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4 May 2011 IAC2011 History of Astronomy 1

Babylonian Astronomy

Teije de Jong

Astronomical Institute ‘’Anton Pannekoek”University of Amsterdam


Literature

1. A. Pannekoek, De Groei van ons Wereldbeeld, Wereld-bibliotheek, Amsterdam (1951) [sections 1 - 6 treat Babylonian astronomy]

2. O. Neugebauer, The Exact Sciences in Antiquity, Brown University Press, Providence (1957) [stimulating reading on Babylonian mathematics and astronomy in chapters I,II, III & V]

3. A. Pannekoek, A History of Astronomy, Interscience Publishers, New York (1961) [this is an English translation of ref. 1 with notes and references added; the Babylonian astronomy chapters 1 - 6 are somewhat outdated]

4. B.L. van der Waerden, Science Awakening II, Noordhoff, Leiden (1974) [very clear presentation of Babylonian astronomy in chapters 2, 3, 4, 6 & 7]

5. O. Neugebauer, History of Ancient Mathematical Astronomy (3 Vols.), Springer, Berlin (1975) [Vol. 1, Book II presents detailed account of Babylonian Astronomy]

6. J.P. Britton and C.B.F. Walker, Astronomy and Astrology in Mesopotamia, in Astronomy before the Telescope, ed. C.B.F.Walker, British Museum Press, London (1996) [quite readable overview of Babylonian astronomy]

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Literature (continued)

7. J. Evans, The history and practice of ancient astronomy, Oxford University Press (1998) [sections on Babylonian astronomy interspersed throughout the book]

8. H. Hunger and D. Pingree, Astral sciences in Mesopotamia, Brill, Leiden (1999) = HP [authorative textbook on Babylonian astronomy]

9. J.M. Steele, A Brief Introduction to Astronomy in the Middle East, Saqi, London (2008) [first two chapters deal with Babylonian astronomy]

Required reading for the Exam:Steele (ref. 9), p. 9 - 65


Outline

• Introduction• Time keeping: the calendar• Diaries: the observational database• Astronomical intermezzo• Sirius and the solar year• Cycles:

– the Saros– Planetary periods

• Normal stars: the ecliptic coordinate system• Planetary theory: ephemerides

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Introduction

• Mesopotamia (Babylonia and Assyria) = area between the rivers Euphrates and Tigris (present-day Iraq; see #6)

• Mesopotamian culture documented on ~ ½ million clay tablets containing texts in cuneiform script

• Cultural continuity over more than three millennia in spite of many changes in rulership (see slide #7)

• City of Babylon main centre of astronomical activity (see slide #8), but also Nineveh, Uruk and Sippar

• At present ~ 2000 tablets with astronomical texts translated and interpreted


The Near East in Antiquity

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Historical Overview

• < 2000 BC: Sumerians, Mesopotamian city-states• ~2000 BC – ~1500 BC: Old-Babylonian period, terminated

by Hittite invasion• ~1500 BC – ~1200 BC: Cassite rule• ~1200 BC – 627 BC: Assyrian period; terminated with

destruction of Nineveh (royal library of Assurbanipal)• 627 BC – 538 BC: New Babylonian period• 538 BC – 330 BC: Persian rule• 330 BC – 311 BC: Greek rule• 311 BC – 129 BC: Seleucid period• 129 BC – 226 AD: Arsacid period


Babylon during Nebuchadnezzar II (604 – 561 BC)

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Introduction (continued)

• Astronomy plays a central role in Babylonian religion; the priesthood attempts to read the will of the Gods from the heavens, and advises the king accordingly

• This provides a strong motivation to study the sky (see “ziggurat’ in slide #10)

• Astronomical knowledge and astrological lore intermingled and practiced by the same priests/scholars

• Climatic conditions excellent for astronomical observation(2π steradian sky and < 100 mm precipitation/year

• Continuity in astronomical activity over > 2000 years• Exceptional mathematical talent (sexagesimal place-value number

system)


