Hipparchic cycle
The Greek astronomer Hipparchus introduced three cycles that have been named after him in later literature.
Calendar cycle
Hipparchus proposed a correction to the 76-year-long Callippic cycle, which itself was proposed as a correction to the 19-year-long Metonic cycle. He may have published it in the book "On the Length of the Year" (Περὶ ἐνιαυσίου μεγέθους), which has since been lost.
From solstice observations, Hipparchus found that the tropical year is about 1⁄300 of a day shorter than the 365+1⁄4 days that Calippus used (see Almagest III.1). So he proposed to make a 1-day correction after 4 Calippic cycles, i.e. 304 years = 3,760 lunations = 111,035 days.
- Error implicit in the cycle
This is a very close approximation for an integer number of lunations in an integer number of days (with an error of only 0.014 days). However, it is in fact 1.37 days longer than 304 tropical years. The mean tropical year is actually about 1⁄128 day (11 minutes 15 seconds) shorter than the
Indeed, from the values of the tropical year (365.2421896698 days) and the synodic month (29.530588853) cited in the respective articles of Wikipedia, it follows that the length of 228=12×19 tropical years is about 83,275.22 days, shorter than the length of 12×235 synodic months—namely about 83,276.26 days—by one day plus about one hour. In fact, an even better correction would be two days every 437 years, rather than one day every 228 years. The length of 437=23×19 tropical years (about 159,610.837 days) is shorter than that of 23×235 synodic months (about 159,612.833 days) by almost exactly two days, up to only six minutes.
Eclipse cycles
An eclipse cycle constructed by Hipparchus is described in Ptolemy's Almagest IV.2:
For from the observations he set out he [Hipparchus] shows that the smallest constant interval defining an ecliptic period in which the number of months and the amount of [lunar] motion is always the same, is 126007 days plus 1 equinoctial hour. In this interval he finds comprised 4267 months, 4573 complete returns in
Gerald Toomer[1]
Actually, dividing 126007 days and one hour by 4267 would give 29;31,50,8,10 in
Ptolemy points out that if one divides this cycle by 17, one obtains a whole number of synodic months (251) and a whole number of
But if one were to look for the number of months [which always cover the same time-interval], not between two lunar eclipses, but merely between one conjunction or opposition and another syzygy of the same type, he would find an even smaller integer number of months containing a return in anomaly, by dividing the above numbers by 17 (which is their only common factor). This produces 251 months and 269 returns in anomaly.
— Book IV, Chapter 2
The Hipparchic eclipse cycle is made up of 25 inex minus 21 saros periods. There are only three or four eclipses in a series of eclipses separated by Hipparchic cycles. For example, the solar eclipse of August 21, 2017 was preceded by one in 1672 and will be followed by one in 2362, but there are none before or after these.[3]
It corresponds to:
- 4,267 synodic months
- 4,630.531 draconic months
- 363.531 eclipse years (727 eclipse seasons)
- 4,573.002 anomalistic months
There are other eclipse intervals that also have the properties desired by Hipparchus, for example an interval of 81.2 years (four of the 251-month cycles, or 19 inex minus 26 saros) which is even closer to a whole number of anomalistic months (1076.00056), and almost equally close to a half-integer number of draconic months (1089.5366). The "tritrix" eclipse cycle,[4] consisting of 1743 synodic months, 1891.496 draconic months, or 1867.9970 anomalistic months (140.925 years, equivalent to 3 inex plus 3 saros) is about as accurate as the interval of Hipparchus, but repeats many more times, around 20. An exceptionally accurate eclipse cycle is one of 1154.5 years (43 inex minus 5 saros), which is much closer to a whole number of anomalistic months (15303.00005) than the interval of Hipparchus. At the solar eclipse of October 17, 1781, the moon had an anomaly of 0°,[5] and similar eclipses have occurred every 1054.5 years for more than 4000 years and will continue at least 13,000 more years.[6]
Ptolemy says that Hipparchus also came up with a period of 5458 synodic months, equal to 5923 draconic months (441.3 years). This is called the Hipparchian Period, and more recently the Babylonian Period, but the latter is a misnomer as there is no evidence that the Babylonians were aware of it.[4] It is equivalent to 14 inex plus 2 saros periods and therefore repeats many more times than the 345-year cycle. The solar eclipse of July 11, 2010, for example, is the latest in a series that has been going for more than 13,000 years and will continue for more than 8000 more.[6]
References
- ^ Ptolemy’s ALMAGEST Translated and Annotated by G. J. Toomer (PDF). 1984. pp. 175–6. In Greek, "ἀποδείκνυσι γάρ, δι' ὧν ἐξέθετο τηρήσεων, ὅτι ὁ πρῶτος ἀριθμὸς τῶν ἡμερῶν, δι' ὅσων πάντοτε ὁ ἐκλειπτικὸς χρόνος ἐν ἴσοις μησὶν και ἐν ἴσοις κινήμασιν ἀνακυκλείται, ιβ μ ἐστιν καὶ ἔτι ͵ςζ ἡμερῶν καὶ μιᾶς ὥρας ἰσημερινῆς, ἐν αἵς μῆνας μὲν ἀπαρτιζομένους εὐρίσκει ͵δσξζ, ὅλας δὲ ἀνωμαλίας ἀποκαταστάσεις ͵δφογ, ζῳδιακοὺς δὲ κύκλους ͵δχιβ λείποντας μοίρας ζ∠ʹ ἔγγιστα, ὅσας καὶ ὁ ἥλιος εἰς τοὺς τμε κύκλους λείπει, πάλιν ὡς τῆς ἀποκαταστάσεως αὐτῶν πρὸς τοὺς ἀπλανεῖς ἀστέρας θεωρουμένης. ὅθεν εὐρίσκει καὶ τὸν μηνιαῖον μέσον χρόνον ἐπιμεριζομένου τοῦ προκειμένου τῶν ἡμερῶν πλήθους εἰς τοὺς ͵δσξζ μήνας ἡμερῶν συναγόμενον κθ λα ν η κ ἔγγιστα." Heiberg's Edition of Ptolemy, VOL I, partes I & II, pp. 270-1. Also available here.
- ^ Franz Xaver Kugler (1900). Die Babylonische Mondrechnung (PDF). pp. 8–21.
- ^ See "Five Millennium Catalog of Solar Eclipses". NASA.
- ^ a b Rob van Gent. "A Catalogue of Eclipse Cycles - List of Eclipse Cycles". Utrecht University.
- ^ Giovanni Valsecchi, Ettore Perozzi, Archie Roy, Bonnie Steves (Mar 1993). "Periodic orbits close to that of the Moon". Astronomy and Astrophysics: 311.
{{cite journal}}: CS1 maint: multiple names: authors list (link) - ^ a b Saros-Inex Panorama. Data in Solar eclipse panaorama.xls.