Radial orbit instability as a dissipation-induced phenomenon
aa r X i v : . [ a s t r o - ph . GA ] N ov Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 7 April 2019 (MN L A TEX style file v2.2)
Radial orbit instability as a dissipation-inducedphenomenon
L. Mar´echal, J. Perez
Laboratoire de Math´ematiques Appliqu´ees, ENSTA, 32 Bd Victor, Paris, France
Accepted . Received ; in original form
ABSTRACT
This paper is devoted to Radial Orbit Instability in the context of self-gravitatingdynamical systems. We present this instability in the new frame of Dissipation-InducedInstability theory. This allows us to obtain a rather simple proof based on energeticsarguments and to clarify the associated physical mechanism.
Key words: gravitation – stellar dynamics – methods: analytical – instabilities
Instabilities in self-gravitating systems are fundamental pro-cesses to understand the shape and physical properties ofobjects such as galaxies or globular clusters. So far, only afew of such mechanisms are described in literature, namelyJeans instability, which governs the collapse of homogeneoussystems; gravothermal catastrophe, which concerns isother-mal spheres; and radial orbit instability, which occurs inanisotropic, strongly radial spherical systems. If the firsttwo are well understood, and have taken their place in thestudy of dynamical stellar systems (see Binney & Tremaine1987, sections 5.2 and 7.3), as a fact, the situation of ra-dial orbit instability is less clear. A complete story of thisphysical process, spanning almost forty years, is presented inMar´echal & Perez (2009). Three main points stick out (seethe review for all detailed references): there is as yet no sim-ple analytical proof of this phenomenon; there is no globalconsensus about its actual physical mechanism; and yet it isa fundamental process which affects the phase space distri-bution of primordial galaxies and contributes to produce theradial density profile of evolved systems. The present paperwill address the first two of these points.
We consider a system constituted of a large number N ofgravitating particles interacting together. We will assumethat all those particles have the same mass m . We denoteas q and p the position and the associated impulsion of aparticle with respect to some Galilean frame R , and Γ =( q , p ) the corresponding point in the phase space R .We assume that the statistical state of the system is de-scribed at each instant t by a distribution function f ( Γ , t ),with f ( Γ , t ) d Γ representing the number of particles con-tained in the elementary phase space volume d Γ locatedaround Γ . If the influence of collisions on the overall dynamics isneglected , this distribution function solves the CollisionlessBoltzmann–Poisson system (hereafter CBP) ∂f∂t + p m · ∇ q f − m ∇ q ψ · ∇ p f = ∂f∂t + { f, E } = 0 ∇ q ψ = 4 πGm Z f d p with boundary conditions ψ = | q |→ + ∞ O (cid:0) r − (cid:1) andlim | q | , | p |→ + ∞ f = 0. The function ψ ( q , t ) is the gravitationalpotential created by the particles, E ( q , p , t ) := p m + mψ is the one-particle Hamiltonian, and { ., . } denotes the Pois-son Bracket defined by { f , f } = ∇ q f · ∇ p f − ∇ q f · ∇ p f Any stationary solution f ( Γ ) of the CBP system is as-sociated to an equilibrium state of the particles distribution.It is now well known that CBP system is Hamiltonian withrespect to a Poisson bracket of non-canonical form arisingfrom the fact that a distribution function does not consti-tute a set of canonical field variables (see Kandrup 1990;Perez & Aly 1996): the set of distribution functions is aninfinite-dimensional space. The total energy associated to adistribution function f can be written as H [ f ] = Z d Γ p m f ( Γ , t ) − Z d Γ Z d Γ ′ f ( Γ , t ) f ( Γ ′ , t ) | q − q ′ | For any two functionals A [ f ] and B [ f ] of the distri-bution function, let h A, B i denote the Morrison bracket –introduced in the context of plasma physics by Morrison For a self-gravitating system with large values of N , this hy-pothesis is justified: see Binney & Tremaine (1987), part 1.2.1.c (cid:13) L. Mar´echal, J. Perez (1980) – defined by h A, B i = Z d Γ (cid:26) δAδf , δBδf (cid:27) f where δAδf stands for the functional derivative of A , which isthe linear part of A [ f + δf ] − A [ f ]. One can easily obtainthe Hamiltonian formulation of CBP systemd F d t = h F, H i where F [ f ] is any functional of f . The stability of equilibrium states is a very old problem oftheoretical stellar dynamics, and a large variety of meth-ods has been used to tackle it. A clear consensus was foundabout the global stability