Epicycles. In this picture, the Earth is at the cen-ter.The planet, P, doesn't simply orbit the Earth. It goes around in a circle, which in turn orbits the Earth. If the planet's motion along the epicycle is faster than the epicycle's motion around the Earth, then the planet can appear to go backward for parts of each orbit. More layers of epicycles can be added to this picture.

An opposing picture was supported by the 16th century Polish astronomer, Nicholas Copernicus. In the Copernican system, the Sun is at the center of the planetary system. This picture is therefore called the heliocentric model. Copernicus showed that the retrograde motion is an artifact, caused by the motion of the Earth. This is illustrated in Fig. 22.5. The Copernican system had the planets in circular orbits, not ellipses. Therefore, detailed predictions of planetary positions had small errors. To correct those errors, epicycles had to be added to the Copernican model, taking away from the simplicity of the picture.

When Galileo Galilei turned his newly invented telescope to the planets, he found that Venus does not appear as a perfect disk. It goes through a series of phases, similar to those of the Moon. The size of the disk also changes as the phase

P'5 P'4 P'3 P'2 P'l circles, called epicycles. As shown in Fig. 22.4, each planet was supposed to move around its epicycle as the center of the epicycle orbits the Earth. To obtain a closer fit to the observed motions, higher order epicycles were added.

Retrograde motion in the heliocentric system.

(a) The Sun is at the center.We consider the Earth at five positions E| through E5 with the planet at P| through P5 at the same times.We use the line of sight from the Sun through E3 and P3 as a reference direction.The dashed lines are all parallel to that direction, and the angles 61 through 65 keep track of the differences between the line of sight from Earth to the planet and the reference direction.We see that since the Earth is moving faster than the planet, the line of sight goes from being ahead of the dashed line to being behind the dashed line.

(b) The view from Earth.The apparent position of the planet on the sky is indicated by P'1 through P'5. During this part of their orbits the planets appears to move backward on the sky.

changes. These observations can be explained easily in the heliocentric model, because Venus would not always be at the same distance from Earth. The phases result from the fact that we see differing amounts of the illuminated surface. There was no similar explanation in the Earth-centered system. Though Galileo was persecuted for holding that the heliocentric picture is the true one, his work had great influence on future scientific thought. Work switched from trying to find what was at the center of the planetary system to trying to understand how the planets, the Earth included, move around the Sun.

Extensive observations were carried out by Tycho Brahe, at Uraniborg, Denmark, late in the 16th century. Brahe moved to Prague in 1597, and died four years later. His results were taken over by an assistant, Johannes Kepler. Based on his analysis, Kepler published two laws of planetary motion in 1609, and a third law nine years later. Together, these are known as Kepler's laws. It is important to remember that these laws were based on observations, not on any particular theory.

Before discussing Kepler's laws, we look briefly at how we survey the Solar System, measuring the periods and sizes of orbits. We take advantage of certain geometric arrangements. These are shown in Fig. 22.6. We first look at planets that are closer to the Sun than the Earth. When the planet is between the Earth and the Sun, we say that it is at inferior conjunction, and it appears too close to the Sun in the sky to observe. As the planet moves in its orbit, the angle between it and the Sun (as seen from Earth) becomes larger. The planet appears farther and farther from the Sun. Eventually, since its orbit is smaller than the Earth's, it reaches a maximum apparent separation from the Sun. This is called the greatest elongation. At that point, the Earth, the Sun and the planet make a right triangle, with the planet at the right angle. After that the planet appears to get closer to the Sun, and when it is on the far side it is at superior conjunction. The pattern then repeats on the other side of the line from the Earth to the Sun. When the planet is on one side of the Sun it will appear east of the Sun in the sky, and when it is on the other side it appears west of the Sun. When it is west of the Sun, it rises and sets before the Sun, and it is therefore most easily visible in

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