IS THERE A PLANET AROUND BETA-PICTORIS - PERTURBATIONS OF A PLANET ON A CIRCUMSTELLAR DUST DISK .2. THE ANALYTICAL MODEL

The evolution of micrometer-sized circumstellar grains orbiting beta Pictoris is studied, taking into account the combined effects of first order resonances due to a hypothetical planet and the dissipative effect due to Poynting-Robertson drag. We first derive the averaged equations of motion of the grain near a resonance, and we describe qualitatively and quantitatively the capture into the resonance (mechanism of entrance, time scales for capture, etc.). It appears that the probability of capture cannot be derived analytically, because of the nonadiabaticity of the motion at the entrance into the resonance, at least for micrometer-sized particles and planet masses smaller than about one saturnian mass. We show that the capture of a grain into a resonance critically depends (i) on the orbital eccentricity and (ii) on the value of the critical argument of resonance just at the entrance into the resonance. Maps of capture/noncapture regions vs these two parameters are derived numerically for the 1:2, 2:3, and 3:4 resonances. They show the complexity of the capture regions, and indicate that uranian or larger planets are able to trap most of the grains into the 1:2 resonance, while approximately 5 Earth masses are sufficient to trap grains into the 3:4 resonance for any grain with initial eccentricities smaller than a few percent. These results underline the dynamical importance of small planetary objects embedded in circumstellar dust disks. (C) 1994 Academic Press, Inc.