Sinusoidal Sun
We all know that the highest point the Sun reaches in the sky —aka
upper culmination— depends on both the day of the year and the latitude at
which we make the observation.
At the time of the winter solstice (≈ December 21 in the northern hemisphere; ≈ June 21 in the
southern hemisphere), the Sun is at its lowest in the sky. At the time of the summer solstice
(≈ June 21 in the northern hemisphere; ≈ December 21 in the
southern hemisphere), the Sun is at its highest in the sky. In between these extremes the Sun's
height in the sky gradually waxes and wanes. This waxing and waning, however, is not constant.
Close to the summer and winter solstices the waxing and waning is smallest, whereas during the time of
the spring and fall equinoxes (≈March 21 and ≈ September 21), the waxing and waning is the largest.
All in all, when plotted day-to-day over an entire year (see below; left), the curve of the Sun's
highest point in the sky has a sinusoidal shape that (roughly) follows the function
height (in degrees) = 90 - | L - δ |, where
- L: geographical latitude (in degrees)
- δ: the angle that the Sun makes with the equator; aka solar declination. The formula for it is
δ = 23.45 * sin(360 / 365 * (D - 81))*, where
- D: day of the year (with January 1 as D = 1)
If we substitute δ in the first formula we get:
height (in degrees) = 90 - | L - (23.45 * sin(360 / 365 * (D - 81))) |, where
- L: latitude (in degrees)
- D: day of the year (with January 1 as D = 1)
We all also know that during the times that the sun is high in the sky, days are longest, whereas when
the sun is low in the sky, days are shortest.
Again, when plotted day-to-day over an entire year (see below; right), the curve of the day's
duration — hours of daylight— has a sinusoidal shape and (roughly) follows the function
hours of daylight = 24 * acos(-tan(L) * tan(δ)), where
- L: latitude (in degrees)
- δ: solar declination
Below we plot the curve of the Sun's highest point in the sky over the period of a year (left)
and the corresponding length of day (right) for three different latitudes. Note that here the Sun is
modeled as a point rather than a disk, so the numbers shown here will be a few minutes off.
You can play with these latitudes yourself. To find the latitude of a place near you, try
batchgeo.com or
gps-coordinates.net.
*Alternative versions of this function exist;
see here for more on this.
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