Navigation Exercise 2

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No matter where you are the Sun will follow the same path through the sky at the same time each year. There are a series of easy formulae you can use to find the bearing and altitude of the Sun at any time, and the time and bearing of sunrise and sunset for the current day.

You will need

• Current Latitude (negative for Southern Hemisphere)
• Days from Equinox (negative during Northern Winter)
• Current Time of Day
• Alusia's Elliptic Tilt (23.5°)

Seasonal Declination

Here are three methods for finding the Seasonal Declination. The simpler the approximation, the greater the error. The last method may be up to 1.4 days, 25 miles or 0.4 degrees out.

• Declination = the value from Chapter Five.
• Declination = ArcSine(Sine(DaysFromEquinox/91*90)*Sine(Tilt))
• Declination ≈ Sine(DaysFromEquinox)*Tilt

First Results

• Hours of Sunlight = 12 + (ArcSine(Tan(Declination)*Tan(Latitude)))/7.5
• Dawn (Hours) = 12 - HoursOfSunlight/2
• Dusk (Hours) = 12 + HoursOfSunlight/2
• Sun Bearing at Dawn = ArcCosine(-Sine(Declination)/Cosine(Latitude))
• Sun Bearing at Dusk = 360 - Sun Bearing at Dawn
• Sun Elevation at Noon = (90-Latitude)+Declination

Second Results

• Day Progression = (Number of hours from Zenith) * 360 / 24
• Sun X-Axis = Cosine(DayProgression)*Cosine(Declination)*Sine(Latitude) - Sine(Declination)*Cosine(Latitude)
• Sun Y-Axis = -Sine(DayProgression)*Cosine(Declination)
• Sun Z-Axis = Cosine(DayProgression)*Cosine(Declination)*Cosine(Latitude) + Sine(Declination)*Sine(Latitude)
• Note Sun X-Axis² + Sun Y-Axis² + Sun Z-Axis² = 1

Final Results

• Sun Bearing = ArcTan(Sun Y-Axis / Sun X-Axis)
• Sun Elevation = ArcSin(Sun Z-axis)