Fractional Derivative Approach to the Self-gravitation Equation

TL;DR

This paper introduces a fractional derivative formalism for deriving equilibrium distribution functions in axisymmetric systems, broadening applicability to various models including 3D and flat systems.

Contribution

It generalizes existing methods by using fractional derivatives, allowing analysis of a wider range of models without pseudo-volume density complications.

Findings

Derived distribution functions for Binney's logarithmic model

Obtained distribution functions for the Mestel disc

Demonstrated applicability to both 3D and flat systems

Abstract

A new formalism is presented for finding equilibrium distribution functions for axisymmetric systems. The formalism, obtainded by using the concept of fractional derivatives, generalizes the methods of Fricke (1952), Kalnajs (1972) and Jiang & Ossipkov (2007), and has the advantage that can be applied to a wider variety of models. We found that this approach can be applied both to tridimensional systems and to flat systems, without the necessity of dealing with a pseudo-volume mass density. As an application, we obtain the distribution functions of the Binney's logarithmic model and of the Mestel disc.