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Johannes Kepler, a 29-year-old German, became one of Brahe s assistants when he moved to Prague Brahe trained his assistants to use instruments, such as those shown in Figure 7-1 Upon his death in 1601, Kepler inherited 30 years worth of Brahe s observations He studied Brahe s data and was convinced that geometry and mathematics could be used to explain the number, distance, and motion of the planets Kepler believed that the Sun exerted a force on the planets and placed the Sun at the center of the system After several years of careful analysis of Brahe s data on Mars, Kepler discovered the laws that describe the motion of every planet and satellite Kepler s first law states that the paths of the planets are ellipses, with the Sun at one focus An ellipse has two foci, as shown in Figure 7-2 Like planets and stars, comets also orbit the Sun in elliptical orbits Comets are divided into two groups long-period comets and short-period comets based on orbital periods, each of which is the time it takes the comet to complete one revolution Long-period comets have orbital periods longer than 200 years, and short-period comets have orbital periods shorter than 200 years Comet Hale-Bopp, with a period of 2400 years, is an example of a long-period comet Comet Halley, with a period of 76 years, is an example of a short-period comet

Figure 7-2 Planets orbit the Sun in elliptical orbits with the Sun at one focus (Illustration not to scale)

Kepler found that the planets 11 12 move faster when they are closer to the Sun and slower when they Planet are farther away from the Sun Thus, Kepler s second law states that an imaginary line from the Sun to a planet sweeps out equal areas in Sun equal time intervals, as illustrated in 1 Figure 7-3 Kepler also found that there is a mathematical relationship between periods of planets and their mean 2 distances away from the Sun 3 Kepler s third law states that the square of the ratio of the periods of any two planets revolving about the Sun is equal to the cube of the ratio of their average distances from the Sun Thus, if the periods of the planets are TA and TB, and their average distances from the Sun are rA and rB, Kepler s third law can be expressed as follows

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Figure 7-3 A planet moves fastest when it is close to the Sun and slowest when it is farther from the Sun Equal areas are swept out in equal amounts of time (Illustration not to scale)

Crystal Zheng lives in Versailles But, this time, the measure has not changed in value It s the same as the total at the Postal Code level in the previous query This would seem to suggest that either Versailles has only one postal code (78000) or that it has only one (78000) with any sales Now that you know how, you could test this by using children on Versailles

The squared quantity of the period of object A divided by the period of object B, is equal to the cubed quantity of object A s average distance from the Sun, divided by object B s average distance from the Sun

FIGURE 8-3

Note that the first two laws apply to each planet, moon, and satellite individually The third law, however, relates the motion of several objects about a single body For example, it can be used to compare the planets distances from the Sun, shown in Table 7-1, to their periods about the Sun It also can be used to compare distances and periods of the Moon and artificial satellites orbiting Earth

Table 7-1

Planetary Data Name Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Average Radius (m) 696 244 605 638 340 715 603 256 248 108 106 106 106 106 107 107 107 107 Mass (kg) 199 330 487 598 642 190 569 868 102 1030 1023 1024 1024 1023 1027 1026 1025 1026 Mean Distance From Sun (m) 579 108 150 228 778 143 287 450 587 1010 1011 1011 1011 1011 1012 1012 1012 1012

120 106