Explanation of the Leap-Day System of the Gregorian Calendar

The time difference between the mean solar day and the mean sidereal day is exactly 86400 s - 86164.0905382 s = 235.9094618 s. When a calendar year ends, precisely 365 × 86400 s = 31536000 s have passed. However, relative to the fundamental sidereal point of reference (vernal equinox) 366 × 86164.0905382 s = 31536057.137 s have elapsed, i.e. a time difference of about 57.14 s occurs.

After four calendar years this time difference, if ignored, accumulates to 4 × 57.137 s = 228.548 s. But every four years (365.25 day calendar cycle) we have a leap day which gives us the fundamental difference of 235.9094618 seconds to compensate for the previously accumulated time difference of 228.548 s. Due to a slight "over-compensation", another time difference occurs every four years: 235.90946 s - 228.548 s = 7.36 seconds.

It is these crucial 7.36 s that will accumulate again close to the fundamental solar-sidereal day difference of 235.90946 s, and hence another leap day is needed in roughly 128 years (235.90946 × 4 ÷ 7.36 = 128.18). The disadvantage of the Gregorian calendar is its long leap cycle. Besides its regular four year leap cycle, it has a 400 year leap-cycle (time-keepers are trying to come as close as possible to the 128.18 year period, i.e. 400 ÷ 3 = 133.33 years). The Gregorian leap-day-system requires therefore, another leap day every 3319.88 year to keep the calendar in synch with the tropical year.

Time Deviation and Adjustment of the Civil Calendar to the Tropical and Sidereal Year

Note: For reason of simplicity the time-functions begin 1600 AD instead of 1582 AD, the year of the calendar reform.

x1 = Tropical year of 365.24219878 mean solar days of 86,400s

x1 = Sidereal year of 366.24219878 mean sidereal days of 86,164.09054s

x1a = Sirius (Sothis) year of 366.24219878 sidereal days; i.e. the mean transit period of Sirius of 86,164.09071s

x2 = x1 plus the annual time-deviation of 50.26"/tropical year (the apparent annual motion of the fixed-stars)

x3 = Gregorian Calendar of 365.2425 days, since the calendar-reform of 1582 AD (because 0.2425 = 97/400; see also x4)

Note: After 3,319.88 years, x3 reaches the y-value of '24hrs' and has to be adjusted to x1 by means of one leap-day

x4 = Civil Calendar of 365.2500 mean solar days including a 400year-leap-cycle (97 leap days in 400 years - i.e. in centuries that cannot be evenly divided by 400 the leap day is omitted, e.g. 1700, 1800 and 1900 AD)

Note: The 400year-cycle of time-function x4 results into the time-function x3

x4a = Julian Calendar of 365.2500 mean solar days (prior to the calendar reform of 1582 AD)

x5 = The so-called sidereal year (fixed-star to fixed-star) of 365.256361 mean solar days

Note: x5 deviates from the time-function x1 noticeable faster than x4a - the ineffectual time-measure of the Julian year that was, in effect, the cause for the calendar reform of 1582 AD