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At the exterior $(p+q):p$ mean-motion resonance,Using the Kepler third law for $M_{\ast }\gg M_1$ givesThe disturbing function is a Fourier series in integer combinations of the orbital angles. The D'Alembert characteristic permits the eccentric termAway from resonance, terms with rapidly circulating angles average away. Here, however,so ${\varphi }_1$ is a slow resonant argument. Successive astronomical conjunctions then act coherently, making this term dominate the long-period resonant dynamics even though a $q$th-order resonance has coefficient proportional to $e^q$ at small eccentricity.

The disturbing function may be expanded in harmonics of the planets' orbital angles. Its terms fall into three useful classes:

Well-separated planets far from resonance are governed on long timescales mainly by the secular terms.

The free precession rate $A(a)$ tends to zero far inside the inner planet, diverges on approaching $a_1$, diverges on both sides of $a_2$, and tends to zero far outside the outer planet. Between $a_1$ and $a_2$ it diverges at both ends and has at least one minimum. A horizontal line $A=g_1$ therefore crosses once inside $a_1$ and once outside $a_2$, plus zero, one tangent, or two times between the planets. There are consequentlylocations of the corresponding secular resonance. This argument uses the smooth Laplace-Lagrange secular theory away from the immediate neighborhoods of the planets and from mean-motion resonances.

Very near one planet, that planet dominates both the particle's free-precession coefficient and its forcing coefficient, with the leading terms arranged approximately as $\dot{z}\simeq iA(z-z_p)-z/\tau$. Since $|A|\tau \gg 1$ close to the planet, the late-time forced solution approachesThe particle's orbit therefore tends toward the planet's eccentricity and apsidal direction. Extremely close to the planet, close encounters, co-orbital dynamics, and individual mean-motion resonances invalidate the orbit-averaged linear approximation.

The body now returns on a bound, highly eccentric orbit whose periapsis crosses more deeply into the planet's neighborhood. Repeated planetary scattering produces a sequence of energy and angular-momentum kicks rather than smooth secular evolution. Possible endpoints include ejection onto an unbound orbit, collision with the planet or star, tidal disruption, or diffusion onto a detached orbit that no longer encounters the planet. Temporary protection by a mean-motion resonance is also possible. The approximate Tisserand parameter constrains weak separated encounters, but the close-encounter sequence is chaotic and does not select a unique final orbit.

The radiation-pressure coefficient reduces the dust grain's effective stellar gravitational parameter to $(1-\beta )G{M}_{\ast }$. At the exterior 5:4 mean-motion resonance, $n_d/n_p=4/5$, so mean motion givesThis is a first-order mean-motion resonance. Its leading disturbing-function term is therefore linear in the small orbital eccentricity:Thus its dimensionless strength is of order $(M_p/M_{\ast })e_d$, up to the Laplace-coefficient combination $f_{5:4}$, and its resonant argument varies slowly near commensurability.