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A Modern Almagest an Updated Version of Ptolemy’S Model of the Solar System

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A Modern Almagest an Updated Version of Ptolemy’S Model of the Solar System A Modern Almagest An Updated Version of Ptolemy’s Model of the Solar System Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 1.1 Euclid’sElementsandPtolemy’sAlmagest . ......... 5 1.2 Ptolemy’sModeloftheSolarSystem . ..... 5 1.3 Copernicus’sModeloftheSolarSystem . ....... 10 1.4 Kepler’sModeloftheSolarSystem . ..... 11 1.5 PurposeofTreatise .................................. .. 12 2 Spherical Astronomy 15 2.1 CelestialSphere................................... ... 15 2.2 CelestialMotions ................................. .... 15 2.3 CelestialCoordinates .............................. ..... 15 2.4 EclipticCircle .................................... ... 17 2.5 EclipticCoordinates............................... ..... 18 2.6 SignsoftheZodiac ................................. ... 19 2.7 Ecliptic Declinations and Right Ascenesions. ........... 20 2.8 LocalHorizonandMeridian ............................ ... 20 2.9 HorizontalCoordinates.............................. .... 23 2.10 MeridianTransits .................................. ... 24 2.11 Principal Terrestrial Latitude Circles . ......... 25 2.12 EquinoxesandSolstices. ....... 25 2.13 TerrestrialClimes .................................. ... 26 2.14 EclipticAscensions .............................. ...... 27 2.15 AzimuthofEclipticAscensionPoint . .......... 29 2.16 EclipticAltitudeandOrientation. .......... 30 3 Dates 63 3.1 Introduction...................................... .. 63 3.2 Determination of Julian Day Numbers . .... 63 4 Geometric Planetary Orbit Models 67 4.1 Introduction...................................... .. 67 2 MODERN ALMAGEST 4.2 ModelofKepler.................................... .. 67 4.3 ModelofHipparchus ................................. .. 69 4.4 ModelofPtolemy ................................... .. 71 4.5 ModelofCopernicus ................................. .. 72 5 The Sun 75 5.1 DeterminationofEclipticLongitude . ........ 75 5.2 ExampleLongitudeCalculations . ........ 77 5.3 DeterminationofEquinoxandSolsticeDates . ........ 78 5.4 EquationofTime ................................... .. 80 6 The Moon 87 6.1 DeterminationofEclipticLongitude . ........ 87 6.2 ExampleLongitudeCalculations . ........ 88 6.3 DeterminationofEclipticLatitude . ........ 90 6.4 LunarParallax..................................... .. 91 7 Lunar-Solar Syzygies and Eclipses 101 7.1 Introduction...................................... 101 7.2 DeterminationofLunar-SolarElongation . ........101 7.3 ExampleSyzygyCalculations. .102 7.4 SolarandLunarEclipses ............................ .104 7.5 ExampleEclipseCalculations. ........107 7.6 EclipseStatistics................................ .108 8 The Superior Planets 115 8.1 DeterminationofEclipticLongitude . ........115 8.2 Mars ............................................119 8.3 Determination of Conjunction, Opposition, and Station Dates.............121 8.4 Jupiter.......................................... 124 8.5 Saturn ...........................................125 9 The Inferior Planets 141 9.1 DeterminationofEclipticLongitude . ........141 9.2 Venus............................................ 142 9.3 Determination of Conjunction and Greatest Elongation Dates . ............145 9.4 Mercury ..........................................147 10 Planetary Latitudes 157 10.1 Introduction..................................... .157 10.2 Determination of Ecliptic Latitude of Superior Planet . .............157 10.3 Mars ............................................159 10.4 Jupiter......................................... 160 10.5 Saturn ........................................... 161 10.6 Determination of Ecliptic Latitude of Inferior Planet . ..............162 10.7 Venus........................................... 164 10.8 Mercury ..........................................166 CONTENTS 3 11 Glossary 179 12 Index of Symbols 185 13 Bibliography 187 4 MODERN ALMAGEST Introduction 5 1 Introduction 1.1 Euclid’s Elements and Ptolemy’s Almagest The modern world inherited two major scientific treaties from the civilization of Ancient Greece. The first of these, the Elements of Euclid, is a large compendium of mathematical theorems con- cerning geometry, proportion, and number theory. These theorems were not necessarily discovered by Euclid himself—being largely the work of earlier mathematicians, such as Eudoxos of Cnidus, and Theaetetus of Athens—but were arranged by him in a logical manner, so as to demonstrate that they can all ultimately be derived from five simple axioms. The Elements is rightly regarded as the first, largely succesful, attempt to construct an axiomatic system in mathematics, and is still held in high esteem within the