Paper 4, Section II, C

A particle moves in the gravitational field of the Sun. The angular momentum per unit mass of the particle is $h$ and the mass of the Sun is $M$. Assuming that the particle moves in a plane, write down the equations of motion in polar coordinates, and derive the equation

$\frac{d^{2} u}{d \theta^{2}}+u=k$

where $u=1 / r$ and $k=G M / h^{2}$.

Write down the equation of the orbit ( $u$ as a function of $\theta$ ), given that the particle moves with the escape velocity and is at the perihelion of its orbit, a distance $r_{0}$ from the Sun, when $\theta=0$. Show that

$\sec ^{4}(\theta / 2) \frac{d \theta}{d t}=\frac{h}{r_{0}^{2}}$

and hence that the particle reaches a distance $2 r_{0}$ from the Sun at time $8 r_{0}^{2} /(3 h)$.

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