In astronomy, air mass or airmass is a measure of the amount of air along the line of sight when observing a star or other celestial source from below Earth's atmosphere (Green 1992). It is formulated as the integral of air density along the light ray. 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 zenith. So, by definition, the relative 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 expressions for air mass do not include the effects of the observer's elevation, so adjustment must usually be accomplished by other means. Tables of air mass have been published by numerous authors, including Bemporad (1904), Allen (1973), and Kasten & Young (1989).
Definition The absolute air mass is defined as:
σ = ∫ ρ d s . {\displaystyle \sigma =\int \rho \,\mathrm {d} s\,.}
where ρ {\displaystyle \rho } is volumetric density of air. Thus σ {\displaystyle \sigma } is a type of oblique column density. In the vertical direction, the absolute air mass at zenith is:
σ z e n = ∫ ρ d z {\displaystyle \sigma _{\mathrm {zen} }=\int \rho \,\mathrm {d} z}
So σ z e n {\displaystyle \sigma _{\mathrm {zen} }} is a type of vertical column density. Finally, the relative air mass is:
X = σ σ z e n {\displaystyle X={\frac {\sigma }{\sigma _{\mathrm {zen} }}}}
Assuming air density to be uniform allows removing it from the integrals. The absolute air mass then simplifies to a product:
σ = ρ ¯ s {\displaystyle \sigma ={\bar {\rho }}s}
where ρ ¯ = c o n s t . {\displaystyle {\bar {\rho }}=\mathrm {const.} } is the average density and the arc length s {\displaystyle s} of the oblique and zenith light paths are:
s = ∫ d s s z e n = ∫ d z {\displaystyle {\begin{aligned}s&=\int \,\mathrm {d} s\\s_{\mathrm {zen} }&=\int \,\mathrm {d} z\end{aligned}}}
In the corresponding simplified relative air mass, the average density cancels out in the fraction, leading to the ratio of path lengths:
X = s s z e n . {\displaystyle X={\frac {s}{s_{\mathrm {zen} }}}\,.}
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 zenith angle (in astronomy, commonly referred to as the zenith distance). A body's angular position can also be given in terms of altitude, the angle above the geometric horizon; the altitude h {\displaystyle h} and the zenith angle z {\displaystyle z} are thus related by
h = 90 ∘ − z . {\displaystyle h=90^{\circ }-z\,.}
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.
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 X {\displaystyle X} then is simply the secant of the zenith angle z {\displaystyle z} :
X = sec z . {\displaystyle X=\sec \,z\,.}
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 X = sec z {\displaystyle X=\sec \,z} 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:
X = sec z t [ 1 − 0.0012 ( sec 2 z t − 1 ) ] , {\displaystyle X=\sec \,z_{\mathrm {t} }\,\left[1-0.0012\,(\sec ^{2}z_{\mathrm {t} }-1)\right]\,,}
where z t {\displaystyle z_{\mathrm {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 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. Hardie (1962) introduced a polynomial in sec z − 1 {\displaystyle \sec \,z-1} :
X = sec z − 0.0018167 ( sec z − 1 ) − 0.002875 ( sec z − 1 ) 2 − 0.0008083 ( sec z − 1 ) 3 {\displaystyle 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 air mass reaches a maximum, and then approaches negative infinity at the horizon. Rozenberg (1966) suggested
X = ( cos z + 0.025 e − 11 cos z ) − 1 , {\displaystyle X=\left(\cos \,z+0.025e^{-11\cos \,z}\right)^{-1}\,,}
which gives reasonable results for high zenith angles, with a horizon air mass of 40. Kasten & Young (1989) developed
X = 1 cos z + 0.50572 ( 6.07995 ∘ + 90 ∘ − z ) − 1.6364 , {\displaystyle X={\frac {1}{\cos \,z+0.50572\,(6.07995^{\circ }+90^{\circ }-z)^{-1.6364}}}\,,}
which gives reasonable results for zenith angles of up to 90°, with an air mass of approximately 38 at the horizon. Here the second z {\displaystyle z} term is in degrees. Young (1994) developed
X = 1.002432 cos 2 z t + 0.148386 cos z t + 0.0096467 cos 3 z t + 0.149864 cos 2 z t + 0.0102963 cos z t + 0.000303978 {\displaystyle 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 {\displaystyle z_{\mathrm {t} }} , for which he claimed a maximum error (at the horizon) of 0.0037 air mass. Pickering (2002) developed
X = 1 sin ( h + 244 / ( 165 + 47 h 1.1 ) ) , {\displaystyle X={\frac {1}{\sin(h+{244}/(165+47h^{1.1}))}}\,,}
where h {\displaystyle h} is apparent altitude ( 90 ∘ − z ) {\displaystyle (90^{\circ }-z)} in degrees. Pickering claimed his equation to have a tenth the error of Schaefer (1998) near the horizon.