scientific community. The second treatise, the Almagest1 of Claudius Ptolemy, is an attempt to find a simple geometric explanation for the apparent motions of the sun, the moon, and the five visible planets in the earth’s sky. On the basis of his own naked-eye observations, combined with those of earlier astronomers such as Hipparchus of Nicaea, Ptolemy proposed a model of the solar system in which the earth is stationary. According to this model, the sun moves in a circular orbit, (nearly) centered on the earth, which maintains a fixed inclination of about 23◦ to the terrestrial equator. Furthermore, the planets move on the rims of small circles called epicycles, whose centers revolve around the earth on large eccentric circles called deferents—see Fig. 8.2. The planetary deferents and epicycles also maintain fixed inclinations,2 which are all fairly close to 23◦, to the terrestrial equator. The scientific reputation of the Almagest has not fared as well as that of Euclid’s Elements. Nowadays, it is a commonly held belief, even amongst scientists, that Ptolemy’s mistaken adher- ence to the tenets of Aristotelian philosophy—in particular, the immovability of the earth, and the necessity for heavenly bodies to move uniformly in circles—led him to construct an overcompli- cated, unwieldy, and faintly ridiculous model of planetary motion. As is well-known, this model was superseded in 1543 CE by the heliocentric model of Nicolaus Copernicus, in which the planets revolve about the sun in circular orbits.3 The Copernican model was, in turn, superseded in the early 1600’s CE by the, ultimately correct, model of Johannes Kepler, in which the planets revolve about the sun in eccentric elliptical orbits. The aim of this treatise is to re-examine the scientific merits of Ptolemy’s Almagest. 1.2 Ptolemy’s Model of the Solar System Claudius Ptolemy lived and worked in the city of Alexandria, capital of the Roman province of Egypt, during the reigns of the later Flavian and the Antonine emperors. Ptolemy was heir—via the writings of Euclid, and later mathematicians such as Apollonius of Perga, and Archimedes of Syracuse—to the considerable mathematical knowledge of geometry and arithmetic acquired by 1The true title of this work is Syntaxis Mathematica, which means something like “Mathematical Treatise”. The name Almagest is probably an Arabic corruption of the work’s later Greek nickname, H Megiste (Syntaxis), meaning “The Greatest (Treatise)”. 2Actually, Ptolemy erroneously allowed the inclinations of the deferents and epicycles to vary slightly. 3In fact, the planets revolve on small circular epicycles, whose centers revolve around the sun on eccentric circular deferents. 6 MODERN ALMAGEST the civilization of Ancient Greece. Ptolemy also inherited an extensive Ancient Greek tradition of observational and theoretical astronomy. The most important astronomer prior to Ptolemy was undoubtedly Hipparchus of Nicaea (second century BCE), who developed the theory of solar mo- tion used by Ptolemy, discovered the precession of the equinoxes, and collected an extensive set of astronomical observations—some of which he made himself, and some of which dated back to Babylonian times—which were available to Ptolemy (probably via the famous Library of Alexan- dria). Other astronomers who made significant contributions prior to Ptolemy include Meton of Athens (5th century BCE), Eudoxos of Cnidus (5th/4th century BCE), Callipus of Cyzicus (4th cen- tury BCE), Aristarchus of Samos (4th/3rd century BCE), Eratosthenes of Cyrene (3rd/2nd century BCE), and Menelaus of Alexandria (1st century CE). Ptolemy’s aim in the Almagest is to construct a kinematic model of the solar system, as seen from the earth. In other words, the Almagest outlines a relatively simple geometric model which describes the apparent motions of the sun, moon, and planets, relative to the earth, but does not attempt to explain why these motions occur (in this respect, the models of Copernicus and Kepler are similar). As such, the fact that the model described in the Almagest is geocentric in nature is a non-issue, since the earth is stationary in its own frame of reference. This is not to say that the heliocentric hypothesis is without advantages. As we shall see, the assumption of heliocentricity allowed Copernicus to determine, for the first time, the ratios of the mean radii of the various planets in the solar system. We now know, from the work of Kepler, that planetary orbits are actually ellipses which are confocal with the sun. Such orbits possess two main properties. First, they are eccentric: i.e., the sun is displaced from the geometric center of the orbit. Second, they are elliptical: i.e., the orbit is elongated along a particular axis. Now, Keplerian orbits are characterized by a quantity, e, called the eccentricity, which measures their deviation
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