http://hpiers.obspm.fr/eop-pc/models/constants.html states the tropical year is $365.242190402$ days.

The Gregorian calendar's average year is $365.2425$ days (every 4th year a leap year, except every 100 years, except every 400 years).

The difference is $0.000309598$ days, which amounts to $26.7493$ seconds.

This is much more than the 1 leap second we add every 2-3 years or so.

What am I missing?


The leap year and the leap second serve different purposes, so there's no reason for them to line up exactly necessarily. The leap second keeps the Sun in the right place (its yearly average position at 12:00:00 PM at the center of a time zone is directly on the meridian). The leap year-day keeps the solstices and equinoxes falling on the correct day, so it need not be precise to the second.

  • 1
    $\begingroup$ Right. In other words, leap seconds correct for amount by which a solar day differs from 24 hours; leap days correct for the amount by which a solar year differs from 365 solar days. (And "24 hours" is defined as 86400 seconds, where the second is defined as "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom" (reference). $\endgroup$ Oct 12 '11 at 0:50

What you're "missing" is that you're basically right! The Gregorian calendar is slowly losing some accuracy. I've previously heard the error quoted as about 3300 yr, which lines up with your measurement because 1 day / 26.75s $\approx$ 3230.

From Wikipedia's Leap Year article:

The marginal difference of 0.000125 days between the Gregorian calendar average year and the actual year means that, in 8,000 years, the calendar will be about one day behind where it is now. But in 8,000 years, the length of the vernal equinox year will have changed by an amount that cannot be accurately predicted (see below[vague]). Therefore, the current Gregorian calendar suffices for practical purposes, and the correction suggested by John Herschel of making 4000 a non-leap year will probably not be necessary.

Not sure why Wiki has 8000 yr but the point is the same.


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