= Secular forcing by a distant outer planet
{title2=$Z_f=5\alpha e_p/4,\quad A\propto\alpha^{3/2}$}
For a test particle interior to a distant eccentric <planet>, write $\alpha=a/a_p\ll1$. The leading <Laplace coefficients> are $b_{3/2}^{1}=3\alpha$ and $b_{3/2}^{2}=15\alpha^2/4$. Thus its <forced eccentricity> is $Z_f=5\alpha e_p/4$ and its secular frequency is $A=(3/4)n_p(m_p/M)\alpha^{3/2}$. More distant inner particles have larger forced <orbital eccentricities> and faster secular precession, producing a growing gradient of <complex eccentricity>.
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