Constraining the Mass Profiles of Stellar Systems: Schwarzschild Modeling of Discrete Velocity Data Sets

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Abstract
(ABRIDGED) We present a new Schwarzschild orbit-superposition code designed to model discrete datasets composed of velocities of individual kinematic tracers in a dynamical system. This constitutes an extension of previous implementations that can only address continuous data in the form of (the moments of) velocity distributions, thus avoiding potentially important losses of information due to data binning. Furthermore, the code can handle any combination of available velocity components, i.e., only line-of-sight velocities, only proper motions, or a combination of both. It can also handle a combination of discrete and continuous data. The code finds the distribution function (DF, a function of the three integrals of motion E, Lz, and I3) that best reproduces the available kinematic and photometric observations in a given axisymmetric gravitational potential. The fully numerical approach ensures considerable freedom on the form of the DF f(E,Lz,I3). This allows a very general modeling of the orbital structure, thus avoiding restrictive assumptions about the degree of (an)isotropy of the orbits. We describe the implementation of the discrete code and present a series of tests of its performance based on the modeling of simulated datasets generated from a known DF. We find that the discrete Schwarzschild code recovers the original orbital structure, M/L ratios, and inclination of the input datasets to satisfactory accuracy, as quantified by various statistics. The code will be valuable, e.g., for modeling stellar motions in Galactic globular clusters, and those of individual stars, planetary nebulae, or globular clusters in nearby galaxies. This can shed new light on the total mass distributions of these systems, with central black holes and dark matter halos being of particular interest.