A&A 383, 338–351 (2002) Astronomy DOI: 10.1051/0004-6361:20011713 & c ESO 2002 Astrophysics
Quasi-linear theory of the Jeans instability in disk-shaped galaxies
E. Griv1, M. Gedalin1,andC.Yuan2
1 Department of Physics, Ben-Gurion University of the Negev, PO Box 653, Beer-Sheva 84105, Israel 2 Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), PO Box 23, Taipei 106, Taiwan
Received 4 September 2001 / Accepted 29 November 2001
Abstract. We analyse the reaction between almost aperiodically growing Jeans-unstable gravity perturbations and stars of a rotating and spatially inhomogeneous disk of flat galaxies. A mathematical formalism in the approx- imation of weak turbulence (a quasi-linearization of the Boltzmann collisionless kinetic equation) is developed. A diffusion equation in configuration space is derived which describes the change in the main body of equilibrium distribution of stars. The distortion in phase space resulting from such a wave–star interaction is studied. The theory, applied to the Solar neighborhood, accounts for the observed Schwarzschild shape of the velocity ellipsoid, the increase in the random stellar velocities with age, and the essential radial migration of the Sun from its birth-place in the inner part of the Galaxy outwards during its lifetime.
Key words. galaxies: kinematics and dynamics – galaxies: structure – waves – instabilities
1. Introduction investigation by studying the nonlinear effects. The prob- lem is formulated in the same way as in plasma kinetic The theory of spiral structure of rotationally supported theory. galaxies has a long history, but, as we emphasize below, is not yet complete. Even though no definitive answer can In the Solar neighborhood the random velocity distri- 8 be given at the present time, the majority of experts in bution function of stars with an age t & 10 yr is close the field is yielded to opinion that the study of the stabil- to a Schwarzschild distribution – a set of Gaussian distri- ity of collective vibrations in disk galaxies of stars is the butions along each coordinate in velocity space, i.e., close first step towards an understanding of the phenomenon. to equilibrium along each coordinate (Binney & Tremaine This is because the bulk of the total optical mass in the 1987; Gilmore et al. 1990). In addition, older stellar popu- Milky Way and other flat galaxies is in stars. Recent mea- lations have a higher velocity dispersion than younger ones surements of the local dynamical density by Hipparcos (Mayor 1974; Wielen 1977; Str¨omgren 1987; Meusinger rule out any disk-shaped dark matter (Cr´ez´e et al. 1998). et al. 1991; Fuchs et al. 1994; Dehnen & Binney 1999). Hipparcos data indicate a moderate contribution of un- High-quality Hipparcos data for a complete sample of seen matter to the local potential (Haywood et al. 1997). nearly 12 000 main-sequence and subgiant stars show that In addition, spiral arms are smoother in images of galaxies the velocity dispersion of a coeval group is found to in- in the near IR (Rix & Zaritsky 1995) indicating that the crease with time (Binney et al. 2000). The latter means old disk stars participate in the pattern. Therefore, the that an unknown mechanism increases the velocity disper- spiral structures are intimately connected with the stel- sion of stars in the Galaxy’s disk after they are born. It lar constituent of a galaxy. We regard spiral structure in is argued that the increase in velocity dispersion (and the most flattened galaxies of stars as a wave pattern, which diffusion of stellar orbits in coordinate space) with time does not remain stationary in a frame of reference rotat- is predominantly a gradual process. Grivnev & Fridman ing around the center of the galaxy at a proper speed, (1990) have shown, by using the observed stellar velocities, excited as a result of the Jeans instability of gravity per- that the random velocity distribution of youngest stars is turbations (those produced by a bar or oval structure in a close to a δ-like one. As the age of stars of spectral types ≈ × 7 ≈ × 8 galactic center, a spontaneous spiral perturbation, and/or B5–A9 increases from t 5 10 yr to t 5 10 yr, a companion galaxy). It is our purpose to extend the the distribution functions of the residual velocities of stars along each coordinate in the momentum space approach Send offprint requests to:E.Griv, a normal distribution. On the other hand, a simple cal- e-mail: [email protected] culation of the relaxation time of the local disk of the
Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20011713 E. Griv et al.: Quasi-linear theory of the Jeans instability 339
Galaxy due to pairwise star–star encounters brings the interaction of the gravity fluctuations with the bulk of the value ∼1014 yr (Chandrasekhar 1960; Binney & Tremaine particle population, and the dynamics of Jeans perturba- 1987, p. 187), which considerably exceeds the lifetime of tions can be characterized as a fluidlike interaction. The the universe. Thus, observations indicate that the local gravitational Jeans-type instability does not depend on disk is well relaxed, so some form of scattering process is the behavior of the particle distribution function in the going on most likely in the form of collisionless processes. neighborhood of a particular speed, but the determining During the years several mechanisms have been pro- factors of the instability are macroscopic parameters like posed to explain the growth of the velocity dispersion: the random velocity spread, mean density, and angular encounters of stars with massive gaseous clouds (Spitzer velocity of regular rotation. The Jeans instability associ- & Schwarzschild 1951, 1953), heating of the galactic disk ated with departures of macroscopic quantities from the by transient spiral waves (Barbanis & Woltjer 1967), a thermodynamic equilibrium is hydrodynamical in nature systematic increase in the velocity dispersion of protostel- and has nothing to do with any explicit resonant effects; lar gas clouds as the region of space increases (Larson a relatively simple hydrodynamical model can be used to 1979; Myers 1983), heating by proposed black holes in the investigate the instability (Lovelace & Hohlfeld 1978; Lin Galactic dark halo (Lacey & Ostriker 1985) or by dwarf & Lau 1979; Drury 1980). galaxies (Quinn & Goodman 1986). To date, a convincing The criterion for a rotationally flattened stellar disk mechanism to explain the observed amount of the disk to be Jeans-unstable to substructure formation by self- heating has not been found. For instance, it is difficult to gravitation is that Toomre’s (1964) stability parameter explain the observed relaxation by the usual binary en- Q be less than Qcrit = 2–2.5 (see Sect. 3.3 below). Here couters between stars and giant molecular clouds. This is Toomre’s Q-value, Q = cr/cT, is a measure of the ratio because Binney & Lacey (1988) have shown that binary of thermal and rotational stabilization to self-gravitation, encounters with giant clouds of an interstellar medium cr is the radial dispersion of residual (random) velocities result in a stellar velocity dispersion which will increase of stars, cT ≈ 3.4Gσ0/κ is the Toomre (Toomre 1964; 1/4 1/2 with age as t , instead of the observed roughly cr ∝ t see also Safronov 1960) critical velocity dispersion, σ0 is law (Wielen 1977). Also, if only giant molecular clouds are the local projected surface density of stars, and κ is the responsible for heating the ratio of the vertical-to-radial local epicyclic frequency. Combined with the Lin–Shu type velocity dispersion