Int. J. Communications, Network and System Sciences, 2011, 4, 483-486 doi:10.4236/ijcns.2011.48059 Published Online August 2011 (http://www.SciRP.org/journal/ijcns) Astronomical Algorithms: Amended Multi-millennia Calendar Boris S. Verkhovsky Computer Science Department, New Jersey Institute of Technology, University Heights, Newark, USA E-mail: [email protected], [email protected] Received May 21, 2011; revised June 16, 2011; accepted June 20, 2011 Abstract Three new worldwide calendars are proposed and compared in this paper. None of them requires any depar- ture from an existing tradition to divide years on lean and leap. Although all three are pretty accurate, it is demonstrated that the Julian calendar with one additional amendment is the simplest and the most suitable for implementation. Keywords: Astronomic Algorithm, Julian Calendar, Gregorian Calendar, Error Accumulation, Amendments, Multi-millennia Calendar, Synchronization 1. Introduction and Gregorian Calendar land by an Act of Parliament. French Republican calen- dar was adopted in 1790s. Although Gregorian calendar Modern computing and communication systems require was adopted in 1923 in the USSR, the Russian Orthodox immensely high degree of synchronization, which in its Church does not recognize it and still celebrates Christ- turn requires measurement of time with great precision. mas thirteen days later. More details about various cal- Nowadays, atomic clock can measure time with accuracy endars are provided in [1-4]. Great mathematician and smaller than one billionth of a second. Without such a astronomer Carl F. Gauss proposed an algorithm that precision numerous navigation systems including the calculates Easter Sundays in the Gregorian calendar [5]. GPS and other equivalents will not be able to properly The Gregorian calendar is proposed by astronomers C. operate. Clavius and L. Lilio. It is basically a Julian calendar with Yet, modern calendars are highly inaccurate. Human- two amendments, {historic details are provided below}. kind has used and is currently using many calendars: It is reasonable to assume that when the Gregorian Egyptian Calendar was introduced in the V Millennium calendar has been introduced in XVI century, one of BCE; the first year of Jewish calendar is 3760 BCE; considerations was that such amendments would be easy Mayan chronology started from 3372 BCE. Six hundred to implement. Indeed, even with the generally low level years later (in 2772 BCE) Egypt adopted a calendar of of education at that time, there was no difficulty to rec- 365 days without adjustments. Babylonian and Chinese ognize what year was divisible by 100 and what was astronomers learned about planetary movements in VIII divisible by 400. With this pattern, a modified Julian century BCE. Julian calendar with leap years was calendar requires not two, but either three additional adopted in 46 BCE, although the idea has been proposed amendments {as demonstrated in (10)-(14)} or five addi- 193 years earlier by Aristarchus from Alexandria. The tional amendments {as demonstrated in (17)-(19)}. average length of a year in the Julian calendar is 365.25 The existing Western calendar basically fluctuates days. This is significantly different from the “real” length between lean and leap years. A year is leap if it is divisi- of the solar year. An error accumulates so fast that after ble by four. However, a year ending with 00 is leap only about 131 years the calendar is out of sync by one day. if it is divisible by 400. For instance, 1900 was lean year, By the 16th Century this affected the determination of but 1600 and 2000 were leap years. the date of Easter. Let y be the y-th year of New Era. In order to find how Gregorian calendar was introduced in 1582 initially in many days Ny are in the y-th year, we use formula France and the Netherlands, and 170 year later in Eng- specified by Gregorian calendar: Copyright © 2011 SciRes. IJCNS 484 B. VERKHOVSKY Ny365f ya , f yb , f yab , , (1) the discrepancy between the number of rotations around Sun and the corresponding number of days computed on where zaba,; 4,100 b ; (2) the basis of Gregorian calendar. and let The first ten millennia: For y = 10,000 1, ifyz mod 0 D10000 = 3,652,422. (9) fyz,: . (3) Yet, from (4) 0, ifyz mod 0 G 10000 = 3,650,000 + 2500 100 + 25 = 3,652,425. Therefore, Equations (1)-(3) specify that leap years must be added every four