Minimum-entropy constraints on galactic potentials

A tracer sample in a gravitational potential, starting from a generic initial condition, phase-mixes towards a stationary state. This evolution is accompanied by an entropy increase, and the final state is characterized by a distribution function (DF) that depends only on integrals of motion (Jeans’ theorem). We present a method to constrain a gravitational potential assuming a stationary (phase mixed) sample by minimizing the entropy the sample would have if it were allowed to phase-mix in trial potentials. This method avoids modeling the DF, and is applicable to any sets of integrals. We provide expressions for the entropy of DFs depending on energy, $f(E)$, energy and angular momentum, $f(E,L)$, or three actions, $f(\vec{J})$, and investigate the bias and statistical uncertainties in their estimates. We show that the method correctly recovers the parameters for spherical and axisymmetric potentials. We also present a methodology to characterize the posterior probability distribution of the parameters with an Approximate Bayesian Computation, indicating a pathway for application to observational data. Using ${10}^4$ tracers with $10\mathrm{\%}(20\mathrm{\%})$-uncertainties in the 6D coordinates, we recover the flattening parameter $q$ of an axisymmetric potential with ${\sigma }_q\mathrm{/}q\sim 5\mathrm{\%}(10\mathrm{\%})$. The python module for the entropy estimators, $\mathtt{\text{tropygal}}$, is made publicly available.