Motion in the Heavens

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1 Motion in the Heavens Most ancient cultures believed that the earth was the centre of the universe. Most felt that the planets, stars, moon and sun revolved around the earth. This is known as a geocentric or Ptomelaic universe. The first major challenge to this view was in the 1500s when Nicolaus Copernicus proposed that the planets revolved around the sun, a heliocentric universe. This was popularized by Galileo, who was put on trial for his support of the Copernican theory. In the late 1500s, Tycho Brahe made a very detailed study of the positions of the stars and planets in the sky over a period of many years. In the early 1600s Johannes Kepler used Brahe s data to propose three laws describing the orbits of the planets: 1. The paths of the planets are ellipses with the centre of the sun at one focus.

2 Focus Focus Planet Sun. An imaginary line from the sun to a planet sweeps out equal areas in equal time interval. Planets move fastest when closest to the sun, and slowest when farthest away.. 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. For example, consider the Earth and Mars:

3 T earth T Mars = r earth r Mars T Mars = 688 days Earth Sun r Earth r Mars T Earth = 65 days Mars 65days 688days = 1.49x108 km.8x10 8 km 0.81 = 0.79 Within reasonable measurement errors, the ratios are equal to each other.

4 Kepler s Third Law works for any orbiting system, as long both objects are orbiting the same object. The units used do not matter, as long as both r and both T units are the same. Kepler s Constant For any object orbiting another, the ratio: r T = K is known as Kepler s constant. The value of K will be the same for all objects in any given orbital system, such as the solar system. EX: Use the data for Earth and for Mars, calculate the value of Kepler s constant for our solar system. Earth: r T = 1.49x108 km = K 65days.48 x km /days = K Mars: K =.50 x km /days

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6 Problem Solving using Kepler s Laws Kepler s Laws can be used to solve many problems about the motion of objects in the solar system. This is how the distance from the sun was first determined for many of the planets. The laws can also be generalized to any system of orbiting objects, not just the solar system. EX. The asteroid Icarus, only a few hundred metres across, orbits the sun with a period of 410 days. Find it s average distance from the sun. The third law can be used with any other object whose orbit is known, such as earth. T earth T Icarus = r earth r Icarus 65days 410days = 1.49x108 km r Icarus r Icarus = ( 1.49x108 km) 65days 410days r Icarus = 1.61 x 10 8 km

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8 EX: In a solar system far, far away, the planet Plaav orbits it s sun once every 69 bleebs, at a distance of 4.9 x 10 5 klas. The planet Wuqu orbits the same star at a distance of.77 x 10 6 klas. Find the length of the a Wuqu year in bleebs. T Plaav T Waqu = r Plaav r Waqu 69bleeb T Waqu = 4.9x105 klas.77x10 6 klas T Waqu = ( 69bleeb) 4.9x10 5 klas.77x10 6 klas T Waqu =.59 x 10 bleeb

9 EX: Determine Kepler s constant for objects orbiting the earth, in km /hr. Note that the orbital period of the moon around the earth is 7. days, at an average distance of.84 x 105 km. r T = K K = 1. x km /hr Use this constant to determine the orbital period of an object with a radius of 7.00 x 10 km. r \ T = K T = r K T = 1.61 hr

10 EX: Use the planetary data on page 159, and Kepler s laws to determine the orbital period of each of the planets, in earth years. T 1 T = r 1 r Let T 1 = one earth year r 1 = x 1011 m T = orbital period of planet other than earth r = planet s orbital radius Results: r (m) T (yr) Mercury 5.80E+10.41E-01 Venus 1.081E E-01 Mars.78E E+00 Jupiter 7.781E E+01 Saturn 1.47E+1.95E+01 Uranus.87E E+01 Neptune 4.5E E+0 Pluto 5.9E+1.48E+0

11 Practise Problems 1. Suppose a small planet was discovered whose orbital period was twice that of earth. How many times further from the sun than earth would this planet be. (Hint: for earth let r = 1, and T = 1). Titan, Saturn s largest moon, and the largest moon in the solar system, orbits Saturn once every 15.6 days, with a mean radius of 1. Gm. Astronomers determined that Saturn s moon Phoebe has an orbital period of 549 days. Find Phoebe s mean orbital radius.. Ceres, the largest known asteroid, 744 km across, orbits at a mean distance of 4.14 x 10 8 km from the sun. Find its orbital period. Complete the Practice Problems on page 160. Answers: times further than earth from the sun. Phoebe s r = 1. Gm. orbital period of Ceres = 4.61 years

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