Density wave formation in differentially rotating disk galaxies: Hydrodynamic simulation of the linear regime

Highlights

• The behavior of a galactic disk is examined numerically by a hydrodynamic code.

• The linear regime of density wave formation are explored.

• The simulation results are compared with the generalized “fluid”–wave theory.

• The disk evolution is fairly well described by the local approximation of the theory.

Abstract

Most rapidly and differentially rotating disk galaxies, in which the sound speed (thermal velocity dispersion) is smaller than the orbital velocity, display graceful spiral patterns. Yet, over almost 240 yr after their discovery in M51 by Charles Messier, we still do not fully understand how they originate. In this first paper of a series, the dynamical behavior of a rotating galactic disk is examined numerically by a high-order Godunov hydrodynamic code. The code is implemented to simulate a two-dimensional flow driven by an internal Jeans gravitational instability in a nonresonant wave–“fluid” interaction in an infinitesimally thin disk composed of stars or gas clouds. A goal of this work is to explore the local and linear regimes of density wave formation, employed by Lin, Shu, Yuan and many others in connection with the problem of spiral pattern of rotationally supported galaxies, by means of computer-generated models and to compare those numerical results with the generalized fluid-dynamical wave theory. The focus is on a statistical analysis of time-evolution of density wave structures seen in the simulations. The leading role of collective processes in the formation of both the circular and spiral density waves (“heavy sound”) is emphasized. The main new result is that the disk evolution in the initial, quasilinear stage of the instability in our global simulations is fairly well described using the local approximation of the generalized wave theory. Certain applications of the simulation to actual gas-rich spiral galaxies are also explored.

Introduction

The dynamics of highly flattened, rapidly and differentially rotating self-gravitating systems, in which the sound speed (thermal velocity dispersion) is smaller than the orbital velocity, has now been studied quite thoroughly since the pioneering works by Toomre (1964) for collision-free stellar disks and Goldreich and Lynden-Bell (1965) for gas sheets. The research is aimed above all to explain the origin of circular and spiral structures in galaxies, the fragmentation of the rotationally flattened solar nebula, rapid lunar and planetary formation, structures in accretion disks around massive objects, fine-scale structures of Saturn’s main rings and, finally, the enhanced angular momentum and mass redistribution in astrophysical disk configurations. One of the main trends has therefore been to analyze the perturbation dynamics in such systems, in both linear and nonlinear regimes. It has been shown that the evolution of all these systems are dominated by the internal instabilities of gravity perturbations (e.g. those produced by a spontaneous disturbance or, in rare cases, a companion system). In particular, unstable, i.e. amplitude-growing compression type waves, or density waves, can be self-excited in the main domain of the disk via the internal Jeans gravitational instability in a nonresonant wave–“fluid” interaction. The evolution of such self-gravitating disks is primarily driven by angular momentum redistribution so that growing gravity perturbations carry angular momentum from the inner parts to the outer parts, and gravitational forces are predominant. The system may then fall toward the lower potential energy configuration and use the energy so gained to increase its coarse grained entropy (Lynden-Bell and Kalnajs, 1972, Griv and Gedalin, 2004, Griv et al., 2008). In a general sense, the instability represents the ability of a gravitating system to relax from a nonthermal state by collective processes in much less time than the ordinary binary collision time (Morozov, 1978, Griv et al., 2001, Griv et al., 2002).

