Low-eccentricity correction to a first-order resonance center ($3(p+1{)}^{2}n{e}^3-C_r$)

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 $3(p+1{)}^{2}n{C}_{r}e-C_r^2/e^2$. Hence zero is a saddle for $3(p+1{)}^{2}n{e}^3>C_r$, but is a center when this inequality reverses. The usual constant-eccentricity pendulum approximation of a mean-motion resonance omits this term and requires $C_r/(n{e}^3)\ll 1$.