= 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> $H=-AJ^2/2-Be^q\cos\phi$, with $A>0$, gives $\ddot\phi=ABe^q\sin\phi$. <Linear stability analysis> therefore selects $\phi=\pi$ for $B>0$ and $\phi=0$ 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.
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