Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 316 / 1

Put ${\mu }_{\star }=G{M}_{\star }$. Conservation of specific orbital energy and specific angular momentum givesThe last comet expression is the vis-viva equation at the planet's orbit. The planet has only tangential velocity, while the comet's tangential component is $v_t=h/a_p$. Consequently its prograde relative velocity is exactly, before planetary deflection,In the near-parabolic encounter limit $Q\gg a_p$, this reduces toThe displayed approximation also requires $Q\gg a_p$. The assumptions $Q\gg q$ and $q\ll a_p$ alone do not guarantee it: an orbit with $Q=a_p$ encounters the planet near apoapsis and has $v_{\mathrm{rel}}\simeq v_p$, not $\sqrt{3}{v}_p$.

If all coplanar near-parabolic planet-crossing comets are admitted, so that $0\le q\le a_p$, the tangential component ranges from $-\sqrt{2}{v}_p$ to $+\sqrt{2}{v}_p$ when retrograde orbits are included. ThusFor prograde orbits alone the upper limit is $\sqrt{3}{v}_p$; for the stated deep-crossing limit $q\ll a_p$, both prograde and retrograde encounters approach $\sqrt{3}{v}_p$. These are the speeds outside the planet's gravitational well. The surface impact speed is $v_{\mathrm{imp}}=\sqrt{v_{\mathrm{rel}}^2+v_{\mathrm{esc},\mathrm{p}}^2}$, where $v_{\mathrm{esc},\mathrm{p}}$ is the planetary escape velocity; under negligible gravitational focusing the two speeds agree.

For an unperturbed Kepler orbit, successive revolutions advance the planet-relative orientation by $-n_p{T}_c$ modulo $2\pi$, where $T_c$ is the comet's orbital period. An irrational period ratio makes these orientations dense; sufficiently distant mean-motion resonance and adequate observation time justify the corresponding phase mixing approximation.

At each radius between its pericentre distance and apocentre distance, the comet eventually visits every azimuth in the rotating reference frame. The spatial projection is thereforeIt is an annulus, with a nonuniform radial residence probability, rather than a uniformly filled area. Strictly, the full position-velocity phase space is not this annulus: at each radius the energy and angular momentum constrain the velocity, with inward and outward branches. The annulus is its position-space projection. This description assumes the orbital elements have not yet been substantially changed by planetary scattering.

The radial velocity of a Kepler orbit follows from the vis-viva equation after subtracting the tangential component:Each radial interval is crossed twice per orbital period, so its fraction of time isThis is the phase-mixed radial probability of a Kepler orbit, normalized to one between $q$ and $Q$. Dividing by the annular area $2\pi rdr$ gives the phase-mixed comet surface density per comet:For $Q\gg a_p>q$,The printed density has the powers of $Q$ and $a_p$ interchanged. Its expression is too large by $Q/a_p$ in this limit. The normalized residence-time derivation fixes the corrected expression, which is also required for the later $Q^{1/2}$ ejection-time scaling. For a population of $N_c$ independent identical comets, multiply ${\Sigma }_c$ by $N_c$ to obtain a number surface density of a disk.

With negligible comet radius and mass, the planet's radius and escape velocity areGravitational focusing changes the limiting collision impact parameter from $R_p$ to $R_p\sqrt{1+v_{\mathrm{esc},\mathrm{p}}^2/v_{\mathrm{rel}}^2}$. Thus it is negligible when $v_{\mathrm{esc},\mathrm{p}}^2\ll v_{\mathrm{rel}}^2$.

Put $s=q/a_p$ and $\gamma =\sqrt{3-2\sqrt{2s}}$ in the $Q\gg a_p$ prograde approximation. Substituting $v_{\mathrm{rel}}=\gamma \sqrt{G{M}_{\star }/a_p}$ givesEquivalently $2(M_p/M_{\star })(a_p/R_p)\ll {\gamma }^2$. For $q\ll a_p$, use ${\gamma }^2\simeq 3$. If negligible gravitational focusing is required for every coplanar near-parabolic orbit, including nearly tangential prograde encounters, use the smaller value ${\gamma }^2=(\sqrt{2}-1{)}^2$.

A coplanar target sweeps a strip of width $2R_p$ through the comet's local position probability. Hence the relevant geometric collision cross-section is a length, and the rare collision rate per comet is ${\Gamma }_{\mathrm{col}}=2R_p{\Sigma }_c(a_p)v_{\mathrm{rel}}$.

Using the corrected phase-mixed comet surface density and the notation $s=q/a_p$, $\gamma =\sqrt{3-2\sqrt{2s}}$, the coplanar comet collision time isFor $q\ll a_p$, this becomes ${\pi }^2{Q}^{3/2}{a}_p/(2\sqrt{3}{R}_p\sqrt{G{M}_{\star }})$, so $t_{\mathrm{col}}\propto Q^{3/2}{M}_p^{-1/3}{a}_p$ at fixed stellar mass and planetary mass density. This is the mean waiting time for one orbiting comet; the mean interval between impacts from $N_c$ such independent comets is $t_{\mathrm{col}}/N_c$. It assumes rare encounters and orbital phase mixing, not a Poisson process derived from a single deterministic trajectory.

