Solar calendar

The solar calendar is based on the duration of the tropical year - 365.24220 days. It immediately shows that the calendar year can contain either 365 or 366 days. The theory should indicate the order of the alternation of simple( in 365 days) and leap years( 366 days) in a certain cycle so that the average duration of a calendar year per cycle is as close as possible to the duration of a tropical year.

Decomposition of the fractional part of the tropical year into a continued fraction has the form

The corresponding fractions have the following values: 1/4;7/29;8/33;31/128;132/545;. ..,

and the average duration of the calendar year is 1) 365.25000, 2) 365.24138, 3) 365.24242, 4) 365.24219,. ..

In the first case, the cycle consists of fouryears, and during this cycle, one insert is made. In other words, out of every four years, three years have 365 days, the fourth 366 days. Such a system of leap-ups existed in the Julian calendar. On average, the duration of such a calendar year is 0.0078 days longer than the duration of the tropical year, and this difference is approximately 24 hours a day.

The cycle of 29 years with 7 leap years has not been used once. The third system of leap years( 8 leap years in 33 years) was developed by the Persian scientist and poet Omar Khayyam( 1048-1123) and formed the basis for the Persian calendar, introduced in 1079 and operating in Iran until the middle of the 19th century. The leap years in this calendar were the 3rd, 7th, 11th, 15th, 20th, 24th, 28th and 32nd cycles. The period of 128 years with 31 leap years was proposed in 1864 by the German astronomer I. Medler( 1794-1874), then a professor at the University of Dorpat( now Tartu).The draft of this calendar, however, was not adopted. Never considered longer cycles.

Two calendar systems were proposed "out of the rules" of suitable fractions solely from the convenience of remembering the order of inserting additional days. Since 1582 the countries of Western Europe, and later many other peoples of the world, have moved to the account of time according to the Gregorian calendar, the draft of which was developed by the Italian scientist Luigi Lilio( 1520-1576).The duration of the calendar year here is equal to 365.24250 days. In accordance with the value of the fractional part of the year, K = 0.2425 = 97/400 in the time interval of 400 years, an additional 366th day of the year is inserted 97 times, ie, in comparison with the Julian calendar here three days in 400 years is thrown out.

The second calendar system is the New Julian calendar, proposed by the Yugoslav astronomer Milutin Milankovic( 1879-1956).In this case, the average duration of the calendar year is 365.24222 or 365 218/900 days. The insertions of an additional 366th day of the year here should be made 218 times every 900 years. This means that in comparison with the Julian calendar in the New-Julian calendar, 7 days are thrown out every 900 years. It is suggested to consider leap years old ones in which the number of hundreds in division by 9 gives in the remainder 2 or 6. The next such years, since 2000, will be 2400, 2900, 3300 and 3800. The average duration of the New-Julian calendar year is longer than the duration of the yeartropical at 0.000022 mean solar days. And this means that the discrepancy for the whole day is such a calendar only for 44,000 years.

The main characteristics of various solar calendar systems are given in Table.

Table. Solar calendar systems

No.	Fraction To	Length of year in days	Errors in 24 hours	Period Accumulation errors in one day	Name of calendar	Author
1	1/4	365,25000	+0,00780	128 years	Julian	Sozigen
2	7 /29	365,24138	-0,00082	1 220 »	-	-
3	97/400	365,24250	+0.00030	3 300»	Gregorian
	Lilio
4	8/33	365,24242	+0.00022	4 500 »	Persian	Lobster
	Hayam
5	218/900	365,24222	+0.00002	43 500»	new-Julian
	M.Mi-
	Lankovic
6	31/128	365,24219	-0,00001	80,000 »	-	I. Medler