13.08, etc.

They describe merely the interval of successive passages of the Sun (on average) through the Moon's orbital node. The 177d interval, then, is an eclipse interval, and so after another eclipse season, 173d later, or 346d after the first solar node passage, an eclipse does not occur for a further 8d interval. The progression, starting at a solar eclipse when the Sun is precisely at a node of the Moon's orbit, is given in Table 5.3. The differences between the first and third columns are given in the fourth, and tell us whether an eclipse will occur. For solar eclipses, an eclipse may occur if the difference is less than or equal to 18d and must occur if it is less than or equal to 11d. Note, therefore, that not all the intervals of column 3 in the upper section of the table are eclipse intervals, the intervals of 30-48 lunations overshoot the interval; by taking the previous lunation in these (and two earlier) cases, eclipse intervals are again found. The negative values indicate that the other side of the node is involved. The intervals in the latter part of the table are tabulated further, and their possible use in the Maya calendar is described by Spinden (1930, Table VI, pp. 52-53). For lunar eclipses, which are half lunations off from the solar cases, there may be an eclipse if the "Difference" in Table 5.3 is less than 15d, and there must be an eclipse if the "Difference" is less than 10d. These intervals follow from the eclipse limits. Note that at the 24th lunation, the difference between the eclipse season accumulation and the eclipse intervals accumulation is 15d49, which is very close to half a lunation (14d77). Therefore, at full moon, just two weeks prior to the 24th new moon, a lunar eclipse must occur. Furthermore, after 48 lunations, the accumulated intervals are different by slightly more than one full lunation, ~31d resulting in another solar eclipse, 47 lunations after the base eclipse.

There must be at least two solar eclipses in any given eclipse year. There may be three: two at the same eclipse season with the first starting at the western eclipse limit. It is also the case that the 177d eclipse interval is not the shortest interval between successive solar eclipses in different seasons: From the end of one ecliptic limit to the beginning of the next is only about 173d - 33d or 140d, although an eclipse cannot occur until slightly later17 than this interval. The celestial longitude range for a total solar eclipse is only 19.8°-23.7° (twice the minor and major central ecliptic limits), and so it is possible for there to be no total (or annular) solar eclipses in any one calendar year.

The global frequency of lunar eclipses is less than solar eclipses (according to Oppolzer 1887/1962, there are 4000 solar eclipses in 1684 years and 4000 lunar eclipses in 2583 years), but because of their greater visibility, lunar eclipses can be seen more often at any single location on Earth. Smither (1986) calculated that viewed from a particular location (the Maya region), lunar eclipses were 4x as numerous as solar eclipses of any type. There cannot be two umbral lunar eclipses a month apart, but there can be two successive eclipses if at least one lunar eclipse is penumbral (which are more difficult to detect).

There can be no more than a total of seven eclipses— lunar and solar—in any one year. Total solar eclipses at a particular spot on Earth may not repeat for 300 years, although partial eclipses seen from that spot are more frequent.

The basic requirement for a repetition of a particular type of eclipse is that the interval between the two eclipses be an integer multiple of mean synodic months.18 There are many possible eclipse intervals, but some are more interesting than are others. The following five conditions on the interval between eclipses will ensure repetition in several key ways:

(1) An integral number of years will give an eclipse at the same time of the year.

(2) An integral number of anomalistic periods will give an eclipse of the same duration.

(3) An integral number of nodical periods (with condition 1) will give an eclipse over the same latitude region and will also ensure a long series of repetitions.

(4) An integral number of solar days will give an eclipse at the same time of day and therefore over the same longitude region of Earth.

17 The interval of 140d is not a possible eclipse interval for the same reason that 173d interval is not. Suppose an eclipse did occur at the end of the previous eclipse season, i.e., at the eclipse limit. After 140d, the Moon will not be at the right phase for an eclipse to occur, because 140d/29.53059d = 4.74 synodic months. Therefore, a new Moon will not occur again until ~8d later or 148d from the eclipse that occurred at the end of the previous eclipse season. Note further that the second eclipse (148d later) could not occur if the first were at the beginning of the previous eclipse season.

