1 Introduction

From a stellar dynamical point of view, the most complete description of a stellar system is the distribution function $F({\vec{r}},{\vec{v}})$, which gives the probability density for the stars in phase space. In this paper, we will concentrate on the problem of constructing anisotropic equilibrium distribution functions that self-consistently generate a given spherical mass density profile $\rho(r)$. In the assumption of spherical symmetry, the mass density of a stellar system can easily be derived from the observed surface brightness profile, at least if we assume that the mass-to-light ratio is constant and that dust attenuation is negligible. And as the surface brightness of a galaxy (or bulge or cluster) is fairly cheap and straightforward to observe, compared to other dynamical observables which require expensive spectroscopy, the problem we will deal with is relevant and important.

The first step in the construction of self-consistent models is the calculation of the gravitational potential $\psi(r)$, which can immediately be determined through Poisson's equation. The second step, the actual construction of the distribution function, is less straightforward. Basic stellar dynamics theory (see e.g. Binney & Tremaine 1987) learns that steady-state distribution functions for spherical systems can generally be written as a function of binding energy and angular momentum. We hence have to determine a distribution function $F({\cal{E}},L)$, such that the zeroth order moment of this distribution function equals the density, i.e. we have to solve the integral equation

$$\rho(r) = \iiint F({\cal{E}},L)~{\rm d}{\vec{v}}$$ (1)

for $F({\cal{E}},L)$. Hereby we have to take into account that not every function $F({\cal{E}},L)$ that satisfies this equation is a physically acceptable solution: an acceptable solution has to be non-negative over the entire phase space. In general, the problem of solving the integral Eq. (1) is a degenerate problem, because there are infinitely many distribution functions possible for a given potential-density pair.

Particularly interesting are models for which the distribution function and its moments can be computed analytically. Such models have many useful applications, which can roughly be divided into two classes. On the one hand, they can improve our general understanding of physical processes in galaxies in an elegant way. For example, they can serve as simple galaxy models, in which it is easy to generate the starting conditions for N-body or Monte Carlo simulations, or to test new data reduction or dynamical modelling techniques. A quick look at the overwhelming success of simple analytical models, such as the Plummer sphere (Plummer 1911; Dejonghe 1987), the isochrone sphere (Hénon 1959, 1960), the Jaffe model (Jaffe 1983) and the Hernquist model (Hernquist 1990), provides enough evidence. On the other hand, analytical models are also useful for the detailed dynamical modelling of galaxies. For example, in modelling techniques such as the QP technique (Dejonghe 1989), a dynamical model for an observed galaxy is built up as a linear combination of components, for each of which the distribution function and its moments are known analytically. As a result, the distribution function and the moments of the final model are also analytical, which obviously has a number of advantages.

Unfortunately, the number of dynamical models for which the distribution function is known analytically is rather modest. Moreover, most of them consist of distribution functions that are isotropic or of the Osipkov-Merritt type, and therefore basically one-dimensional. An exception is the completely analytical family of anisotropic models described by Dejonghe (1987). These models self-consistently generate the Plummer potential-density pair, a simple yet useful model for systems with a constant density core.

During the last decade, however, it has become clear that, at small radii, elliptical galaxies usually have central density profiles that behave as $r^{-\gamma}$ with $0\leqslant\gamma\leqslant2.5$(Lauer et al. 1995; Gebhardt et al. 1996). Such galaxies can obviously not be adequately modelled with a constant density core. This has stimulated the quest for simple potential-density pairs, and corresponding distribution functions, with a central density cusp. The first effort to construct such models was undertaken by Ciotti (1991) and Ciotti & Lanzoni (1997), who discussed the the dynamical structure of stellar systems following the R^1/m law (Sérsic 1968), a natural generalization of the empirical R^1/4 law of de Vaucouleurs (1948). A major drawback of this family, however, is that the spatial density and the distribution function can not be written in terms of elementary functions (see Mazure & Capelato 2002). A more useful family is formed by the so-called $\gamma$-models (Dehnen 1993; Tremaine et al. 1994), characterized by a density proportional to r^-4 at large radii and a divergence in the center as $r^{-\gamma}$ with $0\leqslant\gamma\leqslant3$. The dynamical structure of models with this potential-density pair has been extensively investigated (e.g. Carollo et al. 1995; Ciotti 1996; Meza & Zamorano 1997), but only for isotropic or Osipkov-Merritt type distribution functions. Simple analytical models with a more general anisotropy structure are still lacking.

In this paper we construct a number of families of completely analytical anisotropic dynamical models that self-consistently generate the Hernquist (1990) potential-density pair. It is a special case of the family of $\gamma$-models, corresponding to $\gamma=1$. In dimensionless units, the Hernquist potential-density pair is given by 010

  

$$\psi(r) \frac{1}{1+r}$$ = (2a)

$$\rho(r) \frac{1}{2\pi}~\frac{1}{r(1+r)^3}\cdot$$ = (2b)

As the density diverges as 1/r for $r\rightarrow0$, the surface brightness I(R) will diverge logarithmically for $R\rightarrow0$. More precisely, the surface brightness profile has the form

$$I(R) = \frac{1}{2\pi}~ \frac{(2+R^2)~X(R)-3}{(1-R^2)^2},$$ (3)

with X(R) a continuous function defined as

$$X(R) =\left\{ \begin{array}{l} (1-R^2)^{-1/2}~{\rm arcsech... ...rm for} \ 1\leqslant R\leqslant\infty. \\ \end{array}\right.$$ (4)

The paper is organized as follows. The general theory on the inversion of the fundamental integral Eq. (1) is resumed in Sect. 2. Each of the subsequent sections is devoted to special cases of this inversion technique and the corresponding family of Hernquist models. Isotropic models are the most simple ones; Hernquist (1990) showed that, in the special case of isotropy, the distribution function and its moments can be calculated analytically. We repeat the most important characteristics of the isotropic Hernquist model in Sect. 3. In Sect. 4 we construct a one-parameter family of models with a constant anisotropy. In Sect. 5, a two-parameter family of Hernquist models is constructed by means of the Cuddeford (1991) inversion technique. These models have an arbitrary anisotropy in the center and are radially anisotropic at large radii. On the contrary, in Sect. 6, a two-parameter family is constructed that has a decreasing anisotropy profile, with arbitrary values for the anisotropy in the center and the outer halo. Finally, Sect. 7 sums up.