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 s {\displaystyle s} of a light ray at zenith angle
z {\displaystyle z} through a radially symmetrical atmosphere of height y a t m {\displaystyle y_{\mathrm {atm} }} above the Earth is given by
s = R E 2 cos 2 z + 2 R E y a t m + y a t m 2 − R E cos z {\displaystyle s={\sqrt {R_{\mathrm {E} }^{2}\cos ^{2}z+2R_{\mathrm {E} }y_{\mathrm {atm} }+y_{\mathrm {atm} }^{2}}}-R_{\mathrm {E} }\cos \,z\,}
or alternatively,
s = ( R E + y a t m ) 2 − R E 2 sin 2 z − R E cos z {\displaystyle 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 {\displaystyle R_{\mathrm {E} }} is the radius of the Earth. The relative air mass is then:
X = s y a t m = R E y a t m cos 2 z + 2 y a t m R E + ( y a t m R E ) 2 − R E y a t m cos z . {\displaystyle 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\,.}
Homogeneous atmosphere If the atmosphere is homogeneous (i.e., density is constant), the atmospheric height y a t m {\displaystyle y_{\mathrm {atm} }} follows from hydrostatic considerations as:
y a t m = k T 0 m g , {\displaystyle y_{\mathrm {atm} }={\frac {kT_{0}}{mg}}\,,}
where k {\displaystyle k} is the Boltzmann constant, T 0 {\displaystyle T_{0}} is the sea-level temperature, m {\displaystyle m} is the molecular mass of air, and g {\displaystyle 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% (1/e) 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 {\displaystyle T_{0}=\mathrm {288.15~K} } , m = 28.9644 × 1.6605 × 10 − 27 k g {\displaystyle m=\mathrm {28.9644\times 1.6605\times 10^{-27}~kg} } , and g = 9.80665 m / s 2 {\displaystyle g=\mathrm {9.80665~m/s^{2}} } gives y a t m ≈ 8435 m {\displaystyle y_{\mathrm {atm} }\approx \mathrm {8435~m} } . Using Earth's mean radius of 6371 km, the sea-level air mass at the horizon is
X h o r i z = 1 + 2 R E y a t m ≈ 38.87 . {\displaystyle X_{\mathrm {horiz} }={\sqrt {1+2{\frac {R_{\mathrm {E} }}{y_{\mathrm {atm} }}}}}\approx 38.87\,.}
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
R E y a t m = X 2 − 1 2 ( 1 − X cos z ) ; {\displaystyle {\frac {R_{\mathrm {E} }}{y_{\mathrm {atm} }}}={\frac {X^{2}-1}{2\left(1-X\cos z\right)}}\,;}
matching Bemporad's value of 19.787 at z {\displaystyle z} = 88° gives R E / y a t m {\displaystyle R_{\mathrm {E} }/y_{\mathrm {atm} }} ≈ 631.01 and
X h o r i z {\displaystyle X_{\mathrm {horiz} }} ≈ 35.54. With the same value for R E {\displaystyle R_{\mathrm {E} }} as above, y a t m {\displaystyle y_{\mathrm {atm} }} ≈ 10,096 m. 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. 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 mean sea level. The absolute air mass for the geometrical light path discussed above, becomes, for a sea-level observer,
σ = ∫ 0 y a t m ρ ( R E + y ) d y R E 2 cos 2 z + 2 R E y + y 2 . {\displaystyle \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+2R_{\mathrm {E} }y+y^{2}}}}\,.}
Isothermal atmosphere Several basic models for density variation with elevation are commonly used. The simplest, an isothermal atmosphere, gives
