Damped perturbations in stellar systems: Genuine modes and Landau-damped waves
aa r X i v : . [ a s t r o - ph . GA ] J a n MNRAS , ?? – ?? (2020) Preprint 22 January 2021 Compiled using MNRAS L A TEX style file v3.0
Damped perturbations in stellar systems:Genuine modes and Landau-damped waves
E. V. Polyachenko, ‹ I. G. Shukhman, : O. I. Borodina, ; Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskya St., Moscow 119017, Russia Institute of Solar-Terrestrial Physics, Russian Academy of Sciences, Siberian Branch, P.O. Box 291, Irkutsk 664033, Russia
22 January 2021
ABSTRACT
This research was stimulated by the recent studies of damping solutions in dynamically stable spherical stellarsystems. Using the simplest model of the homogeneous stellar medium, we discuss nontrivial features of stellarsystems. Taking them into account will make it possible to correctly interpret the results obtained earlier and willhelp to set up decisive numerical experiments in the future. In particular, we compare the initial value problem versusthe eigenvalue problem. It turns out that in the unstable regime, the Landau-damped waves can be represented as asuperposition of van Kampen modes plus a discrete damped mode, usually ignored in the stability study. This modeis a solution complex conjugate to the unstable Jeans mode. In contrast, the Landau-damped waves are not genuinemodes: in modes, eigenfunctions depend on time as exp p´ i ωt q , while the waves do not have eigenfunctions on the real v -axis at all. However, ‘eigenfunctions’ on the complex v -contours do exist. Deviations from the Landau damping arecommon and can be due to singularities or cut-off of the initial perturbation above some fixed value in the velocityspace. Key words: galaxies: kinematics and dynamics, galaxies: star clusters: general, physical data and processes: insta-bilities
It is well known that spherical systems, in contrast to stellarsystems of other geometry, have a fair amount of stability(e.g., Fridman & Polyachenko 1984). A fundamental resultis the proof of the stability of isotropic spheres (Antonov1960, 1962; Doremus et al. 1971). In this vein, the choice ofspheres for studying damped oscillations is obvious. However,the difficulties one encounters explain low activity on thistopic.The state-of-art was presented recently by Heggie et al.(2020). Utilizing correlation analysis, the authors attemptto reproduce the real part of frequency for a damped modeobtained earlier by Weinberg (1994). In the latter, using amatrix method, damped dipole and quadrupole modes forergodic King models with parameters W “ ... W “ N -body simulations.The modes referred to in the cited paper were obtained bythe analytic continuation of the left side of dispersion equa-tion (DE) D p ω q “
0. For the perturbed functions depend-ing on time as exp p´ i ωt q , the continuation is made from theupper complex frequency half-plane ω to the lower one, by ‹ E-mail: [email protected] : E-mail: [email protected] ; E-mail: [email protected] deforming the integration contour so that it passes below thesolution ω “ ω L of the DE. There are infinitely many suchdamped solutions, known as Landau-damped waves.Matrix equations for spherical systems are cumbersome,which often makes it difficult to understand the physical sideof the problem. Leaving aside for a while technical difficultiesassociated with the continuation of the matrix DE for spheresto the lower half-plane and subsequent modelling, we wantto make a few remarks about the damped solutions, on theexample of the homogeneous stellar medium.The goal of this paper is to demonstrate that: (i) a dampedmode may indeed exist, but it cannot be found from a DEcontinued to the lower half-plane; (ii) no eigenmodes are de-pending on the real phase variables corresponding to the Lan-dau solutions; (iii) to study the Landau-damped waves, it isnecessary to use the initial functions without singularitiesin the complex plane, i.e. so-called ‘entire’ functions, or atleast function with singularities located low enough (in thecomplex ω -plane) in order not to interfere with the Landaudamping.Our analysis compares the initial value problem and eigen-value approaches, for Maxwell background distribution func-tion (DF). Contrary to plasma physics, where this model isstable at all scales, the stellar medium is unstable for large-scale perturbations.The paper is organised as follows. Section 2 contains ba-sic equations and proves the existence of the damped mode © E. V. Polyachenko et al. in the case when the stellar medium is unstable. Section 3brings examples of deviations from standard Landau expo-nential damping. In Section 4 we give analytical argumentsto support our numerical findings. Final Section 5 discussesimplications and outlines our plans in this field. In the end,we give two Appendices in which some more specific issuesare addressed.
