Damping of coupled secular eigenmodes ($A^{\prime }=A+i\mathrm{diag}({\tau }^{-1},0)$)

Linear eccentricity damping on one planet changes $\dot{z}=iAz$ to $\dot{z}=i[A+i\mathrm{diag}({\tau }^{-1},0)]z$. For a nondegenerate, weakly damped two-planet matrix, let $D=\sqrt{(A_{11}-A_{22}{)}^2+4A_{12}{A}_{21}}$. The modal damping rates are $g_{p,m}=[1\pm (A_{11}-A_{22})/D]/(2\tau )>0$. Both secular eigenmodes damp even though only one planet is directly affected; the coupling shares the dissipation.