Ruin of the ziggurat of Borsippa

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Cuneiform script and number system• Cuneiform script was developed by the Sumerians in southern

Mesopotamia over the period 3300 – 2500 BC• Initially used for administrative purposes; later also for literary texts• Written with reed stylus on wet clay tablets • Gradually evolving from pictographic signs to the well-developed

script that was used in Mesopotamia until ~ 0 AD (see slide #12)• The Sumerian logograms were kept for writing Akkadian (Semitic

language) in Babylonia and Assyria when Sumerian was no longer spoken (after ~1800 BC)

• Akkadian was gradually replaced as a spoken language by Aramaic during the 1st millennium BC but the cuneiform script was kept for writing religious and scientific texts until ~ 75 AD

• Sumerian and Babylonian number system (see slides #13 & 14)– Place-value system– Fractions– Base 60 (sexagesimal)


Development of cuneiform script

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The sexagesimal number system


A numerical example

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Time keeping: the calendar


The Babylonian calendar

• Calendar of importance for civil organisation, agriculture and religion

• Babylonian calendar is lunar calendar with 12 months of 29 or 30 days

• New month begins on day of first (re-)appearance of lunar crescent after sunset

• Day is reckoned from sunset to sunset (still the case in derivative Jewish and Arabic calendars)

• Month I synchronized with Spring Equinox (barley harvest)

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The Baylonian calendar - 2

• Year of 12 lunar months is about 11 days shorter than solar year, so seasons run through the year

• Solution: intercalation = occasionally adding of extra month to correct for this (month VI2 or XII2)

• Until about 600 BC irregular intercalation by decree of the King

• King Hammurabi (~1800 BC) to Sin-iddinam:“This year has an additional month. The coming month should be designated as the second month Ululu [month VI], and wherever the annual tax had been ordered to be brought to Babylon on the 24th of the month Tashritu [month VII] it should now be brought to Babylon on the 24th of the second month Ululu.”

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Babylonian calendar - 3

• On tablet II of astronomical compendium MUL.APIN (ca. 1000 BC) we find the following intercalation prescription:“If KAK.SI.SÁ (= the arrow = Sirius) becomes visible on the 15th of Du’uzu (month IV) this year is normal”.“If KAK.SI.SÁ (= the arrow = Sirius) becomes visible on the 15th of Abu (month V) this year is a leap year”.

• Commentary:– Heliacal rising of Sirius occurs each year on the same

solar day (same position w.r.t. Sun)– Babylonian lunar calendar synchronized to solar year

by using date of heliacal rising of Sirius to decide about intercalation of additional month


The Babylonian calendar - 4• From ~ 500 BC onwards regular intercalation in 19-year

cycle with 7 additional months (after year2/monthXII, y5/mXII, y8/mXII, y10/mXII, y13/mXII, y16/mXII & y19/mVI)

• In earlier periods often intercalation of month VI rather than month XII (see slide #21)

• Intercalation scheme of 19-year cycle such that heliacal rising of Sirius always falls in month IV

• From the equality 19 tropical years (365.2422 days) = 235 synodic months (29.53059 days) we find that the 19-year cycle is a little more than 2 hours too long after 400 years only 2 days!

• 19-year cycle provides calendar that is comparable in accuracy to the Julian calendar (43 BC – 1582 AD)

• Continuous calendar record from ~750 BC onwards

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The Babylonian calendar - 5

Dates (Julian calendar) of the 1st day of the Babylonian month (see slide #22) from604 – 595 BC; 4/2 = 2 April, 5/1 = 1 May, etc. (from Parker & Dubberstein 1956).Notice that around 600 BC most intercalations involved second Ululu’s (month VI)


Babylonian calendar - 6

The logograms give the reading of the cuneiform symbols representing the month names. They consist of the abbreviations of the Sumerian month names of the Nippur calendar.

