On Wednesday, 2006 March 29, a total eclipse of the Sun will be visible from within a narrow corridor which traverses half the Earth. The path of the Moon's umbral shadow begins in Brazil and extends across the Atlantic, northern Africa, and central Asia, where it will end at sunset in northern Mongolia. A partial eclipse will be seen within the much broader path of the Moon's penumbral shadow, which includes the northern two thirds of Africa, Europe, and central Asia (Figures 1-5).
The central eclipse track begins in eastern Brazil, where the Moons umbral shadow first touches down on Earth at (in Universal Time) 08:36 UT, Figure 6. Along the sunrise terminator, the duration is 1min 53s from the center of the 129km wide path. Traveling over 9 km/s, the umbra quickly leaves Brazil and races across the Atlantic Ocean (with no landfall) for the next half hour. After crossing the equator, the Moons shadow enters the Gulf of Guinea and encounters the coast of Ghana at 09:08UT (Figure 2). The Sun stands 44° above the eastern horizon during the 3min 24s total phase. The path width has expanded to 184km while the shadows ground speed has decreased to 0.958km/s. Located about 50km south of the central line, the 1.7 million inhabitants of Accra , Ghanas capital city (Figure 7), can expect a total eclipse lasting 2min 58s (09:11UT).
Moving inland, the umbra enters Togo at 09:14 UT (Figure 4a). Unfortunately, the capital city Lome lies just outside the southern limit so its inhabitants will only witness a grazing partial eclipse. Two minutes later, the leading edge of the umbra will reach Benin whose capital Porto-Novo experiences a deep partial eclipse of magnitude 0.985. Continuing northeast, the shadow's axis enters Nigeria at 09:21 UT (Figure 8). At this Time the central duration has increased to 3 min 40 s, the Sun's altitude is 52°, the path of totality is 188 km wide and the velocity is 0.818 km/s. Because Lagos is situated about 120 km outside the umbra's southern limit, its population of over 8 million will witness a partial eclipse of magnitude 0.968.
The umbra's axis takes about 16min to cross western Nigeria before entering Niger at 09:37 UT (Figure 9). The central duration is 3 min 54 s as the umbra's velocity continues to decrease (0.734 km/s). During the next hour, the shadow traverses some of the most remote and desolate deserts on the planet (Figures 9 - 12). When the umbra reaches northern Niger (10:05 UT), it briefly enters extreme northwestern Chad before crossing into southern Libya (Figures 4b and 11).
The instant of greatest eclipse† occurs at 10:11:18 UT, when the axis of the Moon's shadow passes closest to the center of Earth (gamma‡ = +0.384) where gamma is the minimum distance of the Moon's shadow axis from Earth's center in units of equatorial Earth radii). Totality reaches its maximum duration of 4min 7s, the Sun's altitude is 67°, the path width is 184km and the umbra's velocity is 0.697 km/s. Continuing on a northeastern course, the umbra crosses central Libya and reaches the Mediterranean coast at 10:40 UT. Northwestern Egypt also lies within the umbral path where the central duration is 3min 58s (Figure 12).
Passing directly between Crete and Cyprus, the track reaches the southern coast of Turkey at 10:54 UT (Figures 5a and 13). With a population of nearly 3/4 million people, Antalya lies 50 km northwest of the central line The coastal city's inhabitants are positioned for a total eclipse lasting 3 min 11 s, while observers on the central line an additional 35s of totality. Konya is 25km from path center and experiences a 3 min 36 s total phase beginning at 10:58 UT. Crossing mountainous regions of central Turkey, the Moon's shadow intersects the path of the 1999 Aug 11 total eclipse. A quarter of a million people in Sivas have the opportunity of witnessing a second total eclipse from their homes in less than five years.
At 11:10 UT, the shadow axis reaches the Black Sea along the northern coast of Turkey (Figure 14). The central duration is 3min 30s, the Sun's altitude is 47°, the path width is 165 km and the umbra's velocity is 0.996 km/s. Six minutes later, the umbra encounters the western shore of Georgia (Figure 15). Moving inland, the track crosses the Caucasus Mountains, which form the highest mountain chain of Europe. Georgia's capital, Tbilisi, is outside the path and experiences a magnitude 0.949 partial eclipse at 11:19 UT. As the shadow proceeds into Russia, it engulfs the northern end of the Caspian Sea and crosses into Kazakhstan (Figure 16). At 11:30 UT, the late afternoon Sun's altitude is 32°, the central line duration is 2min 57s and the umbral velocity is 1.508 km/s and increasing.
