Hipparchic cycle

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The Greek astronomer Hipparchus introduced three cycles that have been named after him in later literature.

Contents

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 1300 of a day shorter than the 365+14 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 1128 day (11 minutes 15 seconds) shorter than the Julian calendar year of 365+14 days. These differences cannot be corrected with any cycle that is a multiple of the 19-year cycle of 235 lunations; it is an accumulation of the mismatch between years and months in the basic Metonic cycle, and the lunar months need to be shifted systematically by a day with respect to the solar year (i.e. the Metonic cycle itself needs to be corrected) after every 228 years.[ citation needed ]

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 anomaly, and 4612 revolutions on the ecliptic less about 7½° which is the amount by which the sun’s motion falls short of 345 revolutions (here too the revolution of sun and moon is taken with respect to the fixed stars). (Hence, dividing the above number of days by the 4267 months, he finds the mean length of the [synodic] month as approximately 29;31,50,8,20 days).

Book IV, Chapter 2, translation of Gerald Toomer [1]

Actually, dividing 126007 days and one hour by 4267 would give 29;31,50,8,9 in sexagesimal, whereas 29;31,50,8,20 was already used in Babylonian astronomy, possibly found by Kidinnu in the fourth century BC. This period is a multiple of a Babylonians unit of time equal to one eighteenth of a minute (3+1/3 seconds), which in sexagesimal is 0;0,0,8,20 days. (The true length of the month, 29.53058885 days, comes to 29;31,50,7,12 in sexagesimal, so the Babylonian value was correct to the nearest 3+1/3-second unit.)

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 anomalistic months (269):

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

Franz Xaver Kugler in his Die Babylonische Mondrechnung claimed that the Chaldaeans could have known about this cycle of 251 months, because it falls out of their system of calculating the speed of the moon, seen in a tablet from around 100 BC. [2] In their system, the speed of the moon at new moon varies in a zigzag, with a period of one full moon cycle, changing by 36 arc minutes each month over a span of 251 arc minutes (see graph), and this implies that after 251 months the pattern repeats, and 269 anomalistic months will have gone by. So it is possible that Hipparchus constructed his 345-year cycle by multiplying this 20-year cycle by 17 so as to closely match an integer number of synodic months (4,267), anomalistic months (4,573), years (345), and days (a bit over 126,007). It is also close to a half-integer number of draconic months (4,630.53...), making it an eclipse period. By comparing his own eclipse observations with Babylonian records from 345 years earlier, he could verify the accuracy of the various periods that the Chaldean astronomers used.[ citation needed ]

Comparison of speed of moon with values given in a Babylonian tablet, 104-103 BC Speed of moon in 104-103 BC.png
Comparison of speed of moon with values given in a Babylonian tablet, 104-103 BC

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:

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 in terms of anomalistic months, but repeats many more times, around 20. An exceptionally accurate eclipse cycle from this point of view 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]

Comparison of length variation for three eclipse periods of exceptionally small length variability. The x-axis gives the date of the second eclipse, between 2001 and 2050. The y-axis shows the fraction of a day left after subtracting 539848, 132592, and 126007 days respectively from the 1478-, 363-, and 345-year periods. The axis on the right shows more or less the same thing, expressed as hours and minutes. Length variation in three eclipse periods.png
Comparison of length variation for three eclipse periods of exceptionally small length variability. The x-axis gives the date of the second eclipse, between 2001 and 2050. The y-axis shows the fraction of a day left after subtracting 539848, 132592, and 126007 days respectively from the 1478-, 363-, and 345-year periods. The axis on the right shows more or less the same thing, expressed as hours and minutes.

The period of Hipparchus is also accurate in the sense of always having the same length to within an hour. This is due to the fact that it is close to a whole number of anomalistic years as well as to a whole number of anomalistic months. Its average length is actually 126007.023 days, half an hour less than what Ptolemy says. This is equivalent to 345 Julian years minus 4.227 days (implying that in the Gregorian calendar the date usually goes back by just one or two days, sometimes by three), which is only about 8 days less than 345 anomalistic years. There are few eclipse periods that are so constant the semester for example (six synodic months) can vary by a day in each direction.

