Minimum-entropy Constraints on Galactic Potentials

A tracer sample in a gravitational potential, starting from a generic initial condition, phase-mixes toward 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 that 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(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 10⁴ tracers with 10%(20%) uncertainties in the 6D coordinates, we recover the flattening parameter q of an axisymmetric potential with σ_q/q ∼ 5%(10%). The python module for the entropy estimators, tropygal, is made publicly available.