Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 316 / 1

For the relative position $\mathbf{r}={\mathbf{r}}_1-{\mathbf{r}}_2$, Newton's law of universal gravitation gives the two-body problem in its reduced one-body form,Taking the dot product with the relative velocity $\mathbf{v}=\dot{\mathbf{r}}$ givesIntegration therefore yields conservation of specific orbital energy:

Let $h=r^2\dot{f}$ be the specific angular momentum. The radial Kepler orbit equation has semi-latus rectum $p=h^2/\mu$, so comparison withgives $h^2=\mu a(1-e^2)$ for an elliptic orbit and $h^2=\mu a(e^2-1)$ for a hyperbolic Kepler orbit. At either apsis, $\dot{r}=0$ and $v=h/r$. Substituting the apsidal radius and angular momentum into part (i), or equivalently using the vis-viva equation, givesThus the signs are $C=\mp \mu /(2a)$ in the convention of the question.

At the encounter radius $r=a_p$, the planet's circular Kepler orbit has speedThe comet's speed follows from the vis-viva equation:Its component parallel to the planet's prograde tangential velocity is fixed by its specific angular momentum,The relative velocity therefore satisfiesand hence

In the planetocentric hyperbolic Kepler orbit, the speed at infinity is $\Delta v_{pc}$ and the impact parameter is $b$. Conservation of specific angular momentum givesConservation of specific orbital energy at the moon's radius giveswhere $v_m^2=G{M}_p/a_m$. ThusThe moon moves tangentially at $v_m$, soTherefore the correctly dimensioned reading of the displayed result is

For some orbital phase of the moon to permit an impact, the comet's planetocentric hyperbolic Kepler orbit must at least cross the moon's circular orbit. At the limiting case $a_m$ is the comet's periapsis, so the radial velocity in part (iv) vanishes. ConsequentlyThis is gravitational focusing by the planet; an actual collision additionally requires the moon to occupy the crossing point at the right time.

The limiting trajectory grazes the planet at periapsis $R_p$. Equating the asymptotic and periapsis values of specific angular momentum and using the vis-viva equation givesHence the comet avoids the planet when

An encounter can occur only near an orbital node of the comet's inclined orbit. Its tangential velocity projected into the planet's plane is reduced by $\cos I_c$, so the relative-speed calculation becomesThe missing projected component is a vertical relative velocity, so the moon encounter is three-dimensional: $b$ becomes a vector in the impact plane, and a collision also requires the trajectory to pass close to the moon's orbital plane. Once the total asymptotic speed is used, the two-body gravitational focusing formulae in parts (v) and (vi) retain their form, but the geometrical collision probability is lower.