In the remaining 17min, the shadow rapidly accelerates across central Asia while the duration dwindles (Figures 3 and 5b). It traverses northern Kazakhstan (Figures 16 and 17) and briefly re-enters Russia (Figures 18 and 19) before lifting off Earth's surface at sunset along Mongolia's northern border at 11:48 UT. Over the course of 3h12min, the Moon's umbra travels along a path approximately 14,500 km long and covers 0.41% of Earth's surface area.
†The "instant of greatest eclipse" occurs when the distance between the Moon's shadow axis and Earth's geocenter reaches a minimum. Although the instant of greatest eclipse differs slightly from the instant of greatest magnitude and the instant of greatest duration (for total eclipses), the differences are usually quite small.
‡Gamma is the minimum distance of the Moon's shadow axis from Earth's center in units of equatorial Earth radii.
Figure 1 is an orthographic projection map of Earth (adapted from Espenak 1987) showing the path of penumbral (partial) and umbral (total) eclipse. The daylight terminator is plotted for the instant of greatest eclipse with north at the top. The sub-Earth point is centered over the point of greatest eclipse and is indicated with an asterisk symbol. The subsolar point (Sun in zenith) at that instant is also shown.
The limits of the Moon's penumbral shadow define the region of visibility of the partial eclipse. This saddle shaped region often covers more than half of Earth's daylight hemisphere and consists of several distinct zones or limits. At the northern and/or southern boundaries lie the limits of the penumbra's path. Partial eclipses have only one of these limits, as do central eclipses when the shadow axis falls no closer than about 0.45 radii from Earth's center. Great loops at the western and eastern extremes of the penumbra's path identify the areas where the eclipse begins and ends at sunriseand sunset, respecTimey. In the case of the 2006 eclipse, the penumbra has both a northern and southern limit so that the rising and setting curves form two separate, closed loops. Bisectting the "eclipse begins and ends at sunrise and sunset" loops is the curve of maximum eclipse at sunrise (western loop) and sunset (eastern loop). The exterior tangency points P1 and P4 mark the coordinates where the penumbral shadow first contacts (partial eclipse begins) and last contacts (partial eclipse ends) Earth's surface. The path of the umbral shadow bisects the penumbral path from west to east.
A curve of maximum eclipse is the locus of all points where the eclipse is at maximum at a given time. They are plotted at each half hour in Universal Time and generally run from northern to southern penumbral limits, or from the maximum eclipse at sunriseor sunset curves to one of the limits. The outline of the umbral shadow is plotted every 10min in Universal Time Curves of constant eclipse magnitude† delineate the locus of all points where the magnitude at maximum eclipse is constant. These curves run exclusivey between the curves of maximum eclipse at sunriseand sunset. Furthermore, they are quasi-parallel to the northern and southern penumbral limits and the umbral paths of central eclipses. Northern and southern limits of the penumbra may be thought of as curves of constant magnitude of 0.0, while the adjacent curves are for magnitudes of 0.2, 0.4, 0.6, and 0.8. The northern and southern limits of the path of total eclipse are curves of constant magnitude of 1.0.
At the top of Figure 1, the Universal Time of geocentric conjunction between the Moon and Sun is given followed by the instant of greatest eclipse. The eclipse magnitude is given for greatest eclipse. For central eclipses (both total and annular), it is equivalent to the geocentric ratio of diameters of the Moon and Sun. Gamma is the minimum distance of the Moon's shadow axis from Earth's center in units of equatorial Earth radii. The shadow axis passes south of Earth's geocenter for negative values of Gamma. Finally, the Saros series number of the eclipse is given along with its relative sequence in the series.
Eclipse magnitude is defined as the fraction of the Sun's diameter occulted by the Moon. It is strictly a ratio of diameters and should not be confused with eclipse obscuration, which is a measure of the Sun's surface area occulted by the Moon. Eclipse magnitude may be expressed as either a percentage or a decimal fraction (e.g., 50% or 0.50).
Figures 2 and 3 are maps using an equidistant conic projection chosen to minimize distortion, and which isolate the African and Asian portions of the umbral path. Curves of maximum eclipse and constant eclipse magnitude are plotted and labeled at intervals of 30min and 0.2, respecTimey. A linear scale is included for estimating approximate distances (in kilometers). Within the northern and southern limits of the path of totality, the outlineof the umbral shadow is plotted at intervals of 10min. The duration of totality (minutes and seconds) and the Sun's altitude correspond to the local circumstances on the central line at each shadow position.
