Axisymmetric Three‐Integral Models for Galaxies

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Abstract
We describe an improved, practical method for constructing galaxy models that match an arbitrary set of observational constraints, without prior assumptions about the phase-space distribution function (DF). Our method is an extension of Schwarzschild's orbit superposition technique. As in Schwarzschild's original implementation, we compute a representative library of orbits in a given potential. We then project each orbit onto the space of observables, consisting of position on the sky and line-of-sight velocity, while properly taking into account seeing convolution and pixel binning. We find the combination of orbits that produces a dynamical model that best fits the observed photometry and kinematics of the galaxy. A new element of this work is the ability to predict and match to the data the full line-of-sight velocity profile shapes. A dark component (such as a black hole and/or a dark halo) can easily be included in the models. In an earlier paper (Rix et al.) we described the basic principles and implemented them for the simplest case of spherical geometry. Here we focus on the axisymmetric case. We first show how to build galaxy models from individual orbits. This provides a method to build models with fully general DFs, without the need for analytic integrals of motion. We then discuss a set of alternative building blocks, the two-integral and the isotropic components, for which the observable properties can be computed analytically. Models built entirely from the two-integral components yield DFs of the form f(E, Lz), which depend only on the energy E and angular momentum Lz. This provides a new method to construct such models. The smoothness of the two-integral and isotropic components also makes them convenient to use in conjunction with the regular orbits. We have tested our method by using it to reconstruct the properties of a two-integral model built with independent software. The test model is reproduced satisfactorily, either with the regular orbits, or with the two-integral components. This paper mainly deals with the technical aspects of the method, while applications to the galaxies M32 and NGC 4342 are described elsewhere (van der Marel et al.; Cretton & van den Bosch).