Instability of infinite homogeneous stellar medium can be ap-proached by applying a small amplitude plane-wave pertur-bation f p x, v, t q “ f p v, t q e i kx (2.1)to the unperturbed background DF, F p v q . Throughout thepaper, the unperturbed DF is one-dimensional Maxwell dis-tribution, F p v q “ ρ M σ p v q , M σ p v q ” ? πσ exp ´ ´ v σ ¯ . (2.2)It is known that the perturbation is unstable to Jeans insta-bility, if k ă k J ” p πGρ q { { σ (e.g., Binney & Tremaine2008, hereafter BT). Here we use standard notation: coor-dinate x -axis is directed along wavevector k , wavenumber k “ | k | , k J is a so-called Jeans wavenumber, ρ and σ areconstant background density and velocity dispersion, G isthe gravitation constant. It is convenient to adopt units inwhich 4 πG “ ρ “ σ “
1, so that the wavenumber are nowmeasured in k J ; velocities c , v , and u in σ ; frequencies andgrowth/damping rates in Jeans frequency Ω J ” p πGρ q { . By linearising the collisionless Boltzmann and Poisson equa-tions, it is easy to obtain the equation governing time evolu-tion of the perturbed DF: B f p v, t qB t “ ´ i k ” v f p v, t q ` η k p v q ρ p t q ı , (2.3)where η k p v q ” πGk F p v q , (2.4)and ρ p t q is a perturbed density, ρ p t q ” ż ´8 d u f p u, t q . (2.5)A similar initial problem with f p v, q “ g p v q in plasmawas first treated by Landau (1946) analytically using inverseLaplace transform. Rewriting his eqs. (10) and (12), one canhave: ρ p t q “ π i ic ˚ ż ´8` ic ˚ d c ρ c e ´ i ckt , (2.6) where c “ ω { k is the complex phase velocity of the wave, ρ c “ D ` k p c q ż ô d u g p u q u ´ c , (2.7) D ` k p c q ” ` ż ô d u η k p u q u ´ c , (2.8) c ˚ is a constant chosen so that all singularities of ρ c are lo-cated in the half-plane Im p c q ă c ˚ , symbol ‘ ô ’ denotes theLandau integration contour passing below the singularity at u “ c . Superscript ‘+’ in (2.8) indicates that this functionis defined in the upper half-plane and continued analyticallyto the lower half-plane.If the initial perturbation is given by an entire function g p v q , i.e. it has no singularities for finite complex v , the in-tegral in (2.7) has no singularities in complex c -plane, andbehaviour of ρ p t q is determined by zeros of the denominator.Thus, we obtain the well-know dispersion relation (DR): D ` k p c q “ . (2.9)Solution to this relation is given in Fig. 1 (Ikeuchi et al. 1974,BT). For k ă
1, it consists of an aperiodic growing modewith growth rate γ k ” Im ω and many so-called Landau-damped waves describing exponential decay for density (butnot for the perturbed DF, see Sect. 2.2). In the stable domain, k ą
1, the growing mode is replaced by aperiodic Landausolution. For convenience, we shall refer to the damping rateof the aperiodic Landau-damped solution, which continuesthe growing mode in the stable domain, as γ p q L , and all other(oscillating) Landau-damped solutions as γ p j q L , j “ , ... , (all γ p j q L ă
0) in ascending order of the damping rate.
In the eigenvalue problem, we seek for solutions in the form: f p v, t q “ ˜ f p v q e ´ i ckt (2.10)with complex phase velocity c to be determined. From (2.3)and (2.5) one obtains: c ˜ f p v q “ v ˜ f p v q ` η k p v q ż ´8 d u ˜ f p u q , (2.11)which can be easily solved using matrix approach (see, e.g.Polyachenko 2004, 2005, 2018). The solution for a given k consists of spectrum of modes, presented in Fig.2. In the un-stable k -domain, there are two discrete modes: the growingmode c ` “ i γ k { k already known from Fig. 1, a complex con-jugate damped mode c ´ “ ´ i γ k { k , and a continuum spec-trum of so-called van Kampen modes (Van Kampen 1955).The discrete modes are absent in the stable domain k ą
1. Itis believed that the Landau-damped waves can be regardedas a superposition of van Kampen modes (e.g., BT, p. 415).This is true only for k ą
1, but not in general, see Sect. 3.2.Let’s turn our attention now to the damped mode markedby a red circle in Fig. 2. For a long time this mode was re-garded as an extraneous solution, because it satisfies a rela-tion D ´ k p c q “ , (2.12) In BT, this integration contour is denoted as L .MNRAS , ?? – ?? (2020) amped perturbations in stellar systems −2−1012 R e ω −2−1012 I m ω growingdampingLandau Figure 1.
Unstable (solid black) and Landau-damped (thin blue)solutions, in terms of ω “ ck , of the dispersion relation (2.9), andthe damped mode (red dots), cf. BT, Fig. 5.2. The stellar mediumis unstable to Jeans instability for k ă −
10 0 10 Re c − . . . I m c Figure 2.
Spectrum of modes in the complex phase velocity space c obtained from the matrix equation for infinite homogeneous stel-lar medium, k “ .
9. The unstable discrete mode of Jeans insta-bility is marked by the blue circle. The concealed damped modeis marked by the red circle. The overlaping black dots on the real c -axis present a continuum spectrum of van Kampen modes. rather than (2.9), with D ´ k p c q ” ` ż ñ d u η k p u q u ´ c (2.13)and the integration contour now passes above the singularityat u “ c (e.g., along the real u -axis for Im p c q ă g p v q “ b ` g ` p v q ` b ´ g ´ p v q ` g vK p v q . (2.14)In our notations, an eigenfunction corresponding to the un-stable mode c ` is g ` p v q “ ´ η k p v q v ´ c ` . (2.15)Since c ` obeys the relation (2.9) and Im p c ` q ą
0, the eigen-function is normalised to unity. An eigenfunction correspond-ing to the damped mode c ´ is a complex conjugate to g ` : g ´ p v q “ g ˚` p v q “ ´ η k p v q v ´ c ´ , (2.16)which obviously means that it shares the same normalisation,i.e.: ż ´8 d u g ˘ p u q “ . (2.17)Function g vK p v q ” ż ´8 d c B p c q g c p v q (2.18)represents a superposition of van Kampen modes, g c p v q ” ´ P η p v q v ´ c ` λ p c q δ p v ´ c q , (2.19)where P denotes the Cauchy principal value, δ p v q is the Diracdelta function, λ p c q is needed to satisfy normalisation of g c p v q to unity, i.e.: ż ´8 d u g c p u q “ , (2.20)from where λ p c q “ ` P ż ´8 d u η k p u q u ´ c . (2.21)Given an initial profile for the perturbation g p v q , the expan-sion coefficients are obtained from the following expressions: b ˘ “ ´ C ˘ 8 ż ´8 d u g p u q u ´ c ˘ , C ˘ “ ż ´8 d u η k p u qp u ´ c ˘ q , (2.22) B p c q “ λ p c q ` π η k p c q »– λ p c q g p c q ´ η k p c q P ż ´8 d u g p u q u ´ c fifl . (2.23)In particular, it can be shown that B p c q “ b ` “
0, if g p v q “ g ´ p v q .The eigenfunction of the damped mode obtained from thematrix equation (2.11) coincides with function (2.16).Second, it can be shown numerically that initial condition g p v q “ g ´ p v q gives rise to f p v, t q “ g ´ p v q exp p´ γ k t q . In otherwords, the shape of the perturbed DF is preserved (Fig. 3), MNRAS , ?? – ????
0, if g p v q “ g ´ p v q .The eigenfunction of the damped mode obtained from thematrix equation (2.11) coincides with function (2.16).Second, it can be shown numerically that initial condition g p v q “ g ´ p v q gives rise to f p v, t q “ g ´ p v q exp p´ γ k t q . In otherwords, the shape of the perturbed DF is preserved (Fig. 3), MNRAS , ?? – ???? (2020) E. V. Polyachenko et al. −6 −4 −2 0 2 4 6v−0.4−0.20.00.20.4 d a m p i n g e . f . g − ReIm
Figure 3.
Eigenfunction of the damped mode g ´ p v q , k “ .
9. Nu-merical solution of (2.3) with this initial condition gives f p v, t q “ g ´ p v q exp p´ γ k t q , i.e. the shape of the perturbed DF is preserved. which is a characteristic of a genuine mode. Physically, itis quite obvious that eigenfunctions of the discrete modesused as initial states for eq. (2.3) lead to exponential densitygrowth/decay with rate ˘ γ k . In Appendix A we show thisexplicitly using (2.6) and (2.7).The two reasons considered above demand to complementFig. 1 by the damped mode (see red dots in both panels).Note that exponential density change for genuine modesmanifests itself differently in the behaviour of the DFs fromthe Landau-damped waves. For the formers, shapes of the DFprofiles do not change, but its amplitude varies proportionallyto exp p˘ γ k t q . In the latter, the amplitudes do not change,but the shape becomes more and more jagged, see Fig. 4. Todisentangle from genuine modes, we call them below ‘quasi-modes’.Similarly to the initial Maxwell DF, the initial state ofthe form (2.15) with c ` replaced by the aperiodic Landausolution c p q ” ´ i | γ p q L |{ k gives rise to DF shearing , andasymptotically to density decay exp p γ t q . This takes placeif the DF and other entries of (2.3) are defined on the real v -axis. Now consider the task on the complex v -contour passingbelow c p q , e.g.: v I “ ´ | c p q | exp ` ´ v ˘ , (2.24)where v R and v I are the real and imaginary parts of velocity v . The corresponding DF is shown in Fig. 5 with solid lines.Its time evolution is just decreasing of the amplitude pre-serving the shape of the function. To demonstrate this, wegive DF at t “
50 multiplied by exp p| c p q | kt q (‘ ˝ ’-marks). Weconclude that constructed DF is a genuine eigenmode, buton the complex contour! Corresponding ‘density’ defined asintergal ş f p v, t q dv over this complex contour decays strictlyexponentially from the very beginning.To sum up, this section argues that: ‚ the matrix method on the real v -axis gives discrete com-plex conjugate pairs and a proxy to van Kampen modes. Thecorresponding eigenfunctions do not change their shapes –a characteristic of genuine modes. Any initial perturbationscan be expanded over these modes; ‚ Landau-damped waves are not true modes – they don’thave eigenfunctions on the real v -axis. A perturbation decays − . − . . . . R e f ( v , t ) t = 0 t = 10 − − − v − . − . . . . R e f ( v , t ) t = 0 t = 20 Figure 4.
Deformation of the initial Maxwell distribution (thinblack lines) with time ( k “ . f p v, t q at time t “ t “
20. The DF is gradually shearedout whiledensity (2.5) decreases exponentially in accordance with Laudaudamping rate. − − Re ( v ) − . − . . . . Re f (0) Im f (0) Re f (50) Im f (50) Figure 5.
Time evolution of the ‘eigenfunction’ of the form (2.15)with c ` replaced by the aperiodic quasi-mode c p q “ ´ . k “ . c ` . Theperturbed DF at t “
50 is multiplied by exp p| c p q | kt q “ f p q . in mean, i.e. exponential decay takes place for the perturbeddensity, not for the perturbed DF; ‚ Landau-damped waves do have ‘eigenfunctions’ on acomplex v -contour passing below the corresponding zero of D ` k p c q . MNRAS , ?? – ?? (2020) amped perturbations in stellar systems -20 -15 -10 -5 ρ(t)LandauGauss Figure 6.
Decay of the initial state given by Maxwell distributionof van Kampen waves, B p c q “ M σ p c q . f p v, t q given by (3.1) decaysas a Gaussian in time exp p´ σ k t { q (black dash-dotted line), notlike the Landau-damped wave (black dots) for σ “ k “ . In this section, we give numerical evidence of deviations fromexpected exponential decay of Landau-damped waves. In par-ticular, we show that superposition of van Kampen modesmay lead to density decay of various types.The solution shown in Fig. 4 is for initial Maxwell distri-bution g p v q “ M p v q , i.e. for an entire function. The Landaudamping would appear as usual if singularity of g p v q was be-low ´ i γ { k . A more peculiar decays occur when the initial g p v q is set using the expansion function B p c q for van Kampenmodes. The needed expression reads: g p v q “ B p v q ` P ż ´8 d c B p v q η k p c q ` B p c q η k p v q c ´ v . (3.1)In a sense, it is the inverse of eq. (2.23). k ą k J First of all, we apply eq. (3.1) to typical profiles of choice –Maxwell and Lorentz. Fig. 6 shows density decay for B p c q “ M p c q , which turns out to be perfect Gaussian in time. TheLandau damping (black dots) is much slower. This numericalresult can be easily confirmed analytically. Indeed, each of thevan Kampen modes evolves with its own frequency ω “ ck ,so f p v, t q “ ż ´8 d c B p c q g c p v q exp p´ i ckt q . (3.2)For density (2.5), one obtains using normalisation of g c p v q from (2.20): ρ p t q “ ż ´8 d c B p c q exp p´ i ckt q . (3.3)Substituting Maxwell distribution B p c q “ M σ p c q , one ob-tains the found fit. For Lorentz distribution: B p c q “ π σc ` σ (3.4) -6 -4 -2 |ρ(t)|Landau∝ t −1 Figure 7.
Decay of Maxwell initial state g p u q “ M p u q with 3 σ cut-off, k “ .
1: Landau damping (at rate γ p q L “ ´ . one finds an exponential density decay with rate kσ , ratherthan Landau damping rate γ .Next, Fig. 7 presents the density decay in the case wheninitial Maxwell distribution is cutted above v ˚ : g p v q “ M p v q for | v | ă v ˚ “ , (3.5)and zero otherwise. After a short, barely visible transitionperiod ∆ t „
1, the decay starts with Landau damping rate,but eventually power-law decay sin p kv ˚ t q{ t overtakes. Notethat a similar asymptotical power-law behaviour accompa-nied by oscillations was found by Barr´e et al. (2011). The au-thors studied evolution of perturbations in one-dimensionalnon-homogeneous medium with artificial potential when ac-tion J varies in semi-finite interval J a ă J , and obtainedthe density decay ρ exp r´ i m Ω p J a q t s{ t n , where the integerpositive index n ( n “
1, 2, or 3) depends on the form of theinitial disturbance. k ă k J In general, initial state (2.14) contains the exponentiallygrowing mode leading to the density change in time: ρ p t q “ b ` e γt ` b ´ e ´ γt ` ż ´8 d c B p c q e ´ i kct (3.6)(see Appendix A). Study of the damped solutions thus re-quires elimination of this mode from the initial state. So, foran arbitrary g p v q , we consider the initial distribution: f p v, q “ g p v q ´ b ` g ` p v q “ g vK p v q ` b ´ g ´ p v q . (3.7)It still consists of contributions of van Kampen modes andthe discrete damped mode.One would naturally expect that if the damping rate of thediscrete mode is smaller than the Landau damping rate, i.e. | γ k | ă | γ | (it holds for k ą k ˚ « . exp p´ γ k t q . On the other hand,it is believed that pure superposition of van Kampen modesleads to density decay with the Landau damping rate (e.g.,BT, p. 415).Direct evaluation of (2.3) for k “ . MNRAS , ?? – ????
