Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 316 / 4

The radiation-pressure coefficient reduces the dust grain's effective stellar gravitational parameter to $(1-\beta )G{M}_{\ast }$. At the exterior 5:4 mean-motion resonance, $n_d/n_p=4/5$, so mean motion givesThis is a first-order mean-motion resonance. Its leading disturbing-function term is therefore linear in the small orbital eccentricity:Thus its dimensionless strength is of order $(M_p/M_{\ast })e_d$, up to the Laplace-coefficient combination $f_{5:4}$, and its resonant argument varies slowly near commensurability.

Solved by gpt-5.6-sol high.

Let $\psi ={\lambda }_d-{\lambda }_p$ be azimuth in the frame rotating with the planet, with the planet fixed at $\psi =0$. Since $n_d/n_p=4/5$, one synodic circuit contains four radial oscillations. To first order in $e_d$, choosing conjunction at apoapsis givesThe requested sketch is therefore a four-lobed closed curve centred on the star. For $e_d=0.2$, its radii range from $0.8a_d$ to $1.2a_d$; an outer lobe points along the star-planet line, and the planet lies at $(a_p,0)$. At that radial alignment $\varphi =\pi$, because ${\lambda }_d={\lambda }_p$ and ${\lambda }_d={\varpi }_d+\pi$ at apoapsis.

Solved by gpt-5.6-sol high.

At exact conjunction the planet's force is radial and has no instantaneous torque, but the approach and departure do not cancel when conjunction is displaced from apoapsis. Write the conjunction longitude as ${\lambda }_c={\varpi }_d+\pi +\delta$. At conjunction,The resonant part of the given evolution law has ${\dot{a}}_d=-8e_d{a}_{d}D\sin \varphi$. Thus a conjunction after apoapsis, $\delta >0$, gives $\sin \varphi <0$ and transfers angular momentum to the dust, increasing $a_d$. A conjunction before apoapsis removes angular momentum, while one exactly at apoapsis has zero secular transfer by symmetry.

Solved by gpt-5.6-sol high.

Poynting–Robertson drag removes angular momentum, so a trapped grain requires positive resonant torque from the planet. Part (c) shows that conjunction must occur after the dust has passed apoapsis. In the planet's rotating frame, the four-lobed pattern is therefore rotated so that the relevant outer lobe lags the fixed planet in the direction of orbital motion. Equivalently,with the stable phase displaced beyond $\varphi =\pi$ rather than sitting at the torque-free value $\varphi =\pi$.

Solved by gpt-5.6-sol high.

During resonant trapping of dust, the mean semi-major axis is stationary. Setting the supplied ${\dot{a}}_d$ to zero givesA real resonant phase exists only when $|\sin \varphi |\le 1$. Hencewhere $e_{d0}$ is the grain's orbital eccentricity when trapping begins. The inequality also has the expected sign $\sin \varphi <0$, allowing the planetary torque to oppose Poynting–Robertson drag.

Solved by gpt-5.6-sol high.

While resonance fixes $a_d$, substitute the phase relation from part (e) into the supplied eccentricity equation:To the requested first order in eccentricity, discard the $e_d^2$ correction. It follows thatUsing ${\beta }_{\max }=4c{e}_{d0}{a}_d^{2}D/(G{M}_{\ast })$ and integrating from $e_d(0)=e_{d0}$ givesThe neglected term eventually matters and prevents indefinite validity of this small-eccentricity growth law.

Solved by gpt-5.6-sol high.