11-12-08LectPHY2048

# 11-12-08LectPHY2048 - Elliptical orbits The circular orbit...

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The circular orbit of satellites we discussed last time is actually the limit of a more general type of orbit that takes the form of an ellipse . Elliptical orbits The ellipse is a curve that has two foci, and , that obeys the rule that for every point on the curve the sum of the distances to the two foci is a constant. F F The body of mass M about which the satellite (mass m << M) orbits is at one of the foci , F , while the other at , , is empty. F

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The distance from the center of the ellipse to the furthest point of the orbit is the semi-major axis labeled: a . The distance from the center of the ellipse to the foci is labeled ea where e is a dimensionless number from zero to 1, called the eccentricity , that characterizes how much the orbit deviates from a circle. If e = 0 , ea=0 , the foci coincide, the orbit is a perfect circle and a = r . The eccentricity of earth’s orbit about the sun is small ( e = 0.0167 ) while that of Pluto’s is so much greater ( e = 0.248 ) that at times Pluto's distance to the sun is smaller than Neptune's.

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The distance from the sun to the point of closest approach of a planet is called the perihelion , R p , and the furthest distance is called the aphelion R a . (These have nothing to do with the seasons on earth). The angular momentum of the orbiting body must be conserved so, Lr pr m v s i n = ×= φ Since r shrinks on going from the aphelion to the perihelion the orbital speed v must increase for the angular momentum to remain constant. This speed is maximum at the perihelion and minimum at the aphelion .
At the perihelion and aphelion φ = 90 o so p pa a LR m v R m v == Lr m v s i n p a Rv = a p a p R vv R = Since

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## This note was uploaded on 08/25/2011 for the course PHY 2048 taught by Professor Field during the Fall '08 term at University of Florida.

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11-12-08LectPHY2048 - Elliptical orbits The circular orbit...

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