In a geometric-horizon model, the half-day arc H₀ satisfies cos H₀ = −tan φ·tan δ; day length is T = 2H₀/15 hours when H₀ is in degrees and normal sunrise/sunset occur. If |φ| + |δ| ≥ 90°, signs matter: matching signs (φ·δ > 0) indicate polar day and opposite signs (φ·δ < 0) indicate polar night. Equality is a horizon-tangency case, not automatically a full day of daylight or darkness. The source condition |δ| − |φ| ≤ −90° is not a valid general polar-night criterion. Near equinoxes, day and night are approximately 12 hours; conventional sunrise and sunset are problematic at the exact poles. Calendar boundaries are approximate.
H₀: half-day arc (°) T: day length (hours) δ: solar declination (°) φ: observer latitude (°)
Concept explanation
—
Current day length
—hours
Solar declination δ—
Half-day arc H₀—
Sunrise (local time)—
Sunset (local time)—
Polar day/nightNo
Day-length heatmap (latitude × date)
0h6h12h18h24h
✕ Current position
Latitude comparison (current date)
Latitude
Day length (h)
Status
Counterintuitive insight
Drag latitude to 90°N (North Pole) — at the June solstice, daylight lasts 24h, and at the December solstice it lasts 0h. Polar day and night occur inside the polar circles; days are not merely "a little longer".