of isotropic spherical systems withdistribution function f ( E ) monotonically decreasing: afterthe pioneering works by Antonov (1961), linear stability wasobtained using energy methods after a long series of papersby Kandrup & Sygnet (1985) (and references within) or seealso Perez & Aly (1996) for a comparison of the differentresults; using direct normal mode techniques, complicatedproofs were also obtained (Fridmann & Polyachenko 1984;Palmer 1994). Non-linear stability of such spherical isotropicsystems was also proven for some specific models (see Rein2002, and references within).The stability of anisotropic spherical systems is a moredifficult problem. The distribution function depends bothon the one-particle energy E and on the squared one par-ticle angular momentum L := p q − ( p · q ) . The mostgeneral result in this context was obtained by Perez & Aly(1996) and concerns linear stability for the restricted case ofpreserving perturbations, which includes radially symmet-ric ones. Non-linear stability is assured for some classes ofgeneralized polytropes for which f (cid:0) E, L (cid:1) = E k L p withadapted values of k and p (see Rein (2002) and referenceswithin).Some very technical approaches using normal modesclaim linear instability for anisotropic systems composedonly of radial orbits (see Fridmann & Polyachenko 1984;Palmer 1994): this is known as the radial orbit instability.See Merritt & Aguilar (1985) for one of the first relevantnumerical approaches, Perez et al. (1996) for an intermedi-ate position or Barnes et al. (2009) and references within forthe most recent situation of this problem.A complete historical account of radial orbit instabilityis given in Mar´echal & Perez (2009). In the next section, wepresent some key features of this process. Radial orbit instability (hereafter ROI) appears in self-gravitating system dynamics with the pionnering worksof Antonov (1973) and H´enon (1973). A decade laterPolyachenko & Shukhman (1981) propose a stability crite-rion based on the ratio of radial over tangential kinetic en-ergies, when it is to small the system must leave its spher-ical symmetry and form a bar. This work is criticized byPalmer & Papaloizou (1987) which suggest, using normalmodes techniques, that ROI can occur for arbitrary small values of the Russian ratio, provided the distribution func-tion of the system is unbounded for orbits with zero angularmomentum. This paper is also the first one to propose arelevant physical mechanism to understand ROI based onresonant trapped orbits; we note that this mechanism needsa coupling between orbits.Several factors show that a radial system needs a “seed”from which ROI can appear. This is developed in detail inRoy & Perez (2004). The general idea is that there has tobe a near-equilibrium state, so that coupling between or-bits has the time to develop, and the instability to grow. InMac Millan et al. (1999), density profiles in a power law areconsidered, which means that the core has the time to sta-bilize before outer zones collapse, causing ROI to appear;it also shows that adding clumps tends to accelerate theprocess. A counterexample can be found in Trenti & Bertin(2006), which shows that homogeneous spherical haloes donot undergo ROI, as the system tends to isotropy beforereaching equilibrium. For this reason, our study will focuson ROI emerging from equilibrium states.Although ROI is a natural candidate to produce tri-axiality which can occur in self-gravitating systems, it wasnoted ( e.g.
Katz 1991) that this spatial counterpart of ROIcould disapear during the merging process of the galaxyformation. However, Huss et al. (1999) and more recentlyMac Millan et al. (1999) have shown that ROI is a funda-mental initial process which shapes the phase space of thegalaxy progenitor and allows it to get the good final massdensity profile. It is therefore important to understand fullythe nature of ROI.In this context, the objective of this paper is twofold. Onthe first hand, in section 2, we present a general method forinvestigate instability of self-gravitating systems. This ap-proach couples a general mathematical result by Bloch et al.(1994) which generalizes Lyapunov theory, and the symplec-tic approach of the stability problem of CBP system (seeBartolomew 1971; Kandrup 1990; and Perez & Aly 1996) –it must be noted that Kandrup (1991) has already used thistechnique for non spherical systems without the completemathematical background.On the second hand, in section 3 we apply this methodto obtain a direct energy proof of the radial orbit instabilitywhen the system can dissipate energy.
