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JRASC October 2004 - The Royal Astronomical Society of Canada

Orbital OdditiesFading Foursomeby Bruce McCurdy, Edmonton Centre (bmccurdy@telusplanet.net)As far as the laws of mathematics referto reality, they are not certain; and as faras they are certain, they do not refer toreality.- Albert EinsteinWith these words, Einsteineloquently paraphrased, onthe grandest scale, theIncompleteness Theorem of hisPrinceton colleague and frequentwalking companion, Kurt Gödel.Simultaneously a breakthrough inmathematics, philosophy, and logic,Gödel’s insight has forever changedhumankind’s approach to understandingall this. Simply put, we now know wecan’t (Hofstadter 1979).That we can understand theUniverse to the remarkable extent thatwe do, can surely be credited to thecosmic science of mathematics. Themost fundamental laws of physics arepaved with elegant and profoundstatements such as Einstein’s owne=mc 2 , Newton’s inverse squares, andKepler’s equal areas.Occasionally, numericalrelationships arise that can’t be easilyexplained. Why, for example, should afavourite plaything from the wonderfullyweird world of (pardon the oxymoron)“recreational mathematics” appearacross nature, from the structure ofsunflowers to spiral galaxies, from thegrowth of honeybee populations to thedescription of eclipse cycles?“Eclipse cycles?” you ask. Let meexplain.Solar and lunar eclipses arefundamentally different, to the degreethat the 16th century astronomer Jean-Pierre de Mesmes proposed twounrelated terms, namely “obstructions”and “fades” (Brunier & Luminet 2000).Alas, his common-sense suggestionfell on the deaf ears of those responsiblefor scientific terminology, presumablythe same people who permitted suchpaired terms as “immersion” and“emersion” to test our collective sanity.The result is a dizzying array of eclipsetypes: solar and lunar; total and partial;central and non-central; annular andhybrid and penumbral. Some of thesecategories apply to only one of thetypes, some to both. To give obstructionsand fades the same name is as dumban idea as splitting the day into two12-hour clocks. Grrr...However, both types rely on thealignment of the same three bodies,and do share some characteristics,particularly common cycles. Eacheclipse season features at least one ofeach, separated by a Fortnight. Foreclipses of similar type - and here we’llprimarily focus on lunar eclipses - themost obvious period is that of sixmonths (the Semester), at which intervalseries of about 8 eclipses can be found.The true period, however, is 5.87lunations per half eclipse year, so theSemester series soon phases out to bereplaced by another, one month “early.”These calendar shifts are critical toshort-term eclipse cycles.Each Semester series begins andends with penumbral and partial fades,and has a central “sweet spot” of 2-4total eclipses some seven or eightSemesters after its predecessor. Fromthese intervals come the Hepton andOcton, of 3.3 years and 3.8 yearsrespectively. The two bracket the trueratio, so one of each combines to forma more-accurate period known as theTzolkinex, so named because its 7.1-year period very nearly equals 10Tzolkins (a mysterious 260-day periodon the Mayan calendar). Octon andTzolkinex together make a 10.9-yearperiod called the Tritos, presumablynamed because it includes three calendarshifts. Each of these periods can beused to predict eclipses with increasingdegrees of certainty (Meeus 1997;www.phys.uu.nl/~vgent/calndar/eclipsecycles.htm).But all shrivel in comparison tothe next periodicity, the mighty Saros.After 18 calendar and 19 eclipse years,an immense lap has been completed.The Sun, Earth, and Moon all returnto very near their original positionsafter 6585.32 days - 18 years plus 10(or 11 or 12 days) plus 8 hours - afterwhich eclipses “repeat” with a highdegree of similarity. The period of 223lunations can be considered as 19 eclipseyears of 12 months, minus 5 calendarshifts forward. Thus the “magic” number5 is a key contributor; the shorter-termcycles described above merely subdividethe Saros into fifths.A Saros cycle consists of 69-87eclipses, covering an average period ofsome 1300 years. Yet that is smallpotatoes compared to the lesser-knownInex. This was the name given to the202JRASC October / octobre 2004