Figures 4 and 5 are maps using an oblique equidistant cylindrical projection and are centered on the eclipse track in Africa (Figures 4a and 4b) and Asia (Figures 5a and 5b). The positions of many cities within and near the path of totality are plotted along with the outlineof the umbral shadow at 10min intervals. Once again, the duration of totality, the Sun's altitude and Timeof central eclipse is shown. The size of each city is logarithmically proportional to its population using 1990 census data (Rand McNally 1991). The scale of these maps is approximately 1:11,100,000.
The path of totality is plotted on a series of detailed maps appearing in Figures 6 - 19. The maps were chosen to isolate small regions along the enTimeland portion of the eclipse path. Curves of maximum eclipse are plotted at 4min intervals along the track and labeled with the central line duration of totality and the Sun's altitude. The maps are constructed from the Digital Chart of the World (DCW), a digital database of the world developed by the U.S. Defense Mapping Agency (DMA). The primary sources of information for the geographic database are the Operational Navigation Charts (ONC) and the Jet Navigation Charts (JNC) developed by the DMA.
The scale of the detailed maps varies from map to map depending partly on the population density and accessibility. The approximate scale of each map is as follows:
| Figure 6 | 1:2,000,000 | |
| Figures 7 - 9 | 1:3,000,000 | |
| Figures 10 - 12 | 1:7,000,000 | |
| Figures 13 - 14 | 1:3,000,000 | |
| Figure 15 | 1:4,000,000 | |
| Figures 16 - 19 | 1:7,000,000 |
The scale of the maps is adequate for showing roads, villages, and cities, required for eclipse expedition planning. The DCW database used for the maps was assembled in the 1980s and contains names of places that are no longer used in some parts of Africa and Asia. Where possible, modern names have been substituted for those in the database, but this correction could not be applied to all sites.
While Tables 1 - 6 deal with eclipse elements and specific characteristics of the path, the northern and southern limits, as well as the central line of the path, are plotted using data from Table 7. Although no corrections have been made for center of figure or lunar limb profile, they have little or no effect at this scale. Atmospheric refraction has not been included, as it plays a significant role only at very low solar altitudes. The primary effect of refraction is to shift the path opposite to that of the Sun's local azimuth. This amounts to approximately 0.5° at the extreme ends, i.e., sunrise and sunset, of the umbral path. In any case, refraction corrections to the path are uncertain because they depend on the atmospheric temperature-pressure profile, which cannot be predicted in advance. A special feature of the maps are the curves of constant umbral eclipse duration, i.e., totality, which are plotted within the path at 1 min increments. These curves permit fast determination of approximate durations without consulting any tables.
No distinction is made between major highways and second class soft-surface roads, so caution should be used in this regard. If observations from the graze zones are planned, then the zones of grazing eclipse must be plotted on higher scale maps using coordinates in Table 8. See Sect. 3.6 "Plotting The Path On Maps" for sources and more information. The paths also show the curves of maximum eclipse at 4min increments in Universal Time These maps are also available at the NASA Web site for the 2006 total solar eclipse.
The geocentric ephemeris for the Sun and Moon, various parameters, constants, and the besselian elements (polynomial form) are given in Table 1. The eclipse elements and predictions were derived from the DE200 and LE200 ephemerides (solar and lunar, respectivey) developed jointly by the Jet Propulsion Laboratory and the U.S. Naval Observatory for use in the Astronomical Almanac beginning in 1984. 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 International Astronomical Union (IAU) convention through the use of a smaller constant for the mean lunar radius k for all umbral contacts (see Sect.1.10 "Lunar Limb Profile"). Times are expressed in either Terrestrial Dynamical Time (TDT) or in Universal Time where the best value of ΔT (the difference between Terrestrial Dynamical Time and Universal Time, available at the Time of preparation, is used.
From the polynomial form of the Besselian elements, any element can be evaluated for any Timet1 (in decimal hours) via the equation:
t = t1 - t0 (decimal hours) and
t0 = 10.000 TDT.
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 7.00 < t1 < 13.00 TDT.
Table 2 lists all external and internal contacts of penumbral and umbral shadows with Earth. They include TDT 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).
P2 - Instant of first internal tangency of penumbral shadow cone with Earth's limb.
P3 - Instant of last internal tangency of penumbral shadow cone with Earth's limb.
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 northern and southern extremes of the penumbral and umbral paths, and extreme limits of the umbral central line are given. The IAU 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.
