perplexus.info :: Science : Let there be Light!

Let there be Light! (Posted on 2023-07-04)

It’s common knowledge that the length of a day (time from sunrise to sunset) varies over the year and with one’s location on the earth. Given only the following, can you derive an approximate equation for the length of a day (in hours) of any specific location on the earth, on any given day of the year?

1) The inclination of the earth’s rotational axis is 23.45 degrees

2) The length of a day is exactly 24 hours

3) The length of a year is exactly 365 days

4) Location on earth is given by the latitude

You are allowed to ignore secondary (but real!) effects such as the earth’s non-circular orbit, the sun being a disc, refraction of sunlight by the atmosphere, etc. To allow easier comparison of different solutions, let’s also assume that North Latitude is positive, and that the Winter Solstice in the Northern Hemisphere is “day 0” (Hint: therefore also day 365!) of the year.

Submitted by Kenny M

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Solution: (Hide)

I admit I did not derive this myself, as my spherical geometry skills are lacking, but given the assumptions of the problem, daylight hours (time from sunrise to sunset) =
2/15 *arccosine [-tan(latitude angle) * tan(D)]
where D is the “declination angle” on the day in question. “D” is the angle between the earth’s equator and a line drawn from the center of the earth to the center of the sun. For the Northen Hemisphere, this angle would be -23.45 degrees (north pole pointed somewhat away from the sun) on the Winter Solstice (day 0 of the year), 0 degrees for the Spring and Autumnal Equinox(s) and +23.45 degrees for the Summer Solstice (day 365/2=182.5 of the year). Given the above assumptions, the equation for “D” is then
D= -23.45cosine(360/365*dn)
where “dn” is the day number of the year based on the convention in the problem statement.

This derivation even handles the Arctic/Antarctic circle. When the argument of the arccosine is out of range, it implies that the length of the day is either 0, or maxed out at 24 hours. Lots of resources on the web explain all this nicely.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(2): Solution (spoiler)	Curtis	2023-11-15 19:55:38
re: Solution (spoiler)	Curtis	2023-11-15 19:54:59
Solution (spoiler)	Charlie	2023-07-04 17:31:41

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