Mean-motion resonance

A mean-motion resonance has $j{n}_2-k{n}_1\simeq 0$ for small integers $j,k$. Its slow resonant angle combines mean longitudes with apsidal or nodal longitudes.

For an eccentric inner body and circular outer perturber near the reduced $(m+n):m$ period ratio, the leading resonant argument is $\varphi =(m+n){\lambda }_{\mathrm{out}}-m{\lambda }_{\mathrm{in}}-n{\varpi }_{\mathrm{in}}$. Its disturbing function amplitude is of order $e_{\mathrm{in}}^n$. At a mean conjunction, $\varphi =n(\Lambda -{\varpi }_{\mathrm{in}})$; at actual alignment the exact expression is $(m+n)x-my$ with $x=f_{\mathrm{in}}$ and $y=M_{\mathrm{in}}(x)$.

An inclination-type mean-motion resonance has a leading resonant argument containing a longitude of ascending node. Reflection symmetry forces the total nodal coefficient in each disturbing-function harmonic to be even, so the leading resonant strength is an even power of $\sin (I/2)$.

In an exterior mean-motion resonance, the resonant small body orbits outside the perturbing planet and therefore has lower mean motion. A $(p+q):p$ exterior resonance has $n/n_p\simeq p/(p+q)$.

The nominal semi-major axis of an exterior $(p+q):p$ mean-motion resonance follows from $(p+q)n_r=p{n}_p$ and Kepler's third law. Apsidal precession displaces the actual libration center because stationarity of the resonant argument requires $(p+q)n-p{n}_p-q\dot{\varpi }=0$.

A grain with radiation-pressure coefficient $\beta <1$ feels effective central gravitational parameter $G{M}_{\star }(1-\beta )$. Combining its mean motion with the planet's mean motion at an exterior-type $(j+1):j$ frequency commensurability gives the displayed semi-major axis. The commensurability lies geometrically exterior only for $j<\sqrt{1-\beta }/(1-\sqrt{1-\beta })$. Radiation changes the grain's attraction, so the same frequency ratio need not imply the usual radial ordering.

A first-order mean-motion resonance has $|j-k|=1$. For small eccentricity its leading eccentric resonant term is linear in eccentricity.

A local weak gravitational assist estimate for a nonradiating circular grain gives $\delta ϵ\simeq 32{\mu }^2{ϵ}^{-5}$, where $\mu =M_p/M_{\star }$. Conjunction-longitude sensitivity to an encounter then gives $|\delta \Lambda |\simeq 128\pi {\mu }^2/(3{ϵ}^7)$. Requiring the change to exceed one full turn and using $ϵ\simeq 2/(3j)$ gives $j_c\simeq (2/729{)}^{1/7}{\mu }^{-2/7}$. This motivates loss of coherent resonant conjunctions, rather than proving that every initial phase escapes. The coefficient depends on the encounter convention and the approximations; near-planet Hill equations require a consistent rotating-frame shear instead of mixing local inertial and rotating velocities.

A resonant argument is a slow angular combination whose libration diagnoses resonance. For an exterior $j:(j-1)$ eccentric resonance it can be $\varphi =j\lambda -(j-1){\lambda }_p-\varpi$.

A resonant argument is an integer combination of orbital angles whose slow variation can diagnose mean-motion resonance. Its libration, rather than circulation through every angle, characterizes trapping in the resonance.

For an eccentric outer body and circular inner perturber, the exterior resonant argument is $\varphi =(p+q){\lambda }_{\mathrm{out}}-p{\lambda }_{\mathrm{in}}-q\varpi$. At equal mean longitudes $\Lambda$, conjunction phases areThe synodic period and coprime $p,q$ determine the visitation order, with advance $2\pi p/q$. For actual true alignment, write $x=f$ and $y=M(f)$ for the eccentric body's true anomaly and mean anomaly. Then $\varphi =(p+q)y-px$ instead of $qx$. The evenly spaced directions are a leading-small-eccentricity approximation.

In a high-orbital eccentricity interior $k:1$ mean-motion resonance, the resonant argument $\varphi =k{\lambda }_p-\lambda -(k-1)\varpi$ places successive rotating-frame apocentres at $\pi -[\varphi +(2j+1)\pi ]/k$. Avoiding astronomical conjunction at these slow outer excursions favors $\varphi =0$ for $k=2$ and $\varphi =\pi$ for $k=3$. This geometrical phase protection suggests stable resonant-argument libration centres; actual dynamical stability requires the resonant perturbation problem.

With constant mean motions and $P_c/P_b=(1+\delta )(p+q)/p$, mean conjunctions advance by $\Delta \Lambda =2\pi /(P_c/P_b-1)$ each synodic period. Tracking one branch every $q$ conjunctions and subtracting $2\pi p$ givesThe usual formula $-2\pi p\delta /(q{P}_b)$ keeps only first order in $\delta$. A labelled branch takes $q{P}_b(1+\delta )/(p|\delta |)$ to turn once, whereas the full set of $q$ interchangeable directions repeats after $P_b(1+\delta )/(p|\delta |)$. Finite orbital eccentricity introduces true-conjunction corrections.

For an exterior circular grain with radiation-pressure coefficient $\beta$, set $q=\sqrt{1-\beta }$ and $ϵ=a/a_p-1$. Its next conjunction occurs after the synodic period, during which the planet's unwrapped mean longitude advances by $\Lambda =2\pi /[1-q(1+ϵ{)}^{-3/2}]$. A weak planetary scattering encounter changes this by $\delta \Lambda \simeq -3\pi q(1+ϵ{)}^{-5/2}\delta ϵ/[1-q(1+ϵ{)}^{-3/2}{]}^2$. Unwrapped longitude counts turns; reducing it modulo $2\pi$ before comparison would erase this measure of timing sensitivity.

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.

Resonance capture occurs when slow orbital migration carries a body into a resonance and the resonant argument begins to librate, so resonant torque can balance the migration torque.