## 2. Constructing a model with a tangentially biased pressure tensorFollowing the procedure devised by Shu (1969) for the case of
rotating disks, we consider systems made of stars for which the orbits
are well described by the epicyclic approximation. In this case, for a
given potential , the radial coordinate One possible choice for the distribution function is a quasi-Maxwellian function in the peculiar velocities with respect to the circular motion at speed : with for . Here
Here is the standard epicyclic frequency. ## 2.1. Models made of quasi-circular orbitsThe Maxwellian distribution of peculiar velocities of the star
orbits (with respect to the circular motions) associated with choice
(1) for the distribution function can be recovered in the following
way. First consider the distribution function expressed as a function
. Then, introduce a transformation to the
variables , where Thus, by expanding around , the argument of the exponential in (1) becomes: By approximating by where all the functions appearing on the right hand side are now meant to be functions of . The choice of the function can be made so as to generate some desired anisotropy profile. Indeed, if we introduce the local anisotropy parameter as we can write the hydrostatic equilibrium equation in the form: In particular, this shows that, for a given density distribution, the local anisotropy parameter is . In conclusion, Eqs.(1) and (4), together with condition (2), are
expected to form the basis for a distribution function that supports a
cold distribution of quasi-circular orbits for a given choice of
density and potential .
When the coldness parameter is taken to be of
order unity, the underlying physical picture that has guided our
choice of ## 2.2. The problem at small radiiThe physical picture outlined so far is based on an epicyclic
approximation; the radial range of its validity depends on the choice
of . If we assume that condition (2) applies
uniformly in To be more specific, if we refer to models characterized at large radii by Keplerian forces and constant anisotropy, a choice of of the form gives rise to a singular distribution function, while a choice of the form is going to lead to an isotropic core. In the above expressions is a reference angular momentum. Note from Eqs. (2) and (8) that the epicyclic approximation is expected to break down at radii . Based on choice (8), we may improve the definition of the
normalization factor with at small values of the angular
momentum, so that at the center it matches the normalization of an
isotropic Maxwellian (see Eq. (29) and the discussion at the end of
Appendix A). One relatively smooth choice for the function Note that since Eq. (4) (or Eq. (9)) is derived from a saddle-point
method which breaks down at small radii (see Appendix A), the actual
density given by may differ from the density
appearing in Eq. (4) (or Eq. (9)). In order to
resolve this discrepancy, we may then consider two viable options. The
first option (see following Sect. 2.3) is that of taking the integral
of ## 2.3. Self-consistencyThe definition of the distribution function (1) and (4) (or Eq. (9)) assumes the knowledge of the basic potential (explicitly, in the energy dependence, and implicitly through the definition of the functions , , , and ). If we follow the first option mentioned above, we may proceed iteratively as follows. We calculate the density from then we evaluate a corrected potential from the Gauss theorem A few steps should bring convergence to a self-consistent function. At each iteration step, self-consistency can be judged by testing the virial theorem. In Sect. 3 we shall illustrate the properties of the models that are found following the asymptotic procedure described above by applying it to the case of isochrone potentials. As we shall show there in detail, in the case of small values of the models that are found present some undesired features in their density profile close to the center. ## 2.4. Inversion by contour integrationA different approach from the one outlined so far is that of
looking for an approximate inversion procedure able to determine the
normalization function derived from the definition in Appendix A. Thus, the starting point of the inversion is the following equation: where we have introduced the variable and the functions with the effective potential. In Appendix B we show that by multiplying Eq. (15) by and by integrating along the imaginary axis in the complex variable (for real ) we obtain the expression: which, for small values of , gives the
desired inversion, i.e. an expression for the normalization factor
leading to the specified density distribution
. Here the contour
corresponds to the path of integration in the variable In the following we focus on a specific demonstration for the case where the basic density corresponds to an isochrone model, but the method appears to be fairly general with respect to the choice of the density distribution (see Appendix B). Here we may recall that in the different context of finding exact inversion procedures, but within the general goal of constructing self-consistent stellar dynamical models, the use of contour integrals has found a recent interesting application by Hunter & Qian (1993). The first method (fixed functional form of distribution function)
leads to a normalization function © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |