Accurate Luni-Solar Eclipse Cycles

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Accurate Luni-Solar Eclipse Cycles

Helios
Dear Calendar People,

Here is a collection of some accurate luni-solar eclipse cycles. Any of them could serve to parameterize a calendar without fault. There is more play left in mixing these cycles and the list could have more members.

--------------------------------------------------------------------------------
1154 years = 421490 days = 14273 months = 2432 seasons
4160 years = 1519408 days = 51452 months = 8767 seasons
4551 years = 1662218 days = 56288 months = 9591 seasons
5314 years = 1940898 days = 65725 months = 11199 seasons
5705 years = 2083708 days = 70561 months = 12023 seasons
6468 years = 2362388 days = 79998 months = 13631 seasons
7101 years = 2593583 days = 87827 months = 14965 seasons
8255 years = 3015073 days = 102100 months = 17397 seasons
8711 years = 3181626 days = 107740 months = 18358 seasons
9865 years = 3603116 days = 122013 months = 20790 seasons
--------------------------------------------------------------------------------

Handy to know are; Saros = 358*E - 61*M, Inex = 38*M - 223*E

Also, I have had success beginning calenders on Julian date = 97.5
Here we can test a few cycles from the list

97.5 + 2083708 = 2083805.5
February 24, 993
97.5 + 2362388 = 2362485.5
March 1, 1756
97.5 + 2593583 = 2593680.5
February 26, 2389
which are all eclipses.
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Re: Accurate Luni-Solar Eclipse Cycles

Irv Bromberg
A problem with such long cycles is that the saros won't remain constant for so long due to secular changes in the underlying lunar month lengths, so that limits the long-term usefulness of calendars based upon such cycles.

-- Irv Bromberg, University of Toronto, Canada

http://www.sym454.org/lunar/


From: East Carolina University Calendar discussion List [[hidden email]] on behalf of Helios [[hidden email]]
Sent: Friday, October 02, 2015 10:51

Here is a collection of some accurate luni-solar eclipse cycles. Any of them
could serve to parameterize a calendar without fault. There is more play
left in mixing these cycles and the list could have more members.

1154 years = 421490 days = 14273 months = 2432 seasons
4160 years = 1519408 days = 51452 months = 8767 seasons
4551 years = 1662218 days = 56288 months = 9591 seasons
5314 years = 1940898 days = 65725 months = 11199 seasons
5705 years = 2083708 days = 70561 months = 12023 seasons
6468 years = 2362388 days = 79998 months = 13631 seasons
7101 years = 2593583 days = 87827 months = 14965 seasons
8255 years = 3015073 days = 102100 months = 17397 seasons
8711 years = 3181626 days = 107740 months = 18358 seasons
9865 years = 3603116 days = 122013 months = 20790 seasons

Handy to know are; Saros = 358*E - 61*M, Inex = 38*M - 223*E

Also, I have had success beginning calenders on Julian date = 97.5
Here we can test a few cycles from the list

97.5 + 2083708 = 2083805.5
February 24, 993
97.5 + 2362388 = 2362485.5
March 1, 1756
97.5 + 2593583 = 2593680.5
February 26, 2389
which are all eclipses.
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Re: Accurate Luni-Solar Eclipse Cycles

Karl Palmen
In reply to this post by Helios
Dear Helios, Irv and Calendar People

I mentioned an approximation of the number of eclipse seasons in a lunisolar calendar cycle. It gives the correct number of eclipse cycles in the Metonic cycle, Gregoriana & Grattan-Guinness cycles as well as the unnamed(353).
I expect Helios's examples have slightly fewer eclipse cycles than this approximation.

The approximation can be expressed as 200*Years - 16*Months.

I put in [] this approximation less the number of eclipse seasons after each of Helios's cycles.


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 02 October 2015 15:51
To: [hidden email]
Subject: Accurate Luni-Solar Eclipse Cycles

Dear Calendar People,

Here is a collection of some accurate luni-solar eclipse cycles. Any of them
could serve to parameterize a calendar without fault. There is more play
left in mixing these cycles and the list could have more members.

