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Advances in Space Research 38 (2006) 47–56

Spiral galaxies as gravitational plasmas

E. Griv a,*, M. Gedalin a, C. Yuan b

a Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israelb Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 106, Taiwan

Received 7 July 2004; received in revised form 23 December 2004; accepted 31 December 2004

Abstract

This article reviews recent studies of dynamics of disk-shaped galaxies with special emphasis on their spiral structures. In a localversion of kinetic stability theory, the very existence and the value of the critical wavelength of the spiral structure arising due tononresonant Jeans-type instability of gravity perturbations is explained. A formal analogy between the collective oscillations in a rotatingself-gravitating disk and the oscillations of a hot nonneutral plasma in a magnetic field is explored.� 2006 Published by Elsevier Ltd on behalf of COSPAR.

Keywords: Galaxies: Kinematics and dynamics; Galaxies: Structure; Galaxies: Spiral galaxies; Waves and instabilities

1. Introduction

Self-gravitating disk systems are of great interest inastrophysics because of their widespread appearance, e.g.,disks in spiral galaxies, pancakes and accretion disksaround massive objects, the protoplanetary clouds, and,finally, the main rings of Saturn. Such nonuniformly rotat-ing disks are highly dynamic and are subject to various col-lective instabilities of gravity perturbations (e.g., thoseproduced by a spontaneous perturbation or a companionsystem). This is because their evolution is primarily drivenby angular momentum redistribution. The system maythen fall toward the lower energy configuration and usethe energy so gained to increase its coarse grained entropy.

The theory of spiral structure of rotationally supportedgalaxies has a long history but is not yet complete. Eventhough no definitive answer can be given at the presenttime, the majority of experts in the field is yielded to opin-ion that the study of the stability of collective vibrations indisk galaxies of stars is the first step towards an under-standing of the phenomenon. This is because the bulk ofthe total optical mass in the Milky Way and other flat gal-

0273-1177/$30 � 2006 Published by Elsevier Ltd on behalf of COSPAR.

doi:10.1016/j.asr.2006.05.019

* Corresponding author.E-mail address: griv@bgu.ac.il (E. Griv).

axies is in stars, and therefore stellar dynamical phenomenaplay the main role. In the spirit of Lin and Shu (1964,1966), Lin (1967), Lin et al. (1969), Yuan (1969a,b), andShu (1970), we regard spiral structure in most disk galaxiesof stars as a wave pattern, which does not remain station-ary in a frame of reference rotating around the center of thegalaxy at a proper speed, excited as a result of the classicalJeans instability of gravity perturbations.

In this paper, a connection between several plasmaphysics phenomena and the dynamics of disk galaxies isestablished. Similarities between self-gravitating systemsand ordinary plasmas arise from the common long-range � 1/r2 nature of the basic forces, whereas differencesarise from the opposite signs of these forces. It seems thatsuch a connection deepens our understanding of the natureof spiral structure phenomenon and broadens the readeraudience. Because of the long-range nature of the gravita-tional forces between particles, a self-gravitating systemexhibits collective (or cooperative) modes of motions – modesin which the particles in large regions move coherently or inunison (Sweet, 1963; Lynden-Bell, 1967; Marochnik, 1968).

We are concerned with low amplitude Jeans-type grow-ing oscillations and their stability by studying the spiralstructure of galaxies as the collective effect in a self-gravi-tating system. Generally, the term ‘‘Jeans instabilities’’identifies gravitational nonresonant instabilities associated

1 Self-gravitating evolution of a thick disk is generally very similar tothat of a razor-thin disk, because the induced motions are almost planar.One qualitative difference the disk’s finite but small thickness makes is thatit tends to be stabilizing by reducing self-gravity at the midplane(Vandervoort, 1970; Morozov, 1981a; Shu, 1984).

48 E. Griv et al. / Advances in Space Research 38 (2006) 47–56

with almost aperiodically growing accumulations of mass(gravitational collapse). The instability is driven by astrong nonresonant interaction of the gravity fluctuationswith the bulk of the particle population, and the dynamicsof Jeans perturbations can be characterized as a fluid-likewave–particle interaction. The Jeans instability does notdepend on the behavior of the particle distribution functionin the neighborhood of a particular speed, but the deter-mining factors of the instability are macroscopic parame-ters like the mean random velocity spread, mean density,and angular velocity of regular galactocentric rotation(Griv et al., 1994, 1999, 2001, 2002, 2003; Griv and Peter,1996a,b; Griv, 1998; Griv and Gedalin, 2003, 2004). Thusa kinetic description yields results almost no different fromthose obtained hydrodynamically (Lau and Bertin, 1978;Lin and Lau, 1979). In plasma physics an instability ofthe Jeans type is known as the negative-mass instabilityof a relativistic charged particle ring or the diocotron insta-bility of a nonrelativistic ring in a nonneutral plasma thatcaused azimuthal clumping of beams in synchrotrons, beta-trons, and mirror machines (Davidson, 1972).

One has to realize, however, that in many respects gravi-tating systems differ strongly from laboratory plasmas. Thedifficulties for a satisfactory understanding of the dynamicsof gravitating particulate systems are due to the well-knownfact that in a system of N� 1 (generally, N � 1011) gravita-tionally interacting particles Debye screening, as distinctfrom plasma, is absent. Also, a principal difference betweensystems of electrically charged particles and self-gravitatingsystems is that the latter, because of the nature of the gravi-tation force, are always spatially inhomogeneous. These cir-cumstances make the statistical description of gravitationalsystems much more complicated.