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Astronomical Diaries: the observational database


Astronomical Diaries

• “Astronomical Diaries” is a class of texts containing records of observations and computations made during a period of half a year (6 or 7 months)

• The oldest Diary so far dated was recorded in 652 BC; the latest Diary dates from 61 BC

• The systematic recording probably started in 747 BC, the first year of the reign of Nabû-nāsir, since cuneiform “reports” of lunar eclipses grouped in 18-year (“Saros”) periods and extracted from the Diaries begin in that year

• About 1200 (fragments of) tablets with in total ~ 400 months of recorded observations from 652 BC – 61 BC (~5% of all nights in that period) are preserved

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Astronomical Diaries - 2

The Diaries contain the following information:– Dates of planetary phases (first/last visibility, stationary

points)– Dates on which moon and planets pass “normal stars”– Times of rising and setting of the moon– Dates of solstices and equinoxes (computed)– Dates of Sirius phenomena (often computed)– Dates of sign entries of planets (> 3rd century BC)– Atmospheric phenomena (wind, rain, halos, etc.) – Miscellaneous information (historic events, market

prices, sickness, river levels, etc.)


VAT 4956: an Astronomical Diary from 568 BC

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VAT 4956: an Astronomical Diary for 568 BC - 2

#1 Year 37 of Nebukadnezar, king of Babylon. Month I, (the 1st of which was identical with) the 30th (of the preceding month), the moon became visible behind the Bull of Heaven; [sunset to moonset:] .... [....]

#2 Saturn was in front of the Swallow. The 2nd, in the morning, a rainbow stretched in the west. Night of the 3rd, the moon was 2 cubit in front of [....]

#3 it rained ?. Night of the 9th (error for: 8th), beginning of the night, the moon stood 1 cubit in front of the hind leg of the Lion (= β Virginis). The 9th, the sun in the west [was surrounded] by a halo [.... The 11th]

#4 or 12th, Jupiter’s acronychal rising. On the 14th, one god was seen with the other; sunrise to moonset: 4®. The 15th, overcast. The 16th, Venus [....]

#5 The 20th, in the morning, the sun was surrounded by a halo. Around noon, ...., rain PISAN. A rainbow stretched in the east. [....]

#6 From the 8th of month XII2 to the 28th, the river level rose 3 cubits and 8 fingers, 2/3 cubit [were missing] to the high flood [....]

#7 were killed on order of the king. That month, a fox entered the city. Coughing and a little rišūtu-disease [....]


22 April 568 BC 18:40, 15 minutes after sunset

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29 April 568 BC 19:30, 1 hr after sunset


Astronomical Diaries - 3

Hunger & Pingree (1999) about the Diaries:“That someone in the middle of the eighth century BC conceived of such a scientific program and obtained support for it is truly astonishing; that it was designed so well is incredible; and that it was faithfully carried out for at least 700 years is miraculous.”

[Assignment: Use the astronomical information in line #3 of Tablet VAT4956 to determine the equivalent in angular degrees of 1 cubit]

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Astronomical Intermezzo


Planetary phases

The Babylonians recorded the dates at which the planets reached the following “phases” for the outer planets:– Morning First (MF = Γ)– Eastern stationary point (ES = Φ)– Acronychal rising (≈ opposition = Θ)– Western stationary point (WS = Ψ)– Evening Last (EL = Ω)

and for the inner planets:– Morning First (MF = Γ)– Morning Last (ML = Σ)– Evening First (EF = Ξ)– Evening Last (EL = Ω)

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The phases of Venus

From Σ (ML) Ξ (EF) and from Ω (EL) Γ (MF) Venus is invisible (dashed lines) because of its proximity to the Sun


Orbit of the inner planet Venus seen from the Earth

•Like the Moon Venus has different phases (crescent full)•Its distance to the Earth varies from 0.3 – 1.7 AU (Earth-Sun distance) •Its brightness varies over its orbit from V = −3.8 to −4.2 mag•It is never more than 48º away from the Sun•As seen from the Earth it moves back and forth with respect to the Sun in loops

