Figure 1 shows the time evolution of tir for several different cases. We have used different initial disk masses (the total mass of comets in the disk). Also we have performed calculations with different collision velocities. High collision velocities indicate that the orbits of the comets are highly eccentric and/or inclined. Such orbits lead to a small sweeping time ts and therefore to a short lifetime of the comets. The left pane in Figure 1 assumes collision velocities of the same order as the Kepler velocities at 50 AU from the star. The resulting decrease in tir agrees well with a lifetime of a few times 108 years. Note also that the different initial masses converge to the same solution after about 108 years. This result could already be seen from eq. (3). It is therefore clear that the old stars with Vega-like disks cannot be explained by a higher initial mass of the disk. The destruction time is the same for all masses.
However, the right panel in Figure 1 shows a similar calculation for collision velocities which are only one tenth of the Kepler velocities at 50 AU from the star. It is immediately evident that these disks last for much longer and that in fact less initial mass is required. Only 1M® of comets is needed to reproduce the observed values of tir ss 10-4. Higher initial masses at these low collisional velocities are in fact inconsistent with the observations.
3.1. Stellar Collisions?
Another possibility to explain a Vega-like excess of old stars is to assume that the star has a part of the disk which has been quiescent for most of its life, but only recently became active. It is known from dynamical calculations that any Kuiper Belt in the solar system outside 50 AU would have remained undisturbed by the gravitational influence of
Neptune and could have survived in a pristine state, with nearly circular and collision-free orbits.
How could such a disk be disturbed into a state where it produces copious amounts of dust? A possibility is that of a stellar encounter. Even though direct stellar collisions do not occur at the stellar densities in the solar neighbourhood, encounters within d = 200 AU can happen and have been suggested to explain the structure of the ¡3 Pictoris disk -
The time between collisions of two stellar systems is given by
1 ^ lpc"3 /200AU\2 30km/s te =-= llGyr—- -;— -— 4)
where n* is the stellar density in the solar neighbourhood, < v > is the average relative velocity between stars and d is the maximum encounter distance at which the disturbing influence of the passing star is sufficient to stir up the cometary cloud.
Using eq. (4) at face value, we find with reasonable assumptions for the stellar density and the required encounter distance that about 5% of all 10 Gyr old stars have had such an encounter within the last 500 Myrs. The density of stellar objects in the solar neighbourhood is at least 0.1 pc-3, but maybe higher due to an unknown number of brown dwarfs. It is, therefore, not entirely unreasonable to conclude that one or two of the old Vega-like stars discovered in the ISO study may be due to such close encounters.
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