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

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 ${\rm \mu m}$ 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 ${\rm \mu m}$ 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.