On Friday, 2008 August 01, a total eclipse of the Sun is visible from within a narrow corridor that traverses half of Earth. The path of the Moon’s umbral shadow begins in northern Canada and extends across Greenland, the Arctic, central Russia, Mongolia, and China (Espenak and Anderson, 2006). A partial eclipse is seen within the much broader path of the Moon’s penumbral shadow, which includes northeastern North America, most of Europe, and Asia (Figures 1 and 2).
The path of totality begins in northern Canada (Figure 3), where the Moon’s umbral shadow first touches down in the territory of Nunavut at 09:21 UT (Universal Time). Along the sunrise terminator in Queen Maud Gulf, the duration is 1 min 30 s from the center of the 206 km wide path. Traveling over 9 km/s, the umbra quickly sweeps north across southern Victoria Island, Prince of Wales Island, and Northern Somerset Island (Figure 4). The shadow’s northern limit clips the southeastern corner of Cornwallis Island and just misses the high Arctic town of Resolute. The ~200 residents of this isolated settlement will witness a partial eclipse of magnitude 0.997 at 09:26 UT with the Sun 7° above the horizon.
Continuing on its northeastern trajectory, the umbra crosses Devon Island and reaches the southern coast of Ellesmere Island where it engulfs the tiny hamlet of Grise Fiord. The duration of total eclipse here is 1 min 38 s. The central line cuts across Nares Strait as the shadow straddles Ellesmere Island and Greenland (Figure 5). Canada’s remote outpost Alert, the northernmost permanently inhabited place on Earth, lies near the northern limit of the eclipse track and experiences 43 s of totality with the Sun at 16° altitude at 09:32 UT.
The northern half of the path encounters the open Arctic, while the southern half cuts across the many fjords of northern Greenland. Leaving the coast of Greenland, the shadow reaches its northernmost latitude (83° 47´) at 09:38 UT as it traverses the landless Arctic Ocean. Slowly curving to the southeast, the track passes between Franz Josef Land and Svalbard where George Land and Kvitoya Island are cut by its northern and southern limits, respectively (Figure 6). By the time the central line reaches the northern coast of Novaya Zemlya (10:00UT), the duration is 2 min 23 s with the Sun at 31° (Figures 7 and 8). The track crosses both the island and the Kara Sea before reaching the Yamal Peninsula and the Russian mainland at 10:10 UT (Figure 8).
The instant of greatest eclipse occurs at 10:21:07 UT (latitude 65° 39´N, longitude 72° 18´E) when the axis of the Moon’s shadow passes closest to the center of Earth (gamma = +0.8307). When totality reaches its maximum duration of 2 min 27 s, the Sun’s altitude is 34°, the path width is 237 km, and the umbra’s velocity is 0.908 km/s. The Russian city of Nadym (pop.ulation ~46,000) lies nearby and only loses 1 s of totality because of its short distance (~14 km) from the central line (Figure 9).
During the next hour, the Moon’s umbra works its way across central Asia. The shadow gradually picks up speed and its course changes from south-southeast to nearly east at its terminus (Figure 7). Central Russia is sparsely populated, however, there are a few small cities in the path of totality including Megion, Nizhnevartovsk, and Strezhevoy (Figure 10).
Novosibirsk, Russia’s third most populous city (pop. ~1.4 million), lies only 18 km from the central line. The midpoint of Novosibirsk’s 2 min 18 s total eclipse occurs at 10:45 UT with the Sun’s altitude at 30° (Figures 10 and 11). Three and a half minutes later, Barnaul (pop. ~600,000) is plunged into a 2 min 16 s total eclipse.
As the umbral shadow exits Russia, it briefly encompasses the intersection of four nations: Russian, Kazakhstan, China, and Mongolia. After crossing the Altay Mountains, the center of the track follows the China-Mongolia border for several hundred kilometers while the central duration and the Sun’s altitude decrease (Figure 12). During this period, the central line crosses from Mongolia to China to Mongolia and finally back to China where it remains until the end of the eclipse track.
