Long-term collisional evolution of debris disks

IR surveys indicate that the dust content in debris disks gradually declines with stellar age. We simulated the long-term collisional depletion of debris disks around solar-type (G2V) stars with our collisional code. The numerical results were supplemented by, and interpreted through, a new analytic model. General scaling rules for the disk evolution are suggested. The timescale of the collisional evolution is inversely proportional to the initial disk mass and scales with radial distance as r(4.3) and with eccentricities of planetesimals as e(-2.3). Further, we show that at actual ages of debris disks between 10 Myr and 10 Gyr, the decay laws of the dust mass and the total disk mass are different. The reason is that the collisional lifetime of planetesimals is size dependent. At any moment, there exists a transitional size, which separates larger objects that still retain the primordial'' size distribution set in the growth phase from smaller objects whose size distribution is already set by disruptive collisions. The dust mass and its decay rate evolve as that transition affects objects of ever larger sizes. Under standard assumptions, the dust mass, fractional luminosity, and thermal fluxes all decrease as t(xi) with xi = 0.3 to -0.4. Specific decay laws of the total disk mass and the dust mass, including the value of xi, largely depend on a few model parameters, such as the critical fragmentation energy as a function of size, the primordial size distribution of largest planetesimals, and the characteristic eccentricity and inclination of their orbits. With standard material prescriptions and a distribution of disk masses and extents, a synthetic population of disks generated with our analytic model agrees quite well with the observed Spitzer MIPS statistics of 24 and 70 mu m fluxes and colors versus age.