Because of the nature of the gravitation force, self-gravitating systems are always spatially inhomogeneous and rotate nonuniformly, i.e. the angular velocity of their rotation is a function of distance. The observed arms in these flat systems trail behind with respect to the direction of disk rotation (Pasha and Smirnov, 1982, Clampin et al., 2003, Fukagawa et al., 2004, Porco et al., 2005, Hedman et al., 2007). In the spirit of Lin and Shu, 1964, Lin and Shu, 1966, Lin et al., 1969, Yuan, 1969, Roberts and Yuan, 1970, Shu, 1970, in this paper we regard the spiral structure in astrophysical disks as a Lin–Shu type density wave pattern, which does not remain stationary in a frame of reference rotating around the disk center at a proper speed, excited as a result of the Jeans instability of small-amplitude gravity perturbations. The classical Jeans gravitational instability is set in when the destabilizing effect of the self-gravity in the rotating disk exceeds the combined restoring action of the pressure and Coriolis forces. The instability is one of the most frequent and most important instabilities in the stellar and in the planetary cosmogony (Polyachenko and Fridman, 1972, Goldreich and Ward, 1973, Sekiya, 1983, Griv et al., 2003) and galactic kinematics and dynamics (Yuan, 1969, Shu et al., 1972, Rohlfs, 1977, Bertin et al., 1989, Griv et al., 2002, Griv et al., 2006), and deals with the question of whether initial density fluctuations will be amplified or will die down. Jeans instability identifies nonresonant instabilities of fluctuations associated with almost aperiodically growing accumulations of mass. In other words, the instability associated with departures of macroscopic quantities from the thermodynamic equilibrium is hydrodynamical in nature and has nothing to do with any explicit resonant $\omega =\mathit{k}·\mathit{v}$ effects, where $\omega$ is the oscillation frequency, $\mathit{k}$ is the wavenumber of excited oscillations and $\mathit{v}$ is the particle’s velocity. We do not discuss in the present paper the density wave structures that are associated with spatially limited wave–particle resonances (e.g. Lynden-Bell and Kalnajs, 1972, Griv et al., 2000).

Lin and Shu, 1964, Lin and Shu, 1966, Lin et al., 1969, Roberts and Yuan, 1970, Yuan, 1969, Shu, 1970 stated clearly the concept of quasi-stationary density waves in spiral galaxies. See Rohlfs, 1977, Fridman and Polyachenko, 1984, Binney and Tremaine, 2008 for reviews of the original Lin–Shu density wave theory and its astronomical implications. Our present-day concept is somewhat different. This generalized wave theory was elaborated concurrently on the basis of the Lin and Shu original ideas by a number of authors, using both kinetic and fluid-dynamical approaches (e.g. Bertin and Mark, 1978, Lin and Lau, 1979, Bertin, 1980, Morozov et al., 1985, Bertin et al., 1989, Griv et al., 2001, Griv et al., 2002, Griv et al., 2006, Lou et al., 2001, Griv, 2011). Accordingly, a self-gravitating system of stars or gas clouds in a galaxy exhibits collective, gravitationally unstable modes of motions. Assuming the weakly inhomogeneous system, the behavior of small perturbations of equilibrium parameters is seeked in the form of a superposition of different Fourier harmonics corresponding to different normal modes of oscillation of the system, stable or otherwise,

$$X_1(\mathit{r}\text{,}t)={\displaystyle \sum _{\mathit{k}}}\mathfrak{R}\left\{{\stackrel{\sim }{X}}_{\mathit{k}}(r\text{,}z)e^{\text{i}\mathit{k}·\mathit{r}-\text{i}{\omega }_{\mathit{k}}t}\right\}\text{,}$$

