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Appendix III. The Method Used in Checking Recorded Eclipses
For the checking of the eclipses recorded in Chinese records with astronomical computations there is the monumental Canon der Finsternisse, by Th. von Oppolzer, published in Vienna, in 1887, in which are calculated all the eclipses from 1208 B.C. to A.D. 2161. This book of tables gives the day, hour, and minute when eclipses occurred, the longitude of the sun at the time of the conjunction, and charts the paths of the central eclipses. Since and during Oppolzer's time, improvements in astronomical computations have shown that his calculations may be slightly in error. In the case of some Chinese eclipses, corrections of Oppolzer may be found in F. K. Ginzel, Spezieller Kanon der Sonnenund Mondfinsternisse für das Ländergebiet der klassischen Altertumswis-senschaften und den Zeitraum von 900 vor Chr. bis 600 nach Chr., Berlin, 1899. Ginzel has again been corrected. The most recent work is P. V. Neugebauer, Astronomische Chronologie, Berlin, 1929. For Han times, Oppolzer is correct to within about half an hour. 1 As no one since Oppolzer has produced a set of tables and charts covering Chinese territory, so that his book is the only one convenient to use, and as so little correction of his results is necessary, we may take Oppolzer's calculations as being as reliable now as ever, except at regions near the limit of visibility and when great exactness is necessary.
Oppolzer charts only the paths of umbral (i.e., total, annular, and annular-total) eclipses (more exactly, only central eclipses). In most cases, the Chinese however observed the sun as only partially eclipsed. The region in which an umbral eclipse may be viewed as partial extends in all directions from the umbral path. Oppolzer did not calculate this area. We have allowed, on Oppolzer's charts, 15/16 to 1 inch at right angles to the path of an umbral eclipse and 3/8 of an inch from the ends in the direction of that path, as the area in which an umbral eclipse may be seen as partial, remembering however that when we approach those limits, we may need to make allowance for Oppolzer's errors and also that the region of partiality may extend for much greater distances. 2
More than one-third of the total number of solar eclipses---about 35.3 per cent---are nowhere umbral. These "partial eclipses" should be distinguished from umbral eclipses which are viewed from some points as partial. Since in partial eclipses the moon's umbra does not touch the earth, Oppolzer has not indicated them on his charts. Yet they figure among the eclipses recorded in Chinese history.
Astronomical chronology has sometimes neglected partial eclipses; J. Fr. Schroeter's Spezieller Kanon der zentralen Sonnen- und Mondfinsternisse, Kristiania, 1923, which claims to chart all eclipses visible in Europe from 600 to 1800 A.D., omits partial eclipses entirely, even though many such were visible within that region. Such eclipses are sometimes quite conspicuous, sometimes being visible even from the polar circle as far south as five degrees to the other side of the equator, so that historians have recorded them. In dealing with these partial eclipses, we are aided by certain empirical principles, known to the Greeks, but not much used in modern times. Eclipses recur at intervals of 6585.3 days (18 years and 10 or 11 days), which period is called a saros. At intervals of three saroi, 19,756 days (approximately 54 years 1 month), called by Ptolemy an exeligmos, solar eclipses recur at approximately the same place. If the eclipses occurring at intervals of an exeligmos are plotted on a map, it will be found that they form a series which has certain definite characteristics. Such a series invariably begins with a run of partial eclipses visible in the neighborhood of one of the earth's poles, followed by a long run of umbral eclipses, which gradually shift to the other pole of the earth, and ends with a run of partial eclipses. 3 A solar exeligmos series covers a period of from 1244 to 1532 years or 23 to 28 exeligmoi, with an initial and terminal run of 2 to 8 exeligmoi during which only partial eclipses occur. If the noon points of the umbral eclipses are plotted for such a series, the resulting curves 4 show periods of shifting and periods of quietude, in the latter of which the curve usually describes a loop. In periods of shifting, the noon points during a period of 6 exeligmoi may shift as much as 180 degrees of longitude and 90 degrees of latitude; whereas, in a period of quiescence, at a node, the noon points during as much as 8 exeligmoi may be located within an area of 60° of longitude and 30° of latitude or less.
Since partial eclipses are those in which the moon's umbra (or its prolongation) passes over one of the earth's poles, and since successive eclipses in the same exeligmos series are located close to each other, it is possible to determine definitely that a partial solar eclipse was not visible in China by counting backwards or forwards by exeligmoi to the first umbral eclipse. If that umbral eclipse was visible in the south polar regions, the partial eclipse cannot have been visible in China. In this manner half of the partial eclipses may be disposed of. A simpler method of discovering whether a partial eclipse is visible in China is by noting Oppolzer's calculated value of γ for that eclipse. If it is negative and between -1.0 and -1.6, the eclipse was partial and visible only in the southern hemisphere. By the above methods the regions of visibility for about five-sixths of all eclipses may be approximated without calculation.