The path of the umbral shadow is delineated at 5min 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. Table 4 presents a physical ephemeris for the umbral shadow at 5min intervals in Universal Time The central line coordinates are followed by the topocentric ratio of the apparent diameters of the Moon and Sun, the eclipse obscuration (defined as the fraction of the Sun's surface area occulted by the Moon), 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.
Local circumstances for each central line position listed in Tables 3 and 4 are presented in Table 5. The first three columns give the Universal Timeof 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 time 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 or annular eclipse begins).
Third Contact: Instant of last internal tangency between the Moon and Sun (central or umbral eclipse ends; total or annular eclipse ends).
Fourth Contact: Instant of last external tangency between the Moon and Sun (partial eclipse ends)
The position angles P and V (whereP is defined as the contact angle measured counterclockwise from the north point of the Sun's disk and V is defined as the contact angle measured counterclockwise from the zenith point of the Sun's disk) identify the point along the Sun's disk where each contact occurs. Second and third contact altitudes are omitted because they are always within 1° of the altitude at maximum eclipse.
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 counterclockwise from the north point of the Moon. In addition, corrections to the path limits due to the lunar limb profile are listed (minutes of arc in latitude). 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 Sect. 1.11 "Limb Corrections To The Path Limits: Graze Zones." Corrections to central line durations due to the lunar limb profile are also included. When added to the durations in Tables 3, 4, 5, and 7, a slightly longer central total phase is predicted along most of the path because of the high topography along the Moon's northeastern limb.
To aid and assist in the plotting of the umbral path on large scale maps, the path coordinates are also tabulated at 1° intervals in longitude in Table 7. The latitude of the northern limit, southern limit, and central line for each longitude is tabulated to the nearest hundredth of an arc minute (~18.5m at the Equator) along with the Universal Time of maximum eclipse at each position. Finally, local circumstances on the central line at maximum eclipse are listed and include the Sun's altitude and azimuth, the umbral path width, and the central duration of totality.
In applications where the zones of grazing eclipse are needed in greater detail, Table 8 lists these coordinates over land-based portions of the path at 1° intervals in 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 Sect. 1.11 "Limb Corrections to the Path Limits: Graze Zones"). Expanded versions of Tables 7 and 8 using longitude steps of 7.5' are available at the NASA 2006 Total Solar Eclipse Web site.
Local circumstances for approximately 350 cities; metropolitan areas; and places in Brazil, Africa, Europe, and Asia are presented in Tables 9 - 19. The tables give the local circumstances at each contact and at maximum eclipse for every location. (For partial eclipses, maximum eclipse is the instant when the greatest fraction of the Sun's diameter is occulted. For total eclipses, maximum eclipse is the instant of mid-totality.) The coordinates are listed along with the location's elevation (in meters) above sea level, if known. If the elevation is unknown (i.e., not in the database), then the local circumstances for that location are calculated at sea level. The elevation does not play a significant role in the predictions unless the location is near the umbral path limits or the Sun's altitude is relatively small (<10°).
The Universal Time of each contact is given to a tenth of a second, along with position angles P and V and the altitude of the Sun. The position angles identify the point along the Sun's disk where each contact occurs and are measured counterclockwise (i.e., eastward) from the north and zenith points, respectively. Locations outside the umbral path miss the umbral eclipse and only witness first and fourth contacts. The Universal Time of maximum eclipse (either partial or total) is listed to a tenth of a second. Next, the position angles P and V of the Moon's disk with respect to the Sun are given, followed by the altitude and azimuth of the Sun at maximum eclipse. Finally, the corresponding eclipse magnitude and obscuration are listed. For umbral eclipses (both annular and total), the eclipse magnitude is identical to the topocentric ratio of the Moon's and Sun's apparent diameters.
Two additional columns are included if the location lies within the path of the Moon's umbral shadow. The "umbral depth" is a relative measure of a location's position with respect to the central line and path limits. It is a unitless parameter which is defined as:
where:
| u | is the umbral depth, |
| x | is the perpendicular distance from the central line in km, and |
| W | is the width of the path in km |
The umbral depth for a location varies from 0.0 - 1.0. A position at the path limits corresponds to a value of 0.0, while a position on the central line has a value of 1.0. The parameter can be used to quickly determine the corresponding central line duration; thus, it is a useful tool for evaluating the trade-off in duration of a location's position relative to the central line Using the location's duration and umbral depth, the central line duration is calculated as:
where:
| D | is the duration of totality on the center line (seconds) |
| d | is the duration of totality at location (seconds) |
| u | is the umbral depth |
The final column gives the duration of totality. The effects of refraction have not been included in these calculations, nor have there been any corrections for center of figure or the lunar limb profile
Locations were chosen based on general geographic distribution, population, and proximity to the path. The primary source for geographic coordinates is The New International Atlas (Rand McNally 1991). Elevations for major cities were taken from Climates of the World (U.S. Dept. of Commerce 1972). In this rapidly changing political world, it is often difficult to ascertain the correct name or spelling for a given location; therefore, the information presented here is for location purposes only and is not meant to be authoritative Furthermore, it does not imply recognition of status of any location by the United States Government. Corrections to names, spellings, coordinates, and elevations should be forwarded to the authors in order to update the geographic database for future eclipse predictions.
