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Length of a day at a given latitude

My answer to: How can you calculate the length of the day on Earth at a given latitude on a given date of the year? (To vote and comment on the answer, visit the link here.)

The formula comes below in a while. First, the illustration of the earth’s orbit:

The earth moves in its orbit counter-clockwise as shown in the diagram. Different dates denote the position of earth in the orbit on that date. During northern winter (November, December, January), the northern hemisphere is not facing the sunlight directly, so it is the winter season.

Since there is a tilt of the axis, some parts do end up receiving sunlight but the days are shorter. Due to the tilt, it is only the polar regions do not receive any sunlight during the winters. Consider the north pole during northern winter. It does not receive any sunlight at all during the winter, so it spends a lot of time in total darkness (day length = 0 hours). On the contrary, in the northern summer, the north pole has no sunset for months (day length = 24 hours).

So, we see that the day length theoretically depends on these parameters:

 

  • the date (which should appear in the formula in the form of the angular position δ in the elliptical orbit; δ is called the solar declination),
  • the latitude φ.


(Practically, it depends on other effects too such as atmospheric refraction, weather, etc.)

Then there is a quantity called hour angle (h) which is the angular representation of time. The solar noon refers to hour angle h = 0 degree. Since the earth rotates 15° per hour, a time such as 1pm refers to the hour angle h = +15°. (Here 1pm is the local solar time, not the local standard time). And a time such as 10am refers to the hour angle h = –30°. (Sunrise has a negative value of h, and sunset positive.)

At sunrise, the hour angle is given by:

cos h = – tan φ · tan δ

δ is zero at equinoxes, positive in the northern summer, and negative in the northern winter. φ is zero at the equator, positive in northern hemisphere, and negative in the southern hemisphere.

We know the hour angle is directly a measure of time; the sunrise was h/15 hours before solar noon, and the sunset is h/15 hours after solar noon (since earth rotates at 15 degrees an hour). So the day length is

Day Length = 2h/15 hours

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Please note that this is valid for the latitudes between the Arctic Circle and the Antarctic Circle. For the polar regions, the day length is either 0 or 24 hours (explained already).

Information compiled from:

Sunrise equation
Day length

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