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

1. Introduction

In N -body simulations of stellar systems the gravitational interaction is modified for curing the Newtonian divergence at short distances. Basically, such modifications are introduced through a soft cut-off: the softening length s. However, the precise form in which they are implemented can vary. Since gravity plays a fundamental role in these systems and the gravitational interaction is modified precisely where it becomes singular, the dynamical effects of softening should be well understood when designing experiments and interpreting their results. This dynamical problem has recently stimulated considerable interest (e.g., Hernquist & Barnes 1990; Hernquist & Ostriker 1992; Kandrup et al. 1992; Pfenniger 1993; Pfenniger & Friedli 1993; Gurzadyan & Pfenniger 1994; Romeo 1994, hereafter Paper I; Byrd 1995; Gerber 1996; Merritt 1996; Weinberg 1996; Sommer-Larsen et al. 1997; Theis 1997; see also Goodman et al. 1993; Farouki & Salpeter 1994). For extensive overviews see the above-mentioned Pfenniger & Friedli (1993), Gurzadyan & Pfenniger (1994) and Paper I.

In Paper I we have investigated the stability problem in the case of 2-D models with Plummer softening, which are commonly employed in simulations of disc galaxies. The basic message is that the effect of softening becomes strongly artificial for , being the typical radial wavelength, which means half an order of magnitude below the expected value. The major results are summarized in the form of a criterion of approximate physical consistency for s and a stability criterion for the Toomre parameter. (Other important aspects of the stability problem have been considered by Byrd 1995.)

In the present paper we carry out five extensions, as is discussed below.

1. We generalize the stability analysis of Paper I to an arbitrary isotropic form of softening. This is a natural extension since types of softened gravity different from the standard Plummer softening are becoming more and more commonly employed (e.g., Combes et al. 1990; Palou et al. 1993; Shlosman & Noguchi 1993). In particular, the alternatives proposed by Hernquist & Katz (1989) and Pfenniger & Friedli (1993) reflect an interesting idea, viz. that softening should be as localized as possible since there is no clear reason for modifying the gravitational interaction at long distances, and it is tempting to explore its dynamical consequences. From a more general point of view, this and the following extensions provide the tools for comparing experiments that employ different types of softened gravity.
2. We investigate the implications of our stability analysis for the classical relaxation problem (Rybicki 1972; White 1988). Relaxation and stability ¹ are intimately related in self-gravitating systems, and even simple treatments reveal their strong coupling through random motions. On the other hand, the contribution of velocity dispersion to the relaxation time has been understood only in part and, because of that, the classical argument favouring the choice of large values of s is wrong. We revise this argument and conclude that neither small nor large values of s are convenient. Surprisingly, there exists an intermediate choice of s that optimizes the `dynamical resolution' of the model, i.e. its faithfulness in simulating the dynamics of 3-D discs with Newtonian gravity, especially in situations near to the stability threshold. We identify the optimal characteristics, and show how to evaluate them for a given type of softened gravity. In addition to investigating this aspect of the relaxation problem, we explain how effectively softening reduces noise on various scales.
3. We complete the examination of 2-D models with isotropic softening by investigating the equilibrium problem for an axisymmetric state with epicyclic motions. In particular, we explain how significantly the circular speed and related quantities deviate from their Newtonian behaviours at various distances from the centre.
4. We consider 3-D models with isotropic softening and examine two limiting cases: discs and the simple, yet instructive, Jeans problem. An extension to 3D has been encouraged by Friedli (1994) and Junqueira & Combes (1996, see the interesting remarks in Sect. 2.2). Real stellar systems have several gravitationally interacting components. Both N -body simulations and theoretical works are forced to use simplified models, which do not necessarily provide faithful representations of the complexity of such systems. Our motivation is to understand the basic differences between the dynamical effects of softening in 3D and 2D, in the presence of a single stellar component (3-D vs. 2-D discs and Jeans problem vs. discs). In particular, we point up the strong modifications introduced by a homogeneous geometry and the absence of rotation.
5. We complete the examination of 3-D models by discussing the basic dynamical effects of softening anisotropy. The idea underlying this extension is that softening should be anisotropic in simulations of stellar systems where significantly higher spatial resolution is required along a certain direction, such as in disc galaxies. The alternative family of softening recently proposed by Pfenniger & Friedli (1993) reflects such an important idea, and it is tempting to explore the dynamical relations between its members. (An analogous idea has been discussed in the context of smoothed particle hydrodynamics by Shapiro et al. 1994, 1996 and Fulbright et al. 1995; Hernquist, private communication, has remarked that in models with anisotropic smoothing there may be a significant tendency for angular momentum not to be conserved.)

These extensions all together form a method for exploring the dynamical effects of softening in N -body simulations of stellar systems. Our method is described in Sect. 2, and is structured as in the previous discussion. The two applications mentioned in the same context are shown in Sect. 3 (see also Appendix A). The conclusions of this paper are drawn in Sect. 4, where we present our contribution in a more general perspective and motivate future applications.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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