Nodetides

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Nodetides

Helios
Dear Calendar People,

There's a basic period between when a node alignment and the solar year nearly coincide. This period averages about 9.3 years. I call this a nodetide.

mean nodetide = 1/[ ( y/e ] - 2 )

Should we subtract from the year two eclipse seasons, the remaining 19 days is the portion of the year within which we can expect a node alignment to occur every nodetide.
A luni-solar eclipse cycle will always contain a whole number of nodetides. For a luni-solar eclipse cycle, we can subtract from the eclipse seasons the number of years doubled to find the number of nodetides in the cycle.

N = E - 2*Y

Beginning from a node alignment at a certain time in year 0, we can predict the following nodetide years. I'm satisfied with the following accumulator function;

( 75*Y + 34 )MOD( 698 ) < 75

9, 19, 28, 37, 47, 56, 65, 74, 84, 93, 102, 112, 121, 130, 140, 149, 158, 168, 177, 186, 195, 205, 214, 223, 233, 242, 251, 261, 270, 279, 289, 298, 307, 316, 326, 335, 344, 354, 363, 372, 391, . . .

The function seems to test whether or not a luni-solar cycle is an eclipse cycle.
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Re: Nodetides

Karl Palmen
Dear Helios and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 27 October 2016 08:43
To: [hidden email]
Subject: Nodetides

Dear Calendar People,

There's a basic period between when a node alignment and the solar year
nearly coincide. This period averages about 9.3 years. I call this a
nodetide.

mean nodetide = 1/[ ( y/e ] - 2 )

KARL REPLIES: The nodetide is half the precession period of nodes, the time it takes for the two nodes to interchange.
The Wikipedia article for 'Lunar node' says:

"The plane of the lunar orbit precesses in space and hence the lunar nodes precess around the ecliptic, completing a revolution (called a draconic or nodal period, the period of nutation) in 6798.3835 days or 18.612958 years"

Therefore the nodetide is 9.306479 years, which Helios has approximated to 9.3 years.


HELIOS CONTINUED:
Should we subtract from the year two eclipse seasons, the remaining 19 days
is the portion of the year within which we can expect a node alignment to
occur every nodetide.
A luni-solar eclipse cycle will always contain a whole number of nodetides.
For a luni-solar eclipse cycle, we can subtract from the eclipse seasons the
number of years doubled to find the number of nodetides in the cycle.

N = E - 2*Y

Beginning from a node alignment at a certain time in year 0, we can predict
the following nodetide years. I'm satisfied with the following accumulator
function;

( 75*Y + 34 )MOD( 698 ) < 75

9, 19, 28, 37, 47, 56, 65, 74, 84, 93, 102, 112, 121, 130, 140, 149, 158,
168, 177, 186, 195, 205, 214, 223, 233, 242, 251, 261, 270, 279, 289, 298,
307, 316, 326, 335, 344, 354, 363, 372, 391, . . .

The function seems to test whether or not a luni-solar cycle is an eclipse
cycle.

KARL REPLIES: The end of Helios's list is interesting:

316, 326, 335, 344, 354, 363, 372, 391, . . .
Compare with lunisolar year/month cycles
315, 334, 353, 372, 391, . . . and one gets a perfect match for 372 & 391 years as expected. 315, 334 & 353 are just 1 year out.

Helios here approximated the nodetide to 698/75 = 9.306666666666... years, which is closer to the value of 9.306479 years got from Wikipedia. I don't understand why Helios put 34 in the accumulator express. I'd expect 37.

One could instead use the approximation 9.3065 years. There is no need for an accumulator expression, one can multiply the integers by 9.3065 and then round to nearest integer and starting the starting at 34, even integers yield:
34 316.421 316
36 335.034 335
38 353.647 354
40 372.260 372
42 390.873 391
These are the same as Helios's figures.

Finally, the differences between 1824 and the years in Helios's most recent magic square appear in Helios's figures:
0, 9, 19, 28, 37, 47, 56,
65, 74, 84, 93, 102, 112, 121,
130, 140, 149, 158, 168, 177, 186,
195, 205, 214, 223, 233, 242, 251,
as the 1st, 3rd and 6th in each row.

Karl

16(02(26
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Re: Nodetides

Helios
Dear Karl and Calendar People,

For longer luni-solar eclipse cycles, at least the even-year ones, it seems that they can be decomposed into quantities of 214-year and 242-year cycles.

214 years = 2647 months = 451 eclipse seasons = 23 nodetides = 13 Inex - 9 Saros
242 years = 2993 months = 510 eclipse seasons = 26 nodetides = 4 Inex + 7 Saros

We can either combine the cycles aiming for an accurate luni-solar cycle and end up with an accurate eclipse cycle, or conversely, combine for an eclipse cycle and end up with an accurate luni-solar cycle.

