The structure and stability of disk galaxies with finite thickness

In this paper we calculate the effect of finite disk thickness on the structure and stability of a differentially rotating three-dimensional disk galaxy with stars and gases. The problem is solved exactly in terms of the rigorous mathematical method of Green function and Bessel-Fourier transform. In order to bring out the physical significance of the effect of finite thickness clearly, we concentrate on the axially symmetric modes with azimuthal symmetry. The corresponding reduction factor for these modes are investigated in detail. In this way we avoid the WKB approximation generally adopted for the spiral modes. Our reduction factor depends not only on the surface density and the epicyclic frequency (which depicts the effect of differential rotation) but also explicitly on the disk thickness and the location. In the limit of an infinitely thin disk, our result naturally reduces to Toomre's criterion for a stellar disk and to Kalnajs's result for a gases disk. In addition, the significant role played by the finite thickness of the galactic disk on the mass of the galactic halo required for the Milky Way is also briefly discussed in Appendices A and B.