ay45c4-page15

# ay45c4-page15 - The orbit in this case is an ellipse If we...

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Unformatted text preview: The orbit in this case, is an ellipse. If we put x = r cos( 0) y = r sin( 0) and eliminate  and r, the equation of the orbit becomes   1 1 1+ x (x2 + y2)1=2 = r (x2 + y2)1=2 0 which simpli es to x + a 2 y2 + b2 = 1 a where a = 1 r0 2 ; b = (1 r02)1=2 are semi-major and semi-minor axes. y θ = θ 0 + π/2 b x = a(1 − ε ), r = r0 /(1 + ε), r0 εa y=0 θ=θ x a x = − a(1 + ε), y = 0 r = r0 /(1 − ε), θ = θ 0 + π The axial ratio of the ellipse is b=a = (1 2)1=2. The quantitiy r0 is called the \semi-latus rectum." Note that  < 1 implies, from the de nition 2EL2 2  1 + (GM )23 that E < 0, i.e. the total energy of a bound system is negative. 74 ...
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