From Altay, China, the total eclipse begins at 10:59 UT and lasts 1 min 25 s with the Sun 25° above the horizon. Across the border, western Mongolia is very sparsely populated and the Altan Mountains bring cloudiness to the area. Ten minutes later, the umbra just misses Hami, China (pop. ~137,000) where a deep partial eclipse of magnitude 0.998 occurs at 11:10 UT. About 140 km east of Hami, Yiwu lies just 25 km southwest of the central line (Figure 13). Inhabitants of this small town witness a total eclipse lasting 1 min 56 s with the Sun at an altitude of 19°. This region in northwest China is noteworthy because it offers some of the most promising weather prospects along the entire eclipse path. Its position between the Gobi Desert to the east and the Talikmakan Desert to the west spares it from the monsoon systems that affect much of Southeast Asia during the summer months.
During the final 10 min of the umbra’s track, it quickly sweeps across northern China as the duration of totality and the Sun’s altitude continue to decrease. Juiquan (pop. ~73,000) lies in the path near the southern limit, but it still experiences a total eclipse lasting 1 min 08 s at 11:15 UT (Figure 14). Further east, the major city of Xi’an (pop. ~3.9 million) straddles the southern limit where maximum eclipse occurs with the Sun just 4° above the horizon (Figure 15). From the central line, 106 km to the north, the duration of totality still lasts 1 min 35 s. Seconds later, the Moon’s shadow lifts off Earth and the total eclipse ends (11:21 UT). Over the course of 2 h, the Moon’s umbra travels along a path approximately 10,200 km long and covers 0.4% of Earth’s surface area.
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 southern boundary lies the limit of the penumbra’s path. Great loops at the western and eastern extremes of the penumbra’s path identify the areas where the eclipse begins and ends at sunrise and sunset, respectively. Bisecting 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 travels west to east within the penumbral path.
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 in a north-south direction. The outline of the umbral shadow is plotted every 10 min 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 exclusively between the curves of maximum eclipse at sunrise and sunset. Furthermore, they are quasi-parallel to the southern penumbral limit. This limit may be thought of as a curve 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 for equatorial and ecliptic coordinates followed by the instant of greatest eclipse. The eclipse magnitude is given for greatest eclipse. 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. Finally, the Saros series number of the eclipse is given along with its relative sequence in the series.
The stereographic projection of Earth in Figure 2 depicts the path of penumbral and umbral eclipse in greater detail. The map is oriented with north up. International political borders are shown and circles of latitude and longitude are plotted at 30° increments. The region of penumbral or partial eclipse is identified by its southern limit, curves of eclipse begins or ends at sunrise and sunset, and curves of maximum eclipse at sunrise and sunset. Curves of constant eclipse magnitude are plotted for magnitudes 0.20, 0.40, 0.60, and 0.80, as are the limits of the path of total eclipse. Also included are curves of greatest eclipse at every half hour Universal Time.
Figures 1 and 2 may be used to quickly determine the approximate time and magnitude of maximum eclipse at any location within the eclipse path.
Figures 3 and 7 are maps using an equidistant conic projection chosen to minimize distortion, and that isolate the Arctic and Asian portions of the umbral path. Curves of maximum eclipse and constant eclipse magnitude are plotted and labeled at intervals of 30 min and 0.2 magnitudes, respectively. A linear scale is included for estimating approximate distances (in kilometers). Within the northern and southern limits of the path of totality, the outline of the umbral shadow is plotted at intervals of 5 min or 10 min. 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.