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]

Related Research Articles

<span class="mw-page-title-main">Eclipse cycle</span> Calculation and prediction of eclipses

Eclipses may occur repeatedly, separated by certain intervals of time: these intervals are called eclipse cycles. The series of eclipses separated by a repeat of one of these intervals is called an eclipse series.

The saros is a period of exactly 223 synodic months, approximately 6585.321 days, or 18 years plus 10, 11, or 12 days, and 8 hours, that can be used to predict eclipses of the Sun and Moon. One saros period after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, a near straight line, and a nearly identical eclipse will occur, in what is referred to as an eclipse cycle. A sar is one half of a saros.

The inex is an eclipse cycle of 10,571.95 days. The cycle was first described in modern times by Crommelin in 1901, but was named by George van den Bergh who studied it in detail half a century later. It has been suggested that the cycle was known to Hipparchos. One inex after an eclipse of a particular saros series there will be an eclipse in the next saros series, unless the latter saros series has come to an end.

An exeligmos is a period of 54 years, 33 days that can be used to predict successive eclipses with similar properties and location. For a solar eclipse, after every exeligmos a solar eclipse of similar characteristics will occur in a location close to the eclipse before it. For a lunar eclipse the same part of the earth will view an eclipse that is very similar to the one that occurred one exeligmos before it. The exeligmos is an eclipse cycle that is a triple saros, three saroses long, with the advantage that it has nearly an integer number of days so the next eclipse will be visible at locations and times near the eclipse that occurred one exeligmos earlier. In contrast, each saros, an eclipse occurs about eight hours later in the day or about 120° to the west of the eclipse that occurred one saros earlier.

<span class="mw-page-title-main">Tritos</span> Eclipse cycle

The tritos is an eclipse cycle of 3,986.628 days . It corresponds to:

<span class="mw-page-title-main">Solar eclipse of March 30, 2052</span> Total eclipse

A total solar eclipse will occur at the Moon's descending node of the orbit on Saturday, March 30, 2052. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. A total solar eclipse occurs when the Moon's apparent diameter is larger than the Sun's, blocking all direct sunlight, turning day into darkness. Totality occurs in a narrow path across Earth's surface, with the partial solar eclipse visible over a surrounding region thousands of kilometres wide. The path of totality will cross central Mexico and the southeastern states of the United States. Almost all of North America and the northern edge of South America will see a partial eclipse. It will be the 2nd total eclipse visible from the Florida Panhandle and southwest Georgia in 6.6 years. It will be the first total solar eclipse visible from Solar Saros 130 in 223 synodic months. It will be the last total solar eclipse visible in the United States until May 11, 2078.

<span class="mw-page-title-main">Solar eclipse of May 11, 2078</span> Total eclipse

A total solar eclipse will occur on Wednesday, May 11, 2078. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. A total solar eclipse occurs when the Moon's apparent diameter is larger than the Sun's, blocking all direct sunlight, turning day into darkness. Totality occurs in a narrow path across Earth's surface, with the partial solar eclipse visible over a surrounding region thousands of kilometres wide.

<span class="mw-page-title-main">Solar eclipse of June 11, 2048</span> Future annular solar eclipse

An annular solar eclipse will occur on Thursday, June 11, 2048 with a magnitude of 0.9441. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide.

<span class="mw-page-title-main">Solar eclipse of September 9, 1904</span> Total eclipse

A total solar eclipse occurred on Friday, September 9, 1904. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. A total solar eclipse occurs when the Moon's apparent diameter is larger than the Sun's, blocking all direct sunlight, turning day into darkness. Totality occurs in a narrow path across Earth's surface, with the partial solar eclipse visible over a surrounding region thousands of kilometres wide. Totality was visible from German New Guinea on September 10 and Chile on September 9.

<span class="mw-page-title-main">Solar eclipse of May 31, 2049</span> Future annular solar eclipse

An annular solar eclipse will occur on Monday, May 31, 2049. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide.

