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

Abstract

We derive a new analytical model for the evolution of a collisional cascade in a thin annulus around a single central star. In this model, r max the size of the largest object changes with time, , with 0.1–0.2. Compared to standard models where r max is constant in time, this evolution results in a more rapid decline of M d , the total mass of solids in the annulus, and L d , the luminosity of small particles in the annulus: and . We demonstrate that the analytical model provides an excellent match to a comprehensive suite of numerical coagulation simulations for annuli at 1 au and at 25 au. If the evolution of real debris disks follows the predictions of the analytical or numerical models, the observed luminosities for evolved stars require up to a factor of two more mass than predicted by previous analytical models.