where $(r\text{,}\phi \text{,}z)$ are the cylindrical coordinates with the origin at the galactic center and the axis of disk rotation is taken oriented along the z-axis, ${\stackrel{\sim }{X}}_{\mathit{k}}(r\text{,}z)$ is an amplitude, the phase $\mathit{k}·\mathit{r}$ is a large quantity, $\mathit{k}$ is the real wave vector, $k=\sqrt{k_r^2+k_{\phi }^2}\text{,}{k}_r$ is the radial wavenumber, $k_{\phi }$ is the azimuthal wavenumber, ${\omega }_{\mathit{k}}=\mathfrak{R}{\omega }_{\mathit{k}}+\text{i}\mathfrak{I}{\omega }_{\mathit{k}}$ is some complex frequency of excited collective oscillations and suffixes $\mathit{k}$ denote the $\mathit{k}$th Fourier component. The amplitude ${\stackrel{\sim }{X}}_{\mathit{k}}$ is a slowly varying in space and time, and the rapidly varying part of $X_1$ is absorbed in its phase, $k_{r}r\gg 1$ where $k_r$ is the radial wavenumber of the pattern. In a local WKB approximation we are actually exploring, both ${\stackrel{\sim }{X}}_{\mathit{k}}=\text{const}$ and $\mathit{k}=\text{const}$. The constant phase velocity of spiral density waves is ${\mathrm{\Omega }}_{\text{p}}=\mathfrak{R}{\omega }_{\mathit{k}}/m$ called the pattern rotation speed, where m is the positive azimuthal mode number (=number of spiral arms for a given harmonic). Since ${\mathrm{\Omega }}_{\text{p}}$ is a constant, independent of time or radius, each component will remain identical with time and therefore spirals do not wind up by the general differential rotation. In a linear approximation, a perturbation is considered to be a superposition of different oscillation modes, and the coexistence of several spiral (and circular) waves is possible. This disturbance in the disk will grow until it is limited by some nonlinear effect. Thus, the imaginary part of the wavefrequency $\mathfrak{I}{\omega }_{\mathit{k}}$ corresponds to a growth or decay of the components in time, $\propto \exp (\mathfrak{I}{\omega }_{\mathit{k}}t)$, and $\mathfrak{R}{\omega }_{\mathit{k}}$ corresponds to a rotation with constant angular velocity ${\mathrm{\Omega }}_{\text{p}}$. Unstable normal modes with higher $\mathfrak{I}{\omega }_{\mathit{k}}$ are more likely to achieve a higher amplitude and play more important roles. The gravitational field of a traveling pattern results in the perturbed gas and stars motion in addition to the mean circular motion. A longitudinal Lin–Shu density wave is associated with compression and decompression in the direction of travel, which is the same process as the ordinary sound waves in gases (Fig. 1(b)).

Perturbations with $m=1$ prove especially interesting. This so-called lopsidedness, or the $m=1$ asymmetry, is oftenly seen in the distribution of stars and gas in the outer disks of many galaxies (e.g. van Eymeren et al., 2011).

Thus, when $\mathfrak{I}{\omega }_{\mathit{k}}>0$, the medium transfers its energy to the growing wave and oscillation buildup occurs. The wave propagation is a rigid rotation process at a fixed phase velocity, despite the general differential rotation of the system. At the same time the amplitude of the wave grows exponentially in the linear regime of the instability. As a result, the alternating density enhancements (circles and/or spiral arms) and depletion zones (interarm regions) consist of different material at different times. Lin–Shu unforced density waves cause a temporary bunching together of orbiting disk’s particles and the material stream across the arms; spiral arms (interarm regions) are genuine indicators of higher-than-average (lower-than-average) galactic matter density.¹ This density wave structure may be excited by real instabilities of gravity perturbations. Low-m (say, $m⩽4$) waves are the most important because they are associated with large-scale phenomena (Lin et al., 1969, Lin et al., 1978, Shu, 1970, Rohlfs, 1977). Following Lin and Shu, 1964, Lin and Shu, 1966, Lin et al., 1969, Yuan, 1969, Roberts and Yuan, 1970, Shu, 1970, in the present paper we restrict the analysis to a treatment of even Jeans perturbations (Fig. 1(b)) which are symmetric with respect to the equatorial $z=0$ plane. It seems likely that these perturbations are associated with such phenomena as, for example, the appearance of the spiral structure of galaxies and protoplanetary disks (Rohlfs, 1977, Fridman and Polyachenko, 1984, Binney and Tremaine, 2008, Boss, 2008, Alexander et al., 2008, Cuzzi et al., 2008, Cossins et al., 2009, Boley et al., 2010, Griv et al., 2002, Griv et al., 2006, Griv and Gedalin, 2012) and the fine-scale $\sim 100$ m structure of Saturn’s A and B rings (Griv et al., 2003, Griv, 2007). It was already proposed that all planets and planetesimals in the solar system, giant extrasolar planets, multiple star systems and small moonlets embedded in Saturn’s rings form due to Jeans gravitational fragmentation of parts of astrophysical disks (Polyachenko and Fridman, 1972, Goldreich and Ward, 1973, Boss, 1997, Ward, 2000, Snytnikov et al., 2004, Griv and Gedalin, 2011, Forgan and Rice, 2011, Michael et al., 2012).