cz/cr will approach 0.75 and, if spi- dispersion relation for density waves, this is a venerable ral structure also contributes to the disk heating the ra- suggestion as to why disk galaxies almost always exhibit tio will be lower. The most accurate measurements using spiral structure. the Hipparcos data indicate a ratio cz/cr of 0.53 0.07 The reaction of the distribution function of stars to (Dehnen & Binney 1999). This behaviour is again consis- the Jeans-unstable field fluctuations is such that the ran- tent with the predictions of disk-heating by spiral struc- dom velocity dispersion (or “temperature”) increases un- ture. Scattering by giant molecular clouds plays only a til the difference Q − Qcrit → 0, and the system tends modest role (Binney & Tremaine 1987, p. 470; Gilmore toward marginal stability. Hence, in differentially rotat- et al. 1990, p. 174; Binney 2001)1. ing galaxies, once the entire disk has been heated to val- ≈ In this paper a statistical mechanism of smoothing out ues Q Qcrit, no further spiral waves can be sustained stellar plane velocities, necessary to make them agree with by virtue of the Jeans instability – unless some “cooling” a Schwarzschild distribution is suggested. This mechanism mechanism is available leading to Toomre’s Q-value, un- explains the observable increase in stellar velocity disper- der approximately 2. By using N-body simulations, first sion with age. In our approach, collisionless relaxation by Miller et al. (1970), Hohl (1971), and then Sellwood & virtue of interaction between Jeans-unstable density waves Carlberg (1984), Tomley et al. (1991), and Griv & Chiueh and stars does play a determining role in the evolution of (1998) have shown that the process of formation of fresh stellar populations of the Galactic disk. In this regard, particles, which move on nearly circular orbits, plays a vi- recent data from the Hipparcos satellite already made it tal role in prolonging spiral activity in the plane of the disk clear that the Galaxy is by no means in a steady state by reducing the random velocity dispersion of the entire (Dehnen & Binney 1999; Binney 2001). There are pre- stellar component. Also, in good conformity with obser- liminary indications that we see in the local phase-space vations (van der Kruit & Freeman 1986; Bottema 1993) distribution the dynamical footprints of long-dissolved un- and the theory outlined above, both N-body simulations stable waves. (Sellwood & Carlberg 1984; Tomley et al. 1991) and nu- The classical Jeans instability of gravity disturbances merical solutions of the collisionless Boltzmann equation is one of the most frequent and most important instabil- (Nishida et al. 1984) showed that the stability number Q ities in the stellar and in the planetary cosmogony, and of Toomre in relaxed equilibrium disks does not fall be- galactic dynamics. The instability is driven by a strong low a critical value, which lies about 2–2.5. Liverts et al. (2000) used computer simulations to test the validity of 1 Only the combined effect of star–cloud interactions and the modified stability criterion Qcrit. collective effects (unstable spiral waves with a growth time The investigations carried out in the linear approxi- ∼<109 yr) will be able to explain the observations (Binney & mation allow us to determine only the spectrum of the Tremaine 1987, p. 484; Griv & Peter 1996c; Binney 2001). excited oscillations and their growth rates during the first 340 E. Griv et al.: Quasi-linear theory of the Jeans instability stages of the excitation. Arbitrary perturbations can be nonstationary process of formation of the equilibrium expressed as a superposition of eigenmodes, with each state. A very nonequilibrium initially configuration was eigenmode evolving independently. Other problems, which studied (the virial theorem is strongly violated). Kulsrud can be treated within the framework of nonlinear theory, (1972) also discussed the inverse effects of different insta- are account of the reaction of the oscillations on the equi- bilities of gravity oscillations on the averaged velocity dis- librium parameters of the system and the determination tribution function of stars by collective interaction. It was of the amplitude of the oscillations that are produced. The stated that, because of its long-range Newtonian forces, a quasi-linear approach to nonlinear plasma theory is usu- self-gravitating medium (a stellar “gas”, say) would pos- ally referred to as the theory of weak turbulence, i.e., the sess collective motions in which all the particles of the sys- case when the dynamics of the system can be described tem participate. These properties would be manifested in in the language of weakly interacting linear waves. That the behavior of small gravity perturbations arising against is, there are many random collective oscillations present the equilibrium background. Collective processes are com- in the system and it is permissible to treat the phases of pletely analogous to two-body collisions, except that one these oscillations as being random in some sense. It can particle collides not with another one but with many be justified if the energy in the excited spectrum is small which are collected together by some coherent process such compared with the total mechanical energy in particles as a wave. The collective processes are random, and usu- but large compared to thermal noise. The theory of strong ally much stronger than the ordinary two-body collisions turbulence is still far from complete. and leads to a random walk of the particles that takes the As applied to the fluidlike Jeans instabilities connected complete system toward thermal quasi-steady state. Thus, with a “thermodynamic nonuniformity” of the stellar disk relaxation in stellar systems could occur without binary (the system is not sufficiently “hot” in the equatorial star–star enconters through the influence of collective mo- plane), the nonlinear effects appear in the following fash- tions of the stellar gas upon the particle distribution. ion. The velocity dispersion of a young stellar population Barbanis & Woltjer (1967) studied almost circular or- is small, and their space and velocity distribution are not bits of stars of a rotating galaxy in the gravitational field completely relaxed. As the result of the reaction of the os- of spiral arms on the basis of both epicyclic theory (see cillations, the velocity dispersion of the main part of the also Binney & Tremaine 1987, p. 478) and numerical inte- distribution function of young stars would be expected gration of the equations of motion. Lynden-Bell & Kalnajs to increase in the field of unstable waves with an ampli- (1972) treated the resonant regions separately. Carlberg & tude increasing with time. Because the Jeans instability Sellwood (1985) re-derived Dekker’s (1975) basic equation is characterized by the critical value of velocity disper- governing the response of a rotating stellar disk to any sion ccrit = cTQcrit, the rise in temperature in turn leads transient-perturbing potential, and then calculated nu- to a decrease in the growth rate of the wave amplitude. merically the resonant response to a model slowly-varying Eventually, as a result of such “heating”, the Jeans in- wave in the vanishing growth rate limit. Binney & Lacey stability will be switched off and finally the spiral can- (1988), Jenkins & Binney (1990), and Jenkins (1992) de- not be maintained (Binney & Tremaine 1987, p. 479). veloped a formalism to describe a heating of the local disk This process of the self-suppression of instability