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The impact gives fragments a spread of orbital energy and specific angular momentum. Their resulting spread of mean motion lets Keplerian shear stretch a compact dust clump into an arc and then a ring, while the spread of orbital frequencies causes phase mixing. Size-dependent radiation-pressure coefficients immediately place small grains on different eccentric or even radiation-pressure blowout orbits; Poynting–Robertson drag and stellar-wind drag then alter their orbits on longer timescales. Further collisions grind or disperse the clump, and planetary perturbations can accelerate mixing.

These processes depend strongly on $D_f$. Small grains have larger radiation-force-to-gravity ratios and generally shorter collisional or drag lifetimes, while larger fragments remain closer to the parent orbit but can preserve a velocity-dispersion-driven clump for longer. The lifetime also depends on collision location, ejection velocities, optical depth, and orbital radius. A universal fixed $\Delta t$ is therefore a useful population-model approximation, not a literal property of every collision; a size- and event-dependent lifetime distribution is more realistic.

The linear term $-(3/2){\Omega }_{K}x{\mathbf{e}}_y$ is the local Keplerian shear in the shearing sheet. The remaining constantis the gas's azimuthal velocity relative to the local Keplerian frame. The outwardly decreasing midplane pressure found in part (a)(iii) makes the gas Sub-Keplerian, so

For $u_x=0$ and $u_y=u_y(x)$, the material acceleration vanishes. The radial component of force balance isThereforeThe first term is the background Keplerian shear; the second is the geostrophic balance correction produced by the surface-density gradient.

Let $\delta ={\Sigma }^{\prime }/{\Sigma }_0$ and define the derivative following the background Keplerian shear byThe linearized continuity and momentum equations areThe coefficient $\Omega /2$ combines the background shear with the Coriolis acceleration.