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

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3. Application to isochrone potentials

We now investigate in detail the application to the isochrone potential. The isochrone models (Hénon 1959) are convenient in that they have a realistic outer density profile, while involving simple analytical expressions. Therefore they have often been used in studies of stellar dynamics, mainly for systems with radially biased velocity dispersion (e.g., Gerhard 1991; Saha 1991).

3.1. Basic analytical relations

The isochrone potential is defined as:

[EQUATION]

In the following we shall provide formulae where the units for [FORMULA] and length are [FORMULA] and b respectively. Thus the quantities [FORMULA] and [FORMULA] entering the distribution function (1) become:

[EQUATION]

[EQUATION]

and the quantities [FORMULA] and [FORMULA] that appear in the definition (9) of [FORMULA] are:

[EQUATION]

[EQUATION]

Note that in these units the value of the total mass is [FORMULA] and the half-mass radius is given by [FORMULA].

Finally, in the following models we keep the scale-length for the onset of tangentially biased pressure fixed by taking [FORMULA] in the definition (8) of the quantity [FORMULA]. Thus [FORMULA] is the only parameter that will be changed in our model survey, varying from 0.05 up to 0.30.

3.2. Models with fixed functional form of the distribution function

We now consider the first option described in the last paragraph of Sect. 2.2, and we do so by keeping all quantities appearing in the definition (9) of [FORMULA] fixed by the expressions given in the previous subsection. Thus the only updated quantity in the form of the distribution function subject to the iteration procedure described in Sect. 2.3 is the potential [FORMULA] entering E and [FORMULA] in the exponent of Eq. (1).

The computation of the integral over tangential velocities has been performed using cubic splines on a logarithmic mesh, with the smallest step-size at [FORMULA], as also suggested by the saddle point analysis of Appendix A.

Convergence is reached typically after 5 to 10 iterations, with the virial theorem verified at the 0.01 percent accuracy level.

In the central region, the resulting density exhibits a significant deviation from the isochrone profile (Fig. 1). As expected, the overall deviation is larger for larger values of [FORMULA]. On the other hand, the [FORMULA] limit is not trivial (see the last paragraph of Appendix A); this is illustrated by the fact that the deviations from the isochrone density occur on a smaller and smaller radial scale but, for the smallest values of [FORMULA] ([FORMULA]), the density profile develops a significant non-monotonicity. All models present a small "bump" ([FORMULA]) at very small radii ([FORMULA]).

[FIGURE] Fig. 1. Density profiles of the models with fixed functional form of distribution function (i.e., with [FORMULA] given by Eq. (9)), for various values of the [FORMULA] parameter. The thick line is the isochrone density. The inset shows the density profiles for the corresponding models based on the contour integration method

The profiles of the anisotropy parameter [FORMULA] defined by Eq. (5) are shown in Fig. 2. Close to the center (Fig. 2b), in the region where the deviations in the density distribution occur, these profiles differ from the asymptotic estimates given by Eq. (6). In addition, the cases with the largest values of [FORMULA] may even show a significant radial range where the velocity dispersion is radially biased.

[FIGURE] Fig. 2. Profiles of the anisotropy parameter [FORMULA] for the models with fixed functional form of distribution function (a and b) and for the models based on the contour integration method (c). Frames b and c focus on the central regions of the models

In order to appreciate the structure of velocity space associated with our models better, in Fig. 3 we show some relevant sections of the distribution function at two different radii. While the central cuts indicate that the models are reasonably isotropic (Fig. 3b), a two-horned structure in the tangential velocity plane is clearly seen at [FORMULA] (Fig. 3a). Again, for the cases with the largest values of [FORMULA] an additional feature at [FORMULA] develops which may be traced to the population of radial orbits originating from the center.

[FIGURE] Fig. 3a-c. Sections of the distribution function [FORMULA] for the models with fixed functional form of distribution function, a at the half-mass radius [FORMULA] and b near the center at [FORMULA] ; c sections of the distribution function [FORMULA] for the models based on the contour integration method, near the center at [FORMULA]

In Fig. 4 the anisotropy structure of our sequence of models is illustrated in a different form that gives a simple way of appreciating the size of the anisotropy involved with cases of direct astrophysical interest (in particular, compare the case of [FORMULA] with Fig. 4 of the paper by Bertin et al. 1994a).

[FIGURE] Fig. 4a-c. Profiles of the radial velocity dispersion [FORMULA] and of the velocity dispersion in one tangential direction [FORMULA] for the models with fixed functional form of distribution function (a and b), and for the models based on the contour integration method (c). Frames b and c focus on the central regions of the models

3.3. Models based on the contour integration method

Proceeding with the second option described in the last paragraph of Sect. 2.2, we consider the models constructed following the method outlined in Sect. 2.4. The isochrone density profile is then reproduced rather accurately down to the center. Judging from the closeness between the density computed by the inversion and the isochrone density, this method appears to work reasonably well even for relatively large values of [FORMULA].

In Figs. 2(c), 3(c), and 4(c) we show the various profiles for comparison with the models obtained with the other method.

Numerical properties of the two classes of models described in this and in the previous subsection are summarized in Table 1. Table 2 lists the kinematic properties, and, in particular, the values of the global anisotropy parameter [FORMULA] (cf. Bertin et al. 1994b), where [FORMULA] and [FORMULA] are the total kinetic energies relative to radial and tangential motions respectively, and of the scale-length [FORMULA] defined as the location where [FORMULA].


[TABLE]

Table 1. Numerical parameters of the models compared to those of the initial isochrone potential, based on a construction that keeps the central density fixed: models based on Eq. (9) for [FORMULA] (left) and models based on the contour integration method (right). The value of the virial ratio is indicated only when different from unity by more than [FORMULA]



[TABLE]

Table 2. Kinematical parameters of the models: models based on Eq. (9) for [FORMULA] (left) and models based on the contour integration method (right)


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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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