Spirals and bars in linear theory

Summary

Introduction

We have recently shown (Tagger et al. 1989a,b and references therein) that the linear theory of density waves in a flat self-gravitating disc contains, in addition to the usual tightly wound waves, another type of perturbation which is essentially bar-like and which dominates the mode structure in the vicinity of the co-rotation radius. We briefly discuss this analytical result, which can be illustrated by numerical calculations, and by its relationship to the disc response to external forcing.

Two different descriptions of density waves have been used in the past: steady waves and shearing perturbations. The difficulty of a unique description stems from the flat disc geometry where the solution of the Poisson equation in the vertical dimension involves an integral operator. The WKBJ approximation, in practice the assumption of tightly wound spirals, allows us to calculate waves with well defined physical properties and has the important advantage of incorporating a properly defined boundary condition at infinity, but it cannot be used to describe the efficient swing amplification mechanism.

Spirals and Bars

Swing amplification is most simply described in the shearing sheet model, where the relevant equations can be Fourier transformed very easily. The solution φ(k) can be easily computed for “large” radial wavenumber k, but we found that a problem arises when one transforms back to real space. It has already been noted that when one computes the inverse Fourier transform the integrand oscillates rapidly at large k, except at saddle points K_j. The contributions of these saddle points C_i exp(ik_jx) to φ(x) can be identified with the usual short and long, leading and trailing spiral waves (Goldreich & Tremaine 1978).