A&A 396, 309–313 (2002) DOI: 10.1051/0004-6361:20021395 & c ESO 2002

Jeans’ gravitational instability and nonextensive kinetic theory

J. A. S. Lima, R. Silva, and J. Santos

Departamento de F´ısica, UFRN, C.P. 1641, 59072-970 Natal, RN, Brazil

Received 18 March 2002 /Accepted 27 August 2002

Abstract. The concept of Jeans gravitational instability is rediscussed in the framework of nonextensive statistics and its asso- ciated kinetic theory. A simple analytical formula generalizing the Jeans criterion is derived by assuming that the unperturbed self-gravitating collisionless gas is kinetically described by the q-parameterized class of power law velocity distributions. It is found that the critical values of wavelength and mass depend explicitly on the nonextensive q-parameter. The standard Jeans wavelength derived for a Maxwellian distribution is recovered in the limiting case q = 1. For power-law distributions with cut- off, the instability condition is weakened with the system becoming unstable even for wavelengths of the disturbance smaller than the standard Jeans length λJ.

Key words. gravitation – hydrodynamics

1. Introduction On the other hand, some recent studies involving the sta- tistical description of a large variety of physical systems re- In a seminal paper, James Jeans discussed the conditions un- vealed that some extension of the standard Boltzmann-Gibbs der which a fluid becomes gravitationally unstable under the approach should be needed. Particular examples are systems action of its own gravity (Jeans 1929). Nowadays, for any rel- endowed with long duration memory, anomalous diffusion, evant length scales in the (, galaxies, clusters, turbulence in pure-electron plasma, self-gravitating systems etc.), such an instability has been recognized as the key mech- or more generally systems endowed with long range interac- anism to explain the gravitational formation of structures and tions. In order to work with such problems, Tsallis (1988) their evolution in the linear regime. proposed the following generalization of the Boltzmann-Gibbs A noteworthy conclusion of Jeans’ work is that perturba- (BG) entropy formula for statistical equilibrium (see also tions with mass greater than a critical value MJ (Jeans’ mass) http.tsallis.cat.cbpf.br/biblio.htm for an extensive may grow thereby producing gravitationally bounded struc- and updated list of problems and references) tures, whereas perturbations with a mass smaller than M do   J    not grow and behave like acoustic waves. This gravitational  q S = −k 1 − p  /(q − 1), (1) instability criterion remains basically valid and plays a fun- q B i i damental role even in the expanding Universe (Peebles 1993; Coles & Lucchin 1995). In terms of the wavelengths of a fluc- where kB is the Boltzmann constant, pi is the probability of the tuation, Jeans’ criterion says that λ should be greater than a ith microstate and q is a parameter quantifying the degree of 2 → critical value λJ = πvs /(Gρ0), which is named the Jeans’ nonextensivity. In the limit q 1 the celebrated BG extensive length. In this formula G is the gravitational constant, ρ0 is formula, namely the unperturbed matter and v is the speed for  s = − , adiabatic perturbations. As is widely known, the same crite- S kB pi ln pi (2) rion also holds for a collisionless self-gravitating cloud of gas, i v except that the sound speed s is replaced by the velocity dis- is recovered. One of the main properties of S is its pseudoad- σ q persion . As explained in many textbooks (see, for instance, ditivity. Given a composite system A + B, constituted by two Binney & Tremaine 1987), such a result follows naturally from subsystems A and B, which are independent in the sense of fac- a kinetic theoretical approach based on the the Vlasov equa- torizability of the microstate probabilities, the Tsallis measure tion, where the evolution of the collisionless gas is ultimately −1 verifies S q(A+B) = S q(A)+S q(B)+(1−q)k S q(A)S q(B). In the described by perturbing the equilibrium Maxwellian velocity B limit q → 1, S q reduces to the standard logarithmic measure, distribution. and the usual additivity of the extensive BG statistical mechan- ics and thermodynamics is also recovered. Thus, if q differs Send offprint requests to:J.A.S.Lima, from unity, the entropy becomes nonextensive (superextensive e-mail: [email protected] if q < 1, and subextensive if q > 1), with the Boltzmann factor

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20021395 310 J. A. S. Lima et al.: Jeans’ gravitational instability and nonextensive kinetic theory generalized into a power law. In other words, |q − 1| quantifies the “Jeans Swindle”, the resulting particle distribution function the lack of extensivity of the system. may be approximated as A large portion of the experimental evidence supporting = v + u, r, ,  , Tsallis proposal involves a non-Maxwellian (power-law) equi- f f0( ) f1( t) f1 f0 (3) librium distribution function associated with the thermostatis- where f is the corresponding perturbation in the distribution tical description of the classical N-body problem. This equilib- 1 function. Following standard lines, the dynamic behavior of rium velocity q-distribution may be derived from at least three this system can be described by the linearized Vlasov and different methods: (i) through a simple nonextensive general- Poisson equations. By neglecting up to second-order terms in ization of the Maxwell ansatz, which is based on the isotropy the expansion of the distribution function one obtains of the velocity space (Silva et al. 1998); (ii) within the nonex- tensive canonical ensemble, that is, maximizing Tsallis entropy ∂ f ∂ f ∂ f ∂ f 1 + u. 1 = ∇φ . 1 + ∇φ . 0 , (4) under the constraints imposed by normalization and the en- ∂t ∂r 0 ∂u 1 ∂u ergy mean value (Abe 1999) (iii) using a more rigorous treat- ment based on the nonextensive formulation of the Boltzmann ∇2φ = 4πG f (r, u, t)d3v, (5) H-theorem (Lima et al. 2001). 1 1 In the astrophysical context, the nonextensive equilibrium φ r φ r velocity distribution related to Tsallis’ statistics has been ap- where 0( )and 1( ) are, respectively, the unperturbed plied to stellar collisionless systems (Plastino & Plastino 1993), gravitational potential and its first order correction. As as well as to the peculiar velocity function of galaxies clus- is well known, if the “Jeans Swindle” is assumed, one φ = ters (Lavagno 1998). In the former paper, the entropy nonex- may set the unperturbed potential 0 0. In this case, tensive formula was maximized taking into account the con- the solutions of the above equations can be written as r, u, = u { k · r − ω } φ r, = φ { k · r − straints imposed by the total mass and energy density. It was f1( t) F( )exp i( t) and 1( t) exp i( ω } u φ found that the solutions behave like stellar and that t) provided that F( )and satisfy the relations the mass is finite if the nonextensive parameter is bounded by ∂ f ≤ / k · u − ω u − φk. 0 = , q 9 7. More recently, a more detailed account for spherically ( )F( ) ∂u 0 (6) symmetric systems involving either the gravothermal instabil- ity (Taruya & Sakagami 2002; Chavanis 2002), or the complete 2 3 determination of the density profiles has been presented in the k φ = −4πG F(u)d v. (7) literature (Lima & de Souza 2002). Applications in plasma physics are also important in their Combining these expressions one obtains the dispersion rela- own right, both from a methodological and physical viewpoint. tion between ω and k In this case, some metastable states in pure electron plasmas, 4πG k · ∂ f /∂u and the dispersion relations for an electrostatic plane-wave 1 + 0 d3v = 0. (8) 2 k · u − ω propagation in a collisionless thermal plasma (including un- k damped Bohm-Gross and Landau damped waves) have also The standard instability analysis follows from the above disper- been studied and compared to the standard results (Boghosian sion relation when one takes f0(u) as the Maxwellian velocity 1996; Lima et al. 2000). In particular, Liu et al. (1994) showed distribution. a reasonable indication for the non-Maxwellian velocity distri- Consider now the q-nonextensive framework proposed by bution from plasma experiments. All this empirical evidence Tsallis. In this case, the equilibrium distribution function f0(u) deal directly or indirectly, with the q-distribution of velocities can be written as (Silva et al. 1998; Lima et al. 2000) for a nonrelativistic gas. 1 2 q−1 In the present work, we quantify to what extent the nonex- ρ0Bq v f u = − q − , tensive effects modify the gravitational instability criterion es- 0( ) 3/2 1 ( 1) 2 (9) 2πσ2 2σ tablished by Jeans. For this enlarged framework, we deduce a new analytical expression for the Jeans dispersion relation where the normalization constant reads which gives rise to an extended instability criterion in accor- − Γ 1 + 1 dance with Tsallis thermostatistics. (3q − 1)(q + 1) q 1 q−1 2 Bq = (10) 4 Γ 1 q−1 2. Nonextensive gravitational instability for q ≥ 1, and In what follows, we focus our attention on the kinetic de- − 1 − q Γ 1 scription of a many particle system within the nonrelativistic 3q 1 1−q Bq = (11) 2 Γ 1 − 1 gravitational context. More precisely, we consider an infinite 1−q 2 self-gravitating collisionless gas described by a distribu- u, r, 1 < < σ ρ tion function f ( t) slightly depart from equilibrium. If for 3 q 1. Here is the velocity dispersion and 0 is the f0(v) corresponds to the unperturbed homogeneous and time- equilibrium density. As one may check, for q < 1/3, the q- independent equilibrium distribution, sometimes referred to as distribution (9) is unnormalizable, while for q > 1, it exhibits a J. A. S. Lima et al.: Jeans’ gravitational instability and nonextensive kinetic theory 311 thermal cutoff on the maximum value allowed for the velocity of the particles, which is given by q = 0.7 q = 0.9 q = 1.0 σ2 2 1 q = 1.1 vmax = · (12) q − 1 q = 2.0

This thermal cut-off is absent when q ≤ 1, that is, vmax is also unbounded for these values of the q-parameter. In this con- nection, it is worth noticing that the spirit of the H-theorem

is totally preserved for this nonextensive velocity distribution. (v) 0 As a matter of fact, by introducing a generalized collisional f term, Cq( f ), it has been shown that the entropy source is def- 0.5 inite positive for q > 0, and does not vanish unless the q- equilibrium distribution function assumes the above power-law form (Lima et al. 2001). In addition, taking into account that a lim|z|→∞ Γ(z + a)/[z Γ(z)] = 1, as well as that limq→1[1 − (q − 1)x2]1/(q−1) = exp(−x2) (Abramowitz & Stegun 1972), the ex- pressions (10) and (11) defining Bq reduce to B1 = 1, and as should be expected the distribution function f0 reduces to the 0 exponential Maxwellian distribution −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 ρ v2 v f (u) = 0 exp − · (13) 0 3/2 σ2 u 2πσ2 2 Fig. 1. Nonextensive velocity distribution function f0( ) for typical values of the q parameter. Power law distributions with q > 1 exhibits In Fig. 1 we have plotted the nonextensive distribution as a a thermal cutoff on the maximum value allowed for the velocity of the function of v for some selected values of the q parameter. As ff 1 < ≤ particles. This cuto is absent for 3 q 1, and the distribution is > ff < 1 can be seen, for q 1 the distribution exhibits a cuto on the unnormalizable for q 3 . maximum value allowed for the velocity of the particles. In the limit q → 1 we see from (12) that vmax →∞and the power-law behavior reduces to the exponential Maxwellian distribution. where MJ is the critical Jeans mass. Similarly, we see that for For q < 1thecutoff is also absent with the curves decreasing q > 1(1/3 < q < 1) the critical mass is decreased (increased) / less rapidly than in the Maxwellian case. Note also the conver- by the factor [2/(3q − 1)]3 2. Therefore, in comparison with gence from power-law to the standard Gaussian curve as long a Maxwellian gas (q = 1), a nonextensive collisionless self- as the q parameter approaches unity. gravitating system is more (or less) unstable depending on the Now, substituting the nonextensive equilibrium distribution valueassumedbytheq-parameter. given by (9) into the dispersion relation (8) one finds Let us now discuss in more detail the unstable modes. As

2−q happens in the perturbed fluid theory, fluctuations with wave- 2 − v − − v q 1 λ>λ 4πGρ Bq x 1 (q 1) σ2 lengths J will be unstable. In order to check what happens − 0 2 3v = ω = γ γ 1 5 3/2 d 0 (14) in the present kinetic approach we set i ,where is real σ (2π) kvx − ω positive, and insert this into the dispersion relation (14). As be- v k where the x axis has been chosen in the direction of . fore, choosing the vx axis in the direction of k, the following In the study of Jeans instability, the boundary between sta- dispersion relation is obtained ble and unstable solutions is achieved by setting ω = 0 in (14). For this value of ω the above integral can be easily evaluated k2 √ = − πβ2I β , 2 1 q( ) (16) and the result is a q-parameterized family of critical wavenum- kq bers kq given by √ β = γ/ σ β − where 2k and Iq( )isaq-dependent integral given by 2 2 (3q 1) kq = kJ (15) 2 − Γ 1 + 1 q 1 q−1 2 q + 1 2 β = k = πGρ /σ2 Jeans wavenum- Iq( ) where J 4 0 is the classical Γ 1 π ber (Binney & Tremaine 1987). From (15) we have the q- q−1 1 λ = π/ = λ / − λ − wavelength q 2 kq J 2 (3q 1) where J is the √1 − − 2 q 1 q−1 1 (q 1)x Jeans wavelength. Note that the classical value of the Jeans × dx (17) β2 + 2 wavenumber and wavelength as obtained from fluid theory 0 x = > / < < are recovered only if q 1. For q 1(13 q 1) for q > 1and λ the q-wavelength q is decreased (increased) by the factor 1 2/(3q − 1). Associated with λ there is a critical mass M , q−1 q q 1 − qΓ 1 ∞ 1 − (q − 1)x2 defined as the mass contained within a sphere of diameter λ , 1−q 2 q Iq(β) = dx (18) 4π / 2 2 = ρ λ / 3 = / − 3 2 Γ 1 − 1 π 0 β + x which is given by Mq 3 0( q 2) MJ[2 (3q 1)] , 1−q 2 312 J. A. S. Lima et al.: Jeans’ gravitational instability and nonextensive kinetic theory

0.0 0.0

-0.2 -0.2 0 ρ -0.4 0 -0.4 ρ G π G

Fluid theory π /4

2 Fluid theory /4 ω

q=0.8 2 q=0.8 -0.6 q=0.9 ω -0.6 q=1.0 q=1.0 q=1.5 q=1.2 q=1.5 -0.8 -0.8

-1.0 -1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 2 k2/k 2 k /k J q Fig. 2. Unstable branches of the dispersion relations for an infinite Fig. 3. Unstable branches of the dispersion relations as a function of self-gravitating collisionless gas obeying the nonextensive Tsallis q- the critical wavenumbers kq. Notice that the magnitude of kq is related statistics. For comparison the straight line representing the dispersion to the critical Jeans wavenumber by Eq. (15). As in Fig. 2, the straight relation for an infinite homogeneous fluid has also been plotted. We solid line stands to the case of an infinite homogeneous fluid. see that for q > 1 (power-law distributions with cutoff), the system is unstable even for wavelengths smaller than the standard Jeans length value (λJ). instability mechanism within Tsallis’ nonextensive thermosta- tistical context. The standard Jeans criterion is thus modified with critical values of mass and length depending explicitly on for 1/3 < q < 1. It should be noticed that the limit q → 1of the nonextensive q-parameter. both expressions is In Fig. 3 new plots of the dispersion relation (unstable ∞ −x2 β2 branches) are presented. Different from Fig. 2, instead of the β = 2 e = e − β , lim Iq( ) dx [1 erf( )] (19) ratio involving the critical Jeans wavenumber (k ), the behav- q→1 π β2 + x2 β J 0 ior of the unstable branches are analyzed as a function of the with the expression (16) reducing to ratio k/kq. 2 √ As should be expected at first sight, the curves are conver- k β2 = − πβ − β , gent for the critical ratio (k/kq = 1), but the growth rates are 2 1 e [1 erf( )] (20) kJ slightly different for each value of q. Note also that for all val- ff which is the standard dispersion relation for stellar systems ues of q, the results are quite di erent to what happens in the (Binney & Tremaine 1987). For an arbitrary value of q,the macroscopic fluid approach for gravitational instability (solid dispersion relation is given explicitly by the integral form (16) straight line). which can be numerically evaluated. In Fig 2 we plot the un- stable branches for several values of q along with the disper- 3. Final remarks sion relation for an infinite homogeneous fluid which is given by the solid straight line (see Binney & Tremaine 1987 for We have discussed the Jeans gravitational instability along the comparison). lines of the nonextensive statistical formalism proposed by As can be seen from these plots, for q > 1 the system Tsallis. In this extended kinetic framework, the Gaussian phase presents instability even for wavenumbers of the disturbance space density is replaced by a family of power law distributions greater than the standard critical Jeans value (kJ). In other parameterized by the nonextensive q-parameter. An attractive words, this kind of q-gaseous system is more unstable than the feature of Tsallis thermostatistics is that the models are analyti- one described by the standard Maxwellian curve, and therefore, cally tractable in such a way that a detailed comparison with the structures are more easily formed in this nonextensive context. extensive Maxwell-Boltzmann approach is readily achieved. However, for q < 1 the system may remain stable even for dis- The main interest of our results rests on the fact that the turbance with a wavenumber smaller than the Jeans wavenum- Maxwellian curve may provide only a very crude description ber. This yields a generalization of the Jeans gravitational of the velocity distribution for a self-gravitating gas, or more J. A. S. Lima et al.: Jeans’ gravitational instability and nonextensive kinetic theory 313 generally for any system endowed with long range interactions. References As we have seen, a well determined criterion for gravitational Abramowitz, M., & Stegun, I. A. 1972, Handbook of Mathematical instability is not a privilege of the exponential velocity distri- Functions (Dover Publications) bution function, but is shared by an entire family of power-law Abe, S. 1999, Phys. A, 269, 403 functions (named q-exponentials) which includes the standard Boghosian, B. M. 1996, Phys. Rev. E, 53, 4754 Jeans result for the Maxwellian distribution as a limiting case Binney, J., & Tremaine, S. 1987, Galactic Dynamics (Princeton U. (q = 1). In general, the basic instability criterion is maintained: Press, Princeton) perturbations with k > kq do not grow (or are damped even Chavanis, P. H. 2002 [astro-ph/0207080] considering that the collisionless q-gas is a time reversible sys- Coles, P., & Lucchin, F. 1995, Cosmology – The Origin and Evolution of Cosmic Structure (John Wiley & Sons) tem), while instability takes place for k < kq. However, unlike Jeans’ treatment, in this context there exists a family of growth Jeans, J. H. 1929, Astronomy and Cosmogony (University Press, rates parameterized by the nonextensive q-parameter. In partic- Cambridge) Lavagno, A., Kaniadakis, G., Rego-Monteiro, M., Quarati, P., & ular, for q > 1 (power-law distributions with cutoff) the system Tsallis, C. 1998, Astroph. Lett. and Comm., 35, 449 presents instability even for wavenumbers of the disturbance Lima, J. A. S., & de Souza, R. 2002, submitted for publication greater than the standard critical Jeans value kJ. Lima, J. A. S., Silva, R., & Plastino, A. R. 2001, Phys. Rev. Lett., 86, Finally, since the gravitational instability Jeans’ criterion 2938 remains basically true for an expanding Universe (Coles & Lima, J. A. S., Silva, R., & Santos, J. 2000, Phys. Rev. E, 61, 3260 Lucchin 1995), the adoption of a power law distribution and Liu, J. M., De Groot, J. S., Matte, J. P., Johnston, T. W., & Drake, R. P. the resulting nonextensive effects may have interesting conse- 1994, Phys. Rev. Lett., 72, 2717 quences for the galaxy formation process. Such implications Plastino, A., & Plastino, A. R. 1993, Phys. Lett. A, 177, 177 will be examined in a forthcoming communication. Peebles, P. J. E. 1993, Principles of Physical Cosmology (Princeton U. Press, Princeton) Acknowledgements. The authors are grateful to Dr. I. Szapudi for Silva, R., Plastino, A. R., & Lima, J. A. S. 1998, Phys. Lett. A, 249, many valuable comments. This work was partially supported by the 401 project Pronex/FINEP (No. 41.96.0908.00), FAPESP (00/06695-0), Taruya, A., & Sakagami, M. 2002, Physica A, 307, 185 and CNPq (62.0053/01-1-PADCT III/Milenio). Tsallis, C. 1988, J. Stat. Phys., 52, 479