

Lesson 1: Motion Characteristics for Circular MotionThe Centripetal Force Requirement Mathematics of Circular Motion 
Lesson 4: Planetary and Satellite MotionKepler's Three LawsIn the early 1600s, Johannes Kepler proposed three laws of planetary motion. Kepler was able to summarize the carefully collected data of his mentor  Tycho Brahe  with three statements which described the motion of planets in a suncentered solar system. Kepler's efforts to explain the underlying reasons for such motions are no longer accepted; nonetheless, the actual laws themselves are still considered an accurate description of the motion of any planet and any satellite. Kepler's three laws of planetary motion can be described as follows:
Kepler's first law  sometimes referred to as the law of ellipses  explains that planets are orbiting the sun in a path described as an ellipse. An ellipse can easily be constructed using a pencil, two tacks, a string, a sheet of paper and a piece of cardboard. Tack the sheet of paper to the cardboard using the two tacks. Then tie the string into a loop and wrap the loop around the two tacks. Take your pencil and pull the string until the pencil and two tacks make a triangle (see diagram at the right). Then begin to trace out a path with the pencil, keeping the string wrapped tightly around the tacks. The resulting shape will be an ellipse. An ellipse is a special curve in which the sum of the distances from every point on the curve to two other points is a constant. The two other points (represented here by the tack locations) are known as the foci of the ellipse. The closer together which these points are, the more closely that the ellipse resembles the shape of a circle. In fact, a circle is the special case of an ellipse in which the two foci are at the same location. Kepler's first law is rather simple  all planets orbit the sun in a path which resembles an ellipse, with the sun being located at one of the foci of that ellipse. Kepler's second law  sometimes referred to as the law of equal areas  describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time. For instance, if an imaginary line were drawn from the earth to the sun, then the area swept out by the line in every 31day month would be the same. This is depicted in the diagram below. As can be observed in the diagram, the areas formed when the earth is closest to the sun can be approximated as a wide but short triangle; whereas the areas formed when the earth is farthest from the sun can be approximated as a narrow but long triangle. These areas are the same size. Since the base of these triangles are longer when the earth is furthest from the sun, the earth would have to be moving more slowly in order for this imaginary area to be the same size as when the earth is closest to the sun.
Kepler's third law  sometimes referred to as the law of harmonies  compares the orbital period and radius of orbit of a planet to those of other planets. Unlike Kepler's first and second laws which describe the motion characteristics of a single planet, the third law makes a comparison between the motion characteristics of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets. As an illustration, consider the orbital period and average distance from sun (orbital radius) for Earth and mars as given in the table below.
Observe that the T^{2}/R^{3 }ratio is the same for Earth as it is for mars. In fact, if the same T^{2}/R^{3 }ratio is computed for the other planets, it can be found that this^{ }ratio is nearly the same value for all the planets (see table below). Amazingly, every planet has the same T^{2}/R^{3} ratio.
Kepler's third law provides an accurate description of the period and distance for a planet's orbits about the sun. Additionally, the same law which describes the T^{2}/R^{3} ratio for the planets' orbits about the sun also accurately describes the T^{2}/R^{3} ratio for any satellite (whether a moon or a manmade satellite) about any planet. There is something much deeper to be found in this T^{2}/R^{3} ratio  something which must relate to basic fundamental principles of motion. In the next part of Lesson 4, these principles will be investigated as we draw a connection between the circular motion principles discussed in Lesson 1 and the motion of a satellite.

Jupiter's Moon 
Period (s) 
Radius (m) 
T^{2}/R^{3} 
Io 
1.53 x 10^{5} 
4.2 x 10^{8} 
a. 
Europa 
3.07 x 10^{5} 
6.7 x 10^{8} 
b. 
Ganymede 
6.18 x 10^{5} 
1.1 x 10^{9} 
c. 
Callisto 
1.44 x 10^{6} 
1.9 x 10^{9} 
d. 
5. Determine the T^{2}/R^{3} ratio (last column) for Jupiter's moons.
6. What pattern do you observe in the last column of data? Which law of Kepler's does this seem to support?
7. Use the graphing capabilities of your TI calculator to plot T^{2} vs. R^{3} (T^{2} should be plotted along the vertical axis) and to determine the equation of the line. Write the equation in slopeintercept form below.
See graph below.
8. How does the T^{2}/R^{3} ratios for Jupiter (as shown in the last column of the data table) compare to the T^{2}/R^{3} ratio found in #7 (i.e., the slope of the line)?
9. How does the T^{2}/R^{3} ratio for Jupiter (as shown in the last column of the data table) compare to the T^{2}/R^{3} ratio found using the following equation? (G=6.67x10^{11} N*m^{2}/kg^{2} and M_{Jupiter} = 1.9 x 10^{27} kg)
Return to Question #6
Lesson 4: Planetary and Satellite Motion
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19962007