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Suntan (And Other Solar Tigonometric Functions), John Adam
Suntan (And Other Solar Tigonometric Functions), John Adam
Mathematics & Statistics Faculty Publications
Question 1: If I₀ is the solar irradiance (power per unit area, W/m²) reaching my head, express the intensity on the side of my face (Is) in terms of θ. Assume for now that the irradiance is independent of path length through the atmosphere and that my face is normal to the direction θ = 90°.
Using the 1962 U.S. Standard Atmosphere,² Hottel (1976)³ expressed the solar irradiance using the formula
I = I₀(a₀ + a₁e−k sec θ), where A is the elevation in kilometers and
a₀ = 0.4237 − 0.00821(6 − A)²; a₁ = 0.5055 …
A Question Of Transparency, John Adam
A Question Of Transparency, John Adam
Mathematics & Statistics Faculty Publications
Question 1: Why is it easier to see through rain than fog?
Start thinking about this by imagining a fixed volume (V) of water being dispersed into, say, N identical droplets of diameter d. Surface area and volume considerations should lead to the answer in terms of V and d.
Question 2: (a) How "long" (in meters) might such a rain shower or fog bank be?
Hint: Suppose you are looking along a linear stack of S cubes with 1-m sides (through the rain or fog). Each cube contains N drops. If p is the visibility (i.e., the fraction of …
A Question Of Transparency: Solutions For Fermi Questions, March 2025, John Adam
A Question Of Transparency: Solutions For Fermi Questions, March 2025, John Adam
Mathematics & Statistics Faculty Publications
Question 1: Why is it easier to see through rain than fog?
Start thinking about this by imagining a fixed volume (V) of water being dispersed into, say, N identical droplets of diameter d. Surface area and volume considerations should lead to the answer in terms of V and d.
Solution to Question 1: N = VI(πd³/6) = 6V/πd³ ≈ 2V/d³.
The cross-sectional area A of each drop is πd²/4 ≈ 3d²/4, so the total area blocked off (assuming no overlapping drops—so this is an upper bound) is NA ≈ 1.5V/d, so the area blocked off is inversely proportional to …