Eclipse contact times, the magnitude and the duration of annularity all ultimately depend on the angular diameters and relative velocities of the Sun and the Moon. Unfortunately, these calculations are limited in accuracy by the departure of the Moon's limb from a perfectly circular figure. The Moon's surface exhibits a rather dramatic topography which manifests itself as an irregular limb when seen in profile. Most eclipse calculations assume some mean lunar radius which averages high mountain peaks and low valleys along the Moon's rugged limb. Such an approximation is acceptable for many applications, but if higher accuracy is needed, the Moon's actual limb profile must be considered. Fortunately, an extensive body of knowledge exists on this subject in the form of Watt's limb charts [Watts, 1963]. These data are the product of a photographic survey of the marginal zone of the Moon and give limb profile heights with respect to an adopted smooth reference surface (or datum). Analyses of lunar occultations of stars by Van Flandern [1970] and Morrison [1979] have shown that the average cross-section of Watts' datum is slightly elliptical rather than circular. Furthermore, the implicit center of the datum (i.e. - the center of figure) is displaced from the Moon's center of mass. In a follow-up analysis of 66000 occultations, Morrison and Appleby [1981] have found that the radius of the datum appears to vary with libration. These variations produce systematic errors in Watts' original limb profile heights which attain 0.4 arc-seconds at some position angles. Thus, corrections to Watts' limb profile data are necessary to ensure that the reference datum is a sphere with its center at the center of mass.

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 eclipse path, the Moon's topocentric libration
(physical + optical
libration) in longitude ranges from l=-0.7deg. to l=-2.5deg..
Thus, a limb
profile with the appropriate libration is required in any
detailed analysis of
contact times, central duration's, etc.. Nevertheless, a
profile with an
intermediate libration is valuable for general planning for
any point along the
path. The center of mass corrected lunar limb profile
presented in
**Figure 7**
is for the center line at the instant of greatest eclipse
(17:11:27 UT). At that
time, the Moon's topocentric librations are l=-1.65deg., b=-
0.12deg. and
c=-17.18deg., and the apparent topocentric semi-diameters of
the Sun and Moon
are 950.3 and 896.2 arc-seconds respectively. The Moon's
angular velocity is
0.289 arc-seconds per second with respect to the Sun.

The radial scale of the profile in
**Figure 7**
(see scale
to upper left) 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 along with the mean limb with
respect to the
center of mass. Note that all the predictions presented in
this paper 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 second and third
contact are clearly
marked along with the north pole of the Moon's axis of
rotation and the
observer's zenith at mid-annularity. 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 Moon's mean lunar
radius **k**
(expressed in Earth equatorial radii), topocentric semi-
diameter **SD** and
horizontal parallax **HP**. As discussed in the section
MEAN LUNAR
RADIUS,
the Moon's mean radius **k** (k=0.2722810) is smaller than
the adopted
IAU
value (k=0.2725076). To the upper right of the profile are
the Sun's
semi-diameter **SUN SD**, the angular velocity of the Moon
with respect
to
the Sun **VELOC.** and the position angle of the path's
northern/southern
limit axis **LIMITS**. In the lower right are the
Universal Times of the
four contacts and maximum eclipse. The geographic
coordinates and local
circumstances at maximum eclipse are given along the bottom
of the figure.

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 which 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 7**,
the Sun's limb can be represented as an epicyclic
curve which is
tangential to the mean lunar limb at the point of contact and
departs from the
limb by **h** as follows:

**h** = **S** (**m**-1) (1-
cos[**C**]) [6]

where: **S** = the Sun's semi-
diameter

**m** = the eclipse magnitude

**C** = the 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 **Delta_** by:

**Delta_** = **d**
**v**
cos[**X** - **C**] [7]

where: **d** = the distance of Solar limb
from mean
lunar limb (arc-sec)

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

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

**C** = the 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 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 which is straight forward 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 7. In interpreting these curves, the circumference of the central protractor functions as the nominal or mean contact time (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 7 for any position angle by using the radial scale in the upper right corner (units in seconds of time). Time corrections external to the protractor (most 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.

Across most of North America, the Moon's topocentric
libration in
longitude at
maximum eclipse is within half a degree of its value at
greatest eclipse.
Therefore, the limb profile and contact correction time
curves in
**Figure 7**
may
be used in all but the most critical investigations.

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