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Umbral and total eclipse intervals seen from a certain location

• the number of visible eclipses at a certain location on the earth is on average not depending on the longitude
  
• the number of visible eclipses at a certain location on the earth is on average not really depending on the latitude (certainly as long one is well within the tropics)
  
• An (primary) interval is defined as the time interval between two successive eclipse. Composed intervals (e.g. eclipse families: so adding multiple primary intervals together) are not evaluated (e.g. Saros cycle is not included)
• Two concepts need to be handled distinct:
  
1. individual eclipses and
   
2. primary intervals between two successive eclipse. This web pages is about the distribution of these primary intervals.
• In the below rules for the intervals are tried to be found and not rules for the individual eclipses, but it is hoped that the interval rules will provide help in predicting the next individual eclipses correctly anywhere on earth.
  So sea/land travel could help in finding (faster, if properly understood/handled the changes in individual eclipses at different locations) these interval rules.
• The interval between two successive eclipses is not really depending on the epoch. So an interval seen in 3000 BCE and 2000 CE would be comparable (influence of DT on this is near zero).
• Intervals between successive eclipses (primary intervals) are discrete and only certain interval lengths will happen. They will be represented as LunX, where X is the number of synodic months between the two eclipses (as proposed by Meeus, pers. comm. 2005).
  In this web page one can see other words for certain eclipse intervals.
• Only visible umbral/total eclipses with umbral/total phases from a certain location are included.
  
• It is assumed that penumbral phases/eclipses where less/not experienced by neolithic people. Lunar penumbral eclipse much be larger than a magnitude of 0.7 to be witnessed. (McGraw-Hill Science & Technology Encyclopedia: Eclipse)
• The sigma (error) of the primary interval length is around 0.05 Days, this is somewhat smaller then the expected sigma of a synodic month.
• These intervals are most of the time short (shorter then say ~Lun100: 8 years [which is more or less a generation in neolithic times]), which perhaps can be found in some way in the megalithic art/monuments.

• Lunar eclipses between 1768 CE and 1854 CE have been checked. This is a relative short period (86 years), but overall the behaviour can be evaluated. Perhaps one has to extend this to some 800 years to get a better distribution (in the case of many intervals and/or only looking at total lunar eclipses).
• I think that it is possible that over a region of 10 degrees longitude (say Ireland plus UK) the experience of individual at-least-umbral-phase lunar eclipses will be comparable. For total lunar eclipse this might give a different view.
• In the below I only look at the distributions of the intervals and the chance that a certain interval follow-ups an earlier interval.

Distribution of intervals between two visible lunar eclipses with at-least-umbral-phase.

Lunar eclipses with at-least-umbral-phase at a location

All lunar eclipses (139) seen somewhere on earth

Asking Meeus about the above lunation periods, he discusses (Chapter 27, [2007]) his findings on umbral lunar eclipses in the period 1000 BCE to 4000 CE. He has comparable findings as done above, interestingly though is that before 900 CE Lun5 could happen due to different eccentricity of the Earth's orbit (after that date it will not happen anymore in the near future [6000 CE]).

Lunar eclipses (85) seen from Newgrange, Ireland

Lunar eclipses (88) seen from Rio de Janeiro, Brazil

Lunar eclipses (98) seen from Jerusalem, Israel

Conclusions on at-least-umbral-phase lunar eclipses

• Looking at all lunar eclipse with at-least-umbral-phase visible from earth;
  
• the intervals are mostly 6 synodic months (Lun6: ~83%) and
  
• sometimes a longer interval (Lun17: ~15%). This longer interval is to skip the dates that have only penumbral eclipses.
• there is an occasional Lun11 (~2%)
  
• Looking at all lunar eclipse with at-least-umbral-phase visible from a specific location;
  
• the intervals are mostly 6 synodic months (Lun6: say up to ~51%)
  
• several Lun6 periods (from all earth location) are broken up, because some are not visible from the specific location, thus making it Lun12/Lun11 appear (~21%).
  
• the number of  Lun17/Lun18 intervals (from all earth location) is more or less doubled (~12%), also because some are not visible from the specific location.
• all the other longer intervals (from Lun23 to Lun35, max. 20%) are happening now, because shorter intervals are gone away from all earth location.
• for some locations some interval don't happen (like Lun24 for Newgrange and Lun35 for Jerusalem), but could be due to the short period taken (86 years).
  
• The above pie charts for the specific locations are somewhat different, but they are not really significantly different (variability of percentage with a sigma of  ± 2%).
  
• The global intervals are totally different from the location specific intervals.
  
• So I see the intervals Lun6, Lun12, Lun17, Lun23, Lun29 and possible Lun35 as occurring regularly.
  
• I am wondering if these intervals could be seen/tallied in some way from megalithic art/monuments.
• The Lun6 might have been recognised by the Maya as they used Glyph C to record where a specific moon was in a cycle of six lunations (M.D. Coe and M. van Stone, Reading the Maya glyphs, 2001, page 52).
  

Lunar eclipses with a total phase at a location

All total lunar eclipses (53) seen somewhere on earth

Total lunar eclipses (32) seen from Newgrange, Ireland

Conclusions on total lunar eclipses

The following can be seen from the above charts in this example:

• Compared to the at-least-umbral-phase intervals, much longer intervals (Lun35 to Lun82) are happening.
• A fewer number of different intervals happen at specific locations compared to the at-least-umbral-phase eclipses, but again much longer.
• The global intervals are totally different from the location specific intervals.
• So I see the intervals Lun6, Lun41, Lun47 and possibly Lun53 as occurring regularly.
• So if these total eclipses would have been studied in Neolithic times beside the at-least-umbral-phase eclipses their possible representation would be quite distinct.
  

Follow-up intervals

At-least-umbral-phase lunar eclipses at Newgrange

• say we just had an lunar eclipse interval of 6 synodic months (Lun6), the chance of follow-up at-least-umbral-phase eclipse being a Lun6 (~40%), Lun12 (~24%), Lun23 (~16%), etc.
  
• if one just had a Lun11 (or Lun 18) interval, the next interval is a Lun6.
• etc.

Total lunar eclipses at Newgrange

• say we just had an lunar eclipse interval of 6 synodic months (Lun6), the following total lunar eclipse will happen 41 synodic month later (in this example one knows this for 100%!)
• if one just was in a Lun41 interval (happening 52%) the chances to have a next interval of Lun6 or Lun41 is ~37% and 19% for Lun47 and 6% for Lun53.
• if one just had a Lun47 interval (happens 13%); a 25% chance for a Lun6 and 75% for Lun41

Conclusions of follow-up intervals

• One has to remember that the number of lunar at-least-umbral-phase eclipses is not much (85) in this period and that the number of discrete intervals is large, so the above occurrences are just an indication; one would need a larger period of eclipses to get a more reliable picture.
• For other locations (Jerusalem and Rio de Janeiro), one gets a somewhat different relational graph (although Jerusalem and Rio de Janeiro are quite similar), but they have more or less the same behavior of transitions. I think due to the low number of occurrences in each cell, this discrepancy can be explained. It would be good to test this with a larger period (like 800 years instead of 86 years).

Acknowledgments

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