Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 316 / 4 / v

Ignoring the distant planet at first, the inner two-by-two block has eigenvalues zero and $-(B_{12}+B_{21})$. Therefore its fast differential secular eigenmode hasFor the slow mode, strong inner coupling makes $y_1\simeq y_2=y_{\mathrm{in}}$. Add the first two evolution equations weighted by angular momenta and divide by $L_{12}=L_1+L_2$. The rigid inner pair in hierarchical secular inclination dynamics has equationswithThe two-plane matrix has eigenvalues zero and $-c(1+L_{12}/L_3)$. The architecture implies $L_3\gg L_{12}$ for comparable masses, so its leading slow frequency isThe neglected factor $1+L_{12}/L_3$ gives a relative correction of order $\sqrt{\beta }$; imperfect locking of the inner pair supplies higher-order corrections in its weak coupling to the outer planet.