We consider a model galaxy that differs from that used by Lin and Shu, 1964, Lin and Shu, 1966, Lin et al., 1969, Yuan, 1969, Roberts and Yuan, 1970, Shu, 1970: namely, we consider a more realistic galaxy to be a two-dimensional young stellar or gas disk in hydrostatic equilibrium in the external static potential of a bulge, an old relaxed stellar disk and a dark matter halo (if it exists at all). A similar model has been suggested by Lin and Lau, 1979, Bertin and Mark, 1978. In fact, actual disk galaxies are made of various subsystems with different dynamical roles and with different kinematic properties and types of mass distribution. Spiral galaxies are not “razor thin” but contain a nuclear bulge as well as a large halo that extends far beyond the visible disk. The youngest stars and the interstellar gas and dust form the plane subsystem. This thin disk component of galaxies has a thickness that is typically $10\%$ or even less of their diameters. Massive dark halos create flat rotation curves and may affect the dynamics and evolution of galaxies. When we speak of the spiral structure of a galaxy, we usually have in mind the spiral pattern of precisely the thin subsystem. Modern near-infrared observations have also detected the spiral structure in the old thick stellar disk. Our simulations are, however, inadequate to describe the evolution of the thick subsystem, because they do not have the old relaxed stellar disk as an active component. It is important to note once again that the aim of this paper is to model the spirals in the extremely thin subsystem of the galaxy.

Thus, in our model, gravity includes the self-gravity of the dynamically active thin disk plus that of inactive external components, while the pressure is given by an equation of state $P=P(\mathrm{\Sigma }\text{,}T)$, with P being the gas pressure, $\mathrm{\Sigma }$ the gas density and T the gas temperature.

In Appendix A below, we show that the Jeans instability is sensitive to the spread of particle random velocities (“temperature”). The ordinary Safronov–Toomre (Safronov, 1960, Safronov, 1980, Toomre, 1964) thermal velocity dispersion (the ordinary Safronov–Toomre sound speed $c_{\text{T}}$)

$$c_{\text{T}}=\frac{\pi G{\mathrm{\Sigma }}_0}{\kappa }$$

should stabilize only axisymmetric gravity perturbations, i.e. circular waves, where ${\mathrm{\Sigma }}_0(r)$ is the local equilibrium surface mass density,

$$\kappa =2\mathrm{\Omega }{\left(1+\frac{r}{2\mathrm{\Omega }}\frac{\text{d}\mathrm{\Omega }}{\text{d}r}\right)}^{1/2}\gtrsim \mathrm{\Omega }$$

is the epicyclic frequency and $\mathrm{\Omega }(r)$ is the angular velocity. The differentially rotating disk ($\text{d}\mathrm{\Omega }/\text{d}r\ne 0$) is still unstable against nonaxisymmetric (spiral) perturbations. The modified sound speed against arbitrary but not only circular perturbations $c_{\text{crit}}$ is larger than $c_{\text{T}}$ (although still of the order of $c_{\text{T}}$), and is given by $c_{\text{crit}}\gtrsim (2\mathrm{\Omega }/\kappa )c_{\text{T}}$ at all radii. In galaxies, $2\mathrm{\Omega }/\kappa =1.5-1.8$, and thus the critical sound speed is given finally by the following inequality:

