Using Kepler S Third Law

Using Kepler’s Third Law

Of Kepler’s Three Laws of Planetary Motion, the third law is the one that is most useful today in putting up satellites. It compares the motions of two satellites (planets, moons, etc.) around the same central body.

The key is to find the constant for each central body. For example, let’s use 2 3 (Tearth )= ( k sun )( r earth- sun ) the Sun. We know the Earth goes around the Sun in 1 year, and that the 2 2 2 average distance from the Earth to the Sun is 1 AU (= 1.5 x 1011 m). So we can T (1 yr) yr ksun = = = 1 use those two numbers to find the constant for the Sun. r3(1 AU) 3 A U 3

So as long as you use years and AU for your units, anything going around the Sun has to follow the rule that the period squared divided by the orbital radius cubed has to equal 1. Let’s use this rule to find how long it takes Jupiter to make one orbit. (The orbital radius for Jupiter is 5.2 AU.) 2 (T jup ) 3 = 1 (rjup- sun ) T 2= 1 (5.2 AU) 3 T = 11.9 yr

PROBLEMS 1. The constant for things going around the Sun is not the same as the constant for things going around the Earth. Use the data for the moon (which goes around the Earth) to find the constant for things going around the Earth. (The moon goes around the Earth in 27 days; it is 3.84 x 105 km from the Earth.) 1.29 x 10-14 d2/km3 2. The mean distance of the planet Pluto from the Sun is 39.6 AU. What is the orbital period of Pluto? 249.2 yr 3. (a) Uranus has an orbital period of 84 years. What is its average distance from the sun? (b) Neptune’s orbit is 50% bigger than the orbit of Uranus. What is the orbital period for Neptune? (a) 19.2 AU; (b) 154 yr 4. How long does it take a satellite to orbit the earth if its average orbital radius is 18000 km? 6 hr 35 min 5. A satellite already in orbit around a planet is put into a new orbit whose radius is 4 times as large as the old radius. How many times longer is the new period than the old? 8 times 6. What is the average distance of a satellite from the center of the earth if the period of its orbit is 15 days? 2.6 x 105 km 7. For the planet Mercury, the perihelion distance (closest approach to the sun) has been found to be about 45.8 x 106 km, and the aphelion distance (greatest distance from the sun) about 70.0 x 106 km. What is the eccentricity of the orbit of Mercury? 0.21 8. Halley’s comet has a period of 76 years, and its orbit has an eccentricity of 0.97. (a) What is its average distance from the Sun? (b) What is its greatest distance from the Sun? (c) What is its least distance from the Sun? (a) 17.9 AU; (b) 35.3 AU; (c) 0.5 AU