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 Incoming links: Secular forcing of a test particle

Put

B_{i} = \sum_{j} A_{j} e_{ji}

. For

A \neq = g_{i}

z (t) = e_{p} e^{i (A t + β_{p})} + z_{f} (t), z_{f} (t) = i \sum \frac{B _{i}}{g _{i} - A} e^{i (g_{i} t + β_{i})} .

e_{p} = ∣ z (0) - z_{f} (0) ∣

and

β_{p} = ar g [z (0) - z_{f} (0)]

e_{p}

is the constant amplitude of the homogeneous response; its proper longitude of periapsis is

A t + β_{p}

z_{f} (t)

, not necessarily a constant magnitude.

z - z_{f}

describes a circle of radius

e_{p}

. Its center

z_{f}

can itself move under the planetary modes, so

z

need not trace one fixed circle in the inertial complex plane. At a secular resonance

A = g_{i}

, that mode instead produces

i B_{i} t e^{i (A t + β_{i})}

, and the undamped linear response grows until the approximation fails.