The Watts charts have been digitized by Her Majesty's Nautical Almanac Office in Herstmonceux, England, and transformed to grid-profile format at the U. S. Naval Observatory. In this computer readable form, the Watts limb charts lend themselves to the generation of limb profiles for any lunar libration. Ellipticity and libration corrections may be applied to refer the profile to the Moon's center of mass. Such a profile can then be used to correct eclipse predictions which have been generated using a mean lunar limb.

Along the 1998 eclipse path, the Moon's topocentric libration (physical + optical) in longitude ranges from l=ñ1.7° to l=ñ3.4°. Thus, a limb profile with the appropriate libration is required in any detailed analysis of contact times, central durations, etc.. Since the land based portions of the path occur over a narrow range of librations (i.e.: l=ñ2.3° to l=ñ3.1°), a profile with an intermediate value is useful for general planning purposes and may even be adequate for most applications. The lunar limb profile presented in Figure 14 includes corrections for center of mass and ellipticity [Morrison and Appleby, 1981]. It is generated for 18:00 UT, which corresponds to northern Colombia near Venezuela. The Moon's topocentric libration is l=ñ2.81°, and the topocentric semi-diameters of the Sun and Moon are 969.1 and 1011.1 arc-seconds, respectively. The Moon's angular velocity with respect to the Sun is 0.358 arc-seconds per second.

The radial scale of the limb profile in
Figure 14
(at bottom)
is greatly exaggerated so that the true limb's departure from
the mean lunar limb is readily apparent. The mean limb with respect
to the center of figure of Watts' original data is shown (dashed)
along with the mean limb with respect to the center of mass (solid).
Note that all the predictions presented in this publication are
calculated with respect to the latter limb unless otherwise noted.
Position angles of various lunar features can be read using the
protractor marks along the Moon's mean limb (center of mass).
The position angles of all four contact points are clearly marked
along with the north pole of the Moon's axis of rotation and the
observer's zenith at mid-totality. The dashed line with arrows
at either end identifies the contact points on the limb corresponding
to the northern and southern limits of the path. To the upper
left of the profile are the Sun's topocentric coordinates at maximum
eclipse. They include the right ascension **R.A.**, declination
**Dec.**, semi-diameter **S.D.** and horizontal parallax
**H.P.**. The corresponding topocentric coordinates for the
Moon are to the upper right. Below and left of the profile are
the geographic coordinates of the center line at 18:00 UT while
the times of the four eclipse contacts at that location appear
to the lower right. Directly below the profile are the local circumstances
at maximum eclipse. They include the Sun's altitude and azimuth,
the path width, and central duration. The position angle of the
path's northern/southern limit axis is **PA(N.Limit) **and
the angular velocity of the Moon with respect to the Sun is **A.Vel.(M:S).
**At the bottom left are a number of parameters used in the
predictions, and the topocentric lunar librations appear at the
lower right.

In investigations where accurate contact times are needed, the
lunar limb profile can be used to correct the nominal or mean
limb predictions. For any given position angle, there will be
a high mountain (annular eclipses) or a low valley (total eclipses)
in the vicinity that ultimately determines the true instant of
contact. The difference, in time, between the Sun's position when
tangent to the contact point on the mean limb and tangent to the
highest mountain (annular) or lowest valley (total) at actual
contact is the desired correction to the predicted contact time.
On the exaggerated radial scale of
Figure 14,
the Sun's limb can
be represented as an epicyclic curve that is tangent to the mean
lunar limb at the point of contact and departs from the limb by
**h** through:

where: **h** = departure of Sun's limb from mean lunar
limb

**S** = Sun's semi-diameter

**m** = eclipse magnitude

**C** = angle from the point of contact

Herald [1983] has taken advantage of this geometry to develop
a graphical procedure for estimating correction times over a range
of position angles. Briefly, a displacement curve of the Sun's
limb is constructed on a transparent overlay by way of equation
[6]. For a given position angle, the solar limb overlay is moved
radially from the mean lunar limb contact point until it is tangent
to the lowest lunar profile feature in the vicinity. The solar
limb's distance **d** (arc-seconds) from the mean lunar limb
is then converted to a time correction
by:

where: = correction to contact time (seconds)

**d** = distance of Solar limb from Moon's mean limb (arc-sec)

**v** = angular velocity of the Moon with respect to the
Sun (arc-sec/sec)

**X** = center line position angle of the contact

**C** = angle from the point of contact
This operation may be used for predicting the formation and location
of Baily's beads. When calculations are performed over a large
range of position angles, a contact time correction curve can
then be constructed.

Since the limb profile data are available in digital form, an analytical solution to the problem is possible that is quite straightforward and robust. Curves of corrections to the times of second and third contact for most position angles have been computer generated and are plotted in Figure 14. The circular protractor scale at the center represents the nominal contact time using a mean lunar limb. The departure of the contact correction curves from this scale graphically illustrates the time correction to the mean predictions for any position angle as a result of the Moon's true limb profile. Time corrections external to the circular scale are added to the mean contact time; time corrections internal to the protractor are subtracted from the mean contact time. The magnitude of the time correction at a given position angle is measured using any of the four radial scales plotted at each cardinal point.

For example, Table 17 gives the following data for Maracaibo, Venezuela:

Second Contact = 18:04:03 UT P2=103°

Third Contact = 18:06:53 UT P3=198°

Using Figure 14, the measured time corrections and the resulting contact times (to the nearest second) are:

C2=ñ2.5 seconds; Second Contact = 18:04:03 ñ2s = 18:04:01 UT

C3=ñ1.0 seconds; Third Contact = 18:06:53 ñ1s = 18:06:52 UT

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