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The phases of the outer planet Saturn

Saturn moves from right to left in this diagram (increasing ecliptic longitude)For clarity sake the motion in latitude is amplified by a factor 4Saturn is invisible (dashed lines) from Ω (EL) Γ (MF), reaches its

stationary points at Φ (ES) and Ψ (WS) and is in opposition (with the Sun) in Θ (Acronychal rising)


Orbit of an outer planet seen from the Earth

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Planetary phases

Γ: Morning First Ω: Evening Last

Σ: Morning Last Ξ: Evening First


Arcus Visionis

• The concept of “arcus visionis” (the arc h in slide #33) was introduced in Greek Astronomy by Ptolemy in ~150 AD (Almagest VIII,6 & XIII,7)

• The arcus visionis is defined as the minimum angular distance between the star/planet and the Sun measured perpendicular to the horizon for the star/planet to be visible

• The fainter the star/planet the larger the arcus visionis• For his Handy Tables Ptolemy used the following values:

– Saturn: 13º– Jupiter: 9º– Mars: 14º 30’– Venus: 5º (EL, MF) & 7º (ML, EF)– Mercury: 12º

• These values are somewhat better than those given in the Almagestand are still used today for computations of planetary visibility

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The position of Venus in the sky in Holland at different dates just before sunrise and just after sunset in 2001.Mercury is also shown.Ref: De Sterren-gids.


Venus in Holland in 2001

• Venus disappeared as evening star at the Western horizon (EL) on 22 March 2001N.B. date depends on geographical latitude of observer

• Venus reappeared as Morning star (MF) in the East on 9 April 2001

• Its invisibility period thus amounted to 18 days• How to carry out the observations:

– MF: On the day of first visibility the planet is just visible in the East for about 5-10 minutes while it rises in the brightening morning twilight about 1 hour before sunrise

– EL: On the day of last visibility the planet is just visible in the West for about 5-10 minutes while it sets in the darkening evening twilight about 1 hour after sunset. The next day is the first day of the invisibility period.

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Sirius and the solar year: the Uruk scheme


Sirius and the solar year

• A scheme to compute the dates of first/last visibility of Sirius and the equinox/solstice dates is known from a number of tablets from Urukand Babylon, based on the 19-year calendar cycle (Sachs 1952, Journal of Cuneiform Studies 6, 105-114)

• In this so-called “Uruk scheme” seasons of 93 tithis and an “epact” (= increment) of 11;03,10 tithis are used to proceed from year to year

• A “tithi” (named as such in Indian astronomy; the Babylonians refer to days) is a computational “day” defined as 1/30 of one lunar month so that one lunar month contains always 30 tithis while in reality it lasts 29 or 30 days (1 synodic month = 29.530588 days)

• By using tithis as a computational ”day” the Babylonians avoided having to keep track of the number of days in a month; the errorintroduced in computed dates than never exceeds one day because the first day of the month was determined observationally

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The Uruk scheme


The Uruk scheme

• The 19-year cycle contains 235 lunar months so that 1 year = 12;22,06,18,56,.. months equivalent to 12 months + 11;03,09,28,.. tithis; the latter was approximated to 11;03,10 tithis for practical computational purposes

• The resulting scheme is periodic with a period of 19 years and 18 increments of 11 tithis and one increment of 12 tithis

• Since this scheme is anchored in the 19-year cycle it has a similar accuracy with an error of about 1 day in 300 years

• According to the Uruk scheme the invisibility period of Sirius lasts 65 tithis

• In the Uruk scheme the Sirius dates are related to the summer solstice (SS) dates by MF = SS + 21t and EL = SS + 327t. The winter solstice date and the equinox dates then follow from the season length of 93 tithis (i.e. 3 lunar months + 3 days)

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The Uruk scheme

• All Sirius and equinox/solstice dates in Babylonian astronomical texts (including the Diaries) from the last four centuries BC turn out to comply with dates computed according to this scheme; these datesare thus computed and not observed.

