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 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.

ApJ, in press, 22 pages of text and 14 figures

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• From Pebbles and Planetesimals to Planets and Dust: the Protoplanetary Disk--Debris Disk Connection
• Collisional Fragmentation is Not a Barrier to Close-in Planet Formation
• A Pluto-Charon Concerto: An Impact on Charon as the Origin of the Small Satellites
• A Pluto-Charon Sonata III. Growth of Charon from a Circum-Pluto Ring of Debris
• Rapid destruction of planetary debris around white dwarfs through aeolian erosion
• Seeking echoes of circumstellar disks in Kepler light curves
• Takeout and Delivery: Erasing the Dusty Signature of Late-stage Terrestrial Planet Formation