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Shifted inverse-radius potential

For

V (r) = - \frac{km}{r - r _{0}}, r > r_{0},

Binet’s equation is

u^{''} + u = \frac{K}{( 1 - r _{0} u ) ^{2}}, K = \frac{k m ^{2}}{ℓ ^{2}} .

Its circular orbits solve

K = u (1 - r_{0} u)^{2}

; the branch

u < 1/ (3 r_{0})

is stable and the branch

u > 1/ (3 r_{0})

is unstable.

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Solution