• The Uruk scheme illustrates the extent to which the Babylonians trusted their theories


Cycles: Saros and planetary periods

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Cycles: the Saros and Planetary Periods

• An essential element in the development of Babylonian theoretical astronomy is the recognition of periodic behaviour of lunar and planetary motion

• Based on the periods of these cycles the Babylonians eventually developed arithmetical schemes by which they were able to predict lunar and planetary positions with an accuracy of order 1º over timescales of hundreds of years


The Saros

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The Saros• Lunar and solar eclipses are already mentioned in the old-

Babylonian omen series “Enuma Anu Enlil” consisting of about 70 tablets (2nd millennium BC)

• A continuous lunar eclipse record was probably available from Assyrian times onwards (~ 750 BC)

• Reports and letters sent by Assyrian and Babylonian astronomers to the Assyrian kings Esarhaddon and Assurbanipal in the 7th century BC show awareness that lunar eclipse possibilities occurred at intervals of 6 or, occasionally, 5 months

• From the available texts it appears that about one century later (early 6th century BC) a detailed scheme to predict lunar eclipses based on an 18-year cycle (the so-called Saros) had been worked out


The Saros - 2

• The Saros (a misnomer due to English astronomer Edmund Halley in 1691) derives from the equality: 223 synodic months (29.53089d) ≈ 242 draconitic months (27.212220d) ≈ 239 anomalistic months (27.554550d) ≈241 sidereal months (27.321661d) ≈ 18 years

• In Babylonian texts the Saros is simply referred to as the “18 years”

• The tablet on the next slide contains part of the so-called “Saros canon” (Aaboe et al. 1991, Transactions of the Americn Philosophical Society 81, part 6)

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BM 34597 Obv.


BM 34597 Obv. (copy)

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BM 34597 Obv. (translation)

Line year mm year mm1 ArtIII 12 IV 5 ITU AlexIII 2 IV 5 ITU 347 BC 21-Jul 329 BC 31-Jul

X X 346 14-Jan 328 24-Jan13 IV 3 IV 10-Jul 20-Jul

intercal X X 345 03-Jan 327 14-Jan5 14 III 4 IV 28-Jun 09-Jul

IX intercal X 23-Dec 326 03-Jan15 III 5 III 344 17-Jun 28-Jun

IX IX 12-Dec 23-Dec16 II 5 ITU 6 II 5 ITU 343 08-May 325 19-May

10 intercal VIII VIII 02-Nov 12-Nov17 I 7 II 342 28-Apr 324 08-May

VII intercal VIII 22-Oct 01-Nov18 I Phil 1 I 341 17-Apr 323 28-Apr

VII VII 10-Oct 21-Oct15 XII2 2 I 340 06-Apr 322 17-Apr

19 VI VII 26-Sep 07-OctXI 5 ITU intercal XII 5 ITU 339 24-Feb 321 07-Mar

20 V 3 V 21-Aug 31-AugXI XI 338 BC 14-Feb 320 BC 24-Feb

20 21 V 4 V 10-Aug 20-Aug21-38 continued on reverse side

column II column III column IIJulian date

column III


The Saros - 3

• The Saros has the following structural elements (some first explained by Pannekoek in 1917):– each Saros contains 38 eclipse possibilities, grouped in clusters of

7 or 8 eclipse possibilities– eclipse possibilities within a cluster are separated by 6 months

intervals– clusters are separated by 5-month intervals– cluster are arranged in an 8-7-8-7-8 sequence– eclipses at the same location within the Saros (on the same line in

the Table) have approximately the same magnitude and duration• In the Table eclipse possibilities explicitly mentioned in Babylonian

astronomical texts (either discussed or observed) are underlined• In the Table Lunar eclipses visible in Babylon are printed bold face

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The Saros - 4

• Necessary conditions for a lunar eclipse:– Earth in between Sun and Moon (full moon)– Moon close to its nodes = intersection of lunar and solar orbit