Suppose the inclined population samples its arguments of periapsis and nodal orientations, or that sufficiently slow precession supplies this sampling. Near the planet the vertical amplitude is $H\simeq a_{p}I$. For uniformly distributed vertical phase, $z=H\sin \psi$ has probability density function $[\pi \sqrt{H^2-z^2}{]}^{-1}$, so the midplane volume probability is ${\Sigma }_c/(\pi H)$.

In the genuinely three-dimensional regime $R_p\ll H\ll a_p$, the geometric collision cross-section is $\pi R_p^2$. The relative velocity differs from the coplanar value only at order $I^2$ for the specified high-speed encounters. Therefore the inclined comet collision time isAt fixed ${\rho }_p,M_{\star },I$ the leading scaling is $Q^{3/2}{M}_p^{-2/3}{a}_p^2$, with no leading $q$ dependence when $q\ll a_p$. The displayed factor retains the first dependence on $q/a_p$. Also $t_{\mathrm{col},\mathrm{I}}/t_{\mathrm{col}}\simeq 2a_{p}I/R_p$ in this averaging convention.

Inclination alone does not specify a collision time for one fixed orbit. If its orbital nodes miss the planet's orbit, a tilted Kepler orbit can have no collisions at all. The finite result is an orientation-averaged population result, or requires precession or scattering to sample those orientations. When $a_{p}I\lesssim R_p$, the three-dimensional estimate crosses over to the coplanar result instead of tending to zero.

In a planetary gravitational assist, the incoming and outgoing relative speeds have the same magnitude. If $\mathbf{u}$ is the planet-frame velocity, the heliocentric specific orbital energy change is consequentlynot an arbitrary addition of $|\Delta \mathbf{u}|$ to the heliocentric speed. The initial binding energy per unit mass is $G{M}_{\star }/(Q+q)\simeq G{M}_{\star }/Q$.

For a weak deflection, $b\gg b_{90}=G{M}_p/v_{\infty }^2$, the hyperbolic scattering formula gives $|\Delta \mathbf{u}|\simeq 2G{M}_p/(b{v}_{\infty })$. Its direction is perpendicular to the incoming relative velocity. At the planet, in the near-parabolic prograde limit,On the favorable side of the planet the energy gain is approximately $2G{M}_p{v}_p\kappa /(b{v}_{\infty })$. Equating this to the binding energy gives the ejection impact parameterOnly one sign of impact parameter gains energy for each radial branch. Noncolliding ejections have $R_p<|b|<b_{\mathrm{ej}}$, giving width $(b_{\mathrm{ej}}-R_p{)}_{+}$, where $x_{+}=\max (x,0)$. For $b_{\mathrm{ej}}\gg R_p$, the planetary ejection of a comet rate is approximately ${\Gamma }_{\mathrm{ej}}={\Sigma }_c{v}_{\infty }{b}_{\mathrm{ej}}$. With ${\mu }_p=M_p/M_{\star }$ this givesThe scaling applies when $b_{90}\ll b_{\mathrm{ej}}\ll r_H$ so that the local two-body encounter and weak-deflection approximations are consistent. The favorable-side coefficient assumes a coplanar orbit; averaging over other encounter geometries changes it. Cumulative diffusion by many subthreshold encounters is not included in this one-encounter estimate.

Exclude collisions from the ejection count. The effective collision width is $2R_p$, while the favorable noncolliding ejection width is $(b_{\mathrm{ej}}-R_p{)}_{+}$. Define the first flyby-ejection threshold and the equal-rate crossover byThere are no direct noncolliding ejections in this weak-deflection model for $Q\le Q_{\mathrm{hit}}$. Above that threshold, comparison of the two rates gives collision is more likely for $Q<Q_{\mathrm{crit}}$, while ejection is more likely for $Q>Q_{\mathrm{crit}}$. In the deep-crossing limit $Q_{\mathrm{crit}}\simeq [9/(2\sqrt{2})](M_{\star }/M_p)R_p$, so its scaling at fixed density is $M_{\star }{M}_p^{-2/3}{\rho }_p^{-1/3}$. Order-one factors depend on encounter geometry; the robust condition compares $(M_p/M_{\star })Q$ with $R_p$.

Adding the mutually exclusive rare-event loss rates gives the collision-ejection lifetime of a comet:For $Q\le Q_{\mathrm{hit}}$ this is $t_{\mathrm{col}}\propto Q^{3/2}$. For $Q\gg Q_{\mathrm{hit}}$ it approaches $t_{\mathrm{ej}}\propto Q^{1/2}$. A logarithmic sketch therefore bends from slope $3/2$ to slope $1/2$, with no maximum. At the equal-rate crossover the lifetime is half either individual loss time.

These asymptotic branches use the corrected residence density from part iii and require $Q\gg a_p$ and a local encounter scale below the Hill radius. They cannot be extrapolated indefinitely to arbitrarily distant weakly bound orbits.