Basic Air Mass Models

Zenith, Azimuth and Elevation Angles

The position of the Sun relative to a point on the ground is an important input needed to model solar air mass and solar system performance.  The inputs used to describe solar position include:

  • Zenith angle \normalsize\theta_{Z}
  • Azimuth angle \normalsize A relative to the North point on a compass
  • Elevation angle or altitude \normalsize h, where \normalsize h = 90 - \theta_{Z}

The figures below illustrate how these angles are defined, how they change over time and by location:

Angle1Angle2

Angle3

Click to enlarge

Atmospheric Refraction

The atmosphere not only filters sunlight, but it also dims and bends it.  Atmospheric refraction makes objects appear slightly higher than they really are.  Hence, the apparent zenith angle is slightly less than the true zenith angle.  The true zenith angle locates the center of the solar disc if the Earth had no air.

The true zenith angle, \normalsize \theta_{Zt}, is expressed in degrees and can be found using the apparent zenith angle \normalsize \theta_{Za}, the ambient air temperature (T in degrees celsius), and the current air pressure (P in millibars):

(1)   \begin{equation*} \normalsize \theta_{Zt} = \theta_{Za} - A\left(\frac{P}{273 + T}\right) (\theta_{Za} - \arcsin(B\sin(C\theta_{Za})) - D\theta_{Za} \end{equation*}

where A = 0.05717, B = 0.9986, C = 0.99676, D = 0.00058, and P = 1013.25.  The correction defined by equation (1) is relatively small, but its impact is magnified over time, and will clearly affect solar energy production over the 25-year rated life of a solar energy project.  Accuracy is about 0.1 arc-minutes for angles between 0 and 75°, and 1 arc-minute between 75° and 90°.  Near the horizon, the refraction amounts to 0.5°.

The dimming effect, known as atmospheric extinction, varies for different wavelengths of light with the result that the Sun shows false colors near the horizon.

Air Mass Definition

The air mass coefficient (AM) is a ratio that divides the path length of sunlight through the atmosphere at some zenith angle, \normalsize \theta_{Z} to the path length of sunlight when the zenith equals zero.  The phrase “column depth” is also used to define the two path lengths, N().

(2)   \begin{equation*} \normalsize AM = \frac{N(\theta_{Z})}{N(\theta_{0})} \end{equation*}

Simple Air Mass Calc

Optical air mass filters, refracts and dims sunlight.  When the zenith angle is small to moderate, air mass is easily approximated.  The simplest model assumes air density is constant throughout the atmosphere and ignores the Earth’s curvature (i.e. a homogeneous plane-parallel atmosphere).  The air mass coefficient in such an environment is simply the secant of the zenith angle, as defined below:

(3)   \begin{equation*} \normalsize AM_{simple} = \frac{1}{\cos \theta_{Z}} = \sec \theta_{Z} \end{equation*}

For example, at a zenith angle of 60°, AM = 2 (after converting degrees to radians). Hence, the path length of sunlight through the atmosphere at a zenith angle of 60° is twice the path length at a zenith angle of .  Meanwhile, the Earth is not flat, and the simple formula is only valid for zenith angles up to 65°. At greater zenith angles, function accuracy begins to degrade rapidly and AMsimple becomes infinite at the horizon.

Interpolative Formulas 

Interpolative formulas1 attempt to interpolate between observed values of air mass. The observed values are determined from direct measurements or physics-based atmospheric models.  The benefits of interpolative models are alignment to actual data and low computational overhead.  The models can be applied successfully if the supporting data and model limitations are well understood.

Young and Irvine (1967) 2 The first model makes a basic correction to the simple air mass calculation. In practice, the function expands the utility to zenith angles beyond 65° to 85°:

(4)   \begin{equation*} \normalsize AM_{YI} = \sec\theta_{Z}[1 - .0012(\sec^2\theta_{Z}-1)] \end{equation*}

The calculated air mass reaches a maximum at 87° and then rapidly approaches negative infinity.

Hardie (1962) 3 Another basic extension of the simple model is based on a polynomial series of \normalsize (1-\sec\theta_{Z}).  Again, the result is to expand the utility of the simple air mass model to zenith angles from 65° to 85°:

(5)   \begin{equation*} \normalsize AM_{H} = \sec\theta_{Z} - A(1-\sec\theta_{Z}) - B(1-\sec\theta_{Z})^2 -C(1-\sec\theta_{Z})^3 \end{equation*}

where A = 0.0018167, B = 0.002875, C = 0.002875.  Like the previous model, the Hardie model also converges on negative infinity at the horizon.

Rozenberg (1966) 4 The Rozenberg model gives reasonable results at the horizon with a maximum air mass of 40°:

(6)   \begin{equation*} \normalsize AM_{R} = \frac{1}{\cos\theta_{Z} + 0.025e^{-11\cos\theta_{Z}}} \end{equation*}

Kasten and Young (1989) 5 A more recent model provides reasonable estimates across the range of zenith angles, generating an expected value of 38° at the horizon.  Hence, this model is among the best:

(7)   \begin{equation*} \normalsize AM_{KY} = \frac{1}{\cos\theta_{Zrad} + 0.50572(6.07795 + 90 - \theta_{Zdeg})^{-1.6364}} \end{equation*}

Note that the first zenith angle is in radians and the second is degrees.

Young (1994) 6 The following model is based on the true zenith angle, unlike the prior models:

(8)   \begin{equation*} \normalsize AM_{Y} = \frac{A\cos^2\theta_{Zt} + B\cos\theta_{Zt} + C}{\cos^3\theta_{Zt} + D\cos^2\theta_{Zt} + E\cos\theta_{Zt} + F} \end{equation*}

where A = 1.002432, B = 0.148386, C=0.0096467, D = 0.149864, E = 0.0102963, and F = 0.000303978. The published error of the model at the horizon is said to be no more than 0.0037 air mass, which is among the best of the published models.

Pickering (2002) 7 The following model uses the elevation angle, h, measured in degrees:

(9)   \begin{equation*} \normalsize AM_{P} = \frac{1}{\sin\left(h + \frac{244}{165 + 47h^{1.1}}}\right) \end{equation*}

The Pickering model also converges on 38° at the horizon, making it one of the best.

Model Code and Visualization

137a

 Relative vs. Absolute Air Mass

Each equation provides estimates of relative air mass.  Conversion to absolute air mass is required in some applications.  Absolute air mass is nothing more than relative air mass corrected for weather (i.e. pressure).  The following function completes the conversion, using pressure in Pascals (Pa).

B.Horn

Back | Next

Show 7 footnotes

  1. Wikipedia Contributors (2014), AirMass (Astronomy), Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Air_mass_(astronomy)&action=history.
  2. Young, A.T. and W.M. Irvine (1967). Multicolor photoelectric photometry of the brighter planets. I. Program and Procedure, Astronomical Journal, Vol. 72, p. 945 
  3. Hardie, R.H. (1962).  Astronomical Techniques, Univ. of Chicago Press, 184-
  4. Rozenbergm G.V. (1966). Twilight: A Study in Atmospheric OpticsPlenum Press, New York, 160.  Translated from the Russian by R.B. Rodman
  5. Kasten, F. and A.T. Young (1989), Revised Optical Air Mass Tables and Approxiamtion FormulaApplied Optics, 28:4735-4738.
  6. Young, A.T. (1994).  Air Mass and RefractionApplied Optics. 33:1108-1110.
  7. Pickering, K.A. (2002). The Southern Limits of the Ancient Star CatalogThe International Journal of Scientific History at DIO, http://www.dioi.org.