We can decompose the 214 year cycle into 3 and 5 year luni-solar cycles, and the number of 5 year cycles is equal to the number of nodetides. Furthermore, we can decompose the 242 year cycle into heptons and octons and the number of heptons is equal to the number of nodetides.
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Re: Nodetides

Karl Palmen
Dear Helios and Calendar People

I don't think the 214 year and 242 year cycles are a good idea, because they'd miss out many accurate lunisolar cycles that are also eclipse cycles, starting with the 372-year & 391-year cycles.

One needs components that are good lunisolar cycles as well as good eclipse cycles, and to cover all cases one needs three of them.

One could use:
19 years = 235 months = 2 nodetides (10*I - 15*S)
353 years = 4366 months = 38 nodetides  (26*S - 4*I)
and perhaps
4160 years = 51452 months = 447 nodetides (135*I + 14*S)

Helios presented the 4160-year cycle to this list on 6 August 2015.


The first two cycles
19 years = 235 months = 2 nodetides (10*I - 15*S)
353 years = 4366 months = 38 nodetides  (26*S - 4*I)
have exactly 2 nodetides to each Metonic cycle, include any truncated to 11 years, and so the truncation corrects the mean nodetide from 9.5 years to around 9.3 years.

The third cycle
4160 years = 51452 months = 447 nodetides (135*I + 14*S)
has 224 Metonic cycles including 12 that are truncated to 11 years and so the number (447) of nodetides is one less than twice the number (224) of Metonic cycles including truncated.

If some cycle other that 4160 years is chosen as the third cycle, it must also have one less nodetide than twice the number of Metonic cycles including truncated.

If a cycle has Y years and M months, the number of Metonic cycles including those truncated to 11 years is
99*Y - 8*M.

Karl

16(02(27


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 28 October 2016 08:26
To: [hidden email]
Subject: Re: Nodetides

Dear Karl and Calendar People,

For longer luni-solar eclipse cycles, at least the even-year ones, it seems
that they can be decomposed into quantities of 214-year and 242-year cycles.

214 years = 2647 months = 451 eclipse seasons = 23 nodetides = 13 Inex - 9
Saros
242 years = 2993 months = 510 eclipse seasons = 26 nodetides = 4 Inex + 7
Saros

We can either combine the cycles aiming for an accurate luni-solar cycle and
end up with an accurate eclipse cycle, or conversely, combine for an eclipse
cycle and end up with an accurate luni-solar cycle.

We can decompose the 214 year cycle into 3 and 5 year luni-solar cycles, and
the number of 5 year cycles is equal to the number of nodetides.
Furthermore, we can decompose the 242 year cycle into heptons and octons and
the number of heptons is equal to the number of nodetides.




--
View this message in context: http://calndr-l.10958.n7.nabble.com/Nodetides-tp17237p17240.html
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Re: Nodetides

Irv Bromberg
Irv asks:

Doesn't the 1803-year cycle, which is an excellent lunisolar cycle (with 664 leap months) and equal to 100 saros periods (22300 mean synodic months), hence also an excellent eclipse cycle, belong in this list?

Using the 9.306479 years that Karl got from Wikipedia, this cycle contains very nearly 286 nodetides. Since it is a multiple of an excellent eclipse cycle (the saros), we could reverse the calculation and estimate the nodetide interval as 1803/286 = about 6.3041958 (actually the last 6 decimal points repeat) years.

Is the figure from Wikipedia in terms of mean solar years, or northward equinoctial years, or Julian years?

--- Irv Bromberg, Toronto, Canada



From: East Carolina University Calendar discussion List [[hidden email]] on behalf of Karl Palmen [[hidden email]]
Sent: Friday, October 28, 2016 08:11
To: [hidden email]
Subject: Re: Nodetides

Dear Helios and Calendar People

I don't think the 214 year and 242 year cycles are a good idea, because they'd miss out many accurate lunisolar cycles that are also eclipse cycles, starting with the 372-year & 391-year cycles.

One needs components that are good lunisolar cycles as well as good eclipse cycles, and to cover all cases one needs three of them.

One could use:
19 years = 235 months = 2 nodetides (10*I - 15*S)
353 years = 4366 months = 38 nodetides (26*S - 4*I)
and perhaps
4160 years = 51452 months = 447 nodetides (135*I + 14*S)

Helios presented the 4160-year cycle to this list on 6 August 2015.


