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. Our implementation incorporates several new features: (i) we calculate velocity profiles and represent them by a Gauss-Hermite series. This allows us to constrain the orbital anisotropy in the fit. (ii) we take into account the error on each observational constraint to obtain an objective χ 2 measure for the quality-of-fit. Given 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. Constant mass-to-light ratio models 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 Υ * ,B and the characteristic halo velocity, V 200 . The star+halo models provide an excellent fit to the data, with Υ * ,B = 3.35 ± 0.25 (in B-band solar units) and V 200 = 450 ± 100 km/s. The best-fitting potential has a circular velocity V c that is constant to within ∼ 10% between 0.2-3 effective radii and is very similar to the best-fitting logarithmic potential, which has V c = 300 ± 15 km s −1 . In NGC 2434 roughly half of the mass within an effective radius is dark. Models without a dark halo overestimate the mass-to-light ratio of the stellar population by a factor of ∼ 2.

Axisymmetric dynamical models are constructed for the E3 galaxy M32 to interpret high spatial resolution stellar kinematical data obtained with the Hubble Space Telescope (HST). Models are studied with two-integral, f (E, L z ), phase-space distribution functions, and with fully general three-integral distribution functions. The latter are built using an extension of Schwarzschild's approach: individual orbits in the axisymmetric potential are calculated numerically, and populated using non-negative least-squares fitting so as to reproduce all available kinematical data, including line-of-sight velocity profile shapes. The details of this method are described in companion papers by Rix et al. and Cretton et al.Models are constructed for inclinations i = 90 • (edge-on) and i = 55 • . No model without a nuclear dark object can fit the combined ground-based and HST data, independent of the dynamical structure of M32. Models with a nuclear dark object of mass M • = 3.4 × 10 6 M ⊙ (with 1σ and 3σ error bars of 0.7 × 10 6 M ⊙ and 1.6 × 10 6 M ⊙ , respectively) do provide an excellent fit. The inclined models provide the best fit, but the inferred M • does not depend sensitively on the assumed inclination. The models that best fit the data are not two-integral models, but like two-integral models they are azimuthally anisotropic. Two-integral models therefore provide useful low-order approximations to the dynamical structure of M32. We use them to show that an extended dark object can fit the data only if its half-mass radius is r h ∼ < 0.08 ′′ (= 0.26 pc), implying a central dark matter density exceeding 1 × 10 8 M ⊙ pc −3 .The inferred M • is consistent with that suggested previously by ground-based kinematical data. However, radially anisotropic axisymmetric constant mass-to-light ratio models are now ruled out for the first time, and the limit on the dark matter density implied by the HST data is now stringent enough to rule out most plausible alternatives to a massive black hole. Thus, the evidence for a massive black hole in the quiescent galaxy M32 is now very compelling.The dynamically inferred M • is identical to that suggested by existing models for HST photometry of M32 that assume adiabatic growth (over a time scale exceeding 10 6 yr) of a black hole into a pre-existing core. The low activity of the nucleus of M32 implies either that only a very small fraction of the gas that is shed by evolving stars is accreted onto the black hole, or alternatively, that accretion proceeds at very low efficiency, e.g. in an advection-dominated mode.

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).

We have constructed axisymmetric dynamical models of the edge-on S0 galaxy NGC 4342: simple two-integral Jeans models as well as fully general, three-integral models using a modified version of Schwarzschild's orbit superposition technique. The two-integral models suggest a black hole (BH) of 3 or $6\times 10^8 M_\odot$, depending on the data set. The three-integral models can fit all ground-based and HST data simultaneously, but only when a central BH is included. Models without BH are ruled out at better than 99.73% confidence level. We determine a BH mass of $3.0^{+1.7}_{-1.0} \times 10^8 M_\odot$. This corresponds to 2.6% of the bulge mass, making NGC 4342 one of the galaxies with the highest BH mass to bulge mass ratio currently known.Comment: 4 pages, 3 figures (2 color). Contributed Talk at "Galaxy Dynamics", Rutgers University, New Jersey, August 8-12, 199

We present direct observational constraints on the orbital distribution of the stars in the giant elliptical NGC 2320. Long-slit spectra along multiple position angles are used to derive the stellar line-of-sight velocity distribution within one effective radius. In addition, the rotation curve and dispersion profile of an ionized gas disk are measured from the [OIII] emission lines. After correcting for the asymmetric drift, we derive the circular velocity of the gas, which provides an independent constraint on the gravitational potential.To interpret the stellar motions, we build axisymmetric three-integral dynamical models based on an extension of the Schwarzschild orbit-superposition technique. We consider two families of gravitational potential, one in which the mass follows the light (i.e., no dark matter) and one with a logarithmic gravitational potential.Using χ 2 -statistics, we compare our models to both the stellar and gas data to constrain the value of the V-band mass-to-light ratio Υ V . We find Υ V = 15.0 ± 0.6 h 75 for the mass-follows-light models and Υ V = 17.0 ± 0.7 h 75 for the logarithmic models. For the latter, Υ V is defined within a sphere of 15 ′′ radius.Models with radially constant Υ V and logarithmic models with dark matter provide comparably good fits to the data and possess similar dynamical structure. Across the full range of Υ V permitted by the observational constraints, the models are radially anisotropic in the equatorial plane over the radial range of our kinematical data (1 ′′ ∼ < r ∼ < 40 ′′ ). Along the true minor axis, they are more nearly isotropic. The best fitting model has σ r /σ total ≃ 0.7, σ φ /σ total ≃ 0.5 − 0.6 and σ θ /σ total ≃ 0.5 in the equatorial plane.