years with an exception of Hence, a three-day discrepancy will accumulate by the those years that are divisible by 100, but not divisible by year y = 10,000 if the Gregorian calendar is used. 400. Assuming that P1 = 365 and P2 = 365.25, Formula (1) takes into account that the astronomic year, i.e., the 4. Astronomic Algorithm with the Fourth period of rotation of the Earth around Sun is Amendment P3 = 365.242 days [6]. For further verification, we also calculate the number In the past, three amendments were introduced into the of days Gy at the end of the y-th year from the be- World calendar: ginning of the New Era. 1) The Egyptians introduced 365 days instead of 360 Then days (the 1st Amendment); 2) 365.25 days were introduced by Julian calendar (the Gy 365 yyy4 100 y 400. (4) 2nd Amendment); th For instance, at the end of the 100 year 3) Further corrections were introduced by Gregorian th G 100 36,500 25 1 36,524; at the end of the 400 calendar (the 3rd Amendment), {see (1)-(3)}. year G400 36,500 400 100 4 1 144,097; and However, even these amendments do not provide an at the end of the First Millennium accurate account for a large period of time measured in G 1000 365,000 250 10 2 365,242 . Yet, as it millennia. To make the counting of the days more accu- is demonstrated below, Gregorian calendar is correct for rate, an additional (fourth) term should be introduced. a relatively small number of centuries. Step1.1: if 3200 divides y 2. Average Astronomical Year then y-th year is lean; stop; Step2.1: if 400 divides y then y-th year is leap; stop; An average astronomical year is defined as a period with Step3.1: if 100 divides y which the Earth rotates around Sun. The time from one then y-th year is lean; stop; fixed point, such as an equinox, to the next is called a Step4.1: if 4 divides y tropical year. Its current length is 365.242190 days, but then y-th year is leap; stop; it fluctuates. In 1900 the period was 365.242196 days, else y-th year is lean; stop. (10) and in 2100 it will be 365.242184 days. Thus, to get more accurate results, we must consider that in average The 4-th amendment provides a more accurate count- ing of the days for dozens of millennia ahead. From the P = 365.24219, [7,8]. (5) fourth amendment in the astronomic algorithm (10) it follows that the 3200-th year, 6400-th year, 9600-th year 3. Inaccuracy in Gregorian Calendar etc. must be the lean years. As a result, The first five millennia: Let Vy 365 y y 4 y 100 y 400 y 3200 Dy: yP (6) (11) Therefore, using P, we derive by direct computation Finally, let z 4, 100, 400, 3200 ; (12) that for y = 5000 and let D5000= 1,826,211. 1, ifyz mod 0 Yet, by Formula (4) for Gregorian calendar fyz,: (13) 0, ifyz mod 0. G 5000365 5000 1250 50 1 2 1,826,212 . (7) Then Thus, G 5000 D5000= 1. (8) M yfy365 ,4 fy,100 fy , 400 fy ,3200 Hence, as can be seen in (8), even the more accurate period of rotation P 365.24219 does not eliminate (14) Copyright © 2011 SciRes. IJCNS B. VERKHOVSKY 485 However, there is a simpler solution with only one 7. Alternative Algorithm for Calendar amendment to Julian calendar. Step1.3: If 100,000 divides y 5. Julian Calendar with One Amendment then y-th year is lean; stop; Step2.3: if 5000 divides y + a Step1.2: if 128 divides y then y-th year is lean; stop; then y-th year is lean; stop; Step3.3: if 2000 divides y Step2.2: if 4 divides y then y-th year is lean; stop; then y-th year is leap Step4.3: if 400 divides y else y-th year is lean; stop. (15) then y-th year is leap; stop; Mnemonic rule: After thirty one leap years skip one. Step5.3: if 100 divides y then y-th year is lean; stop; 6. Discrepancy Analysis Step6.3: if 4 divides y then y-th year is leap else y-th year is lean; stop. (17) Let By365 y y 4 y 128; (16) In this case then By is the number of days at the end of y-th year Ay Gy y2000 y a 5000 y 100000, from the beginning of the New Era if the amended Julian calendar is used. (18) Let’s compute Dy r foy = 2,000; 3,200; 10,000; where a is an even positive integer such that amod 4 0 50,000; and 100,000; and compare them with corre- or a = 100. sponding values for By, Gy and Vy {see Then Table 1}: Sy Ny f y, 2000 From Table 2, it follows that Julian calendar with one additional amendment, or the Gregorian calendar with an fya ,5000 fy ,100000 . (19) additional amendment, provides the same accuracy. Thus, from our point of view, the former one should be imple- 8.