# Explanation of Solar Eclipse Predictions

## Table 1 - Elements of the Eclipse

The geocentric ephemeris for the Sun and Moon, various parameters, constants, and the Besselian elements (polynomial form) are given in the Table Elements of the Eclipse. The eclipse elements and predictions were derived from the DE200 and LE200 ephemerides (solar and lunar, respectively) developed jointly by the Jet Propulsion Laboratory and the U. S. Naval Observatory for use in the Astronomical Almanac for 1984 and thereafter. Unless otherwise stated, all predictions are based on center of mass positions for the Moon and Sun with no corrections made for center of figure, lunar limb profile or atmospheric refraction. The predictions depart from normal IAU convention through the use of a smaller constant for the mean lunar radius k for all umbral contacts. Times are expressed in either Terrestrial Dynamical Time (TDT) or in Universal Time (UT), where the best value of ΔT1 available at the time of preparation is used.

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 + a2*t2 + a3*t3 (or a = Sum[an*tn]; n = 0 to 3)

where: a = x, y, d, l1, l2, or µ
t = t1 - t0 (decimal hours)
t0 = Terrestrial Dynamical Time (TDT) to the nearest hour of the instant of greatest eclipse

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: (t0-3 hours) ≤ t1 ≤ (t0 +3 hours).

1ΔT is the difference between Terrestrial Dynamical Time and Universal Time.

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## Table 2 - Shadow Contacts and Circumstances

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 ΔT. The contacts are defined:

 P1 Instant of first external tangency of penumbral shadow cone with Earth's limb. (partial eclipse begins) 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 noethern and southern extremes of the penumbral and umbral paths, and extreme limits of the umbra's central line are given. The IAU (International Astronomical Union) 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).

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## Table 3 - Path of the Umbral (or Antumbral) Shadow

The path of the umbral (or antumbral) shadow is delineated at one minute intervals in Universal Time in Table 3. Coordinates of the northern limit, the southern limit and the central 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 central line position.

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## Table 4 - Physical Ephemeris of the Umbral (or Antumbral) Shadow

Table 4 presents a physical ephemeris for the umbral shadow at one minute intervals in UT. The central line coordinates are followed by the topocentric ratio of the apparent diameters of the Moon and Sun, the eclipse obscuration1, 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 central line duration of the umbral phase is given.

1 Eclipse obscuration is defined as the fraction of the Sun's surface area occulted by the Moon. In comparison, eclipse magnitude is defined as the fraction of the Sun's diameter occulted by the Moon.

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## Table 5 - Local Circumstances on the Central Line

Local circumstances for each central line position listed in Table 5. The first three columns give the Universal Time of maximum eclipse, the central 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 Instant of first external tangency between the Moon and Sun. (partial eclipse begins) Second Contact Instant of first internal tangency between the Moon and Sun. (central or umbral eclipse begins; total eclipse begins) Third Contact Instant of last internal tangency between the Moon and Sun. (central or umbral eclipse ends; total eclipse ends) Fourth Contact 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 occurs1. Second and third contact altitudes are omitted since they are always within 1° of the altitude at maximum eclipse.

1 P is defined as the contact angle measured counter-clockwise from the north point of the Sun's disk. V is defined as the contact angle measured counter-clockwise from the zenith point of the Sun's disk.

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## Table 6 - Topocentric Data and Path Corrections Due to Lunar Limb Profile

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 central line durations due to the lunar limb profile are also included. The addition of these corrections to the durations in Tables 3, 4, 5 and 7, typically results in a slightly shorter central duration for total eclipses and a longer central duration for annular eclipses.

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## Table 7 - Mapping Coordinates for the Central Path

To assist in the plotting of the umbral path on large scale maps, Table 7 gives path coordinates in equal 1° steps of longitude. The latitude of the northern limit, southern limit and central 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 (or annularity).

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## Table 8 - Coordinates for the Zones of Grazing Eclipse

In applications where the zones of grazing eclipse are needed, Table 8 lists these coordinates in equal 30´ steps of longitude. The time of maximum eclipse is given at both northern and southern limits as well as the path's azimuth. The elevation and scale factors are also given (see: Limb Corrections to the Path Limits: Graze Zones).

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