Stability and Structure of Spiral Galaxies with Finite Thickness

We calculate the effect of finite disc thickness on the structure and stability of a differentially rotating three-dimensional spiral galaxy with stars and gases. The problem is solved exactly in terms of the rigorous mathematical method of Green function and Bessel--Fourier transform. By using the appropriate three dimensional self-gravitational potential due to the density perturbations of the spiral arms in the galactic disc, the dispersion relation for the quasi-stationary density waves are deduced for the composite disc model
of stars and gases with finite thickness. The stability criterion depicted by the Toomre Q parameter for these spiral modes are investigated in detail. In contrast to the standard Q parameter for an infinitely thin two-dimensional disc, our generalized Toomre Q parameter depends explicitly on the disc thickness and the location. In the limit of an infinitely thin disc, our result naturally reduces to Toomre’s criterion for a stellar disc and to Kalnajs’s result for a gaseous disc. In addition, the significant role played by the finite thickness of the galactic disc on the spatial structure of the stellar arms and the gaseous arms in the galactic plane is also briefly discussed. Application of our theory to the disc models with two stellar arms and four gaseous arms, gas content 15% and the total surface density in the solar neighbourhood σ = 52M₀pc^-2 but with the thickness parameter α in the range of 3.0 < α < 4.0 (250pc < h_z < 325pc) appropriate to our Galaxy yields good results in general agreement with recent observations.