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:

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 degrees (°);

φ is the latitude of the observer on the Earth in degrees;

δ is the sun declination in degrees.

Contents 1 Theory of the Equation 2 Corrections to the Equation 3 Complete calculation on Earth 3.1 Calculate current Julian Cycle 3.2 Approximate Solar Noon 3.3 Solar Mean Anomaly 3.4 Equation of Center 3.5 Ecliptic Longitude 3.6 Solar Transit 3.7 Declination of the Sun 3.8 Hour Angle 3.9 Calculate Sunrise and Sunset 4 See also 5 External links 6 References

[edit] Theory of the Equation

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 0 at the Equator,positive in Northern Hemisphere and negative in Southern Hemisphere. And the value of δ 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.

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.

[edit] Corrections to the Equation

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

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

[edit] 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 already have all of the solar-earth constants put into them with angular constants in degrees.

[edit] Calculate current Julian Cycle

where:

J_date is the Julian date;

l_w is the longitude west (west is positive, east is negative) of the observer on the Earth;

n is the Julian cycle since Jan 1st, 2000.

[edit] Approximate Solar Noon

where:

J* is an approximation of solar noon at l_w.

[edit] Solar Mean Anomaly

where:

M is the solar Mean Anomaly.

[edit] Equation of Center

where:

C is the Equation of the center.

[edit] Ecliptic Longitude

where:

λ is the ecliptic longitude.

[edit] Solar Transit

where:

J_transit is the hour angle for solar transit (or solar noon).

[edit] Declination of the Sun

where:

δ is the declination of the sun.

[edit] Hour Angle

This is the equation from above with the solar disc correction.

where:

ω_o is the hour angle;

φ is the north latitude of the observer (north is positive, south is negative) on the Earth.

This is the main equation from above with the solar disc correction.

[edit] Calculate Sunrise and Sunset

where:

J_set is the actual Julian Date of sunset;

J_rise is the actual Julian Date of sunrise.

[edit] See also

• Sunrise
• Sunset
• Day length

[edit] External links

• Sunrise, sunset, or sun position for any location

[edit] References

• http://www.astro.uu.nl/~strous/AA/en/reken/zonpositie.html

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