Gravitational Instability of a Nonrotating Galaxy *

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1 SLAC-PUB-536 October 25 Gravitational Instability of a Nonrotating Galaxy * Alexander W. Chao ;) Stanford Linear Accelerator Center Abstract Gravitational instability of the distribtion of stars in a galaxy is a well-known phenomenon in astrophysics. This work is a preliminary attempt to analyze this phenomenon sing the standard tools developed in accelerator physics. By applying this analysis, it is fond that a stable nonrotating galaxy wold become nstable if its size exceeds a certain limit that depends on its mass density. Introdction There are some notable examples in the past when developments in astrophysics later are fond to be profondly connected to important topics in accelerator physics. Two major topics in accelerator physics has been nonlinear dynamics and collective effects. It trns ot that each of these topics has its origin traced back to astrophysics. In nonlinear dynamics, Henri Poincaré (854-92) was believed to be the first person who noted the behavior of nonlinear dynamical chaos. In 887, he entered a contest sponsored by the king of Sweden and Norway, and the problem was to prove that the solar system (as a 3-body system) was dynamically stable. He did not scceed in proving it, bt his work won the prize anyway. Poincaré was also the person who introdced the Poincareé section, which accelerator physicists today se everyday. In fact, what a beam position monitor detects in a circlar accelerator is a special case of Poincaré section. Dynamic apertre and chaotic motion are also typically observed as Poincaré sections (this time on a compter printot), and have become daily langage of nonlinear dynamists in accelerator physics. In collective effects, one notable preview was the impressive work by James Clerk Maxwell (83-879). In 857, Maxwell also won a prize the Adams Prize when he proved analytically that the Satrn rings can not be stable nless they consisted of many small solid satellites instead of a single solid piece. Today, we call this mechanism of Maxwell negative mass instability in accelerator physics. Following these pioneering fonders, one might ask if today, after years of evoltion, might there be some stdies that the accelerator physicists have developed, and that can be applied to astrophysics in retrn. One sch attempt is ventred in this paper. We will try to apply modern accelerator techniqes to the well-known problem of a gravitational instability of a nonrotating galaxy. Consider a distribtion of stars in a galaxy described by a distribtion density ρ( x, v,t) in the phase space ( x, v) at time t. We wish to analyze the stability of this distribtion of stars nder the inflence of their collective gravitational force. To simplify the problem, we will se a flat Eclidean space-time and will consider Newtonian, nonrelativistic dynamics only. In other words, we ignore both the special theory and the general theory of relativity. The instability ths does not assme a specific cosmological model other than Newtonian gravity. If this approach trns ot fritfl, a large arsenal of analysis tools can be transported from accelerator physics to this and other problems in astrophysics. The instability we are interested in is selfgenerated, i.e. it occrs spontaneosly. In particlar, it does not reqire an initial seed flctation at the * Department of Energy contract DE-AC2-76SF55 ) achao@slac.stanford.ed Sbmitted to Jornal of High Energy Physics and Nclear Physics (China) SLAC, Stanford University, Stanford, CA 9439

2 birth of the galaxy. The instability growth pattern as well as its rate of growth are intrinsic properties of the system. This gravitational instability is a well-known problem; its first analysis appeared almost a centry ago []. What we do in the following is to treat the same problem sing the standard techniqes developed in the stdy of collective instabilities in circlar accelerators [2]. 2 Dispersion Relation Consider a particlar star in the galaxy. eqations of motion of this star are x = v v = G d v The d x ρ( x, v,t)( x x) x x 3 () G is the gravitational constant. Note that these eqations do not depend on the mass of the star nder consideration. [3] Evoltion of ρ is described by the Vlasov eqation ρ t + x ρ ρ x+ v v = ρ t + x v ρ ρ + v G, v,t)( x x) d v d x ρ( x x x 3 = (2) To examine the stability of the system, let the galaxy distribtion be given by an npertrbed distribtion ρ pls some small pertrbation. Let the npertrbed distribtion ρ depend only on v, ρ = ρ ( v) (3) This npertrbed distribtion is niform in x, i.e. it is niform in the infinite 3-D space. ρ ( v) is so far nrestricted. The fnction We will allow the small pertrbation arond ρ to depend on t and in x althogh the npertrbed distribtion ρ has been assmed not to depend on t or in x. We Forier decompose the pertrbation and write ρ( x, v,t) = ρ ( v)+ ρ( v)e iωt+i k x (4) k is the wavenmber vector and ω is the anglar oscillation freqency of the pertrbation. We anticipate that for a given k, there will be a specific oscillation freqency ω. In general, k is considered to be real, bt for a given k, the corresponding ω can be complex. With the time dependence of the pertrbation given by e iωt, we see that the imaginery part of ω is the instability growth rate (growth rate if Im(ω) >, damping rate if Im(ω) < ). The qantity ρ is considered to be infinetisimal compared with ρ. Sbstitting Eq.(4) into Eq.(2) and keeping only first order in ρ yield i() ρ( v) ( ) +G d v ρ( v ρ ( v) ) q( v k) = (5) q( k) d x ei k ( x x) ( x x) = d y ei k y y (6) x x 3 y 3 is a well-defined qantity depending only on k; it is the Forier transform of the Newton kernel x/ x 3, and might be called the graviton propagator following a terminology in qantm field theory. In fact, aside from the singlarity at the origin k =, it can be shown that q( k) = 4πi k 2 k (7) In accelerator physics, the Newton kernel x/ x 3 stands for the wake fnction while its Forier transform q stands for the impedance. Eq.(5) can be rewritten as ( ) ρ ( v) q( ρ( v) = ig d v ρ( v v ) k) (8) Integrating both sides over v and canceling ot the mtal factor of d v ρ( v ) then gives a dispersion relation that mst be satisfied by ω and k, ρ ( v) q( v = ig d v k) (9) Given ρ ( v), we solve this dispersion relation for ω as a fnction of k. This soltion is then sed to find the most nstable pattern of pertrbation and its corresponding growth rate will be described next. 3 Uniform Isotropic Galaxy We next consider an npertrbed distribtion that depends only on the magnitde of v, i.e., let which gives ρ = ρ ( v 2 ) () ρ v = 2 v ρ ( v 2 ) ()