The first two cycles
19 years = 235 months = 2 nodetides (10*I - 15*S)
353 years = 4366 months = 38 nodetides (26*S - 4*I)
have exactly 2 nodetides to each Metonic cycle, include any truncated to 11 years, and so the truncation corrects the mean nodetide from 9.5 years to around 9.3 years.

The third cycle
4160 years = 51452 months = 447 nodetides (135*I + 14*S)
has 224 Metonic cycles including 12 that are truncated to 11 years and so the number (447) of nodetides is one less than twice the number (224) of Metonic cycles including truncated.

If some cycle other that 4160 years is chosen as the third cycle, it must also have one less nodetide than twice the number of Metonic cycles including truncated.

If a cycle has Y years and M months, the number of Metonic cycles including those truncated to 11 years is
99*Y - 8*M.

Karl

16(02(27


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 28 October 2016 08:26
To: [hidden email]
Subject: Re: Nodetides

Dear Karl and Calendar People,

For longer luni-solar eclipse cycles, at least the even-year ones, it seems
that they can be decomposed into quantities of 214-year and 242-year cycles.

214 years = 2647 months = 451 eclipse seasons = 23 nodetides = 13 Inex - 9
Saros
242 years = 2993 months = 510 eclipse seasons = 26 nodetides = 4 Inex + 7
Saros

We can either combine the cycles aiming for an accurate luni-solar cycle and
end up with an accurate eclipse cycle, or conversely, combine for an eclipse
cycle and end up with an accurate luni-solar cycle.

We can decompose the 214 year cycle into 3 and 5 year luni-solar cycles, and
the number of 5 year cycles is equal to the number of nodetides.
Furthermore, we can decompose the 242 year cycle into heptons and octons and
the number of heptons is equal to the number of nodetides.
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Re: Nodetides

Helios
Dear Irv and Calendar People,

I calculate that this cycle contains:
3799.7345 eclipse seasons
193.7345 nodetides
and therefore is not an eclipse cycle.

The value n = 9.306479 years for a nodetide would indicate that the year equals:
y = 365.24259 days
An eclipse season equals 173.310037942 days.
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Re: Nodetides

Helios
Dear Karl and Calendar People,

I found that 214y and 242y compose the following mixers:
1154 years = 2*214 + 3*242 = 38*I + 3*S
1852 years = 3*214 + 5*242 = 59*I + 8*S

The 1154y is a rather long year but a very accurate eclipse. The 1852y is a short year. Here are some mixes that include some previously mentioned cycles:

1852, 3006, 4160, 5314, 6468, 7622, 8776, 9930

Note that 214 + 242 = 456 = 19*24, or 24 Metons = 17*I - 2*S = 470 12-month years
Still, more cycles remain to be found, including odd-year cycles.
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Re: Nodetides

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

Irv erroneously reckoned 6.3... years to a nodetide instead of 9.3... years.

Nevertheless, this does highlight a weakness of my mixer cycles of 19, 353 & 4160 years. The 1803-year cycle can be made from them 5*353+2*19 with 5*38+2*2=194 nodetides and so has mean nodetide of 9.2938... years, which as shown by Helios, it is not quite accurate enough to make an eclipse cycle.

So 19, 353, & 4160 years need remixing make accurate eclipse cycles. Helios has done this with his 214 & 242 to get 1154 & 1852. A third mixer is needed to ensure that the number of nodetides can be regulated independently of the number of years and months.

Karl

16(03(01

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 31 October 2016 00:15
To: [hidden email]
Subject: Re: Nodetides

Dear Irv and Calendar People,

I calculate that this cycle contains:
3799.7345 eclipse seasons
193.7345 nodetides
and therefore is not an eclipse cycle.

The value n = 9.306479 years for a nodetide would indicate that the year
equals:
y = 365.24259 days
An eclipse season equals 173.310037942 days.




--
View this message in context: http://calndr-l.10958.n7.nabble.com/Nodetides-tp17237p17243.html
Sent from the Calndr-L mailing list archive at Nabble.com.
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Re: Nodetides

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

I check the accuracy of Helios's mixers as lunisolar cycles.

Using year/month ratio of 12.3683 I get
14273.0182 months in 1154 years and
22906.0916 months in 1852 years.

So we get year/month ratios of
14273/1154 = 12.3682842...
22906/1852 = 12.3682505...
For comparison
3896/315 = 12.368254...
4131/334 = 12.368263...
4366/353 = 12.368272... (unnamed 353)
4601/372 = 12.368280... (Gregoriana)

Because the number of years, months and nodetides need regulating two mixers are not enough, one needs three.

