KEPLER\'S LAWS OF PLANETARY MOTION OBJECTIVE: To study and verify Kepler\'s Laws
ID: 1416080 • Letter: K
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KEPLER'S LAWS OF PLANETARY MOTION
OBJECTIVE: To study and verify Kepler's Laws of Planetary Motion. To become acquainted with the elements which are used to describe the shape, positions and orientations of orbits in space.
INTRODUCTION: Johannes Kepler, a student of Tycho Brahe, failed in his attempts to set up a musical relation among the planets. He also failed to find a geometrical relation by the use of regular polygons. But he did find certain relations given by three laws. The first two laws were published in 1609 and the third in 1619. The last was given in a book called The Harmony of the Worlds. Later on, Newton was able to derive all three of Kepler's Laws from his own Law of Universal Gravitation.
NOTES: Data for three-day intervals have been listed. The actual perihelion date is November 8; aphelion is September 25. All positions are for Greenwich midnight. Perihelion is when the planet is closest to the sun; aphelion is when the planet is farthest from the sun.
Make a two-dimensional plot of Mercury's orbit. Your drawing will be slightly larger than 8.5 by 11 inches, so you may fax it to me in two pieces. Place a point at the center of the paper; this is your origin which also represents the Sun. Draw a horizontal line through the origin as shown in Fig. 2. Then, for each date shown in Table I, plot a point at the right angle and also at the correct distance from the sun. For example, for Nov. 3, at an angle of 47º (we will round off angles to the nearest degree) plot a point 10.5 cm from the sun; for Nov. 6, at an angle of 65º, plot a point 10.3 cm from the sun, etc. Do not connect the points as yet. The points should look like the outline of an ellipse. (Fig. 3). Kepler's first law states that all planetary orbits are ellipses with the sun at one focus.
To prove the orbit you've outlined is an ellipse, do the following: First, draw the major axis which is the longest diameter of the ellipse. Clearly, this line should connect the perihelion and aphelion points. Then locate the center of the ellipse, which is a point halfway between the perihelion and aphelion points (Fig. 4). The second focus should be as far from the center as the sun is, but on the opposite side of the center. Having found the two foci, using two tacks and a string, construct the ellipse by the method to be described in (Figure 2.11 page 38 3rd ed. or Figure 2.13 page 46 4th ed.) of your text. The ellipse should pass through all the points plotted, provided the two foci are accurately located. Try changing their positions if you have to until a good fit is found. This verifies Kepler's First Law.
Kepler's Second Law ? Kepler's Second Law says that equal areas are swept out by a planet in equal
times. For example, the shades areas in Fig. 5 should all be equal. Sketch in three such areas (each being 6 days long) on your orbit plot avoiding Sept. 1 or Nov. 30, however. Show that the areas are equal. You may approximate the areas with triangles. Record the areas in a table. Use the formula Area= 1/2 base x height for your approximated areas.
Kepler's Third Law ? To verify Kepler's Third Law, the data for Mercury from your text may be used. Substitute the period P in years and the mean distance A in astronomical units in the equation P2 = A3. If the left side of this equation equals the right side, you have verified the Third Law.
For the Second Law, make a table and include your computations. (It is not necessary to show the arithmetic.) Likewise for the Third Law, show the computation you have performed.
TABLE I Orbital Data for Mercury, 1967 (From American Ephemeris, pp. 165-67) DATE LONG. Degrees LATT. Degrees RADIUS VECTOR AU X 33 DATE LONG. Degrees LATT. Degrees RADIUS VECTOR 9/1 182.7 9/4 193.9 9/7 204.3 9/10 214.1 9/13 223.3 9/16 232.3 9/22 240.6 9/25257.3 9/28265.6 10/1 273.9 10/4 282.5 10/7 291.4 10/10 300.6 10/13 310.4 10/16 320.8 10/19 332.1 10/22 344.4 4.96 3.91 2.80 1.68 0.57 13.1 13.6 14.1 14.5 14.9 15.2 15.5 15.6 15.5 15.4 15.2 14.9 14.5 14.1 13.6 13.0 12.5 10/25 357.9 10/28 12.7 10/31 29.0 11/3 46.6 11/6 65.1 11/9 84.2 11/12 102.9 11/15 120.8 11/18 137.4 11/21 152.5 11/24 166.3 11/27 178.9 11/30 190.4 5.36 4.03 2.27 0.17 2.06 4.13 5.72 6.69 7.00 6.78 6.16 5.29 4.26 11.9 11.3 10.9 10.5 10.3 10.3 10.4 10.8 11.2 11.7 12.3 12.9 13.5 0.50 1.54 3.42 4.27 5.03 5.7 6.26 6.68 6.94 7.00 6.79 6.79Explanation / Answer
Kepler's first law explains that planets are orbiting the sun in a path described as 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. Kepler's first law says that all planets orbit the sun in a path that resembles an ellipse, with the sun being located at one of the foci of that ellipse.
Kepler's second law 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.
Kepler's third law compares the orbital period and radius of orbit of a planet to those of other planets. 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.
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