Symmetric libration centres of an eccentricity resonance

In the weak, non-crossing, leading-eccentricity planar model, exterior orders one to three have positive resonant disturbing function coefficients $B_q$. Interior eccentricity-type coefficients of these orders have sign $(-1{)}^q$. A local resonant Hamiltonian $H=-A{J}^2/2-B{e}^q\cos \varphi$, with $A>0$, gives $\ddot{\varphi }=AB{e}^q\sin \varphi$. Linear stability analysis therefore selects $\varphi =\pi$ for $B>0$ and $\varphi =0$ for $B<0$. Exterior symmetric centres are $\pi$; interior centres are $0,\pi ,0$ for orders one, two and three. These are conditional leading-harmonic predictions, not universal centres at arbitrary orbital eccentricity; asymmetric resonant-argument libration changes the picture.

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 $C_r/(n{e}^3)\ll 1$.