I suggested 19-years, 353-years and 4160-years for this, but the mix of these three has to be right to produce an eclipse cycle, but no eclipse cycle that is a lunisolar cycle should be missed out by these. Therefore these three need remixing like Helios has done with his 214 & 242 to ensure eclipse cycles.

I've found out that the number of nodetides in such a cycle is very near twice the number of Metonic cycles including Metonic cycles truncated to 11 years by removing an octaeteris to correct a long run of Metonic cycles (sadly less than 24). I'll call twice the number of Metonic cycles including truncated minus the number of nodetides, the nodetide difference.
One could then work with nodetide differences instead of counting nodetides.

Helios has not given the number of nodetides in his 1154-year and 1852-year cycles or even the number of eclipse seasons. I'll work them out from the 23 and 26 Helios gave to the 214-year and 242-year cycles.

1154 has 2*23 + 3*26 = 124 nodetides
1852 has 3*23 + 5*26 = 199 nodetides

1154 has 99*1154 - 8*14273 = 62 Metonic cycles including truncated and the number of nodetides is exactly twice this.
1852 has 99*1852 - 8*22906 = 100 Metonic cycles including truncated and the number of nodetides is one less than twice this.

In forming
1852, 3006, 4160, 5314, 6468, 7622, 8776, 9930
Helios added 1154 repeatedly to 1852.
Each cycle listed here therefore has exactly one nodetide less than twice the number of Metonic cycles including truncated (nodetide difference of 1).

A third mixer is still required. Its ratio of 353s to 19s would differ from 3:5 and it would have an odd number of years.
Another criticism of Helios's mixers is that they miss out the Gregoriana and Grattan-Guinness cycles listed in
https://www.staff.science.uu.nl/~gent0113/eclipse/eclipsecycles.htm 


So I proceed with mine, using Helios's as guidance.
I'd now replace 19, 353 & 4601 with something like 372, 391 & 3006. These need checking before a final decision can be made. The nodetide difference, would indicate the number of 3006s needed.
Helios's mixers arise thus:
1154 = 372 + 2*391
1852 = 3006 - 372 - 2*391.

3006 is equal to exactly nine 334-year cycles, but is not equal to 9*(3*372 - 2*391), because the number of nodetides would not agree.

For full reckoning, I'd represent these three 372, 391 & 3006 as vectors:
(372, 1, 0)
(391, 1, 0)
(3006, 9, 1)
where the 2nd component is the number of truncations of the Metonic cycle (to 11 years) and the 3rd component is the nodetide difference.
Then it is obvious that
(3006, 9, 1) is not equal to 9*[ 3*(372, 1, 0) - 2*(391, 1, 0) ].

Karl

16(03(01


-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 31 October 2016 01:05
To: [hidden email]
Subject: Re: Nodetides

Dear Karl and Calendar People,

I found that 214y and 242y compose the following mixers:
1154 years = 2*214 + 3*242 = 38*I + 3*S
1852 years = 3*214 + 5*242 = 59*I + 8*S

The 1154y is a rather long year but a very accurate eclipse. The 1852y is a
short year. Here are some mixes that include some previously mentioned
cycles:

1852, 3006, 4160, 5314, 6468, 7622, 8776, 9930

Note that 214 + 242 = 456 = 19*24, or 24 Metons = 17*I - 2*S = 470 12-month
years
Still, more cycles remain to be found, including odd-year cycles.




--
View this message in context: http://calndr-l.10958.n7.nabble.com/Nodetides-tp17237p17244.html
Sent from the Calndr-L mailing list archive at Nabble.com.
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Re: Nodetides

Irv Bromberg
In reply to this post by Helios
Helios, there must be something wrong with your calculations. How can you conclude that the saros interval, which is long established as the best eclipse cycle, is not an eclipse cycle?

There is some concern that after the passage of 100 saros intervals it may not hold up quite so well, due to drift of the 3 types of lunar month lengths, but that shouldn't yield a large deviation such as you have obtained, even after a full 1803-year cycle has elapsed.

I suspect that there is some confusion between mean solar time vs. atomic time durations in your calculations, or other duration measurement units.

-- 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: Sunday, October 30, 2016 20:15
To: [hidden email]
Subject: Re: Nodetides

Dear Irv and Calendar People,

I calculate that this cycle contains:
3799.7345 eclipse seasons
193.7345 nodetides
and therefore is not an eclipse cycle.

The value n = 9.306479 years for a nodetide would indicate that the year
equals:
y = 365.24259 days
An eclipse season equals 173.310037942 days.
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Re: Nodetides

Karl Palmen

Dear Irv, Helios and Calendar People

 

The Wikipedia page section

https://en.wikipedia.org/wiki/Saros_(astronomy)#Saros_series

gives the lifetime of the Saros cycle and puts it at about 71 or 72 cycles on average. The period that eclipses occur is about 35 days or 0.2 of an eclipse season, so an error of around 100/72*0.2= 0.2778 eclipse seasons is expected. Helios reckoned 0.2655 eclipse seasons.