2. Basic equations

The problem is formulated in the same way as in plasmakinetic theory. In the rotating frame of a disk galaxy, the col-lisionless motion of an ensemble of identical stars in the planeof the system can be described by the collisionless Boltzmannkinetic equation for the distribution function f (r,v, t)

ofotþ vr

oforþ Xþ vu

r

� � ofouþ 2Xvu þ

v2u

rþ X2r � o@

or

!ofovr

� j2vr

2Xþ vrvu

rþ 1

ro@ou

� �ofovu¼ 0; ð1Þ

where the total azimuthal velocity of the stars is representedas a sum of the peculiar (random) vu and the basic rotationvelocity Vrot = rX, vr is the velocity in the radial direction,the epicyclic frequency j (r) is defined by j = 2X [1 +(r/2X) (dX/dr)]1/2, and r, u, z are the galactocentric cylindri-cal coordinates. The quantity X (r) denotes the angular veloc-ity of galactic rotation at the distance r from the center.Random velocities are usually small compared with Vrot.Collisions are neglected here because the collision frequencyis much smaller than the cyclic frequency X. In Eq. (1), @(r, t)

is the total gravitational potential determined self-consis-tently from the Poisson equation

o2@or2þ 1

ro@orþ 1

r2

o2@ou2þ o2@

oz2¼ 4pG

Zf dv: ð2Þ

Eqs. (1) and (2) give a complete self-consistent description ofthe problem for disk modes. The combined system of Eqs. (1)and (2) is the counterpart of the system of Vlasov and Max-well equations in electromagnetic plasmas. Hence the term‘‘gravitational plasma’’ is appropriate for the descriptionof mutually gravitating stars (Lin and Bertin, 1984) [(andparticles of Saturn’s rings; Griv (1998), Griv et al. (2003),Griv and Gedalin (2003))]. Reviews of plasma kinetic theoryare given by Mikhailovsky (1983), Alexandrov et al. (1984),Krall and Trivelpiece (1986), and Swanson (1989).

The equilibrium state is assumed, for simplicity, to be anaxisymmetric and spatially inhomogeneous only along the r-coordinate stellar disk. Second, in our simplified model, theperturbation is propagating in the equatorial plane of thedisk. This approximation of an infinitesimally thin disk is avalid approximation if one considers perturbations with aradial wavelength kr that is greater than the typical diskthickness h = 100–200 pc. We assume here that the starsmove in the disk plane so that vz = 0. This allows us to usethe two-dimensional distribution function f = f (vr,vu, t)d (z)such that � fdvr dvudz = r, where r (r, t) is the surface density.We expect that the waves and their instabilities propagatingin the disk plane have the greatest influence on the develop-ment of structures in the system (Shu, 1970). Limiting our-selves to the case of infinitesimally thin disk simplifies thealgebra without introducing any fundamental changes inthe physical results (Safronov, 1980).1

2.1. Perturbation

Let us suppose that the nonlinear effects in galactic disksare small, so that the linear theory is a good first approxima-tion. We proceed by applying the standard procedure of thelinear approach and decompose the time dependent distribu-tion function of stars f = f0 (r,v) + f1 (r,v, t) and the totalgravitational potential @ ¼ @0 (r) + @1 (r, t) with |f1/f0|� 1and |@1=@0|� 1 for all r and t. The functions f1 and @1 areoscillating rapidly in space and time, while the functions f0

and @0 describe the nondeveloping ‘‘background’’ againstwhich small perturbations rapidly develop.

To determine oscillation spectra, let us consider the sta-bility problem in the lowest (local) WKB approximation:the perturbation scale is sufficiently small for the disk tobe regarded as spatially homogeneous (e.g., Alexandrovet al., 1984, p. 243). This is accurate for short wave pertur-bations only, but qualitatively correct even for perturba-

E. Griv et al. / Advances in Space Research 38 (2006) 47–56 49

tions with a longer wavelength, of the order of the diskradius R � 20 kpc. In the local WKB approximation inequations above, assuming the weakly inhomogeneousdisk, the perturbation is selected in the following form (inthe rotating frame):

ff1;@1g ¼X

k

ffk;@kg expðikrr þ imu� ix�;ktÞ þ c:c:; ð3Þ

where fk and @k are amplitudes that are constants in spaceand time, kr = 2p/kr is the radial wave number, m is thenonnegative (integer) azimuthal mode number (= numberof spiral arms), x*,k = xk � mX is the Doppler-shiftedcomplex frequency of excited waves, suffixes k denote thekth Fourier component, ‘‘c.c.’’ means the complex conju-gate, and |kr|R� 1. The perturbed density r1 (r, t) = � f1 dv

is also of this form. In the linear theory, one can selectone of the Fourier harmonics: {fk,@k}exp (ikrr +imu � ix*t) + c.c.. The solution in such a form representsa spiral wave with m arms or a ring (m = 0). With uincreasing in the rotation direction, we have kr > 0 for trail-ing spiral patters, which are the most frequently observedamong spiral galaxies. With m = 0, we have the densitywaves in the form of concentric rings that propagate awayfrom the center when kr > 0 or toward the center whenkr < 0. The imaginary part of x* corresponds to a growthðIx� > 0Þ or decay ðIx� < 0Þ of the components in time,f1;@1 / exp ðIx�tÞ, and the real part to a rotation withangular velocity Xp ¼ Rx�=m. Thus, when Ix� > 0, themedium transfers its energy to the growing wave and oscil-lation buildup occurs. A galaxy is considered as a superpo-sition of different oscillation modes. A disturbance in thedisk will grow until it is limited by some nonlinear effect.

