Three new families of anisotropic dynamical models have been presented that self-consistently generate the Hernquist potential-density pair. For all models, in particular for the Cuddeford models of Sect. 5, we checked the conditions on the adopted parameters such that the distribution is positive, and hence physically acceptable, in phase space.
They host a wide variety of orbital structures: in general, the models presented can have an arbitrary central anisotropy, and a outer halo with the same anisotropy, a purely radial orbital structure, or an arbitrary, but more tangential, anisotropy. In order to produce models that have an arbitrary anisotropy in the central regions, and a more radial, but not purely radial, anisotropy at large radii, the most cost-effective way seems to construct a linear combination of a number of `component' dynamical models, such as the ones presented here. This technique has been adopted for several years in the QP formalism (Dejonghe 1989, for an overview see Dejonghe et al. 2001), where most of the components in the program libraries have an intrinsically tangential orbital structure.
For all of the presented models, we have analytical expressions for the distribution function and the velocity dispersions in terms of elementary functions. They are hence ideal tools for a wide range of applications, for example to generate the initial conditions for N-body or Monte Carlo simulations. At this point, a number of remarks are appropriate.
First, very few elliptical galaxies are perfectly spherical; actually, various observational and theoretical evidence suggests that many elliptical galaxies are at least moderately triaxial (Dubinski & Carlberg 1991; Hernquist 1993; Tremblay & Merritt 1995; Bak & Statler 2000). Unfortunately, an extension of the presented techniques to construct analytical axisymmetric or triaxial systems is not obvious, because the internal dynamics of such stellar systems is much more complicated than in the spherical case. Nevertheless, our models can be used as an onset to construct numerical axisymmetric of triaxial distribution functions with different internal dynamical structures, for example by the adiabatic squeezing technique presented by Holley-Bockelmann et al. (2001).
Second, the models presented here are self-consistent models,
whereas it is nowadays believed that most elliptical galaxies
contain dark matter, either in the form of a central black hole
(Merritt & Ferrarese 2001 and references therein) and/or a dark
halo (Kronawitter et al. 2000; Magorrian & Ballantyne 2001).
When constructing dynamical models with dark matter, an extra
component must be added to the gravitational potential. For
example, Ciotti (1996) constructed analytical two-component models
in which both the stellar and dark matter components have a
Hernquist density profile and an Osipkov-Merritt type distribution
function. The models presented in this paper can also be extended
to contain a dark halo or a central black hole. Indeed, the
adopted inversion techniques are perfectly suitable for this,
because the augmented density functions
do
not necessarily need to satisfy the self-consistency condition (5).
Adding an extra term to the potential does not
conceptually change the character of the inversion, but it might
complicate the mathematical exercise.
Third, we have not discussed stability issues for the presented
models. The study of the stability of anisotropic stellar systems
is difficult, and a satisfactory criterion can not easily be
given. For stability against radial perturbations, we can apply
the sufficient criterions of Antonov (1962) or Dorémus & Feix
(1973), but numerical simulations have shown that these criteria
are rather crude (Dejonghe & Merritt 1988; Meza & Zamorano
1997). Moreover, the only instability that is thought to be
effective in realistic galaxies is the so-called radial orbit
instability, an instability that drives galaxies with a large
number of radial orbits to forming a bar (Hénon 1973; Palmer &
Papaloizou 1987; Cincotta et al. 1996). The behavior of
galaxy models against perturbations of this kind can only be
tested with detailed N-body simulations or numerical linear
stability analysis. Meza & Zamorano (1997) used N-body
simulations to investigate the radial orbit instability for a
number of spherical models of the Osipkov-Merritt type, including
the Hernquist model. They found that the models are unstable for
,
which significantly restricts the set of models
that correspond to positive distribution functions (see Table 1).
It would be interesting to extend this
investigation to the three families of Hernquist models presented
in this paper, but this falls beyond the scope of this paper.
Acknowledgements
The authors are grateful to Andrés Meza for a careful check on the formulae derived in this paper. Fortran codes to evaluate the internal and projected dynamics of the presented models are available from the authors.
Copyright ESO 2002