• Solar eclipses require similar conditions but their location is much more difficult to predict because the lunar shadow cone traces anarrow path (~100 km) on the earth surface. Therefore the Babylonians were able to predict the date on which a solar eclipse could occur but not whether it actually occurred


The Saros - 5

• The Saros can be explained in terms of the difference between the length of the synodic month (S = 29.530588 days) and the draconiticmonth (D = 27.212220 days) because the Sun and Moon must be opposite each other (full moon) and the Moon must be close enough to the ecliptic, i.e within ~30º of one of its two nodes corresponding to 2.3 days in the draconitic month)

• Then the Saros pattern is provided by the month numbers n that satisfy the condition n x S mod (D/2) < 2.3 days

[Assignment: check this and visualize what happens]

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The Saros - derived parameters

• From the number of synodic months in the Saros the Babylonians derived a value for the length of the solar (sidereal) year of 12;22,07,50,.. synodic months compared to the modern value of 12;22,07,29,.. [accuracy ~10-5]

• In their “system A” lunar and planetary theories the Babylonians use the approximated value 1 yr = 12;22,08 months

• From the number of days in the Saros they derived for the length of the synodic month a value of 29;31,50,8,20 days compared to the modern value of 29;31,50,07,03,.. [accuracy ~10-7]

• The latter value was adopted by Hipparchos and later by Ptolemy for their geometric theory of lunar motion and was widely used by astronomers until the Renaissance


Planetary periods

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Planetary periods - 1

• Periodicity in recurrence of planetary phases• Planetary phases = first and last visibilities (MF & EL for

the outer planets and MF, ML, EF & EL for the inner planets) and stationary points

• Periods explicitly listed on several tablets and implicit in “goal-year” texts and planetary almanac texts

• Example:– Section of 17 lines on tablet BM 41004 (Neugebauer &

Sachs 1967; text E)– Tablet copied by Iddin-Bēl (late 4th century BC)– Tablet gives periods for planets to return to same

sidereal position and to same day in lunar calendar



Part of the text of Tablet BM 41004 (Neugebauer and Sachs 1967, Journal of Cuneiform Studies 21, 183-217)

(10) The passing by of Mars at the Normal Stars. In 32 years it lacks 5 days in your year.(11) Variant: in 47 years, it goes 4 days beyond.(12) Variant: in 64 years it goes 4 days beyond your year. Secret: in 126 years you will find (the same day as before).(13) The passing by of Saturn at the Normal Stars. In 59 years it lacks 6 days to your year.(14) In 30 years it goes 9 days beyond your year.[…](15) In 30 years it passes by its place 7:20º to the east. In 147 (years) you will find the same day as before.

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Summary of main planetary periods in BM 41004:Saturn: 59 years (- 6 days)Jupiter: 71 yearsMars: 47 years (+ 4 days)Venus: 8 years (- 4 days)Mercury: 46 years (- 1 day)


Venus: Amsterdam 23 June 1996, 5 a.m.

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Venus: Amsterdam 21 June 2004, 5 a.m.



• Using their observational database the Babylonians noted that after certain periods of time the phases of the planets recurred on the same day in the lunar calendar and at the same position in the sky

• Among the surviving textual material we find lists of the dates of first and last visibility for all planets which may have been used for this purpose

• We can reproduce these periods by computing the smallest common multiple of the planetary synodic period, the sidereal year (365.256363 days) and the synodic month (29.53089 days)

• The results for the main planetary periods in BM 41004 are listed in the table in the next slide (#65)

[Assignment: Reproduce the periods in the Table by computing thesmallest common multiple of the three periods]

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Planet Bab Per Del(lon) Del(cal) T(syn) Syn rev Period Del(lon) Del(cal)[yrs] [degr] [days] [days] [#] [sid yrs] [degr] [days]

Mercury 46 -1 115,877 145 46 0 -1Venus 8 -4 -4 583,922 5 8 -3 -4Mars 47 4 779,936 22 47 -6 1Jupiter 71 0 398,884 65 71 -1 0Saturn 59 -6 378,092 57 59 0 -6

