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 10 4 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.