Sunrise equation

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:

where:

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 [-1, 1], but the value of the expression on the right side is in the range ${\displaystyle [-\infty ,\infty ]}$. 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