Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 316 / 4

At the exterior $(p+q):p$ mean-motion resonance,Using the Kepler third law for $M_{\ast }\gg M_1$ givesThe disturbing function is a Fourier series in integer combinations of the orbital angles. The D'Alembert characteristic permits the eccentric termAway from resonance, terms with rapidly circulating angles average away. Here, however,so ${\varphi }_1$ is a slow resonant argument. Successive astronomical conjunctions then act coherently, making this term dominate the long-period resonant dynamics even though a $q$th-order resonance has coefficient proportional to $e^q$ at small eccentricity.

At periapsis, $\lambda =\varpi$. If the planet has longitude ${\lambda }_0$ there, thenso ${\varphi }_1/p$ is the angular displacement of periapsis from the planet, modulo $2\pi /p$. At an astronomical conjunction, $\lambda ={\lambda }_1={\lambda }_c$, andso ${\varphi }_1/q$ is the conjunction longitude measured from periapsis, with the $q$ possible branches differing by $2\pi /q$.

The planet's mean motion is $n_1=(p+q)n/p$. Hence the synodic period, or mean interval between conjunctions, iswhere $T=2\pi /n$ is the planetesimal's orbital period.

For the 8:5 resonance, $p=5$ and $q=3$. Five planetesimal periods equal eight planetary periods, so the curve closes afterStarting at conjunction at apoapsis gives ${\varphi }_1=q\pi =3\pi \equiv \pi (\mathrm{mod}2\pi )$. In the frame rotating with the planet, let $\theta$ be the planetesimal's longitude relative to the fixed planet. At periapsis, $5\theta ={\varphi }_1$ modulo $2\pi$, so the five possible periapsis directions areAt apoapsis, ${\varphi }_1=5\theta +3\pi$, givingA sketch should therefore show a five-lobed rotating-frame rosette centered on the star, with the fixed planet on the $\theta =0$ ray, five outer turning points separated by $72^{\circ }$, and five inner turning points halfway between their rays. Successive orbits visit these points in resonance order before the fifth orbit closes the pattern.

If an exterior planetesimal reaches conjunction just before apoapsis, its radius is increasing. The closer pre-conjunction pull from the trailing inner planet removes more specific angular momentum than the more distant post-conjunction pull restores. Its semi-major axis falls, its mean motion rises, and the next conjunction moves later in its orbit toward apoapsis. A conjunction just after apoapsis produces the reverse imbalance: the stronger post-conjunction pull adds angular momentum, raises the semi-major axis, and shifts the next conjunction earlier. The conjunction phase is therefore restored toward apoapsis.

The symmetric configuration has one of the three conjunction branches at apoapsis and the other two symmetrically placed. It hasso the resonant argument librates about $180^{\circ }$.

At conjunction, the planetesimal's mean anomaly isIf ${\varphi }_1=\pi +\delta$ with $|\delta |\le \Delta {\varphi }_1\le \pi$, the conjunction branch that can come closest to periapsis hasLet $E_{min}$ solve Kepler's equationAt that phase the orbital radius is $r_{min,c}=a(1-e\cos E_{min})$. A geometrical close encounter is possible only ifwhere $R_{\mathrm{enc}}$ may be chosen as the planet's Hill radius or another encounter distance. With a point planet, set $R_{\mathrm{enc}}=0$. This implicit inequality is the requested eccentricity constraint as a function of $\Delta {\varphi }_1$.

The weaker necessary condition that the orbits cross isIt becomes sufficient for phase access only as $\Delta {\varphi }_1\to \pi$, when a conjunction can approach periapsis. Smaller libration amplitude keeps conjunctions farther from periapsis and requires a larger eccentricity than this orbit-crossing bound.

For each point $(\Delta {\varphi }_1,e)$, sample $\delta \in [-\Delta {\varphi }_1,\Delta {\varphi }_1]$ and the three conjunction branchesFor every branch, solve Kepler's equation for the eccentric anomaly, evaluate $r=a(1-e\cos E)$, and minimize the planet-planetesimal separation over $\delta$ and $k$. Mark the point as encounter-capable when this minimum is below a chosen $R_{\mathrm{enc}}$, naturally the Hill radius for strong scattering. Repeating this calculation on a grid traces the boundary in the $\Delta {\varphi }_1$--$e$ plane.

Direct integrations of the circular restricted three-body problem can then refine the geometric map by allowing the resonant argument, eccentricity, and conjunction kicks to evolve self-consistently. The integrations distinguish merely orbit-crossing initial data from trajectories that actually enter the encounter region, and reveal chaotic layers near the boundary.