Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S30 / 2

The source expression $E=\frac{1}{2}\mu v^2-GMm/r$ is total mechanical energy in the centre-of-mass frame, not kinetic energy alone. Take $v_0$ as the relative speed at a large initial separation where the mutual potential is negligible. At closest approach, the radial relative velocity is zero, so conservation of angular momentum and conservation of energy giveUsing $Mm/\mu =M+m$ eliminates $v_c$:An impact occurs when $r_c\le R$, where $R=r_M+r_m$. Since the right side increases with positive $r_c$, the collision threshold isThe mutual escape velocity is the relative speed at contact required to reach infinite separation with zero residual speed. Setting total mechanical energy to zero givesThis includes target recoil. If the given initial speed were instead at a finite non-negligible separation, its potential energy would have to be retained; $v_0$ would then not equal the asymptotic speed used in the focusing formula.

The collision cross section in the incoming impact-parameter plane isWithout gravity it is only ${\sigma }_{\mathrm{geom}}=\pi (r_M+r_m{)}^2$, so gravitational focusing increases the cross-section by $1+f$. Trajectories that would miss geometrically can be bent into contact, especially for slow relative encounters. Gravity does not itself guarantee sticking: the following accretion estimate assumes collisions agglomerate rather than fragment, rebound or erode the target.

A uniform incoming projectile mass density ${\rho }_p$ has mass flux ${\rho }_p{v}_0$, soThe formula counts mass swept per unit time and assumes an adequate continuing projectile supply, the stated speed, and negligible many-body or stellar-tidal modification of the two-body cross-section.

In the strong-focusing, small-projectile limit, write $R_M=(3M/(4\pi {\rho }_s){)}^{1/3}$ with constant solid-material density. ThenIntegrating $dM/M^{4/3}=Adt$ from $M_0$ to $M$ givesfor $M\gg M_0$. The fractional growth rate $\dot{M}/M\propto M^{1/3}$ increases with mass: larger bodies gain an increasing advantage, hence gravitationally focused growth of a planetary core is called runaway accretion in this regime.

The formal finite time to unlimited mass is not physical. Growth depletes the feeding region; embryos also stir planetesimal velocities and reduce focusing, while fragmentation, orbital competition and stellar tides alter the encounter geometry. A growing core can enter an oligarchic rather than runaway regime, and eventually reach a supply-limited isolation mass.

Dynamical friction transfers random kinetic energy from a massive body to a swarm and tends toward $M{v}_M^2\sim m{v}_m^2$. Larger bodies therefore have smaller random eccentricities and inclinations, remain concentrated near the planetesimal layer, and can retain favorable encounter conditions. When their own random motion contributes appreciably to relative speed, damping it also strengthens focusing through $f\propto v_0^{-2}$. If the small-body dispersion already dominates $v_0$, equipartition of the core alone does not further lower that encounter speed: the displayed growth law assumed a fixed $v_0$, not an automatically imposed $v_0\propto M^{-1/2}$.