Variations on Debris Disks: Icy Planet Formation at 30-150 AU for 1-3   Solar Mass Main Sequence Stars
  We describe calculations for the formation of icy planets and debris disks at 30-150 AU around 1-3 solar mass stars. Debris disk formation coincides with the formation of planetary systems. As protoplanets grow, they stir leftover planetesimals to large velocities. A cascade of collisions then grinds the leftovers to dust, forming an observable debris disk. Stellar lifetimes and the collisional cascade limit the growth of protoplanets. The maximum radius of icy planets, roughly 1750 km, is remarkably independent of initial disk mass, stellar mass, and stellar age. These objects contain no more than 3% to 4% of the initial mass in solid material. Collisional cascades produce debris disks with maximum luminosity of roughly 0.002 times the stellar luminosity. The peak 24 micron excess varies from roughly 1% of the stellar photospheric flux for 1 solar mass stars to roughly 50 times the stellar photospheric flux for 3 solar mass stars. The peak 70-850 micron excesses are roughly 30-100 times the stellar photospheric flux. For all stars, the 24-160 micron excesses rise at stellar ages of 5-20 Myr, peak at 10-50 Myr, and then decline. The decline is roughly a power law, with f propto t^{-n} and n = 0.6-1.0. This predicted evolution agrees with published observations of A-type and solar-type stars. The observed far-IR color evolution of A-type stars also matches model predictions. 
Scott J. Kenyon and Benjamin C. Bromley