The path of totality is plotted on a series of detailed maps appearing in Figures 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, and 15. The maps were chosen to isolate small regions along the entire land portion of the eclipse path. Curves of maximum eclipse are plotted at 5 min 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:
Figures 4 to 6 | 1:6,000,000 |
Figure 8 | 1:5,600,000 |
Figures 9 to 15 | 1:5,000,000 |
The scale of the maps is adequate for showing the roads, villages, and cities required for eclipse expedition planning. The DCW database used for the maps was developed in the 1980s and contains place names in use during that period. Whenever possible, current names have been substituted for those in the database, but this correction has not been applied in all instances.
While Tables 1, 2, 3, 4, 5, and 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 1/2 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 5 min increments in Universal Time. These maps are also available at the NASA Web site for the 2008 total solar eclipse: http://eclipse.gsfc.nasa.gov/SEmono/TSE2008/TSE2008.html.
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, respectively) 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. 1.11 “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.
The Besselian elements are used to predict all aspects and circumstances of a solar eclipse. The simplified geometry introduced by Bessel in 1824 transforms the orbital motions of the Sun and Moon into the position, motion, and size of the Moon’s penumbral and umbral shadows with respect to a plane passing through Earth. This fundamental plane is constructed in an X-Y rectangular coordinate system with its origin at Earth’s center. The axes are oriented with north in the positive Y direction and east in the positive X direction. The Z axis is perpendicular to the fundamental plane and parallel to the shadow axis.
The x and y coordinates of the shadow axis are expressed in units of the equatorial radius of Earth. The radii of the penumbral and umbral shadows on the fundamental plane are l_{1} and l_{2}, respectively. The direction of the shadow axis on the celestial sphere is defined by its declination d and ephemeris hour angle µ. Finally, the angles that the penumbral and umbral shadow cones make with the shadow axis are expressed as f_{1} and f_{2}, respectively. The details of actual eclipse calculations can be found in the Explanatory Supplement (Her Majesty’s Nautical Almanac Office 1974) and Elements of Solar Eclipses (Meeus 1989).
From the polynomial form of the Besselian elements, any element can be evaluated for any time t_{1} (in decimal hours) during the eclipse via the equation
a = a_{0} + a_{1} t + a_{2} t^{2} + a_{3} t^{3 }(or a = ∑ [a_{n} ^{n}]; n = 0 to 3), | (1) |
where a = x, y, d, l_{1}, l_{2}, or µ; and t = t_{1} – t_{0} (decimal hours) and t_{0 }= 10.00 TDT.
The polynomial Besselian elements were derived from a least-squares fit to elements rigorously calculated at five separate times over a 6 h period centered at t_{0}; thus, the equation and elements are valid over the period 7.0 ≤ t_{1} ≤ 13.0 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:
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 3 min 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 3 min 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 Time of 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 times 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:
The position angles P and V (where P is defined as the contact angle measured counterclockwise from the equatorial north point of the Sun’s disk and V is defined as the contact angle measured counterclockwise from the local 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 equatorial 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, which 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. 1.12 “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, 7A, and 7B, a slightly shorter central total phase is predicted along most of the path because of several deep valleys along the Moon’s western 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 (~18.5 m at the Equator) along with the Universal Time of maximum eclipse at the central line 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 2008 Total Solar Eclipse Web site: http://eclipse.gsfc.nasa.gov/SEmono/TSE2008/TSE2008.html.
Local circumstances for 308 cities; metropolitan areas; and places in Canada, Europe, the Middle East, and Asia are presented in Tables 9, 10, 11, 12, 13, 14, 15, and 16. 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
u = 1 – (2 x/W), | (2) |
where:
u is the umbral depth,
x is the perpendicular distance from the central line in kilometers, and
W is the width of the path in kilometers.
The umbral depth for a location varies from 0.0 to 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
D = d /[1 – (1 – u)^{2}]^{1/2}, | (3) |
where:
D is the duration of totality on the central line (in seconds),
d is the duration of totality at location (in seconds), and
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 site for the 2008 total solar eclipse: http://eclipse.gsfc.nasa.gov/SEmono/TSE2008/TSE2008.html.
