Exact density-potential pairs from complex shifted axisymmetric systems

TL;DR

This paper extends the complex-shift method to generate new analytical axisymmetric and triaxial density-potential pairs from known models, with applications to galaxy structures and new mathematical functions.

Contribution

It introduces an extended complex-shift technique to derive new density-potential pairs from specific galaxy models and identifies limitations of the method for certain discs.

Findings

Miyamoto-Nagai discs can model peanut-shaped bulges or bars.

The method cannot generate certain discs from spherical distributions.

Two new closed-form generating functions for Legendre polynomials were discovered.

Abstract

In a previous paper the complex-shift method has been applied to self-gravitating spherical systems, producing new analytical axisymmetric density-potential pairs. We now extend the treatement to the Miyamoto-Nagai disc and to the Binney logarithmic halo, and we study the resulting axisymmetric and triaxial analytical density-potential pairs; we also show how to obtain the surface density of shifted systems from the complex-shift of the surface density of the parent model. In particular, the systems obtained from Miyamoto-Nagai discs can be used to describe disc galaxies with a peanut-shaped bulge or with a central triaxial bar, depending on the direction of the shift vector. By using a constructive method that can be applied to generic axisymmetric systems, we finally show that the Miyamoto-Nagai and the Satoh discs, and the Binney logarithmic halo, cannot be obtained from the complex-shift of any spherical parent distribution. As a by-product of this study we also found two new generating functions in closed form for even and odd Legendre polynomials, respectively.