1, 2, or 3) depends on the form of theinitial disturbance. k ă k J In general, initial state (2.14) contains the exponentiallygrowing mode leading to the density change in time: ρ p t q “ b ` e γt ` b ´ e ´ γt ` ż ´8 d c B p c q e ´ i kct (3.6)(see Appendix A). Study of the damped solutions thus re-quires elimination of this mode from the initial state. So, foran arbitrary g p v q , we consider the initial distribution: f p v, q “ g p v q ´ b ` g ` p v q “ g vK p v q ` b ´ g ´ p v q . (3.7)It still consists of contributions of van Kampen modes andthe discrete damped mode.One would naturally expect that if the damping rate of thediscrete mode is smaller than the Landau damping rate, i.e. | γ k | ă | γ | (it holds for k ą k ˚ « . exp p´ γ k t q . On the other hand,it is believed that pure superposition of van Kampen modesleads to density decay with the Landau damping rate (e.g.,BT, p. 415).Direct evaluation of (2.3) for k “ . MNRAS , ?? – ???? (2020) E. V. Polyachenko et al. -30 -20 -10 | ρ ( t ) / ρ ( ) | g − b + g + g vK modeLandau Figure 8.
Density decay of two initial states, k “ .
9: (i) Maxwell g p v q “ M p v q with growing mode b ` g ` p v q subtracted (solid red)decays with the Landau damping rate, ω p q L “ ˘ . ´ . g vK p v q decays with a rateof the damped discrete mode, γ k “ ´ . -30 -20 -10 | ρ ( t ) / ρ ( ) | g − b + g + GaussmodeLandau
Figure 9.
Density decay of Maxwell initial state g p v q “ M . p v q with growing mode b ` g ` p v q subtracted (solid red) for k “ .
3. ‘Gauss’ blue dashed curve shows Gaussian in time decayexp p´ σ k t { q , black dashes show dicrete mode decay exp p´ γ k t q ,black dots show Landau-damped decay exp p´ γ p q L t q . Here γ k “ . c k “ . ω p q L “ ˘ . ´ . c p q L “ ˘ . ´ . marking the solution for initial DF g ´ b ` g ` decays with Lan-dau damping rate γ p q L . The oscillations of the density occurbecause the corresponding Landau solution has a nonzero realpart of the frequency. The blue curve for initial DF g vK p v q ,after some transition period, decays exp p´ γ k t q . Note thatalthough the damping rate of the density decay coincides withthe damping rate of the discrete mode, the character of thisdecay is the same as for quasi-modes, see Fig. 4.In domain k ă k ˚ both initial states predictably decaywith rate γ p q L in agreement with Landau theory. Neverthe-less, deviations could happen here as well, for example whenconsidering narrow initial DFs. Fig. 9 shows a long transitionperiod for initial g p v q “ M . p v q (with the growing modesubtracted) for k “ .
3. The transition is approximatelyGaussian decay exp p´ σ k t { q , which is replaced by theLandau damping at t „ | γ p q L | σ ´ k ´ .To sum up, this section shows: ‚ construction an initial perturbation from van Kampenwaves only by defining function B p c q allows to obtain variousdecaying laws that have nothing to do with Landau damping(yet, function g p v q is smooth on the real axis); ‚ cut-off of the initial function above some value in thevelocity space leads to power-law decay (power law is knownto appear also if g p v q is not smooth); ‚ if an initial perturbation is given by an entire function,its van Kampen part does not decay with Landau dampingrate in the unstable k -domain; ‚ a transition process before Landau damping regimecould be lengthy (in our case, we observe Gaussian in timedensity decay because the initial B -distribution is Gaussian). In the previous sections, we saw several astonishing numericalshreds of evidence concerning the time evolution of pertur-bations in the homogeneous stellar medium.
1. The damped mode.
Matrix equation (2.11) predicts theexistence of the damped mode, see Figs. 2, 3. The questionarises why this solution is missed in the standard approach.The task considered by Landau was to find the evolution of anarbitrary initial perturbation given by an entire function. Itevolves as the sum of contributions corresponding to the sin-gularities of (2.7), or zeros of D ` k p c q . These contributions havean exponentially growing component and components corre-sponding to Landau-damped waves, but there is no dampedmode in this expansion. One might think that the dampedmode is incorporated in the Landau-damped waves.Nevertheless, this mode exists, as confirmed by our numeri-cal solution of evolutionary eq. (2.3). It can be found from DEinvolving integration along the real u -axis, or (2.13). The lat-ter provides solutions complex conjugate to (2.9), and playsthe same role as (2.9) under time reversal. The damped modeforward in time appears as the growing mode when integrat-ing backwards in time.