$$c_{\text{crit}}\gtrsim (2\mathrm{\Omega }/\kappa )c_{\text{T}}\approx 2c_{\text{T}}$$

or by the following conventional Toomre’s $Q\equiv c_s/c_{\text{crit}}$-value, which describes the degree of random motion present:

$$Q_{\text{crit}}=c_{\text{crit}}/c_{\text{T}}\gtrsim 2\text{,}$$

respectively, with $c_s$ being the sound speed. It can be seen that only for uniformly rotating disks ($2\mathrm{\Omega }/\kappa =1$), the stability criterion for nonaxisymmetric perturbations is the same as that for axisymmetric perturbations, $c_s⩾c_{\text{T}}$ (Morozov, 1985, Griv et al., 2006). The free kinetic energy associated with the differential rotation of the system is one possible source for the growth of the energy of the spiral gravity perturbations, and appears to be released when angular momentum is transferred outward (Griv et al., 2008).

The growth rate of the gravity-unstable modes has a maximum $\mathfrak{I}{\omega }_{\mathit{k}}\sim \mathrm{\Omega }$ at the wavelength

$${\lambda }_{\text{crit}}\approx 4\pi c_s/\kappa$$

and wave-structures appear on a dynamical time scale $\sim {\mathrm{\Omega }}^{-1}$. As is seen, the most unstable modes of actual differentially rotating three-dimensional disk have radial wavelengths of some $2\pi$ the characteristic thickness $2h$ of the system (this is because $c_s/\kappa \approx h$). See Griv and Gedalin (2011) for a discussion of the problem. The above equation reflects the well-known fact that the thermal velocity spread shifts the threshold of Jeans instability toward a longer wavelength, and the wavelength is proportional to the mean in-plane size of an epicycle $\approx c_s/\kappa$.

At the limit of stability with respect to all perturbations of a differentially rotating disk (including the most unstable spiral ones) the sound speed $c_s\approx 2c_{\text{T}}$ and the critical wavelength becomes therefore approximately equal to

$${\lambda }_{\text{crit}}=2{\lambda }_{\text{JT}}\text{,}$$

where ${\lambda }_{\text{JT}}=4{\pi }^{2}G{\mathrm{\Sigma }}_0/{\kappa }^2$ is the Jeans–Toomre wavelength (Toomre, 1964). As one can see, the finite inclination of spiral arms shifts the threshold of instability towards a longer wavelength than is implied by the ordinary Jeans–Toomre critical wavelength ${\lambda }_{\text{JT}}$ (and larger wavelength will include more mass). Only wavelengths close to ${\lambda }_{\text{crit}}$ are unstable: both small-scale $\lambda \ll {\lambda }_{\text{crit}}$ and large-scale $\lambda \gg {\lambda }_{\text{crit}}$ disturbances are gravitationally stable. It means that of all harmonics of initial perturbation, several dominant perturbations with the maximum of the growth rate $\mathfrak{I}{\omega }_{\mathit{k}}\sim \mathrm{\Omega }$, the wavelength $\lambda \sim {\lambda }_{\text{crit}}$ and number of spiral arms $m\sim m_{\text{crit}}$ will be formed rapidly in time of a single rotation. In simulations, it is essential to keep the ${\lambda }_{\text{crit}}$ as resolved as possible in order to reveal the gravity-unstable structures.

The density wave ought to manifest itself as a characteristic systematic change in the velocity field (Lin et al., 1969, Lin et al., 1978, Yuan, 1969, Rohlfs, 1977). The galactic material will exhibit small azimuthal and radial streaming motions in addition to the basic circular rotation in a coherent structure like a spiral density wave pattern. In particular, the gas and stars move faster on the outer edge of the spiral arm and slower on the inner edge, and normally the point masses in the spiral arm moves toward the galactic center (see Eqs. (A.10), (A.11) below). As a rule, the amplitudes of these azimuthal and radial components are of the order of sound speed and they are small compared with the mean circular motion.

