Stability and Structure of Spiral Galaxies with FiniteThickness

Abstract

We calculate the effect of finite disc thickness on the structure andstability of a differentially rotating three-dimensional spiral galaxywith stars and gases. The problem is solved exactly in terms of therigorous mathematical method of Green function and Bessel–Fouriertransform. By using the appropriate three-dimensionalself-gravitational potential due to the density perturbations of thespiral arms in the galactic disc, the dispersion relation for thequasi-stationary density waves are deduced for the composite disc modelof stars and gases with finite thickness. The stability criteriondepicted by the Toomre Q parameter for these spiral modes areinvestigated in detail. In contrast to the standard Q parameter foran infinitely thin two-dimensional disc, our generalized Toomre Qparameter depends explicitly on the disc thickness and the location. Inthe limit of an infinitely thin disc, our result naturally reduces toToomre's criterion for a stellar disc and to Kalnajs's result for agaseous disc. In addition, the significant role played by the finitethickness of the galactic disc on the spatial structure of the stellararms and the gaseous arms in the galactic plane is also brieflydiscussed. Application of our theory to the disc models with twostellar arms and four gaseous arms, gas content 15% and the totalsurface density in the solar neighbourhood σ = 52M0 pc−2 but with the thickness parameter α in the range of 3.0<α<4.0 (250 pc <hz<325 pc) appropriate to our Galaxyyields good results in general agreement with recent observations.