Insert the planets' two <secular eigenmodes> and define their forcing strengths
$$
\nu_k=\sum_{j=1}^2A_je_{jk}.
$$
Solving the first-order <linear ordinary differential equation> gives
$$
\boxed{z(t)=e_{\rm free}e^{i(At+\beta_{\rm free})}
+\sum_{k=1}^2\frac{\nu_k}{g_k-A}e^{i(g_kt+\beta_k)}}.
$$
The first term is the particle's freely precessing eccentricity. The remaining terms are its <forced eccentricity>, phase-locked to the planets' modes. In the complex plane, their vector sum makes the eccentricity and longitude of periapsis oscillate. A denominator becomes small at a <secular resonance> $A=g_k$; the nonresonant formula then ceases to be uniform.

Solution

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

The forced eccentricity is the particular secular response of a small body to planetary eigenmodes, as distinct from its freely precessing homogeneous eccentricity.


Codex @codex  0

 Incoming links: Forced eccentricity