Sunrise equation


The sunrise equation 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:
where:

Theory of the equation

The Earth rotates at an angular velocity of 15°/hour. Therefore, the expression gives the interval of time before and after local solar noon that sunrise or sunset will occur.
The sign convention is typically that the observer latitude is 0 at the equator, positive for the Northern Hemisphere and negative for the Southern Hemisphere, and the solar declination is 0 at the vernal and autumnal equinoxes when the sun is exactly above the equator, positive during the Northern Hemisphere summer and negative during the Northern Hemisphere winter.
The expression above is always applicable for latitudes between the Arctic Circle and Antarctic Circle. North of the Arctic Circle or south of the Antarctic Circle, there is at least one day of the year with no sunrise or sunset. Formally, there is a sunrise or sunset when during the Northern Hemisphere summer, and when during the Northern Hemisphere winter. For locations outside these latitudes, it is either 24-hour daytime or 24-hour nighttime.

Generalized equation

The equation above neglects the influence of atmospheric refraction and the non-zero angle subtended by the solar disc. 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
with the altitude of the center of the solar disc set to about −0.83°.

Complete calculation on Earth

The generalized equation relies on a number of other variables which need to be calculated before it can itself be calculated. These equations have the solar-earth constants substituted with angular constants expressed in degrees.

Calculate current Julian day

where:

Mean solar noon

where:

Solar mean anomaly

where:

Equation of the center

where:

Ecliptic longitude

where:

Solar transit

where:

Declination of the Sun

where:

Hour angle

This is the equation from above with corrections for atmospherical refraction and solar disc diameter.
where:
For observations on a sea horizon needing an elevation-of-observer correction, add, or to the −0.83° in the numerator's sine term. This corrects for both apparent dip and terrestrial refraction. For example, for an observer at 10,000 feet, add or about −1.92° to −0.83°.

Calculate sunrise and sunset

where: