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

Appendix A:

A.1. The local stability criterion

A highly flattened disk of stars in almost circular orbit about the galactic center will be subject to self-gravity, which will tend to cause clumping of the matter. This tendency will be counteracted by centrifugal force due to the rotational motion of the mass, and stellar "pressure" due to the thermal motion. If the "binding energy"

of a clump of matter of radius in orbit about the galactic center at radius is negative, collapse will occur. If , any perturbation in density will be damped out. Below, through the studying of dispersion relations, we reexamine the theory of small-amplitude gravity oscillations and their stability in a practically collisionless, two-dimensional, and spatially homogeneous galactic disk of stars.

In order to find the dispersion relation describing the collective oscillations of a medium near its metaequilibrium state within the method of particle orbit theory, one must determine first the perturbed particle trajectories. ¹⁰ Therefore, we start by deriving formulae for nearly circular stellar trajectories in the rotating galactic disk with nonaxisymmetric perturbations due to spiral density waves. The perturbation of the main smoothed galactic potential will be assumed small, and the star's motion will be represented, as usual, by epicyclic free oscillations plus additional forced ones under the action of the gravitational field of the waves. Then, the perturbed (or forced) velocities will be used in the continuity equation to determine perturbation of the surface density. Equating the result with the surface density given by the asymptotic solution of the Poisson equation, the dispersion relation will be obtained. Finally, from the dispersion relation the local generalized stability criterion will be derived. The criterion guarantees the lack of arbitrary but not only axisymmetric Jeans-type unstable perturbations in a disk of mutual-gravitating particles.

In the absence of any perturbing gravity, a nearly circular orbit of a star (and a particle in planetary rings) may be represented as an epicyclic motion along the Coriolis ellipse (epicycle) with the simultaneous rotation of the ellipse (the guiding center) about the galactic center (Lindblad 1963; Chandrasekhar 1960; Binney & Tremaine 1987). In the epicyclic approximation the dispersion of random velocities of stars is taken to be small compared to the circular velocity of regular rotation determined by the smooth potential (Eq. [8]). This condition of nearly circular stellar orbits is normally satisfied in disks of flat galaxies that are seen in the sky. With the exception of resonances, the small perturbing gravitational field of a wave causes small forced oscillations in addition to the usual free epicyclic motion. Of course, it is doubtful that the approximation of nearly circular orbits adopted above is valid for the very central regions of flat galaxies. Assuming the nearly axially symmetric model, the vertical, normal to the plane motion in the rapidly rotating self-gravitating disk can be neglected (Shu 1970; Griv & Peter 1996). This assumption is partially supported by the global N-body simulations showing that the inclusion of the vertical motion makes little difference to the evolution of the thin, rapidly rotating disk (Hohl 1978).

The disk is subject to the equation of continuity and the equations of motion along the radial and azimuthal directions. The linearized equations of two-dimensional motion (9)-(10) in the frame of reference rotating with angular velocity can be rewritten in Hill's approximation as (Spitzer & Schwarzschild 1953; Toomre 1990):

where is the radius of the chosen circular orbit in the () plane, , and and are small perturbations of the coordinates. Eqs. (14) and (15) must be solved simultaneously with the continuity equation and the Poisson equation.

In the model described by Eqs. (14) and (15), the case of rare gravitational collisions between particles is considered when

where is the effective collision frequency. That is, collisions are so infrequent that their effects on both unperturbed and perturbed particle motions can be neglected. Evidence in favour of such an almost collisionless galactic model is provided by observations (Chandrasekhar 1960). The evolution of the system described by Eqs. (14) and (15) is determined by pure stellar encounters with collective modes.

Neglecting all the terms containing the small perturbation , the homogeneous differential equations (14)-(15) yield the ordinary Lindblad's expressions for unperturbed coordinates and velocities of a star along the elliptic-epicyclic orbit:

where and , are constants of integration (Spitzer & Schwarzschild 1953). The set of Eqs. (16)-(17) describes the rotation of a star along the epicycle with frequency and the mean epicyclic radius . Griv (Grivnev) (1988) and Griv & Peter (1996) have obtained expressions for the galactic orbits of stars to the second order of the epicyclic theory, when terms proportional to are also retained in the linearized equations of motion.

Now in order to find an inhomogeneous solution of Eqs. (14) and (15) we have to choose a particular form of the gravitational perturbation . Bearing in mind that the equilibrium distribution does not depend on the (and the z) coordinate, in a rotating frame, the perturbation may be expanded in a Fourier series

where is the Doppler-shifted complex frequency of excited waves as seen by the moving star and the term takes into account the possibility of different harmonics in the rotating system (many-armed waves), and and are the real and imaginary parts of the wavefrequency, respectively. Evidently is a periodic function of , and hence the azimuthal number m must be an integer. The criteria for stability differ for each m, and must be determined by a detailed analysis. In the framework of the linear theory, we can select one of the harmonics: , which rotates at a uniform rate and m is the number of spiral arms.

For such a form of the particular solution of the system (14)-(15) is (e.g., Lin & Lau 1979, Sellwood & Kahn 1991, and Griv et al. 1999):

The solutions (18) and (19) describe the forced velocities of a star in the radial and azimuthal directions under the action of the small gravity perturbation, and . Thus, the present theory suggests some systematic radial and azimuthal motions of the stars distributed in the form of a spiral-like flow field which is a small correction to the basic almost circular galactic motion.

To stress, the solutions (18) and (19) define the forced velocities of an individual star. In order to obtain the perturbed density, by using the continuity equation, we shall wish to average Eqs. (18) and (19) over the distribution of initial velocities. Such a distribution (the so-called modified Schwarzschild distribution) has been derived by Shu (1970) as follows:

Here and are well defined integrals of motion, that is, the epicyclic energy integral and the angular momentum integral, and a distance is defined by the relation

Then, a star in circular motion at a distance has precisely the given value of . Such a distribution function for the unperturbed system is particularly important because it provides a fit to observations (Shu 1970).

The continuity equation for a small density perturbation in a spatially homogeneous, two-dimensional disk is

where and we omitted the term , i.e., we neglected the curvature effect. This is a valid approximation if r is large (Lin & Lau 1979; Sellwood & Kahn 1991). To find a solution of Eq. (20) one has to choose an amplitude of the perturbation in the set (18)-(19). If a medium is only weakly inhomogeneous on the scale of the radial oscillation wavelength , i.e.,

where is the radial scale of the spatial inhomogeneity, the wave behaves approximately as a plane one (Alexandrov et al. 1984). In this case, the analysis can be greatly simplified by using the convenient WKB approximation. We seek thus the radial variation of the wave amplitude in a form:

where is the radial wavenumber (Shu 1970; Griv & Peter 1996). In Eq. (21), is a slowly varying amplitude, while the rapidly varying part of resides in the phase, i.e., . Since the amplitude and the wave vector depends weakly on the coordinates, we can construct the solutions of dynamic problems for weakly inhomogeneous disks in the form of an expansion in the parameter ; when calculating the terms of higher order one can simultaneously solve the field equations with any desired degree of accuracy (Alexandrov et al. 1984, p. 243). Further, by applying the zero-order or the so-called local approximation of the WKB method we shall assume that and are homogeneous, and . In other words, in the local WKB approximation the wave is considered plane: all terms of the order and of higher order are fully neglected (or all derivatives of and are neglected).

Thus, from here on we consider localized dispersion relations only. The reason for doing so is that localized solutions seem to describe the physical situation in what follows in a natural way. The meaning of localized dispersion relation has been discussed in plasma physics (Krall & Rosenbluth 1963; Alexandrov et al. 1984, p. 243; Krall & Trivelpiece 1986, p. 418).

Utilizing the above expansion of , we can approximate by substituting the unperturbed orbits from Eqs. (16) and (17). ¹¹ Such a substitution is permissible in the framework of the linear theory. Then by averaging over initial random velocities with the equilibrium Schwarzschild distribution , the integral in Eq. (20) can be approximated as:

where is the Bessel function of imaginary argument of the order l. Its argument is with the effective wavenumber defined by . To obtain Eq. (22) we introduced the polar coordinates in wavenumber space and . The integral in Eq. (20) was estimated using the forced coordinates of stars and (Eqs. [18] and [19]), the identity

and the formula:

where is the Bessel function of the first kind of the order l. Note that analogous integrals appear in the theory of magnetic plasma oscillations when one integrates the perturbed phase-space distribution function along the unperturbed particle trajectories (Krall & Rosenbluth 1963; Alexandrov et al. 1984, p. 110; Krall & Trivelpiece 1986, p. 402).

In Eq. (22) the denominators vanish when . At these values one gets hydrodynamic-type "wave-fluid" resonances, and thereby this solution obtained in the framework of linear approximation cannot be used. The most important resonances are the corotation one, for which and correspondingly , and the inner and outer Lindblad's resonances, for which and . Resonances of a higher order, , are dynamically less important (Griv & Peter 1996). It is obvious that all the terms except in the sum over the Bessel functions in Eq. (22) can be ignored for the most important long-wavelength oscillations, for which . (But, of course, in order to be appropriate for a WKB wave approximation we consider the perturbations with ; typically, in galaxies .) For example, comparing the contributions of to that of , in the long-wavelength limit one obtains (see below):

As we shall see later, one has to consider the case of perturbations only. Therefore the above ratio is of order and in accordance with the earlier assumption terms with can be neglected.

In Eq. (22) we should consider the low-frequency perturbations only. Indeed, in the opposite case of the high perturbation frequencies, , the effect of the disk rotation (or of magnetic field in plasmas) is negligible and therefore not relevant to us. This is because in this case the star motion is approximately rectilinear on the time and length scales of interest which are the wave growth/damping periods and wavelength, respectively. In this rotationless case instead of Eq. (22) another expression for the perturbed surface density can be found. In plasma physics the analogous problem has been described, e.g., by Alexandrov et al. (1984, p. 110).

To summarize, starting from equations of motion and the continuity equation we obtained the perturbed surface density (Eq. [22]). Self-consistency requires that it should be equal to the solution of the Poisson equation. Such an improved solution of the Poisson equation in the two-dimensional case in which we are interested has been obtained to the second order of the Lin-Shu asymptotic approximation of moderately tightly-wound spirals ( or , respectively):

(e.g., Lin & Lau 1979 and Bertin 1980).

Equating the "in-phase" parts of Eq. (22) and Eq. (24), we get the generalized Lin-Shu local dispersion relation for low-frequency oscillations with near a certain arbitrary radius r in the following form:

It is valid even for relatively open spirals and barlike structures throughout a disk excluding the resonance zones. Only the principal part of the disk between the inner (where ) and outer (where ) wave-fluid Lindblad's resonances considered. Note that Morozov (1980) by using a kinetic approach numerically calculated the contributions of the terms and found them to be smaller than (see also Griv et al. 1999, Fig. 1 in their paper).

The basic dispersion relation above is highly nonlinear in the frequency . Following the plasma physics method (Lifshitz & Pitaevskii 1981, p. 128), let us consider various limiting cases of perturbations described by some simplified variations of Eq. (25), that have a special interest for us. For instance, we solve this equation by successive approximations. In the first approximation, one can omit all terms which depend on and . Under this condition, the zeroth-order approximation solution is

Such a form for the trivial solution seems fairly straightforward. Indeed, when , that is, when the self-gravitation of the disk is neglected, from the generalized dispersion relation (25) we have ordinary epicyclic oscillations:

where is a small perturbation of the radius of the initially circular orbit, , at the motion in the central field with the effective potential energy (Griv & Peter 1996).

Using the elementary solution (26), in the next approximation the squared wavefrequency is

where is the square of the so-called Jeans frequency. This is the required simplified dispersion relation, which describes the physics and the condition of the gravitational (Jeans) modes in the two-dimensional disk. The hydrodynamic-type Jeans instability occurs when .

Generally, there are two branches to our solution (27): the case of long waves, or , in which we are especially interested, and the opposite case of short waves, . The short-wavelength instabilities (those with ) are not dangerous in the problem of the galactic disk stability, since they lead to the very small-scale perturbations of the density only. Therefore from now on, we consider just the long-wavelength (or the hydrodynamical) limit , for which the following expansions can be used

In the short-wavelength limit,

while in a more rigorous approximation is a monotonically decreasing function of l for a fixed .

The local dispersion relation in the simple form (27) generalizes that of the Lin-Shu one (Lin et al. 1969; Shu 1970). This type of the dispersion relation for spiral waves, derived in a similar form, e.g., by Morozov (1980, 1981b) who used a kinetic approach, takes into account effects of azimuthal forces (m and ). It goes beyond the original Lin-Shu relation in that it is now applicable to the critically important case of the nonaxisymmetric perturbations concerning spiral structures. This relation is qualitatively similar to the standard dispersion relation of Lin-Shu in that both in the long-wavelength, or fluid limit , and in the short-wavelength limit . Similar dispersion relation can also be derived from the Lynden-Bell & Kalnajs (1972, Eq. [A11] in their paper) dispersion relation for open spirals. Unlike Lynden-Bell & Kalnajs, Morozov, and Griv & Peter, we used here a simplified method of particle orbit theory.

In Eq. (27), is the so-called reduction factor, which is approximately equal to unity in dynamically cold systems () and is always smaller than unity in dynamically hot disks (). Lin & Shu (1966) first introduced such a reduction factor; they have already pointed out that the high-dispersion stars would not participate in the spiral pattern in full, and this effect can be described with the help of the reduction factor. Different forms of the reduction factor are given by Athanassoula (1984). The existence of solutions of the dispersion relation with implies the aperiodic Jeans instability. In this case of gravitational instability the wavefrequency is purely imaginary, so that the wave propagation cannot occur. The solutions with describe long-lived natural (harmonic) oscillations. The marginal condition between these cases is given by . To emphasize, this instability is hydrodynamical in nature and has nothing to do with any resonant effects. In a general sense, the instability represents the ability of a gravitating disk to relax from a nonthermal (or an almost nonthermal) state by collective collisionless processes in much less time than the binary collision time.

Apart from the obvious replacement of by k, which originates from the consideration of the nonaxisymmetrical modes, the relation (27) differs from the corresponding standard Lin-Shu expression by the appearance of the factor . This factor indicates an extra clumping associated with the azimuthal forces in the differentially rotating media: spiral perturbations, in contrast with radial ones, are subject to the influence of the nonuniform character of the rotational motion. Lau & Bertin (1978) first obtained a somewhat similar expression for the extra clumping in a gas dynamical model (see also Bertin & Mark 1978, Lin & Lau 1979, Bertin 1980, and Lin & Bertin 1984).

Let us further analyze the consequences of this simple dispersion relation on the dynamical behavior of disks of stars. First, by using the condition for all possible k to second order in asymptotic theory, a generalized stability criterion can be immediately obtained. Indeed, if the nonaxisymmetric Jeans-type perturbations are to be stable, the value of the stellar radial-velocity dispersion should be greater or at least equal to that given by Eq. (2). ¹² To repeat, it is clear from the criterion (2) that stability of the nonaxisymmetric perturbations in a nonuniformly rotating disk requires a larger velocity dispersion than the ordinary Toomre's critical value (cf. Fridman & Polyachenko 1984, Vol. 1, p. 323). It is crucial to realize that the various dynamical properties of the perturbations with different are peculiarities of the differentially rotating disks only. In a way of contrast, in the rigidly rotating disk and the critical velocity dispersion (2) is in fact equal to .

Second, according to the dispersion relation (27), the growth rate of the axisymmetric gravitational modes has a maximum at the radial wavenumber or at the radial wavelength

The above equation reflects the well-known fact that the velocity dispersion shifts the threshold of gravitational stability toward a longer wavelength. At the limit of stability with respect to axisymmetric gravity perturbations the critical radial velocity dispersion and the critical wavelength becomes approximately equal to . This reproduces the usual Toomre's stability criterion to have a stable disk against axisymmetric collapse and the usual Jeans-Toomre critical radial wavelength (Toomre 1964, 1977).

On the other hand, in the case of nonaxisymmetric perturbations of a differentially rotating disk, the critical wavelength is a slightly longer:

where in galaxies as a rule .

Third, the growth rate of the Jeans instability is

Generally, . That is, the instability growth rate is high and the instability develops rapidly on the dynamical timescale (which is the time of one galactic rotation ). Eq. (30) indicates that open "barlike" modes are seem to be the most unstable, . It is important to point out that the growth rate decreases as the radial velocity dispersion grows approximately as . It is also interesting that in the case of differentially rotating disks the growth rate is dependent on the mode number m; it is only in a rigidly rotating disk that the growth rate is independent of the mode number m. In addition, for the Jeans-unstable perturbations () the wavefrequency is purely imaginary, and , and therefore the instability develops aperiodically.

