Bill Jefferys wrote:
>John Baez wrote: >>For example, remember how Keith Ramsay and Phillip Helbig pointed >>out the existence of two kinds of day? There's the 24-hour >>"solar day" and the 24-hour, 56-minute and 4-second "sidereal >>day", depending on whether you watch the sun or the stars going >>around. >But note: The "sidereal" day is really reckoned with respect to the vernal >equinox, which moves relative to the stars! It's dependent upon the >precession rate.Yuck. I didn't know that. Is there a name for the "true sidereal day", where we keep track of how the earth rotates relative to the fixed stars, rather than the vernal equinox?
>None of the other sidereal periods are so reckoned. (e.g., >the tropical year is reckoned with respect to the vernal equinox, not the >sidereal year, which is really reckoned with respect to the stars, as best >as we know how).Yeah, that makes me not like this use of the term "sidereal day". But then astronomy, being the oldest science, seems full of strange terminology that goes way back and is too deeply rooted to change.
1) The SIDEREAL MONTH is the time it takes for the moon to orbit >>360 degrees around the earth as viewed from the fixed stars. It's >>27.321661 days long. (Don't ask me if these are solar or sidereal days - >>I assume the former, but the thing I'm reading doesn't say.) >Your assumption is correct. Days of 86,400 SI seconds are the standard unit >for timekeeping in astronomy when "days" are mentioned without qualifier.Whew, good - I was starting to get paranoid, with all these different kinds of days, months and years floating around.
>>This means that eclipses occur in a pattern which depends >>on how the synodic and draconic months drift in and out of >>phase. >> >>And *this* means that to predict eclipses, you need to find >>a number which gives you something close to an integer when >>you divide it by 27.321661 and also when you divide it by >>27.212220. >Except that you've used sidereal and draconitic months in your calculation >here.Yikes! Thanks for catching that slip!
>Interestingly enough, the number 6585 + 1/3 is what you get for all >three months (sidereal, synodic, draconitic), as well as for the >anomalistic month!Cool! That's why I didn't notice I'd slipped up - the calculation seemed to work pretty well even though I accidentally used the sidereal month. But redoing the calculation, it turns out to work better with the synodic month than it did with the sidereal one. I.e., 6585 + 1/3 days is closer to being an integral number of synodic months than sidereal ones.
Let me see how well it works with all 4 kinds of month:
6585 + 1/3 days is (6585 + 1/3)/29.530589 = 223.000403 synodic months
and it's also (6585 + 1/3)/27.212220 = 241.999121 draconitic months
and it's also (6585 + 1/3)/27.321661 = 241.029758 sidereal months
and it's also (6585 + 1/3)/27.554549 = 238.992601 anomalistic months
Yeah, you're right, it even works pretty well with the anomalistic month!
>This means that the Saros duplicates not only the fact >of an eclipse (Sun at node, Moon and Sun in conjunction) but also the >_kind_ of eclipse (annular or total in the case of solar eclipses). The >Moon will be nearly the same distance from the Earth after one Saros period.Yes. It seems like a rather lucky coincidence.
>It's nice to point out that these near-integers are conveniently calculated >using continued fraction expansions.You know, I've been reading a book called "The Mathematics of Plato's Academy: A New Reconstruction" by D. H. Fowler, and he makes the case that lots of Greek mathematics was secretly (or not so secretly) about continued fraction expansions. For example, the Euclidean algorithm to determine the greatest common divisor is based on the same idea as continued fraction expansions: apparently the Greeks called it "anti-hypo-hairesis", meaning "reciprocal subtraction". There's also a nice simple proof that the golden ratio is irrational, based on the fact that its continued fraction expansion:
1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + .................... )))))
never ends. I don't know if the Greeks knew this proof, but there's a nice geometrical version of this proof based on chopping a golden rectangle into a square and another golden rectangle, then chopping *that* golden rectangle into a square and a golden rectangle, etc. - and it's so simple that I think some people argue the Greeks *had* to know it. Anyway, I'm really curious now in how much of this continued fraction technology was known to the Babylonians, and how much it came from work on astronomy, and so on.
It's also amusing how important continued fraction expansions are in the modern theory of dynamical systems - for essentially the same reason, namely that they let us study how long we have to wait for two different periodic motions to come within epsilon of drifting back into phase. Chris Hillman has written cool things about this stuff....
>I do this for my "TIME" course (for Freshmen nonscience students).Gee, maybe I should sit in on that. :-)
>Also, the 1/3 day means that the eclipse happens 1/3 of the way >around the world.Yeah, and that makes me wonder: did the Babylonians really know about this 6585 + 1/3 day Saros cycle, or did they just know about the cycle that's 3 times as long (which is about 54 years)? If your empire doesn't stretch 1/3 around the world, you're not gonna see the eclipses repeat until this longer cycle rolls around. It seems you can only appreciate the shorter cycle if have some real understanding of the draconistic month, which unlike the synodic month is not a terribly simple concept. Did the Babylonians know about the nodes used to define the draconistic month?
>Eclipses are wonderful, just >knowing the fact that an ancient eclipse occurred at a particular place >gives very accurate information about earth rotation and the loss of spin >angular momentum of the Earth.Neat.
>The ancient idea was that a dragon ate the Sun or Moon, causing an eclipse.
>There are two nodes, the ascending node and the descending node. They are
>the "head of the dragon" and the "tail of the dragon", respectively. The
>symbol for the node looks like \Omega; (except that the serifs are replaced
>by one or two little circles in the old-style symbol). That's for the
>ascending node. To get the symbol for the descending node ("tail of the
>dragon"), turn the symbol upside down!
Oh, I've seen those symbols but I never knew what they meant.
>The Babylonians indeed knew the [Metonic] cycle.Aha. Not surprising, since it's so blitheringly obvious compared to the Saros.
>If you look at the development of >the Babylonian calendar--see Parker and Dubberstein's book--you see that >they gradually regularized the intercalations of the 13th month over the >centuries. By the 3rd century BCE the intercalation scheme was the same >same sequence as in the modern Hebrew calendar (and of course, the Hebrew >calendar was derived from Jewish experience during the Babylonian >captivity).I'll have to get ahold of that book - I guess you mean:
Richard A. Parker and Waldo H. Dubberstein, Babylonian chronology 626 B.C.-A.D. 75, Brown U. Press, Providence, Rhode Island, 1956.
I'm feeling a need to get in touch with my roots as a mathematical physicist.
>>Q: Hey! The Metonic cycle is pretty close to the Saros cycle: >>19 years versus 18 years plus a bit. And these are both close to the >>period of the nutation of the earth - 18.6 years. Is there something >>interesting going on here? >> >>A: Hmm, that's a *very* good question. Coincidence, phase-locking, >>or some sort of mathematical a priori relationship? >Just a coincidence, I fear.Oh shucks.