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KEPLERS LAWS With some Law of Universal Gravitational Examples thrown in for good measure How much would a 70.0-kg person weigh on Mercury? Explore Kepler’s Laws of Plenary Motion on your own for a few minutes ◦ http://www.regentsprep.org/Regents/physics/phys06/keplers/default.htm Law 1: All planets move in elliptical orbits with the sun as one focus Anatomy of an Ellipse An ellipse is defined as the set of points that satisfies the equation: R + R’ = 2a F, F’ – Foci of the ellipse A = semi major axis B = semi minor axis Eccentricity ◦ Eccentricity of an ellipse is the ratio between the distance between the foci and the major axis of the ellipse ◦ 𝐸𝑐𝑐𝑒𝑛𝑡𝑟𝑖𝑐𝑖𝑡𝑦 = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑓𝑜𝑐𝑖 𝐿𝑒𝑛𝑔ℎ𝑡 𝑜𝑓 𝑚𝑎𝑗𝑜𝑟 𝑎𝑥𝑖𝑠 𝑜𝑓 𝑒𝑙𝑙𝑖𝑝𝑠𝑒 How to create an ellipse Go to http://www.mathwarehouse.com/animated-gifs/images/how-to-create-anellipse-animation-of-locus-and-focus.gif Law of Equal Areas ◦ When the planet is closer to the sun, it moves faster, sweeping through a longer path in a given time. Implications ◦ Although the orbit is symmetrical, the motion is not The planet speeds up and slows down along its orbit ◦ A planet speeds up as it approaches the Sun, gets its greatest velocity when passing closest, then slows down again. ◦ As the planet moves away from the Sun (or the satellite from Earth), it loses energy by overcoming the pull of gravity, and it slows down, like a stone thrown upwards. And like the stone, it regains its energy (completely--no air resistance in space) as it comes back. Perihelion and Aphelion Kepler’s Third Law (Law of Harmonics) ◦ The ratio of the square of the periods T of any two planets revolving about the Sun is equal to the ratio of the cubes of the mean distances, s, from the Sun: ◦ 𝑇1 2 𝑇2 = 𝑠1 3 𝑠2 ◦ Note that s is the length of the Semi-Major Axis Astronomical Units (AU) ◦ IF the distance from the earth to the Sun is set = 1, then we can scale distances relative to the distance from this value ◦ Example ◦ Distance from Mercury to the Sun (59 x 106 km) ◦ Distance from Earth ot Sun (149.6 x106 km) ◦ If Earth to Sun = 1, Earth to Mercury = (59/149.6 = 0.39 AU) Derivation of Kepler’s Laws ◦ Notes Jupiter’s Moons ◦ Use graphic of Jupiter’s moons to confirm Kepler’s Third Law General equation of an ellipse (for you mathematicians in the group)