If the nearest umbral eclipse in the same exeligmos series was visible near the north pole, we cannot say with any certainty by this method whether the partial eclipses in that series were visible in China. Less than one-third of such partial eclipses are visible there, since only the eclipse at one of the three saroi in an exeligmos can usually be visible in China, and since the farther partial eclipses are from the nearest umbral eclipse in the same exeligmos series, the more they tend to move northwards, outside of Chinese latitudes. Because of the shifting in longitude shown by some series, a determination of the approximate circumstances of a partial eclipse from umbral eclipses in the same exeligmos series is not reliable.
The visibility of partial eclipses in the northern hemisphere and of umbral eclipses that lie near the limits of visibility must be calculated. For that purpose the method used in Neugebauer, Astronomische Chronologie, has been used. A useful preliminary step is to use the elements given by Oppolzer with Neugebauer's tables as the latter directs (Op. cit. I, p. 103). In this way little more than a mere inspection of Neugebauer's tables is necessary. But the results thus obtained are only approximate, and are useful only to eliminate eclipses that are plainly invisible in China. In all cases where exact results are required, the computation should be carried through as directed by Neugebauer. The elements given in Oppolzer furnish a useful check upon such computations.
1. In my computations of Han eclipses by Neugebauer's method I have found Oppolzer's times for eclipses in error by varying amounts, from less than a minute to a maximum of 36 minutes.
2. To determine approximately the region in which an umbral eclipse may be seen as partial, I transferred the plotted charts for some 35 recent eclipses from the nautical almanacs to polar coordinates, like those used by Oppolzer in his charts, and measured the areas of partial visibility. These eclipses belonged to 26 different saros series and included all the umbral eclipses of the 19 years preceding 1938, in order to include all the saros series now running which produce umbral eclipses. While this is only a small proportion of the 152 saros series in Oppolzer's Kanon, the uniformity of results makes them probably typical. The areas of visibility for 23 of these eclipses were conchoid and for 12 were cylindrical.The curves for the areas in which an umbral eclipse is viewed as partial are of two sorts: cylindrical and conchoid. In those cases in which the moon's umbra reaches regions in the vicinity of the earth's equator, the limits of partial visibility for the eclipse are roughly parallel to the umbral path, so that the area of visibility swept by the moon's penumbra forms a sausage-shaped curve. I shall call such eclipses "cylindrical" in default of a better term. When the moon's umbra comes closer to the poles, its penumbra reaches beyond the northern (or southern) limits to which the sun is visible, so that the eclipse is visible only as far north (or south) as the sun is visible and the northern (or southern) border of the area of partial visibility is cut off. Then the area of visibility assumes a shell-like shape. I shall call such eclipses "conchoid." In the balance of this discussion, for simplicity's sake, I shall consider only the northern hemisphere, in which China is located. These results however apply equally well to the southern hemisphere, with directions reversed.Conchoid curves are obtained for those eclipses in which the moon's penumbra reaches the northern limits to which the sun is visible; such eclipses are those near the ends of a saros or exeligmos series. Cylindrical curves are obtained for those eclipses in which the penumbra does not reach that far north. This limit depends, of course, upon the season of the year: at the summer solstice, the sun reaches "north" as far as the polar circle on the opposite side of the globe; at the winter solstice, it reaches only to the polar circle on this side; at the equinoxes, it reaches to the poles; in between, the northern limits of the sun depend upon the date in the year.For cylindrical eclipses, the area of partial visibility, as measured on charts similar to and of the same dimensions as those of Oppolzer, extended for a distance of 15/16 to 2 inches at right angles to the path of umbral eclipse, with a mode of 1(1/4); inches, and also extended from 3/8; to 1 inch in the same direction as and beyond the ends of that path.For conchoid eclipses, outside the polar regions, that area extended from 1 to 2(5/8); inches at right angles to the path of the umbral eclipse and from 3/8; to 1 inch from the ends in the direction of that path. Within the polar regions, greater variations occurred. The area of visibility sometimes extended only for 3/8; of an inch at right angles to the path of centrality or only 1/16 inch from the ends in the direction of the path. Since however China does not extend that far north, we may neglect these limits.The area of visibility reaches its greatest size when the line joining the ends of the umbral path lies north and south and the convex side of that path faces west; then the eclipse is visible for a distance of 1(1/2); to 2 inches south from the umbral path (except near the ends), for 1(1/2); to 1(3/4); inches westwards from the ends of that path, and from 1/16 to (1/2); inch in the direction opposite to south from the ends of that path (which direction may also be south but across the pole). In proportion as the umbral path is tilted away from a general east and west direction, these unusual conditions are likely to be realized.
3. The data upon which this part of the discussion is based will be found in the papers of Dr. Alexander Pogo in Popular Astronomy, vol. 43, 1935.
4. For such curves, cf. Neugebauer, Astronomische Chronologie, I, p. 24; Pogo, Popular Astronomy, Jan. 1935; W. Hartner, "Das Datum der Shih-ching-finsternisse," T'oungpao, 1935, diagram, p. 202-3.
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