= Dissipative resonant eccentricity equilibrium
{title2=$e_{\rm eq}^2$}
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> gives
$$
\frac{d(e^2)}{dt}=\frac A{a^2}\left[\frac{2q}{p+q}-\frac{5p-q}{p+q}e^2\right],
\qquad e_{\rm eq}^2=\frac{2q}{5p-q},\qquad
\tau=\frac{a^2(p+q)}{A(5p-q)}.
$$
For $5p>q$, 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.
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