Axisymmetric Three‐Integral Models for Galaxies

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 SchwarzschildÏs orbit superposition technique. As in SchwarzschildÏs 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 Ðnd the combination of orbits that produces a dynamical model that best Ðts 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 proÐle 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 Ðrst 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 twointegral 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, which depend L z ), only on the energy E and angular momentum This provides a new method to construct such models. L z . 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).

The Halo Mass Distribution of Field and Cluster Early-Type Galaxies

“…The no dark halo model is strongly excluded by the data. The best fitting model has an orbital structure which is slightly radial throughout most of the galaxy (similar to what Cretton et al (1999) find in their sample) and then becomes tangential at large radii.…”

Orbit-based dynamical models of the Sculptor dSph galaxy

“…Note however that for this non-parametric problem, the parameter space is very large, and a solution will often yield an unrealistically spiky df. To effectively reduce the parameter space and yield a smoother df, we add a regularisation constraint, in analogy to Cretton et al (1999) and , by including a penalty term to the total χ 2 . This term has the form:…”

“…Since then many codes have been developed (e.g. Richstone & Tremaine 1984;Rix et al 1997;van der Marel et al 1998;Cretton et al 1999;Valluri et al 2004;. While first only the lowest moments of the line of sight velocity distribution (mean velocity and velocity dispersion) were fitted, better data have led to the inclusion of higher moments in the fits.…”

An iterative method for the construction of N-body galaxy models in collisionless equilibrium

“…In the latter case, finding such models is a main component of the reverse-engineering process aimed at constructing self-consistent three-dimensional systems that reproduce the observations. They can then be examined in great detail, allowing insights into properties that are not directly observable (Cretton et al 1999; van den Bosch et al 2012). …”