The orbital PDF: general inference of the gravitational potential from steady-state tracers

TL;DR

This paper introduces two new methods based on the orbital probability density function (oPDF) to infer the gravitational potential from steady-state tracers, applicable to spherical and potentially non-spherical systems, without assuming specific distribution functions.

Contribution

The paper develops and tests two novel, assumption-free methods for gravitational potential inference using steady-state tracers, based on the oPDF, applicable to arbitrary orbit distributions.

Findings

Methods are unbiased and efficient in dark matter haloes.

The likelihood estimator fits analytical potentials accurately.

The phase-mark method reconstructs potential profiles non-parametrically.

Abstract

We develop two general methods to infer the gravitational potential of a system using steady-state tracers, i.e., tracers with a time-independent phase-space distribution. Combined with the phase-space continuity equation, the time independence implies a universal Orbital Probability Density Function (oPDF) $\mathrm{d} P(\lambda|{\rm orbit})\propto \mathrm{d} t$, where $\lambda$ is the coordinate of the particle along the orbit. The oPDF is equivalent to Jeans theorem, and is the key physical ingredient behind most dynamical modelling of steady-state tracers. In the case of a spherical potential, we develop a likelihood estimator that fits analytical potentials to the system, and a non-parametric method ("phase-mark") that reconstructs the potential profile, both assuming only the oPDF. The methods involve no extra assumptions about the tracer distribution function and can be applied to tracers with any arbitrary distribution of orbits, with possible extension to non-spherical potentials. The methods are tested on Monte Carlo samples of steady-state tracers in dark matter haloes to show that they are unbiased as well as efficient. A fully documented \textsc{C/Python} code implementing our method is freely available at a GitHub repository linked from \url{http://icc.dur.ac.uk/data/#oPDF}.