= Low-eccentricity correction to a first-order resonance center
{title2=$3(p+1)^2ne^3-C_r$}
In the planar exterior first-order constant-coefficient model, differentiating the apsidal-precession term retains a contribution proportional to $C_r^2/e^2$. At the zero-phase fixed point its linear angular curvature is $3(p+1)^2nC_re-C_r^2/e^2$. Hence zero is a saddle for $3(p+1)^2ne^3>C_r$, but is a center when this inequality reverses. The usual constant-eccentricity <pendulum approximation of a mean-motion resonance> omits this term and requires $C_r/(ne^3)\ll1$.
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