a Ptolemaic values are in the top line and the Copernican on the lower for each planetary entry.

a Ptolemaic values are in the top line and the Copernican on the lower for each planetary entry.

ascending node, W (and, below, its variation in arc-seconds per year), the argument of perihelion, w (and, below, its variation in arc-seconds per year), the mean motion in degrees per day, n, the average sidereal period in mean solar days, and the average synodic period in mean solar days (and the number of integral Earth sidereal years, ©, and remainder in days). The mean motion is not independent of other elements, but it directly indicates the orbital motion of the planet; so we include it here. As we have noted, a combination of angles, the longitude of perihelion (ro) is sometimes given in place of the argument of perihelion (w): ro = W + w. The data for the telescopic planets Neptune and Pluto are included only for completeness. Uranus is marginally visible to the unaided eye. It is conceivable that the motion of Uranus could have been noticed during an appulse or close approach to a star, but its motion is so small, only 20 arc-minutes per month, that this is unlikely to have been noticed in antiquity. Whether it was or was not noticed by someone (see Hertzog 1988), to the present day, no evidence for early nontelescopic observations of Uranus has been found.

The data of Table 2.9 can be used to find the position of a planet in its orbit at subsequent times and its position in the ecliptic and equatorial systems. The mean longitude, €, is related to the mean anomaly through the relation, M = € - ffl = € - w- W [(2.16) to (2.18) in §2.3.5]. A full discussion of the required procedures is beyond the scope of this book, but is provided by several sources.41 Appendix A provides lists of published tables of planetary positions for the remote past, as well as some of the currently available commercial software packages for computing them.

Some of the elements of Table 2.9 may be compared with those of Table 2.10, which lists planetary parameters as reck-

41 For example, Brouwer and Clemence (1961), Danby (1962/1988), or for less-critical determinations, Schlosser et al. (1991/1994, pp. 70-76 and Appendix E).

oned by Ptolemy (2nd century) and by Copernicus (16th century), extracted from values provided by Gingerich (1993, p. 128, fn. 38; p. 214, Table 4). The Ptolemaic values are on the top line, and the Copernican are on the lower, for each planetary entry. The solar distance parameter a is given in units of the average Earth-Sun distance and is tabulated only for the heliocentric model; n, e, and w follow. The parameters that were used to characterize orbits in antiquity are not always the same as the modern elements. All orbits were circular, but a planet's orbit was not centered on the Earth (or, in the Copernican model, on the Sun), so that the Copernican "eccentricity," e, for instance, is the mean distance between the center of the orbit and the Sun and expressed in units of a. In Copernicus's model, this "eccentricity" varies with time, because the center of the orbit moves on a circle (the mean value is given in Table 2.10). As a consequence, the argument of perihelion also varies and adds to the perturbation-induced variation. Altogether, the model of Copernicus required at least six parameters to compute each planet's longitude and five additional parameters to include the effects of his (incorrect) theory of precession (see §§3.1.6, 7.7).

The periodicities that were most noticable and most noted by ancient astronomers were the synodic periods of the planets and those that were commensurate with the solar calendar or other calendars. The formulation of Kepler's Third Law, which relates the sidereal period to the semimajor axis, had to await understanding of the difference between the synodic and sidereal periods, correct planetary distances from the Sun, and, of course, the heliocentric perspective.

Finally, we supply positions of a planet at a particular configuration. Table 2.11 (based on information provided by Jet Propulsion Laboratory astronomer E. Myles Standish) is a partial list42 of dates of inferior conjunctions of Venus. The dates indicated are Julian Day Numbers and decimals thereof and Julian calendar (36,525 days in a century) dates and hours; the uncertainty is about 3 hours. There is a cycle of 251 tropical years for Venus conjunction events. Purely boldfaced dates indicate entries for one such series, and the bold-italicized dates those for another; the latter is carried forward into the 20th century at the end of the table. The 20th century dates, however, are given in the Gregorian calendar (see §4.2.3). Note that the difference in JDN (an accurate indication of the number of days between the conjunctions) is only about 0.03d/cycle.43 Although they certainly did not use the Gregorian or Julian calendars, Mayan astronomers were well aware of these sorts of periodicities of Venus, and of the tropical year, and tied some of them into their sacred calendar (see §12, where the repetitions of Venus phenomena are discussed in the context of the Mesoamerican calendar). Calendrical and iconographic evidence strongly suggests that the complicated series of motions of Venus in the sky over many years were observed carefully. The motion of the perihelion of a planet means

42 This is an updated version of part of a table from Spinden 1930, pp. 82-87.

43 This can be seen as follows: 251 x 365.2422 = 91675.79, while 1955664.29 - 1863988.47 = 91675.82, for example.

Table 2.11. Venus inferior conjunctions.
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