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I am doing some calculations to see how many hours of light does a specific location (identified by latitude and longitude) has in a specific day of the year. Contrary to my expectations I get a slight variance for different years. See the data calculated for this point in Svalbard (77.553604,23.670272) in the same day for different years. The values are calculated using the same formula and the same coordinates.

2005-02-22  5.84500
2006-02-22  5.76472
2007-02-22  5.68389
2008-02-22  5.60222
2009-02-22  5.85472
2010-02-22  5.77472
2011-02-22  5.69389
2012-02-22  5.61250

Is this normal and why so?

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It would be very helpful if you provide calculations also for the previous or the next day, e.g. for 23-2-XX (so we can estimate error) –  Pygmalion May 21 '12 at 13:58
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up vote 8 down vote accepted

Of course it's normal. Most of the variation in your table is controlled by the leap years. Note that in 75% of the years, the number of hours drops by $-0.08\pm 0.02$ hours relatively to the previous year; it's really because the day is $365.24219-365 = 0.24219$ solar days closer to the summer than the day of the same date in the previous year.

The exceptions are the first years after the leap years such as 2009; in those years, the number of hours moves in the opposite i.e. positive direction, by $+0.25\pm 0.03$ or so relatively to the previous year, in order to approximately compensate the drop in the previous three years. It's because the leap day such as February 29th, 2008 is in between.

Compare the Februaries 22nd in 2008 and 2009, the most extreme ones in your table. In 2009, the day had 5.85 hours of light but in 2008, it was just 5.60 hours. A significant difference. Why is it so? It's because the day in 2009 is more summer-like than the day in 2008, by almost one day. Try to express February 22nd as a day in March. In 2008, March 0th was February 29th so February 22nd was March -7th. However, in 2009, March 0th was February 28th so February 22nd was March -6th. The number $-6$ is larger than the number $-7$ so it's later in the season, closer to the summer by 3/4 of a solar day or so (if I compute the most appropriate weighted compromise between the schemes to describe the day as a day in February or day in March), which is why there was more light on February 22nd, 2009.

To treat the variations above properly, you need to understand that years divisible by 100 are not leap years unless they are multiples of 400 as well. For a discussion of leap years and the length of the years in solar days, see e.g.

http://motls.blogspot.com/2012/01/happy-new-leap-year-2012.html

This 4-100-400 rule makes the average year's length in solar days to be 365.2425, close to the actual figure close to 365.24219 (one day of deviation accumulates in 3,000+ years or so with our rules) but the number 365.24219 is somewhat variable anyway so it doesn't make sense to define the leap years more accurately than the 4-100-400 rule does.

If you deal with the leap years correctly, it will still leave some variations. There are interannual irregularities in the Earth's motion due to the gravity and mass of the Moon and due to the attraction from ("randomly located") Jupiter and other planets as well as very slow corrections taking thousands of years due to precession of the Earth's axis and changing eccentricity of the "approximately elliptic" orbit of the Earth (some of these variations of eccentricity were already counted with Jupiter etc.).

The Solar System is very close to a set of regularly moving planets but it's not quite accurate due t othe internal dynamics of the gyroscopes, deviations of the positions of the planets due to their moons, and due to the gravitational influence of other bodies different from the Sun.

Incidentally, Wolfram Alpha seems to yield significantly shorter days in Svalbard than you but the logic doesn't change. In Wolfram Alpha, the day is between 4.9 and 5.15 hours or so:

http://www.wolframalpha.com/input/?i=length+of+day+February+22nd%2C+2008+in+Svalbard
http://www.wolframalpha.com/input/?i=length+of+day+February+22nd%2C+2009+in+Svalbard

But maybe the difference is due to the large size of Svalbard and different latitudes picked by you and Wolfram Alpha.

Aside from the impact on the calendar and leap years, irregularities and non-integralities in the motion of the Earth also influence the fact that one solar day isn't exactly 86,400 seconds as most of our clocks assume. With a more accurate definition of one second we have today – linked to atomic clocks – the typical solar day is usually a bit longer. In order to keep the highest-Sun moment near 12:00:00, people got used to insert leap seconds, see

http://motls.blogspot.com/2012/01/leap-seconds-may-be-abolished-in-2015.html

Since 1972, 25 leap seconds have been inserted. This literally means that at the end of June or December, some most accurate of atomic clocks show an excessive, otherwise non-existent time 23:59:60 for a second. Some officials are negotiating possible cancellation of this policy. This would have advantages as well as disadvantages.

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Nice comprehensive answer. I would guess the discrepancy between OP and Wolfram Alpha have to do with exactly how the daylight hours are calculated (center vs top of sun visible, atmospheric refraction accounted for, etc). –  tmac May 21 '12 at 16:26
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