Download Babylonian Astronomy Literature

Babylonian Astronomy
Teije de Jong
Astronomical Institute ‘’Anton Pannekoek”
University of Amsterdam
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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added; the Babylonian astronomy chapters 1 - 6 are somewhat outdated]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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1
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
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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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2
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
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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
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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)
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Ruin of the ziggurat of Borsippa
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5
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)
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Development of cuneiform script
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6
The sexagesimal number system
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A numerical example
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7
Time keeping: the calendar
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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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8
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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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9
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
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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 Æ 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) from
604 – 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)
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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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11
Astronomical Diaries: the observational database
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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.)
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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 [....]
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22 April 568 BC 18:40, 15 minutes after sunset
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14
29 April 568 BC 19:30, 1 hr after sunset
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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
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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) Æ
Ξ (EF) and from
Ω (EL) Æ Γ (MF)
Venus is
invisible (dashed
lines) because of
its proximity to
the Sun
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Orbit of the inner planet Venus seen from the Earth
•Like the Moon Venus has different phases (crescent Æ 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
ƒSaturn moves from right to left in this diagram (increasing ecliptic
longitude)
ƒFor clarity sake the motion in latitude is amplified by a factor 4
ƒSaturn is invisible (dashed lines) from Ω (EL) Æ Γ (MF), reaches its
stationary points at Φ (ES) and Ψ (WS) and is in opposition (with the
Sun) in Θ (Acronychal rising)
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Orbit of an outer planet seen from the Earth
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Planetary phases
Ω: Evening Last
Γ: Morning First
Ξ: Evening First
Σ: Morning Last
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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 Almagest
and are still used today for computations of planetary visibility
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19
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
Sterrengids.
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Venus in Holland in 2001
• Venus disappeared as evening star at the Western
horizon (EL) on 22 March 2001
N.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
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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 Uruk
and 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 error
introduced 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
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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 dates
are thus computed and not observed.
• The Uruk scheme illustrates the extent to which the Babylonians
trusted their theories
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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
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The Saros
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The Saros
• Lunar and solar eclipses are already mentioned in the oldBabylonian 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
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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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25
BM 34597 Obv.
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BM 34597 Obv. (copy)
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BM 34597 Obv. (translation)
Line
1
5
10
15
20
21-38
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column II
column III
year mm
year mm
ArtIII 12 IV 5 ITU AlexIII 2 IV 5 ITU
X
X
13 IV
3 IV
intercal X
X
14 III
4 IV
IX
intercal X
15 III
5 III
IX
IX
16 II
5 ITU
6 II
5 ITU
intercal VIII
VIII
17 I
7 II
VII
intercal VIII
18 I
Phil 1 I
VII
VII
XII2
2I
19 VI
VII
XI 5 ITU intercal XII 5 ITU
20 V
3V
XI
XI
21 V
4V
continued on reverse side
column II
column III
Julian date
347 BC 21-Jul 329 BC 31-Jul
24-Jan
346
14-Jan 328
10-Jul
20-Jul
345
03-Jan 327
14-Jan
28-Jun
09-Jul
23-Dec 326
03-Jan
344
17-Jun
28-Jun
12-Dec
23-Dec
343
08-May 325
19-May
02-Nov
12-Nov
342
28-Apr 324
08-May
01-Nov
22-Oct
341
28-Apr
17-Apr 323
10-Oct
21-Oct
340
06-Apr 322
17-Apr
26-Sep
07-Oct
339
24-Feb 321
07-Mar
31-Aug
21-Aug
338 BC 14-Feb 320 BC 24-Feb
10-Aug
20-Aug
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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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27
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 a
narrow 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
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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 draconitic
month (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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28
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
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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
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Planetary periods - 2
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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Planetary periods - 3
Summary of main planetary periods in BM 41004:
Saturn:
59 years (- 6 days)
Jupiter:
71 years
Mars:
47 years (+ 4 days)
Venus:
8 years (- 4 days)
Mercury:
46 years (- 1 day)
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Venus: Amsterdam 23 June 1996, 5 a.m.
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31
Venus: Amsterdam 21 June 2004, 5 a.m.
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Planetary periods - 4
• 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 the
smallest common multiple of the three periods]
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Planetary periods - 5
BM 41004 (Sect. 3, l. 1-17) The passings of the Planets by the Normal Stars
Babylonian
Modern
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
-1
Venus
8
-4
-4 583,922
5
8
-3
-4
Mars
47
4 779,936
22
47
-6
1
Jupiter
71
0 398,884
65
71
-1
0
Saturn
59
-6 378,092
57
59
0
-6
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Planetary periods - 6
• 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
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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 δ
Cap (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 an
essential 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
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BM 46083
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Normal stars 1 - 18
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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 Æ 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)
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Planetary theory: ephemerides
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37
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
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AO 6457 – hand-written copy
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AO 6457 – translation
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Jupiter – System A theory
• Parameters:
– Angular velocity:
30º per “time step” for λ = 85º - 240º, and
36º per “time step” for λ = 240º - 85º
– Transition from slow to fast arc by linear interpolation so that
accuracy of the scheme is retained (no accumulating errors)
– “Time step” Δt (“tithis”) = Δλ (º) + 12;05,10
• 1 tithi = 1/30 of lunar month Æ 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
Babylonian Calendar
Year Month Day Fraction
Julian Calendar
Year Month Day Time
Jupiter longitude
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
Note: Vernal equinox in 200 BC at Babylonian longitude ~ 5:50o Aries
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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 successive
heliacal risings/settings by enforcing strict periodicity of the
phenomena over long periods of time.
• Next to the step function approach of system A the Babylonian
astronomers also developed a numerically even more sophisticated
system B based on the same principles but now using so-called zigzag functions (a crude numerical approximation to our trigonometric
functions).
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Development of Babylonian astronomy - 1
2000 – 1500 BC
Text: 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 BC
Text: 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
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Development of Babylonian astronomy - 2
750 BC
Start of Astronomical Diaries
700 – 500 BC
Solstices and Equinoxes
27-year Sirius cycle
Saros and Planetary Periods
500 – 300 BC
19-year calendar cycle (Uruk scheme)
360º Zodiac
Lunar and Planetary Theories
300 – 0 BC
Extensive computational and continued observational activity
Dissemination 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 (Æ 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
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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.
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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.
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