Dear Helios and Calendar People
I've named the mean period of time each pentalunex occurs later than the previous pentalunex, an eclimpiad, because it is nearly 4 years. I have previously calculated this as follows. Mean eclimpiad = m/( 6*m/e - 1), where e is the mean eclipse season and m the mean synodic month. For a mean synodic month of 29.530588 days and a mean eclipse season interval of 173.31004 days, we get: mean eclimpiad = 44.7426.... months = 1321.276... days = 3.61753... Gregorian years = 7.62377... eclipse seasons. Helios came up with the approximation of 123/34 years from a continued fraction method applied to an unknown value. This works out as 3.617647... years. Given it has 1321.276 days this implies a year of 365.23077.. days, which is somewhat short for this value of the eclimpiad. Helios gives the 4551-year cycle as an example that uses 123/34, it has 56288/1258 = 44.7440... months per eclimpiad. For a mean month of 29.530588 days this works out as 1321.31775... days and so the mean year is then 365.24231... days, which is accurate. I now do calculations for some of the lunisolar eclipse cycles I recently listed: 372 years, 4601 months, 784 seasons, 103 eclimpiads; Eclimpiad = 3.61165... years 391 years, 4836 months, 824 seasons, 108 eclimpiads; Eclimpiad = 3.62037... years 1154 years, 14273 months, 2432 seasons, 319 eclimpiads; Eclimpiad = 3.61755... years 4160 years, 51452 months, 8767 seasons, 1150 eclimpiads; Eclimpiad = 3.61739... years 4551 years, 56288 months, 9591 seasons, 1258 eclimpiads; Eclimpiad = 3.61765... years I see that while 4551 has 123/34, 4160 has 416/115. The 416/115 is a mix of three 123/34 and one 47/13. In an earlier note, I worked out that by using the approximation Years = 18.03*S + 28.945*I, where S is the number of Saros cycles and I the number of Inex cycles. That a Saros has a mean eclimpiad of 3.606 years and an Inex had a mean eclimpiad of 3.618125 years. This led me to find simpler approximations, 65/18 and 47/13, both have a mean eclimpiad less than 123/34. There is no larger one with a smaller denominator less than 76/21 = 3.619..., which exceeds the Saros-Inex range reckoned above. Karl 16(04(01 -----Original Message----- From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios Sent: 03 December 2015 10:33 To: [hidden email] Subject: Year/Pentalunex Ratio Dear Calendar People, A number I've meant to calculate is the Year/ Pentalunex Ratio. For example, 19 / 5 = 3.8 years. After inspecting the recent "accurate eclipse" list, I approximated this ratio and looked at the continued fraction make-up. It turns out that there's really no significant improvement better than 123/34. There are 34 Pentalunex every 123 years. This is within one of the listed; 4551 years = 1662218 days = 56288 months = 9591 E-seasons total Pentalunexs = 1258 = ( 34/123 )*4551 P = 6*E - M -- View this message in context: http://calndr-l.10958.n7.nabble.com/Year-Pentalunex-Ratio-tp16360.html Sent from the Calndr-L mailing list archive at Nabble.com. |
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