Linear <eccentricity damping> adds $-z/\tau$ to the complex equation:
$$
\dot z=(iA-\tau^{-1})z+i\sum_k\nu_ke^{i(g_kt+\beta_k)}.
$$
Its solution is
$$
\boxed{z(t)=C e^{(iA-1/\tau)t}
+\sum_{k=1}^2
\frac{\nu_k}{g_k-A-i/\tau}e^{i(g_kt+\beta_k)}}.
$$
Unlike the undamped free eccentricity, the homogeneous term decays exponentially. The forced response survives, acquires a phase lag, and has finite amplitude
$$
\frac{|\nu_k|}{\sqrt{(g_k-A)^2+\tau^{-2}}}.
$$

Solution

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

Linear eccentricity damping contributes $\dot z=-z/\tau$ to the complex-eccentricity equation, exponentially suppressing the free mode and regularizing a secular resonance.


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