The <radiation-pressure coefficient> reduces the dust grain's effective stellar gravitational parameter to $(1-\beta)GM_*$. At the exterior 5:4 <mean-motion resonance>, $n_d/n_p=4/5$, so <mean motion> gives
$$
\boxed{a_d\simeq a_p(1-\beta)^{1/3}\left(\frac54\right)^{2/3}}.
$$
This is a <first-order mean-motion resonance>. Its leading disturbing-function term is therefore linear in the small <orbital eccentricity>:
$$
\mathcal R_{\rm res}\sim
\frac{GM_p}{a_d}f_{5:4}(a_p/a_d)e_d\cos\phi,
\qquad
\boxed{\phi=5\lambda_d-4\lambda_p-\varpi_d}.
$$
Thus its dimensionless strength is of order $(M_p/M_*)e_d$, up to the Laplace-coefficient combination $f_{5:4}$, and its <resonant argument> varies slowly near commensurability.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-316/4/a/solution