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Kepler’s Laws of Planetary Motion © David Hoult 2009
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The eccentricity of an ellipse gives an indication of the difference between its major and minor axes © David Hoult 2009
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The eccentricity depends on the distance between the two points, f (compared with the length of the piece of string) The eccentricity of an ellipse gives an indication of the difference between its major and minor axes © David Hoult 2009
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eccentricity = distance between foci / major axis © David Hoult 2009
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The eccentricity of the orbits of the planets is low; their orbits are very nearly circular orbits. eccentricity = distance between foci / major axis © David Hoult 2009
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Law 1 Each planet orbits the sun in an elliptical path with the sun at one focus of the ellipse. © David Hoult 2009
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Mercury 0.206 © David Hoult 2009
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Mercury 0.206 Venus 0.0068 © David Hoult 2009
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Mercury 0.206 Venus 0.0068 Earth 0.0167 © David Hoult 2009
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Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 © David Hoult 2009
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Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 © David Hoult 2009
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Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556 © David Hoult 2009
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Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556 Uranus 0.0472 © David Hoult 2009
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Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556 Uranus 0.0472 Neptune 0.0086 © David Hoult 2009
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Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556 Uranus 0.0472 Neptune 0.0086 Pluto 0.25 © David Hoult 2009
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...it can be shown that... © David Hoult 2009
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minor axis major axis = 1 - e 2 where e is the eccentricity of the ellipse © David Hoult 2009
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minor axis major axis = 1 - e 2 where e is the eccentricity of the ellipse which means that even for the planet (?) with the most eccentric orbit, the ratio of minor to major axis is only about: © David Hoult 2009
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minor axis major axis = 1 - e 2 where e is the eccentricity of the ellipse which means that even for the planet (?) with the most eccentric orbit, the ratio of minor to major axes is only about: 0.97 © David Hoult 2009
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In calculations we will consider the orbits to be circular © David Hoult 2009
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Eccentricity of ellipse much exaggerated © David Hoult 2009
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Law 2 A line from the sun to a planet sweeps out equal areas in equal times. © David Hoult 2009
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Law 3 The square of the time period of a planet’s orbit is directly proportional to the cube of its mean distance from the sun. © David Hoult 2009
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T2T2 r3r3 = a constant © David Hoult 2009
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F = G r2r2 Mm © David Hoult 2009
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F = m r 2m r 2 F = G r2r2 Mm © David Hoult 2009
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F = G r2r2 Mm F = m r 2m r 2 © David Hoult 2009
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F = G r2r2 Mm F = m r 2m r 2 r2r2 G M mG M m m r 2m r 2 = © David Hoult 2009
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F = G r2r2 Mm F = m r 2m r 2 = T 2 2 r2r2 G M mG M m m r 2m r 2 = © David Hoult 2009
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T2T2 r3r3 = 4242 GM © David Hoult 2009
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T2T2 r3r3 = 4242 GM in which we see Kepler’s third law © David Hoult 2009
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