by the due to gravitational scattering by an imposed weak, time- growing wave amplitude is reminiscent of the nonreso- varying perturbing potential. They showed that such a nant relaxation in a plasma, which can effectively heat the heating process could be described by a diffusion equa- medium even in the absence of collisions between particles tion in action space and solved the diffusion equation by (Alexandrov et al. 1984, p. 420; Krall & Trivelpiece 1986, Monte Carlo simulation for cloud and spiral wave scat- p. 520). Apparently, Toomre (1964, p. 1237) first advanced terers. Although all of these studies showed that the disk the idea that the stars in the galactic disk would tend perturbations affect the dynamical evolution significantly, to develop random motion from the gravitational energy they focused mainly on a test particle responce to a given via this tendency toward gravitational instability. In addi- field of waves having adopted the model form for spi- tion, the nonlinear relaxation causes the diffusion of stars ral perturbation and assumptions as to the typical wave- in coordinate space, that is, mass re-distribution. Romeo length, etc. of the spiral waves (Carlberg & Sellwood 1985, (1990) already discussed the role played by instabilities in p. 81; Jenkins 1992, p. 623). The exact velocity depen- the disk’s secular evolution. The collective relaxation of dence of the velocity diffusion tensor, and the resultant stars may be done by their interaction with the gravita- evolution in velocity space, were not completely pinned tional field of unstable waves, solely. In interaction with down. standing waves, with the exception of spatially small reso- We present a self-consistent quasi-linear theory of dy- nant regions, there is no energy exchange in the wave–star namical relaxation of two-dimensional self-gravitating, ro- system (Binney & Tremaine 1987, p. 482). tating stellar disks toward a thermal quasi-steady state At the qualitative level, Goldreich & Lynden-Bell via collective effects. In the process a star “collides” with (1965) and Marochnik (1968) have suggested instabil- almost aperiodically growing inhomogeneities of a galac- ities as a cause of enhanced relaxation in galaxies. tic gravitational field which result from the development Lynden-Bell (1967) and later Shu (1978) considered the of the fierce Jeans instability. The dominant interactions, problem of the collisionless relaxation in a vigorously which change the velocities and orbits of stars, are those E. Griv et al.: Quasi-linear theory of the Jeans instability 341 with transient, rapidly-varying gravity perturbations. We the collisionless Boltzmann equation (Lin et al. 1969) find that the theory successfully accounts for several basic 2 observations of the Galaxy, given that the growth rate of ∂f ∂f vϕ ∂f vϕ + v + Ω+ + 2Ωv + +Ω2r the perturbations is comparable to the epicyclic period of ∂t r ∂r r ∂ϕ ϕ r stars. In particular, we find that the off-resonant charac- 2 ∂Φ ∂f κ vrvϕ 1 ∂Φ ∂f ter of the interaction leads to a velocity diffusion tensor − − vr + + =0, (1) that is independent of velocity, and leads to an anisotropic ∂r ∂vr 2Ω r r ∂ϕ ∂vϕ Maxwellian distribution whose velocity dispersion grows where the total azimuthal velocity of the stars was repre- with time. Nonlinear effects near resonances in a Jeans- sented as a sum of vϕ and the basic circular velocity rΩ. stable system deserve separate research (e.g., Rauch & Here vr and vϕ are the residual velocities in the radial Tremaine 1996). To emphasize it again, unlike Carlberg and azimuthal directions, and r and ϕ are the galacto- & Sellwood (1985), Binney & Jenkins (1988), and Jenkins centric cylindrical coordinates. As a rule, in spiral galax- & Binney (1990) we solve a self-consistent Boltzmann– ies |vr| and |vϕ| rΩ. In Eq. (1), Φ(r,t) is the total Poisson system of equations. The relaxation mechanism gravitational potential determined self-consistently from suggested in the present paper has an essential depen- the Poisson equation ∆Φ = 4πGσδ(z). dence on the equilibrium of the disk matter, and it has As the equilibrium state an axisymmetric, weakly in- nothing to do with Lynden-Bell’s violent relaxation. Brief homogeneous in the radial direction stellar disk is adopted. first reports have been published by Griv et al. (1994) and The system in the equlibrium is described by the equation: Griv et al. (2001). ! 2 2 ∂fe vϕ ∂fe κ vrvϕ ∂fe vr + 2Ωvϕ + − vr + =0 (2) ∂r r ∂vr 2Ω r ∂vϕ 2. Basic equations or ∂f /∂t = 0, and the angular velocity of rotation Ω(r) A thin rotating disk is taken as a model of the flat galaxy e at a distance r from the center is such that the neces- in many papers for analysis of the gravity perturbations. sary centrifugal acceleration is exactly provided by the This is because stars of the nonrotating spherical-like sub- central gravitational force, rΩ2 = ∂Φ /∂r. In these equa- system, if it exists at all, which have large random veloci- e tions, f (r, v)andΦ(r) are the equilibrium distribution ties, will make a relatively small contribution to the wave e e function and the gravitational potential. field (Marochnik & Suchkov 1969)2. In the spirit of Griv In the quasi-linear theory, one may follow the proce- & Peter (1996a), we solve the system of the collisionless dure of linearization by writing f = f (r, v,t)+f (r, v,t) Boltzmann equation and the Poisson equation describing 0 1 and Φ = Φ (r, t)+Φ(r,t) with |f /f | 1and the motion of a self-gravitating ensemble of stars in such a 0 1 1 0 |Φ /Φ | 1 for all r and t. The functions f and Φ system within an accuracy of up two orders of magnitude 1 0 1 1 are functions oscillating rapidly in space and time, while with respect to small parameters 1/|k |r and c /rΩfor r r the functions f and Φ describe the slowly developing radial wavenumber k , radius r, and angular velocity Ω, 0 0 r “background” against which small perturbations develop; looking for waves which propagate in a two-dimensional f (t =0)≡ f and Φ (t =0)≡ Φ . Linearizing Eq. (1) galactic disk. This approximation of an infinitesimally thin 0 e 0 e and separating fast and slow varying variables one obtains disk is a valid approximation if one considers perturba- the equation for the fast developing distribution function tions with a radial wavelength λ =2π/kr that is greater h, the typical disk thickness. In actual spiral galaxies for a df1 ∂Φ1 ∂f0 1 ∂Φ1 ∂f0 ≈ = + , (3) subsystem of young, low-dispersion stars, h 200 pc. The dt ∂r ∂vr r ∂ϕ ∂vϕ dimensionless parameters 1/|kr|r and cr/rΩ are small, and in addition 1/|kr|r ∼ cr/rΩ. The fact is took into account where d/dt means total derivative along the star orbit that because of the nature of the gravitational force, disks and f0 is a given equilibrium distribution function deter- of spiral galaxies are always spatially inhomogeneous and mined from Eq. (2). The equation for the slow part of the are far from uniform rotation. distribution function is D E ∂f ∂Φ ∂f 1 ∂Φ ∂f 0 = 1 1 + 1 1 , (4) 2.1. Boltzmann and Poisson equations ∂t ∂r ∂vr r ∂ϕ ∂vϕ h···i We assume that the stars move in the disk plane so that where denotes a time average. vz = 0. This allows us to use the two-dimensional distri- ˜ butionR function f(r, ϕ,R vr,vϕ,t) such that f = fδ(z)δ(vz), 2.2. Equilibrium distribution f = f˜dzdvz,and fdvrdvϕ = σ,whereσ(r,t)isthe surface density. In the rotating frame of a disk galaxy, the Making use of expressions for the unperturbed epicyclic local distribution function of stars f(r, v,t)mustsatisfy trajectories of stars in the equilibrium central field Φe, r = r + v⊥ [sin φ − sin(φ − κt)] ,u =dr/dt, 2 The stellar halo can account only for about 1% of the local 0 κ 0 0 r 2Ω v⊥ − − dϕ dark halo density (Robin et al. 1999). ϕ =Ωt + κ rκ [cos(φ0 κt) cos φ0] ,uϕ = r dt , (5) 342 E. Griv et al.: Quasi-linear theory of the Jeans instability where v⊥, φ0 are constants of integration, v⊥/κr ≈ 3. Linear theory ρ/r 1, and ρ ∼ v⊥/κ is the mean epicyclic radius, we To determine oscillation spectra, let us consider the sta- can choose the Schwarzschild (the anisotropic Maxwellian) bility problem in the lowest (or local) WKB approxima- distribution function f (r, |v|,t= 0) satisfying the unper- 0 tion; this is accurate for short wave perturbations only, but turbed part of the kinetic equation, that is, Eq. (2)3.In qualitatively correct even for perturbations with a longer Eqs. (5), u and u are the components of the star’s veloc- r ϕ wavelength, of the order of the disk radius R. In galaxies, ity relative to the disk center. To integrate Eq. (3) over t, R ≈ 15 kpc. In the local WKB approximation in Eqs. (3) we need to determine the components of the star’s velocity and (4), assuming the weakly inhomogeneous disk, the at each point relative to the local standard of rest: perturbation of equilibrium parameters is selected in the vr = ur = v⊥ cos(φ0 − κt), form of a planeP wave (in the circular rotating frame) ik r+imϕ−iω∗t ℵ1(r,t)=0.5 ℵk e r +c. c. ,whereℵk is vϕ = uϕ − rΩ ≈ (κ/2Ω)v⊥ sin(φ0 − κt)(6) k an amplitude that is a constant in space and time, m is (Spitzer & Schwarzschild 1953). As is seen, the motion the nonnegative azimuthal mode number (= number of of a star in the disk of a rotating galaxy is represented spiral arms), ω∗ = ω − mΩ is the Doppler-shifted complex as in epicyclic motion along the small Hipparchus epicy- wavefrequency, ω∗,k = <ω∗,k + i=ω∗,k, |kr|R 1, suffixes cle with a simultaneous circulation of the epicenter about k denote the kth Fourier component, and c. c. means the the galactic center. The problem of epicyclic motion in complex conjugate. In the local WKB approximation it its most general form is equivalent to the problem of is assumed that the wave vector and the wavefrequency the motion of a charged particle in a given electromag- vary continuously. By utilizing the more accurate nonlo- netic field, in which the solution can be decomposed into cal WKB approximation, it may be shown that in fact the guiding center motion and the epicyclic motion. The the characteristic oscillation frequencies of an inhomoge- epicycle radius is analogous to the gyroradius in a plasma neous disk must be quantized, i.e., must pass through a (Marochnik 1966). The Schwarzschild distribution func- discrete series of values. In galaxies, discrete spiral modes tion, which is a function of the two epicyclic constants of were already found in stellar population by Rix & Zaritsky 2 2 motion v⊥/2andr0Ω(r0), has been given by Shu (1970): (1995), Zaritsky & Rix (1997), Rudnick & Rix (1998), and 2 Block & Puerari (1999). In the near-infrared, the morphol- 2Ω(r0) σ0(r0) − v⊥ · ogy of older star-dominated disk indicates a simple classi- f0 = 2 exp 2 (7) κ(r0) 2πcr (r0) 2cr (r0) fication scheme: the dominant Fourier m-mode. Fridman et al. (1998) and Burlak et al. (2000) detected m =1−9 In Eq. (7), r0 is the radius of the circular orbit, which is chosen so that the constant of areas for this circular orbit spiral modes in relatively young stellar population of the 2 2 2 nearly face-on galaxies from observations in the Hα line. r0ϕ˙ 0 is equal to the angular momentum integral r ϕ˙, v⊥ = 2 2 2 2 ≡ 2 A ubiquity of low-m (m =1−4) modes was confirmed. In vr +(2Ω/κ) vϕ,˙ϕ0 Ω (r0)=(1/r0)(∂Φ0/∂r)0,andΩ as well as κ and and c are evaluated at r . In the equation the linear theory, one can select one of the Fourier har- r 0 ℵ − above the fact is used that in a rotating reference frame monics: 0.5 [ k exp(ikrr + imϕ iωt)+c. c.]. The solution in such a form represents a spiral wave with m arms. the radial velocity dispersion cr and the azimuthal velocity dispersion cϕ are not independent but connected through ≈ (Eqs. (6)) cr (2Ω/κ)cϕ. In the Solar vicinity a velocity 3.1. Perturbed distribution function distribution of the early O and B stars with ages t<(5 − 7)×107 yr is almost the spherical Maxwellian distribution Using the transformation of the partial derivatives ∂/∂vr − with the velocity dispersion near 8 km s 1 (Grivnev & and ∂/∂vϕ given by Eqs. (9), the solution of the linearized Fridman 1990). Such a distribution function f0(t =0)for kinetic Eq. (3) may be written immediately: Z the unperturbed system is particularly important because t it fits observations for all stars in the Galaxy (Shu 1970). 0 v⊥ · ∂Φ1 ∂f0 2Ω 1 ∂Φ1 ∂f0 f1 = dt + 2 , (10) It is this metaequilibrium that is examined for stability. −∞ v⊥ ∂r ∂v⊥ κ r ∂ϕ ∂r The partial derivatives in Eqs. (3) and (4) transform where f is given by Eq. (7), and f (t →−∞)=0,so as follows (Shu 1970; Morozov 1980; Griv & Peter 1996a): 0 1 by considering only growing perturbations we neglected ∂ vr ∂ − 2Ω vϕ ∂ the effects of the initial conditions. Paralleling the analysis = 2 , (8) ∂vr v⊥ ∂v⊥ κ v⊥ ∂φ0 leading to Eq. (13) of Griv & Peter (1996a), from Eq. (10) it is straightforward to show that 2 " ∂ 2Ω vϕ ∂ 2Ω ∂ 2Ω vr ∂ ∞ ∞ ≈ + + · (9) X X i(n−l)(φ0−ζ) ∂v κ v⊥ ∂v⊥ κ2 ∂r κ v2 ∂φ κ ∂f0 e Jl(χ)Jn(χ) ϕ ⊥ 0 f1 =Φ1 l v⊥ ∂v⊥ lκ − ω∗ −∞ n=−∞ 3 Equation (2) does not determine the equilibrium distribu- l= # X∞ X∞ i(n−l)(φ −ζ) tion uniquely: there are, in fact, many solutions f0 of Eq. (2) 2Ω m ∂f e 0 J (χ)J (χ) + 0 l n , that satisfy ∂f0(t =0)/∂t = 0. In plasma physics, these states κ2 r ∂r ω∗ − lκ are often called metaequilibria, since they are only equilibria l=−∞ n=−∞ on a time scale short compared with collision times. (11) E. Griv et al.: Quasi-linear theory of the Jeans instability 343 where Jl(χ) is the Bessel function of the first kind, χ = Λ0(0) = 1; and (c) Λl(x)forl =6 0 start from Λl(0) = 0, 2 k∗v⊥/κ,tanζ =(2Ω/κ)tanψ,andk∗ = k{1+[(2Ω/κ) − reach maxima, and then decrease. 2 }1/2 1] sin ψ is the effective wavenumber. In Eq. (11) the Equation (12) replaces the standard Lin–Shu disper- − denominators vanish when ω∗ lκ = 0. This occurs near sion relation (Lin & Shu 1966; Lin et al. 1969; Shu 1970), corotation and other resonances. The Lindblad resonances to take into account all terms up to 2 orders in small pa- occur at radii where the field (∂/∂r)Φ1 resonates approx- rameters ρ/r and 1/|k |r. Also, the terms ∝ L−1 were − r imately with the harmonics l = 1 (inner resonance) and omitted in Toomre (1964), Lin & Shu (1966), Lin et al. l = 1 (outer resonance) of the epicyclic (radial) frequency (1969), Shu (1970), Mark (1977), Lynden-Bell & Kalnajs of equilibrium oscillations of stars κ. Clearly, the location (1972) studies. The main difference from the original Lin– of these resonances depends on the rotation curve and Shu dispersion relation is in the factor ψ =6 0, which makes < the spiral pattern speed ω∗/m; the higher the m value, the generalized dispersion relation to be correct even in the closer in radius the resonances are located (Lin et al. the regime of open perturbations (see Griv & Peter 1996a 1969). The corotation resonance occurs at a radius where for a discussion). Actually, Lin & Shu (1966), Lin et al. l = 0 in Eq. (11). Resonances are places where linearized (1969), Shu (1970), and Mark (1977) allowed for a de- equations describing the motion of particles do not apply. parture from axial symmetry of the perturbations only In the vicinity of the resonances it is necessary to use non- partially by introducing