Accuracy Near The Horizon
The above approximation overlooks the curvature of the Earth, and is reasonably accurate for values of up to around 75°. A number of refinements have been proposed to more accurately model the path thickness towards the horizon, such as that proposed by Kasten and Young (1989):
-
(A.2)
A more comprehensive list of such models is provided in the main article Airmass, for various atmospheric models and experimental data sets. At sea level the air mass towards the horizon ( = 90°) is approximately 38.
Modelling the atmosphere as a simple spherical shell provides a reasonable approximation:
-
(A.3)
where the radius of the Earth = 6371 km, the effective height of the atmosphere ≈ 9 km, and their ratio ≈ 708.
These models are compared in the table below:
| Flat Earth | Kasten & Young | Spherical shell | |
|---|---|---|---|
| degree | (A.1) | (A.2) | (A.3) |
| 0° | 1.0 | 1.0 | 1.0 |
| 60° | 2.0 | 2.0 | 2.0 |
| 70° | 2.9 | 2.9 | 2.9 |
| 75° | 3.9 | 3.8 | 3.8 |
| 80° | 5.8 | 5.6 | 5.6 |
| 85° | 11.5 | 10.3 | 10.6 |
| 88° | 28.7 | 19.4 | 20.3 |
| 90° | 37.9 | 37.6 |
This implies that for these purposes the atmosphere can be considered to be effectively concentrated into around the bottom 9 km, i.e. essentially all the atmospheric effects are due to the atmospheric mass in the lower half of the Troposphere. This is a useful and simple model when considering the atmospheric effects on solar intensity.
Read more about this topic: Air Mass (solar Energy)
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