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

The evolution of micrometer-sized circumstellar grains orbiting beta P ictoris is studied, taking into account the combined effects of first order resonances due to a hypothetical planet and the dissipative effe ct due to Poynting-Robertson drag. We first derive the averaged equati ons of motion of the grain near a resonance, and we describe qualitati vely and quantitatively the capture into the resonance (mechanism of e ntrance, time scales for capture, etc.). It appears that the probabili ty of capture cannot be derived analytically, because of the nonadiaba ticity of the motion at the entrance into the resonance, at least for micrometer-sized particles and planet masses smaller than about one sa turnian mass. We show that the capture of a grain into a resonance cri tically depends (i) on the orbital eccentricity and (ii) on the value of the critical argument of resonance just at the entrance into the re sonance. Maps of capture/noncapture regions vs these two parameters ar e 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 la rger planets are able to trap most of the grains into the 1:2 resonanc e, while approximately 5 Earth masses are sufficient to trap grains in to 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) 1 994 Academic Press, Inc.