Measuring Cosmic Distances (Major Trends In The History of Astronomy)

Early Measurements of The Universe

A spherical Earth was accepted very early, although the first conclusive physical argument in favour came from Aristotle. He pointed out that during a lunar eclipse the shadow cast by the Earth on the Moon was circular, which would not be the case unless the Earth were spherical. Aristotle alluded to certain measure¬ments of the size of the Earth in his writings, but the first fully-recorded account, although second-hand, is that of Eratosthenes, (276-195 BC). Using only a gnomon (shadow-stick) he measured the length of the shadow cast by the noon-day Sun on the summer solstice at Alexandria, while the Sun was directly overhead (in the zenith) at Syene, a city some 5000 stades’ (850km) due south of Alexandria. He found the angle of the Sun at Alexandria to be 7° 12′ away from the zenith. From this observation he arrived at an Earth circumference of 250000 stades (equivalent to a diameter of about 42 500 km), only 5 per cent lower than the modem value. For some reason this value was later ignored by Ptolemy and the less accurate result (180 000 stades) of Posidonius adopted.

Speculative estimates of the scale of the Solar System had also been made by the third century BC. The first-known estimate is that of Anaximander (611-546 BC), a philosopher of the Ionian school, who put the Sun’s distance at 27 times the radius of the Earth and that of the Moon at 18 times the radius of the Earth. The Pythagorean school of philosophers also made estimates of the sizes of planetary orbits in the early fifth century, based on their analogy of the orbits with musical scales. They sought a harmony of the Universe in the distances of the planets, and from musical proportions arrived at the relative distances of the orbits. There are other references to distances in the classical sources, but very little remains of the methods adopted. During the third century BC, however, a series of measurements were made by Aristarchus of Samos, in an account entitled ‘On the sizes and distances of the Sun and Moon’, which is still extant. Aristarchus is now famous for his anticipation of the heliocentric system, which we know of only through second-hand sources.

To obtain the relative distances of the Moon and Sun, Arist¬archus observed the Moon at quadrature, when the Earth, Moon and Sun form a right-angled triangle. His measurements gave the angle Moon-Earth-Sun as 87°, which corresponded to a triangle in which the Earth-Sun distance was 19 times the Earth-Moon distance. He also reported that since the Sun and Moon subtend the same angle at the Earth, the Sun must then be 19 times larger than the Moon. Modern measurements give the ratio of distances as 400:1 rather than 19:1. This discrepancy would have arisen from the difficulty of the measurement. In practice it is extremely tricky to determine the exact moment of quadrature (when the Moon is at precisely half phase), and even a slight miscalculation produces a large error in the final result.

The observation mentioned above gives only the relative measures of the Sun-Moon distances. Aristarchus was also able to arrive at an absolute’ distance by using his observations of the Earth’s shadow during a lunar eclipse. By measuring the time elapsed for the Moon first to appear totally eclipsed, and the total time of obscuration, he concluded that the Earth’s shadow at the Moon’s distance was twice the size of the Moon. Since he knew the actual diameter of the Earth, by use of simple geometrical argu¬ments he was able to arrive at the absolute distances of the Sun and Moon. His value of the Sun’s distance, of the order of six million kilometres, was grossly underestimated primarily because of the difficulties in the initial measure of the relative distances, but the method which he used was ingenious.

A more tenuous method of obtaining the scale of the Solar System was through a synthesis of the Aristotelian mechanistic scheme of the Universe with the Ptolemaic mathematical model of the planets. Many attempts were made in the post-Ptolemaic period, particularly among the Arabs, to fit the epicyclic theory more satisfactorily into the concept of thick spherical shells. This combination allowed the relative dimensions of the Ptolemaic scheme to be transformed into absolute distance by utilizing known distance of the innermost sphere, the sphere of the Moon. Each shell was thick enough to allow both for each planet’s nearest and farthest approaches to the Earth. From a knowledge of the relative sizes of the epicycle and deferent, they calculated the ratio of the inside and outside diameters of each sphere. Since the spheres were all nested together, the outer diameter of one determined the inner diameter of the next so that knowledge of the distance of one sphere sufficed to determine the distances of all the others. Implicit in this whole scheme was the belief that no empty spaces existed between the spheres. A void was not permissible.

After the fifth century AD, estimates based on the concept of space-filling spheres became quite common. Al-Farghani, the ninth-century Arabic astronomer, provided a well-known set of cosmological dimensions using this method.

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