--------------------------------------------------------------------------------
1154 years = 421490 days = 14273 months = 2432 seasons [0]
4160 years = 1519408 days = 51452 months = 8767 seasons [1]
4551 years = 1662218 days = 56288 months = 9591 seasons [1]
5314 years = 1940898 days = 65725 months = 11199 seasons [1]
5705 years = 2083708 days = 70561 months = 12023 seasons [1]
6468 years = 2362388 days = 79998 months = 13631 seasons [1]
7101 years = 2593583 days = 87827 months = 14965 seasons [3]
8255 years = 3015073 days = 102100 months = 17397 seasons [3]
8711 years = 3181626 days = 107740 months = 18358 seasons [2]
9865 years = 3603116 days = 122013 months = 20790 seasons [2]
--------------------------------------------------------------------------------

Handy to know are; Saros = 358*E - 61*M, Inex = 38*M - 223*E

Also, I have had success beginning calenders on Julian date = 97.5
Here we can test a few cycles from the list

97.5 + 2083708 = 2083805.5
February 24, 993
97.5 + 2362388 = 2362485.5
March 1, 1756
97.5 + 2593583 = 2593680.5
February 26, 2389
which are all eclipses.

KARL CONTINUES:
Since the approximation always gives a number of eclipse seasons divisible by 8 and the differences (in []) are less than 8 for a cycle less the 10,000 years, one can work out the difference from the remainder of dividing the number of eclipse seasons by 8 ( 0[1] 7[1] 6[2] 5[3]).

One thing that surprises me is that the cycles with [3] are shorter than the cycles with [2].

I now look at the differences between the cycles to gain insight of the composition of these cycle from shorter less accurate cycles.
Let 'Gr' refer to the Gregoriana and 'GG' refer to the Grattan-Guinness cycle. Then we have
--------------------------------------------------------------------------------
1154 years = 421490 days = 14273 months = 2432 seasons [0] Gr+2GG
4160 years = 1519408 days = 51452 months = 8767 seasons [1] 4160
4551 years = 1662218 days = 56288 months = 9591 seasons [1] 4160+GG
5314 years = 1940898 days = 65725 months = 11199 seasons [1] 4160+Gr+2GG
5705 years = 2083708 days = 70561 months = 12023 seasons [1] 4160+Gr+3GG
6468 years = 2362388 days = 79998 months = 13631 seasons [1] 4160+2Gr+4GG
7101 years = 2593583 days = 87827 months = 14965 seasons [3] 7101
8255 years = 3015073 days = 102100 months = 17397 seasons [3] 7101+Gr+2GG
8711 years = 3181626 days = 107740 months = 18358 seasons [2] 2*4160+GG
9865 years = 3603116 days = 122013 months = 20790 seasons [2] 2*4160+Gr+3GG
--------------------------------------------------------------------------------

Also the numbers of days in both the Gregoriana and the Grattan-Guinness cycle are divisible by 10. This is why the last digit of the number of days is the same for cycles of the same [] number ( [0] 0, [1] 8, [2] 6, [3] 3).

Helios has not given the mean years, months and eclipse seasons of these cycles. I work them out for the two that have the dodgiest difference of 633 years [2], both of which Helios has 'tested'.
6468-years 365.24242424 29.5305882 173.309955 (= two 3234-year cycles of 98 33-year cycles)
7101-years 365.24193775 29.5305885 173.309923

This indicates that the cycles are excessively accurate to today's values of the mean month and mean eclipse season and most of the accuracy would be lost over one cycle. It would be better to select shorter cycles that are a little less accurate. All shorter cycles would need to follow my approximation to be reasonably accurate. i.e.,

Eclipse seasons = 200*Years - 16*Months

This leads to the number of months per year being a little high, for accurate eclipse cycles.


