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

## TL;DR

This paper introduces an improved analytical model for collisional cascades in debris disks, showing that the largest objects shrink over time, leading to faster declines in disk mass and luminosity, aligning well with numerical simulations.

## Contribution

The paper presents a new analytical model accounting for decreasing maximum object size, enhancing the understanding of debris disk evolution beyond previous constant-size assumptions.

## Key findings

- The maximum object size declines as $r_{max} \, \propto \, t^{-\gamma}$ with $\gamma$ between 0.1 and 0.2.
- Total mass and luminosity decline faster than in standard models, with $M_d \propto t^{-(\gamma + 1)}$ and $L_d \propto t^{-(\gamma/2 + 1)}$.
- The model matches numerical simulations at 1 AU and 25 AU, suggesting more mass is needed to explain observed luminosities of evolved stars.

## 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 declines with time (t); $r_{max} \propto t^{-\gamma}$, with $\gamma$ = 0.1-0.2. Compared to standard models where $r_{max}~$ is constant in time, this evolution results in a more rapid decline of $M_d$, the total mass of solids in the annulus and $L_d$, the luminosity of small particles in the annulus: $M_d \propto t^{-(\gamma + 1)}~$ and $L_d \propto t^{-(\gamma/2 + 1)}~~$. 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.