Finally, in Sect. 4 of the present paper, we confirmed the generalized local stability criterion (2) for the case of the most unstable spiral perturbations - barlike ones with - by local N-body computer simulations.

A.2. The effect of interparticle collisions

Thus far, we have studied the dynamics of the collisionless disk. Let us here estimate the influence of interparticle collisions on the dispersion law of Jeans perturbations using the simple method of particle orbit theory. Of course, the Boltzmann kinetic equation provides a more rigorous but much more complicated treatment of the problem of a collisional disk oscillations (Griv & Chiueh 1996; Griv & Yuan 1996; Griv et al. 1997a).

Including non-physical elastic (gravitational) interparticle collisions, the equations of motion (6)-(7) for an individual star in inertial frame with the origin at the disk center take the form:

where the friction term approximates the force produced by collisions, is the effective collision frequency, n is the number density of particles, s is the effective "radius" of a particle, denotes the average over particles of all random velocities v in a Maxwellian distribution, and the terms with are small corrections (in the case of rare, , and weak collisions, , in which we are especially interested). This is just the opposite of the procedure in ordinary gas dynamics, where collisions are the dominant effect. This approach is valid for high temperatures and low densities, when the mean potential between neighboring particles is small compared with the thermal energy. The collision model of the form (31)-(32) does not take into account the detailed mechanism of the gravitational long-range interaction such as the spatial distribution of particles, non-rectilinear orbits of particles in a rapidly rotating system, etc. (Griv et al. 1997a). It seems that this model can give qualitatively correct results in considered rarefied disks where the detailed effects of gravitational collisions may be ignored.

The linearized Eqs. (31) and (32), and , take the form:

where is the area constant and . Equations above describe the small departure of the actual radius from , which is chosen so that the constant of areas for the circular orbit is equal to the angular momentum integral . From these equations we get

where and for all and t.

The homogeneous differential Eqs. (33) and (34) yield the ordinary Lindblad's elliptic-epicyclic orbits:

where (Spitzer & Schwarzschild 1953).

The particular solutions yield the expressions for perturbed velocities (cf. Eqs. [18] and [19])

where and only the "in-phase" terms are included. As we can see from the equations above, in comparison with the collisionless disk in the collisional system one needs to replace the wavefrequency by ; thus if is small enough we can ignore these collisions.

Paralleling the analysis leading to Eq. (27) and making use of Eqs. (35) and (36), it is straightforward to show that the simplified dispersion relation can now be expressed as

where as usual is the squared Jeans frequency. The solution of Eq. (37) is

where for Jeans-unstable perturbations () and for Jeans-stable ones (), , and .

Eq. (38) describes the weak damping of Jeans-stable perturbations, . Such a stabilizing influence is quite obvious, because in general the effect of collisions is to disrupt the organized wave motion (Alexandrov et al. 1984). Accordingly, as a result of collisions, a Jeans-stable wave tends to be damped on a timescale of the order of the mean time between collisions . Clearly, however, these rare, , and weak, , gravitational collisions between particles do not affect the local stability criterions (2)-(3).

It follows from Eq. (38) that the collisional effects do not depend on the wavenumber k. The latter contradicts our recent results obtained with the exact Landau integral of collisions (Griv et al. 1997a). Therefore, gravitational collisions are poorly represented by an approximate method presented here. The results obtained in this Appendix indicate only a tendency of Jeans-stable perturbations to be damped in a colisional system, and the damping rate given by Eq. (38) is correct only to the order of magnitude.

Thus, it is found that rare and weak collisions between particles lead to the weak stabilization of Jeans-stable modes in a stellar disk. The effect is small: the time necessary for the wave amplitude to fall to of its initial value is about the collision time, . We have assumed and . This is much longer than the characteristic time of a single revolution of a disk .

According to observations, in the disk of the Galaxy the frequency of gravitational collisions between stars and giant molecular clouds yr^-1 (Grivnev & Fridman 1990). Therefore, even though the time is longer than the characteristic time of a single revolution of the Galaxy in the solar vicinity, it is quite sufficient to damp the standard Lin-Shu quasi-stationary density waves on the Hubble time yr. By this way, the effects of even rare (and weak) encounters may become essential.

