Answers to Chapter 2. Motion of the Planets
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ANSWERS TO CHAPTER 2 Motion of the Planets 33. Because of their apparent motion relative to the star field. 34. Because their positions in the star field change continuously with time. 35. Any three of the following twelve Zodiacal constellations: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Sagittarius , Capricornus, Aquarius, and Pisces. 36. Note observed date, time, and position on the star field and brightness. 37. (a) Copernicus. 38. Both concepts are purely geometric in nature. The Ptolemaic hypothesis (geocentric) in its simplest form envisions the uniform motion of a planet or the Sun along a circular orbit called an epicycle whose center in turn moves uniformly along a larger circle called the deferent centered on the Earth. The combination of epicycle and deferent is different for each body. 128 Motion of the Planets / 129 The diagram gives a schematic view of this concept. Note that the centers of the epicycles of Mercury and Venus are attached to a radial line from the Earth to the Sun and that no epicycle is shown for the Sun. JUPITER IfSTAR ~~~~~~------~----~--~~-+~~--~~* PTOLEMY'S SCHEME MOTION OF THE PLANETS - 140 A.D. ( SCHEMATIC: NOT TO SCALE) The Copernican hypothesis (heliocentric) in its simplest form envisions the uniform motion of the Earth and all of the other planets along circles of various radii centered on the Sun. Both schemes (more elaborate in detail) undertake to describe the observed motion of the planets and the Sun on the star field. 39. It is totally incompatible with Newton's law of gravitation and his three laws of motion. 40. The motion of planets and the Sun along the epicycles and deferents of Ptolemy's scheme has no relationship whatever to the masses, forces, and accelerations that are the heart of physical reality as recognized by Galileo and Newton. 130 / Motion of the Planets 41. The heliocentric hypothesis has at least a general resemblance to the physical reality of forces, masses, and accelerations whereas the geocentric hypothesis has no such resemblance. 42. y • The first diagram depicts the Copernican scheme. Suppose that the initial configuration is an opposition with the Sun (S), the Earth (Eo) and Mars (Mo) aligned as shown along the Xoaxis . After a lapse of time t, the Earth is at E and Mars at M. A translated coordinate system XY is centered at E with the X-axis parallel to the Xo-axis. By a sequence of carefully scaled drawings, the Earth-Mars distance R and the geocentric longitude of Mars a can be tabulated as a function of t and then assembled as a polar plot in the XY coordinate system. The analytical solution is as follows with t in years: rl = SE = 1.0 Angle Eo SE = 360 t r2 =SM = 1.59 Angle Mo SM = 180 t Coordinates of M: x = 1.59 cos 180 t - 1.0 cos 360 t Y= 1.59 sin 180 t - 1.0 sin 360 t R = ";x2 + y2 sin a = y/R cos a = x/R y DEFERENT Q'---+---t----=... Motion of the Planets / 131 • The second diagram depicts the Ptolemaic scheme. Q is the center of the epicycle, a circle of radius rl. Q moves along the deferent, a circle of radius r2, such that the angle QEX = 180 t. Mars is located on the epicycle and moves along its circumference so that angle Q'QM = 360 t. The X-Y coordinate system is centered on the Earth and the X-axis points at a distant star. QQ' is parallel to the X-axis. The initial configuration at t = 0 is with Mars and the center of the epicycle on the +X-axis. The diagram shows the configuration after a lapse of time t. Again it is evident that a sequence of carefully scaled drawings will yield a tabulation of R and a as a function of time t. 132 / Motion of the Planets The analytical solution is as follows: Comparing the first and second diagrams, one notes (1) that angle ESM in the first diagram is equal to angle EQM in the second; (2) that SE in the first equals QM in the second, each being rl; and (3) that SM in the first equals EQ in the second, each being r2. Hence, triangle ESM in the first diagram is identical to triangle MQE in the second diagram. It follows that M has the same x,y coordinates in the second diagram as it had in the first and hence the...