Linearizing Eq. (1), one obtains the equation for thedeveloping perturbation

df1

dt¼ o@1

orof0

ovrþ 1

ro@1

ouof0

ovu; ð4Þ

where d/dt means total derivative along the star orbit andf0 is a given equilibrium distribution.

2.2. Equilibrium stellar trajectories

The left-hand side of the collisionless Boltzmann Equa-tion (1) represents the total time rate of change of the dis-tribution function f along a star trajectory in (r,v) space asdefined by Lagrange’s system of characteristic equations:

dr=dt ¼ v and dv=dt ¼ �o@=or: ð5ÞFrom Eqs. (1) and (5), one defines the trajectories of starsin the equilibrium central field @0(r) (in the circular rotatingreference frame):

r ¼ v?j½sin /0 � sinð/0 � jtÞ�; vr ¼ v? cosð/0 � jtÞ; ð6Þ

u ¼ � 2Xj

v?rj½cos /0 � cosð/0 � jtÞ�; vu

¼ rðdu=dtÞ � rX � ðj=2XÞv? sinð/0 � jtÞ; ð7Þ

where v^, /0 are constants of integration, v^/jr0 � q/r0� 1, and q is the mean epicycle radius. In Eqs. (6) and(7), r0 is the radius of the circular orbit, which is chosenso that the constant of areas for this circular orbitr2

0ðdu0=dtÞ is equal to the angular momentum integralr2(du/dt), v2

? ¼ v2r þ ð2X=jÞ2v2

u, u0 is the position angleon the circular orbit, and _u2

0 X2ðr0Þ ¼ ð1=r0Þðo@0=orÞ0.The quantities X, j, and cr are evaluated at r0. Accordingto Eqs. (6) and (7), the motion of a disk star is representedas in epicyclic motion along the small Coriolis ellipse (epi-cycle) with a simultaneous circulation of the epicenterabout the galactic center (Lindblad, 1963). Note that theproblem of epicyclic motion in its most general form isequivalent to the problem of the motion of a charged par-ticle in a given electromagnetic field, in which the solutioncan be decomposed into two parts: the guiding center mo-tion and the epicyclic motion. The role of the magnetic partof the Lorentz force is assumed by the Coriolis force, andthe epicycle radius used here is analogous to the gyroradiusin a plasma (Marochnik, 1966; Griv and Peter, 1996a). Ofcourse, the epicyclic approximation (q/r0� 1) may be ap-plied only when the actual star motion is nearly circular, asin spiral galaxies.

Thus, zeroth-order approximation is simply a circle onwhich the star moves with angular velocity X = Vrot/rand rotational velocity V rotðrÞ ¼ ðrd@0=drÞ1=2

0 . The epicy-clic theory superposes on this rotation harmonic oscilla-tions both in the radial and azimuthal directions with acharacteristic frequency called the epicyclic frequencyj(r). In the proper rotating frame the particle will movein a retrograde sense around a small ellipse with axial ratioj/X, called an epicycle (Fig. 1). The resulting motion in theinertial frame is a closed ellipse if j = X or j = 2X, or anunclosed rosette orbit.

2.3. Equilibrium distribution

From Eq. (1), the disk in the equilibrium is described bythe following equation:

vrðof0=orÞ þ r@0 ðof0=ovÞ ¼ 0 ð8Þor of0/ot = 0, where the angular velocity of rotation is suchthat the necessary centrifugal acceleration is exactly provid-ed by the central gravitational force, rX2 = o@0/or. Eq. (8)does not determine the equilibrium distribution f0 uniquely.The random velocity is assumed to follow a two-dimensionalGaussian distribution. We choose f0 in the form of theSchwarzschild (anisotropic Maxwellian) distribution (Shu,1970; Morozov, 1980, 1981b; Griv and Peter, 1996a):

f0 ¼r0ðr0Þ

2pcrðr0Þcuðr0Þexp � v2

r

2c2r ðr0Þ

�v2

u

2c2uðr0Þ

" #

¼ 2Xj

r0ðr0Þ2pc2

r ðr0Þexp � v2

?2c2

r ðr0Þ

� �: ð9Þ

The Schwarzschild distribution function is a function of thetwo epicyclic constants of motion v2

?=2 and r20Xðr0Þ, where

Fig. 1. A schematic representation of epicyclic motion in a disk with nearly circular star motions. The disk angular velocity is X (r) and the epicyclicfrequency is j (r). The current particle radius vector is r = r0 + r1, where vector r0 uniformly rotates with angular velocity X = X (r0) and |r1/r0|� 1. In acoordinate system that rotates with velocity X (r0), the star moves along the epicycle in a retrograde sense. Particle orbit: (a) epicyclic orbit in a rotatingframe (j = X), (b) closed orbit in an inertial frame (j = X), (c) rosette orbit in an inertial frame (X > j), and (d) rosette orbit in an inertial frame (X < j).The resulting motion in the inertial frame is a rosette orbit, generally not closed.

50 E. Griv et al. / Advances in Space Research 38 (2006) 47–56

r0 = r + (2X/j2)vu.These constants of motion are related tothe unperturbed epicyclic star orbits (e.g., Griv and Peter,1996a).The equilibrium distribution function f0 hasbeen normalized according to

R1�1R1�1 f0 dvr dvu ¼

2pðj=2XÞR1

0v? dv?f0 ¼ r0, where r0 (r) is the mean (local)

surface density, cr � (2X/j)cu (Eqs.(6)–(7)), and cr, cu arethe equilibrium radial and azimuthal dispersions of ran-dom velocities, respectively.Such a distribution functionfor the unperturbed system is particularly important be-cause it provides a fit to observations (Shu, 1970).Accord-ing to observations, for all types of stars in the Galaxy wehave the classical result that the dispersion of randomvelocities (‘‘temperature’’) is larger in the radial directionthan in the azimuthal and that the latter is systematicallylarger than the dispersion perpendicular to the Galacticplane, cr > cu > cz (Binney and Tremaine, 1987).In theclose solar neighborhood, for a subsystem of relativelyold F, K, and M dwarfs with ages t > 109 yr which containmost of the disk mass cu/cr � 0.6 and cz/cu = 0.7–0.8.2 Theessential part of the analytical method used in the paper isto regardq/r0 as a small parameter and to expand the solu-tion in terms of it.

3. Oscillation spectrum

3.1. Perturbed distribution function

Using Eqs. (3), (4), (6), (7), and (9) and paralleling theanalysis leading to Eq. (11) in Griv et al. (2002) or to Eq.(26) in Griv et al. (2003), it is straightforward to show that

f1 ¼ �@1

jv?

of0

ov?

X1l¼�1

lJ 2

l ðvÞx� � lj

� @1

2Xj2

mr

of0

or

X1l¼�1

� J 2l ðvÞ

x� � ljþ c:c:; ð10Þ

where f1(t!�1) = 0, so by considering only growingperturbations we neglected the effects of the initialconditions. In equation above, Jl (v) is the Bessel function

2 Hipparcos data provide that the local dynamical density is aboutq0 = 0.076 Mx pc�3 a value well below all previous determinationsleaving no room for any disk-shaped component of dark matter (Crezeet al., 1998).

of the first kind with its argument v = k*v^/j, k2� ¼

k2f1þ ½ð2X=jÞ2 � 1� sin2 wg is the squared effective wave

number, k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

r þ m2=r2

qis the total wave number,

w = arctan (m/krr) is the perturbation pitch angle, and

m2=k2r r2 � 1. In Eq. (10) only the most important low-fre-

quency (jx2�jK j2) perturbations developing in the plane

z = 0 between the inner (l = �1) and outer (l = 1) Lindbladresonances should be considered (Griv et al., 1999, 2002,2003; Griv and Gedalin, 2003): below in all equationsx* � lj 6¼ 0. Eq. (10) becomes singular at the Lindblad res-onances and at corotation, showing that special mathemat-ical methods should be applied at those radii (Griv et al.,2000). The resonances, however, have only limited radial

extent. Note that the condition m2=k2r r2 � 1 is equivalent

to the requirement |tanw| < 1, with ‘‘<’’ in place of ‘‘�’’in contrast to the first approximation (Lin and Shu, 1964,1966; Lin et al., 1969; Yuan, 1969a; Shu, 1970) of theLin–Shu asymptotic theory. The distortion of the wavepacket due to the disk inhomogeneity is included throughthe second term on the right-hand side in Eq. (10).

3.2. Asymptotic solution of the Poisson equation

For such a form of perturbation, the Poisson Equation(2) becomes

d2

dz2� k2

� �@1 ¼ 4pGr1dðzÞ: ð11Þ

In the vacuum (z > 0 and z < 0), Eq. (11) is reduced to theLaplace equation D@2 = 0, and, therefore, in these regionsthe solutions are

@þ @1;z>0 ¼ C1 e�jkjz; @� @1;z<0 ¼ C2 ejkjz; ð12Þ

where C1 and C2 are constants. By integrating Eq. (11) overz, the boundary conditions relating @1 to @2 on the surfaceof the disk–vacuum partition (z = 0) are found:

@þ ¼ @�jz¼0;o@þoz� o@�

oz

� �z¼0

¼ 4pGr1: ð13Þ

On substituting the solutions (12) into the boundary condi-tions (13), one obtains the required connection between the

E. Griv et al. / Advances in Space Research 38 (2006) 47–56 51

perturbed potential @1 (r, t) and the perturbed surface den-sity r1 (r, t) of the infinitesimally thin disk

@1 ¼ �ð2pGr1=jkjÞe�jkjjzj þ c:c:; ð14Þwhich reduces to @1 = �2pGr1/|k| + c.c. on z = 0 where thedisk lies, showing that density maxima correspond to po-tential minima. See also Lau and Bertin (1978), Lin andLau (1979), and Bertin (1980) for details. Thus Eq. (2), anintegral relation for arbitrary values of |kr|r, becomes a localrelation (14) in the short-wave, or WKB limit |kr|r� 1.

