Resolving the mass--anisotropy degeneracy of the spherically symmetric Jeans equation II: optimum smoothing and model validation

TL;DR

This paper introduces an optimized smoothing technique combined with MCMC methods to effectively resolve the mass-anisotropy degeneracy in the spherical Jeans equation, validated through synthetic data with accurate recovery of mass profiles.

Contribution

It presents an improved algorithm that uses optimal smoothing and MCMC exploration to uniquely determine mass and anisotropy profiles from observational data.

Findings

Accurately recovers theoretical mass and anisotropy profiles from synthetic data.

Demonstrates robustness of the method across systems with varying mass-to-light ratios.

Achieves excellent fits to observables with unique solutions.

Abstract

The spherical Jeans equation is widely used to estimate the mass content of a stellar systems with apparent spherical symmetry. However, this method suffers from a degeneracy between the assumed mass density and the kinematic anisotropy profile, $\beta(r)$. In a previous work, we laid the theoretical foundations for an algorithm that combines smoothing B-splines with equations from dynamics to remove this degeneracy. Specifically, our method reconstructs a unique kinematic profile of $\sigma_{rr}^2$ and $\sigma_{tt}^2$ for an assumed free functional form of the potential and mass density $(\Phi,\rho)$ and given a set of observed line-of-sight velocity dispersion measurements, $\sigma_{los}^2$. In Paper I (submitted to MNRAS: MN-14-0101-MJ) we demonstrated the efficiency of our algorithm with a very simple example and we commented on the need for optimum smoothing of the B-spline representation; this is in order to avoid unphysical variational behaviour when we have large uncertainty in our data. In the current contribution we present a process of finding the optimum smoothing for a given data set by using information of the behaviour from known ideal theoretical models. Markov Chain Monte Carlo methods are used to explore the degeneracy in the dynamical modelling process. We validate our model through applications to synthetic data for systems with constant or variable mass-to-light ratio $\Upsilon$. In all cases we recover excellent fits of theoretical functions to observables and unique solutions. Our algorithm is a robust method for the removal of the mass-anisotropy degeneracy of the spherically symmetric Jeans equation for an assumed functional form of the mass density.