Consider the first-order variation of an equilibrium f → f + f (1) . It is well-known (see Bartolomew (1971), Kandrup(1990) and Perez & Aly (1996)) that there exists a phasespace function g ( Γ , t ), such that f (1) ( Γ , t ) = − { g, f } (1)This function is called a generator of the perturbation.Written in this form, f (1) is the largest class of physicalperturbations which can be considered as acting on f . Inother words, f (1) is a deformation of f and then there exists This function is clearly not unique.c (cid:13) , 000–000 adial orbits & dissipation-induced instabilities a g such that we have equation (1). Associated to this per-turbation, variation of the total energy – which turns out tobe of second order in g , see for instance a short calculationin Mar´echal & Perez (2009) – is given by H (2) [ f ] = − Z { g, E }{ g, f } d Γ − Gm Z Z { g, f }{ g ′ , f ′ }| q − q ′ | d Γ d Γ ′ (2)where Γ ′ refers to ( q ′ , p ′ ), f ′ to f ( Γ ′ ) and so on.When H (2) [ f ] is positive for a given set G of acceptablegenerators g , the system is reputed stable against the asso-ciated perturbations. This argument was detailed and usedto prove, in the case when f = f ( E ) and ∂ E f := ∂f ∂E < g ; and when f = f (cid:0) E, L (cid:1) and ∂ E f <
0, stability for all g such that { g, L } = 0 whichare called preserving perturbations (see Perez & Aly 1996for all details).When there are negative energy modes, generators g that cause H (2) [ f ] <
0, they are not necessarily associatedto an instability. Taking into account dissipation in the sys-tem can drastically change its dynamics.In the next section, we will illustrate this point with asimpler, yet instructive example.
Consider a particle of mass m = 1, charge e , let us de-note as q = ( x, y, z ) ⊤ its position with respect to someGalilean frame. This particle is influenced by two forces:one derives from a potential V that is maximal at q = 0.We’ll write V ( q ) = − ω q . The other one is the Lorentzforce generated by a static magnetic field B = B e z = ∇∧ A with A = B ( x e y − y e x ). The Lagrangian of this particle is L = ˙ q + e ˙ q · A + ω q , then the impulsion p conjugateto the position q is given by p = ∇ ˙ q ( L ) = ( p x , p y , p z ) ⊤ ,and thus, with β = eB : p x = ˙ x − βyp y = ˙ y + βxp z = ˙ z The Hamiltonian of the system is H = p · ˙ q −L , the equationsof motion are given by Hamilton’s ones, i.e. ˙ q = ∇ p ( H ) and ˙p = −∇ q ( H ). The behaviour of ( z, p z ) being trivial and in-dependent from movement on the other axes, let us focus onthe system in the reduced phase space of ξ = ( x, y, p x , p y ) ⊤ for which one has˙ ξ = Λ ξ where Λ = (cid:18) βK I αI βK (cid:19) (3)with K = (cid:18) − (cid:19) , α = ω − β (4)Since K = − I one can see that if we split Λ = A + B with A = (cid:18) I αI (cid:19) and B = β (cid:18) K K (cid:19) we have the fundamental property AB = BA , so exp (Λ t ) =exp ( At ) exp ( Bt ), by direct series summation one can findthat exp ( Bt ) = (cid:18) Σ ( t ) 00 Σ ( t ) (cid:19) with Σ ( t ) = (cid:18) cos ( βt ) sin ( βt ) − sin ( βt ) cos ( βt ) (cid:19) ∈ SO ( R )and exp ( At ) = (cid:18) ϕ ( t ) I ϕ ( t ) I αϕ ( t ) I ϕ ( t ) I (cid:19) with (cid:26) ϕ ( t ) = cosh ( √ αt ) ϕ ( t ) = ( α ) − / sinh ( √ αt ) if α > (cid:26) ϕ ( t ) = cos (cid:0) √− αt (cid:1) ϕ ( t ) = ( − α ) − / sin (cid:0) √− αt (cid:1) if α ≤ ξ ( t ) = exp ( At ) · exp ( Bt ) · ξ ( t = 0)and it is stable provided that α = ω − β ≤
0. We note thatthis stability is not asymptotic as all eigenvalues of Λ lie onthe imaginary axis. The mathematical condition on α corre-sponds to the physical case when the effect of the magneticfield B is stronger than the effect of the scalar potential V .We can see on this example that it is possible to have a sta-ble equilibrium even on a point where the potential is at amaximum; negative energy variations around an equilibriumis not a sufficient criterion for an instability . The physicalexplanation is that the magnetic force, which does not de-rive from a scalar potential, tends to ‘curve’ the particle’strajectory, and if the magnetic field is strong enough, thiscan be enough to keep