3.3. Generalized Lin–Shu type dispersion relation

Integrating Eq. (10) over velocity spaceðj=2XÞ

R 2p0

d/0

R10

f1v?dv? r1 and equating the result tothe ‘‘in-phase’’ perturbed surface density given by theasymptotic (k2

r � m2=r2) solution of the Poisson equation(Eq. (14)), r1 = �|k|@1/2pG + c.c., the generalized Lin–Shu type dispersion relation for low amplitude gravity per-turbations developing in the plane z = 0 is easily obtained

k2c2r

2pGr0jkj¼ �j

X1l¼�1

le�xIlðxÞx� � lj

þ 2Xmq2

rL

X1l¼�1

e�xIlðxÞx� � lj

ð15Þ(cf. Mikhailovsky, 1983, p. 594; Krall and Trivelpiece, 1986,p. 409; Swanson, 1989, p. 150). In Eq. (15),x ¼ k2

�c2r=j

2 � k2�q

2, q = cr/j is now the mean epicyclic radi-us, Il (x) is the modified Bessel function of imaginary argu-ment, q2/r|L|� 1, and jLj � jo lnð2Xr0=jc2

r Þ=orj�1 is theradial scale of spatial inhomogeneity. The dispersion rela-tion (15), which determines the spectrum of oscillations, isvalid only sufficiently far from resonant circumferencesand must be substituted by another equation near them,see, e.g., Griv et al. (2000). Because in galaxies j � X andq2/r|L|� 1, for the case of low-m (m < 10) spiral oscillationsthe second term on the right-hand side in Eq. (15) is the smallcorrection. High-m spiral modes (m P 10) are not importantin the problem of spiral structure because in contrast to thelow-m ones, they do not extend essentially over a large rangeof the disk (Lin and Shu, 1966; Lin et al., 1969; Shu, 1970;Bertin, 1980; Griv and Peter, 1996a).

3.4. Simplified dispersion relation

The dispersion relation (15) is complicated: it is highlynonlinear in the wave frequency x*. A further simplificationresults from restricting the frequency range of the wavesexamined by taking the low-frequency limit: the Doppler-shifted wave frequency |x*| less than the epicyclic frequencyof any disk star. In the opposite case of the high perturbationfrequencies, |x*| > j, the effect of the disk rotation is negligi-ble and therefore not relevant to us. This is because in this‘‘rotationless’’ case the particle motion is approximately rec-tilinear on the time and length scales of interest which are thewave growth/damping periods and wavelength, respectively(cf. Alexandrov et al., 1984, p. 113). Thus, the terms in series

(15) for which |l| P 2 may be neglected, and considerationwill be limited to the transparency region between the innerand outer Lindblad resonances. Second, we pay attentionmainly to the long-wavelength oscillations, x � k2

�q2 K 1,

the case of epicyclic radius small compared with wavelength(but, of course, in order to be appropriate for a WKB wavewe consider the perturbations with |kr|r� 1). In this limit,one can use the following asymptotic expansion of the Besselfunctions I0 (x) � 1 + x2/4, I1 (x) � x/2, and exp(�x) �1 � x + x2/2. The short-wavelength perturbations,k2�q

2 > 1, are not as dangerous in the problem of disk stabil-ity as oscillations with k2

�q2 K 1, since they lead only to very

small-scale perturbations of the density with the radial scalekr < 2pq. As a result, the dispersion relation (15) can berepresented in the simplest form

x3� � x�x

2J þ Xj2 mq2

rLI0ðxÞI1ðxÞ

¼ 0; ð16Þ

where the squared Jeans frequency is

x2J � j2 � 2pGr0jkjF ðxÞf1þ ½ð2X=jÞ2 � 1� sin2 wg; ð17Þ

x K 1; F ðxÞ � e�3x=4 is the so-called ‘‘reduction factor’’,F (x)! 1 in a dynamically cold system (cr! 0) anddecreases with increasing x (increasing the velocity spread)in a dynamically hot disk. The reduction factor takes intoaccount the fact that the wave field affects only weaklythe stars with high random velocities. The third term onthe left-hand side in Eq. (16) is the small correction.

Apart from the obvious replacement of kr by k (whichoriginates from the consideration of the nonaxisymmetricalmodes), Eq. (17) differs from the standard Lin–Shu expres-sion (Lin and Shu, 1966; Lin et al., 1969; Shu, 1970; Binneyand Tremaine, 1987, p. 360) by the appearance of the factorA ¼ f1þ ½ð2X=jÞ2 � 1� sin2 wg > 1. This factor indicatesan extra ðA > 1Þ clumping associated with the azimuthal(m or w 6¼ 0) forces in the differentially rotating (2X/j > 1)media. Lau and Bertin (1978), Lin and Lau (1979) haveobtained a somewhat similar expression for the extra clump-ing in a gasdynamical model. See also Bertin (1980) for a dis-cussion. The expression (17) indicates the tendency ofgrowth of the particle clumping with increasing w.

The resulting dispersion relation (16) is a third-orderequation with respect to x* with real coefficients, whichdescribes three branches of oscillations: two ordinary Jeansbranches (short-wavelength, x > 1, and long-wavelength,x [ 1, ones) modified by the inhomogeneity and an addi-tional gradient branch. We named the third branch as thegradient one because it can exist only when there is spatialinhomogeneity of the disk, oð2Xr0=jc2

r Þ=or 6¼ 0. From Eq.(16) in the frequency range

jx3�j � jx3

J j � Xj2 mq2

rjLjI0ðxÞI1ðxÞ

andmq2

rjLjI0ðxÞI1ðxÞ

� 1; ð18Þ

one obtains the dispersion law for the most important long-wavelength (x [1) and low-frequency ðjx2

�j < j2Þ Jeansbranch of oscillations

3 Shu (1970), Vandervoort (1970), Morozov (1981a) modified theordinary Safronov–Toomre criterion cT by taking into account thestabilizing influence of a finite thickness for the disk. See Shu (1984) for adiscussion. Safronov (1960, 1980), Toomre (1964), Genkin and Safronov(1975), Osterbart and Willerding (1995) have also estimated the effect.

4 The random (‘‘thermal’’) motion of the stars weakens the Jeansmechanism of instability growth but it by no means leads to completestabilization of the self-gravitating disk in all cases. The reason for this isthe resonant interaction between the stars and oscillations that ismanifested when there is a large thermal spread. This resonant interactioncan lead to microscopic instabilities. In a stellar disk with a Maxwelliandistribution of the random velocities, the resonant interaction is due to theinverse collisionless damping discovered by Landau (Griv et al., 2000).

52 E. Griv et al. / Advances in Space Research 38 (2006) 47–56

x�1;2 � �pjxJj � Xj2

2x2J

mq2

rLI0ðxÞI1ðxÞ

; ð19Þ

where p = 1 for Jeans-stable perturbations withx2� � x2

J > 0, p = i for Jeans-unstable perturbations withx2� � x2

J < 0, and the term involving L�1 is the small cor-rection. As is seen, a spatial inhomogeneity will not influ-ence the stability condition of Jeans modes.

In the Jeans-unstable case ðx2J < 0Þ, the equilibrium

parameters of the disk and the azimuthal mode numberm determine the spiral pattern speed Xp (in the local rotat-ing frame)

Xp ¼ Rx�1;2=m

¼ signðLÞ � Xj2

2jx2J j

q2

rjLjI0ðxÞI1ðxÞ

and m > 0: ð20Þ

As is seen, the typical pattern speeds of spiral structures areonly a small fraction of some average angular velocity Xav.The theory states that in spatially homogeneous (|L|!1)disks Xp = 0. Also, because Xp does not depend on m, eachFourier component of a gravity perturbation in an inho-mogeneous system will rotate with the same angular veloc-ity. The growth rate of the instability is relatively high,Ix� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pGr0jkjAF ðxÞ � j2

pand in general Ix� � X.

From the relation oxJ2/ok = 0, one finds that the growth

rate has a maximum at the wavelength

kcrit ¼ 2p=kcrit � ð4pX=jÞq � ð2–4Þph: ð21ÞIt means that of all harmonics of initial perturbation, oneperturbation with kcrit � 10h and Ix� � X will be formedasymptotically in time of 2–3 revolutions in galaxies. Forthe parameters of the Galaxy, kcrit � 2 kpc; thus, kcrit� h.The free kinetic energy associated with the differentialrotation is one possible source for the growth of the aver-age wave energy. The spiral arms in nonuniformly rotat-ing systems are a mechanism for angular momentumtransfer (Lynden-Bell and Kalnajs, 1972; Griv and Geda-lin, 2004). Such hydrodynamic, or algebraic (Sagdeev andGaleev, 1969, p. 67) instabilities distinguish themselves inthat a rather large group of the so-called nonresonantparticles (i.e., such that x* � lj 6¼ 0, where l = 0, ±1,. . .) takes part in their generation. The almost aperiodic(jRx�=Ix�j � 1) Jeans instabilities are generated by al-most all the particles of the phase space. The wavelength(the size of spirals and the spacing between them) of themost unstable perturbations is proportional to the meansize of an epicycle: kcrit � cr/j � q, that is, proportionalto the ‘‘temperature.’’

In turn, in the another, opposite to (18) frequency range

jx�j � Xmq2

rjLjI0ðxÞI1ðxÞ

� jxJj andmq2

rjLjI0ðxÞI1ðxÞ

� 1; ð22Þ

that is, |x*|� X, the simplified dispersion relation (16) hasalso another root equal to

x�3 � Xj2

x2J

mq2

rLI0ðxÞI1ðxÞ

: ð23Þ

The root (23) describes the gradient (L�1 6¼ 0) branch ofoscillations. The gradient perturbations are natural onesðIx�3 ¼ 0Þ and are independent of the stability of Jeansmodes. These low-frequency, |x*3|� X, harmonic oscilla-tions are obviously not important in dynamics of galaxies.

3.5. Modified stability criterion

In the previous subsection we have shown that an oscil-lation frequency that satisfies Eq. (19) and the conditionx2

J < 0 can have a positive imaginary part, i.e., the gravityperturbations described by this dispersion equation cangrow in time. These Jeans-unstable perturbations can bestabilized by the random velocity spread. Indeed, if onerecalls that Jeans-unstable perturbations are possible onlywhen x2

�1;2 � x2J < 0, then by using the conditions

dx2J=dk ¼ 0 and x2

J P 0 for all possible k, from Eq.(16) a stability criterion against arbitrary but not only axi-symmetric gravity perturbations can be written in the form(Morozov, 1980, 1981b; Griv and Peter, 1996a; Griv et al.,1999, 2002)

cr P ccrit � cTf1þ ½ð2X=jÞ2 � 1� sin2 wg1=2

� ð2X=jÞcT; ð24Þ

or Toomre’s Q ” cr/cT-value be greater than

Qcrit ¼ 2X=j � 2; ð25Þrespectively, where cT � 3.4Gr0/j is the well-knownSafronov–Toomre (Safronov, 1960, 1980; Toomre, 1964;Genkin and Safronov, 1975) critical radial-velocity disper-sion to suppress the instability of axisymmetric perturba-tions only. Here, Toomre’s Q-value is a measure of theratio of thermal � cr and rotational � X stabilization toself-gravitation � Gr0. A slightly improved modified sta-bility criterion was obtained by Morozov (1981a) byincluding a weak destabilizing effect resulting from spatialinhomogeneity of the disk and a weak stabilizing effectresulting from the small but finite thickness of the disk.3

One should keep in mind that the stability conditionabove is clearly only an approximate one, since it was ob-tained in the framework of the local WKBapproximation.4

In Eq. (24), the parameter 2X/j is an additional sta-bility parameter which depends on the amount of differ-

5 Lau and Bertin (1978) think of Q < 2X/j � 2 (cf. Eq. (24)) as acriterion for appreciable swing amplification rather than as a criterion forlocal nonaxisymmetric instability. The point is that there exists ambiguityin the interpretation of ‘‘swing’’ as a transient temporal amplification ofsingle wavelets, or as a steady amplification of propagating waves thatreflect and form standing patterns (global normal modes). Most workersin the field feel more comfortable with nonaxisymmetric stability criteriathat derive from global normal-mode calculations than local similar tothat presented in this paper (or shearing-sheet; e.g., Goldreich andLynden-Bell, 1965, Fuchs, 2001) analysis.

E. Griv et al. / Advances in Space Research 38 (2006) 47–56 53

ential rotation in the disk dX/dr. Thus, the shear, 2X/j > 1 (and the spatial inhomogeneity, L�1 6¼ 0; Morozov,1981a,b) gives rise to destabilization effect. According toEq. (24), the critical Safronov–Toomre velocity disper-sion cT should stabilize only axisymmetric (radial) pertur-bations of the Jeans type (w = 0). The differentiallyrotating disk is still unstable against nonaxisymmetric(spiral) Jeans perturbations (w 6¼ 0). It would be naturalto suppose that as a disk of mutually gravitating parti-cles evolves it should arrive at a state near the limit ofstability, so that the mean particle radial-velocity disper-sion would come close to the modified critical value ccrit.Thus, for a nonuniformly rotating thin system of mutu-ally gravitating particles the spiral gravity perturbationsare more important in determining the stability thanare the radial perturbations studied by Safronov (1960,1980), Toomre (1964), Lin and Shu (1964, 1966), Linet al. (1969), Shu (1970), Genkin and Safronov (1975).Interestingly, Maxwell (1960) considered just this kindof spiral instabilities with m = 1 in his study concerningthe stability of the Saturnian uniform rings whose radialextent was considerably larger than the average interpar-ticle distance. Maxwell made a correct that, in such asystem, the azimuthal force resulting from azimuthal dis-placements was more important in determining the sta-bility than was the radial force resulting from radialdisplacements.

The fact that self-gravitating disks can exhibit strongnonaxisymmetric responses even when the axisymmetricstability criterion is fulfilled, was most convincinglydemonstrated already by Julian and Toomre (1966),concerning the gravitational effect of any single orbitingmass concentration (such as gas ‘‘lumps’’ in galaxies).Similar result for self-gravitating gas disk was obtainedby Goldreich and Lynden-Bell (1965). Julian and Too-mre’s calculations showed that even a stable stellar sys-tem (possessing a velocity dispersion more thansufficient for local axisymmetric stability) to be remark-ably responsive in a spiral-like manner to localized forc-ing. These forced spiral waves are not to be confused,of course, with Lin and Shu’s fully self-consistent densi-ty wave proposal explored in the present paper. In Too-mre (1981) this amplification was discussed in terms of‘‘swing-mechanism’’, very reminiscent of the way wereach the nonaxisymmetric stability criterion (24). Seealso Morozov (1980, 1981b), Griv and Peter (1996a,b),Griv (1998), and Griv et al. (1999, 2002, 2003) for adiscussion.

In sharp contrast to Julian and Toomre (1966), ourmodel relies on normal modes of collective oscillations ofa particulate disk to maintain spiral coherence at all radii.Our analysis is based on the microscopic (kinetic) descrip-tion of the properties and processes of a particulate disk. Arelatively simple macroscopic (hydrodynamical) model canalso be used to investigate the Jeans instability (e.g., Lauand Bertin, 1978; Lin and Lau, 1979). However, a studyof the kinetic theory adds depth to the concepts of waves

and their stability already explored by Lau and Bertin(1978), Lin and Lau (1979) in the gasdynamical frame-work.5 The gravity disturbances studied in our paper areassociated with such phenomena as, for example, theappearance of the spiral structure of galaxies (Lin et al.,1969; Binney and Tremaine, 1987; Griv et al., 1999, 2002)and Saturn’s rings (Griv et al., 2003).

Summarizing, in the differentially rotating equilibriumdisks, there is a difference between random velocity disper-sions of stars in azimuthal and radial directions; the radialdispersion is larger, cr � (2X/j)cu � 2cu. This difference isconfirmed by observations. In this case stability conditionis different for axisymmetric (ccrit � cT) and nonaxisymmet-ric (ccrit � 2cT) perturbations. The differentially rotatingdisk of stars is more unstable in the case of nonaxisymmet-ric, that is, spiral perturbations.