For countries in the path of totality, expanded versions of the local circumstances tables listing additional locations are available via the NASA Web sitefor the 2006 total solar eclipse.
The time of second and third contact for any location not listed in this publication can be estimated using the detailed maps (Figures 6 - 19). Alternatively, the contact time can be estimated from maps on which the umbral path has been plotted. Table 7 lists the path coordinates conveniently arranged in 1° increments of longitude to assist plotting by hand. The path coordinates in Table 3 define a line of maximum eclipse at 5 minute increments in time. These lines of maximum eclipse each represent the projection diameter of the umbral shadow at the given time; thus, any point on one of these lines will witness maximum eclipse (i.e., mid-totality) at the same instant. The coordinates in Table 3 should be plotted on the map in order to construct lines of maximum eclipse.
The estimation of contact time for any one point begins with an interpolation for the time of maximum eclipse at that location. The time of maximum eclipse is proportional to a point's distance between two adjacent lines of maximum eclipse, measured along a line parallel to the central line. This relationship is valid along most of the path with the exception of the extreme ends, where the shadow experiences its largest acceleration. The central line duration of totality D and the path width W are similarly interpolated from the values of the adjacent lines of maximum eclipse as listed in Table 3. Because the location of interest probably does not lie on the central line, it is useful to have an expression for calculating the duration of totality d (in seconds) as a function of its perpendicular distance a from the central line:
where:
| d | = | duration of totality at desired location (seconds) |
| D | = | duration of totality on the center line (seconds) |
| a | = | perpendicular distance from the center line (kilometers), and |
| W | = | width of the path (kilometers) |
If tm is the interpolated time of maximum eclipse for the location, then the approximate times of second and third contacts (t2 and t3, respectively) are:
| Second Contact: | t2 | = | tm - d/2 | [5] |
| Third Contact: | t3 | = | tm + d/2 | [6] |
The position angles of second and third contact (either P or
where:
A fundamental parameter used in eclipse predictions is
the Moon's radius k, expressed in units
of Earth's equatorial radius. The Moon's actual radius varies as a function of
position angle and libration because of the irregularity in the limb profile.
From 1968 - 1980, the Nautical Almanac Office used two separate values for
k in their predictions. The larger value (k = 0.2724880),
representing a mean over topographic
features, was used for all penumbral (exterior) contacts and for annular
eclipses. A smaller value (k = 0.272281),
representing a mean minimum radius, was reserved exclusively for umbral
(interior) contact calculations of total eclipses (Explanatory
Supplement 1974). Unfortunately, the use of
two different values of k for
umbral eclipses introduces a discontinuity in the case of hybrid or
annular-total eclipses. In August 1982, the IAU General Assembly adopted a value
of k =0.2725076 for the mean lunar
radius. This value is now used by the Nautical Almanac Office for all solar
eclipse predictions (Fiala and Lukac 1983) and is currently the best mean
radius, averaging mountain peaks and low valleys along the Moon's rugged limb.
The adoption of one single value for k eliminates the discontinuity
in the case of annular-total eclipses and
ends confusion arising from the use of two different values; however, the use
of even the best ômean' value for the Moon's radius introduces a problem in
predicting the true character and duration of umbral eclipses, particularly
total eclipses. A total eclipse can be defined as an eclipse in which
the Sun's disk is completely occulted by the Moon. This cannot occur so long as
any photospheric rays are visible through deep valleys along the Moon's limb
(Meeus et al. 1966). The use of the IAU's mean k, however,
guarantees that some annular or hybrid (i.e., annular-total)
eclipses will be misidentified as total. A case in point is the eclipse of 1986
October 03. Using the IAU value for k, the Astronomical Almanac
identified this event as a total eclipse of 3s duration when it was, in fact, a
beaded annular eclipse. Because a smaller value of k is more
representative of the deeper lunar valleys
and hence, the minimum solid disk radius, it helps ensure the correct
identification of an eclipse's true nature. Of primary interest to most observers are the time when
an umbral eclipse begins and ends (second and third contacts, respectively) and
the duration of the umbral phase. When the IAU's value for k
is used to calculate these times, they must be
corrected to accommodate low valleys (total) or high mountains (annular) along
the Moon's limb. The calculation of these corrections is not trivial but is
necessary, especially if one plans to observe near the path limits (Herald 1983).
