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

It is formulated as:

${\displaystyle \cos \omega _{\circ }=-\tan \phi \times \tan \delta }$

where:

${\displaystyle \omega _{\circ }}$ is the solar hour angle at either sunrise (when negative value is taken) or sunset (when positive value is taken);

${\displaystyle \phi }$ is the latitude of the observer on the Earth;

${\displaystyle \delta }$ is the sun declination.

The Earth rotates at an angular velocity of 15°/hour. Therefore, the expression ${\displaystyle \omega _{\circ }/\mathrm {15} ^{\circ }}$, where ${\displaystyle \omega _{\circ }}$ is in degree, gives the interval of time in hours from sunrise to local solar noon or from local solar noon to sunset.

The sign convention is typically that the observer latitude ${\displaystyle \phi }$ is 0 at the equator, positive for the Northern Hemisphere and negative for the Southern Hemisphere, and the solar declination ${\displaystyle \delta }$ 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 ${\displaystyle -90^{\circ }+\delta <\phi <90^{\circ }-\delta }$ during the Northern Hemisphere summer, and when ${\displaystyle -90^{\circ }-\delta <\phi <90^{\circ }+\delta }$ during the Northern Hemisphere winter. For locations outside these latitudes, it is either 24-hour daytime or 24-hour nighttime.

In the equation given at the beginning, the cosine function on the left side gives results in the range , but the value of the expression on the right side is in the range ${\displaystyle }$. An applicable expression for ${\displaystyle \omega _{\circ }}$ in the format of Fortran 90 is as follows:

omegao = acos(max(min(-tan(delta*rpd)*tan(phi*rpd), 1.0), -1.0))*dpr

where omegao is ${\displaystyle \omega _{\circ }}$ in degree, delta is ${\displaystyle \delta }$ in degree, phi is ${\displaystyle \phi }$ in degree, rpd is equal to ${\displaystyle {\frac {\pi }{180}}}$, and dpr is equal to ${\displaystyle {\frac {180}{\pi }}}$.

The above expression gives results in degree in the range ${\displaystyle }$. When ${\displaystyle \omega _{\circ }=0^{\circ }}$, it means it is polar night, or 0-hour daylight; when ${\displaystyle \omega _{\circ }=180^{\circ }}$, it means it is polar day, or 24-hour daylight.

Suppose ${\displaystyle \phi _{N}}$ is a given latitude in Northern Hemisphere, and ${\displaystyle \omega _{\circ N}}$ is the corresponding sunrise hour angle that has a negative value, and similarly, ${\displaystyle \phi _{S}}$ is the same latitude but in Southern Hemisphere, which means ${\displaystyle \phi _{S}=-\phi _{N}}$, and ${\displaystyle \omega _{\circ S}}$ is the corresponding sunrise hour angle, then it is apparent that

${\displaystyle \cos \omega _{\circ S}=-\cos \omega _{\circ N}=\cos(-180^{\circ }-\omega _{\circ N})}$,

which means

${\displaystyle \omega _{\circ N}+\omega _{\circ S}=-180^{\circ }}$.

The above relation implies that on the same day, the lengths of daytime from sunrise to sunset at ${\displaystyle \phi _{N}}$ and ${\displaystyle \phi _{S}}$ sum to 24 hours if ${\displaystyle \phi _{S}=-\phi _{N}}$, and this also applies to regions where polar days and polar nights occur. This further suggests that the global average of length of daytime on any given day is 12 hours without considering the effect of atmospheric refraction.

The equation above neglects the influence of atmospheric refraction (which lifts the solar disc — i.e. makes the solar disc appear higher in the sky — by approximately 0.6° when it is on the horizon) and the non-zero angle subtended by the solar disc — i.e. the apparent diameter of the sun — (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

${\displaystyle \cos <!--\; \; -->{\mathrm{\omega <!--\; \omega \; -->}}_{\circ <!--\; \circ \; -->}={\displaystyle \frac{\sin <!--\; \; -->a-<!--\; -\; -->\sin <!--\; \; -->\mathrm{\varphi <!--\; \varphi \; -->}\times <!--\; \times \; -->\sin <!--\; \; -->\mathrm{\delta <!--\; \delta \; -->}}{\cos <!--\; \; -->\mathrm{\varphi <!--\; \varphi \; -->}\times <!--\; \times \; -->\cos <!--\; \; -->\mathrm{\delta <!--\; \delta \; -->}}}}$Zdroj:https://en.wikipedia.org?pojem=Sunrise_equation
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