With a little geometry, students can study alignments. Archaeoastronomers measure four important angles (see figure 3 and figure 4):
Definition of declination. Declination is the angle between a star and the celestial equator, an imaginary line across the sky that runs parallel to the Earth's equator. To find the celestial equator, hold your arms in an `L' shape and point one arm at the North Star; the other arm will point at the celestial equator.
Definition of azimuth and elevation. Azimuth is the compass direction, measured clockwise, between a star and the north. Elevation is the angle between a star and the horizon.
The geometry determines which objects are visible from a given site and how the Earth's rotation affects their motion. Stars fall into different categories:
Using their knowledge of these angles, archaeoastronomers have analyzed the great pyramid of the pharaoh Cheops (see figure 5). Two air shafts leading to the king's chamber relate to an afterlife, a strong theme found in Egyptian hieroglyphics. The shaft on the north side leads directly to the north celestial pole, which in Cheops' time corresponded to the star Thuban in the constellation Draco. The shaft on the south side is associated with the constellation Orion, which in Cheops' time passed directly through the line of the shaft every day. Orion was a multipurpose god associated with the hereafter.
Shafts to King's chamber of Cheops' pyramid. The pyramids, built nearly 5,000 years ago, incorporated the astronomical knowledge of the ancient Egyptians. The sides of the base line up with north, south, east, and west. Two air shafts slope upwards from the main burial chamber. These shafts are aligned with two stars of religious importance: Thuban (the North Star at the time) and Alnilam (the center star in Orion's belt).
Geometry also sheds light on the great raised earthworks of the Hopewell. The Hopewell were Native Americans who lived in southern Ohio between the second century B.C. and sixth century A.D. Only a few of their earthworks have survived the farmers' plows and developers' bulldozers. Fortunately, surveys of several dozen monuments by George Squier and E.H. Davis in the mid-1800s preserve the knowledge of the Hopewell (see figure 6). These surveys give the azimuths and the latitude of the sites, so that archaeoastronomers can calculate the declinations of the alignments. Nearly all correspond to noteworthy positions of the Sun or Moon on the horizon.
Hopewell earthworks. Right, a diagram of the earthworks at Seal in southern Ohio. Left, the circle and octagon mounds at Newark, Ohio, now a municipal golf course. This photograph looks northeast along what some archaeoastronomers think is a moonrise alignment. Other earthworks have been leveled to build shopping malls. Photo courtesy of E.C. Krupp, Griffith Observatory.
Many of the largest earthworks have squares or octagons associated with circles. The squares and octagons have several openings in their perimeter, but the circles do not. This suggests that ceremonial participants entered the earthworks through the square or octagon openings and proceeded to the center of the circle. Which ceremonies were conducted here, we don't know. Yet nearly all the azimuths defined by the direction of the ceremonial procession are associated with a life hereafter. During the ceremonies, spectators could stand on the surrounding earthworks to view the activities.
Geometry also accounts for the well-known alignment at Stonehenge in England. There, the Sun rises over the Heelstone, as viewed from the center of the site, at the summer solstice. Students can easily verify this for themselves, using figure 7, a protractor, and the equation: sin delta = cos phi cos A.
Layout of Stonehenge in Wiltshire, England. Stonehenge is the most famous of ancient astronomical sites. If you stand in the center of the site and look northeast through the stone arches, you see the Heelstone, which points to the place where the Sun rises in midsummer. Stonehenge was built and rebuilt beginning 5,000 years ago.
With the protractor, measure the azimuth of the Heelstone as seen from the center of the Aubrey Circle. It should be about A = 50 degrees. The direction of geographic north is indicated on the figure. Then, on a map of England, look up the latitude of Stonehenge, near the town of Salisbury. It should be about phi = 51.2 degrees. By substituting A and phi into the equation, find the declination. It should be delta = 23.7 degrees, which is close to 23.9 degrees, the declination of the summer solstice position at the time Stonehenge was built.
The same mathematical expressions demonstrate that this sighting cannot determine the exact time of the solstice. The sighting is of low precision. Because the Sun's path moves slowly at this time of year, there are dozens of days when the positions of the Sun are indistinguishable. From this we conclude that Stonehenge was a great ceremonial center, but not an observatory.
Although naked-eye observations cannot determine the precise time of the solstice, they can determine the precise time of the equinox. At this time of year, the Sun's path changes considerably from day to day, so that the positions on successive days are quite discernible. The length of time it takes the Sun to return to the same equinox is the year. If a long enough baseline in time is used, the year can be determined with precision.
By dividing the year into quarters, ancient peoples predicted the time of the solstices. Whether these predictions agree exactly with our modern calculations does not matter. Generally people don't mind if they get the date of their holidays wrong, as long as everyone agrees to celebrate on the same day. For example, it is unlikely that Christ was born on December 25. If Christians meant to celebrate Christmas on the winter solstice, they are a few days off. But they don't care, since they have agreed to celebrate Christmas on December 25 (in North America and Western Europe).
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