Daylight Calculator

Solar mechanics rely on spherical trigonometry and Earth's axial tilt, which is approximately 23.44 degrees. The fundamental inputs require the day of the year (Day), latitude (Phi), and longitude (Longi). The core algorithm computes the solar declination angle based on the seasonal cycle (dayum). Sunrise and sunset times are then derived by calculating the hour angle where the sun crosses the horizon, factoring in atmospheric refraction (sunriseref and sunsetref) which bends light rays and slightly extends visible daylight. Total day length is calculated simply as the temporal difference between sunset and sunrise.

Consider calculating daylight hours for a location at latitude 45.0 degrees North (Phi = 45.0) and longitude 90.0 degrees West on the Spring Equinox (Day 80 of the year). First, the solar declination is evaluated for that specific day of the year. Using the celestial coordinate equations, the hour angle for the horizon intersection is determined. For this mid-latitude coordinate on an equinox, unrefracted sunrise occurs approximately at 06:00 local time, and sunset occurs at 18:00 local time. Subtracting the sunrise timestamp from sunset gives a total unrefracted day length of 12 hours. Accounting for atmospheric refraction typically adds a few extra minutes to the visible day length.

Ensure your latitude and longitude use correct directional signs, typically positive for North and East, and negative for South and West. Keep time zone (Timzon) offsets aligned with local standard time or daylight saving rules to match observational checks. Remember that extreme polar latitudes may trigger NaN (Not a Number) flags during polar night or midnight sun events, indicating the sun never officially crosses the geometric horizon.