the particle close to the potentialmaximum in spite of the repulsive force.However, this behaviour is only possible as long as thereis no energy dissipation. If the system is able to dissipateenergy, such an equilibrium becomes unstable. For example,assume there is some form of fluid friction force F f = − γ ˙ q ,the system is no longer Hamiltonian. Movement is still trivialin the ( z, p z ) plane; keeping the same variables that we haveused in the non-dissipative case, the equation of motion isnow˙ ξ = Λ γ ξ where Λ γ = Λ + γC, C = (cid:18) − βK − I (cid:19) .The matrix C happens to commute with B , so we have( A + γC ) B = B ( A + γC ): the fundamental matrix of thedissipative system splits intoexp (Λ γ t ) = exp ([ A + γC ] t ) · exp ( Bt )As exp ( Bt ) is a rotation matrix, the stability of the dy-namics is governed by exp ([ A + γC ] t ). The characteristicpolynomial of A γ = A + γC is χ ( λ ) = λ + 2 γλ + ( γ − α ) λ − αγλ + γ β + α roots of which are λ , = 12 h − γ ± p γ + 4 α + 4 iβγ i and λ , = 12 h − γ ± p γ + 4 α − iβγ i .Let us focus on the transition from the stable equilibriumwe have determined towards the dissipative case. We then c (cid:13) , 000–000 L. Mar´echal, J. Perez have α = ω − β < < γ ≪
1. In this limit case, onecan get λ , = γ − ± (cid:18) − ω β (cid:19) − / ! ± i (cid:0) β − ω (cid:1) / + o ( γ )and λ , = − γ − ± (cid:18) − ω β (cid:19) − / ! ± i (cid:0) β − ω (cid:1) / + o ( γ )From our assumption that α <
0, we have (1 − ω β ) − / > (eigenvalues of A γ ) withpositive real parts. From the equation of motion it fol-lows that the system is unstable; this kind of insta-bility, linked to its operator’s spectrum, is called a spectral instability . When γ is not infinitesimal, this insta-bility persists as one can check by direct spectrum calcula-tion or by more elegant approaches. The physical meaning isclear: if the particle loses energy, the magnetic field cannot‘curve’ it back as close to the maximum as it was previously,and it will spiral further and further from the origin. The previous three-dimensional example is a special caseof a general theorem which applies for finite dimensionalsystems: a Hamiltonian dynamical system with a negativeenergy mode (which could be stable without further hypoth-esis) becomes spectrally and hence linearly and non-linearlyunstable when any kind of dissipation is introduced. Thiscounterintuitive result takes its genesis from the classicalworks by Thomson (Lord Kelvin) and Tait (1879), but itwas proven only recently in the case of finite-dimensionalsystems (Bloch et al. 1994 and Krechetnikov & Marsden2007), and, as suggested by references in the latter, ap-pears to be very useful in mechanics. More recent worksby Krechetnikov & Marsden (2009) suggest that the infinite-dimensional case works similarly, although there is no defini-tive proof for the time being. In the context of theoreticalastrophysics, it is interesting to note that H. Kandrup usedsuch kind of arguments to investigate gravitational insta-bilities for triaxial systems (see Kandrup 1991), before anyactual, formal result.As recalled in section 1.1, CBP is a Hamiltonianinfinite-dimensional system, so we can apply this theory ofdissipation-induced instability for stability investigations inthis context of gravitational plasmas. In the next sectionwe will show that, when a spherical and anisotropic self-gravitating system becomes more and more radial, we canchoose a certain class of g for which H (2) [ f ] <
0: this provesthe existence of negative energy modes in such systems. Fol-lowing the dissipation-induced instability theory such kindof gravitating systems will become unstable as soon as anykind of dissipation can appear. As noticed by Kandrup in hisvisionary paper, in physical self-gravitating systems dissipa-tion could take several forms like a little bit of gas, dynam-ical friction or at minimum gravitational radiation! In thecontext of numerical modelizations of self-gravitating sys-tems where radial orbit instability also appears, dissipationis also inevitably introduced by numerical algorithms of timeintegration or by potential computation.
A pure radial orbit system is characterized by particles with L = 0, the corresponding distribution function could thenbe written f ro (cid:0) E, L (cid:1) = ϕ ( E ) δ (cid:0) L (cid:1) where ϕ is any pos-itive smooth normalized function, and δ denotes the Diracdistribution. However, this distribution is very irregular inzero which is quite problematic, in addition to being unre-alistic (orbits can hardly be perfectly radial). So, instead ofactually using the Dirac distribution, we will use functionsthat approach it.The choice we made is to use Gaussian functions. Morespecifically, we will consider an initial distribution functionof the form f a (cid:0) E, L (cid:1) = ϕ ( E ) δ a (cid:0) L (cid:1) , δ a ( L ) = 1 πa exp (cid:18) − L a (cid:19) (5)By direct calculation one can easily check that, for anysmooth function Z defined on the phase space, one has, bylimited development of Z with respect to p θ and p φ , Z Zδ a ( L )dΓ = Z Zδ a ( L ) d p r d p θ d p φ r sin ( θ ) d q = Z r Z + a ∂ Z∂p θ + a ( θ ) ∂ Z∂p φ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L =0 d p r d q + O ( a ) (6)which has a clear limit when a → Z at L = 0 as expected. We could say that f a tends to adistribution function of purely radial orbits when a → a . Our goal in this section is to show that there exist pertur-bation generators g that, for sufficiently small values of a (that is, for systems that are close enough to the purely ra-dial case), give a negative energy variation. To do so, wewill start with a general g , calculate the energy variation H (2) [ f a ] for our quasi-radial systems, and explain along theway what hypotheses we make about g to reach this goal. N.B.: variables p r , p θ and p φ thereafter are the conjugate vari-ables of r , θ and φ , not the projections of p along the base vectors.We have p r = m ˙ r , p θ = mr ˙ θ and p φ = mr sin ( θ ) ˙ φ . Also L = p θ + p φ sin ( θ ) The calculation involves the well-known Gaussian integrals Z e − x r d x = r √ π Z x e − x r d x = 12 r √ π Z x e − x r d x = 34 r √ π c (cid:13) , 000–000 adial orbits & dissipation-induced instabilities Usual Poisson bracket properties give, for f a ( E, L ) { g, f a } = ∂ E f a { g, E } + ∂ L f a { g, L } where ∂ E f a := ∂f a ∂E and ∂ L f a := ∂f a ∂L , hence the secondorder energy variation (2) splits into H (2) [ f a ] = K L + K E − G Z Z ( δρ L + δρ E )( δρ ′ L + δρ ′ E ) | q − q ′ | d q d q ′ (7)where K L := − Z ∂ L f a { g, E }{ g, L } d Γ (8) K E := − Z ∂ E f a { g, E } d Γ (9) δρ L := − m Z ∂ L f a { g, L } d p (10) δρ E := − m Z ∂ E f a { g, E } d p (11)For a general perturbation, it is difficult to say moreabout the sign of H (2) . However, the system