<span class="mw-page-title-main">Solar eclipse of February 5, 2046</span> Future annular solar eclipse

An annular solar eclipse will occur on Monday, February 5, 2046. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide.

<span class="mw-page-title-main">Solar eclipse of April 18, 1977</span> Annular solar eclipse

An annular solar eclipse took place at the Moon's descending node of the orbit on Monday, April 18, 1977. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide. Annularity was visible in South West Africa, Angola, Zambia, southeastern Zaire, northern Malawi, Tanzania, Seychelles and the whole British Indian Ocean Territory.

<span class="mw-page-title-main">Solar eclipse of September 4, 2100</span> Total solar eclipse

A total solar eclipse is forecast to occur on Saturday, September 4, 2100. It will be the last solar eclipse of the 21st century. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. A total solar eclipse occurs when the Moon's apparent diameter is larger than the Sun's, blocking all direct sunlight, turning day into darkness. Totality occurs in a narrow path across Earth's surface, with the partial solar eclipse visible over a surrounding region thousands of kilometres wide.

<span class="mw-page-title-main">Solar eclipse of February 17, 2064</span> Future annular solar eclipse

An annular solar eclipse will occur on Sunday, February 17, 2064. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide.

<span class="mw-page-title-main">Solar eclipse of May 1, 2079</span> Total eclipse

A total solar eclipse will occur on Monday, May 1, 2079, with a maximum eclipse at 10:48:25.6 UTC. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. A total solar eclipse occurs when the Moon's apparent diameter is larger than the Sun's, blocking all direct sunlight, turning day into darkness. Totality occurs in a narrow path across Earth's surface, with the partial solar eclipse visible over a surrounding region thousands of kilometres wide. The eclipse will be visible in Greenland, parts of eastern Canada and parts of the northeastern United States.

<span class="mw-page-title-main">Solar eclipse of August 10, 1934</span> 20th-century annular solar eclipse

An annular solar eclipse occurred on Friday, August 10, 1934, with an eclipse magnitude of 0.9436. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide.

<span class="mw-page-title-main">Solar eclipse of December 13, 1936</span> 20th-century annular solar eclipse

An annular solar eclipse occurred between Sunday, December 13 and Monday, December 14, 1936. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide. Annularity was visible from Australia, New Zealand on December 14 (Monday), and Oeno Island in Pitcairn Islands on December 13 (Sunday).

<span class="mw-page-title-main">Solar eclipse of July 20, 1925</span> 20th-century annular solar eclipse

An annular solar eclipse occurred on Monday, July 20, 1925. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide. Annularity was visible from northern part of Northland Region and the whole Kermadec Islands in New Zealand on July 21 (Tuesday), and Rapa Iti in French Polynesia on July 20 (Monday).

<span class="mw-page-title-main">Solar eclipse of July 9, 1926</span> 20th-century annular solar eclipse

An annular solar eclipse occurred on Friday, July 9, 1926. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide. Annularity was visible from the islands of Pulo Anna and Merir in Japan's South Seas Mandate and Wake Island on July 10 (Saturday), and Midway Atoll on July 9 (Friday).

The tzolkinex is an eclipse cycle equal to a period of two saros minus one inex. As consecutive eclipses in an inex series belongs to the next consecutive saros series, each consecutive Tzolkinex belongs to the previous saros series.

References

  1. 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 κθλανηκ ἔγγιστα." Heiberg's Edition of Ptolemy, VOL I, partes I & II, pp. 270-1. Also available here.
  2. Franz Xaver Kugler (1900). Die Babylonische Mondrechnung (PDF). pp. 8–21.
  3. See "Five Millennium Catalog of Solar Eclipses". NASA.
  4. 1 2 Rob van Gent. "A Catalogue of Eclipse Cycles - List of Eclipse Cycles". Utrecht University.
  5. 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)
  6. 1 2 Saros-Inex Panorama. Data in Solar eclipse panaorama.xls.