Sunrise equation

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The sunrise equation as follows can be used to derive the time of sunrise and sunset for any solar declination and latitude in terms of local solar time when sunrise and sunset actually occur:

cos(ω_o) = -tan(φ)×tan(δ)

where ω_o is the hour angle in degree at either sunrise (when negative value is taken) or sunset (when positive value is taken) in degree (°); φ is the latitude of the Earth in degree; δ is the sun declination in degree.

The Earth rotates at the angular speed of 15°/hour and, therefore, ω_o/15° gives the time of sunrise as the number of hours before the local noon, or the time of sunset as the number of hours after the local noon. Here the term local noon indicates the local time when the sun is exactly to the south or north or exactly overhead.

The convention is usually that the value of φ is positive in Northern Hemisphere and negative in Southern Hemisphere. And the value of δ is positive during the Northern Hemisphere summer and negative during the Northern Hemisphere winter.

Please note that the above equation is applicable only when indeed there is a sunrise or sunset when -90°+δ < φ < 90°-δ during the Northern Hemisphere summer, and when -90°-δ < φ < 90°+δ during the Northern Hemisphere winter. Out of these latitudinal ranges, it is either 24-hour daytime or 24-hour nighttime.

Also note that the above equation neglects the influence of atmospheric refraction (which lifts the solar disk by approximately 0.6° when it is on the horizon) and the non-zero angle subtended by the solar disk (about 0.5°). The times of the rising and the setting of the upper solar limb as given in astronomical almanacs correct for this by using the more general equation

cos(ω_o) = (sin(a) - sin(φ)×sin(δ))/(cos(φ)×cos(δ))

with the altitude (a) of the center of the solar disk set to about -0.83° (or -50 arcminutes).