Dust distributions in debris disks: effects of gravity, radiation pressure and collisions

Abstract

We model a typical debris disk, treated as an idealized ensemble of dust particles, exposed to stellar gravity and direct radiation pressure and experiencing fragmenting collisions. Applying the kinetic method of statistical physics, written in orbital elements, we calculate size and spatial distibutions expected in a steady-state disk, investigate timescales needed to reach the steady state, and calculate mass loss rates. Particular numerical examples are given for the debris disk around Vega. The disk should comprise a population of larger grains in bound orbits and a population of smaller particles in hyperbolic orbits. The cross section area is dominated by the smallest grains that still can stay in bound orbits, for Vega about 10 µm in radius. The size distribution is wavy, implying secondary peaks in the size distribution at larger sizes. The radial profile of the pole-on surface density or the optical depth in the steady-state disk has a power-law index between about −1 and −2. It cannot be much steeper even if dust production is confined to a narrow planetesimal belt, because collisional grinding produces smaller and smaller grains, and radiation pressure pumps up their orbital eccentricities and spreads them outward, which flattens the radial profile. The timescales to reach a steady state depend on grain sizes and distance from the star. For Vega, they are about 1 Myr for grains up to some hundred µm at 100 AU. The total mass of the Vega disk needed to produce the observed amount of micron and submillimeter-sized dust does not exceed several earth masses for an upper size limit of parent bodies of about 1 km. The collisional depletion of the disk occurs on Gyr timescales.

Modelling the huge,Herschel-resolved debris ring around HD 207129

“…Such a radial profile that rises outward with a sharp outer edge contrasts with those derived for the archetypal disks Vega (Su et al 2005;Krivov et al 2006;Müller et al 2010;Sibthorpe et al 2010) and β Pictoris (Golimowski et al 2006;Thébault & Wu 2008;Krivov et al 2009;Vandenbussche et al 2010). In these disks, the derived geometrical optical thickness typically falls off with a moderate r −1.5 relationship beyond the peak value, i.e.…”

Vertical structure of debris discs

“…This is clearly not the case for a Dohnanyi size distribution with dN ∝ s −3.5 ds, as well as for most of the more realistic "wavy" distributions derived from collisional evolution models (e.g., Krivov et al 2006;Thébault & Augereau 2007). However, there is a range of systems for which these numerical studies have shown that they could be depleted of small grains, i.e., those with very low dynamical excitation e dyn .…”

“…Numerical studies of debris discs fall into two main categories: those investigating the collisional and size distribution evolution of the system are usually statistical particle-in-the-box models, of no or poor spatial resolution and very limited dynamical evolution (e.g., Krivov et al 2006;Thébault & Augereau 2007;Löhne et al 2008), while those studying the dynamics and the formation and evolution of spatial structures are mostly N-body type codes, where size distributions and mutual collisions are usually neglected (see for example Reche et al 2008).…”

Kuiper belts around nearby stars

“…With such data, any deviation from the theoretical power law size distribution of observable dust, e.g. the "wavy" size distribution predicted from numerical models of collisional processes in debris disks (Thébault & Augereau 2007;Krivov et al 2006), can be thoroughly investigated. In addition, ALMA will enable detection and imaging of many faint disks, and permit detailed SED modelling of the very coldest Kuiper-belt systems.…”