ON THE POTENTIALS OF GALACTIC DISKS

The standard Bessel function formula for the potential of a thin axisymmetric disc is extended to include arbitrary vertical structure, yielding a formula which reduces the potential to a single quadrature in the important cases of a disc with constant scale-height and either exponential or Gaussian vertical density profiles. The general solution of Poisson's equation for an axisymmetric body is also given as a double integral over a Legendre function. A new formulation for the potential of thin axisymmetric discs is also given. The potential is expressed as a double integral over elementary functions in the most general case, and can usually be reduced to a single quadrature. This yields a more convenient form (from a numerical point of view) for the potentials of discs which are not known completely analytically, including in particular that of the exponential disc. Finally, the potential corresponding to an arbitrary distribution of matter is given as an integral of a four-dimensional Fourier transform, which reduces to a four-dimensional Fourier transform in the case of triaxial discs. The link between the Green's function and Bessel function formulations is shown up explicitly by this formula, which should prove useful in practice for the computation of many triaxial disc potentials. This solution is then illustrated by the analytical calculation of the potential of a particular family of triaxial discs.