Mountain Valley Sunrise

What is modelled

The sun elevation at any hour follows from the latitude and the date. The summit is lit when the sun is above the geometric horizon, but the valley floor is lit only when the sun is above the eastern ridge, so the valley gets the sun later. A lit summit throws a shadow whose length is its height divided by the tangent of the sun elevation. The mist along the valley dims the light from the summit exponentially with the path length inside the mist.

Equations

• Elevation sin a = sin φ sin δ + cos φ cos δ cos ω, sunrise cos ω = (sin a0 − sin φ sin δ) / (cos φ cos δ).
• Shadow L = H / tan a.
• Beer-Lambert I = I0 exp(−∫ β ds), Koschmieder β = 3.912 / V.
• Apparent contrast C = C0 exp(−∫ β ds) with C₀ = 0.5; visible while C is above 2 %.

Checks run in the page

• Sunrise by bisection against the hour-angle formula.
• A step-by-step shadow ray against H / tan a.
• A numerical Beer-Lambert integration along the mist.
• The Koschmieder 2 % rule and the noon elevation.

Honest simplifications

A flat-earth valley, a straight line of sight, a mist layer with a hard top and a single uniform extinction, geometric sunrise without refraction, no scattering of sky light and no real terrain. The scene is a schematic of the 3D landscape, not a map of the Carpathians.

• Press Autumn valley mist and watch the apparent contrast of the summit drop below the 2 % line.
• Press Midwinter, high ridge and see the valley sunrise arrive hours after the summit.
• Raise the summit height and see the shadow it throws at the same sun elevation grow.