At the exterior $(p+q):p$ <mean-motion resonance>,
$$
\frac n{n_1}=\frac p{p+q}.
$$
Using the <Kepler third law> for $M_*\gg M_1$ gives
$$
\boxed{a=a_1\left(\frac{p+q}{p}\right)^{2/3}}.
$$
The <disturbing function> is a Fourier series in integer combinations of the orbital angles. The <D'Alembert characteristic> permits the eccentric term
$$
\mathcal R_{\rm res}=C_{p,q}(a)e^q\cos\phi_1+O(e^{q+2}),
\qquad
\phi_1=(p+q)\lambda-p\lambda_1-q\varpi.
$$
Away from resonance, terms with rapidly circulating angles average away. Here, however,
$$
\dot\phi_1=(p+q)n-pn_1-q\dot\varpi\simeq0,
$$
so $\phi_1$ 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.

Solution

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

The <disturbing function> may be expanded in harmonics of the planets' orbital angles. Its terms fall into three useful classes:

* <short-period orbital perturbations> involve rapidly varying combinations of mean longitudes and produce bounded oscillations associated with conjunctions;
* <mean-motion resonance> terms have a nearly stationary integer combination of mean longitudes and apsidal angles, so repeated conjunctions act coherently and can drive libration, migration, or eccentricity growth;
* <secular perturbations> survive averaging over both mean longitudes and describe slow exchange of eccentricity and apsidal precession.

Well-separated planets far from resonance are governed on long timescales mainly by the secular terms.

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 3 / i / Solution

{title2=$\mathcal R$}
{wiki}

The disturbing function is the perturbing part of the gravitational potential used to derive the evolution of orbital elements in celestial mechanics.


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