a Doppler-shifted in a rotating linear equations, or to include terms of higher orders into reference frame wavefrequency ω∗ = ω − mΩ but omitting the approximate form of the equations. The former ap- all other m and ψ-dependent terms in Eq. (12), on the proach was adopted by Contopoulos (1979) and the latter grounds that they were interested in almost axisymmet- one was adopted by Griv et al. (2000a,b). ric perturbations. Therefore, in fact these authors as well as Toomre (1964) obtained a criterion for an instability 3.2. Generalized dispersion relation of axisymmetric perturbations of the Jeans kind only – the widely used Toomre critical radial velocity dispersion. Integrating Eq. (11) over velocity space and equating the It says nothing about the stability of Jeans modes which result to the perturbed surface density given by the im- are not tightly wound, particularly the dominant open in- −| | proved solution of the Poisson equation σ1 = k Φ1/2πG stabilities of the differentially rotating disks. The effects (Lin & Lau 1979; Griv & Peter 1996a), the generalized of the azimuthal forces have been analyzed by Lin & Lau dispersion relation may easily be obtained (1979) by adopting the hydrodynamical model. Important ∞ − ∞ − conclusions were obtained about the enhanced amplifica- k2c2 X e xI (x) mρ2 X e xI (x) r = κ l l +2Ω l ,(12) tion of nonaxisymmetric density waves in a differentially 2πGσ0|k| lκ − ω∗ rL ω∗ − lκ l=−∞ l=−∞ rotating system (Bertin & Mark 1978; Lin & Lau 1979;
2 2 2 2 2 Bertin 1980). The free kinetic energy associated with the where κ ∼ 2Ω, ω∗ =6 lκ, x = k c /κ ≈ k ρ , ρ = c /κ ∗ r ∗ r differential rotation of the system under study is one pos- is now the mean epicyclic radius, and I (x) is the Bessel l sible source for the growth of the energy of these spiral function of imaginary argument. In the second term on the − perturbations, and appears to be released when angular | |≈ 2 1 right-hand side, L ∂ ln(2Ωσ0/κcr )/∂r is the radial momentum is transferred outward. scale of a spatial inhomogeneity, and in the local WKB In disk-shaped galaxies, L ∼ (d ln σ /dr)−1 < 0. In approximation ρ2/r|L| 1. The dispersion relation (12) 0 the Solar vicinity of the Galaxy the value of the radial connects the frequency of excited oscillations ω∗ with the scale length |L| is about 3 kpc, which is a typical value for wavenumber k for every r and describes the ordered be- the radial scale length of the exponential component of havior of a medium near its equilibrium state. the disk when compared with external galaxies of similar The asymptotic expansion of the Bessel function Il(x) 2 2 morphological types (Porcel et al. 1998). in the short-wavelength limit, x ≈ k∗ρ 1 (the case of epicyclic radius that is large compared with wavelength): The dispersion relation (12) is valid for relatively open spirals throughout a weakly inhomogeneous disk excluding x e 1 resonance zones4. The existence of solutions of Eq. (12) Il(x) ' √ 1+O . (13) 2πx x ω∗ = ω∗(k∗) with <ω∗ =0,6 |=ω∗/<ω∗| 1, and 2 2 =ω∗ > 0 implies oscillating instability, while the solu- In the opposite long-wavelength limit, x ≈ k∗ρ ∼< 1: tions with <ω∗ =0and6 =ω∗ < 0 describe the absorp- ∞ X x l+2n 1 tion of waves. The solutions with =ω∗ =0and<ω∗ =06 I (x)= · (14) l 2 n!(n + l)! describe the natural (harmonic) oscillations, and the so- n=0 lutions with =ω∗ > 0, |=ω∗/<ω∗| 1 describe the Jeans In the later limit the epicyclic radius is small (or compa- rable to) compared with wavelength. 4 Griv & Peter (1996a) and Griv et al. (2000a,b) have found a −x In Eq. (12), the functions Λl(x)=e Il(x) appear peculiar instability of collective oscillations of stellar disks that commonly in a theoretical treatment of Maxwellian plas- is different in nature from the Jeans instability. It was shown mas in a magnetic field. It is instructive to note: (a) that in the Jeans-stable differentially rotating disk a resonant 0 ≤ Λl(x) ≤ 1; (b) Λ0(x) decreases monotonically from Landau-type oscillating instability (overstability) may develop. 344 E. Griv et al.: Quasi-linear theory of the Jeans instability
3 3 density waves. This problem has been studied by Morozov (a) (b) 2 2 (1980, 1981a), Polyachenko (1989), Griv & Peter (1996a), 1 1 ←2 ←2 Polyachenko & Polyachenko (1997), Griv et al. (1999). In ←1 ←1 0 ←1 0 ←1 ←2 ←2 Sect. 4 of the present paper, the problem is studied in the −1 −1 , wavefrequency , wavefrequency framework of the quasi-linear theory. ν ν −2 −2 The generalized dispersion relation (12) is compli- −3 −3 0 1 2 3 4 5 0 1 2 3 4 5 cated: the basic dispersion relation above is highly non- k ρ, wavenumber k ρ, wavenumber * * linear in the frequency ω∗. Therefore, in order to deal with the most interesting oscillation types analytically, 3 3 (c) (d) only various limiting cases of perturbations described by 2 2 some simplified variations of dispersion relation may be 1 1 ←1 ←1 ←2 ←2 considered. 0 0 ←1 ←2 ←2 −1 −1 ←1 , wavefrequency , wavefrequency ν ν −2 −2 3.3. Simplified dispersion relation
−3 −3 0 1 2 3 4 5 0 1 2 3 4 5 Let us first restrict ourselves to consideration of the princi- k ρ, wavenumber k ρ, wavenumber * * pal part of a system between the inner and outer Lindblad Fig. 1. The generalized Lin–Shu dispersion relation of a ho- resonances, |l|≤1, by considering low-frequency pertur- mogeneous√ (|L|→∞) stellar disk in the case 2Ω/κ = bations with |ω∗| ∼< κ (which dispersion laws are given | | 2and sin ψ = 1 for the different Toomre’s Q-values: in Fig. 1 by curves 1 and 2). That is, |ω∗| less than the a) Q =0.5(2Ω/κ), b) Q =0.8(2Ω/κ), c) Q =2Ω/κ,and epicyclic frequency of any disk stars, and the consider- d) Q =1.5(2Ω/κ). The solid curves represent the real part ation is limited to the transparency region between the of the dimensionless Doppler-shifted wavefrequency of low- turning points in a disk. In the opposite case of the high frequency, |ω∗| <κ, long-wavelength (1) and short-wavelength perturbation frequencies, |ω∗| >κ, the effect of the disk (2) Jeans oscillations we are interested in. The dashed curves represent the imaginary part of the dimensionless wavefre- rotation is negligible and therefore not relevant to us: in quency of low-frequency vibrations. The dot-dashed curves this “rotationless” case the star motion is approximately represent the wavefrequencies of additional high-frequency, rectilinear on the time and length scales of interest which |ω∗| >κ, Jeans modes. Long-wavelength vibrations (those with are the wave growth/damping periods and wavelength (cf. 2 2 k∗ρ ∼< 1) are the most unstable ones. Alexandrov et al. 1984, p. 110). Resonances of a higher order, l = 2, 3,..., are dynamically less important. Secondly, in Eq. (12) one can consider two asymptotic instability. The quantity Ωp = <ω∗/m characterizes the limits: the limit of long-wavelength perturbations, x ∼< 1, rate of rigid-body rotation of the wave pattern. and the opposite limit of short-wavelength perturbations, A general impression of how the spectrum of non- x 1. Finally, we consider the weakly inhomogeneous axisymmetric Jeans perturbations behaves in a homo- system and the most important for the problem of spiral geneous nonuniformly rotating disk can be gained from structure low–m perturbations: from now on in all equa- Fig. 1, which shows the dispersion curves in the cases of tions 2Ω(mρ2/rL) 2Ω ∼ κ and m ∼ 1. Additionally, in −1 Jeans-unstable systems ((a) and (b)), a marginally Jeans- the local version of the WKB method |kr| < |L|
0.45 deformed and the additional low-frequency (|=ω∗3|/κ
0.4 1) gradient oscillations appear in an inhomogeneous disk.
