Spiral galaxies as gravitational plasmas

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<ul><li><p>ra</p><p>ed</p><p>ersi</p><p>sics,</p><p>23</p><p>galueineda</p><p>xies:</p><p>produced by a spontaneous perturbation or a companionsystem). This is because their evolution is primarily driven</p><p>standing of the phenomenon. This is because the bulk ofthe total optical mass in the Milky Way and other at gal-</p><p>physics phenomena and the dynamics of disk galaxies isestablished. Similarities between self-gravitating systems</p><p>unison (Sweet, 1963; Lynden-Bell, 1967; Marochnik, 1968).We are concerned with low amplitude Jeans-type grow-</p><p>ing oscillations and their stability by studying the spiralstructure of galaxies as the collective eect in a self-gravi-tating system. Generally, the term Jeans instabilitiesidenties gravitational nonresonant instabilities associated</p><p>* Corresponding author.E-mail address: griv@bgu.ac.il (E. Griv).</p><p>Advances in Space Researchby angular momentum redistribution. The system maythen fall toward the lower energy conguration and usethe energy so gained to increase its coarse grained entropy.</p><p>The theory of spiral structure of rotationally supportedgalaxies has a long history but is not yet complete. Eventhough no denitive answer can be given at the presenttime, the majority of experts in the eld is yielded to opin-ion that the study of the stability of collective vibrations indisk galaxies of stars is the rst step towards an under-</p><p>and ordinary plasmas arise from the common long-range 1/r2 nature of the basic forces, whereas dierencesarise 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 ofmotionsmodesin which the particles in large regions move coherently or in1. Introduction</p><p>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,nally, 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., those</p><p>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.</p><p>In this paper, a connection between several plasmaSpiral galaxies as g</p><p>E. Griv a,*, M. Ga Department of Physics, Ben-Gurion Univ</p><p>b Institute of Astronomy and Astrophy</p><p>Received 7 July 2004; received in revised form</p><p>Abstract</p><p>This article reviews recent studies of dynamics of disk-shapedversion of kinetic stability theory, the very existence and the vanonresonant Jeans-type instability of gravity perturbations is explaself-gravitating disk and the oscillations of a hot nonneutral plasm 2006 Published by Elsevier Ltd on behalf of COSPAR.</p><p>Keywords: Galaxies: Kinematics and dynamics; Galaxies: Structure; Gala0273-1177/$30 2006 Published by Elsevier Ltd on behalf of COSPAR.doi:10.1016/j.asr.2006.05.019vitational plasmas</p><p>alin a, C. Yuan b</p><p>ty of the Negev, Beer-Sheva 84105, Israel</p><p>Academia Sinica, Taipei 106, Taiwan</p><p>December 2004; accepted 31 December 2004</p><p>laxies with special emphasis on their spiral structures. In a localof the critical wavelength of the spiral structure arising due to. A formal analogy between the collective oscillations in a rotatingin a magnetic eld is explored.</p><p>Spiral galaxies; Waves and instabilities</p><p>www.elsevier.com/locate/asr</p><p>38 (2006) 4756</p></li><li><p>is the total gravitational potential determined self-consis-tently from the Poisson equation</p><p>o2@ 1 o@ 1 o2@ o2@ Z</p><p>pace Research 38 (2006) 4756with almost aperiodically growing accumulations of mass(gravitational collapse). The instability is driven by astrong nonresonant interaction of the gravity uctuationswith the bulk of the particle population, and the dynamicsof Jeans perturbations can be characterized as a uid-likewaveparticle 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 dierent 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).</p><p>One has to realize, however, that in many respects gravi-tating systems dier strongly from laboratory plasmas. Thediculties for a satisfactory understanding of the dynamicsof gravitating particulate systems are due to the well-knownfact that in a system ofN 1 (generally, N 1011) gravita-tionally interacting particles Debye screening, as distinctfrom plasma, is absent. Also, a principal dierence 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.</p><p>2. Basic equations</p><p>The problem is formulated in the same way as in plasmakinetic theory. In the rotating frame of a disk galaxy, the col-lisionlessmotionof 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)</p><p>ofot vr ofor X</p><p>vur</p><p> ofou</p><p> 2Xvu v2ur X2r o@</p><p>or</p><p> !ofovr</p><p> j2vr2X</p><p> vrvur 1</p><p>ro@ou</p><p> ofovu</p><p> 0; 1</p><p>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 dened by j = 2X [1 +(r/2X) (dX/dr)]1/2, and r, u, z are the galactocentric cylindri-cal coordinates. ThequantityX (r) denotes the angular veloc-ity of galactic rotation at the distance r from the center.Random velocities are usually small compared with Vrot.