3 This is the case of a niform isotropic (niform in x, isotropic in v) galaxy. Normalization condition is gives 4πv 2 dv ρ (v 2 ) = ρ m (2) = mass density of stars per nit volme Sbstitting Eqs.(7) and () into Eq.(9) then = 8πG k 2 v k d v ρ ( v 2 ) (3) Let k = (,,k), and choose coordinates so that v = v(sinθ cosφ,sinθ sinφ,cosθ), Eq.(3) becomes, with a change of variable = cosθ, = 6π2 G k v 3 dv ρ (v 2 ) d ω kv (4) One mst refrain from performing the integration over at this time becase that integral involves a singlarity. Proper treatment of the singlarity follows the standard techniqe sed in accelerator physics (and plasma physics) on Landa damping [4]. Omitting the details, the treatment amonts to adding an infinitesimal positive imaginary part to ω, i.e. ω ω +iɛ, I(ω,kv) d ω kv d ω +iɛ kv = P.V. d ω kv iπω ( k 2 v H ω ) 2 kv = 2 kv ω k 2 v ln ω kv 2 ω+kv iπω ( k 2 v H ω ) 2 kv (5) P.V. means taking the principal vale of the integral, and H(x) = for x > and for x < is the step fnction. By taking P.V., the singlarity is avoided in a well-defined manner. To be specific, we next take a niform distribtion of ρ (niform in x, isotropic in v, and ρ is constant p to v ), 3ρ m if v 2 < v 2 ρ (v 2 4πv ) = 3 otherwise (6) This distribtion is called the waterbag model in accelerator physics. The qantity v 2 is related to the temperatre of the stars. Sbstitting Eq.(6) into Eq.(4) gives the dispersion relation λ = 2+xln x x+ λ = 6πGρ m k 2 v 2 +iπxh( x ) and (7) x = ω kv (8) In accelerator physics, λ is replaced by the impedance. One simplification for the gravitational instability is that λ is a real qantity, while the impedance is complex in general. 4 Stability Condition We next need to compte the instability growth rate, which is given by the imaginary part of ω, as a fnction of k. The star distribtion ρ ( v) wold be nstable if, for any k, its corresponding ω is complex with a positive imaginary part. We need to compte x as a fnction of λ sing Eq.(7) in order to obtain ω as a fnction of k. Unfortnately Eq.(7) gives λ as a fnction of x, and its inversion to give x as a fnction of λ is difficlt. Here we apply another techniqe of accelerator physics as follows. In general x is complex, bt at the edge of instability, x is real. The edge of stability can therefore be seen by plotting the RHS of Eq.(7) as x is scanned along the real axis from to. Fig. shows the real and imaginary parts of the RHS of Eq.(7) in sch a scan. The horizontal and vertical axes are the real and imaginary parts of the RHS of Eq.(7) respectively. As x is scanned from to, the RHS of Eq.(7) traces ot a cherry-shaped diagram, inclding the stem of the cherry rnning from to along the real axis. If λ lies inside this cherry diagram (inclding the stem), the galaxy distribtion is stable. Since λ is necessarily real and positive, the stability condition therefore reads λ < 2 (9)

