The Theory of Concentric Spheres

Greek philosophers began a new approach that reached beyond astrology, attempting to explain in a rational manner why the planets move as they do. The sphere and circular motion was their preferred ideal for celestial motion (remaining so for two millennia). As geometric forms, the sphere and the circle were much investigated in Greek mathematics. Also, perfect circular motion, always returning back its original point, seemed to be suitable for celestial bodies held to be divine beings or at least eternally existing objects, and in fact the celestial sphere does seem to rotate perfectly uniformly.

Plato asked his pupils what kind of simple motions could explain the complicated movements of planets. Eudoxus (ca. 408-355 BC) took up the challenge. Eudoxus' other achievements include a method to derive formulae for areas and volumes similar to modern integral calculus.

Fig. 3.1 The reversal of motion of Mars in 2003. Its synodic period of 780 days separates any two retrograde loops, which happen in different constellations of the ecliptic. This was a key phenomenon that the ancients and later (more successfully) Copernicus attempted to explain by models of planetary motion. (Credit: NASA/JPL-Caltech)

Eudoxus' theory of spheres concentric on the Earth was the first mathematical model explaining in some detail the motions of the sky, including the puzzling retrograde loops. The model was based on spheres rotating with different but uniform speeds around their axes. These axes connected an inner sphere to the next one and were inclined with fixed angles relative to each other. Beyond all the planets was the celestial sphere of fixed stars revolving uniformly once a day around the immobile Earth. We hope our brief description will not make the reader dizzy! Sets

 Planet Synodic period Sidereal period Mercury 116 days 88 days Venus 584 245 days Mars 780 687 days = 1.88 years Jupiter 399 4,333 days = 11.8 years Saturn 378 10,744 days = 29.4 years Uranus 370 30,810 days = 84.4 years Neptune 368 60,440 days = 165.5 years Plutoa 367 91,750 days = 251.2 years

aAccording to the modern definition Pluto is not a major planet, but a dwarf planet. Note how the synodic period approaches the length of our year for increasingly long sidereal periods (can you figure out why?)

Fig. 3.2 A simplified diagram of Eudoxus' concentric spheres. Spheres rotate with different but uniform speeds around their axes. The axes connect an inner sphere to the next one and were inclined with fixed angles relative to each other. Therefore, the path of the planet as seen from Earth is more complex than a circle

Fig. 3.2 A simplified diagram of Eudoxus' concentric spheres. Spheres rotate with different but uniform speeds around their axes. The axes connect an inner sphere to the next one and were inclined with fixed angles relative to each other. Therefore, the path of the planet as seen from Earth is more complex than a circle

of interconnected spheres provided each planet with its own specific motions. The more uniformly moving Sun and Moon could be dealt with just using three each. The basic idea is shown schematically for the Moon in Fig. 3.2. The first rotated on a north-south axis to create the daily motion. The second was tilted relative to the first to include the tilt of the ecliptic relative to the celestial equator, turning once every sidereal period. Finally, the third turned with the path tilted to include deviations from the ecliptic by the Moon.1

With his model, Eudoxus could explain fairly well the planetary motions known at the time. However, Mars proved to be a thorny case whose motions were next to impossible to match with the model. Eudoxus does not seem to have imagined the model as representing a real physical structure but instead as a purely mathematical construction with one planet's set of spheres not affecting the motion of another's even though they were nested concentrically.

Aristotle's planetary model was an expanded version of Eudoxus' using a total of 56 spheres centered on the Earth. Aristotle may have viewed the spheres as physical entities (a sort of celestial crystal). However, he rejected Pythagoras's idea about the music of the spheres. On the contrary, he regarded the silence of the heavens as a proof of the sphere-carriers - noise would be expected if the celestial bodies would rush through some medium. The number of spheres was larger because Aristotle wanted to link together the sets of spheres belonging to each planet with additional spheres so that the fundamental daily motion of the outer sphere of the fixed stars was transferred from up to down.

1 The planets having retrograde loops (Mercury, Venus, Mars, Jupiter, Saturn) required sets of four spheres each to explain their more complicated motions. Hence the number of planetary spheres is 26 (= (2 x 3) + (5 x 4)) all nested concentrically.

Fig. 3.3 A schematic illustration of the epicycle model. The planet moves on a small circle (epicycle) whose center moves along a large circle (deferent) centered on the Earth

Fig. 3.3 A schematic illustration of the epicycle model. The planet moves on a small circle (epicycle) whose center moves along a large circle (deferent) centered on the Earth

Planet

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