From the polynomial form of the Besselian elements, any element
can be evaluated for any time 't1' (in decimal
hours) via the equation:
a = a0
+ a1*t +
(or a = Sum [an*tn];
n = 0 to 3)
where: a = x, y, d, l1, l2, or µ
t = t1 ñ t0
(decimal hours) and t0 = 17.000 TDT
The polynomial Besselian elements were derived from a least-squares fit to elements rigorously calculated at five separate times over a six hour period centered at t0. Thus, the equation and elements are valid over the period 14.00 .le. t0 .le. 20.00 TDT.
Table 2 lists all external and internal contacts of penumbral and umbral shadows with Earth. They include TDT times and geodetic coordinates with and without corrections for ÆT. The contacts are defined:
P1 - Instant of first external tangency of penumbral shadow cone with Earth's limb.
(partial eclipse begins)
P2 - Instant of first internal tangency of penumbral shadow cone with Earth's limb.
P3 - Instant of last internal tangency of penumbral shadow cone with Earth's limb.
P4 - Instant of last external tangency of penumbral shadow cone with Earth's limb.
(partial eclipse ends)
U1 - Instant of first external tangency of umbral shadow cone with Earth's limb.
(umbral eclipse begins)
U2 - Instant of first internal tangency of umbral shadow cone with Earth's limb.
U3 - Instant of last internal tangency of umbral shadow cone with Earth's limb.
U4 - Instant of last external tangency of umbral shadow cone with Earth's limb.
(umbral eclipse ends)
Similarly, the northern and southern extremes of the penumbral and umbral paths, and extreme limits of the umbral center line are given. The IAU longitude convention is used throughout this publication (i.e. - for longitude, east is positive and west is negative; for latitude, north is positive and south is negative).
The path of the umbral shadow is delineated at five minute intervals in Universal Time in Table 3. Coordinates of the northern limit, the southern limit and the center line are listed to the nearest tenth of an arc-minute (~185 m at the Equator). The Sun's altitude, path width and umbral duration are calculated for the center line position. Table 4 presents a physical ephemeris for the umbral shadow at five minute intervals in UT. The center line coordinates are followed by the topocentric ratio of the apparent diameters of the Moon and Sun, the eclipse obscuration, and the Sun's altitude and azimuth at that instant. The central path width, the umbral shadow's major and minor axes and its instantaneous velocity with respect to Earth's surface are included. Finally, the center line duration of the umbral phase is given.
Local circumstances for each center line position listed in Tables 3 and 4 are presented in Table 5. The first three columns give the Universal Time of maximum eclipse, the center line duration of totality and the altitude of the Sun at that instant. The following columns list each of the four eclipse contact times followed by their related contact position angles and the corresponding altitude of the Sun. The four contacts identify significant stages in the progress of the eclipse. They are defined as follows:
First Contact ñ Instant of first external tangency between the Moon and Sun.
(partial eclipse begins)
Second Contact ñ Instant of first internal tangency between the Moon and Sun.
(central or umbral eclipse begins; total or annular eclipse begins)
Third Contact ñ Instant of last internal tangency between the Moon and Sun.
(central or umbral eclipse ends; total or annular eclipse ends)
Fourth Contact ñ Instant of last external tangency between the Moon and Sun.
(partial eclipse ends)
The position angles P and V identify the point along the Sun's disk where each contact occurs. Second and third contact altitudes are omitted since they are always within 1° of the altitude at maximum eclipse.
Table 6 presents topocentric values from the central path at maximum eclipse for the Moon's horizontal parallax, semi-diameter, relative angular velocity with respect to the Sun, and libration in longitude. The altitude and azimuth of the Sun are given along with the azimuth of the umbral path. The northern limit position angle identifies the point on the lunar disk defining the umbral path's northern limit. It is measured counter-clockwise from the north point of the Moon. In addition, corrections to the path limits due to the lunar limb profile are listed. The irregular profile of the Moon results in a zone of 'grazing eclipse' at each limit that is delineated by interior and exterior contacts of lunar features with the Sun's limb. This geometry is described in greater detail in the section LIMB CORRECTIONS TO THE PATH LIMITS: GRAZE ZONES. Corrections to center line durations due to the lunar limb profile are also included. When added to the durations in Tables 3, 4, 5, and 7, a slightly shorter central total phase is predicted.
To aid and assist in the plotting of the umbral path on large scale maps, the path coordinates are also tabulated at 1° intervals in longitude in Table 7. The latitude of the northern limit, southern limit and center line for each longitude is tabulated to the nearest hundredth of an arc-minute (~18.5 m at the Equator) along with the Universal Time of maximum eclipse at each position. Finally, local circumstances on the center line at maximum eclipse are listed and include the Sun's altitude and azimuth, the umbral path width and the central duration of totality.
In applications where the zones of grazing eclipse are needed in great detail, Table 8 lists these coordinates over land based portions of the path at 7.5' intervals in longitude. The time of maximum eclipse is given for both northern and southern grazing zones as well as the Sun's center line circumstances (altitude and azimuth) and the azimuth of the umbral path. The Elevation Factor and Scale Factor are also given (See: LIMB CORRECTIONS TO THE PATH LIMITS: GRAZE ZONES).
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