Lesson 4: Planetary and Satellite Motion

Kepler's Three Laws

In 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 sun-centered 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:

• The path of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)
• An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
• The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)

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 will be constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if a 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 that line were drawn from the earth to the sun, then the area swept out by the line in every month would be the same. This is depicted in the diagram below. As can be noted 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 (0.5*base*height) 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 the 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 T²/R³ ratio is the same for earth as it is for mars. In fact, if the same T²/R³ ratio is computed for the other planets, it will be found that this ratio shows approximate agreement with the same value for both earth and mars (see table below). Amazingly, every planet has the same T²/R³ 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 T²/R³ ratio for the planets' orbits about the sun also accurately describes the T²/R³ ratio for any satellite (whether a moon or a man-made satellite) about any planet. There is something much deeper to be found in this T²/R³ ratio 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.

1. Today it is widely believed that the planets travel in elliptical paths around the sun.

1. Who gathered the data to support this fact?
2. Who analyzed the data to support this fact?
3. Who provided the accurate explanation for why this occurs?

2. Galileo discovered four moons of Jupiter. One moon - Io - which he measured to be 4.2 units from the center of Jupiter and had an orbital period of 1.8 days. Galileo measured the radius of Ganymede to be 10.7 units from the center of Jupiter. Use Kepler's third law to predict the orbital period of Ganymede.

3. If a small planet were located eight times as far from the sun as the Earth's distance from the sun (1.5 x 10¹¹ m), how many years would it take the planet to orbit the sun. GIVEN: T²/R³ = 2.97 x 10^-19 s²/m³

4. On average, the planet Mars is 1.52 times further from the sun as is Earth. Given that the Earth orbits the sun in approximately 365 earth days, predict the time required for Mars to orbit the sun.

The information below represents the R-T data for the four biggest moons of Jupiter. The mass of the planet Jupiter is 1.9 x 10²⁷ kg. Use this information to answer the next five questions.

5. Fill in the last column of the data table.

6. What do you notice about the values in the last column? What law is this?

7. Use the graphing capabilities of your TI calculator to plot T² vs. R³ (T² should be plotted along the vertical axis) and to determine the equation of the line. Write the equation in slope-intercept form below.

See graph below.

8. How does the T²/R³ ratios for Jupiter (as shown in the last column of the data table) compare to the T²/R³ ratio found in #7 (i.e., the slope of the line)?

9. How does the T²/R³ ratio for Jupiter (as shown in the last column of the data table) compare to the T²/R³ ratio found using the following equation? (G=6.67x10^-11 N*m²/kg² and M_Jupiter = 1.9 x 10²⁷ kg)

Return to Question #6

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