ρ = ρ 0 e − y / H , {\displaystyle \rho =\rho _{0}e^{-y/H}\,,}
where ρ 0 {\displaystyle \rho _{0}} is the sea-level density and H {\displaystyle H} is the density scale height. When the limits of integration are zero and infinity, the result is known as Chapman function. An approximate result is obtained if some high-order terms are dropped, yielding (Young 1974, p. 147),
X ≈ π R 2 H exp ( R cos 2 z 2 H ) e r f c ( R cos 2 z 2 H ) . {\displaystyle X\approx {\sqrt {\frac {\pi R}{2H}}}\exp {\left({\frac {R\cos ^{2}z}{2H}}\right)}\,\mathrm {erfc} \left({\sqrt {\frac {R\cos ^{2}z}{2H}}}\right)\,.}
An approximate correction for refraction can be made by taking (Young 1974, p. 147)
R = 7 / 6 R E , {\displaystyle R=7/6\,R_{\mathrm {E} }\,,}
where R E {\displaystyle R_{\mathrm {E} }} is the physical radius of the Earth. At the horizon, the approximate equation becomes
X h o r i z ≈ π R 2 H . {\displaystyle X_{\mathrm {horiz} }\approx {\sqrt {\frac {\pi R}{2H}}}\,.}
Using a scale height of 8435 m, Earth's mean radius of 6371 km, and including the correction for refraction,
X h o r i z ≈ 37.20 . {\displaystyle X_{\mathrm {horiz} }\approx 37.20\,.}
Polytropic atmosphere The assumption of constant temperature is simplistic; a more realistic model is the polytropic atmosphere, for which
T = T 0 − α y , {\displaystyle T=T_{0}-\alpha y\,,}
where T 0 {\displaystyle T_{0}} is the sea-level temperature and α {\displaystyle \alpha } is the temperature lapse rate. The density as a function of elevation is
ρ = ρ 0 ( 1 − α T 0 y ) 1 / ( κ − 1 ) , {\displaystyle \rho =\rho _{0}\left(1-{\frac {\alpha }{T}}_{0}y\right)^{1/(\kappa -1)}\,,}
where κ {\displaystyle \kappa } is the polytropic exponent (or polytropic index). The air mass 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.
Layered 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 (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.
Refracting radially symmetrical atmosphere When atmospheric refraction is considered, ray tracing becomes necessary (Kivalov 2007), and the absolute air mass integral becomes
σ = ∫ r o b s r a t m ρ d r 1 − ( n o b s n r o b s r ) 2 sin 2 z {\displaystyle \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 o b s {\displaystyle n_{\mathrm {obs} }} is the index of refraction of air at the observer's elevation y o b s {\displaystyle y_{\mathrm {obs} }} above sea level, n {\displaystyle n} is the index of refraction at elevation y {\displaystyle y} above sea level, r o b s = R E + y o b s {\displaystyle r_{\mathrm {obs} }=R_{\mathrm {E} }+y_{\mathrm {obs} }} ,
r = R E + y {\displaystyle r=R_{\mathrm {E} }+y} is the distance from the center of the Earth to a point at elevation y {\displaystyle y} , and r a t m = R E + y a t m {\displaystyle r_{\mathrm {atm} }=R_{\mathrm {E} }+y_{\mathrm {atm} }} is distance to the upper limit of the atmosphere at elevation y a t m {\displaystyle y_{\mathrm {atm} }} . The index