BM 41004 (Sect. 3, l. 1-17) The passings of the Planets by the Normal StarsModernBabylonian



• It turns out that in most cases the periods selected by the Babylonians are quite accurate: a remarkable achievement

• For Mars which is the most difficult to model because of the relative large eccentricity of its orbit the Babylonian numbers are a compromise between accuracy in time and position

• These periods were used to construct ephemerides for a future year x by using observations of year (x - period) for each planet separately (“goal-year” method)

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Normal stars: the ecliptic coordinate system


Normal stars: the ecliptic coordinate system

• In the Astronomical Diaries hundreds of observations of the Moon and planets passing by “Normal Stars” are recorded

• The same list of 32 Normal Stars - all lying close to the ecliptic - was used throughout the period covered by the preserved Diaries (650 – 0 BC)

• One of the great achievements of Babylonian astronomy is the introduction of a theoretical zodiac, consisting of 12 “signs” of 30 UŠ(degrees) each

• A schematic mathematical coordinate system is a pre-requisite (probably simultaneously developed) for the creation of a mathematical theory of planetary motion

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Normal stars - 2

• The Diaries provide direct evidence that the 360º ecliptic was anchored in the list of Normal Stars

• For 12 listings of “sign entries” in Gemini, Cancer and Aquarius we have a simultaneous record of the passage of ζ Tau, β Gem and δCap (Huber 1958; Jones 2003)

• The oldest reference to zodiacal signs in the Diaries dates from 385 BC

• The 360º zodiac was probably introduced in the 5th century BC as anessential step in the development of lunar and planetary theories

• BM 46083 (Sachs 1952) was until recently the only (badly damaged) tablet containing (part of) a list of Normal Star positions in the Babylonian ecliptic coordinate system


BM 46083

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Normal stars 1 - 18


Normal stars 19 - 32

Positions in ecliptic coordinates computed for 100 BC; from HP p.148-149

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Zero-point of Babylonian zodiac

• The zero-point of the Babylonian zodiac is fixed with respect to the stars (close to the star η Piscium).

• The position of the zero-point can be derived by using the sign entry dates of planets and the dates of Normal Star passages of planets recorded in a number of texts the position of the Vernal equinox in 100 BC at Babylonian ecliptic longitude λ = 4º28’ ± 20’ (P.J. Huber, Centaurus 5, 192-208, 1958)

• N.B.– Precession was not known to the Babylonians– Ecliptic latitude was incorporated in the theory of the Moon

(eclipses!) but played a minor role in planetary theories (only one text known so far: BM 37266)


Planetary theory: ephemerides

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Babylonian theory of planetary motion

• Based on the observed periodicity in the dates of heliacal rising/setting and of the stationary points of the planets

• Taking account in an empirical way of the variation in angular velocity of the Sun, Moon and planets (due to the eccentricity of the planetary orbit)

• Numerically described by so-called “step” functions in System A and “zig-zag” functions (approximations to trigonometric functions) in System B

• Example: Computation of Eastern stationary points of Jupiter for 199 – 139 BC recorded on tablet AO 6457


AO 6457 – hand-written copy

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AO 6457 – translation


Jupiter – System A theory

• Parameters:– Angular velocity:

30º per “time step” for λ = 85º - 240º, and36º per “time step” for λ = 240º - 85º

– Transition from slow to fast arc by linear interpolation so thataccuracy of the scheme is retained (no accumulating errors)

– “Time step” Δt (“tithis”) = Δλ (º) + 12;05,10• 1 tithi = 1/30 of lunar month greatly simplifies computation and is

always accurate within ½ day• The beauty of this system lies in its:

– Numerical simplicity– Constant “accuracy” over hundreds of years

• How were these parameters determined from observation?

