Astron. Astrophys. 347, 821-840 (1999)

1. Introduction

1.1. Local stability criterion

Spirals are common in rapidly (and nonuniformly) rotating galaxies. The origin and maintenance of the spiral structure of such highly flattened systems has proved to be a difficult problem in galactic dynamics. Even though no definitive answer can be given at the present time, the study of the stability of small-amplitude waves in disk-shaped galaxies of stars is the first step towards an understanding of the phenomena. This is because in the Milky Way Galaxy and many other giant galaxies the bulk of the optical mass, probably , is composed of stars, and therefore stellar dynamical phenomena play a basic role.

More than three decades ago, Lindblad (1963, and earlier references) proposed that spiral arms of a galaxy are quasi-stationary density waves propagating through differentially rotating parts of a collisionless disk of stars with a constant phase velocity. Subsequently, Lin & Shu (1966), Lin et al. (1969), Shu (1970), and others (e.g., Nakamura et al. 1975) further developed the density wave theory by studying collective effects in self-gravitating stellar systems; see reviews by Toomre (1974, 1977) and Athanassoula (1984). It seemed reasonable to attribute galactic spiral arms to Lin-Shu type small-wavelength density waves driven by the classical Jeans instability in a rapidly rotating system of young, dynamically cold stars.

Initially, in the asymptotic Lin-Shu density wave theory of tightly-wound spirals, important effects of the azimuthal gravitational forces in nonuniformly rotating systems were not properly taken into account. As a result of this simplification, the well-known Toomre's (1964) local criterion for stability against only axially symmetric (radial) Jeans-type perturbations of the gravitational potential can be derived from the original Lin-Shu-Kalnajs dispersion relation (Lin & Shu 1966; Lin et al. 1969; Shu 1970; Toomre 1977). ¹

The original Toomre's criterion states that the radial residual (random)-velocity dispersion of stars , which is proportional to the square root of the "temperature" of the system, will suppress the axisymmetric Jeans perturbations in the rapidly rotating, nearly homogeneous, and very thin (, where h is a typical thickness and R is a characteristic radius of the system) disk, if

In Eq. (1), G is the gravitational constant and is the equilibrium surface mass density. The local epicyclic frequency is given by where the quantity denotes the angular velocity of rotation at the distance r from the galactic center. The epicyclic frequency decreases from for the rigid body rotation to for the Keplerian one. The disk with the velocity dispersion is on the verge of the gravitational Jeans-type instability with respect to the short-scale, , purely radial or ringlike perturbations only. The local criterion (1) gives a necessary condition for radial stability. It does not obviously address the stability of nonaxisymmetric, relatively large-scale modes, , particularly open spiral modes in the bar form of the differentially rotating () disks.

In a series of papers Morozov (1980, 1981a, 1981b) extended the works of Toomre (1964), Lin & Shu (1966), Lin et al. (1969), and Shu (1970) by including the azimuthal forces. It was demonstrated by Morozov that the presence of the differential rotation (or shear) results in quite different dynamical properties of the axisymmetric and nonaxisymmetric (spiral) perturbations. A dispersion relation for arbitrary perturbations which propagate in the plane of a differentially rotating stellar disk is derived using a kinetic approach. This generalized Lin-Shu type dispersion relation leads to the following modified local stability criterion obtained by Morozov:

where the condition always holds in the differentially rotating system. In flat galaxies,

, and . The pitch angle between the direction of the wave front and the tangent to the circular orbit of a star in Eq. (2) is , where the nonnegative azimuthal mode number m is the number of spiral arms, while and are the radial and the azimuthal wavenumbers, respectively. The parameter is an additional stability parameter which depends on both the pitch angle and the amount of differential rotation in the galaxy (cf. the parameter introduced by Lau & Bertin 1978, Lin & Lau 1979, and Bertin 1980, 1994).

It is clear from the modified criterion (2) that in a nonuniformly rotating disk, namely when , for nonaxisymmetric perturbations () the modified velocity dispersion of a marginally Jeans-stable system is larger than (although still of the order of ). Moreover, Morozov took into account the additional weak destabilizing effect of a density inhomogeneity, and stabilizing effects of a radial gradient of a velocity dispersion and of a finite disk's thickness. The result is that these effects practically cancel out each other, at least in the solar vicinity of our own Galaxy. In the present study, we therefore neglect these small corrections. In addition to Morozov's studies, Griv (1992) has obtained a value of critical dispersion to the next leading order in the asymptotic expansion by including higher-order terms in the epicyclic amplitude. Recently, Griv (1996) and Griv & Peter (1996) clarified the basic assumptions of the asymptotic approximation and rederived the criterion (2) by using the kinetic approach. A relationship exists between Eq. (2) and what Toomre (Toomre 1981; Binney & Tremaine 1987, p. 375) called "swing amplification" in which the material at radius is pulled forward by the azimuthal forces of the material ar that it trails.

