Axisymmetric three-integral models for galaxies

We describe an improved, practical method for constructing galaxy models th at match an arbitrary set of observational constraints, without prior assum ptions about the phase-space distribution function (DF). Our method is an e xtension of Schwarzschild's orbit superposition technique. As in Schwarzsch ild's original implementation, we compute a representative library of orbit s in a given potential. We then project each orbit onto the space of observ ables, 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 th e observed photometry and kinematics of the galaxy. A new element of this w ork 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 da rk halo) can easily be included in the models. In an earlier paper (Rix et al.) we described the basic principles and impl emented them for the simplest case of spherical geometry. Here we focus on the axisymmetric case. We first show how to build galaxy models from indivi dual 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 o f alternative building blocks, the two-integral and the isotropic component s, for which the observable properties can be computed analytically. Models built entirely from the two-integral components yield DFs of the form f(E, L-z), which depend only on the energy E and angular momentum L-z. This pro vides a new method to construct such models. The smoothness of the two-inte gral and isotropic components also makes them convenient to use in conjunct ion with the regular orbits. We have tested our method by using it to reconstruct the properties of a tw o-integral model built with independent software. The test model is reprodu ced satisfactorily, either with the regular orbits, or with the two-integra l components. This paper mainly deals with the technical aspects of the met hod, while applications to the galaxies M32 and NGC 4342 are described else where (van der Marel et al.; Cretton & van den Bosch).