Astron. Astrophys. 324, 523-533 (1997)

4. Conclusions

Modelling gravity is a fundamental problem that must be tackled in N -body simulations of stellar systems, and satisfactory solutions require a deep understanding of the dynamical effects of softening. This problem has deserved special attention both in the past (e.g., Miller 1970, 1976) and in more recent times (e.g., Efstathiou et al. 1985; Hernquist & Katz 1989; Pfenniger & Friedli 1993). Viewed in this general perspective, our contribution has a threefold practical importance in addition to the points emphasized in Sect. 1.

1. The two present applications of our method reveal the dynamical differences between the most representative types of softened gravity: the isotropic P, CS and HS (abbreviated as in Sect. 3.1), and the anisotropic HOS and HPS (abbreviated as in Sect. 3.2). The major conclusions concerning their dynamical resolution, i.e. their faithfulness in simulating the Newtonian dynamics, are summarized below. 1 As regards the isotropic types, the dynamical resolution is comparable. This results from the fact that, even though the spatial resolution and the effectiveness in reducing noise differ significantly in P, CS and HS for the same nominal softening length s, those differences can largely be removed by considering a more appropriate softening length of reference. 2 As regards the anisotropic types, the dynamical resolution is significantly coupled parallel and perpendicular to the plane. In the plane, it decreases in quality from HPS to HOS, and the transition occurs in the oblate members for a softening axial ratio (HPS is dynamically similar to P). These disadvantages result from the finite-sized particle implementation of softening anisotropy. On the other hand, they have less importance than the advantage of introducing such a degree of freedom into 3-D simulations of disc galaxies, which has been emphasized by Pfenniger & Friedli (1993) and in our method. 3 Last but not least, when employing these types of softened gravity in simulations of disc galaxies, we should recall that the dynamical resolution depends critically on two quantities: s, or for a given , and the Safronov-Toomre parameter Q (cf. Paper I). The choice of s or should be checked vs. the profiles of the characteristic values , and , which are tabulated in the applications. The choice of Q should be checked vs. the profiles of the stability threshold and level , which can be evaluated as is explained in the method.
2. Our method can be applied for testing new ideas about softening. There are two features that encourage such future applications. 1 One is the unified approach adopted for investigating stability, relaxation and equilibrium. As a result, full information about the dynamical effects of softening is contained in a single quantity: the reduction factor . 2 The other is the modular structure of the method. We describe step by step how to extract detailed information concerning the dynamical properties, starting from and pointing out the quantities of major interest.
3. But our method can be applied in another, more fruitful, way: for developing new ideas about softening. Indeed, it opens a direct route to the discovery of optimal types of softened gravity for given dynamical requirements, and thus to the accomplishment of a physically consistent modelling even in the presence of a cold interstellar gaseous component. Such a future application will be the objective of a `twin' paper.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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