4 Figre Stability diagram for the galaxy distribtion. Eq.(9) indicates that a hot niverse (high temperatre, i.e. large v ) is more stable than a cold niverse. This is expected de to the Landa damping mechanism. It also indicates that the star distribtion is nstable for long-wavelength pertrbations (small k). The threshold wavelength is given by x th = 2π k th (2) 2πGρm k th = (2) v Pertrbations with wavelength longer than x th are nstable. One might expect that the galaxy will have a dimension of the order of x th becase if the galaxy had a larger dimension, it wold have broken p de to the instability ntil it is redced to the stable size. When y +, we obtain Eq.(7) as it shold. We will need to solve Eq.(24) for x and y for given λ >. It trns ot that in this range there is always one 2 soltion with prely imaginary ω, i.e. x =, and therefore or, written ot explicitly, 6πGρ m k 2 v 2 λ = 2 2y tan y (25) = 2τ (26) 2 kv tan kv τ We need to find τ as a fnction of k. To do so, we first scale the variables by = kv, v = τ (27) 6πGρm 6πGρm and then = v tan v Fig.2 shows the reslt (28) 5 Spontaneos Gravitational Instability When λ > /2, ω will be complex. The instability growth rate is determined by the imaginary part of ω, τ = Im(ω) (22) We need to modify Eq.(7) slightly for complex ω. In the nstable region, let Eq.(7) reads λ = 2+ x+iy 2 ω kv = x+iy, (y > ) (23) + (ix y) tan x+ y ln (x )2 +y 2 (x+) 2 +y 2 (24) tan x y Figre 2 v vs according to Eq.(28). As seen from Fig.2, the growth rate vanishes (v = ) when = 2, corresponding to λ = /2, i.e. at the stability bondary. This is of corse expected. Fig.2 also shows that instability occrs fastest for small, i.e. small k and large wavelength of the pertrbation. The growth rate is maximm at = with v = 2/3. This means the maximm growth rate occrs for pertrbation of infinite wavelength, and is given by (τ ) max = 4πGρ m (29) Note that the growth rate is independent of v, even thogh that for instability, there is still the condition λ > /2, which does depend on v and can be cast into the form (see Eq.(2)) k < 3 v (τ ) max (3)

5 The fastest instability corresponds to k =, or an instability of infinite wavelength. According to Eq.(3), all stable galaxies mst have a dimension smaller than a critical vale, i.e. galaxy dimension < 2πv 2πGρm (3) The stability is provided throgh Landa damping. When the temperatre v Eqs.(29) and (3) are or main reslts. 6 Nmerical Estimates, no galaxies can be stable. For a nmerical application, we take estimates from the Milky Way, ρ m = 2 23 g/cm 3 v = 2 km/s We obtain a maximm growth time of τ max = 7 6 years for pertrbations with very large wavelengths. For stability, the galaxy dimension mst be smaller than 4 light-years, which seems to be consistent with the size of the Milky Way. 7 Discssions The case stdied so far is that of a galaxy with niform distribtion of stars. One direction of generalization is to consider galaxies with a finite extent. One attempt was made in [5]. It is fond that a spherically symmetric distribtion of the Haissinski type [6] (potential-well distortion in accelerator physics) does not exist. An attempt to extend to a planar galaxy, still nonrotating, had also been made in [5]. The npertrbed Haissinski distribtion does exist. However, this planar distribtion is fond to be always stable against pertrbations that do not involve transverse strctres. Any instability of the planar galaxy will therefore have to have a sfficiently complex pattern. It is conceivable that the same analysis can be applied to the dynamics of galaxies in a galaxy clster, instead of stars in a galaxy. In that case, ρ( x, v,t) describes the distribtion of galaxies in the galaxy clster. We might then take the corresponding nmerical vales ρ m = 3 g/cm 3 v = km/s We obtain a growth time of τ max = years. The galaxy clster dimension shold be smaller than 9 light-years. These vales do not seem to be nreasonable. For more detailed applications, we will have to inclde the rotation of the galaxy into the analysis. The npertrbed distribtion will then involve also the anglar momentm. The analysis will be more involved. Still frther extensions might inclde the special relativity and general relativity to replace Newtonian gravity and to avoid the action-at-a-distance problem. 8 Acknowledgments This work was spported by Department of Energy contract DE-AC2-76SF55. I wold like to thank Pisin Chen, Gennady Stpakov, Ron Rth, Ernest Corant, Alan Krisch, and Larry Jones for several enlightening discssions. References See e.g., P.L. Palmer, Stability of Collisionless Stellar Systems, Klwer Academic Pblishers, The Netherlands, 994, and references qoted therein. 2 See e.g., Alexander W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, John Wiley & Sons, New York, 993, and references qoted therein. 3 A.A. Vlasov, J. Phys. USSR 9, 25 (945); S. Chandrasekhar, Plasma Physics, Univ. Chicago, L.D. Landa, J. Phys. USSR, 25 (946). 5 Alex Chao, Proc. Workshop on Qantm Aspects of Beam Physics, Hiroshima, Japan, 23, SLAC-PUB-9. 6 J. Haissinski, Novo Cimento 8B, 72 (973). This reslt depends on or assmption of Newtonian dynamics of action-at-a-distance. Under this assmption, pertrbation at one location instantly affects locations infinitely far away. If this assmption is appropriately removed, it is expected that the instability for pertrbations with very large wavelengths will be weakened.

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