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Astron. Astrophys. 321, 703-712 (1997)

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1. Introduction

Anisotropic equilibrium models of spherical, non-rotating stellar systems have been extensively investigated in recent years. Largely guided by physical arguments, N-body experiments, and observations, attention has generally been focused on systems with an excess of radial orbits (e.g., see Fridman & Polyachenko 1984; Bertin & Stiavelli 1993, and the many references quoted there).

A wide choice of self-consistent tangentially anisotropic models would be desirable for at least three reasons. It would be an important tool to further explore the stability of spherical collisionless stellar systems, for which the main interest so far has focused on the issue of the radial-orbit instability (Palmer 1994). Furthermore, it could provide a natural basis for fitting velocity dispersion profiles in elliptical galaxies when the observed radial decline is relatively slow; it is well known (Tonry 1983) that these cases can be interpreted as evidence for the existence of gradients in the mass-to-light ratios (see Bertin et al. 1994a), but that in principle they could be modelled as tangentially anisotropic equilibrium configurations with a constant mass-to-light ratio, as for example shown by Dejonghe (1989) through the use of polynomial distribution functions. Finally, in a related objective, a wide choice of tangentially anisotropic models would be interesting in order to construct synthetic line-of-sight velocity distribution functions for comparison with the observed line profiles (see Bender et al. 1994 and references therein).

In practice, there is only a rather limited choice of self-consistent tangentially anisotropic models. In particular, distribution functions dependent on one relatively straightforward combination of integrals of the motion (Osipkov 1979; Merritt 1985a, b) are well suited to the construction of radially anisotropic systems, but often fail when tried for the tangentially biased case (for example, they do not work for the isochrone models - see also Saha 1991). One useful assumption is to consider distribution functions that are separable in E (or J) and some other simple variable depending on E and J (Hénon 1973; Gerhard 1991; Cuddeford 1991; Louis 1993). For example, Gerhard (1991) has devised a general method for the construction of anisotropic stellar systems, based on a distribution function factorized as the product of a function of energy and a circularity function ; the latter depends on a combination of energy and angular momentum and controls the distribution of orbits in the outer parts. For a given simple choice of the circularity function, such a factorization allows for an inversion determining the distribution function from a given density profile. Gerhard's models are isotropic in the central regions and reach constant anisotropy in their envelopes. In general, the models that are anisotropic at large radii have an isotropic core, while those that have tangential anisotropy in the center (Cuddeford 1991) are radially anisotropic in their envelopes.

In this paper we describe an alternative method specially aimed at the construction of self-consistent spherical stellar systems characterized by tangentially biased velocity dispersion with a given density distribution. This method appears to be very simple from the physical point of view and fairly flexible. The construction starts with the case of systems populated mostly by quasi-circular orbits and identifies sequences of models with varying degrees of pressure anisotropy; these sequences can then be explored also in regimes where the pressure tensor becomes relatively close to being isotropic. In addition, there is significant freedom in the choice of the relevant anisotropy profile. The method is illustrated in detail for the case of an isochrone potential.

The main idea of our method is that of adapting the epicyclic procedure originally developed (Shu 1969) for rotating disks with a modified Schwarzschild (i.e., quasi-Maxwellian) distribution function to the different geometry of spherical, non-rotating stellar systems. Thus the method is physically intuitive. It turns out that the whole procedure can be carried through analytically, to the extent that the exact behavior of the models at small radii is judged to be unimportant. When one tries to complete the description by following the properties of the models all the way down to the center, one is forced to consider significant departures from the initially assumed density profile, and often some peculiar gradients are noted. Alternatively, a solution with the assumed density profile can be implemented, at the cost of completing the definition of the distribution function numerically. This is obtained by a suitable contour integration in the complex plane. Both methods are based on the smallness of the epicyclic parameter. The solutions that are found can be used as an input for stability analyses of the type described in a previous paper (Bertin et al. 1994b), i.e. linear modal analyses that require the knowledge of the dependence of the distribution function on the integrals of the motion.

In stellar dynamics many articles address the question of whether a given density profile can be generated exactly by a (positive definite) distribution function and some investigate all the possible forms of the distribution function that can serve such purpose (the "inversion" problem [FORMULA]). It should be stressed that, in this respect, this paper addresses a completely different problem. In fact, since the stability of collisionless systems is known to depend primarily on the gradients of the distribution function in phase space and on the degree of pressure anisotropy (see Fridman & Polyachenko 1984), we give priority to a physically reasonable choice of the distribution function and show that wide classes of self-consistent solutions that yield the desired density profile, with a tangentially biased velocity pressure anisotropy, can be constructed with this procedure. The focus on Maxwellian-like distributions in the relevant velocity space serves the purpose of avoiding systems that might be affected a priori by "spurious" kinetic instabilities. For the same reason, we choose to avoid the construction of solutions with singular behavior at [FORMULA]. Such a behavior occurs naturally if we insist on loading circular orbits all the way down to the center. In fact, if we consider a distribution function made of purely circular orbits [FORMULA] and integrate over velocity space to solve for [FORMULA], given a density [FORMULA], a difficulty arises at small radii for any regular density-potential pair, since [FORMULA] so that [FORMULA], and thus the supporting distribution function, must be singular. Since singular models would be artificial and, again, might be subject to "spurious" instabilities, in this paper we avoid such a central singularity by allowing for an isotropic core.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998