 

Karl

 

16(03(02

 

From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Irv Bromberg
Sent: 01 November 2016 14:55
To: [hidden email]
Subject: Re: Nodetides

 

Helios, there must be something wrong with your calculations. How can you conclude that the saros interval, which is long established as the best eclipse cycle, is not an eclipse cycle?

There is some concern that after the passage of 100 saros intervals it may not hold up quite so well, due to drift of the 3 types of lunar month lengths, but that shouldn't yield a large deviation such as you have obtained, even after a full 1803-year cycle has elapsed.

I suspect that there is some confusion between mean solar time vs. atomic time durations in your calculations, or other duration measurement units.

-- 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: Sunday, October 30, 2016 20:15
To: [hidden email]
Subject: Re: Nodetides

Dear Irv and Calendar People,

I calculate that this cycle contains:
3799.7345 eclipse seasons
193.7345 nodetides
and therefore is not an eclipse cycle.

The value n = 9.306479 years for a nodetide would indicate that the year
equals:
y = 365.24259 days
An eclipse season equals 173.310037942 days.

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Re: Nodetides

Irv Bromberg
That duration is that of the saros series, not the saros interval itself.
It means that eclipses will repeat under almost the same conditions (including a fade-in and fade-out) on average for 71 or 72 times at intervals of the saros period. Sometimes the number of repeats is longer, sometimes shorter.

This has little to do with the accuracy of the saros interval, rather it depends on the specific conditions of the eclipse's saros series and how stable those conditions are over the long term. Some saros series are considerably longer because their circumstances are more stable.

It doesn't mean that the 100 saros cycle is excessive in duration, but it does mean that an eclipse that happened under certain circumstances most likely won't repeat under similar circumstances 100 saros intervals later.

Does Helios agree that a single saros interval is an eclipse cycle? Perhaps he is looking only for cycles that contain an integer number of mean solar days?

-- Irv Bromberg, University of Toronto, Canada

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


From: East Carolina University Calendar discussion List [[hidden email]] on behalf of Karl Palmen [[hidden email]]
Sent: Tuesday, November 01, 2016 12:21
To: [hidden email]
Subject: Re: Nodetides

Dear Irv, Helios and Calendar People

 

The Wikipedia page section

https://en.wikipedia.org/wiki/Saros_(astronomy)#Saros_series

gives the lifetime of the Saros cycle and puts it at about 71 or 72 cycles on average. The period that eclipses occur is about 35 days or 0.2 of an eclipse season, so an error of around 100/72*0.2= 0.2778 eclipse seasons is expected. Helios reckoned 0.2655 eclipse seasons.

 

Karl

 

16(03(02

 

From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Irv Bromberg
Sent: 01 November 2016 14:55
To: [hidden email]
Subject: Re: Nodetides

 

Helios, there must be something wrong with your calculations. How can you conclude that the saros interval, which is long established as the best eclipse cycle, is not an eclipse cycle?

There is some concern that after the passage of 100 saros intervals it may not hold up quite so well, due to drift of the 3 types of lunar month lengths, but that shouldn't yield a large deviation such as you have obtained, even after a full 1803-year cycle has elapsed.

I suspect that there is some confusion between mean solar time vs. atomic time durations in your calculations, or other duration measurement units.

-- 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: Sunday, October 30, 2016 20:15
To: [hidden email]
Subject: Re: Nodetides

Dear Irv and Calendar People,

I calculate that this cycle contains:
3799.7345 eclipse seasons
193.7345 nodetides
and therefore is not an eclipse cycle.

The value n = 9.306479 years for a nodetide would indicate that the year
equals:
y = 365.24259 days
An eclipse season equals 173.310037942 days.

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Re: Nodetides

Helios
Dear Irv and Calendar People,
Who says the Saros is the best eclipse cycle and since when has this been long established? This eclipse dies after about 72 occurrences. That's different from "not holding up quite so well", it's dead, deceased, kaput. You even had the opposite opinion in 2009.

"Irv replies: As interesting as that observation is, it would be much more interesting for a short leap cycle, containing just a few saros periods per cycle, because in the present era the longest solar eclipse saros series have only about 85 eclipses and more typically the count is in the low 70s, so for a 100-saros cycle if an eclipse marked the start of the cycle then that saros series would have expired well before the end of the 1803-year leap cycle. Lunar eclipse saros series are appreciably shorter."

If you find just one single example of a 100-fold Saros on the lunar tables I will buy you a cannoli!
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Re: Nodetides

Irv Bromberg
Helios, you've taking my quotation out of context, and/or I'm not really understanding what you are after in this thread.

You won't find any long cycles that you would consider acceptable, and none that could be repeated many times and still keep their initial relationship to eclipses. The saros is an excellent eclipse cycle -- the fact that the NASA Eclipses web site uses it exclusively for reckoning the saros series of eclipses indicates how excellent it is -- but no eclipse series lasts forever.

The saros works because 223 synodic months are nearly equal to 242 draconic months and 239 anomalistic months, but they aren't perfectly equal, so eventually the eclipse series dies out.

Besides the small differences here, there are also long term "secular" changes that confound using any simple eclipse cycle for calendrical purposes.

There are always many simultaneously active solar and lunar saros eclipse series in progress. Although each dies after about 72 occurrences, new series start and others still in progress continue along their way. So it would be with any type of eclipse series, but the saros has proven to be most useful for eclipse purposes.

It seems that you don't like the 100 saros cycle because no saros series can last that long, but you could consider the 25-saros cycle, which also corresponds to a useful lunar cycle having 5575 months, of which 2958 would be "full" (30 days), for a mean month of 29+2958/5575 days = 29 days 12 hours 44 minutes 2+82/223 seconds.

-- Irv Bromberg, Toronto, Canada

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

________________________________________
From: East Carolina University Calendar discussion List [[hidden email]] on behalf of Helios [[hidden email]]
Sent: Tuesday, November 01, 2016 14:56
To: [hidden email]
Subject: Re: Nodetides

Dear Irv and Calendar People,
Who says the Saros is the best eclipse cycle and since when has this been
long established? This eclipse dies after about 72 occurrences. That's
different from "not holding up quite so well", it's dead, deceased, kaput.
You even had the opposite opinion in 2009.

"Irv replies: As interesting as that observation is, it would be much more
interesting for a short leap cycle, containing just a few saros periods per
cycle, because in the present era the longest solar eclipse saros series
have only about 85 eclipses and more typically the count is in the low 70s,
so for a 100-saros cycle if an eclipse marked the start of the cycle then
that saros series would have expired well before the end of the 1803-year
leap cycle. Lunar eclipse saros series are appreciably shorter."

If you find just one single example of a 100-fold Saros on the lunar tables
I will buy you a cannoli!




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Re: Nodetides

Karl Palmen

Dear Irv and Helios and Calendar People

 

The 100-Saros cycle of 1803 year is not an eclipse cycle, precisely because no Saros series can last 100 Saros cycles.  

 

I don’t think the 25-Saros cycle will be considered by Helios in this topic because it is not near a whole number of years or nodetides, but he might consider 50-Saros cycle, because it can be thought of as having a whole number (97) of nodetides. Hence half-year cycles could be considered within the scope of the “nodetides” topic. The unnamed (176.5) is one of them. This is all I’ll say about them in this note.

 

The eclipse cycles listed in https://www.staff.science.uu.nl/~gent0113/eclipse/eclipsecycles.htm very close to a whole number of years have

19, 353, 372, 391, 725 & 1154 years.

I’ve eliminated those whose number of days differ significantly from the number of days in the same number of Gregorian years.

 

All these have number of eclipse seasons equal divisible by eight and equal to

8*(25*Years – 2*Months).

This is equivalent to the number of nodetides being equal to exactly twice the number of Metonic cycles including truncated Metonic cycles (to 11 years of 136 months), which is determined by

99*Years – 8*Months.

If the Saros were accurate enough to make a series of 100 Saros cycles, this too would apply to the 1803-year cycle.

 

The eclipse catalogue of eclipse cycles linked, lists eclipse cycles up to 3300 years, but none over 1154 years is very close to a whole number of years. This because the number of eclipse seasons or nodetides in a lunisolar cycle of such length is no longer close to an integer, but is a significant fraction less than the value given by my formula.

 

For longer lunisolar cycles the difference between the actual number of eclipse seasons and the value given by my formula will reach 1 and then more lunisolar eclipse cycles will occur such as 4160 years.

These cycles also have 1 less nodetide than twice the number of Metonic cycles including truncated. The number of eclipse seasons has remainder 7 when divided by 8.

 

Karl

 

16(03(03

 

 

From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Irv Bromberg
Sent: 01 November 2016 23:04
To: [hidden email]
Subject: Re: Nodetides

 

Helios, you've taking my quotation out of context, and/or I'm not really understanding what you are after in this thread.

