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

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2. Constructing a model with a tangentially biased pressure tensor

Following 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 [FORMULA], the radial coordinate r may be essentially identified with the angular momentum coordinate J, since stars are confined to a small annulus around the guiding center radius [FORMULA], with [FORMULA] a monotonic increasing function of J. Thus we look for distribution functions that are significantly different from zero only for values of the energy E close to the minimum energy [FORMULA] characteristic of the circular orbit at [FORMULA].

One possible choice for the distribution function is a quasi-Maxwellian function in the peculiar velocities with respect to the circular motion at speed [FORMULA]:

[EQUATION]

with [FORMULA] for [FORMULA]. Here J denotes the magnitude of the star specific angular momentum and the star specific energy is [FORMULA]. This distribution function depends on two functions [FORMULA] and [FORMULA] of the angular momentum, which should be chosen in order for the system to be consistent with the potential [FORMULA] and with the assumed epicyclic approximation. The latter requirement may be met if the quantity [FORMULA] is taken to be small: [FORMULA] with [FORMULA] a small dimensionless parameter such that

[EQUATION]

Here [FORMULA] is the standard epicyclic frequency.

2.1. Models made of quasi-circular orbits

The 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 [FORMULA]. Then, introduce a transformation to the variables [FORMULA], where w is a peculiar tangential velocity defined by the relation [FORMULA]. Therefore, in the epicyclic approximation we have [FORMULA]

Thus, by expanding [FORMULA] around [FORMULA], the argument of the exponential in (1) becomes:

[EQUATION]

By approximating [FORMULA] by r in the function [FORMULA] and integrating over velocity space, based on a saddle point method (under certain assumptions that turn out to fail at small radii - see Sect. 2.2 and Appendix A), we find an approximate relation between the function P and the mass density [FORMULA]: [FORMULA] where all the functions appearing on the right hand side are meant to be functions of r. Thus we can start from this approximate relation and define the function [FORMULA] as:

[EQUATION]

where all the functions appearing on the right hand side are now meant to be functions of [FORMULA].

The choice of the function [FORMULA] can be made so as to generate some desired anisotropy profile. Indeed, if we introduce the local anisotropy parameter [FORMULA] as

[EQUATION]

we can write the hydrostatic equilibrium equation in the form:

[EQUATION]

In particular, this shows that, for a given density distribution, the local anisotropy parameter is [FORMULA].

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 [FORMULA] and potential [FORMULA]. When the coldness parameter [FORMULA] is taken to be of order unity, the underlying physical picture that has guided our choice of f is bound to fail and problems are expected to be encountered when the Poisson equation is considered.

2.2. The problem at small radii

The physical picture outlined so far is based on an epicyclic approximation; the radial range of its validity depends on the choice of [FORMULA]. If we assume that condition (2) applies uniformly in J, then the function [FORMULA] must become vanishingly small at [FORMULA] for any assumed regular density distribution [FORMULA]. This implies a singular behavior of the distribution function (1). [Note that this behavior does not depend on the specific choice of distribution function made, as is briefly shown in the Introduction.] In the following, we relax condition (2) at small radii, so that the distribution function f is regular, but then the choice of the function P given by Eq. (4) should be improved in order to take into account the orbits that are no longer quasi-circular at small values of the angular momentum. Note that if we require [FORMULA] to remain finite at [FORMULA], the epicyclic parameter appearing in (2) formally diverges, indicating that the central parts of the model will no longer be characterized by tangentially biased pressure and are presumably going to be isotropic.

To be more specific, if we refer to models characterized at large radii by Keplerian forces and constant anisotropy, a choice of [FORMULA] of the form

[EQUATION]

gives rise to a singular distribution function, while a choice of the form

[EQUATION]

is going to lead to an isotropic core. In the above expressions [FORMULA] is a reference angular momentum. Note from Eqs. (2) and (8) that the epicyclic approximation is expected to break down at radii [FORMULA].

Based on choice (8), we may improve the definition of the normalization factor P as obtained from the saddle-point method by defining

[EQUATION]

with [FORMULA] 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 g is:

[EQUATION]

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 [FORMULA] may differ from the density [FORMULA] 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 f to be the true density and, by an iterative procedure, to correct the value of the potential [FORMULA] in order to satisfy self-consistency as required by the Poisson equation in terms of the true density. In this case, the final model turns out to be associated with a density-potential pair that may be appreciably different from that used to initialize the procedure. An alternative option gives priority to the initial choice of density distribution (and thus of the corresponding potential) and aims at improving definition (4) so that self-consistency is automatically guaranteed; in this case (see Sect. 2.4) the analytical definition (4) is replaced by one that involves a numerical evaluation of an integral.

2.3. Self-consistency

The definition of the distribution function (1) and (4) (or Eq. (9)) assumes the knowledge of the basic potential [FORMULA] (explicitly, in the energy dependence, and implicitly through the definition of the functions [FORMULA], [FORMULA], [FORMULA], and [FORMULA]). If we follow the first option mentioned above, we may proceed iteratively as follows. We calculate the density [FORMULA] from

[EQUATION]

then we evaluate a corrected potential [FORMULA] from the Gauss theorem

[EQUATION]

with

[EQUATION]

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 [FORMULA] the models that are found present some undesired features in their density profile close to the center.

2.4. Inversion by contour integration

A different approach from the one outlined so far is that of looking for an approximate inversion procedure able to determine the normalization function P guaranteeing support to the assumed density distribution [FORMULA] uniformly at all radii. In other words, we would like to identify a function P of the angular momentum such that

[EQUATION]

derived from the definition [FORMULA] in Appendix A. Thus, the starting point of the inversion is the following equation:

[EQUATION]

where we have introduced the variable [FORMULA] and the functions

[EQUATION]

and

[EQUATION]

with [FORMULA] the effective potential.

In Appendix B we show that by multiplying Eq. (15) by [FORMULA] and by integrating along the imaginary axis in the complex variable [FORMULA] (for real [FORMULA]) we obtain the expression:

[EQUATION]

which, for small values of [FORMULA], gives the desired inversion, i.e. an expression for the normalization factor [FORMULA] leading to the specified density distribution [FORMULA]. Here the contour [FORMULA] corresponds to the path of integration in the variable u, which is reduced to a small circle in the complex plane in the vicinity of [FORMULA] (residue). Such an integral can be performed numerically.

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 P that does not guarantee uniform support to a given [FORMULA] at all radii, while the second method (contour integration) does. From a technical point of view, the reason why this occurs is explained in detail in the last paragraph of Appendix A and the first paragraph of Appendix B.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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