Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 4

Treat the just-bound orbit as a parabolic Kepler orbit with $q\simeq a_p$. Its specific angular momentum is $h=\sqrt{2G{M}_{\ast }q}$, so its instantaneous angular speed is $h/r^2$. Equating this to the planet's mean motion $n_p=\sqrt{G{M}_{\ast }/a_p^3}$ givesThe parabolic orbit equation $r=2q/(1+\cos f)$ then gives $f\simeq 47^{\circ }$, or $50^{\circ }$ to the nearest ten degrees.

If the bodies were in conjunction at periapsis, Barker equation givesThe planetesimal is therefore at its maximum lead, approximatelyahead of the planet as seen from the star.

In the frame rotating with the planet, put the planet at fixed radius $a_p$. At periapsis the planetesimal moves prograde faster than the planet and passes just outside it. Its relative longitude initially increases, reaches a turning point near $r=2^{1/4}{a}_p$ at a lead of about $10^{\circ }$, and then decreases because its angular speed has fallen below the frame rate. Meanwhile its radius continues to grow along the outgoing parabolic branch. A sketch should therefore show an exterior cusp-like closest passage, a short prograde sweep to maximum lead, and a long outward trail that bends retrograde in the rotating frame.

Let $v_K=\sqrt{G{M}_{\ast }/a_p}$. At periapsis the nearly parabolic planetesimal and planet have parallel speeds $\sqrt{2}{v}_K$ and $v_K$, soFor a passage just outside the planet, the impact parameter is $b\simeq \delta a_p$. In the small-angle limit of the supplied gravitational scattering angle,Thus

In the planetocentric frame, the incoming relative velocity ${\mathbf{w}}_i$ is prograde and the encounter outside the planet bends it inward through $\theta$, without changing $w=|\mathbf{w}|$. In the inertial frame $\mathbf{v}={\mathbf{v}}_p+\mathbf{w}$, soSubstitution of the previous result yields the negative gravitational assist energy kick

The specific orbital energy changes by $\Delta \epsilon =\Delta (v^2)/2<0$, so a marginally bound orbit becomes more tightly bound. Neglecting its tiny initial binding energy,

To obtain the new periapsis, put $c=\sqrt{2}-1$ and $s=\mu /\delta$. Immediately after the encounter,Using the post-encounter energy and angular momentum in the Kepler orbit relations and retaining $O(s^2)$ givesThe inward radial kick therefore lowers the next periapsis as well as shrinking the semi-major axis.

The body now returns on a bound, highly eccentric orbit whose periapsis crosses more deeply into the planet's neighborhood. Repeated planetary scattering produces a sequence of energy and angular-momentum kicks rather than smooth secular evolution. Possible endpoints include ejection onto an unbound orbit, collision with the planet or star, tidal disruption, or diffusion onto a detached orbit that no longer encounters the planet. Temporary protection by a mean-motion resonance is also possible. The approximate Tisserand parameter constrains weak separated encounters, but the close-encounter sequence is chaotic and does not select a unique final orbit.