An empirical formula for the distribution function of a thin exponential disc

TL;DR

This paper introduces an empirical formula for the Shu distribution function that accurately models thin exponential galactic discs, simplifying computations for large surveys and simulations.

Contribution

It presents a new empirical formula with two free parameters that reproduces exponential surface density profiles without iterative algorithms.

Findings

Works for flat, rising, and falling rotation curves

Enables efficient model fitting for large datasets

Extends to velocity dispersion profiles as exponential functions

Abstract

An empirical formula for a Shu distribution function that reproduces a thin disc with exponential surface density to good accuracy is presented. The formula has two free parameters that specify the functional form of the velocity dispersion. Conventionally, this requires the use of an iterative algorithm to produce the correct solution, which is computationally taxing for applications like Markov Chain Monte Carlo (MCMC) model fitting. The formula has been shown to work for flat, rising and falling rotation curves. Application of this methodology to one of the Dehnen distribution functions is also shown. Finally, an extension of this formula to reproduce velocity dispersion profiles that are an exponential function of radius is also presented. Our empirical formula should greatly aid the efficient comparison of disc models with large stellar surveys or N-body simulations.