Advanced Calendrics for a New Epoch

It should be noted that the mean tropical year for the epoch of J2010.0 is about 365.24219 days, or more precisely the deliberately generated 18-digit figure of:

http://www.wolframalpha.com/input/?i=N[%28365.2421896698-0.00000615359+0.1-7.29E-10+0.1^2%2B2.64E-10+0.1^3%29%2C35]

The value of 365.242189054433974 SI days is, in turn, extremely close to the value of this equation,

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 Mean year in days 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:

365 + 97/400 = 365.2425 days

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