CHAPTER THIRTEEN SYZYGIES A conjunction or an opposition of the Sun with the Moon occurs when the elongation between the two luminaries is 0° or 180°, respec- tively. Syzygy is an astronomical term that refers both to conjunction and opposition. At mean syzygy, the double elongation, 2η ̄ = 0°, where ̄ ̄ ̄ ̄ η ̄ = λm – λs, and λm and λs are the mean longitudes of the Moon and the Sun. Analogously, at true syzygy, the double elongation, 2η = 0°, where η = λm – λs, and λm and λs are the true longitudes of the Moon and the Sun. The determination of the time from mean to true syzygy, ∆t, an essential step in the calculation of eclipses, was historically one of the major problems considered in computational astronomy (Chabás and Goldstein 1992 and 1997). Figure 20 represents a mean conjunction of the Sun and the Moon (indicated by the direction of S ̄ and M̄ ) and its corresponding true conjunction (indicated by the direction of S´ and M´). In this case, the ̄ ̄ mean conjunction (when λs = λm), which takes place at time t, comes M S̅ M̅ S S’ M’ λ’ = λ’s O m Aries 0° ̅̅ λλm = s Figure 20: Mean and true syzygies 140 chapter thirteen Table 13.1A: Some historical values of the mean synodic month Mean synodic month 29;31,50,7,37,27,8,25d Parisian Alfonsine Tables 29;31,50,7,54,25,3,32d Levi ben Gerson 29;31,50,8,9,20d al-Ḥajjāj’s Arabic trans. of the Almagest, Copernicus 29;31,50,8,9,24d Ibn Yūnus, al-Bitrūjị̄ 29;31,50,8,14,38d Ibn al-Kammād 29;31,50,8,19,50d al-Battānī 29;31,50,8,20d Almagest, Toledan Tables after the true conjunction λ( s´ = λm´), which occurs at time t´, so that ∆t = t´ – t < 0. The mean synodic month is the time between two consecutive mean syzygies of the same kind, that is, between two successive mean con- junctions or two successive mean oppositions. The length of the mean synodic month was determined in antiq- uity by the Babylonians. The standard parameter in the Almagest is 29;31,50,8,20d, but many variants of it are found in the astronomi- cal literature (see Goldstein 2003; see also Mancha 2002–2003). A few medieval values (some given explicitly and others derived from medi- eval tables by modern computation) are displayed in Table 13.1A. 1. Mean Syzygies In Almagest VI.3 Ptolemy displayed tables for mean conjunctions and oppositions of the Sun and the Moon (see Toomer 1984, pp. 278–280). In fact, there are four different tables, all of them with the same struc- ture. The first is for mean conjunctions, the second for mean opposi- tions, and the other two for the yearly and monthly increments for mean syzygies. The tables proceed in steps of 25 Egyptian years (of 365 days each) and have five columns. The first column is for the argument (periods of 25 years, single years, or months); the second indicates the number of days elapsed for a certain mean syzygy in the periods used as argument; the third is the mean solar anomaly (in degrees); the fourth is the mean lunar anomaly (in degrees); and the fifth is the mean argument of lunar latitude (in degrees). With this tabulated information one can easily derive the time of mean syzygy for any specific date..