0.35 4. Quasi-linear equations 0.3 We anticipate that the fluidlike Jeans-unstable oscillations
0.25 must influence the distribution function of the main part of stars in such a way as to hinder the wave excitation, i.e.,
growth rate 0.2 to increase the random velocity spread ultimately at the expense of circular motion or gravitational energy. This 0.15 is because the Jeans instability, being essentially a grav-
0.1 itational one, tends to be stabilized by random motions of stars (Eqs. (16) and (17)). Therefore, along with the 0.05 growth of the oscillation amplitude the velocity dispersion must increase, and finally in the disk there can be estab- 0 0 2 4 6 8 10 12 14 16 lished a stationary distribution so that the Jeans-unstable m, azimuthal mode number perturbations are completely vanishing. Eventually the Fig. 2. The growth rate of the Jeans instability of a homo-√ disk evolves toward a quasi-stationary, marginally Jeans- geneous stellar disk (arbitrary units) in the case 2Ω/κ = 2, stable distribution. (In turn, the Jeans-stable perturba- | | | | r0 =8, kr =1,ρ =1,and sin ψ = 1 for the different val- tions are subject to a weak Landau-type oscillating insta- ues of the azimuthal mode number m. The growth rates have bility; Griv et al. 2000a,b.) maxima with respect to the critical mode number m ∼ 1. crit Next, we substitute the solution (11) into Eq. (4) and average the latter over time. Taking into account that the terms l =6 n vanish for axially symmetric functions f , the galaxy NGC 4254 already revealed the dominance of 0 after averaging over φ we obtain the equation the m = 1 component (Iye et al. 1982). It was stressed 0 ∞ that theories of the origin of spiral structure of galax- X X 2 2 2 2 ∂f0 π E ∂ k∗κ l Jl (χ) − l Jl (χ) ies must explain the asymmetric structure of this galaxy = i k ∗ ∂t 2 ∂v⊥ v⊥χ ω∗ − lκ ω∗ − lκ k l=−∞ without the help of spiral exciters such as tidal compan- X X∞ 2 ions or a bar. About 20% of all late-type spiral galaxies ∂f0 π 2Ω m ∂ 2Ω m Jl (χ) × + i Ek and 54 early-type disk galaxies have the strong m =1az- ⊥ 2 2 ∗ − ∂v 2 −∞ κ r ∂r κ r ω lκ imuthal component of the surface brightness in the near k l= J 2(χ) ∂f IR–band and the R–band, confirming lopsidedness as a − l 0 ∗ , (20) dynamical phenomenon (Zaritsky & Rix 1997; Rudnick ω∗ − lκ ∂r & Rix 1998). A similar fraction of galaxies ≈30%−50% E | |2 = ∗ has lopsided HI distributions or kinematic asymmetries where k = Φk exp(2 ω∗t)andω∗ is the complex con- (Baldwin et al. 1980; Bosma 1981; Richter & Sancisi 1994; jugate wavefrequency. Swaters et al. 1999; Kornreich et al. 2000). In the single- As usual in the quasi-linear theory, in order to close E arm galaxy NGC 4378 the spiral arm can be traced over the system one must engage an equation for k. Averaging 1 over the fast oscillations, we obtain most 1 4 revolutions (Kormendy & Norman 1979). Disk ellipticity (m = 2) may also be common (Andersen et al. (∂/∂t)E =2=ω∗E . (21) 2001). k k The shape and the number of spiral arms depend on Equations (20) and (21) form the closed system of quasi- the equilibrium parameters of a galaxy. For the Galaxy linear equations for Jeans oscillations of the rotating in- the most unstable pattern is that of 1−4 arms, the radial homogeneous disk of stars, and describe a diffusion in ve- distance between the arms being about 2 kpc. locity and coordinate space. The distortion of the wave In the another frequency range, |ω∗| |ωJ|, Eq. (15) packet due to the disk inhomogeneity is included through has another root, which describes the gradient, L−1 =6 the second term on the right-hand side in Eq. (20). The 0, branch of oscillations. The gradient perturbations are spectrum of oscillations and their growth rate are given stable and are independent of the stability of Jeans modes by Eqs. (12) and (19), respectively. In the Solar vicinity, −8 −1 (Griv & Peter 1996b). These low-frequency, |ω∗3| Ω, =ωJ ∼ Ω ≈ 2 × 10 yr . Equations (20) and (21) are oscillations are not important in dynamics of galaxies. clearly very approximate, since the local WKB, epicyclic, Although Eq. (15) can be analyzed directly and even and weakly nonlinear approximations were used. solved analytically in the case |l|≤1, graphical represen- Two general physical conclusions can be deduced with- tation of the roots is much more convenient. A graphic out solving Eq. (20). First, the initial distribution of stars method of solution of Eq. (15) is indicated in Fig. 3. It will change upon time only under the action of growing is seen in Fig. 3 how the usual Jeans oscillations with perturbations (=ω∗ > 0). Therefore, only transient, gravi- |ω∗|/κ ∼< 1 as shown in Fig. 1 by curves 1 and 2 are tationally unstable patterns heat the disk and cause star’s E. Griv et al.: Quasi-linear theory of the Jeans instability 347
1 1 1 (a) (b) (c) 0.5 ←2 0.5 0.5 ←1 ←3 κ κ κ / / / * 0 * 0 * 0 ω ω ←1 ω
−0.5 −0.5 ←2 −0.5
−1 −1 −1 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 k ρ k ρ k ρ * * *
1 1 1 (d) (e) (f) 0.5 0.5 ←2 0.5 ←1 κ κ κ / / / * 0 * 0 * 0
ω ω ω ←3 ←1 −0.5 −0.5 −0.5 ←2 −1 −1 −1 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 k ρ k ρ k ρ * * * Fig. 3. The behavior√ of the simplified dispersion relation of an inhomogeneous disk for reasonable galactic parameters L = −0.01, 2Ω/κ = 2, | sin ψ| = 1 and for the different Toomre’s Q-values (upper panels – Q =0.5(2Ω/κ)andlowerpanels– Q =1.5(2Ω/κ)). Unlike in Fig. 1, only low-frequency modes are shown, |l| < 2or|ω∗/κ| ∼< 1. The solid curves represent the real part of the dimensionless Doppler-shifted Jeans wavefrequency (1 and 2) and the additional gradient frequency (3), and the dashed curves represent the imaginary part of the wavefrequency. A feature of this dispersion relation is the existence of the gradient branch of oscillations both in the case of the Jeans-unstable disk (panel c)) and in the case of the Jeans-stable one (panel f)). The gradient perturbations are stable, =ω∗3 ≤ 0, and are low-frequency ones, |ω∗3| Ω. guiding centers to diffuse radially6. Secondly, the distri- 5. Astronomical implications bution function of random velocities ∝ (∂/∂v⊥)(∂f /∂v⊥) 0 As an application of the quasi-linear theory we investi- will change under the action of both the radial and the tan- q gate the relaxation of low-frequency and Jeans-unstable ∝ 2 2 6 gential forces (this is because in Eq. (20) k∗ kr + kϕ = | | 2 ( ω∗ <κand ωJ < 0, respectively) oscillations in the 0). But the mean surface mass density (that is, star’s guid- weakly inhomogeneous (ρ2/r|L| 1) disk of the Galaxy. ing centers positions) ∝ (∂/∂r)(∂f0/∂r) will change under the action of only the azimuthal forces (m =6 0): the re- 5.1. Dynamical evolution of the stellar disk quirement that m =6 0 is required for shifts in guiding cen- ters. Since the guiding center encodes angular momentum, In the following, we restrict ourselves to the fastest grow- this is merely a statement of angular-momentum conserva- ing long-wavelength, χ2 and x2 1, and low-frequency, tion. As angular momentum is transferred outward, inner |ω∗| ∼< κ perturbations (see the explanation just at the material falls to the center and outer material migrates end of Sect. 4 and Fig. 1). Then in Eqs. (12) and (20) one 2 ≈ 2 2 ≈ 2 outward (Lynden-Bell & Pringle 1974). Thus, the diffu- can use the expansions J0 (χ) 1, J1 (χ)=J−1(χ) χ /4 −x 2 3 sion of star’s guiding centers leads to the core-dominated and e I1(x) ≈ (1/2)x−(1/2)x +(5/16)x . Equation (20) mass density profile in rotating galaxies, together with the takes the simple form buildup of an extended outer envelope. 