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Daylength For sunrise and sunset, θ = 90 , so cosθ = sinφ sinδ + cosφ cosδ cosτ becomes cosτ = -tanφ tanδ Daylength = 2 cos -1 (-tanφ tanδ) What is the daylength at Fairbanks, Alaska (65 N, 148 W) at the winter solstice? Daylength = 2 cos -1 (-tan(65 ) tan(-23.5 ))
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Daylength • For sunrise and sunset, θ = 90 , so • cosθ = sinφ sinδ + cosφ cosδ cosτ becomes • cosτ = -tanφ tanδ • Daylength = 2 cos-1(-tanφ tanδ) • What is the daylength at Fairbanks, Alaska (65 N, 148 W) at the • winter solstice? • Daylength = 2 cos-1(-tan(65) tan(-23.5)) • = 2 cos-1(-tan(65) tan(-23.5)) • =42.4 / (15hr-1) = 2hr 49min • Sunrise: τ = -21.2 Sunset: τ = +21.2 • (T – 12)*15 + ( - ) + F / 4 - D (T – 12)*15 = τ - ( - ) - F / 4 + D Sunrise: (T – 12)*15 = -21.2 -(-148+135) – 0 + 0 (T – 12)*15 = -8.2 T =11:27am
Azimuth Angle A = 180 º + sin-1 (cos δ sin τ /sin θ) Example 1 : Find the azimuth angle at sunrise in Fairbanks at the winter solstice. A = 180 º + sin-1 (cos δ sin τ) A =180 º + sin-1 (cos (-23.5 º) sin (-21.2 º)) A =180 º + (-19. 4 º ) = 160.6 º Example 2 : Find the azimuth angle at τ = -120 in Fairbanks at the summer solstice. A = 180 º + sin-1 (cos δ sin τ) A =180 º + sin-1 (cos (23.5 º) sin (-120 º)) A =180 º + sin-1 (-0.79) A = 180 º + (-52 º ) = 128 º ???? No. sin-1 (-0.79) has 2 values. A = 180 º + (-128 º ) = 52 º
A = 180 º + sin-1 (cos δ sin τ) At the equinox, cos δ = 1 At sunset, sin τ = 1 A = 180 º + 90 δ = 270 º A = 180 º + sin-1 (cos δ sin τ) At the equinox, cos δ = 1 At sunrise, sin τ = -1 A = 180 º + -90 º = 90 º For δ > 0, i.e. March 20-Sept. 22, sunrise A < 90 º and sunset A > 270 º in the Northern Hem. For δ < 0, i.e. Sept. 22-March 20, sunrise A > 90 º and sunset A < 270 º in the Northern Hem.
Shortcut for determining Noon Zenith Angle cosθ = sinφ sinδ + cosφ cosδ cosτ At “noon”, cosτ = 1, so cosθ = sinφ sinδ + cosφ cosδ 3 Trig. Identities cos(x+y) = cosxcosy – sinxsiny sin(-y) = -siny cos(-y) = cosy These yield cos(x-y) = cosxcosy + sinxsiny Therefore, cosθ = cos(φ-δ) or θ = |φ-δ| Denver for Feb. 1: θ = 40 º – (-17 º ) = 57 º Buenos Aires for Feb. 13: θ = |-35 º – (-13 º )| = 22 º Fairbanks for Dec. 21: θ = 65 º – (-23.5 º ) = 88.5 º
