Dust production from collisions in extrasolar planetary systems

Dust particles observed in extrasolar planetary discs originate from undetectable km-sized bodies but this valuable information remains uninteresting if the theoretical link between grains and planetesimals is not properly known. We outline in this paper a numerical approach we developed in order to address this issue for the case of dust producing collisional cascades. The model is based on a particle-in-a-box method. We follow the size distribution of particles over eight orders of magnitude in radius taking into account fragmentation and cratering according to different prescriptions. A very particular attention is paid to the smallest particles, close to the radiation pressure induced cut-off size Rpr, which are placed on highly eccentric orbits by the stellar radiation pressure. We applied our model to the case of the inner (< 10 AU) β Pictoris disc, in order to quantitatively derive the population of progenitors needed to produce the small amount of dust observed in this region (≃ 10 g). Our simulations show that the collisional cascade from kilometre-sized bodies to grains significantly departs from the classical dN ∝ RdR power law: the smallest particles (R ≃ Rpr) are strongly depleted while an overabundance of grains with size ∼ 2Rpr and a drop of grains with size ∼ 100Rpr develop regardless of disc’s dynamical excitation, Rpr and initial surface density. However, the global dust to planetesimal mass ratio remains close to its dN ∝ RdR value. Our rigorous approach thus confirms the depletion in mass in the inner β Pictoris disc initially inferred from questionable assumptions. We show moreover that collisions are a sufficient source of dust in the inner β Pictoris disc. They are actually unavoidable even when considering the alternative scenario of dust production by slow evaporation of km-sized bodies. We obtain an upper limit of ∼ 0.1M⊕ for the total disc mass below 10AU. This upper limit is not consistent with the independent mass estimate (at least 15M⊕) in the frame of the Falling Evaporating Bodies (FEB) scenario explaining the observed transient features activity. Furthermore, we show that the mass required to sustain the FEB activity implies a so important mass loss that the phenomena should naturally end in less than 1Myr, namely in less than one twentieth the age of the star (at least 2 10 years). In conclusion, these results might help converge towards a coherent picture of the inner β Pictoris system: a low-mass disc of collisional debris leftover after the possible formation of planetary embryos, a result which would be coherent with the estimated age of the system.