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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 l.jpg
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 / (15hr-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


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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


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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.


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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


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