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Binet equation

For
$$
V(r)=-\frac{km}{r-r_0},
\qquad r>r_0,
$$
Binet’s equation is
$$
u''+u=\frac{K}{(1-r_0u)^2},
\qquad K=\frac{km^2}{\ell^2}.
$$
Its circular orbits solve $K=u(1-r_0u)^2$; the branch $u<1/(3r_0)$ is stable and the branch $u>1/(3r_0)$ is unstable.

Shifted inverse-radius potential

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 10C / iii / Solution

Shifted inverse-radius potential

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Shifted inverse-radius potential

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