2. Density decay in the unstable k -domain . It is natu-rally expected that a packet of van Kampen modes g vK p v q “ ż ´8 d c B p c q g c p v q “ g p v q ´ b ` g ` p v q ´ b ´ g ´ p v q (4.1)decays with Landau damping rate. However this is not thecase when | γ k | ă | γ p q L | , as is evident in Fig. 8. We shall showthat this is due to a singularity of B p c q in c “ c ´ .Since each of the van Kampen modes, g c , evolves in time exp p´ i kct q , their contribution to the total density is: ρ vK p t q “ ż ´8 d c B p c q exp p´ i kct q . (4.2)For positive kt , the integration can be performed over a con-tour closed in the lower half-plane and replaced by a sum ofresidues: ρ vK p t q “ ´ π i ÿ n Res p c n q exp p´ i kc n t q . (4.3)Here c n are all poles of B p c q in the lower half-plane. In (2.23)this function is defined on the real c -axis. Our goal is to prove MNRAS , ?? – ?? (2020) amped perturbations in stellar systems that an analytic continuation of B p c q to the lower half-plane c , apart from poles of Landau quasi-modes, contains also apole due to the damped mode c “ c ´ , and near this pole B p c q has the form: B p c q « π i b ´ c ´ c ´ , (4.4)where b ´ “ ´ ż ´8 g p u q d uu ´ c ´ »– ż ´8 g p u q d u p u ´ c ´ q fifl ´ . (4.5)For real c , expression (2.23) can be decomposed as follows: B p c q “ B ` p c q ` B ´ p c q , (4.6)where B ˘ p c q ” g ˘ p c q ε ˘ p c q , (4.7) g ˘ p c q “ g p c q ˘ π i P ż ´8 g p u q d uu ´ c , (4.8) ε ˘ p c q “ ` π i »– ˘ η k p c q ` π i P ż ´8 η k p u q d uu ´ c fifl . (4.9)Now analytic continuation is obvious, since ε ˘ p c q can be re-placed by D ˘ k p c q off the real axis, and g ˘ are replaced by g ` p c q “ π i ż ô g p u q d uu ´ c , g ´ p c q “ ´ π i ż ñ g p u q d uu ´ c , (4.10)where, depending on signs ‘ ô ’ or ‘ ñ ’, integration contourpasses below or above the singularity at u “ c . Note that g ` p c q ` g ´ p c q “ π i ”ż ô g p u q d uu ´ c ´ ż ñ g p u q d uu ´ c ı “ π i ¿ g p u q d uu ´ c “ g p c q , (4.11)i.e. this is a decomposition of function g p v q . Since D ´ k p c q hasone zero in the lower half-plane c “ c ´ , B ´ p c q has a poleat this point leading to exp p´ γ k t q contribution to densitydecay. Expansion D ´ k p c q near c “ c ´ gives: D ´ k p c q « d D ´ k p c q d c ˇˇˇˇ c ´ p c ´ c ´ q “ p c ´ c ´ q ż ñ d u η k p u qp u ´ c ´ q . (4.12)Finally, substitution to B ´ p c q “ g ´ p c q D ´ k p c q (4.13)leads to the desired result (4.4), if we choose the integrationcontour along the real u -axis. The input from this pole gives adensity decay slower that the Landau damping for k ą k ˚ « . C Figure 10.
Integration contour in (2.6) in the complex c -plane(blue dashes). ’+’ signs mark the Landau-damped quasi-modesobtained as zeros of D ` k p c q . A wavy line marks the cut on real axis c : ´ v ˚ ă c ă v ˚ . is possible to decompose B p c q so that parts of new decom-position have no singularities in the upper/lower half-planes(Appendix B).Gaussian in time density decay occurred for Maxwell B p c q (Fig. 6) is simply a reflection of the fact that B p c q has nosingularities in the lower half-plane.
