Self-consistent dynamical models with a finite extent -- IV. Wendland models based on compactly supported radial basis functions

TL;DR

This paper introduces Wendland models based on compactly supported radial basis functions for self-consistent dynamical models with finite extent, supporting diverse orbital structures and offering analytical simplicity.

Contribution

It presents the first family of models supporting both radial and tangential Osipkov--Merritt distribution functions with analytical density and potential profiles.

Findings

All Wendland models can be supported by isotropic distribution functions.

Models support a continuum of Osipkov--Merritt orbital structures.

Models are characterized by a parameter controlling smoothness at the truncation radius.

Abstract

We present a new step in our systematic effort to develop self-consistent dynamical models with a finite radial extent. The focus is on models with simple analytical density profiles allowing for analytical calculations of many dynamical properties. In this paper, we introduce a family of models, termed Wendland models, based on compactly supported radial basis functions. The family of models is characterised by a parameter $k$ that controls the smoothness of the transition at the truncation radius. In the limit $k\to\infty$, the Wendland model reduces to a non-truncated model with a Gaussian density profile. For each Wendland model, the density, mass and gravitational potential are simple truncated polynomial functions of radius. Via the SpheCow tool we demonstrate that all Wendland models can be supported by isotropic distribution functions. Surprisingly, the isotropic distribution function exhibits varied behaviour across different Wendland models. Additionally, each model can be supported by a continuum of Osipkov--Merritt orbital structures, ranging from radially anisotropic to completely tangential at the truncation radius. To the best of our knowledge, the Wendland models presented here are the first family of models accommodating both radial and tangential Osipkov--Merritt distribution functions. Using linear superposition, these models can easily be combined to generate Wendland models with even more diverse orbital structures. While the Wendland models are not fully representative of real dynamical systems due to their Gaussian-like density profile, this study lays important groundwork for constructing more realistic models with truncated density profiles that can be supported by a range of orbital structures.