Age Dependence of the Vega Phenomenon: Theory

In a separate paper, we have reexamined the observations of IR excess obtained with the Infrared Space Observatory satellite and discussed the ages of stars with excess. The amount of dust (measured by the luminosity fraction f_d=L_IR/L_*) seen around main-sequence stars of different ages shows several interesting trends. To discuss these results in the context of a physical model, we develop in this paper an analytical model for the dust production in Vega-type systems. Previously it has been claimed that a power-law slope of about -2 in the diagram plotting amount of dust versus time could be explained by a simple collisional cascade. We show that such a cascade in fact results in a power law f_d~t^-1 if the dust removal processes are dominated by collisions. A power law f_d~t^-2 only results when the dust removal processes become dominated by Pointing-Robertson drag. This might be the case in the Kuiper Belt of our own solar system, but it is certainly not the case in any of the observed disks. A steeper slope can, however, be created by including continuous stirring into the models. We show that the existence of both young and old Vega-like systems with large amounts of dust (f_d~=10^-3) can be explained qualitatively by Kuiper Belt-like structures with delayed stirring. Finally, the absence of young stars with intermediate amounts of dust might be due to the fact that stirring due to planet formation may not be active in young low-mass disks. The considerations in this paper support the picture of simultaneous stirring and dust production proposed by Kenyon and Bromley.