</p><p>48 E. Griv et al. / Advances in SCollisions are neglected here because the collision frequencyis much smaller than the cyclic frequencyX. In Eq. (1), @(r, t)or2</p><p>r or</p><p>r2 ou2</p><p>oz2 4pG fdv: 2</p><p>Eqs. (1) and (2) give a complete self-consistent description ofthe problem for diskmodes. The combined systemofEqs. (1)and (2) is the counterpart of the system of Vlasov and Max-well equations in electromagnetic plasmas. Hence the termgravitational plasma is appropriate for the descriptionof mutually gravitating stars (Lin and Bertin, 1984) [(andparticles of Saturns rings; Griv (1998), Griv et al. (2003),Griv andGedalin (2003))]. Reviews of plasma kinetic theoryare given by Mikhailovsky (1983), Alexandrov et al. (1984),Krall and Trivelpiece (1986), and Swanson (1989).</p><p>The equilibrium state is assumed, for simplicity, to be anaxisymmetric and spatially inhomogeneous only along the r-coordinate stellar disk. Second, in our simplied model, theperturbation is propagating in the equatorial plane of thedisk. This approximation of an innitesimally thin disk is avalid approximation if one considers perturbations with aradial wavelength kr that is greater than the typical diskthickness h = 100200 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 fdvrdvudz = r, where r (r, t) is the surface density.We expect that the waves and their instabilities propagatingin the disk plane have the greatest inuence on the develop-ment of structures in the system (Shu, 1970). Limiting our-selves to the case of innitesimally thin disk simplies thealgebra without introducing any fundamental changes inthe physical results (Safronov, 1980).1</p><p>2.1. Perturbation</p><p>Let us suppose that the nonlinear eects in galactic disksare small, so that the linear theory is a good rst approxima-tion. We proceed by applying the standard procedure of thelinear approach anddecompose the time dependent distribu-tion function of stars f = f0 (r,v) + f1 (r,v, t) and the totalgravitational potential @ @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 f0and @0 describe the nondeveloping background againstwhich small perturbations rapidly develop.</p><p>To determine oscillation spectra, let us consider the sta-bility problem in the lowest (local) WKB approximation:the perturbation scale is suciently 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-</p><p>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 dierence the disks nite but small thickness makes is that</p><p>it tends to be stabilizing by reducing self-gravity at the midplane(Vandervoort, 1970; Morozov, 1981a; Shu, 1984).</p></li><li><p>pactions 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):</p><p>ff1;@1g Xk</p><p>ffk;@kg expikrr imu ix;kt c:c:; 3</p><p>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, suxes k denote thekth Fourier component, c.c. means the complex conju-gate, and |kr|R 1. The perturbed density r1 (r, t) = f1dvis 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 &gt; 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 &gt; 0 or toward the center whenkr &lt; 0. The imaginary part of x* corresponds to a growthIx &gt; 0 or decay Ix &lt; 0 of the components in time,f1;@1 / exp Ixt, and the real part to a rotation withangular velocity Xp Rx=m. Thus, when Ix &gt; 0, themedium transfers its energy to the growing wave and oscil-lation buildup occurs. A galaxy is considered as a superpo-sition of dierent oscillation modes. A disturbance in thedisk will grow until it is limited by some nonlinear eect.</p><p>Linearizing Eq. (1), one obtains the equation for thedeveloping perturbation</p><p>df1dt</p><p> o@1or</p><p>of0ovr</p><p> 1ro@1ou</p><p>of0ovu</p><p>; 4</p><p>where d/dt means total derivative along the star orbit andf0 is a given equilibrium distribution.</p><p>2.2. Equilibrium stellar trajectories</p><p>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 asdened by Lagranges system of characteristic equations:</p><p>dr=dt v and dv=dt o@=or: 5From Eqs. (1) and (5), one denes the trajectories of starsin the equilibrium central eld @0(r) (in the circular rotatingreference frame):</p><p>r v?jsin/0 sin/0 jt; vr v? cos/0 jt; 6</p><p>u 2X v? cos/0 cos/0 jt; vu</p><p>E. Griv et al. / Advances in Sj rj rdu=dt rX j=2Xv? sin/0 jt; 7where 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 orbitr20du0=dt is equal to the angular momentum integralr2(du/dt), v2? v2r 2X=j2v2u, u0 is the position angleon the circular orbit, and _u20 X2r0 1=r0o@0=or0.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 eld, 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.</p><p>Thus, zeroth-order approximation is simply a circle onwhich the star moves with angular velocity X = Vrot/rand rotational velocity V rotr rd@0=dr1=20 . 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.</p><p>2.3. Equilibrium distribution</p><p>From Eq. (1), the disk in the equilibrium is described bythe following equation:</p><p>vrof0=or r@0 of0=ov 0 8or 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 randomvelocity 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):</p><p>f0 r0r02pcrr0cur0 exp </p><p>v2r2c2r r0</p><p> v2u</p><p>2c2ur0</p><p>" #</p><p> 2Xj</p><p>r0r02pc2r r0</p><p>exp v2?</p><p>2c2r r0 </p><p>: 9</p><p>e Research 38 (2006) 4756 49The Schwarzschild distribution function is a function of thetwo epicyclic constants of motion v2?=2 and r</p><p>20Xr0, where</p></li><li><p>rlyctoe epinclos</p><p>50 E. Griv et al. / Advances in Spacr0 = r + (2X/j2)vu.These constants of motion are related to...</p></li></ul>