Sunrise equation

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 degrees 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 degrees; δ is the sun declination in degrees.

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 disc by approximately 0.6° when it is on the horizon) and the non-zero angle subtended by the solar disc (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 disc set to about -0.83° (or -50 arcminutes).

ee also

*Sunrise
*Sunset
*Day length

References

*http://physicsweb.org/articles/world/17/10/2