POTENTIAL-DENSITY PAIRS FOR FLAT GALAXIES

The potential-density pairs of an infinite number of families of non-axisymmetric discs are presented. The qth family corresponds to surface density distributions that fall off not faster than r(-2q) at large radii r. The special case of q=3/2 comprises the set of Stackel discs. The families are all generated by superposing a class of exactly elliptical surface density distributions, which we designate the elliptic discs. For any chosen short-axis surface density profile, an integral equation for the weight-function - which describes the superposition of the elliptic discs - is solved by Stieltjes transform methods to yield the potential-density pair of the constructed disc as quadratures. This extends and generalizes Newtonian potential theory of discs away from axisymmetry. All the families of discs are shown to satisfy generalized Kuzmin formulae relating the surface density at a general point to the surface density on the short axis. The physical properties of the models as well as the scale-free limit are described. In the axisymmetric limit, the method reduces to decomposing any disc into constituent Toomre discs. As the potential and density of the Toomre discs are elementary, this is simpler than the conventional Fourier-Bessel methods using Hankel transforms. As examples, we investigate a family of axisymmetric discs that includes the Mestel and Kalnajs isochrone discs as special cases.