could receiveany kind of perturbations. In order to go further, we haveto make some assumptions about g . We already know, fromsection 1.2, that a radial function will not lead to an in-stability, and from section 2.1, that dependency on E and L plays no part. To find a g function that works, we thushave to consider a non-radial perturbation. We can considera perturbation that is axisymmetric around the z axis: g ( Γ ) = g ( E, L , θ, p θ ) (12)With this hypothesis: { g, L } = 2 p θ ∂g∂θ + 2 p φ cos( θ )sin ( θ ) ∂g∂p θ (13) { g, E } = 12 mr { g, L } (14)With f a = ϕ ( E ) δ a (cid:0) L (cid:1) , we get ∂ E f a = ϕ ′ ( E ) δ a ( L ) (15) ∂ L f a = − a ϕ ( E ) δ a ( L ) (16)With the previous results, and using (6), we can calculateexplicitly the four terms K L , K E , δρ L and δρ E in a powerseries of a for a →
0. If we consider only the first term in a , which corresponds to the term of lower power in p θ and p φ , a long but straightforward calculation eventually leadsto the following results: K L = 1 m Z r ϕ ( E ) (cid:18) ∂g∂θ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L =0 d p r d q (17) K E = − a m Z r ϕ ′ ( E ) (cid:18) ∂g∂θ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L =0 d p r d q (18) δρ L = mr Z ϕ ( E ) (cid:18) ∂ g∂θ∂p θ + cos( θ )sin( θ ) ∂g∂p θ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) L =0 d p r (19) δρ E = − ma r Z ϕ ′ ( E ) (cid:18) ∂ g∂θ∂p θ + cos( θ )sin( θ ) ∂g∂p θ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) L =0 d p r (20)As one can see K E and δρ E are of order a for a → K L and δρ L do not depend on a in the sameregime. Therefore, for near radial orbit systems one can ne-glect K E in front of K L and δρ E in front of δρ L . The second-order energy variation of perturbed near ra-dial orbit systems is then H (2) [ f a ] = 1 m Z ϕ ( E ) r (cid:18) ∂g∂θ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L =0 d p r d q − G Z Z δρ L δρ ′ L | q − q ′ | d q d q ′ (21)The first term is clearly positive as an integral of a pos-itive function, while the second is clearly negative owing tothe negativeness of the Laplacian operator: introducing µ ( q ) := − Z δρ ′ L | q − q ′ | d q ′ (22)one has ∆ µ = 4 πδρ L , hence − Z Z δρ L δρ ′ L | q − q ′ | d q d q ′ = 14 π Z µ ∆ µ d q = − π Z ( ∇ µ ) d q < H (2) is thus unclear, unless we make anotherassumption about g . To be able to say more, we can considera generating function verifying ∀ E, θ : ∂g∂θ (cid:12)(cid:12)(cid:12)(cid:12) L =0 = 0 while ∂g∂p θ (cid:12)(cid:12)(cid:12)(cid:12) L =0 = 0 (23)For such perturbations one can easily check that K L =0 and δρ L = 0, therefore H (2) [ f a ] = − G Z Z δρ L δρ ′ L | q − q ′ | d q d q ′ < Result 1.