Helios has found four independent cycles, none of which can be constructed from the others and so all accurate cycles can be constructed from the four. The shortest four independent cycles are:

1154 years = 421490 days = 14273 months = 2432 seasons [0] Gr+2GG
4160 years = 1519408 days = 51452 months = 8767 seasons [1] 4160
4551 years = 1662218 days = 56288 months = 9591 seasons [1] 4160+GG
7101 years = 2593583 days = 87827 months = 14965 seasons [3] 7101

For shorter cycles, the following three are independent
372 years = 135870 days = 4601 months = 784 seasons [0] Gr
391 years = 142810 days = 4836 months = 824 seasons [0] GG
706 years = 257861 days = 8732 months = 1488 seasons [0] 4Gr-2GG+1day
A 4th independent cycle would not follow the approximation [0], so would not accurate number of eclipse seasons if short.

Karl


15(04(22


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Re: Accurate Luni-Solar Eclipse Cycles

Helios
In reply to this post by Irv Bromberg
Dear Irv & calendar people,

From the nasa Five Millennium Catalog of Solar Eclipses, I am only able to confirm the 4551 cycle, so there's no problem up to this point. When does a calendar cycle become too long? We can describe the cause of the defect, and this is interesting, but we still don't know the beginning or meaning of too long. I don't discount any of these cycles on account of these gradual effects.
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Re: Accurate Luni-Solar Eclipse Cycles

Helios
Dear Karl, Irv, and Calendar People,
With the Nasa atlases, it's not hard to find the 4551 cycle. Here a whole season is shown between -1931 and 2620.


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Re: Accurate Luni-Solar Eclipse Cycles

Karl Palmen
Dear Helios and Calendar People

Are there many examples besides -1931 and 2620?

I think this can be estimated by looking at the Saros and Inex numbers of these two eclipses and other eclipses around their times.
If both occupy about the same position in the range of Saros and Inex numbers of their respective times, then there'll be many examples and you'll have a good cycle (for at least one cycle).

I note the absence of cycles between the length of 1154 and 4551 years arises from the need of the number of eclipse seasons to differ from my approximation by a whole number. The difference is 0 for 1154 and 1 for 4551.

Karl

15(04(24

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 07 October 2015 10:18
To: [hidden email]
Subject: Re: Accurate Luni-Solar Eclipse Cycles

Dear Karl, Irv, and Calendar People,
With the Nasa atlases, it's not hard to find the 4551 cycle. Here a whole
season is shown between -1931 and 2620.


<http://calndr-l.10958.n7.nabble.com/file/n16216/es4551.jpg>



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Re: Accurate Luni-Solar Eclipse Cycles

Karl Palmen
In reply to this post by Karl Palmen
Dear Helios and Calendar People

I was looking for this thread, when Helios first mentioned his 214-year & 242-year cycles.
In the course of this note, I suggested building eclipse cycles from the cycles of 372, 391, 4160, & 7101 years and show how all of Helios's examples are made from them. Here there are four of them because the number of days is also taken into account.

I believe the number I put in [] is also equal to the nodetide difference.


Karl

16(03(01

-----Original Message-----
From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 05 October 2015 13:15
To: 'East Carolina University Calendar discussion List'
Subject: RE: Accurate Luni-Solar Eclipse Cycles

Dear Helios, Irv and Calendar People

I mentioned an approximation of the number of eclipse seasons in a lunisolar calendar cycle. It gives the correct number of eclipse cycles in the Metonic cycle, Gregoriana & Grattan-Guinness cycles as well as the unnamed(353).
I expect Helios's examples have slightly fewer eclipse cycles than this approximation.

The approximation can be expressed as 200*Years - 16*Months.

I put in [] this approximation less the number of eclipse seasons after each of Helios's cycles.


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 02 October 2015 15:51
To: [hidden email]
Subject: Accurate Luni-Solar Eclipse Cycles

Dear Calendar People,

Here is a collection of some accurate luni-solar eclipse cycles. Any of them
could serve to parameterize a calendar without fault. There is more play
left in mixing these cycles and the list could have more members.