3. Power-law decay.
To understand an appearance of thepower-law term (Fig. 7), we should reexamine derivation ofeqs. (2.7, 2.8). They are obtained as a result of analytic con-tinuation of the integrals over the real u -axis by deformationof the integration contour to the lower half-plane. Since g p v q is zero for | v | ą v ˚ , the cut in the complex plane is finite(Fig. 10), and deformation of the contour is not needed forthis integral. On the either side of the cut g ˚ p c q “ ż ´8 d u g p u q u ´ p c ˘ i0 q “ ˘ i πg p c q ` P v ˚ ż ´ v ˚ d u g p u q u ´ c . (4.14)The cut leads to an additional input to density (2.6): ρ cut p t q “ ¿ C d c g ˚ p c q D ` k p c q e ´ i ckt “ π i v ˚ ż ´ v ˚ d c g p c q D ` k p c q e ´ i ckt . (4.15)Integrating by parts, assuming g p v q is even and real, one ob-tains for large t : ρ cut p t q “ ´ πkt g p v ˚ q Im ” e ´ i v ˚ kt D ` k p v ˚ q ı ` O p t ´ q . (4.16)Expression in the square brackets gives sinusoidal oscillationswith some phase shift due to the presence of D ` k p v ˚ q in thedenominator. If g p v ˚ q is small, the power-law decay revealsitself only after a while. In this article, we analysed numerical solutions of equationsdescribing perturbations of the homogeneous stellar medium
MNRAS , ?? – ????
MNRAS , ?? – ???? (2020) E. V. Polyachenko et al. and provided explanations for the observed numerical results.The unexpected behaviour of damped solutions in the un-stable domain is associated with the presence of a genuinedamped mode, which is complex conjugate to the unstableJeans mode. The damped mode cannot be found as a solu-tion to the analytic continuation of DE from upper complexfrequency half-plane ω to the lower half-plane. It is a solutionto another DE, in which the integration is carried out overreal phase variables (velocity u in the case of the homoge-neous medium and action variables in the case of spheres).This mode is usually ignored (cf. e.g., BT, Fig. 5.2 and ourFig. 1).We have shown that Landau-damped waves possess their‘eigenfunctions’, but defined on complex contours passing be-low the corresponding solution of continued DE. No eigen-functions of Landau solutions exist on the real axis. There-fore, we call these solutions not ‘genuine modes’, but ‘quasi-modes’, emphasizing that exponential decay takes place onlyon average, i.e. for density and potential. In genuine dampedmodes, DF is decreasing exponentially (along with densityand potential).Weinberg (1994) numerically investigated the evolution ofthe initial perturbation of DF taken as ‘eigenfunction’ corre-sponding to Landau-damped quasi-mode. It contains denomi-nator similar in shape to our solution for damped mode g ´ p v q (2.16), but instead of velocity, real action variables appear init. We have argued that Landau-damped solutions have noeigenfunctions as functions of real variables. Besides, we haveseen that the use of non-analytic functions to describe the ini-tial perturbations leads to artefacts – the appearance of expo-nential decay with given characteristics. In numerical experi-ments on damping in spheres, this could lead to a prescribedresult. We consider the initial conditions given by entire func-tions or the use methods that are not related to the choiceof initial conditions with singularities to be more appropri-ate. An example of such a study is provided by Heggie et al.(2020).In connection with the aforesaid, only solutions found with-out analytic continuation to the lower half-plane turn out tobe genuine modes, e.g., for real values of the velocity u in theintegrals of Laplace transform (2.7).In the future, we plan to extend our research from the ho-mogeneous medium to spherical systems using our matrixapproach for spheres (Polyachenko et al. 2007) and investi-gate the discreteness effects that inevitable in any N-bodymodels (e.g., Polyachenko et al. 2020). ACKNOWLEDGMENTS
The reported study was funded by Foundation for the ad-vancement of theoretical physics and mathematics “Basis”,grant
DATA AVAILABILITY
Data underlying this article will be shared on reasonable re-quest to the authors via [email protected].
