Dynamical Modeling of Velocity Profiles: The Dark Halo around the Elliptical Galaxy NGC 2434

We describe a powerful technique to model and interpret the stellar line-of-sight velocity profiles of galaxies. It is based on Schwarzschild's approach to build fully general dynamical models. A representative library of orbits is calculated in a given potential, and the non-negative superposition of these orbits is determined that best fits a given set of observational constraints. The most significant new feature of our implementation is that we calculate and fit the full velocity profile shapes, represented by a Gauss-Hermite series. This allows us to constrain the orbital anisotropy in the fit. We also use an objective χ2 measure for the quality of fit, taking into account the error on each observational constraint. Given χ2 from the observational constraints, the technique assesses the relative likelihood of different orbit combinations in a given potential, and of models with different potentials. In our implementation only projected, observable quantities are included in the fit, aperture binning and seeing convolution of the data are properly taken into account, and smoothness of the models in phase space can be enforced through regularization. This scheme is valid for any geometry. In a first application of this method, we focus here on spherical geometry; axisymmetric modeling is described in companion papers by Cretton et al. and van der Marel et al. We test the scheme on pseudo-data drawn from an isotropic Hernquist model and then apply it to the issue of dark halos around elliptical galaxies. We model radially extended stellar kinematical data for the E0 galaxy NGC 2434 obtained by Carollo et al. This galaxy was chosen because it may be nearly round, in which case the present spherical modeling is applicable. Models with constant mass-to-light ratio are clearly ruled out, regardless of the orbital anisotropy. To study the amount of dark matter needed to match the data, we considered a sequence of cosmologically motivated "star + halo" potentials. These potentials are based on the CDM simulations by Navarro et al., but also account for the accumulation of baryonic matter; they are specified by the stellar mass-to-light ratio *,B and the characteristic halo velocity, V200. The star + halo models provide an excellent fit to the data, with *,B = 4.35 ± 0.35 (in B-band solar units) and V200 = 450 ± 100 km s-1. The best-fitting potential has a circular velocity Vc that is constant to within ~10% between 0.2 and 3 effective radii and is very similar to the best-fitting logarithmic potential, which has Vc = 300 ± 15 km s-1. In NGC 2434 roughly half of the mass within an effective radius is dark. In comparison, our models without a dark halo estimate a mass-to-light ratio for the stellar population that is twice as large. If NGC 2434 is a significantly flattened system seen nearly face-on, it would be considerably more difficult to limit the gravitational potential without further observational constraints.