Spiral structure generated by major planets in proto-planetary disks: the role of periodic orbits near resonance

Abstract

In this paper I describe numerical calculations of the motion of particles in a disk about a solar-mass object perturbed by a planet on a circular orbit with mass greater than 0.001 of the stellar mass. A simple algorithm for simulating bulk viscosity is added to the ensemble of particles, and the response of the disk is followed for several planet orbital periods. A two-arm spiral structure forms near the inner resonance (2-1) and extends to the planetary orbit radius (corotation). In the same way for gaseous disks on a galactic scale perturbed by a weak rotating bar-like distortion, this is shown to be related to the appearance of two perpendicular families of periodic orbits near the resonance combined with dissipation which inhibits the crossing of streamlines. Spiral density enhancements result from the crowding of streamlines due to the gradual shift between families. The results, such as the dependence of pitch-angle on radius and the asymmetry of the spiral features, resemble those of sophisticated calculations that include more physical effects. The morphology of structure generated in this way clearly resembles that observed in objects with well-defined two-arm spirals, such as SAO 206462. This illustrates that the process of spiral formation via interaction with planets in such disks can be due to orbital motion in a perturbed Keplerian field combined with kinematic viscosity.}