2 2 ∂f0 −D ∂ f0 −D ∂ f0 | |≤ v 2 r 2 =0, (22) In the most important low-frequency limit, l 1, ∂t ∂v⊥ ∂r the function Λ(x) = exp(−x)I (x) starts from Λ(0) = 0, P 1 D 2 2= E D reaches a maximum Λ < 1atx ≈ 0.5 (Figs. 1 and 3), whereP v =(π/2κ ) k k∗ ω∗ k, r = max 2E 2= = E and then decreases. Hence, the growth rate has a maxi- π k m k/η ω∗,both ω∗ and k are functions 2 of t, and we took into account the fact that in actual mum at x ≈ 0.5 < 1(ork∗ = kcrit ≈ κ /4πΩcr). spiral galaxiesp with a flat rotation curve 2Ω/κ2 ≈ r/η, where η ∼ (GM/r0)ln(r/r0)andM is a galaxy’s 6 mass. The term ∝D(t) describes a diffusion of stellar The stationary (quasi-stationary) density waves, =ω∗ → 0, r could serve to heat the disk stars and to change star’s guiding guiding centers in coordinate space. As is seen, the centers positions at the wave–particle resonances (Lynden-Bell velocity diffusion coefficient for stars Dv(t) is independent & Kalnajs 1972; Binney & Tremaine 1987, p. 482). These res- of v⊥ (to lowest order). This is a qualitative result of onances, however, have only limited radial extent. the nonresonant character of the star’s interaction with 348 E. Griv et al.: Quasi-linear theory of the Jeans instability collective aggregates. Whereas Binney & Lacey (1988), of the bulk of stars in configuration space takes place be- Jenkins & Binney (1990), and Jenkins (1992) attributed cause they gain wave energy as the instability develops. the effect to some unspecified time behavior of the The instability-induced diffusion, however, presumably ta- spiral density wave, we attribute the effect to the term pers off as Jeans stability is approached: the radial velocity resulting from wave growth of Jeans-unstable almost dispersion cr becomes greater than the critical one ccrit. aperiodic perturbations. Hence we obtain a velocity Observations already convincingly indicated a secular dy- diffusion coefficient that is independent of velocity v⊥, namical evolution in spiral galaxies on the Hubble time without any further assumptions. In addition, in actual scale or even smaller (Martinet 1995). spiral galaxies Dr is independent of r. By contrast, Thus, the true time scale for relaxation in the Galaxy Lynden-Bell & Kalnajs (1972) and Carlberg & Sellwood may be much shorter than its standard value ∼1014 yr for (1985) considered a model slowly-varying density wave in the Chandrasekhar collisional relaxation; it may be of the = → −1 −1 8 the vanishing growth rate limit, ω∗ 0. Therefore, the order (=ωJ) ∼> Ω ∼> 3 × 10 yr, i.e., comparable with density wave and the basic state stars could interact only 2−3 periods of the Galaxy rotation. This short relaxation at the limited radially wave–particle resonances. time is in agreement with both observations (Binney & An expression for f0 at t →∞can be deduced from Tremaine 1987, p. 473; Grivnev & Fridman 1990) and N- Eq. (22). We introduce the substitution dτ/dt = Dv(t), body simulations (Hohl 1971; Morozov 1981a; Sellwood & d/dt =(dτ/dt)(d/dτ). Then Eq. (22) can be re-written Carlberg 1984; Tomley et al. 1991; Griv & Chiueh 1998).
2 ∂f0 ∂ f0 Dr ∂f0 ∂Dv − − = ∗ 2 2 =0; =2 ω . (23) 5.2. Time evolution of the velocity dispersion ∂τ ∂v⊥ Dv ∂r ∂τ We have shown above that the squared plane velocity dis- The particular solutions of the system (23) are persion of stars increases with time t as − 2 const v⊥ 2 2 2 f0(v⊥,τ)=p exp (24) hv i≈(π/2κ )k E ∝ exp(=ω∗t), (26) 2 2 ∗ k τ + cr /2 4(τ + cr /2) where k∗ and =ω∗ ∼ Ω are the effective wavelength and and the growth rate of the most unstable oscillations. For const −r2 young stellar populations, say, with ages t<109 yr, f (r, τ )≈ p exp ·(25) 0 2 2 D D =ω∗t<1 and we see that heating by a Jeans-unstable r +(Dr/Dv)τ 4[r0 +( r/ v)τ] 0 density wave can produce the observed hv2i∝t law for (The observations have taken into account that most the age–velocity dispersion rate (Wielen 1977). However, spiral and S0 galaxies have an exponential disk with over a long time span, the unstable mode itself would radial surface distribution of oldest stellar populations, change its properties as the basic state evolves. That is, 2 σ0 ∝ exp[−(r/r0) ].) Accordingly, during the develop- the wave–star interaction is a nonlinear process, which ment of the Jeans instability, the Schwarzschild distri- should take into account the fact that the wave field bution of random velocities (i.e., Gaussian spread along affects only weakly the stars with high peculiar veloc- vr,vϕ coordinates in velocity space) is established. As ities. Mathematically, this fact is expressed by the in- the perturbation energy increases, the initial distribution verse dependence of the reduction factor F (x) in Eqs. (15) 2 spreads (f0(v⊥) becomes less peaked), the effective tem- and (19) on the velocity dispersion: only the young stars, perature grows with time (i.e., Gaussian spread increases), with their small velocity dispersion, are extremely sensi- and the effectiveR velocity dispersionP attains the value: tive to any gravity perturbation. Thus, we expect the de- h 2i D 2 2E v =2τ =2 v(t)dt =(P π/2κ ) k k∗ k. The en- crease in the growth of unstable modes, and, consequently, 2 2 E ergy of the oscillation field k(k∗/κ ) k thus plays the the decrease in the growth of the velocity dispersion with h 2i 2 role of a “temperature” T in the particle distribution. age of stars, and the growth will not lead to v >vobs −1 Subsequently, sufficient velocity dispersion prevents the in less than a Hubble time. Here vobs ≈ 40 km s is the Jeans instability from occuring. The diffusion in configu- observed velocity dispersion of the oldest disk stars. The ration space is due entirely to the growth of the Jeans- latter is consistent with the observed age–velocity correla- unstable modes in a self-gravitating collisionless system tion in the Galaxy which tapers off beyond a certain age subject to a time-dependent potential. ∼3 Gyr (Wielen 1977; Str¨omgren 1987; Haywood et al. This mechanism increases (a) the velocity dispersion of 1997; Dehnen & Binney 1999; Binney 2001). This process stars in the Galaxy’s plane after they are born (Eq. (24)), of diffusion in phase-space, with the velocity distribution and (b) the radial spread of the disk (Eq. (25)). The col- function scattering into a smoother and hotter distribu- lisionless relaxation mimics thermal relaxation in a two- tion, lead naturally to definitions of young disk, of age dimensional stellar disk, leading to Maxwell–Boltzmann less than about 1−2 Gyr, and old disk, of age older than type velocity distributions with an effective temperature about 3 Gyr. that increases with time. To repeat ourselves, sufficient One concludes that the proposed mechanism of wave– random velocity spread prevents the Jeans instability from star interaction is able to account for both the shape of occuring (Griv et al. 1994; Griv et al. 1999). The diffusion the velocity ellipsoid (the anisotropic Maxwellian, that E. Griv et al.: Quasi-linear theory of the Jeans instability 349 is, Schwarzschild distribution) and the