Resonant-argument libration

Resonant-argument libration is bounded oscillation of a resonant argument about a stable equilibrium value instead of circulation through all angles.

Keeping one resonant harmonic and treating orbital eccentricity as fixed reduces mean-motion resonance dynamics to a physical pendulum equation. For the inner first-order term $\mathcal{R}=(G{M}_2/a_2)f(\alpha )e\cos \varphi$, leading Lagrange planetary equations give $\ddot{\varphi }\simeq 3j^2{\mu }_2{n}^2\alpha fe\sin \varphi$. Around its stable phase, the small-amplitude libration frequency is $jn\sqrt{3{\mu }_2\alpha |f|e}$. Evolution of eccentricity, other harmonics, and approach to a separatrix change this approximation.

For $\ddot{\varphi }={\omega }_0^2\sin \varphi$ with constant ${\omega }_0$, multiplication by $\dot{\varphi }$ gives the conserved energy $E={\dot{\varphi }}^2/2+{\omega }_0^2(1+\cos \varphi )$. The stable center is $\pi$; the separatrix has $E=2{\omega }_0^2$. Below it the resonant argument librates and above it the angle circulates. This is an invariant of the constant-coefficient pendulum approximation of a mean-motion resonance, not of arbitrary coupled orbital element evolution.

Libration about a resonant phase distinct from the symmetric values zero and $\pi$. For example, in a negative-Kepler-curvature reduction with disturbing potential $R=A\cos \varphi +B\cos 2\varphi$, $A>0$, $B>A/4$, stationary points obey $\sin \varphi (A+4B\cos \varphi )=0$. At $\pi$, $R^{\prime \prime }=A-4B<0$, so it is unstable, whereas at $\cos \varphi =-A/(4B)$ one has $R^{\prime \prime }=4B-A^2/(4B)>0$, giving two stable asymmetric centres. Actual branch thresholds depend on the resonance coefficients and evolving orbital elements.

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

The libration amplitude is the maximum angular displacement of a librating resonant argument from its equilibrium value.

Resonance protection confines conjunctions to orbital phases that avoid close encounters even when the osculating orbits geometrically cross.