Self-consistent size and velocity distributions of collisional cascades

TL;DR

This paper develops a self-consistent model for collisional cascades that simultaneously determines size and velocity distributions, revealing steeper size spectra and enabling better interpretation of observational data.

Contribution

It introduces a model that accounts for velocity evolution in collisional cascades, improving upon previous steady-state size distribution theories.

Findings

Steeper size distributions than traditional models predict.

Velocity-dependent scale heights can be quantitatively predicted.

Observations can constrain large body populations and internal strengths.

Abstract

The standard theoretical treatment of collisional cascades derives a steady-state size distribution assuming a single constant velocity dispersion for all bodies regardless of size. Here we relax this assumption and solve self-consistently for the bodies' steady-state size and size-dependent velocity distributions. Specifically, we account for viscous stirring, dynamical friction, and collisional damping of the bodies' random velocities in addition to the mass conservation requirement typically applied to find the size distribution in a steady-state cascade. The resulting size distributions are significantly steeper than those derived without velocity evolution. For example, accounting self-consistently for the velocities can change the standard q=3.5 power-law index of the Dohnanyi (1969) differential size spectrum to an index as large as q=4. Similarly, for bodies held together by their own gravity, the corresponding power-law index range 2.88<q<3.14 of Pan & Sari (2005) can steepen to values as large as q=3.26. Our velocity results allow quantitative predictions of the bodies' scale heights as a function of size. Together with our predictions, observations of the scale heights for different sized bodies for the Kuiper belt, the asteroid belt, and extrasolar debris disks may constrain the mass and number of large bodies stirring the cascade as well as the colliding bodies' internal strengths.