which provides a very good approximation. The number above, by the way, is a transcendental number, and can be given a yet simpler alternate form.
The tropical year is defined as the mean interval between vernal equinoxes; it corresponds to the cycle of the seasons. The vernal equinox is the time in March when the sun passes the equator moving from the southern to the northern hemisphere. Day and night have approximately the same length at this time. The date of the vernal equinox is near 20 March. Our calendar year is linked to the tropical year as measured between two March equinoxes, as originally established by Caesar and Sosigenes. The following expression, based on the orbital elements of Laskar (1986), is used for calculating the length of the tropical year:
365.2421896698 - 0.00000615359 T - 7.29E-10 T2 + 2.64E-10 T3 (days)
where T = (JD - 2451545.0) / 36525 and JD is the Julian day number. The value of the mean tropical year for the epoch of J2010.0 (Julian day of 2455197.5, and thus T = 0.1) was calculated with Laskar's equation above.The main interest of the tropical year value is to keep the calendar year synchronised with the beginning of seasons.
All the progressive solar calendars since Old Egyptian times are arithmetical calendars. This means an easy rule to try to reach the best possible astronomical value.
In the history of solar calendars notably these five
rules (approximations) shown below were used, are used, or are proposed.
| Calendar rule |
|
|
| Old Egyptian | 365 | = 365. 000 000 000 |
| Julian | 365 + ¼ | = 365. 250 000 000 |
| Gregorian | 365 + 97/400 | = 365. 242 500 000 |
| Khayyam | 365 + 8/33 | = 365. 24 24 24 24 |
| Revised Julian | 365 + 218/900 | = 365. 24 22 22 22 |
| von Mädler | 365 + 31/128 | = 365. 242 187 500 |
| Mean tropical year at epoch J2010.0 | = 365. 242 189 054 433 974 | |
The currently widely-used Gregorian calendar has an average year of:
However, according to von Mädler, if the years
3200, 6400, 9600, 12,800,
16,000, and so on are NOT
leap years (an additional rule to the ones already in use with the
Gregorian calendar),
the duration of the mean Gregorian
year will then be 365.2421875 days,
and
this
approaches
very
close
the
real
duration
of the mean tropical
year which lasts 365.24219 days.
This
additional
rule
will
produce
a
very
little
error of only
3.125 - 3.100 = 0.025 days in 10,000 years.
I had figured out a better system than the Gregorian-von Mädler one for leap year
decisions, but I plan on using it only if the Cesidian
calendar becomes adopted worldwide.
HMRD Cesidio Tallini