREFHayesLatham1975" |reference=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. [[Digital object identifier|doi]]: [http://dx.doi.org/10.1086/153548 10.1086/153548]. [[Bibcode]] [http://adsabs.harvard.edu/abs/1975ApJ...197..593H 1975ApJ...197..593H].}}
| |
| * {{wikicite |ref="CITEREFJaniczekDeYoung1987" |reference=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]] [http://adsabs.harvard.edu/abs/1987USNOC.171.....J 1987USNOC.171.....J].}}
| |
| * {{wikicite |ref="CITEREFKastenYoung1989" |reference=Kasten, F., and A. T. Young. 1989. Revised optical air mass tables and approximation formula. ''Applied Optics'' 28:4735–4738. [[Digital object identifier|doi]]: [http://dx.doi.org/10.1364/AO.28.004735 10.1364/AO.28.004735]. [[Bibcode]] [http://adsabs.harvard.edu/abs/1989ApOpt..28.4735K 1989ApOpt..28.4735K]. (payment required)}}
| |
| * {{wikicite |ref="CITEREFPickering2002" |reference=Pickering, K. A. 2002. [http://www.dioi.org/webvols/d112.xml The Southern Limits of the Ancient Star Catalog]. ''DIO'' 12:1, 20, n. 39. Available as [http://www.dioi.org/vols/wc0.pdf PDF] from [http://www.dioi.org DIO].}}
| |
| * {{wikicite |ref="CITEREFRozenberg1966" |reference=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.}}
| |
| * {{wikicite |ref="CITEREFSchaefer1993" |reference=Schaefer, B. E. 1993. Astronomy and the Limits of Vision. ''Vistas in Astronomy'' 36:311–361. [[Digital object identifier|doi]]: [http://dx.doi.org/10.1016/0083-6656(93)90113-X 10.1016/0083-6656(93)90113-X]. [[Bibcode]] [http://adsabs.harvard.edu/abs/1993VA.....36..311S 1993VA.....36..311S]. (payment required)}}
| |
| * {{wikicite |ref="CITEREFSchaefer1998" |reference= Schaefer, B. E. 1998. To the Visual Limits: How deep can you see?. ''Sky & Telescope'', May 1998, 57–60.}}
| |
| * {{wikicite |ref="CITEREFSchoenberg1929" |reference=Schoenberg, E. 1929. Theoretische Photometrie, Über die Extinktion des Lichtes in der Erdatmosphäre. In ''Handbuch der Astrophysik''. Band II, erste Hälfte. Berlin: Springer.}}
| |
| * {{wikicite |ref="CITEREFThide2007" |reference= [[Bo Thidé|Thidé, Bo]]. 2007. [http://arxiv.org/pdf/0707.4506v1 Nonlinear physics of the ionosphere and LOIS/LOFAR] ''Plasma Physics and Controlled Fusion''. 49(12B, December): B103–B107. [[Digital object identifier|doi]]: [http://dx.doi.org/10.1088/0741-3335/49/12B/S09 10.1088/0741-3335/49/12B/S09]. [[Bibcode]] [http://adsabs.harvard.edu/abs/2007PPCF...49..103T 2007PPCF...49..103T].}}
| |
| * {{wikicite |ref="CITEREFThomasonHermanReagan1983" |reference=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. [[Digital object identifier|doi]]: [http://dx.doi.org/10.1175/1520-0469(1983)040%3C1851:TEOAAW%3E2.0.CO;2 10.1175/1520-0469(1983)040<1851:TEOAAW>2.0.CO;2]. [[Bibcode]] [http://adsabs.harvard.edu/abs/1983JAtS...40.1851T 1983JAtS...40.1851T].}} <!-- doi is 10.1175/1520-0469(1983)040<1851:TEOAAW>2.0.CO;2 -- reformatted for URL -->
| |
| * {{wikicite |ref="CITEREFVanDerTolVanDerVeen2007" |reference=van der Tol, S., and A. J. van der Veen. 2007 Ionospheric Calibration for the LOFAR Radio Telescope. International Symposium on Signals, Circuits and Systems, July, 2007. [[Digital object identifier|doi]]: [http://dx.doi.org/10.1109/ISSCS.2007.4292761 10.1109/ISSCS.2007.4292761]. Available as [http://ens.ewi.tudelft.nl/pubs/tol07isscs.pdf PDF].}}
| |
| * {{wikicite |ref="CITEREFDeVosGunstNijboer2009" |reference=de Vos, M., A. W. Gunst, and R. Nijboer. 2009. The LOFAR Telescope: System Architecture and Signal Processing. ''Proceedings of the IEEE''. 97(8): 1431–1437. [[Digital object identifier|doi]]: [http://dx.doi.org/10.1109/JPROC.2009.2020509 10.1109/JPROC.2009.2020509]. [[Bibcode]] [http://adsabs.harvard.edu/abs/2009IEEEP..97.1431D 2009IEEEP..97.1431D]. Available as [http://www.astro.rug.nl/~peletier/BDT_LOFAR-IEEE_main.pdf PDF] from [http://www.astro.rug.nl/~peletier/ www.astro.rug.nl].}}
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| * {{wikicite| ref="CITEREFWynneWorsick1988"|reference= Wynne, C. G., and S. P. Worswick. 1988. [http://www.hq.eso.org/sci/facilities/lasilla/instruments/feros/Projects/ADC/references/wynne_worswick88_primeFocus.ps.gz Atmospheric dispersion at prime focus]. ''Royal Astronomical Society, Monthly Notices'' 230:457–471 (February 1988). [[Bibcode]] [http://adsabs.harvard.edu/abs/1988MNRAS.230..457W 1988MNRAS.230..457W]. [[ISSN]] [[Special:BookSources/01247491210035-8711|01247491210035-8711]].}}
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| * {{wikicite |ref="CITEREFYoung1974" |reference=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. {{Listed Invalid ISBN|0-12-474912-1}}.}}
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| * {{wikicite |ref="CITEREFYoung1994" |reference=Young, A. T. 1994. [http://www.opticsinfobase.org/abstract.cfm?id=41471 Air mass and refraction]. ''Applied Optics''. 33:1108–1110. [[Digital object identifier|doi]]: [http://dx.doi.org/10.1364/AO.33.001108 10.1364/AO.33.001108]. [[Bibcode]] [http://adsabs.harvard.edu/abs/1994ApOpt..33.1108Y 1994ApOpt..33.1108Y]. (payment required)}} <!-- DOI does not seem to work, but is a direct copy from optics infobase -->
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| * {{wikicite |ref="CITEREFYoungIrvine1967" |reference=Young, A. T., and W. M. Irvine. 1967. Multicolor photoelectric photometry of the brighter planets. I. Program and procedure. ''Astronomical Journal'' 72:945–950. [[Digital object identifier|doi]]: [http://dx.doi.org/10.1086/110366 10.1086/110366]. [[Bibcode]] [http://adsabs.harvard.edu/abs/1967AJ.....72..945Y 1967AJ.....72..945Y].}}
| |
| | |
| ==External links==
| |
| * 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:Astronomical imaging]]
| |
| [[Category:Observational astronomy]]
| |