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# Orbital Period (page 2)

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Author: Janice VanCleave

Center of mass is the point on a body where its mass seems to be concentrated. Gravity is a force of attraction between all objects in the universe (everything throughout space including Earth). Two celestial bodies held together by mutual gravitational attraction are called binary bodies, such as the Sun and a planet. The barycenter is the point between binary bodies where their center of mass is located, which is the point around which the bodies rotate.

1. The farther the barycenter of a planet-Sun system is from the Sun, the greater the orbital period of the planet. Use two dowels of different lengths to illustrate this principle. First, determine the center of mass of each dowel by balancing it horizontally on your finger. Where your supporting finger touches the rods is their center of mass. Mark the center of mass on each dowel. Rest one end of each dowel on a level surface, such as a floor or table, with a few inches (cm) between them. The flat surface represents the Sun. Hold the dowels up vertically, steadying them with the tips of your fingers. Lean both dowels forward the same amount so they will fall in the same direction. Release both at the same time. Which hits the surface first? For your display you may wish to make a drawing like Figure 5.2 showing the results of this experiment.
2. a. Design an experiment to measure the orbital periods of one of the visible planets. Note that Mercury is too close to the Sun to be seen and the movements of Jupiter and Saturn are very slow. The visibility and relatively fast movements of Venus and Mars make them the best subjects. One experiment is to measure the average angular motion of the selected planet per day. Use this to calculate the orbital period, which is the time for 1 revolution of 360°. Staring on day 1, use a cross-staff  to measure the angular separation between the planet and a star. Take five measurements on the first day and average them. Take five measurements 7 days later and average them. Record measurements in an Angular Separation Data table like Table 5.2. Calculate the orbital period of the planet using these steps:
• Calculate the difference (D1) between the average angular separation on the first and seventh day.
• Calculate the time (t1) for D1, which is the time between the two sets of measurements. Thus, t1, = 7 days.
• Calculate the portion of the revolutions (D2) the planet moved in 6 days by dividing D1 by the degrees in 1 revolution, which is 360°. D2 = D1/360° .
T = t1/D2
• .
3. b. Kepler's third law of planetary motion states that a planet's orbital period (time of 1 revolution around the Sun) depends on its average distance from the Sun. This equation is:

P2 = k × r3

where P is the orbital period, r the average distance from the Sun, and k a constant. If P is in years and r is in AU, k equals 1. Using the known radius of the visible planets (see Appendix 3), calculate the orbital period of each. Note that the radius (1/2 diameter) must be expressed in astronomical units (AU), which can be determined by taking the average distance of the planet from the Sun and dividing by the average distance of Earth from the Sun. For example, for Mercury the radius in AU is 0.39 AU. See page 33 for AU calculations.

c. Using the known orbital period and the experimentally determined orbital periods in the previous experiment, calculate your experimental percentage error. (See Appendix 2 for information about percentage error.) What is retrograde motion and how would it affect your percentage error? For information about retrograde motion, see Dinah Moche's Astronomy (New York: Wiley, 2000), pp. 197–200.

### Get the Facts

The German astronomer Johann Titius (1729–1796) showed that the distances of planets from the Sun follow a fixed formula when measured in astronomical units. The formula is known as Bode's law. What is this pattern? Why isn't it called Titius's law? What significant role did the pattern play in the discovery of the asteroids and some of the planets? How accurate are the distances using the formula? For information, see Nancy Hathaway's, The Friendly Guide to the Universe (New York: Penguin, 1994), pp. 190–192.

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