18 Recall that the synodic period varies slightly from month to month because the Moon moves on an eccentric (as well as a perturbed) orbit and therefore moves with varying velocity at different locations in its orbit. In the absence of perturbations, the cycle length of lunar velocity would be the anomalistic period (see Chapter 3), Pa « 27?5546, which is about 2d or 7% shorter than the mean synodic period. The time taken to traverse the additional distance to catch the Sun will be smaller if the Moon is closer to perigee and longer if it is closer to apogee, than average.

(5) An integral number of sidereal months will result in eclipses occurring in the same region of the sky. This condition may have had astrological significance for some cultures.

The rules can be summarized as follows:

i x S = ny x Y = j x A = k x N = nd x D = l X0, (5.1)

where i, j, k, l, nd, and ny are integers and S, N, A, Q, D, and Y are the lengths of the synodic, nodical (draconic), anomalistic, and sidereal months and the lengths of the mean solar day and year, respectively. The integers nd and ny are numbers of days and years, respectively. The conditions can be relaxed for any of the criteria, but the eclipses must take place very close to new or full Moon. A rigid application of all the criteria is virtually impossible in practice. An examination of a few examples (Table 5.4) shows, however, that the intervals can approximate at least some of the ideal conditions. Note that successive eclipses in cycle 1 will be visible at nearly the same place on Earth, but will have differing durations and will occur at different times of year. They will also occur in different regions of the sky. One of the better repetition intervals is sample cycle 3, the well-known Saros, encompassing an interval of 18y11d. According to Neugebauer (1969, p. 141ff), there is no clear evidence that the Saros eclipse cycle was used in ancient Mesopotamia.19 It does not succeed in meeting all the criteria in (5.1), but it does meet most of them: The eclipses occur at nearly the same time of year, will be seen from about the same latitudes,20 and have similar durations; moreover, each succeeding eclipse will take place on the sky within half a zodiacal sign (<15°) of the previous eclipse in the series. There is one serious criticism of the idea that eclipses in this series were actually predicted by ancient cultures, however. Each successive eclipse takes place ~1/3 day later than the preceding, so that the Earth rotates ~120° in this extra interval. From such circumstances, only an eclipse warning table could be constructed—one that indicated the possibility, though not the certainty, that an eclipse would occur. This is a distinct possibility for a table of eclipses found in the "Dresden Codex" from Mesoamerica, used by the Maya (see §12.10).

19 Neugebauer (1969, p. 116) states that a column found in linear zigzag functions of system A tables from the Seleucid era relates to the Saros cycle of 223 mean synodic months, but that its purpose was to compute the variable length of the synodic month. The term relates to an interval of 3600y by Berossos (~290 b.c.), following a meaning of the Babylonian sign "sa'r," 3600. The application of the term "Saros" to an eclipse cycle (presumed known to the Babylonians) was due to a mistaken hypothesis about an 11th-century manuscript by Edmund Halley (1691 and 1692). Halley's view, which was strongly attacked by Le Gentil in 1766 cited in Neugebauer (1969, p. 142), nevertheless survived to propagate and repropagate through the literature. Kugler (1900) suggested that eclipses were computed during the Seleucid era from a study of the lunar latitudes relative to that when the Earth, Moon, and Sun were aligned (in syzygy). Neugebauer (1969, p. 142) does, however, suggest that some evidence points to a crude 18-year cycle as the means for a lunar eclipse repetition cycle prior to the Seleucid era.

20 For solar eclipses, there is a slow northern shift from eclipse to eclipse in the Saros series, in which the eclipses occur at a descending node; there is a southern shift in successive Saros eclipses involving the ascending node.

Table 5.4.

Eclipse repetitions table.

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