You won't find any long cycles that you would consider acceptable, and none that could be repeated many times and still keep their initial relationship to eclipses. The saros is an excellent eclipse cycle -- the fact that the NASA Eclipses web site uses it exclusively for reckoning the saros series of eclipses indicates how excellent it is -- but no eclipse series lasts forever.

The saros works because 223 synodic months are nearly equal to 242 draconic months and 239 anomalistic months, but they aren't perfectly equal, so eventually the eclipse series dies out.

Besides the small differences here, there are also long term "secular" changes that confound using any simple eclipse cycle for calendrical purposes.

There are always many simultaneously active solar and lunar saros eclipse series in progress. Although each dies after about 72 occurrences, new series start and others still in progress continue along their way. So it would be with any type of eclipse series, but the saros has proven to be most useful for eclipse purposes.

It seems that you don't like the 100 saros cycle because no saros series can last that long, but you could consider the 25-saros cycle, which also corresponds to a useful lunar cycle having 5575 months, of which 2958 would be "full" (30 days), for a mean month of 29+2958/5575 days = 29 days 12 hours 44 minutes 2+82/223 seconds.

-- Irv Bromberg, Toronto, Canada

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

________________________________________
From: East Carolina University Calendar discussion List [[hidden email]] on behalf of Helios [[hidden email]]
Sent: Tuesday, November 01, 2016 14:56
To: [hidden email]
Subject: Re: Nodetides

Dear Irv and Calendar People,
Who says the Saros is the best eclipse cycle and since when has this been
long established? This eclipse dies after about 72 occurrences. That's
different from "not holding up quite so well", it's dead, deceased, kaput.
You even had the opposite opinion in 2009.

"Irv replies: As interesting as that observation is, it would be much more
interesting for a short leap cycle, containing just a few saros periods per
cycle, because in the present era the longest solar eclipse saros series
have only about 85 eclipses and more typically the count is in the low 70s,
so for a 100-saros cycle if an eclipse marked the start of the cycle then
that saros series would have expired well before the end of the 1803-year
leap cycle. Lunar eclipse saros series are appreciably shorter."

If you find just one single example of a 100-fold Saros on the lunar tables
I will buy you a cannoli!




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Re: Nodetides

Karl Palmen

Dear Helios, Irv and Calendar People

 

I found out more about the nodetide lengths got from Wikipedia.

 

https://en.wikipedia.org/wiki/Lunar_node gives 6798.3835 days or 18.612958 years for two nodetides and this implies that the year unit is 365.25 days, as usually used by astronomers when treating the year as a unit of time.

This leads to a mean nodetide of about 9.306673 years of 365.2424 days or 9.306679 years of 365.2422 days. This implies that Helios’s approximation of 698/75 = 9.3066666…recurring is very accurate.

 

https://en.wikipedia.org/wiki/Year#Draconic_year gives a different figure 18.612815932 Julian years.

This leads to a mean nodetide of 9.3066016 years of 365.2424 days and 9.3066071 years of 365.2424 days, suggesting an approximation of 9.3066 days.

 

Neither figure has a reference.

 

The nodetide measured in years changes faster as a proportion of its length than does the eclipse season.

The number of days or months in a nodetide depends on the length of the year. For a sidereal year it is very near 9.3 Julian years.

 

All this has negligible impact on the 1803-year cycle not being an eclipse cycle. I checked this possibility before realising the reason I gave.

 

Incidentally, four 1803-year cycles do form an eclipse cycle (based on Helios’s calculations multiplied by 4). The resulting 7212-year cycle has one Saros series of eclipses in each series of months 223 months apart, including some that run between successive 7212-year cycles. This arises because although it has the same number of months as 400 Saros cycles, it is one eclipse season short of 400 Saros cycles. It is therefore one nodetide short of two per Metonic cycle including truncated, like the 4160-year cycle.

 

Karl

 

16(03(04

 

 

 

From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 02 November 2016 13:21
To: 'East Carolina University Calendar discussion List'
Subject: RE: Nodetides

 

Dear Irv and Helios and Calendar People

 

The 100-Saros cycle of 1803 year is not an eclipse cycle, precisely because no Saros series can last 100 Saros cycles.  

 

 

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Re: Nodetides

Helios
Dear Karl and Calendar People,

From the equation;

( 75*Y + 34 )MOD( 698 ) < 75

Karl said he expected 37 instead of 34 and I don't know why.

