Lunar Limb Profile

Fred Espenak

Eclipse contact times, magnitude and duration of totality (or annularity) all depend on the angular diameters and relative velocities of the Moon and Sun. 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 radius that 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 Watts' 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. Additional work by Morrison and Appleby [1981] shows that the radius of the datum varies with libration producing systematic errors in Watts' original limb profile heights that attain 0.4 arc-seconds at some position angles. Thus, corrections to Watts' limb 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 and may be used to generate limb profiles for any libration. Ellipticity and libration corrections can 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.

The lunar limb profile in Figure x includes corrections for center of mass and ellipticity [Morrison and Appleby, 1981]. It is generated for one instant along the central line which typically is close to the time of greatest eclipse. The Moon's topocentric libration (physical + optical), the topocentric semi-diameters of the Sun and Moon and the Moon's angular velocity with respect to the Sun are all included in this figure.

The radial scale of the limb profile (at the bottom of the lunar profile figure) 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 web 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 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-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 for the given time 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 the lunar profile figure, the Sun's limb can be represented as an epicyclic curve that is tangent to the mean lunar limb at the point of contact. Using the digitized Watts' datum, an analytical solution is straightforward and robust. Curves of corrections to the times of second and third contact for most position angles have been computer generated and plotted. 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.

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Reproduction of Eclipse Data

All eclipse calculations are by Fred Espenak, and he assumes full responsibility for their accuracy. Permission is freely granted to reproduce this data when accompanied by an acknowledgment:

"Eclipse Predictions by Fred Espenak, NASA's GSFC"

For more information, see: NASA Copyright Information

2007 Apr 26