InterAcademiaal College 2011 Chapters from the history of astronomy. Numerical methods in ancient astronomy

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1 InterAcademiaal College 2011 Chapters from the history of astronomy Numerical methods in ancient astronomy A computation of the positions of the Eastern stationary point of Jupiter in the 2nd century BC according to Babylonian methods Teije de Jong Assignment: Start with the times and positions of Jupiter as given on the reverse of tablet AO 6476 (= ACT No. 600). These are computed according to Babylonian system A theory of Jupiter. Continue the computations beyond the end of the tablet (Seleucid Era 173 = 139 BC) to try to connect to the positions given on another tablet BM (= ACT No. 607) which covers the period SE (= BC). Discuss agreement and difference between the computational results on the two tablets. Table of contents: Copy of AO6476 (= ACT No.600) Reverse side Transcription of AO6476 Rev. Copy of BM (= ACT No.607) Transcription of BM Babylonian month names Babylonian logograms of the signs of the zodiac Babylonian calendar tables SE Cumputational prescription Comparison of BM with modern longitudes References: J. Evans, The History and Practice of Ancient Astronomy, New York (1998), p O. Neugebauer, Astronomical Cuneiform Texts - 3 Vols., London (1955), Vol. II R.A. Parker and W.H. Dubberstein, Babylonian Chronology BC, Providence (1956) B.L. van der Waerden, Science Awakening II. The Birth of Astronomy, Leyden (1974), p

2 AO6476 (= ACT No. 600) Rev. Copy - 2 -

3 AO 6476 (= ACT No. 600) Rev. Transcription Written: Seleucid Era 118, month VII, day 12 (4 October 194 BC) Provenance: Uruk Notes to transcription: Column I: year in Seleucid Era (S.E.) me = 100 kin-a = year with 2 nd Ululu (month VI) a = year with 2 nd Addaru (month XII) Column II: time interval (fractional days) Column III: calendar date (lunar month, fractional days) Column IV: ecliptic longitude (fractional degrees, zodiacal signs) - 3 -

4 BM (= ACT No. 607) Copy - 4 -

5 BM (= ACT No. 607) Transcription Written: date unknown Provenance: Babylon Notes to transcription: [.] indicates restored text Column I: date (S.E. year, month, day) Column II: Longitude of eastern stationary point of Jupiter (fractional degrees, zodiacal sign, unit) Column III: date (S.E. year, month, day) Column IV: Longitude of opposition of Jupiter (fractional degree, zodiacal sign, unit) - 5 -

6 Names of the months in the Babylonian calendar Babylonian logograms of the signs of the Zodiac Logogram Latin (Dutch) name # Longitude lu Aries (ram) 1 0º - 30º múl Taurus (stier) 2 30º - 60º maš Gemini (tweelingen) 3 60º - 90º kušú Cancer (kreeft) 4 90º - 120º a Leo (leeuw) 5 120º - 150º absin Virgo (maagd) 6 150º - 180º rín Libra (weegschaal) 7 180º - 210º gír-tab Scorpio (schorpioen) 8 210º- 240º pa Sagittarius (boogschutter) 9 240º - 270º máš Capricornus (steenbok) º - 300º gu Aquarius (waterman) º - 330º zib Pisces (vissen) º - 360º - 6 -

7 Babylonian Calendar Tables (Parker & Dubberstein 1956) - 7 -

8 Computation of dates and positions of the Eastern stationary point of Jupiter in ACT Definitions: λ1 = longitude of Jupiter at time t1 λ2 = longitude of Jupiter at time t2 Δλ = λ2 λ1, and Δt = t2 t1 2. Computation of longitude λ2 The longitude λ2 follows from where λ2 = λ1 + Δλ Δλ = 30 when 85 < λ1 & λ2 < 240 ("slow" arc), and Δλ = 36 when 0 < λ1 & λ2 < 85, or 240 < λ1 & λ2 < 360 ("fast" arc). In case the newly computed λ2 exceeds the transition point from the "fast" to the "slow" arc, or vice versa, the longitudes have to be adjusted by interpolation λ2 = λ (λ )/6 λ2 = λ (λ )/5 for λ1 < 85 < λ2 for λ1 < 240 < λ2 Notes: 2.1. To keep the computation of this interpolation simple the Babylonians made sure (also for the other planets) that the ratio of the fast and the slow arc was always (reducable to) a simple ratio and that often one of the numbers of this ratio was a multiple of three or six (like here 5 : 6) 2.2. Different velocities in different sectors of the ecliptic are due to the "anomaly" (deviation from uniform circular motion) in the motion of the Earth (Sun) and the planet in its elliptical orbit (eccentricity). This results in different values for the "fast" and "slow" arcs and in different intervals along the ecliptic for each planet. 3. Computation of date t2 The magnitude of the time step follows immediately from the relation Δt (in tithis) = Δλ (in ) + 12;05,10. This relation is basic to the Babylonian computational approach (see Introduction). Using the value of Δλ computed above we obtain Δt and then finally the time at which Jupiter reaches λ2 t2 = t1 + Δt - 8 -

9 Accuracy of Babylonian computed positions ACT 607: Jupiter Eastern stationary point in BC Babylonian date Julian date Longitude SE mo day yr BC mo day Bab Modern B-M 208 II May 2 282;06 277;32 4; II Jun 7 318;06 313;04 5; IV Jul ;06 349;37 4; V Aug 18 30;06 25;55 4; VI Sep 24 66;06 60;47 5; VIII 6 99 Oct 28 99;15 93;46 5; VIII Nov ;15 125;08 4; IX Dec ;15 155;30 3; XI Jan ;15 185;40 3; XI Feb ;15 216;26 2; XII Mar ;06 248;34 2;32 mean 4;10 r.m.s. 0;57 Note 1: Julian date is computed for midnight of Babylonian date (see Parker and Dubberstein, Babylonian Chronology 626 BC- 75 AD, 1956) Note 2: Vernal Equinox at 4;28 Babylonian longitude in 100 BC - 9 -