What Defines Sunrise?

Sunrise is officially defined as the instant when the upper limb (top edge) of the Sun's disk touches the geometric horizon, as observed at sea level. But the Sun is not actually at the horizon when we first see it — atmospheric refraction bends light rays, making the Sun visible when it is still geometrically below the horizon. The standard refraction at the horizon is 34 arcminutes (0.567°). Additionally, we want to detect the upper limb, not the center, so we add the Sun's apparent semi-diameter of 16 arcminutes. The combined threshold: the Sun's geometric center must be at altitude = -0.8333° (-(34' + 16') = -50' = -0.8333°).

The fundamental equation: given the observer's latitude (lat) and the Sun's declination (decl), the hour angle H₀ at which the Sun reaches altitude h₀ = -0.8333° is: H₀ = arccos([sin(h₀) - sin(lat) x sin(decl)] / [cos(lat) x cos(decl)]). This is the cosine rule from spherical trigonometry applied to the astronomical triangle. The sunrise time is then: Sunrise = Solar Noon - H₀/15, where H₀ is in degrees and dividing by 15 converts to hours (since the Earth rotates 15° per hour).