Is there any geometrically derived mathematical expression for the duration of the natural day at any latitude and any time of the year?
Below is the geometrically derived formula for estimating day length based on latitude and time of year:
Formula for Day Length
The day length DDD in hours can be estimated using:
D=24πcos−1(−tan(ϕ)tan(δ))D = \frac{24}{\pi} \cos^{-1}(-\tan(\phi) \tan(\delta))D=π24cos−1(−tan(ϕ)tan(δ))
where:
- ϕ\phiϕ is the latitude of the location.
- δ\deltaδ is the solar declination angle, which varies throughout the year.
Solar Declination Angle (δ\deltaδ)
The solar declination angle δ\deltaδ can be approximated by:
δ=23.44∘⋅sin(360∘365⋅(N+10))\delta = 23.44^\circ \cdot \sin\left(\frac{360^\circ}{365} \cdot (N + 10)\right)δ=23.44∘⋅sin(365360∘⋅(N+10))
where NNN is the day of the year (1 for January 1, 2 for January 2, etc.).
Calculation Steps
This formula captures the variation in day length due to Earth's axial tilt and orbit around the Sun.
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