where ts is the sweeping time, the time the by the cross section of a comet moving at velocity vcoii to sweep the entire volume V.

The solution of eq. (1) is obviously

where jV0 is the initial number of comets. For times t ^ the solution becomes independent or jV0.

We may now proceed to compute the amount of dust grains produced by the collisional cascade of the comets. It has been show that collisional equilibrium produces a size distribution f(m) <x m~I1/'6 [5]. At the small end of the size distribution, particles are removed. The main removal processes are radiation pressure for grains which are small enough to receive an acceleration close to or higher than the gravitational acceleration of the star. Somewhat bigger grains can still be removed from the disk by Poynting Robertson drag. Which process dominates is dependent upon the dust density in the disk. If the density is high, collisional destruction dominates all the way down to small particles which are removed by radiation pressure. If the disk contains less mass, particles of a certain size have Poynting Robertson timescales which are smaller than the collisional time and are being pulled out of the cometary region towards the star (see contribution of Dermott, this volume). It turns out that in a system with as little mass as the solar Kuiper Belt (estimates indicate about 0.1 M®), Poynting Robertson drag dominates for lpm sized grains. In the denser Vega-like systems collisions always dominate.

One can show that in a collisionally dominated disk the number of small grains is proportional to Np while in a PR-drag dominated disk, the number of grains is proportional to The number of grains in the disk can then be calculated analytically and the infrared/bolometric luminosity tir be evaluated.

Dust production in the Kuiper Belt and in Vega-like systems

Figure 1. 7"ir as a function of time. Left panel: a system with v — 1, i.e. highly eccentric orbits. Right panel: a system with v — 0.1, i.e. low-eccentricity orbits. Different curves are for different initial masses of the comet cloud. From bottom to top 0.1, 1, 10, 100, 1000 Mffi.

Figure 1. 7"ir as a function of time. Left panel: a system with v — 1, i.e. highly eccentric orbits. Right panel: a system with v — 0.1, i.e. low-eccentricity orbits. Different curves are for different initial masses of the comet cloud. From bottom to top 0.1, 1, 10, 100, 1000 Mffi.

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