Daylight length vs day-of-year
Rate of change (hours gained/lost per day)
Mathematical Explanation
Solar Declination
The solar declination δ is the angle between the Sun and the celestial equator. It varies throughout the year due to Earth's axial tilt (obliquity ε ≈ 23.44°):
where N is the day of year. The phase shift of 80 days places the zero-crossing near the March equinox (day ~80).
Hour Angle at Sunrise/Sunset
The hour angle H measures time from solar noon in angular units (15°/hour). At sunrise and sunset, the Sun is at altitude h₀ = −0.833° (accounting for atmospheric refraction and the Sun's radius).
From spherical trigonometry, the solar altitude α satisfies:
Setting α = h₀ and solving for the hour angle H₀:
The ± solutions give sunrise (−H₀) and sunset (+H₀).
Daylight Duration
Since the Sun traverses 2H₀ radians from sunrise to sunset, and Earth rotates 2π radians in 24 hours:
Polar day (24h) occurs when cos(H₀) ≤ −1; polar night (0h) when cos(H₀) ≥ 1.
Rate of Change
The derivative dD/dN (hours gained or lost per day) is computed via centered finite difference:
This rate peaks near the equinoxes and approaches zero near the solstices.