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Comparison of Babylonian and modern positions

Note: Vernal equinox in 200 BC at Babylonian longitude ~ 5:50o Aries

Year Month Day Fraction Year Month Day Time Babylonian Modern B-M

113 I 28 41:40 199 BC Apr 23 7:40 278:06 272:43 5:23

117 VI 11 2:20 195 BC Sep 15 15:56 62:06 55:45 6:21

121 XI 3 12:00 190 BC Jan 19 19:48 185:55 180:52 5:03

Babylonian Calendar Julian Calendar Jupiter longitude


Babylonian Theory of Planetary Motion

• Conclusion: Using their observational database and their arithmetic talents Babylonian astronomers derived the parameters of system A from the known planetary periods and from observed variations in the longitude intervals and in the time intervals between successiveheliacal risings/settings by enforcing strict periodicity of thephenomena over long periods of time.

• Next to the step function approach of system A the Babylonian astronomers also developed a numerically even more sophisticatedsystem B based on the same principles but now using so-called zig-zag functions (a crude numerical approximation to our trigonometricfunctions).

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Development of Babylonian astronomy - 1

2000 – 1500 BCText: Enuma Anu Enlil (70 tablets with “astronomical omina” (= signs) and astrological forecasts)– Luni-solar calendar in use– Stellar constellations– Lunar eclipse and planetary “omina”– Regular observational activity

1500 – 1000 BCText: MUL.APIN (2 tablets with astronomical compendium)– Early theoretical schemes (e.g. ideal 360-day calendar)– Risings and settings of stars and constellations– Planetary (in)visibility periods– Intercalation schemes– Solstices and equinoxes– Length of day and night


Development of Babylonian astronomy - 2750 BC

Start of Astronomical Diaries700 – 500 BC

Solstices and Equinoxes27-year Sirius cycleSaros and Planetary Periods

500 – 300 BC19-year calendar cycle (Uruk scheme)360º ZodiacLunar and Planetary Theories

300 – 0 BCExtensive computational and continued observational activityDissemination of Babylonian methods and parameters to Hellenistic world and India

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Astronomical heritage from Mesopotamia

• Most of (the names of) the stellar constellations• Sexagesimal place-value number system• Theoretical Zodiac of 360º divided in 12 signs of 30º each• The 19-year calendar cycle ( later Jewish and Arabic

calendar, “Easter” computus)• Eclipse dates and planetary positions and lunar and

planetary parameters employed by Hipparchos and Ptolemy for Hellenistic geometrical theories of the motion of Sun, Moon and planets (Almagest, 2nd century AD)

• Division of hour in minutes


Exam questions - 1

1. The Babylonian calendar. The Babylonians constructed a lunar calendar that was adapted to the solar year. Describe the evolution of this calendar by answering the following question:- How did they observationally determine the beginning of a new month?- How did they synchronize the lunar calendar to the solar year?- What regular cycle was eventually institutionalized?- What was the role of the star KAK.SI.SA in the synchronization process?- What is the accuracy of the calendar cycle adopted?

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Exam questions - 2

2. Saros. The recognition of the periodic behavior of astronomical phenomena is basic to the Babylonian astronomy and to the development of the arithmetical techniques used to explain and predict astronomical phenomena. The periodicity in lunar eclipses was one of the first such patterns recognized and analyzed. Explain this pattern in terms of our present understanding of the lunar orbit and describe the way in which the Babylonians constructed Saros tables that could be used for predicting future lunar eclipses.


Exam questions - 3• 3. Sirius and the solar year. Compute the first visibility

dates of Sirius in the Babylonian calendar from 199 – 174 BC by using the 19-year calendar cycle and the epact that goes with it (the so-called Uruk scheme). Make use of the fact that in the Astronomical Diary of 196 BC we read: “Month IV, the 1st (which followed the 30th). The 1st, Sirius’ first appearance…” Also compute the dates of the vernal equinox in those years.

babylonian astronomy - astrophysicsfverbunt/iac2011/babastr.pdf · babylonian astronomy ......

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