8.1 Laws. Purpose. Concept and Skill Check. Materials. Procedure EXPERIMENT DATE : PERIOD NAME. Plot a planetary orbit and apply Kepler s Laws.

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1 ATE : PERIO NAME EXPERIMENT 8.1 Laws Purpose Plot a planetary orbit and apply Kepler s Laws. oncept and kill heck The motion of the planets has intrigued astronomers since they first gazed at the stars, moon, and planets filling the evening sky. But the old ideas of eccentrics and equants (combinations of circular motions) did not provide an accurate accounting of planetary movements. Johannes Kepler adopted the opernican theory that Earth revolves around the sun (heliocentric, or suncentered, view) and closely examined Tycho Brahe s meticulously recorded observations on Mars orbit. With these data, he concluded that Mars orbit was not circular and that there was no point around which the motion was uniform. When elliptical orbits were accepted, all the discrepancies found in the old theories of planetary motion were eliminated. From his studies, Kepler derived three laws that apply to the behavior of every satellite or planet orbiting another massive body. 1. The paths of the planets are ellipses, with the center of the sun at one focus. 2. An imaginary line from the sun to a planet sweeps out equal areas in equal time intervals, as shown in Figure The ratio of the squares of the periods of any two planets revolvingabout the sun is equal to the ratio of the cubes of their respective aver age distances from the sun. Mathematically, this relationship can be expressed as Ta 2 Tb 2 ra3 Fb 3 Figure 1. Kepler s law of areas. In this experiment, you will use heliocentric data tables to plot the positions of Mercury on polar graph paper. Then you will draw Mercury s orbit. The distance from the sun, the radius vector, is compared to Earth s average distance from the sun, which is defined as 1 astronomical unit or 1 AU. The angle, or longitude, between the planet and a reference point in space is measured from the zero degree point, or vernal equinox. Materials polar graph paper sharp pencil metric ruler Procedure 1. Orient your polar graph paper so that the zero degree point is on your right as you view the graph paper. The sun is located at the center of the paper. Label the sun without covering the center mark. Move about the center in a counterclockwise direction as you measure and mark the longitude. 2. electan appropriatescaleto representthevaluesforthe radiusvectors ofmercury s positions. ince Mercury is closer to the sun than is Earth, the value of the radius vector will always be less than 1 AU. In this step, then, each concentric circle could represent onetenth of an AU. opyright 1992 by the Glencoe ivision of Macmillan/McGrawHill Publishing ompany LABORATORY MANUAL 49

2 LABORATORY MANUAL opyright 1992 by the Glencoe ivision of Macmillan/McGrawHill Publishing ompany Adapted from The Astronomical Almanac for the Year 199, U.. Government Printing Office, Washington,.., 242, p. E Nov ec Nov Oct. 1, (degrees) (degrees) ate ate ome Heliocentric Positions for Mercury for Oh ynamical Time Table 1 Observations and ata 4. Repeat the procedure, plotting all given longitudes and associated radius vectors. date. Make a small dot at this point to represent Mercury s distance from the sun. Write the orbit of Mercury. out along the longitude line an appropriate distance, in your scale, for the radius vector for this the set of data for October 1 and locate the given longitude on the polar graph paper. Measure date next to this point. 3. Table 1 provides the heliocentric positions of Mercury over a period of several months. elect 5. After plotting all the data, carefully connect the points of Mercury s positions and sketch the 8.lLaws EXPERIMENT NAME

3 EXPERIMENT : Kepler s NAME 8.1 Laws Analysis 1. oes your graph of Mercury s orbit support Kepler s law of orbits? 2. raw a line from the sun to Mercury s position on ecember 2. raw a second line from the sun to Mercury s position on ecember 3. The two lines and Mercury s orbit describe an area swept by an imaginary line between Mercury and the sun during the tenday interval of time. Lightly shade this area. Over a small portion of an ellipse, the area can be approximated by assuming the ellipse is similar to a circle. The equationthat describes this value is where r is the average radius for the orbit. area = (O/36 )irr 2, etermine 8 by finding the difference in degrees between ecember 2 and ecember 3. Measure the radius at a point midway in the orbit between the two dates. alculate the area in AUs for this tenday period of time. 3. elect two additional tenday periods of time at points distant from the interval in Question 2 and shade these areas. alculate the area in AUs for each of these tenday periods. 4. Find the average area for the three periods of time from Questions 2 and 3. alculate the relative error between each area and the average. oes Kepler s law of areas apply to your graph? opyright 1992 by the Glencoe ivision of Macmillan/McGrawHill Publishing ompany LABORATORY MANUAL 51

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5 EXPERIMENT o1 epers O.F Laws NAME Application There has been some discussion about a hypothetical planet X that is on the opposite side of the sun from Earth and that has an average radius of 1. AU. If this planet exists, what is its period? how your calculations. Extension Using the data in Table 2, plot the radius vectors and corresponding longitudes for Mars. oes the orbit you drew support Kepler s law of ellipses? elect three different areas and find the area per day for each of these. oes Kepler s law of areas apply to your model of Mars? Table 2 ome Heliocentric Positions for Mars for Oh ynamical Time* ate (degrees) Jan. 1, Feb Mar Apr May June ate (degrees) July Aug ept Oct Nov ec Adtd Jan. 4, from The Astronomical Almanac for the Year 199, U.. Government Printing Office, Washington,.., 242, p. El 2. opyright 1992 by the lencoe ivision of Macmillan/McGrawHill Publishing ompany LABORATORY MANUAL 53

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Astron 104 Laboratory #5 The Orbit of Mars

Name: Date: Section: Astron 104 Laboratory #5 The Orbit of Mars Section 1.3 Note: Use a pencil with a sharp point! Mark your data as accurately as possible. This table contains measurements by Tycho Brahe.