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We derive a new analytical model for the evolution of a collisional cascadein a thin annulus around a single central star. In this model, $r_{max}~$ thesize of the largest object declines with time (t); $r_{max} \proptot^{-\gamma}$, with $\gamma$ = 0.1-0.2. Compared to standard models where$r_{max}~$ is constant in time, this evolution results in a more rapid declineof $M_d$, the total mass of solids in the annulus and $L_d$, the luminosity ofsmall particles in the annulus: $M_d \propto t^{-(\gamma + 1)}~$ and $L_d\propto t^{-(\gamma/2 + 1)}~~$. We demonstrate that the analytical modelprovides an excellent match to a comprehensive suite of numerical coagulationsimulations for annuli at 1 AU and at 25 AU. If the evolution of real debrisdisks follows the predictions of the analytical or numerical models, theobserved luminosities for evolved stars require up to a factor of two more massthan predicted by previous analytical models.

Variations on Debris Disks IV. An Improved Analytical Model for Collisional Cascades