Paper 4, Section II, C

Dynamics | Part IA, 2007

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

d2udθ2+u=k\frac{d^{2} u}{d \theta^{2}}+u=k

where u=1/ru=1 / r and k=GM/h2k=G M / h^{2}.

Write down the equation of the orbit ( uu as a function of θ\theta ), given that the particle moves with the escape velocity and is at the perihelion of its orbit, a distance r0r_{0} from the Sun, when θ=0\theta=0. Show that

sec4(θ/2)dθdt=hr02\sec ^{4}(\theta / 2) \frac{d \theta}{d t}=\frac{h}{r_{0}^{2}}

and hence that the particle reaches a distance 2r02 r_{0} from the Sun at time 8r02/(3h)8 r_{0}^{2} /(3 h).

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