Greek Astronomy and the Ptolemaic System

ASTRONOMY 9: HISTORYOF COSMOLOGY Handout #11 Notes by I. R. King: Greek Astronomy and the Ptolemaic System Greek Astronomy and the Ptolemaic System 1 Heraclides 1Ve're going to go through toda?- the developments that led to the Ptolemaic system of the motion of ' was planets around the Earth, but the developnlent of Greek thought, and in particular Greekr aclence, far from nlonolithic and undeviating. Two "aberrations" el-en appeared that are part of our modern vie-, of the world: Heraclides suggested that the daily rising and setting of the sun, moon. and stars could be explained by a rotation of the earth on its axis, and Aristarchus (and probably Heraclides too) thought that the Earth went around the Sun. But these ideas attracted little attention or following. (I don't think Ptolemy el-en nlentions Aristarchus or bothers to refute him.) Now it's ver?- easy today to say, .'Oh. those stupid Greeks! They had people who knew the right answer. but they ignored them." Dut this of course is a ridiculously naive view-. They lived in a different world, with views that in lnany ma?-s are just as intelligent as ours, but different fundamental ideas were dominant. In particular, philosophy dominated over scientific inquiry, and Aristotle's ideas about the nature of the earth and the heavens ruled out both of these motions of the earth. And let's not forget the role of co~nnlonsense: everybod?. knows that whenel-er you accelerate you can feel it, and we certainly don't feel the earth spinning or rushing around the sun. But I don't want to get off 011 this side track; let's get back to the developnlent of Greek astronomy. Ilut that brings us right back to i2ristarchus. - 2 Aristarchus His heliocentric picture hasn't survived in ariting; we knom7 of it o11l?- through mention by others. Dut one thing he did was definitely mainstream: measuring the relative distances of the moon and sun. Illagnificent in principle. but Raved in practice, for tw-o reasons. First, it is hard to measure the time of quarter moon precisely. (This is usually excited to explain the poor accuracy of his result.) Second, and rarel?- noted, is the fact that the eccentricity (ellipticity) of the moon's orbit malies it move with varying speed, ahich vitiates the ahole idea, unless you take an average over many moon orbits. The nlonthly 7-ariations are :-lox larger than the effect Aristarchus was trying to measure. 3 Eratosthenes One other quantity that was measured in ancient times is the size of the earth. Around 200 IIC. Eratosthenes, who lived in Alexandria, had heard that in the town of Syene, some distance to the south of him, the sun vias exactly overhead on the summer solstice (when the sun is farthest north). From the length of a shadow in .$lexandria, he measured that the sun mas 7.2' from his zenith at solstice. This angle is 1/50 of a full circle, so the circumference of the earth is 50x the distance from S?-ene to Alexandria. The accuracy of his result is not known with certaint?., because the size of the unit he used is not known. A confusion of units also seems to have affected Columbus, who thought that the earth's circumference was s~nallerthan it really is (and used the erroneous figures to justify his voyage across the ocean). 4 The Problem of Planetary Motions The greatest achievement of ancient science rz-as the aorking out of an accurate description of the co~nplicated motions of the planets (for example, retrograde motion). How- do we get around this? Appollonius (c. eon DC) came up a i t h the idea of the epicycle. Another problem was no11-unifornl speed, ~ v l ~ i cinh the Aristotelian universe vias unacceptable. It 7cas well knoan that the sun doesn't move uniformly throughout the year; the lengths of the four seasons differ by several days. Appollonius had a solution for this one too: the eccentric circle. Motion vias not around the cosmic center (earth) but rather around a displaced point. (Ironic that the greatest ancient authority on ellipses seems never to ha\-e thought of using them for orbits!) 5 Hipparchus Hipparchus (c. 140 DC) vias the greatest observer of ancient times. He cataloged about 8 j 0 stars, and observed the positions of the planets (Hipparchus was to Ptolem?- as q c h o later vias to Iiepler). Magnit u d e s ( a logarithmic measure of brightness) in his catalog established the system of magnitudes that vie use today. Hipparchus also applied the ideas of Appollonius (the eccentric and epicycle) to real planetary data. Ptolemy (c. 150 AD) was the last great natural philosopher of classical times. He vias responsible for putting together a picture of planetary motion that stood until the beginning of modern times. (He also drew the best viorld atlas of ancient times, ~ v l ~ i clater h influenced Colu~llbustoward thinking that China was a reasonably short distance west of Europe!) He 7crote a great compendium of math and astronomy: /ie?strroi, or Almagest in its Arabic version. Ptolemy used observations made by Hipparchus to measure the p a r a l l a x of the moon. He seems to have gotten the right answer. but made two big mistakes which remarkably canceled out! Parallax exists for the stars too (since the earth moves round the sun) but is very small (since the stars are very far aviay). Whereas earlier scientists had been assembling ideas and observations (e.g., epicycle and eccentric of Appollonius; obserl-ations of Hipparchus, whose importance Ptolemy ackno~vledges),it a a s Ptolemy who finally put it all together to nlake a system that worked that is, it would fit observed positions in the past and could be used t o predict them in the future. (Ironically, the greatest practical importance for this system was considered to be its use for astrology, 7chic11 of all ignorant superstitions is the one that modern astrono~llersmost abominate.) The epicycle is the central device of the Ptolemaic system it can explain retrograde nlotion (ahich is much easier to explain in a heliocentric system!), but it's not enough. Ptolem?; had to nlake things really fit ~vell. He used the eccentric, which can account for varying speed. Today. vie know that each planetary orbit has a different eccentricity (that is. degree of ellipticity or non-circularity) of its orbit. So in the Ptolenlaic system, each planet has its circular orbit centered a t a different place. But even on an eccentric circle. he still couldn't get the speeds right ~ v i t huniform inotion ( a philosophical necessity), so he devised a new- trick. the e q u a n t . This is another point. distinct from both earth and the center of the eccentric circle, from which the planet appears to svieep out equal angles in equal times. IVhen you come to study Copernicus, you'll find that he was quite complace~ltabout epicycles and even used some small ones himself, but he despised the equant. In summary, a Ptolemaic orbit is defined by the d e f e r e n t , epicycle, eccentric. and equant. Ptolem?found nunlerical values for each of the seven "planets" (including Sun and bloon), that aould fit the observations of his day as vie11 as those of Hipparchus, nearly 300 years earlier. His model provided a good fit (within a fraction of a degree) to all planetary motions for several centuries! Some medieval Arabic astronomers added subsidiar?. epicycles, greatly complicating the system and leading to the idea that Ptolemy piled epicycles on top of epic?-cles. In fact, he didn't, and it was the (relatively) simple system just described that came into western Europe in the thirteenth century and was used there. Ptolemy had succeeded brilliantly a t solving Plato's old challenge to "save the phenomena". Dut his model had some shortcomings. First, it didn't contain any physics no explanation of why things move the way the?- do. (But note Copernicus also lacked physics, which 7cas only supplied a century and a half later by i'iewton.) Secondly, there were some suspicious coincidences u~lesplainedby Ptolemy's model. One was that the centers of the epicycles of Mercury and \Tenus were fixed on a line between the earth and sun. A second coincidence rz-as that retrograde nlotion al7cays takes place 7chen the planet is in the opposite direction from the sun (at opposition). Ilut it worked, and it fit many Aristotelian principles, so it 71,as just fine in its day.