A two-parameter family of double-power-law biorthonormal potential-density expansions

TL;DR

This paper introduces a versatile two-parameter family of biorthonormal potential-density expansions with closed-form expressions, unifying many known models and including the NFW profile, useful for galactic dynamics and cosmological simulations.

Contribution

It presents a new two-parameter family of biorthonormal expansions that generalize and encompass previous models, with closed-form solutions and a systematic methodology for deriving such expansions.

Findings

Includes all known analytic biorthonormal expansions.

Contains models based on common spherical density profiles.

Reproduces the NFW profile at zeroth order.

Abstract

Biorthonormal basis function expansions are widely used in galactic dynamics, both to study problems in galactic stability and to provide numerical algorithms to evolve collisionless stellar systems. They also provide a compact and efficient description of the structure of numerical dark matter haloes in cosmological simulations. We present a two-parameter family of biorthonormal double-power-law potential-density expansions. Both the potential and density are given in closed analytic form and may be rapidly computed via recurrence relations. We show that this family encompasses all the known analytic biorthonormal expansions: the Zhao expansions (themselves generalizations of ones found earlier by Hernquist & Ostriker and by Clutton-Brock) and the recently discovered Lilley, Sanders, Evans & Erkal expansion. Our new two-parameter family includes expansions based around many familiar spherical density profiles as zeroth-order models, including the $\gamma$ models and the Jaffe model. It also contains a basis expansion that reproduces the famous Navarro-Frenk-White (NFW) profile at zeroth order. The new basis expansions have been found via a systematic methodology which has wide applications in finding further examples. In the process, we also uncovered a novel integral transform solution to Poisson's equation.