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 simplies to x + a 2 y2
+ b2 = 1
a where a = 1 r0 2 ; b = (1 r02)1=2
are semimajor and semiminor 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 \semilatus rectum." Note that < 1 implies, from the denition
2EL2
2 1 + (GM )23
that E < 0, i.e. the total energy of a bound system is negative.
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 Fall '09
 dion
 Energy, Denition, simpli es, semiminor axes, \semilatus rectum

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