Analytical Potentials for Barred Galaxies

We present simple, analytical, prolate, and triaxial potential-density pairs in three dimensions. The potentials are convolutions of the density of a thin "needle" with a softened potential; the density is everywhere positive. Each choice of softening model and needle density generates a family of models that vary between the extremes of a needle and a sphere. With appropriate choices of the needle density and softening model, the potential can be found in terms of elementary functions. The softening potential need not be spherically symmetric; triaxial models can be constructed in this way. We derive a prolate model that reduces to the Plummer sphere and a triaxial model that reduces to the Miyamoto-Nagai disk. Formulae are given for the forces and projected surface densities.