For observers near the central line of a total eclipse, the limb corrections
can be more closely approximated by using a smaller value of k which accounts for the valleys along the profile. This publication uses the IAU's accepted value of k = 0.2725076
for all penumbral (exterior) contacts. In
order to avoid eclipse type misidentification and to predict central durations
which are closer to the actual durations at total eclipses, this document
departs from standard convention by adopting the smaller value of
k = 0.272281 for all umbral (interior) contacts. This is
consistent with predictions in Fifty Year Canon of Solar Eclipses:
1986 - 2035 (Espenak 1987). Consequently, the
smaller k produces shorter umbral
durations and narrower paths for total eclipses when compared with calculations
using the IAU value for k .
Similarly, predictions using a smaller k result in longer umbral
durations and wider paths for annular eclipses
than do predictions using the IAU's k. Eclipse contact time, magnitude, and duration of
totality all depend on the angular diameters and relative velocities of the
Moon and Sun. 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 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 when 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. In a follow-up analysis of 66,000 occultations, Morrison
and Appleby (1981) found that the radius of the datum appears to vary with
libration. These variations produce systematic errors in Watts' original limb
profile heights that attain 0.4arc-sec 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 were 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 profile
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 path, the Moon's topocentric libration
(physical plus optical) in longitude ranges from l = +3.0°
to l = +1.4° thus, a
limb profile with the appropriate libration is required in any detailed
analysis of contact time, central durations, etc. A profile with an
intermediate value, however, is useful for planning purposes and may even be
adequate for most applications. The lunar limb profile presented in
Figure 20 includes corrections
for center of mass and ellipticity
(Morrison and Appleby 1981). It is generated for 10:30 UT, which
corresponds to central Libya, south of Jalu. The Moon's topocentric libration
is l = +2.07°, and the topocentric semi-diameters
of the Sun and Moon are 961.2
and 1010.3 arc sec, respectively. The Moon's angular velocity with respect to
the Sun is 0.404 arc sec/s. The radial scale of the limb profile in Figure 20
(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 curve) along with the mean
limb with respect to the center of mass (solid curve). 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 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 central line at 10:30 UT, while the time of the four
eclipse contacts at that location appear to the lower right. The limb-corrected
time of second and third contacts are listed with the applied correction to
the center on mass prediction. Directly below the limb 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
to 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 time 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 determine 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 20, 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:
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 [9]. 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 Δ by:
where:
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. Because 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 time of second and
third contact for most position angles have been computer generated and are
plotted in Figure 20. 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 profie. 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 11 give the following data for
Jalu, Libya:
Using Figure 20,
the measured time corrections and the resulting contact times are:
The above corrected values are within
0.2 s of a rigorous calculation using the true limb profile. Note the 5 s
correction to third contact due to a deep lunar valley along the western limb. The northern and southern umbral limits provded in this
publication were derived using the Moon's center of mass and a mean lunar
radius. They have not been corrected for the Moon's center of figure or the
effects of the lunar limb profile In applications where precise limits are
required, Watts' limb data must be used to correct the nominal or mean path.
Unfortunately, a single correction at each limit is not possible because the
Moon's libration in longitude and the contact points of the limits along the
Moon's limb each vary as a function of time and position along the umbral path.
This makes it necessary to calculate a unique correction to the limits at each
point along the path. Furthermore, the northern and southern limits of the
umbral path are actually paralleled by a relatively narrow zone where the
eclipse is neither penumbral nor umbral. An observer positioned here will
witness a slender solar crescent that is fragmented into a series of bright
beads and short segments whose morphology changes quickly with the rapidly
varying geometry between the limbs of the Moon and the Sun. These beading
phenomena are caused by the appearance of photospheric rays that alternately
pass through deep lunar valleys and hills behind high mountain peaks, as the
Moon's irregular limb grazes the edge of the Sun's disk. The geometry is
directly analogous to the case of grazing occultations of stars by the Moon.
