Self-Consistent Size and Velocity Distributions of Collisional Cascades

Abstract

The standard theoretical treatment of collisional cascades derives a steadystate 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.

Fomalhaut debris disk emission at 7 millimeters: constraints on the collisional models of planetesimals

“…This predicted value of q is ruled out by our analysis at the 99% confidence level. Our analysis therefore suggests that the variation in body size of the velocity distribution of planetesimals in the Fomalhaut debris disk is weaker than predicted by the model of Pan & Schlichting 2011). This might be explained by the gravitational stirring expected in the Fomalhaut planetesimal belt caused by the known massive exo-planet (Fomalhaut b) detected close to the debris disk (Mustill & Wyatt 2009).…”

“…obtained by our analysis is consistent with the prediction of q = 3.65 of the Gaspar et al (2011) fiducial model, where they adopted a tensile strength curve varying with body size as a −0.38 . Pan & Schlichting (2011) have recently extended the classical treatment of collisional cascades (Donhanyi 1969) by relaxing the hypothesis of a single constant velocity dispersion, and solving self-consistently for the body size and size-dependent velocity distributions at the steady-state. They derived a velocity distribution v(a) ∝ a 0.5 and a correspondingly very steep slope of the body's size-distribution q = 4 in the sizes of interest for the comparison with our results for the Fomalhaut disk.…”

“…For example, collisional models of debris disks provide different predictions for the grain size distribution depending on the dynamical state (velocity dispersion, e.g. Pan & Schlichting 2011) or physical conditions (e.g. tensile strength, Durda & Dermott 1997) of the large bodies.…”

A dearth of small particles in debris disks

“…Some variation in α has been found in different simulations. Pan & Schlichting (2012) find up to α = 4 for cohesion-dominated collisional particles, and up to α = 3.26 for gravity-dominated ones.…”

HD 104860 and HD 192758: Two Debris Disks Newly Imaged in Scattered Light with the Hubble Space Telescope

“…Without a prior on the dust separation to the star and assuming that the disks are in radiative and collisional equilibrium, these blackbody emissions correspond, respectively, to a ∼10 −5 M Moon disk at a radius of 3au from the star, and to a massive ∼1.5 M Moon cold disk at a radius of 366au, based on silicate spherical grain compositions. By measuring the cold disk spectral index in the millimeter from VLA and ATCA observations, MacGregor et al (2016b) estimated a dust size distribution in the system with a power-law q=3.64±0.15, consistent with steadystate collisional cascade models (Dohnanyi 1969;Gáspár et al 2012;Pan & Schlichting 2012). Assuming a disk composed of astro-silicates (Draine 2003), a stellar luminosity of L å =1.16 L e , and mass of M å =1.04 M e , the radiative pressure blowout grain size limit is estimated as a blow ∼0.4 μm, but the minimum grain size is inferred to be ∼7 μm from joint modeling of the system's SED and Herschel images (Pawellek et al 2014).…”