Let f a be a distribution function of nearly-radialorbits, such that f a = ϕ ( E ) 1 πa exp (cid:18) − L a (cid:19) . Let g be a generator of a perturbation, of the form g ( Γ ) = g ( E, L , θ, p θ ), with ∂g∂θ (cid:12)(cid:12) L =0 = 0 while ∂g∂p θ (cid:12)(cid:12)(cid:12) L =0 = 0.Then for sufficiently small values of a , the energy vari-ation H (2) [ f a ] caused by g is negative. It is interesting to analyse the density variation associatedto the generator described in the previous section. The sym-plectic formulation of the problem allows us to write the per-turbation of the distribution function in terms of the gener-ating function g : this is equation (1). From this relation onecan obtain the density ρ ( q ) = ρ + ρ (1) = Z mf d p = Z mf a d p − Z m { g, f a } d p = Z mf a d p + δρ E + δρ L For sufficiently small values of a , δρ E is negligible infront of δρ L , as we have seen in (20), hence δ (1) ρ = δρ L . c (cid:13) , 000–000 L. Mar´echal, J. Perez
Using (19), it can be checked that the first-order variation oftotal mass δ (1) m associated to the perturbation is vanishing. δ (1) m = Z δ (1) ρ d q = m Z δρ L r sin( θ )d r d θ d φ = m Z ϕ ( E ) (cid:18) sin( θ ) ∂ g∂θ∂p θ + cos( θ ) ∂g∂p θ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) L =0 d p r d r d θ d φ = m Z ϕ ( E ) (cid:18)Z π ∂∂θ (cid:18) sin( θ ) ∂g∂p θ (cid:19) d θ (cid:19) d p r d r d φ = 0Without more hypotheses than (23) on the perturbationgenerating function g , and for sufficiently small values of a , the first order induced variations of density are δ (1) ρ = δρ L = mr Z ϕ ( E ) (cid:18) ∂ g∂θ∂p θ + cos( θ )sin( θ ) ∂g∂p θ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) L =0 d p r In order to obtain some physical characteristics of theinstability, we have to make yet another assumption about g . To find a function that verifies condition (23), given theform of g given in (12), we can suppose for example that g is separated in the θ variable, i.e. one can find two functions A and B such that g ( E, L , θ, p θ ) = B ( E, L , p θ ) A ( θ ) (25)Under this assumption, criterion (23) becomes ∀ E, θ : A ′ ( θ ) B ( E, ,
0) = 0 while A ( θ ) ∂B∂p θ ( E, , = 0(26)It is very easy to find a B that verifies this condition. Adirect calculation then gives δ (1) ρ = m D ( θ ) r Z ϕ ( E ) | L =0 ∂B∂p θ ( E, , p r where D ( θ ) = A ′ ( θ ) + cos( θ )sin( θ ) A ( θ )If D ( θ ) is not constant, which corresponds to a wideclass of A , then δ (1) ρ does depend on θ and the sphericalsymmetry of the equilibrium state is broken. We have reached our goal: we have found a class ofperturbations g which leads to a negative energy variation,and which creates a density variation that is not sphericallysymmetric. As per section 2, this means that with the help ofdissipation, the system is unstable against this perturbation:hence a favoured direction will appear in the system, whichwas initially spherical. Result 2.
Consider a self-gravitating system, described bythe CBP system and represented by a distribution function f ( E, L ), that is spherically symmetric and with nearly ra-dial orbits. Assume this system can dissipate energy. The equation D ( θ ) = k can be easily solved and gives A ( θ ) = λ − k cos( θ )sin( θ )where k and λ are θ -free constants. The fact that the resulting perturbation depends only on θ , r and E , and thus is axisymmetric around the z axis, is of coursea consequence of our choice of the form (12) for the generatingfunction. Then there exists perturbations, generated by a func-tion g , against which the system is unstable, and that causeit to lose its spherical symmetry. In this paper we have shown two important points: self-gravitating dynamical systems described by the Colli-sionless Boltzmann–Poisson equations are candidates forDissipation-Induced Instability when they are more andmore radially anisotropic; and this mechanism genericallyintroduces a favoured direction in the spatial part of thesystem’s phase space. In comparison with previous tediousnormal modes techniques used in this context, the detailof the first point gives a simple proof of radial orbit insta-bility based on energetics arguments. Dissipation, which isneeded in our proof, is also implicitly required in the clas-sical intuitive understanding of this instability presented inPalmer (1994) (section 7.3.1). It is a fact that in a pure ra-dial system — which is the most unstable — orbits, whichare frozen in a fixed direction, cannot precess or librate asit is required for the trapping resonance invoked by Palmer.Hence, if two radial orbits actually attain a lower energystate by approaching each other, this mechanism actuallyneeds a way to dissipate excessive energy. Finally, a pointabout time-scales should be stressed: it is well known thatradial orbit instability is effective on a few crossing times,therefore if dissipation appears to be the cornerstone of ra-dial orbit instability, it is clear that it could not act alone.Non-linear and non-local aspects of the gravitational poten-tial clearly amplifies and completes the dissipation-triggeredwork.
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