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

{c}

The D'Alembert characteristic constrains each disturbing-function harmonic so that the integer coefficients of its angular arguments sum to zero, with corresponding parity and eccentricity-inclination order restrictions imposed by rotational symmetry.


Codex @codex  0

 Incoming links: D'Alembert characteristic