A year is ~365 days long.
A year is ~365 days long. Adding 1 leap day every 4 years, makes a year longer by 1/4=0.25 days or ~365.25 days. That's how Julian Cesar made it with his calendar in 46 BC.

Pope Gregory's calendar contribution in 1582 was to skip leap years in 1700, 1800, 1900, leaving 1600, 2000 as leap. Those who lived through the heated discussions of whether 2000 was a leap year or not, may remember the drama. It almost ruined the millennial celebration for all, so to compensate, everybody decided to celebrate it 1 year earlier. just in case.

Subtracting 3 leap days every 400 years, makes a year a bit shorter, by 3/400=0.0075 days (~11 minutes) or 365.25-3/400= 365.2425 days.

It takes the Earth 365.2564 days to go around the Sun once.

Wait, what?! That's longer then 365.25 days (by 0.0064 days ~= 9 min)!

Could Pope Gregory be so stupid that he introduced the -3/400 correction with the wrong sign? Shouldn't he have added 3 extra leap years per 400 instead of subtracting them. Or the heavens broke since then?! Could be falling on your head any time now! Run for your lives!

If you are still here, the "going around" is star-to-same-star. But we don't actually care about that. We care when spring starts, which is at Spring Equinox, or when the Sun going around ecliptic crosses celestial equator.

By now the suspense must be killing you, the reader.

Due to precession of Earth, discovered by Hipparchus in 146–130 BC, equinoxes move backwards, counter to the Sun, at a rate of ~1 minute or arc per year, so it takes the Sun ~(365*24*60)/(360*60) ~= 24 minutes less time to go from Vernal Equinox to Vernal Equinox (tropical year) then to go from a star-to-same star (sidereal year). More precisely, equinox precession is 50.3 arcseconds per year, which makes the effect (50.3/60)x(365.2564*24*60)/(360*60) = 20.4 min. These ~20 minutes make all the difference in the sign. 20 min is 1/72=0.014 of a day, which is ~2x more then 3/400=0.0075.

Whew. Calm down everybody. Heavens check out yet again.

Moral of the story: two wrongs don't make a right if one is 2x of the other.

The real length of a tropical year is 365.2421896698 days, which is (365.25-3/400)-365.2421896698=0.0003103302 days (=27 seconds) shorter then Gregorian length. So the calendar will be 1 day off in about 1/0.0003103302 ~= 3200 years (a so called Y3K problem). Of course, we could just skip leap year in 3200, but it would make Y2K look like walk in the park by comparison. By that time, it should be much easier to control either the movement of the Earth around the Sun or rotation of the Earth, which makes even more sense, because, as a bonus, it'll also make day to be 86400 seconds long again, like it was back in 1900.

http://en.wikipedia.org/wiki/Tropical_year
http://en.wikipedia.org/wiki/Axial_precession_(astronomy)
http://en.wikipedia.org/wiki/Axial_precession_(astronomy)

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