The graze zone is typically 5 - 10 km wide and its interior and exterior
boundaries can be predicted using the lunar limb profile. The interior
boundaries define the actual limits of the umbral eclipse (both total and
annular) while the exterior boundaries set the outer limits of the grazing
eclipse zone. Table 6 provide
topocentric data and corrections to the
path limits due to the true lunar limb profile. At 5min intervals, the table
lists the Moon's topocentric horizontal parallax, semi-diameter, relative
angular velocity with respect to the Sun and lunar libration in longitude. The
Sun's central line altitude and azimuth is given, followed by the azimuth of
the umbral path. The position angle of the point on the Moon's limb, which
defines the northern limit of the path, is measured counterclockwise (i.e.,
eastward) from the north point on the limb. The path corrections to the
northern and southern limits are listed as interior and exterior components in
order to define the graze zone. Positive corrections are in the northern sense,
while negative shifts are in the southern sense. These corrections (minutes of
arc in latitude) may be added directly to the path coordinates listed in
Table 3.
Corrections to the central line umbral durations due to the lunar
limb profile are also included and they are almost all positive thus, when
added to the central durations given in
Tables 3, 4, 5, and 7, a slightly
longer central total phase is predicted. This effect is caused by a significant
departure of the Moon's eastern limb from both the center of figure and center
of mass limbs for the predicted libration during the 2006 eclipse. Detailed coordinates for the zones of grazing eclipse at
each limit for all land based sections of the path are presented in
Table 8. Given the uncertainties
in the Watts data, these predictions
should be accurate to ±0.3 arc seconds. The interior graze coordinates take into
account the deepest valleys along the Moon's limb, which produce the
simultaneous second and third contacts at the path limits; thus, the interior
coordinates that define the true edge of the path of totality. They are
calculated from an algorithm which searches the path limits for the extreme
positions where no photospheric beads are visible along a ±30° segment of the
Moon's limb, symmetric about the extreme contact points at the instant of
maximum eclipse. The exterior graze coordinates are arbitrarily defined and
calculated for the geodetic positions where an unbroken photospheric crescent
of 60° in angular extent is visible at maximum eclipse. In Table 8, the graze zone
latitudes are listed
every 1° in longitude (at sea level) and include the time of maximum eclipse at
the northern and southern limits, as well as the path's azimuth. To correct the
path for locations above sea level, Elev Fact (elevation factor) is a multiplicative factor by which the path must be
shifted north or south perpendicular to itself, i.e., perpendicular to path
azimuth, for each unit of elevation (height) above sea level. The elevation
factor is the product, tan(90-A) x sin(D),
where A is the altitude of the Sun, and D is the difference
between the azimuth of the Sun
and the azimuth of the limit line, with the sign selected to be positive if the
path should be shifted north with positiveelevations above sea level. To calculate
the shift, a location's elevation is multiplied by the elevation factor value.
Negative values (usually the case for eclipses in the Northern Hemisphere)
indicate that the path must be shifted south. For instance, if one's elevation
is 1000m above sea level and the elevation factor value is -0.50, then the
shift is -500 m (= 1000 m x -0.50); thus,
the observer must shift the path coordinates 500 m in a direction perpendicular
to the path and in a negative or southerly sense. The final column of Table 8
lists the Scale Fact (in kilometers per arc second). This scaling factor
provides an indication of the width of the zone of grazing phenomena, because
of the topocentric distance of the Moon and the projection geometry of the
Moon's shadow on Earth's surface. Because the solar chromosphere has an
apparent thickness of about 3arcsec, and assuming a scaling factor value of
2 km/arcsec, then the chromosphere should be visible continuously during
totality for any observer in the path who is within 6 km (=2 x 3) of each interior limit.
The most dynamic beading phenomena, however, occur within 1.5 arcsec of the Moon's
limb. Using the above scaling factor, this translates into the first 3 km inside the
interior limits, but observers should position themselves at least 1 km inside
the interior limits (south of the northern interior limit or north of the
southern interior limit) in order to ensure that they are inside the path
because of small uncertainties in Watts' data and the actual path limits. For applications where the zones of grazing eclipse are
needed at a higher frequency of longitude interval, tables of coordinates every
7.5' in longitude are available via the
NASA
Web site for the 2006 total solar eclipse.