Apparently, Toomre (1964, p. 1222) first noted the different dynamical properties of perturbations with different in the nonuniformly rotating stellar disk. Later the destabilizing effect of the azimuthal forces has been studied using an analysis based both on a gas dynamical model by Lau & Bertin (1978) and Lin & Lau (1979), and a stellar dynamical model by Bertin & Mark (1978) and Bertin (1980) by using an improved potential theory. They have explained the physical origin of the difference between radial and spiral perturbations in a nonuniformly rotating system, e.g., Lau & Bertin (1978, p. 509). Briefly, in order to fit in with the gravitational field in flat systems, galactic rotation has to be differential and such shear has important kinematic and dynamic consequences. They pointed out that the generalized stability criterion in the form of Eq. (2) takes properly into account the combined influence of self-gravity, thermal motions, shear, and azimuthal forces. The reader should consult Lau & Bertin (1978), Bertin (1980, 1994), and Lin & Bertin (1984) for a detailed discussion of the problem. Recently, the problem has been nicely reviewed by Polyachenko & Polyachenko (1997). Note only that the free kinetic energy associated with the differential rotation of the system under study is only one possible source for the growth of the energy of these spiral Jeans-type perturbations, and appears to be released when angular momentum is transferred outward.

As one can see from Eq. (2), the modified critical velocity dispersion grows with . Consequently, in order to suppress the most "dangerous," in the sense of the loss of gravitational stability, nonaxisymmetric perturbations in a form of a bar (), should obey the following generalized criterion:

One should keep in mind that Eq. (3) is clearly only an approximate one, since it was obtained in the framework of the moderately tightly-wound Lin-Shu perturbations approximation (Lin & Lau 1979; Griv 1996; Griv & Peter 1996). Strictly speaking, the above expression (2) for cannot be used when the pitch angle is large, since in the asymptotic theory it is necessarily assumed that

The condition (4) limits the analysis of the actual low-m galaxies (in the standard Fourier analysis of the azimuthal coordinate) with to a consideration of disturbances with a pitch angle smaller than about only (Lin & Lau 1979; Griv & Peter 1996). Such a requirement naturally arises within the WKB approximation we are interesting. Polyachenko (1989) and Polyachenko & Polyachenko (1997) tried to find a stability criterion for arbitrary localized perturbations beyond the limitation of the WKB approximation by considering a hydrodynamical model. Note, however, for the disk with flat rotation curve at least, Polyachenko's marginal stability condition and (3) are practically coincidental.

Although the expression (2) only indicates the tendency of growth of the critical dispersion with increasing , it is clear that the generalized criterion for the local stability of a stellar disk against arbitrary Jeans-type perturbations (including the most unstable barlike ones) should be approximately of the form of Eq. (3). In this case, in a Jeans-stable differentially rotating disk, the widely used Toomre (1964, 1977) critical stability parameter (which guarantees the suppression of arbitrary Jeans-type perturbations in a rapidly rotating disk by the thermal velocities of stars)

where is the critical radial-velocity dispersion, must be greater than 1 and equal to about . (Toomre's Q-value is a measure of the ratio of thermal and rotational stabilization to self-gravitation and is defined below.) In particular, in a gravitating system with the Keplerian rotation (), Toomre's critical parameter is . In the case of the flat rotation curve and hence also. According to Eqs. (2), (3), and (5), the value of Toomre's critical stability parameter becomes only for arbitrary perturbations in the rigidly rotating disk () and/or for axisymmetric perturbations in the differentially rotating one (Bertin & Mark 1978; Lau & Bertin 1978).

It is obvious that in differentially rotating galaxies, disks manage to keep their local stability parameter close to the critical value, or , respectively. In this case, once the entire differentially rotating disk has been heated to values , no further spiral waves can be sustained by virtue of the Jeans instability - unless some "cooling" mechanism is available leading to Toomre's Q-value,

under approximately 2 or to the value of smaller than approximately , respectively (e.g., by the dissipation in the gas and accretion, and/or by the star formation in a "cold" interstellar medium). By using N-body simulations, first Hohl (1971) and then, e.g., Sellwood & Carlberg (1984) and Griv & Chiueh (1998), have already shown that the process of formation of new dynamically cold stars, which move on nearly circular orbits, plays a vital role in prolonging spiral activity in the plane of the disk by reducing the random velocity dispersion of the entire stellar component. Thus, the cold interstellar medium may play a dominant role in determining the observed spiral structure in galaxies because it is the site of the generation of new, dynamically cold stars. In Saturn's rings such a cooling mechanism is also operating: inelastic physical collisions between particles reduce the magnitude of the relative velocity of particles. ² Salo (1992) already investigated numerically the role of the Jeans instability mechanism in long-lived sculpting of Saturn's rings by including inelastic (dissipative) interparticle impacts.

