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KEPLER'S LAWS OF ORBITAL MOTION
As explained in the text, Kepler's first law states that orbits are shaped like ellipses, with the center of mass (usually the object being orbited) at one focus. So, for example, Earth orbits the Sun in an ellipse, with the Sun at one focus of the ellipse. Kepler's second law states that orbits sweep out equal areas in equal times; this is a direct result of the conservation of angular momentum. (See Exercise 33 below.) Kepler's third law, sometimes called the Harmonic Law, states that the square of an orbital period is proportional to the cube of its orbital radius. This can be expressed mathematically as P^2 = kr^3 (orbital period squared = a constant * orbital radius cubed) Johannes Kepler showed that for our solar system, k = 1 for objects orbiting the Sun if P is given in years and r in AU. The constant of proportionality k varies depending on the masses of the objects in the system, so units are important. By using Newton's Law of Gravity and the equations of circular motion, the constant of proportionality can be derived as a function of the masses of the objects in the system, and expressed as k = (4 ?2)/(G (M+m)) where M and m are the masses of the orbiting objects. If one of the masses is very small compared to the other (for example, Earth's mass is about 300,000 times smaller than the Sun's mass), then using just the large mass gives just about the right answer as well. See Exercise 34 below. |