form of the age– the critical value of velocity dispersion ccrit, eventually as velocity dispersion law in the plane of the Galaxy. a result of such heating the gravitational instability will be switched off rapidly. Therefore, from the theoretical point of view, the Jeans-unstable density waves in a collision- 5.3. Velocity diffusion in the Solar vicinity less stellar disk have to be short-lived, and they should Transient spiral arms excite random motions parallel to dissipate after a few rotations of the system. the equatorial plane. According to Eq. (24), the heating N-body experiments have shown similar behavior for a efficiency of unstable density wave features depends on hot system of stars repeatedly. A spiral structure usually their spatial and temporal form. Let us evaluate the heat- develops in a numerical model during the first rotation ing ∆v for a realistic model of the disk of the Galaxy of the system. These spirals are evidently Jeans-unstable in the Solar vicinity. In accordance with the theory as density waves and not material arms, since test particles developed above, we consider the fastest growing mode pass right through them (Miller et al. 1970; Quirk 1972). 2 −1 Sellwood & Carlberg (1984), Sellwood & Athanassoula with k∗ = kcrit ≈ κ /4πΩcr ≈ π kpc and =ω∗ ≈ Ω. E 2 ≈ (1986), and Griv et al. (1999) presented evidences that the According to observations, in the Solar vicinity k/Φ0 −3 ≈ 2 2 ≈ spirals arise from collective processes. The growth of these 10 (Lin et al. 1969; Yuan 1969), Φ0 0.5r0Ω , r0 ≈ −1 ≈ perturbations then saturates due to the increasing velocity 8.5kpc,cr(t =0) 10 km√ s ,andκ 1.5Ω. From −1 dispersion of the particles, and the Jeans-unstable modes Eq. (24), one obtains ∆v ≈ Dvt =20−30 km s ,where t =109 yr. This value of ∆v is in agreement with both decay during the next two to three rotations. The increase estimates based on the observed stellar velocities (Mayor of velocity dispersion in those experiments cannot be ex- 1974; Wielen 1977; Str¨omgren 1987; Meusinger et al. 1991; plained by usual two-body encounters (Hohl 1973; Griv Grivnev & Fridman 1990; Dehnen & Binney 1999) and et al. 1999). It seems likely, the fast heating in the N-body N-body simulations (Hohl 1971; Quirk 1972; Sellwood & models is due to collective effects discussed in the present Carlberg 1984; Tomley et al. 1991). Thus already in the paper, i.e., due to density fluctuations as a result of the first 4−5 galactic revolutions, in say about 109 yr, the stel- Jeans instability. The violent Jeans instability occurs in lar populations see their epicyclic energy vary by a factor numerical models because the Toomre stability criterion, of ten. satisfied for an initial computer model, is not applicable to nonaxisymmetric gravity perturbations in a differentially rotating, inhomogeneous disk. Density perturbations aris- 5.4. Migration of the Sun’s guiding center ing from the instabilities of the nonaxisymmetric pertur- bations grow into spiral structures. These density waves, There is considerable scatter in the metallicities of stars which have growth rates comparable to the mean orbital that have a common guiding center and age. On the other frequency, dynamically heat the disk and exert torques hand, it is widely believed that all interstellar material at a which redistribute both mass and angular momentum. In given time and radius has a common metallicity. The para- a final, quasi-steady state after two to three rotations, dox can be resolved if one assumes that these stars were the stars in acomputer model acquire large random veloc- born at different radii and then migrated to its present ities about 2 times more than Toomre’s criterion predicts locations as a result of a series of uncorrelated scattering (Hohl 1971; Morozov 1981a; Sellwood & Carlberg 1984; events (Wielen et al. 1996; Binney 2001). Tomley et al. 1991; Griv et al. 1999). Thus, in the non- The migration may be explained naturally by linear regime, the stars (and indeed also the gas before “collisions” of stars with the Jeans-unstable density waves. their formation) can continue developing Jeans-unstable Let us estimate the scale of radial migration ∆R of the condensations only if some effective mechanism of cooling Sun’s guiding center. According to observations, we adopt exists, leading to Toomre’s Q-values smaller than 2−2.5. E 2 ≈ −3 2 ≈ 2 2 ≈ ≈ the ratio k/Φ0 10 , η Ω r , m 1, r0 8.5kpc, Cooling of a numerical model through dissipation in = ≈ ≈ and ω∗√ Ω, k∗ kcrit. Then from Eq. (22) we obtain the gas layer, accretion, and/or star formation (injection ≈ D − ∆R rt =2 3kpc.This∆R is in fair agreement of new-born stars which are formed on almost purely cir- with the estimate of Wielen et al. (1996) ∆R =1.9kpc cular orbits) has already been proposed as a mean to pro- based on a radial galactic gradient in metallicity. We con- long collective instabilities in the plane of the stellar disk clude that the Sun has migrated from its birth-place at (Miller et al. 1970; Quirk 1972; Sellwood & Carlberg 1984; − r =6 7 kpc in the inner part of the Galaxy outwards by Tomley et al. 1991; Griv & Chiueh 1998). The cold inter- − ≈ × 9 2 3 kpc during its lifetime of t 4.5 10 yr. stellar medium may play a dominant role in determin- ing the wave-like structure in galaxies with a high star 5.5. Problem of the spiral structure formation rate because it is the site of the generation of dynamically cold objects. The star formation process fu- Under the influence of Jeans-unstable perturbations, the els the Jeans instability, favoring the excitation of short- random velocity dispersion of the main part of the stellar lived spiral modes exponentially growing through a recur- distribution function increases essentially on a dynamical rent instability cycle in the disk of newly formed stars. In time scale (on the time scale of only 2−3 galactic rota- this regard, no prominent spirals are seen in S0 galaxies tions). Because the Jeans instability is characterized by that have little or no interstellar matter. Martinet (1995) 350 E. Griv et al.: Quasi-linear theory of the Jeans instability already pointed out the connection between dynamical Muzafar Maksumov, Frank Shu, Irina Shuster, and Jun-Hui evolution and efficiency of star formation in galaxies of Zhao. This work was supported in part by the Israel Science various morphological types. We expect that stellar disks Foundation and the Israeli Ministry of Immigrant Absorption. of flat galaxies are rife with many transient, chaotic- looking Jeans-unstable wakes. Such short-lived spirals References develop in young stellar population in rapid succession and possible interaction (cf. Sellwood & Carlberg 1984; Alexandrov, A. F., Bogdankevich, L. S., & Rukhadze, A. A. Tomley et al. 1991). Summarizing, multiple armed spiral 1984, Principles of Plasma Electrodynamics (New York: patterns in gas-rich galaxies may result from the simul- Springer) Andersen, D. R., Bershady, M. A., Sparke, L. S., et al. 2001, taneous excitation and superposition of different Jeans- ApJ, 551, L131 unstable modes. The coexistence of several spiral waves is Baldwin, J. E., Lynden-Bell, D., & Sancisi, R. 1980, MNRAS, − possible. The low–m modes (m =1 4) are the dominant 193, 313 ones. 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