1.2. Physics motivation

The value of Toomre's stability parameter Q is critically important for any gravitational theory of spiral structure in galaxies (and for dynamics of planetary rings). The generalized local stability criterion as well as Toomre's critical Q-value has been discussed at length by Morozov (1980, 1981a, 1981b), Polyachenko (1989), and recently by Griv (1996), Griv & Peter (1996), and Polyachenko & Polyachenko (1997). Surprisingly, their ideas on the generalized local stability criterion have not attracted a great deal of attention, and other explanations were involved to confront the observations and N-body simulations. For instance, Bertin & Romeo (1988) invoked the destabilizing effect of a sufficient amount of cold interstellar material to explain the observed large value of the parameter Q for NGC 488. Although this explanation can be accepted for the gas-rich galaxies, it certainly cannot be universal. For example, Cinzano & van der Marel (1994) showed that even in such a gas-poor spiral galaxy like NGC 2974, sometimes classified as E4, the Q value considerably exceeds unity, and probably is larger than 3. The problem seemed so complicated that Bottema (1993) even claims that it is very difficult to relate the pure observational results, that Q between 2 and over a large range of galactic disks, to any existing theoretical concept. ³

Recently, the dynamical behavior of weakly collisional, planetary rings system has been studied via an N-body simulation (Osterbart & Willerding 1995; Salo 1992, 1995). It was found that the stability number Q of Toomre in relaxed equilibrium disks does not fall below a critical value, which lies about 2-2.5. No adequate explanation of the latter fact has been presented. (Interestingly, observational data on the Saturnian rings system, obtained with the Voyager 2 spacecraft, have indicated about the same value of for the densest B ring; Lane et al. 1982, p. 543.)

We conclude that even though the criterion for local stability in a gravitating, rapidly rotating particulate disk is a relatively old issue in galactic and planetary rings dynamics, it is necessary to address the problem again. In the present work, we turn to studies of localized gravity perturbations by using both N-body simulations and an analytical approach. The linear disk's stability theory is reexamined and conditions which guarantee the lack of all Jeans-type unstable perturbations in a disk of stars are found. We restrict ourselves to the simplest case of practically collisionless stellar system which is spatially homogeneous and two-dimensional. The effects of inhomogeneity and three-dimensional motion will be investigated in a forthcoming paper.

The main objective of the current work is to check the generalized local stability criterion (3) numerically using the method of direct many-body simulations. Moreover, the dispersion relations (9) and (20) of Morozov (1980) and Griv & Peter (1996), respectively, and the stability criterion (3) obtained in the framework of the linear kinetic theory do not reveal what kind of structure can emerge due to the gravitational instability. Simulations should be able to identify these structures. As is thought, a very simple model should be taken in the numerical work to compare the analytical stability criterion with the one obtained numerically (see the next section of the paper). In addition, for the sake of completeness in Appendix A.1 of the current paper the local stability criteria (2) and (3) are rederived by employing the Lagrangian formalism of magnetized plasma theory which deals with similar problems. In our theory, the simplest theoretical method of plasma physics is used. This is the so-called method of particle dynamics (or particle orbit theory) in which the motion of an "average" star-"particle" is considered (Rosenbluth & Longmire 1957; Alexandrov et al. 1984, p. 46). The essential part of the method is to regard as a small parameter and to expand the solution in terms of it, where is the mean epicyclic radius of the star and is the epicyclic center (the guiding center in plasma physics) displacement from the galactic center.

Occasionally doubts have been raised about the validity of strictly two-dimensional N-body simulations of stellar disks of galaxies (White 1988; Romeo 1997). For example, White (1988) found a few computer models in the exactly planar simulations which are probably affected by noise and two-body relaxation (see, however, Hohl 1973). The physical effect of relaxation in the N-body simulations is to generate viscosity and heat conduction. One obvious effect of short-term relaxation is a heating of the disk, and therefore some of the two-dimensional N-body simulations probably cannot be trusted (White 1988). We show, however, in Appendix A.2 of the present work that in general the effect of such rare, , say, , elastic gravitational collisions (encounters) is very small, and may be important only on a timescale of the order of the mean time of many galactic rotations, typically rotations. Here is the effective frequency of interparticle collisions. Thus, two-body relaxation effects in such N-body models probably do not yield any interesting physics on a timescale of several first rotations when the gravity perturbation may be already large as a result of Jeans instability, i.e., in weakly collisional systems with the collective effects may be apparent before the collisional timescale is reached.

In Appendix A.3 of the present paper, following Rybicki (1971) and Hohl (1973), we shall use an experimental method of testing a computational procedure by repeating calculations using a mass spectrum. The latter would clearly show whether computations are sensitive to the undesirable particle relaxation effects.

The organization of the paper is as follows. In Sect. 2 the details of the numerical simulation model are discussed. The results of computer simulations are shown in Sect. 3 and compared with the predictions of the basic theory as outlined in Appendix A.1. Sect. 4 is devoted to a discussion of the principal results of the work and their application to observational data. Through Appendix A.2 the effect of interparticle encounters on the dispersion law of Jeans perturbations is estimated. In Appendix A.3 we check if the system is being correctly modeled as a collisionless Boltzmann (Vlasov) system.

© European Southern Observatory (ESO) 1999

Online publication: June 6, 1999
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