A course in Si solar cell design - Lesson on the solar spectrum

Calculations of the air mass

If the sun shines from 90◦ above the horizon – from the zenith – the light goes by definition through the optical air mass 1. The air mass at an angle z from the zenith (or an angle h from the horizon) is longer: to a first approximation, assume that the earth is flat, and you obtain 1 1 am = = , (1) cos z sin h

6

5

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3 3

19.47° 2 2

Optical air mass 1.5 30° 1 41.81°

0 10 20 30 40 50 60 70 80 90 Sun's elevation angle [°]

Figure 1 Calculations of the air mass as a function of the sun’s elevation angle above the horizon, h, using Eq. (1), which is identical to more sophisticated calcu- lation procedures (2) if h > 10◦, see Fig. 2.

Notice that, with this approximation, we would never see the sun rise or sun set in flat country, because the air mass would be infinite. To which angle h is the above equation a good approximation? To get a definite answer, you need to compare the above equation with rigorous calculations that take the curvature of the atmosphere as well as its vertical density profile into account. A widely used model of this sort is the one of Kasten and Young [2], who fitted their rigorous numerical calculations with 1 am = (2a) cos z + 0.50572(96.07995◦ − z)−1.6364 1 = (2b) sin h + 0.50572(6.07995◦ + h)−1.6364

A comparison of (2) with (1) in Figure 2 shows that Eq. (1) is a good approximation down to h ≈ 10◦. 80

Water Planar approximation vapour

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40 Mixed gases

Optical air mass 20 Ozone

0 0 2 4 6 8 10 Apparent sun's elevation angle [°]

Figure 2 Calculations of the air mass at low apparent elevation angles h of the sun above the horizon, as calculated assuming the earth to be flat (1), or taking the curvature of the atmosphere as well as its vertical density profile into account. The latter depends on whether gasses are well mixed (2), or more abundant close to the earth surface (such as water vapour), or more abundant at high altitudes (such as ozone) [1].

Figure 2 also shows that it matters whether a gas is well mixed in the atmosphere (such as oxygen, nitrogen etc, which are distributed according to the atmosphere’s vertical density profile) [2], or wether the gas is more abundant near the earth surface (such as water vapor, aerosols and NO2 from pollution) [1], or whether the gas is more abundant at high altitude (such as ozone and natural NO2) [1]. It also matters that the sunlight is bent on its path through the atmosphere due to refraction: z or h used in Eq. (2) and shown in Figure 2 are the apparent angles (the angles seen from the earth surface) not the angles you would obtain from purely astronomical calculations. These two angles differ by 0.15◦ at h = 45◦ and by 0.5◦ (the sun’s diameter) at h=0◦. You may conclude that such small differences can be neglected; however, they cause an error in air mass by up to 10%. An expression for the air mass in terms of the astronomical angles is [3] 1.003198 cos z + 0.101632 am = . (3) cos2 z + 0.090560 cos z + 0.003198

2 Bibliography

[1] C. A. Gueymard, ‘Parameterized transmittance model for direct beam and circumsolar spectral irradiance’, Solar Energy 71(5), 325–346 (2001). [2] F. Kasten and A. T. Young, ‘Revised optical air mass tables and approximation formula’, Applied Optics 28(22), 4735–4738 (1989). [3] A. T. Young, ‘Air mass and refraction’, Applied Optics 33(6), 1108–1110 (1994).