The times of second and third contact for any location not listed in this publication can be estimated using the detailed maps (Figures 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15). Alternatively, the contact times 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 3 min 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 times 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:
d = D [1 – (2 a/W)^{2}]^{1/2}, | (4) |
where:
d is the duration of totality at location (in seconds),
D is the duration of totality on the central line (in seconds),
a is the perpendicular distance from the central line (in kilometers), and
W is the width of the path (kilometers).
If t_{m} is the interpolated time of maximum eclipse for the location, then the approximate times of second and third contacts (t_{2} and t_{3}, respectively) follow:
Second Contact: t_{2} = t_{m} – d/2; | (5) |
Third Contact: t_{3} = t_{m} + d/2. | (6) |
The position angles of second and third contact (either P or V) for any location off the central line are also useful in some applications. First, linearly interpolate the central line position angles of second and third contacts from the values of the adjacent lines of maximum eclipse as listed in Table 5. If X_{2} and X_{3} are the interpolated central line position angles of second and third contacts, then the position angles x_{2} and x_{3} of those contacts for an observer located a kilometers from the central line are
Second Contact: x_{2}= X_{2} – arcsin (2a/W), | (7) |
Third Contact: x_{3}= X_{3}+ arcsin (2a/W), | (8) |
where:
x_{n} is the interpolated position angle (either P or V) of contact n at location,
X_{n} is the interpolated position angle (either P or V) of contact n on central line,
a is the perpendicular distance from the central line in kilometers (use negative values for locations south of the central line), and
Wis the width of the path in kilometers.
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 to 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, Her Majesty’s Nautical Almanac Office, 1974). Unfortunately, the use of two different values of k for umbral eclipses introduces a discontinuity in the case of hybrid (annular-total) eclipses.
In 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 accepted as 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 hybrid eclipses and ends confusion arising from the use of two different values; however, the use of even the “best” mean value for the Moon’s radius introduces a problem in predicting the true character and duration of umbral eclipses, particularly total eclipses.
During a total eclipse, 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 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 3 s 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 an eclipse’s correct classification.
Of primary interest to most observers are the times 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) and Five Millennium Canon of Solar Eclipses: -1999–3000 (Espenak and Meeus 2006). Consequently, the smaller k value 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 value results in longer umbral durations and wider paths for annular eclipses than do predictions using the IAU’s k value.
Eclipse contact times, 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’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’s 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’s original limb profile heights that attain 0.4 arc-sec at some position angles, thus, corrections to Watts’s 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 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 2008 eclipse path, the Moon’s topocentric libration (physical plus optical) in longitude ranges from l = +4.6° to l = +3.6°; thus, a limb profile with the appropriate libration is required in any detailed analysis of contact times, 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 16 includes corrections for center of mass and ellipticity (Morrison and Appleby 1981). It is generated for 11:00 UT, which corresponds to northern China near Altay. The Moon’s topocentric libration is l = +3.80°, and the topocentric semi-diameters of the Sun and Moon are 945.5 and 980.2 arcsec, respectively. The Moon’s angular velocity with respect to the Sun is 0.539 arcsec/s.
The radial scale of the limb profile in Figure 16 (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’s 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 as are 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 11:00 UT, while the times of the four eclipse contacts at that location appear to the lower right. The limb-corrected times of second and third contacts are listed with the applied correction to the center of 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 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 16, 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
h = S (m–1) (1–cos[C]), | (9) |
where:
h is the departure of Sun’s limb from mean lunar limb,
S is the Sun’s semi-diameter,
m is the eclipse magnitude, and
C is the angle from the point of contact.