A.3. Relaxation time in strictly two-dimensional simulations

Consider a system of mutual-gravitating particles. The local distribution functions must satisfy the Boltzmann kinetic equation

where is the total gravitational potential determined self-consistently from the Poisson equation, is the so-called collisional integral which defines the change of f arising from ordinary interparticle collisions, is the collision frequency, and is the quasi-steady state distribution function.

In plasma physics, Lifshitz & Pitaevskii (1981, p. 115) have discussed phenomena in which interparticle collisions are unimportant, and such a plasma is said to be collisionless (and in the lowest-order approximation of the theory one can neglect the collision integral in the kinetic equation). It was shown that a necessary condition is that : then the collision operator in the kinetic equation (39) is small in comparison with . In Appendix A.1, we have shown that generally speaking the frequency of collective Jeans-type oscillations in a stellar disk . Therefore, in the gravitation case in the lowest-order approximation of the theory we can neglect the effects of collisions between particles on a timescale of many rotations if . Lifshitz & Pitaevskii (1981) have pointed out that collisions may be neglected also if the particle mean free path is large compared with the wavelength of collective oscillations. Then the collision integral in Eq. (39) is small in comparison with the term .

In this Appendix we test numerically if the models used in our N-body simulations are being correctly modelled as collisionless Boltzmann (Vlasov) systems. The direct method of checking if the system is being modelled as a collisionless system is to repeat a calculation using a mass spectrum (Rybicki 1971). It is obvious that as a result of gravitational collisions there is a tendency towards energy equipartition between the various masses. Hohl (1973) has determined the experimental relaxation time and compared it with a theoretical prediction for the collisional relaxation time of a two-dimensional disk by using the method of global simulations; here we do such a comparison by using the method of local simulations.

Let us consider the strictly two-dimensional computer model consisting of stars of mass , stars of mass , and stars of mass . The total number of stars, which are distributed in the rectangular box with , is small, , in comparison with the number of stars in simulations presented in Sect. 3. Initially, the different mass groups of stars are distributed with the same velocity dispersion (with different temperatures).

In Sect. 3.1 of the present paper it has been found that the rigidly rotating disk becomes almost stable gravitationally for . In such a Jeans-stable system collective effects associated with the classical gravitational instability will not affect the random velocity dispersion of particles (Griv et al. 1994; Griv & Peter 1996): the change of velocity dispersion can be explained only by usual two-body encounters. ¹³ For this reason the initial condition was chosen to be a quasi-stable uniformly rotating disk with . Following Hohl (1973), let us define the relaxation time as the time required for the mean change of the kinetic energy per unit mass of the test star to equal the initial kinetic energy.

In Fig. A1 we show the change of the ratio of the mean particle (kinetic) energy, , , and (in units of the total kinetic energy of the system), for the different mass groups, where is the total velocity of a given mass group. As is expected, the two groups of heavy stars lose energy while the group of lightest stars gains an approximately corresponding amount of kinetic energy. Also as is expected, one can see the decrease in the change of the kinetic energy with time. This is because the collisional frequency is inversely proportional to the velocity dispersion (Eq. [12]), and thus the encounters only weakly affect the stars with high random velocities.

Fig. A1. Rate of change of the mean kinetic energy for stars of the three mass groups of the rigidly rotating model with , , and ; 1 - kinetic energy of stars with the mass of a star , 2 - with the mass of a star , and 3 - with the mass of a star . The two groups of heavy stars lose kinetic energy while the group of lightest stars gains an approximately corresponding amount of kinetic energy. The mean slope of the curves will result in energy equipartition after about 20 rotation periods. This result suggests that interparticle collisions do not play a significant role for instabilities studied in the paper.

As one can see, the mean slope of the curves shown in Fig. A1 will result in energy equipartition after about 20 rotation periods. It is crucial to realize that these relaxation times even for this relatively small number of model stars are much longer that the time of a single disk revolution. We conclude that the two-dimensional computer models used in the present study may indeed be considered as collisionless ones to a good approximation at least during the first 8-10 rotations which are of especial interest in spiral-galaxy simulation. Therefore, we argue that the collective effects studied in this paper were apparent before the collisional timescale was reached.

© European Southern Observatory (ESO) 1999

Online publication: June 6, 1999
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