REFERENCES
Antonov V. A., 1960, Azh, 37, 918 (in Russian). Also SvA, 4, 859(in English)Antonov V. A., 1962, Vestnik Leningrad Univ., 19, 96 (in Russian).Also de Zeeuw (1987), 531 (in English)Barr´e J., Olivetti A., Yamaguchi Y. Y., 2011,Journal of Physics A Mathematical General, 44, 405502Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition.Princeton University PressCase K. M., 1959, Annals of Physics, 7, 349Doremus J. P., Feix M. R., Baumann G., 1971, Phys. Rev. Lett.,26, 725Fridman A. M., Polyachenko V. L., 1984, Physics of gravitatingsystems. I - Equilibrium and stability. Springer, New YorkHeggie D. C., Breen P. G., Varri A. L., 2020, MNRAS, 492, 6019Ikeuchi S., Nakamura T., Takahara F., 1974,Progress of Theoretical Physics, 52, 1807Landau L., 1946, J. Phys. USSR, 10, 25Polyachenko E. V., 2004, MNRAS, 348, 345Polyachenko E. V., 2005, MNRAS, 357, 559Polyachenko E. V., 2018, MNRAS, 478, 4268Polyachenko E. V., Polyachenko V. L., Shukhman I. G., 2007,MNRAS, 379, 573Polyachenko E. V., Berczik P., Just A., Shukhman I. G., 2020,MNRAS, 492, 4819Van Kampen N. G., 1955, Physica, 21, 949Weinberg M. D., 1994, ApJ, 421, 481
APPENDIX A: DENSITY EVOLUTION FORINITAL DF g p v q “ g ˘ p v q The eigenfunction of the growing solution is g p v q “ g ` p v q ” ´ η k p v q v ´ c ` , (A1)where c ` “ i γ k { k is a solution of the dispersion relation D ` k p c q “
0, see (2.8). For the integral in (2.7) one obtains: ż ô d u g p u q u ´ c “ ´ c ´ c ` ż ô d u η k p u q ” u ´ c ´ u ´ c ` ı “ ´ c ´ c ` ” ` ż ô d u η k p u q u ´ c ı “ ´ c ´ c ` D ` k p c q , (A2)valid in the whole complex c -plane. In expression for ρ c , func-tions D ` k p c q in the numerator and denominator cancel, so onefinally obtains from (2.6): ρ p t q “ ´ π i ic ˚ ż ´8` ic ˚ d cc ´ c ` e ´ i ckt “ e γ k t . (A3)Now we consider the eigenfunction of the damped mode g p v q “ g ´ p v q ” ´ η k p v q v ´ c ´ , (A4) MNRAS , ?? – ?? (2020) amped perturbations in stellar systems where c ´ “ ´ i γ k { k is a solution of the equation1 ` ż ´8 d u η k p u q u ´ c “ , (A5)i.e. integration is performed above the singularity u “ c ´ . Anexpression similar to (A2) in the upper half-plane is ż ´8 g p u q d uu ´ c “ ´ c ´ c ´ 8 ż ´8 d u η k p u q ” u ´ c ´ u ´ c ´ ı “ ´ c ´ c ´ ” ` ż ´8 d u η k p u q u ´ c ı , Im p c q ą . (A6)Analytic continuation to the lower half-plane is achieved bychanging the last integral over real u -axis to an integral overLandau contour: ż ô g p u q d uu ´ c “ ´ c ´ c ´ ” ` ż ô d u η k p u q u ´ c ı “ ´ c ´ c ´ D ` k p c q , (A7)valid in the whole complex c -plane. Finally, for density weobtain: ρ p t q “ ´ π i ic ˚ ż ´8` ic ˚ d cc ´ c ´ e ´ i ckt “ e ´ γ k t . (A8) APPENDIX B: B -DECOMPOSITION ANALYTICIN UPPER/LOWER HALF PLANE For entire initial DF g p v q , we showed that B ` p c q is not ana-lytic neither in the upper half-plane (discrete growing mode),nor in the lower half-plane (Landau-damped waves). Simi-larly, B ´ p c q contains singularity in the lower half-plane (dis-crete damped mode) and Landau-damped waves in the upperhalf-plane. It is possible however to redefine expansion (4.6)so that new parts B ˘ p c q will be analytic in the upper/lowerhalf-plane, correspondingly. In order to do this, we define g ˘ p c q ” g ˘ p c q ˘ D ˘ k p c q π i „ b ` c ´ c ` ` b ´ c ´ c ´ (B9)and B ˘ p c q ” g ˘ p c q D ˘ k p c q . (B10)It is easy to verify that B p c q “ B ` p c q ` B ´ p c q “ B p c q . (B11)Substitution of (B9) into (B10) gives: B ` p c q “ D ` k p c q „ g ` p c q ` b ` π i D ` k p c q c ´ c ` ` π i b ´ c ´ c ´ , (B12) B ´ p c q “ D ´ k p c q „ g ´ p c q ´ b ´ π i D ´ k p c q c ´ c ´ ´ π i b ` c ´ c ` . (B13)For c near c ` : B ` p c q « D ` k p c q « g ` p c q ` b ` π i d D ` k p c q d c ˇˇˇˇ c ` ff . (B14) From a relation analogous to (4.5), g ` p c ` q “ π i ż ´8 g p u q d uu ´ c ` “ ´ b ` π i d D ` k p c q d c ˇˇˇˇ c ` , (B15)thus B ` p c q is analytic in the upper half-plane. Similarly, B ´ p c q is analytic in the lower half-plane.To evaluate the integral (4.2), we wish to extend the inte-gration path to a closed contour in the lower half-plane. Since B ´ p c q has no singularities there, we obtain using (B12): ρ vK p t q “ ż ´8 d c r B ` p c q ` B ´ p c qs e ´ i kct “ ż ´8 d c B ` p c q e ´ i kct “ ż ´8 d c g ` p c q D ` k p c q e ´ i kct ´ b ´ e ´ γ k t . (B16)This form represents explicitly contributions of the Landaudamping and the damped mode. The Landau term differsfrom (2.6) in integration contour only: here the integral istaken over the real c -axis, while there the integration contourlies above singularity u “ c ` . MNRAS , ?? – ????