Resonant trapping of dust

During inward migration, dust can be captured into an exterior mean-motion resonance with a planet. A phase-shifted conjunction then lets planetary torque replace angular momentum lost to Poynting–Robertson drag.

For an exterior $(p+1):p$ mean-motion resonance, use mean longitude in $\varphi =(p+1)\lambda -p{\lambda }_p-\varpi$, $z=e\exp (i\varphi )$, $x=(a-a_r)/a_r$ and $B=3p{n}_p/2$. Given $\dot{e}=-C\sin \varphi -Ae$ and $\dot{\varpi }=(C/e)\cos \varphi$, the leading Keplerian detuning gives $\dot{\varphi }=-Bx-(C/e)\cos \varphi$. Differentiating $z$ and using $e^{i\varphi }(\sin \varphi +i\cos \varphi )=i$ proves the displayed equation. It retains resonant forcing and eccentricity damping without the coordinate singularity at $e=0$. True longitude adds a fast orbital ripple and does not yield this autonomous averaged equation directly.

For positive locally constant $A,C$, supplement damped first-order resonant complex eccentricity by $\dot{a}/a=-2(p+1)Ce\sin \varphi -4A/5$. Balancing the two dissipative rates gives $e_f^2=2/[5(p+1)]$, $\sin {\varphi }_f=-A{e}_f/C$ and $B{x}_f=-(C/e_f)\cos {\varphi }_f$. Two fixed points exist when $A<C\sqrt{5(p+1)/2}$. The negative-cosine branch has $x_f>0$ and is stable in this leading model: a phase increase raises the positive resonant torque, raises semi-major axis, lowers mean motion, and reverses the phase drift. The positive-cosine branch has the opposite feedback.

For a coupled verification put $z=u+iv$. The characteristic polynomial at either fixed point is ${\lambda }^3+2A{\lambda }^2+[A^2+B^2{x}_f^2-2B(p+1)C{u}_f]\lambda -4AB(p+1)C{u}_f$. For $u_f<0$ its positive coefficients satisfy the cubic Routh-Hurwitz stability criterion, since the remaining inequality reduces to $2A(A^2+B^2{x}_f^2)>0$. For $u_f>0$ its negative constant term forces a positive real root. At the equality threshold the equilibria merge and an eigenvalue is zero. This is local stability for the frozen smooth-coefficient approximation, not a conclusion for arbitrary rapidly varying dissipation laws.

For resonant trapping of dust in an exterior $(p+q):p$ mean-motion resonance, let Poynting–Robertson drag have parameter $A$. Holding semi-major axis $a$ fixed and retaining secular rates through second order in orbital eccentricity givesFor $5p>q$, squared orbital eccentricity approaches this equilibrium exponentially in the truncated model. If the predicted value is not small, higher-order terms are needed for quantitative dynamics. Finite planetary torque also imposes a minimum eccentricity for balance, which alone does not guarantee capture during migration.

At pericentre, mean longitude equals longitude of periapsis, so resonant argument $\varphi =(p+q)\lambda -p{\lambda }_p-q\varpi$ becomes $\varphi =p(\varpi -{\lambda }_p)$. The pericentre angle in the planet's rotating frame is $\varphi /p$ modulo $2\pi /p$. For coprime $p,q$, there are $p$ distinct pericentre directions before the trajectory closes. Resonant-argument libration restricts these directions and supplies resonance protection.