The periodicity and recurrence of solar (and lunar)
eclipses is governed by the Saros cycle, a period of approximately 6,585.3 d
(18 yr 11 d 8 h). When two eclipses
are separated by a period of one Saros, they share a very similar geometry. The
eclipses occur at the same node with the Moon at nearly the same distance from
Earth and at the same time of year, thus, the Saros is useful for organizing
eclipses into families or series. Each series typically lasts 12 - 13 centuries
and contains 70 or more eclipses. The total eclipse of 2006 is the 29th member of Saros
series 139 (Table 20), as defined by
van den Bergh (1955). All eclipses in
the series occur at the Moon's ascending node and the Moon moves southward with
each member in the family, i.e., gamma decreases
and takes on negative values south of the Earth's center. Saros 139 is a
middle-aged series which began with a small partial eclipse at high northern
latitudes on 1501 May 17. After seven partial eclipses each of increasing
magnitude, the first umbral eclipse occurred on 1627 August 11. This event was
of the unique hybrid or annular-total class of eclipses. The nature of such an
eclipse changes from total to annular or vice versa along different portions of
the track. The dual nature arises from the curvature of Earth's surface, which
brings the middle part of the path into the umbra (total eclipse) while other,
more distant segments remain within the antumbral shadow (annular eclipse). Such hybrid eclipses are rather rare and account for
only 5.2% of the 14,283 solar eclipses occurring during the six millennia
period from - 1999 to +4000 (2000 B.C.E. to A.D. 4000). Quite remarkably, the
first dozen central eclipses of Saros 139 were all hybrid with the duration of
totality steadily increasing during each successive event. The first purely
total eclipse of the series occurred on 1843 December 21 and had a maximum
duration of 1 min 43 s. Throughout the 19th and 20th centuries, Saros 139
continued to produce total eclipses with increasing durations. The last two
members of the series were in 1970 and 1988. The 1970 March 07 eclipse lasted
3.5min and was widely visible from Mexico, the eastern seaboard of the United
States, and maritime Canada. The track of the 1988 Mar 18 eclipse began in the
Indian Ocean, extended across the islands of Sumatra, Kalimantan (Borneo), and
Mindanao (Philippines), and ended in the Pacific Ocean. Saros 139 will continue its current trend of producing
total eclipses of increasing duration over the course of the next two
centuries. The trend culminates with the 39th member of the series on 2186 July
16. This remarkable eclipse will produce a total phase lasting as much as 7min
29s. This is very close to a total eclipse's theoretical maximum duration of 7min
32s (Lewis 1931 and Meeus 2003). In fact, calculations show that the 2186
eclipse has the longest duration of any total eclipse during the eight
millennia period from -2999 to +5000 (3000 B.C.E.
to A.D. 5000). Unfortunately, the Moon's shadow will lie in the Atlantic Ocean
900 km north of Brazil at the instant of greatest eclipse. Nevertheless, the
duration will exceed 7min on the central line as the path crosses Colombia and
Venezuela. Long total eclipses from the series will occur
throughout the 23rd century with gradually decreasing durations. By the eclipse
of 2294 September 20, the duration will slip below 5min. Over the succeeding
centuries, the duration of totality steadily dwindles. The last central eclipse
occurs on 2601 March 26 and has a duration of just 36 s. The final nine eclipses
are all partial events visible from the southern Hemisphere. Saros series 139
ends with the partial eclipse of 2763 July 03. In summary, Saros series 139 includes 71 eclipses. It
begins with 7 partials, followed by 12 hybrids, then 43 totals, and finally
ends with 9 more partials. From start to finish, the series spans a period of
1262 years.
Second Contact: x2
= X2 -
arcsin (2a/W)
[7]
Third Contact: x3
= X3 +
arcsin (2a/W)
[8]
xn = interpolated position angle
(either P or V) of contact
n at location
Xn = interpolated position angle
(either P or V) of contact
n on center line
a = perpendicular distance from the center
line (kilometers)
(use negative values for locations south of the center line), and
W = width of the path (kilometers)
1.9 Mean Lunar Radius
1.10 Lunar Limb Profile
h
= S (m - 1) (1 - cosC)
[9]
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
Δ
= dv cos [X - C],
[10]
Δ
= 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
Second Contact = 10:28:29.3 UT
P2=8°
Third Contact = 10:31:43.4 UT
P3=262°
C2 = -0.5 seconds Second Contact = 10:28:29.3 - 0.5 s
= 10:28:28.8 UT
C3 = -5.1 seconds Third Contact = 10:31:43.4 2.7 s
= 10:32:38.3 UT
1.11 Limb Corrections to the Path Limits: Graze Zones
1.12 Saros History
Return To: Total Solar Eclipse of 2006 March 29 (NASA/TP-2004-212762)
