Infrared 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 ( G2 V ) 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 … -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 , as well as 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 µm fluxes and colors versus age .