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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
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The eccentricity of an ellipse gives an indication of the difference between its major and minor axes The eccentricity depends on the distance between the two points, f (compared with the length of the piece of string)
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eccentricity = distance between foci / major axis
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eccentricity = distance between foci / major axis
The eccentricity of the orbits of the planets is low; their orbits are very nearly circular orbits.
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Law 1 Each planet orbits the sun in an elliptical path with the sun at one focus of the ellipse.
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Mercury 0.206
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Mercury 0.206 Venus 0.0068
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Mercury 0.206 Venus 0.0068 Earth 0.0167
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Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934
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Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485
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Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556
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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
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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
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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
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...it can be shown that...
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minor axis = e2 major axis where e is the eccentricity of the ellipse
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minor axis = e2 major axis 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:
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0.97 minor axis = 1 - e2 major axis
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
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In calculations we will consider the orbits to be circular
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Eccentricity of ellipse much exaggerated
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Law 2 A line from the sun to a planet sweeps out equal areas in equal times.
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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.
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r3 = a constant T2
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Mm F = G r2
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Mm F = G F = m r w2 r2
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Mm F = G F = m r w2 r2
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Mm F = G F = m r w2 r2 G M m m r w2 = r2
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Mm F = G F = m r w2 r2 G M m m r w2 = r2 2 p w = T
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r3 GM = T2 4p2
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r3 GM = T2 4p2 in which we see Kepler’s third law
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