--------------------------------------------------------------------------------
1154 years = 421490 days = 14273 months = 2432 seasons [0]
4160 years = 1519408 days = 51452 months = 8767 seasons [1]
4551 years = 1662218 days = 56288 months = 9591 seasons [1]
5314 years = 1940898 days = 65725 months = 11199 seasons [1]
5705 years = 2083708 days = 70561 months = 12023 seasons [1]
6468 years = 2362388 days = 79998 months = 13631 seasons [1]
7101 years = 2593583 days = 87827 months = 14965 seasons [3]
8255 years = 3015073 days = 102100 months = 17397 seasons [3]
8711 years = 3181626 days = 107740 months = 18358 seasons [2]
9865 years = 3603116 days = 122013 months = 20790 seasons [2]
--------------------------------------------------------------------------------

Handy to know are; Saros = 358*E - 61*M, Inex = 38*M - 223*E

Also, I have had success beginning calenders on Julian date = 97.5
Here we can test a few cycles from the list

97.5 + 2083708 = 2083805.5
February 24, 993
97.5 + 2362388 = 2362485.5
March 1, 1756
97.5 + 2593583 = 2593680.5
February 26, 2389
which are all eclipses.

KARL CONTINUES:
Since the approximation always gives a number of eclipse seasons divisible by 8 and the differences (in []) are less than 8 for a cycle less the 10,000 years, one can work out the difference from the remainder of dividing the number of eclipse seasons by 8 ( 0[1] 7[1] 6[2] 5[3]).

One thing that surprises me is that the cycles with [3] are shorter than the cycles with [2].

I now look at the differences between the cycles to gain insight of the composition of these cycle from shorter less accurate cycles.
Let 'Gr' refer to the Gregoriana and 'GG' refer to the Grattan-Guinness cycle. Then we have
--------------------------------------------------------------------------------
1154 years = 421490 days = 14273 months = 2432 seasons [0] Gr+2GG
4160 years = 1519408 days = 51452 months = 8767 seasons [1] 4160
4551 years = 1662218 days = 56288 months = 9591 seasons [1] 4160+GG
5314 years = 1940898 days = 65725 months = 11199 seasons [1] 4160+Gr+2GG
5705 years = 2083708 days = 70561 months = 12023 seasons [1] 4160+Gr+3GG
6468 years = 2362388 days = 79998 months = 13631 seasons [1] 4160+2Gr+4GG
7101 years = 2593583 days = 87827 months = 14965 seasons [3] 7101
8255 years = 3015073 days = 102100 months = 17397 seasons [3] 7101+Gr+2GG
8711 years = 3181626 days = 107740 months = 18358 seasons [2] 2*4160+GG
9865 years = 3603116 days = 122013 months = 20790 seasons [2] 2*4160+Gr+3GG
--------------------------------------------------------------------------------

Also the numbers of days in both the Gregoriana and the Grattan-Guinness cycle are divisible by 10. This is why the last digit of the number of days is the same for cycles of the same [] number ( [0] 0, [1] 8, [2] 6, [3] 3).

Helios has not given the mean years, months and eclipse seasons of these cycles. I work them out for the two that have the dodgiest difference of 633 years [2], both of which Helios has 'tested'.
6468-years 365.24242424 29.5305882 173.309955 (= two 3234-year cycles of 98 33-year cycles)
7101-years 365.24193775 29.5305885 173.309923

This indicates that the cycles are excessively accurate to today's values of the mean month and mean eclipse season and most of the accuracy would be lost over one cycle. It would be better to select shorter cycles that are a little less accurate. All shorter cycles would need to follow my approximation to be reasonably accurate. i.e.,

Eclipse seasons = 200*Years - 16*Months

This leads to the number of months per year being a little high, for accurate eclipse cycles.


Helios has found four independent cycles, none of which can be constructed from the others and so all accurate cycles can be constructed from the four. The shortest four independent cycles are:

1154 years = 421490 days = 14273 months = 2432 seasons [0] Gr+2GG
4160 years = 1519408 days = 51452 months = 8767 seasons [1] 4160
4551 years = 1662218 days = 56288 months = 9591 seasons [1] 4160+GG
7101 years = 2593583 days = 87827 months = 14965 seasons [3] 7101

For shorter cycles, the following three are independent
372 years = 135870 days = 4601 months = 784 seasons [0] Gr
391 years = 142810 days = 4836 months = 824 seasons [0] GG
706 years = 257861 days = 8732 months = 1488 seasons [0] 4Gr-2GG+1day
A 4th independent cycle would not follow the approximation [0], so would not accurate number of eclipse seasons if short.