Notice that Yuan (1969) was the first to examine small systematic velocities of the gas and stellar populations of our own Galaxy and claimed that they can be used as an important test of the Lin–Shu density wave theory; “the existence of the tangential systematic motion of the gas is most galaxies clearly demonstrated by the oscillations of the observed rotation curves” (Yuan, 1969, p. 873).

As we have shown analytically (Section A.3), the equilibrium parameters of the disk and the azimuthal mode number determine the spiral pattern speed of Jeans-unstable perturbations (in a circular rotating frame):

$${\mathrm{\Omega }}_{\text{p}}=\frac{\mathfrak{R}{\omega }_{\mathit{k}}}{m}\approx \frac{\mathrm{\Omega }}{{\mathit{rk}}_{\perp }^{2}L}\text{,}$$

where $k_{\perp }=\sqrt{k_r^2+{(m/r)}^2}$ is the total in-plane wavenumber, $|L|$ is the radial scale of spatial inhomogeneity and in galaxies $\left|L\right|=3''–''4$ kpc. In Eq. (7), ${\mathit{rk}}_{\perp }^{2}L\gg 1$, therefore ${\mathrm{\Omega }}_{\text{p}}\ll \mathrm{\Omega }$. As is seen, the typical pattern speeds of spiral structures in Jeans-unstable ($\mathfrak{I}{\omega }_{\mathit{k}}>0$) disks are only a small fraction of some average angular velocity ${\mathrm{\Omega }}_{\text{ave}}$ (in a rotating frame). The theory states that in homogeneous ($|L|\to \infty$) disks ${\mathrm{\Omega }}_{\text{p}}=0$. Also, because ${\mathrm{\Omega }}_{\text{p}}$ does not depend on m, each Fourier component of a gravity perturbation in a spatially inhomogeneous system will rotate with the same constant angular velocity (but with the different growth rate $\mathfrak{I}{\omega }_{\mathit{k}}$) and any “mixing” of the spirals is therefore absent.

In this series of papers, we present a detailed study of the origin of spiral activity both in hydrodynamic and in N-body simulations. In the current study, we solve numerically a self-consistent system of the hydrodynamical equations and the Poisson equation describing the motion of a self-gravitating ensemble of stars or gas clouds, looking for time-dependent waves which propagate in an infinitesimally thin disk. The approximation of a razor-thin disk is a valid approximation if one considers perturbations with a radial wavelength $\lambda =2\pi /k_r$ that is greater $2h$, the disk thickness (Safronov, 1960, Safronov, 1980, Toomre, 1964, Shu, 1970, Shu, 1984, Griv and Gedalin, 2012). We expect that the waves and their instabilities propagating in the disk plane have the greatest influence on the development of structures in the system under study. Our research addresses the hypothesis that both circular and spiral structures in rapidly rotating, highly flattened and gas-rich galaxies form directly by the internal Jeans instability of spontaneous gravity disturbances.

In this paper, our purpose is to compare the results of simulations with the expectations based on the local approximation of the generalized linear fluid-dynamical theory of disks. We will find that even in the global simulations the theoretical model is consistent with four major features of our simulations. First, the novelty is that we test the validities of the modified Safronov–Toomre criterion (4) for stability of arbitrary but not only axisymmetric perturbations developing in a differentially rotating gas disk (Section 4.1). Second, we address the problems of the characteristic size of spiral structure, that is, the size of spirals and the spacing between them (Eq. (6)), and the superposition of unstable Fourier harmonics of oscillation of the system (Eq. (1)) (Section 4.2). Third, in Section 4.3 we compare the systematic motion of model “particles” with the velocity field of density waves (Eqs. (A.10), (A.11)). And four, in Section 4.4 we verify our analytical findings that the pattern speed of Jeans-unstable perturbations does not depend on m (Eq. (7)). To be consistent with the theory that considers only the linear regime of the instability, we explore here the very first orbital periods of dynamical evolution of numerical models. Clearly, on longer time scales, waves interact with themselves and with fluid elements in a nonlinear fashion, and then this limits the extent of our calculations (Morozov, 1978, Griv et al., 2001, Griv et al., 2002).

