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 1997 eclipse path, the Moon's topocentric libration (physical + optical libration) in longitude ranges from l=+1.7° to l=+1.0°. Thus, a limb profile with the appropriate libration is required in any detailed analysis of contact times, central durations, etc.. Nevertheless, a profile with an intermediate libration is valuable for general planning purposes. The lunar limb profile presented in Figure 5 includes corrections for center of mass and ellipticity [Morrison and Appleby, 1981]. It is generated for 1:00 UT, which corresponds to the eastern Russia near the Mongolian border. The Moon's topocentric libration in longitude is l=+1.55°, and the topocentric semi-diameters of the Sun and Moon are 966.5 and 1005.8 arc-seconds, respectively. The Moon's angular velocity with respect to the Sun is 0.495 arc-seconds per second.

The radial scale of the limb profile in
Figure 5 (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 in the center of the diagram. 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 arrows identify the points on
the limb which define 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 1:00 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 5, 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 the following equation.

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: **Delta** = 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 analytic solution to the problem is possible that is straightforward and quite 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 5. In interpreting these curves, the circumference of the central protractor functions as the nominal or mean contact time (i.e. - calculated using the Moon's mean limb) as a function of position angle. The departure of the correction curve from the mean contact time can then be read directly from Figure 5 for any position angle by using the radial scale at bottom right (units in seconds of time). Time corrections external to the protractor (about half of all second contact corrections) are added to the mean contact time; time corrections internal to the protractor (all third contact corrections) are subtracted from the mean contact time.

Throughout Asia, the Moon's topocentric libration in longitude at maximum eclipse is within 0.3° of its value at 1:00 UT. Therefore, the limb profile and contact correction time curves in Figure 5 may be used in all but the most critical investigations.

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