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

We present a new Schwarzschild orbit superposition code designed to model discrete data sets composed of velocity measurements 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. The code can handle any combination of velocity components, i.e., only line-of-sight velocities, only proper motions, or both. The code determines the combination of orbital mass weights as a function of the three integrals of motion E, L_z, and I₃ that best reproduces, in a maximum likelihood sense, the available kinematic and photometric observations in a given axisymmetric gravitational potential. The fully numerical approach ensures considerable freedom on the form of the distribution function f(E, L_z, I₃) , avoiding restrictive assumptions about the degree of orbital (an)isotropy. We describe the implementation of the discrete code and present a series of performance tests based on the modeling of simulated data sets generated from a known distribution function. Exploring pseudo-data sets with varying degrees of overall rotation and different inclinations on the sky, we study the results as a function of relevant observational variables such as the size of the data set and the type of velocity information available. The discrete Schwarzschild code recovers the original orbital structure, mass-to-light ratio, and inclination of the input data sets to satisfactory accuracy. The code will be valuable, e.g., for modeling stellar motions in Galactic globular clusters and the motions 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 halos being of particular interest.