It is important to note that the simplest description of a particulate system of stars or gas clouds with a self-consistent potential field within the framework of the simple fluidlike hydrodynamic or macroscopic model, similar to that used by Goldreich and Lynden-Bell, 1965, Lin and Lau, 1979, Drury, 1980, Griv et al., 2002, Griv et al., 2003, Griv et al., 2008 (see also Section A), does not exhaust all the possible types of motion. There exist perturbations that can be considered in a consistent fashion only within the framework of kinetic or microscopic theories, based on either the Boltzmann (Vlasov) kinetic equations or particle dynamics equations, taking into account effects of the thermal spread of particle velocities. This comes about because of the strong interaction between a hydrodynamically stable wave and resonant particles with random velocities nearly equal to the phase velocity of the wave. A property of collective waves in a self-consistent system that is predicted by microscopic theories but which is completely outside the scope of macroscopic theory is the collisionless growth of amplitudes of normal modes of oscillations, i.e. Landau type instabilities (the inverse Landau damping), instabilities from particles resonant with waves (Alexandrov et al., 1984, Swanson, 1989). In turn, a resonant wave–particle interaction in a particulate disk can be considered to be a generating mechanism for propagating density waves, thereby leading to long-lived spiral patterns in differentially rotating galaxies. The kinetic theory of such an interaction has been developed and the Landau excitation of spiral density waves in a stellar disk of flat galaxies has been proposed by Griv et al. (2000). This excitation of waves has been suggested as a mechanism for the formation of recurrent spiral arms and the slow dynamical relaxation of galaxies in a regime of hydrodynamical Jeans-stability.

In this connection, Fujii et al., 2011, Baba et al., 2013, D’Onghia et al., 2013 have recently found that – by using particulate (N-body) numerical simulations – spiral structures are induced by initial perturbations and become long-lived, nonlinear and recurrent features of computer-generated galaxies. Similar oscillatory-growing modes have already been discovered by Sellwood and Lin, 1989, Donner and Thomasson, 1994 in collisionless N-body experiments. In those experiments, a new type of small-amplitude long-lived spiral modes in many-body models of galaxies was found. These modes lead to wave amplitudes far greater than can be explained by particle noise, and the instabilities do not scale with N, the number of particles. Therefore, the recurrent instabilities discovered by these authors cannot be a chance superposition of unrelated features; the instabilities appear to be physical. The shapes of the spirals certainly are not shapes predicted by the well-known Lin–Shu “nonresonant” dispersion relation for linear density waves and the trailing spiral shape of the mode transport angular momentum outwards (Sellwood and Lin, 1989, Donner and Thomasson, 1994). As the instability develops through changes to the particle distribution function in the neighborhood of a particular speed, it strongly resembles Landau excitation in plasmas (Sellwood and Lin, 1989).²

In this series of papers, the methods of quantitative description of galactic disks will be developed from simple hydrodynamic models to advanced particulate concepts. In the following publications of the series, by using collisionless N-body simulations, we intend to show that hydrodynamically Jeans-stable disks (see Fig. 6 below) with radially dependent densities, angular velocities and temperatures are nevertheless unstable to resonant Landau type instabilities. It is expected that the larger the angular velocity gradient, the more unstable the system. This microinstability leads to oscillatory-growing waves propagating through the system with phase velocities, which resonate with particles that drift at the same velocities. To emphasize it again, this is the Landau excitation, or collisionless resonant excitation of Jeans-stable oscillations, which cannot be simulated in the framework of the hydrodynamic model. The effect is absent in the latter case because the governing equations lose microstructure (Alexandrov et al., 1984, Swanson, 1989, Griv et al., 2000).