P

syn

= P®

38 The heliacal risings and settings of stars are analogous, but simpler because, unlike planets, their annual changes in position are not detectable to the naked eye.

39 The difference between the mean angular rates wplanet and W® is the relative rate: wrel. Because w = 2p/P and Prel = Psyn, we obtain equations (2.23) and (2.24), after division by 2p.

where Psid is the planet's sidereal period and P® is that of the earth. Note the reciprocal relations among the synodic and sidereal periods. If the periods are taken in units of the Earth's sidereal period of revoution around the Sun, the expressions simplify further.

Neugebauer (1969, p. 172) gives "synodic periods" of Saturn and Jupiter: 28;26,40 and 10;51,40, in the sexigesimal (base-60) notation of the Babylonians used by Neugebauer. These quantities, 28y444 and 10y861 in decimal-based notation, are approximately equal to Psid - 1; by setting P® = 1 in (2.24), one finds that this quantity is the ratio of the two periods, viz., Psid/Psyn when they are expressed in units of the Earth's period of revolution. They are not, therefore, the synodic periods as usually defined in astronomy. They are, however, very interesting nevertheless.

In an ancient astronomy context, one can draw a distinction between the time interval for a planet to come to the same configuration, e.g., from opposition to opposition, and the time for it to reappear in the same asterism or at the same celestial longitude. The former is the synodic period as defined astronomically, whereas the latter is a kind of sidereal period, although the motion of the earth around the Sun creates a moving platform and the observation therefore suffers from parallax. Figure 2.24 illustrates the effect of parallax on the apparent direction to the planet in space.

Even with the complication of parallax, ancient astronomy was capable of giving relatively high precision in the periodicities of the planets; the way they did this was to make use of large numbers of cycles. The number of years required for a planet to reach the same configuration, in the same star field, had to be recorded. The number of times the planet moved around the sky through a particular star field provided an integer multiple of the sidereal period. The number of years required for the planet to reach this point in the sky and have the same configuration (with the Sun) is a multiple of the synodic period. The relationship is one of a ratio: mPsid = nPsyn = N years. Hence, if m and n are observed, the ratio of the two type of periods follows. For Saturn, we have m = 9, n = 256, N = 265y; whence, Psid/Psyn = 256/9 = 28.444. For Jupiter, m = 36, n = 391, N = 427y, so that Psid/Psyn = 391/36 = 10.861. Given the total number of years required for the same configuration to be observed40 at the same place among the stars, we can compute, in theory, both Psid and Psyn. For instance, a complete cycle for Saturn would take 265 years. Therefore, Psid = N/9 = 265/9 = 29.444 y, and Psyn = N/256 = 265/256 = 1.0352y. These results can be compared with the modern values, Psid = 29.458 y and Psyn = 1.0352y (see below). For Jupiter, Psid = N/36 = 427/36 = 11.8611 y, and Psyn = N/391 = 427/391 = 1.0921y, compared with modern values, Psid = 11.8622 y and Psyn = 1.0921 y.

The results are excellent for the synodic periods, and the derived sidereal periods are reasonable approximations, but they are not exact. One of the reasons for deviations from modern values is the effect of the shape of the orbit—the orbital eccentricity (others include the accuracy and preci-

40 Or, as in Mesopotamia, calculated, based on the differences between observed and exact ecliptic longitudes in near-repetitions of the phenomena.

sion of length of the year, and the use, exclusively, of the ecliptic longitude and exclusion of the ecliptic latitude). The time interval between repetition of celestial longitude coordinate values (and the mean sidereal period) depends on the traveled portion of the orbit of the planet involved: Near to a common starfield to a common starfield Figure 2.24. The effect of parallax on the apparent direction to a planet: (a) The shift of an exterior planet against the starry background. (b) Compensating motions of the planet and Earth may reduce the parallax shift: The positions of alignment of earth and planet to a distant star are not unique but may occur at nearly any planetary configuration. The three positions of the outer planet shown here place it in the same star field. See Figure 2.18 for the positions of Mars near an opposition. Drawn by E.F. Milone.

Figure 2.24. The effect of parallax on the apparent direction to a planet: (a) The shift of an exterior planet against the starry background. (b) Compensating motions of the planet and Earth may reduce the parallax shift: The positions of alignment of earth and planet to a distant star are not unique but may occur at nearly any planetary configuration. The three positions of the outer planet shown here place it in the same star field. See Figure 2.18 for the positions of Mars near an opposition. Drawn by E.F. Milone.

perihelion, the interval will be shorter than near aphelion. It also depends on the change in position of the earth in its orbit. The length of the synodic period that is specified in most planetary tables is a period that a planet would have if both it and the earth moved at constant, average rates of motion in their respective orbits. The lap difference involves different portions of the orbit and therefore different velocities, reflected in the change of angular motion of the planet across the sky. Of course, the larger the number of cycles that are involved, the smaller is the effect of the remaining segment of the orbit. The ancients were interested in such problems, and we consider the matter somewhat further in §7. At this point, we need to discuss how to characterize orbits.

Table 2.9 lists the mean sidereal and mean synodic periods as well as other orbital parameters for the planets. The sources of the data in Table 2.9 are the Astronomical Almanac for the year 2000 and earlier editions and Allen's Astrophysical Quantities (Allen 1973, pp. 140-141; updated by Cox 2000). The elements refer to the mean equinox and ecliptic for the year 2000. The rates dW/dt and dw/dt and the values of the periods are long-term average values. The precision in the elements actually exceeds the number of significant figures that are shown, but because of the gravitational perturbations produced by the other planets, the elements will vary with time. Following the modern planetary names are the adopted symbols, the semimajor axis or mean distance to the Sun in units of the astronomical unit, a (and, below, the date of a recent passage through perihelion T0), the orbital eccentricity e (and, below it, the mean longitude t), the orbital inclination, the longitude of the

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