I've also looked at the approximate relation,

373 eclipse season  = 177 years = 19 nodetides

in which we can see, amid a 19-day period, where and what year a node alignment will occur. These modular numbers look like,

047 019 168 140 112 084 056 028 000 149 121 093 065 037 009 158 130 102 074

with year 000 at the center. A correction would be needed every 53 - 54 years. We could also fashion something like a tithi and epact for a 1/177th eclipse season reckoning.
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Re: Nodetides

Karl Palmen
Dear Helios and Calendar People

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 04 November 2016 07:36
To: [hidden email]
Subject: Re: Nodetides

Dear Karl and Calendar People,

From the equation;

( 75*Y + 34 )MOD( 698 ) < 75

Karl said he expected 37 instead of 34 and I don't know why.


KARL REPLIES: I expected 37 because it would produce symmetry about 0

... -56, -47, -37, -28, -19, -9, 0, 9, 19, 28, 37, 47, 56, ...

Here I assume a calendar (x)MOD(y) = x - y*floor(x/y), which is appropriate for calendar use.

Also symmetry would occur around 698 and every multiple of it.


Why did Helios choose 34?


Karl

16(03(05
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Re: Nodetides

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

Helios said in this note
" We could also fashion something like a tithi and epact for a 1/177th
eclipse season reckoning."

I did this on 19 September 2013, but with the tithi normally 1/176 eclipse season. I used the conventional tithi of 1/30 synodic month. A year normally has 371 tithis, but one in every 20 or 21 years on average has an additional tithi, which is called a saltus lunae (jump of the moon).

If every month has 30 tithis and every eclipse season has 176 tithis, the Tzolkinex of 88 months = 15 eclipse seasons would result.
For a more accurate eclipse cycle, we can occasionally have a leap eclipse season of 177 tithis.

If 1 in every 16 eclipse seasons is a leap eclipse season, the calendar would follow the Short Callippic Cycle of 939 months = 160 eclipse seasons.

If 2 in every 31 eclipse seasons is a leap eclipse season, the calendar would follow the Half-Babylonian period of 2729 months = 465 eclipse seasons.

Karl

16(03(05

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 04 November 2016 07:36
To: [hidden email]
Subject: Re: Nodetides

Dear Karl and Calendar People,

From the equation;

( 75*Y + 34 )MOD( 698 ) < 75

Karl said he expected 37 instead of 34 and I don't know why.

I've also looked at the approximate relation,

373 eclipse season  = 177 years = 19 nodetides

in which we can see, amid a 19-day period, where and what year a node
alignment will occur. These modular numbers look like,

047 019 168 140 112 084 056 028 000 149 121 093 065 037 009 158 130 102 074

with year 000 at the center. A correction would be needed every 53 - 54
years. We could also fashion something like a tithi and epact for a 1/177th
eclipse season reckoning.




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Re: Nodetides

Karl Palmen
Dear Helios and Calendar People

I've found out that this tithi scheme would give

2 leap eclipse seasons to a Saros (1 in 19 eclipse seasons)
4 leap eclipse seasons to an Inex (1 in 15.25 eclipse seasons on average)

Karl

16(03(08

-----Original Message-----
From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 04 November 2016 13:10
To: 'East Carolina University Calendar discussion List'
Subject: RE: Nodetides

Dear Helios and Calendar People

Helios said in this note
" We could also fashion something like a tithi and epact for a 1/177th
eclipse season reckoning."

I did this on 19 September 2013, but with the tithi normally 1/176 eclipse season. I used the conventional tithi of 1/30 synodic month. A year normally has 371 tithis, but one in every 20 or 21 years on average has an additional tithi, which is called a saltus lunae (jump of the moon).

If every month has 30 tithis and every eclipse season has 176 tithis, the Tzolkinex of 88 months = 15 eclipse seasons would result.
For a more accurate eclipse cycle, we can occasionally have a leap eclipse season of 177 tithis.

If 1 in every 16 eclipse seasons is a leap eclipse season, the calendar would follow the Short Callippic Cycle of 939 months = 160 eclipse seasons.

If 2 in every 31 eclipse seasons is a leap eclipse season, the calendar would follow the Half-Babylonian period of 2729 months = 465 eclipse seasons.


Karl

16(03(05

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 04 November 2016 07:36
To: [hidden email]
Subject: Re: Nodetides

Dear Karl and Calendar People,

From the equation;

( 75*Y + 34 )MOD( 698 ) < 75

Karl said he expected 37 instead of 34 and I don't know why.

I've also looked at the approximate relation,

373 eclipse season  = 177 years = 19 nodetides

in which we can see, amid a 19-day period, where and what year a node
alignment will occur. These modular numbers look like,

047 019 168 140 112 084 056 028 000 149 121 093 065 037 009 158 130 102 074

with year 000 at the center. A correction would be needed every 53 - 54
years. We could also fashion something like a tithi and epact for a 1/177th
eclipse season reckoning.




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