How to Calculate Sunrise and Sunset Times

The time of sunrise and sunset depends of the time of year and your latitude on Earth. During the summer solstice around June 21, the farther north you are, the earlier the sun rises and the later the sun sets. The farther south you are, the later the sun rises and the earlier the sun sets. The opposite effect is observed during the winter solstice around December 21, in which northern latitudes experience more darkness and southern latitudes experience more daylight.

At the vernal and autumnal equinoxes, March 20 and September 22/23 respectively, all locations on earth experience a day with 12 hours of daylight and 12 hours of night, regardless of latitude.

You can use a trigonometric formula to compute the time of sunrise and sunset, given as a function of the number of hours before and after the local noon. The term "local noon" means the time of day when the sun is directly overhead. Due to the effects of Daylight Savings Time and asymmetric time zone boundaries, the local noon does not always correspond to 12:00 pm.

Sunset Equation

Let L bet the latitude in degrees. (For the formula below, latitudes in the Southern Hemisphere must be entered as negative numbers. For the calculator, do not enter a negative sign for southern latitudes, simply choose "South.") Let D be the day number of the date, starting with D = 1 for January 1 and ending with D = 365 for December 31. For simplicity, you can use either D = 59 or D = 60 for February 29 during leap years. Then the approximate time of sunrise, given as hours before local noon, is

H = | (1/15)*arccos[-tan(L)*tan(23.44*sin(360(D+284)/365))] |.

For the time of sunset, the equation is the same, but it is H hours after the local noon.

Example: Suppose it is April 15 and a person is 40° south of the equator. Thus, L = 40 and D = 105. We find H by computing

| (1/15)*arccos[-tan(-40)*tan(23.44*sin(360(105+284)/365))] |
= | (1/15)*arccos[0.8391*tan(23.44*0.4015)] |
= | (1/15)*arccos[0.1391] |
= (1/15)*82.0056
= 5.467 hours, or 5 hours and 28 minutes.

This means that sunrise will occur 5 hours and 28 minutes before the local noon, and sunset will occur 5 hours and 28 minutes after the local noon. If the local noon occurs at the local time of 12:47 PM in this location, then the sunrise will happen around 7:19 AM and the sunset will be around 6:15 PM.

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