Karl


15(04(22


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FW: Accurate Luni-Solar Eclipse Cycles

Karl Palmen
In reply to this post by Helios
Dear Helios and Calendar People

In October last year, Helios produced a list of accurate lunisolar eclipse cycles.
Yesterday I produced another list based on Helios's 698-year cycle of 75 nodetides and on correcting the Metonic cycle twice every 698-year cycle by 1/19 month.

0(0), 19(2),
372(40), 391(42),
763(82), 782(84),
1135(122), 1154(124), 1173(126),
1526(164), 1545(166), 1564(168),
1917(206),
None with 6, 7 or 8 truncations of Metonic cycle
2987(321), 3006(323), 3025(325),
3378(363), 3397(365), 3416(367),
3750(403), 3769(405),
4141(445), 4160(447), 4179(449),
4532(487), 4551(489), 4570(491),
4923(529), 4942(531),
5295(569), 5314(571), 5333(573),
5686(611), 5705(613), 5724(615),
6077(653), 6096(655).
The number in () is the number of nodetides, which is equal to the number of eclipse seasons less twice the number of years.
Each row corresponds to the number of truncations of the Metonic cycle by one octaeteris and so has numbers 19 apart in it.
In my previous note, I referred to this truncation as a correction and is indeed a correction by 1/19 month.

The following of Helios's cycles occur here:
1154 years = 14273 months = 2432 seasons = 124 nodetides = 3 truncations
4160 years = 51452 months = 8767 seasons = 447 nodetides = 12 truncations
4551 years = 56288 months = 9591 seasons = 489 nodetides = 13 truncations
5705 years = 70561 months = 12023 seasons = 613 nodetides = 16 truncations

The 6468-year cycle does not appear. Its number of years is equal to 186 mod 698 and so does occur in the 698-year cycle.
The number of truncations (235*Years - 19*Months) is 18, which occurs for the range (17.5)*349 to (18.5)*349
= 6108 to 6456. So 6468 is just outside this range.
My method picked 6096 instead of 6468 for year -186 of the 896-year cycle.

My method produces many more examples, but does not take account of the number of days in a cycle and so does not guarantee any cycle is close to a whole number of days. One could do this by counting yerms or yerm eras. For long cycles, this may be futile, because of change in Earth's rotation.


Karl

16(03(24

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 02 October 2015 15:51
To: [hidden email]
Subject: Accurate Luni-Solar Eclipse Cycles

Dear Calendar People,

Here is a collection of some accurate luni-solar eclipse cycles. Any of them
could serve to parameterize a calendar without fault. There is more play
left in mixing these cycles and the list could have more members.

--------------------------------------------------------------------------------
1154 years = 421490 days = 14273 months = 2432 seasons
4160 years = 1519408 days = 51452 months = 8767 seasons
4551 years = 1662218 days = 56288 months = 9591 seasons
5314 years = 1940898 days = 65725 months = 11199 seasons
5705 years = 2083708 days = 70561 months = 12023 seasons
6468 years = 2362388 days = 79998 months = 13631 seasons
7101 years = 2593583 days = 87827 months = 14965 seasons
8255 years = 3015073 days = 102100 months = 17397 seasons
8711 years = 3181626 days = 107740 months = 18358 seasons
9865 years = 3603116 days = 122013 months = 20790 seasons
--------------------------------------------------------------------------------

Handy to know are; Saros = 358*E - 61*M, Inex = 38*M - 223*E

Also, I have had success beginning calenders on Julian date = 97.5
Here we can test a few cycles from the list

97.5 + 2083708 = 2083805.5
February 24, 993
97.5 + 2362388 = 2362485.5
March 1, 1756
97.5 + 2593583 = 2593680.5
February 26, 2389
which are all eclipses.




--
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