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

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

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

Diagonalize the real Laplace-Lagrange secular matrix $\mathbf{A}$. Its eigenvalues areand choose corresponding real eigenvectors ${\mathbf{v}}_k$. If $V=({\mathbf{v}}_1{\mathbf{v}}_2)$, the initial complex eccentricity vector determines complex mode coefficientsThe matrix exponential solution isThuswhere a negative eigenvector component may equivalently be made positive by adding $\pi$ to its phase. The $g_k$ are secular precession frequencies, each eigenvector fixes the planets' eccentricity ratio and relative apsidal orientation, and ${\beta }_k$ fixes the phase selected by the initial conditions.

Insert the planets' two secular eigenmodes and define their forcing strengthsSolving the first-order linear ordinary differential equation givesThe 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.

The free precession rate $A(a)$ tends to zero far inside the inner planet, diverges on approaching $a_1$, diverges on both sides of $a_2$, and tends to zero far outside the outer planet. Between $a_1$ and $a_2$ it diverges at both ends and has at least one minimum. A horizontal line $A=g_1$ therefore crosses once inside $a_1$ and once outside $a_2$, plus zero, one tangent, or two times between the planets. There are consequentlylocations of the corresponding secular resonance. This argument uses the smooth Laplace-Lagrange secular theory away from the immediate neighborhoods of the planets and from mean-motion resonances.

Linear eccentricity damping adds $-z/\tau$ to the complex equation:Its solution isUnlike the undamped free eccentricity, the homogeneous term decays exponentially. The forced response survives, acquires a phase lag, and has finite amplitude

For $t\gg \tau$, the free term has disappeared. Exactly at $A=g_1$, the resonant contribution tends towith finite eccentricity $|{\nu }_1|\tau$ and a quarter-cycle phase shift. Without damping, exact resonance instead giveswhose secular resonance amplitude grows linearly in the ideal linear theory.

Very near one planet, that planet dominates both the particle's free-precession coefficient and its forcing coefficient, with the leading terms arranged approximately as $\dot{z}\simeq iA(z-z_p)-z/\tau$. Since $|A|\tau \gg 1$ close to the planet, the late-time forced solution approachesThe particle's orbit therefore tends toward the planet's eccentricity and apsidal direction. Extremely close to the planet, close encounters, co-orbital dynamics, and individual mean-motion resonances invalidate the orbit-averaged linear approximation.