3.6. Numerical solution of the dispersion equation

Although the complicated dispersion relation (15) canbe analyzed directly and even solved analytically in thecase of low-frequency oscillations as shown above,graphical representation of the roots is much more con-venient. A general impression of how the spectrum ofnonaxisymmetric Jeans perturbations behaves in ahomogeneous nonuniformly rotating disk can be gainedfrom Fig. 2, which shows the dispersion curves in thecases of Jeans-unstable systems ((a) and (b)), a margin-ally Jeans-stable system (c), and a Jeans-stable one (d)for values of a summation index l from l = �4 tol = 4 (as determined on a computer from Eq. (15)). Inthis figure, the ordinate is the effective wave numberk* measured in terms of the inverse epicyclic radius qand the abscissa is m = x*/j, i.e., the dimensionlessangular frequency at which the stars meet with the pat-tern, measured in terms of the epicyclic frequency j. Ingeneral, for fixed dimensionless wave frequency x*/jthere are two solutions in k*q, comprising a long-wave-length wave, k*q [ 1, and a short-wavelength wave,k*q > 1. As is seen, the maximum of instability occursat the wavenumber k* q � 0.5. The latter is in goodagreement with the approximate solution given by Eq.(21). A property of the solution (15) is that in a homo-geneous system the Jeans-stable modes those withQ > 2X/j are separated from each other by frequencyintervals where there is no wave propagation: gapsoccur between each harmonic (cf. the Bernstein modes

Fig. 2. The generalized Lin–Shu dispersion relation (15) of a homogeneous (|L|!1) gravitating disk in the case 2X/j = 2 and |sinw| = 1 for the differentToomre’s Q-values: (a) Q = 0.5(2X/j), (b) Q = 0.8(2X/j), (c) Q = 2X/j, and (d) Q = 1.5(2X/j). The solid curves represent the real part of thedimensionless Doppler-shifted wave frequency of low-frequency, |x

*| < j, long-wavelength (1) and short-wavelength (2) Jeans oscillations we are interested

in. The dashed curves represent the imaginary part of the dimensionless wave frequency of low-frequency vibrations. The dot-dashed curves represent thewave frequencies of additional high-frequency, |x

*| > j, Jeans modes. As is seen, low-frequency and long-wavelength collective vibrations (those with

|x*| < j and k2

�q2 K 1) are the most unstable ones.

a b

Fig. 3. A schematic model of a Jeans-unstable disk; (a) the Safronov–Toomre unstable disk (cr < cT) and (b) the Safronov–Toomre stable disk(cr P cT but cr < (2X/j)cT and in spiral galaxies 2X/j � 2).

6 Axisymmetric and nonaxisymmetric features seen in Fig. 3a veryreminiscent of the rings and spokes in the Cartwheel galaxy, the moststriking example of the small class of ring galaxies (Griv, 2005). Theexpanding ring in the Cartwheel can be considered as the crest of a highamplitude m = 0 Jeans-unstable density wave. In turn, the spokes areprobably nothing but density maxima of spiral (m 6¼ 0) density waves.

54 E. Griv et al. / Advances in Space Research 38 (2006) 47–56

in a magnetized plasma). Note that Lovelace et al.(1997) have pointed out the higher-order, |x*| > j, Jeansbranches of oscillations as shown in Fig. 2 by the dot-dashed curves. These authors have mentioned that thebranches l = 2, 3, . . . are the analogs of the Bernsteinmodes, which propagate across a uniform magnetic fieldin a collisionless plasma without Landau damping.Apparently, Lin et al. (1969) and Nakamura et al.(1975) first pointed out that the stable density wavesin a thin disk derived by Lin and Shu (1966), Linet al. (1969), and Shu (1970) are no other than the rep-resentation of the Bernstein mode in the gravitationalsystem. In contrast to our study, however, Lin et al.,Nakamura et al., and Lovelace et al. considered the sta-bility of axisymmetric oscillations only.

4. Conclusions

We conclude that if the stellar disk is thin, cr� rX, anddynamically cold, cr<cT, then such a model will be Jeans-unstable against both ring and spiral gravity perturbations,and it should almost instantaneously take the form of acartwheel, Eq. (3) (Fig. 3a). One concludes that the ToomreQ-parameter that is <1 suggests that the disk is likely sub-ject to both radial and azimuthal instabilities and might

therefore be clumpy.6 On the other hand, if the disk is thinand dynamically warm, cr P cT (but cr < (2X/j)cT � 2 cT),then such a model will be gravitationally unstable only

E. Griv et al. / Advances in Space Research 38 (2006) 47–56 55

against spiral perturbations (Fig. 3b). An uncooled hot

model with c J 2cT, or Q J 2, respectively, (and with acore-dominated exponential-like mass density profile; Grivand Gedalin, 2004) is Jeans-stable.

Thus, as a result of the Jeans instability of spiral gravityperturbations, a disk of a galaxy is subdivided into spiralwaves with size and the spacing between them of the orderof 2 kpc. It is natural to attribute the observed spiral pat-terns in actual galaxies, in particular, in our own Galaxyto the action of almost aperiodic Jeans instability of nonax-isymmetric gravity perturbations so far discussed in thiswork.

Acknowledgements

We thank Tzi-Hong Chiueh, David Eichler, Yury Lyu-barsky, Frank Shu, Irena Shuster, and Raphael Steinitzfor many helpful discussions. The work was supported, inpart, by the Israel Science Foundation, the Israeli Ministryof Immigrant Absorption in the framework of the pro-gramme ‘‘KAMEA’’, the Binational U.S.–Israel ScienceFoundation, and the Academia Sinica in Taiwan.

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