Herald (1983) takes advantage of this geometry in developing a graphic 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
Δ = dv cos[X – C], | (10) |
where:
Δ is the correction to contact time (in seconds),
d is the distance of solar limb from Moon’s mean limb (in arc seconds),
v is the angular velocity of the Moon with respect to the Sun (arc seconds per second),
X is the central line position angle of the contact, and
C is 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 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 times of second and third contact for most position angles have been computer generated and are plotted in Figure 16. 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 16 gives the following data for Yiwi, China:
Second Contact = 11:08:09.6 UT P_{2} = 129°, and
Third Contact = 11:10:05.7 UT P_{3} = 286°.
Using Figure 16, the measured time corrections and the resulting contact times are
C_{2}= –1.2 s;
Second Contact = 11:08:09.6 –1.2 s = 11:08:08.4 UT, and
C_{3}= –1.1 s;
Third Contact = 11:10:05.7 –1.1 s= 11:10:04.6 UT.
The above corrected values are within 0.2 s of a rigorous calculation using the true limb profile.
The northern and southern umbral limits provided 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’s 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 hide 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 provides topocentric data and corrections to the path limits due to the true lunar limb profile. At 3 min 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 equatorial 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 shorter 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 and to several deep valleys along the Moon’s western limb for the predicted libration during the 2008 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 arcsec. (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 that 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) ´ 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 positive elevations 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 1000 m above sea level and the elevation factor value is –0.50, then the shift is –500 m (= 1000 m ´ –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 3 arcsec, and assuming a scaling factor value of 3.5 km/arcsec, then the chromosphere should be visible continuously during totality for any observer in the path who is within 10.5 km (=3.5 ´ 3) of each interior limit. The most dynamic beading phenomena, however, occurs within 1.5 arcsec of the Moon’s limb. Using the above scaling factor, this translates to the first 5.25 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’s 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 2008 total solar eclipse: http://eclipse.gsfc.nasa.gov/SEmono/TSE2008/TSE2008.html.
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 or 13 centuries and contains 70 or more eclipses.
The total eclipse of 2008 is the 47th member of Saros series 126 (Table 17), as defined by van den Bergh (1955). All eclipses in an even numbered Saros series occur at the Moon’s descending node and the Moon moves northward with each succeeding member in the family (i.e., gamma increases). Saros 126 is a senior-aged series that began with a small partial eclipse at high southern latitudes on 1179 Mar 10. After eight partial eclipses, each of increasing magnitude, the first umbral eclipse occurred on 1323 Jun 04. This event was a central annular eclipse with no southern limit (Espenak and Meeus, 2006).
For the next five centuries, the series produced 27 more annular eclipses. On 1828 Apr 14, the first hybrid or annular total eclipse occurred. The nature of such an eclipse changes from annular to total 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 4.8% of the 11,898 solar eclipses occurring during the five millennium period from –1999 to +3000 (Espenak and Meeus, 2006). The next two eclipses of Saros 126 were also hybrid with a steadily increasing duration of totality at greatest eclipse. The first purely total eclipse of the series occurred on 1882 May 17 and had a maximum duration of 1 min 50 s.
Throughout of the 20th century, Saros 126 continued to produce total eclipses. The maximum central duration gradually increased with each event until the peak duration of 2 min 36 s was reached on 1972 Jul 10. The duration of the following total eclipse on 1990 Jul 22 was 2 min 33 s. This decrease in duration was due to the large value of gamma (0.7595) which produced a high latitude eclipse track along the northern coast of Siberia (Espenak 1989).
Because the 2008 eclipse has an even larger value of gamma (0.8306), its path sweeps over higher latitudes resulting in a shorter duration of 2 min 27 s. With just two more total eclipses left in Saros 126 (2026 and 2044), the series becomes partial again with the eclipse of 2062 Sep 03. Four more centuries of partial eclipses occur before the series terminates with the partial eclipse of 2459 May 03.
In summary, Saros series 126 includes 72 eclipses. It begins with 8 partials, followed by 28 annulars, 3 hybrids, and 10 totals. It ends with a long string of 23 partial eclipses. From start to finish, the series spans a period of 1280 years.