On the Escape of Cosmic Rays from Radio Galaxy Cocoons

A&A 399, 409–420 (2003) Astronomy DOI: 10.1051/0004-6361:20021827 & c ESO 2003 Astrophysics

On the escape of cosmic rays from radio galaxy cocoons

T. A. Enßlin?

Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild-Str.1, 85740 Garching, Germany Department of Physics, University of Toronto, 60 St. George Street, Toronto M5S1A7, Canada

Received 2 November 2000 / Accepted 12 November 2002

Abstract. The escape rate of cosmic ray (CR) particles from radio galaxy cocoons is a problem of high astrophysical relevance: e.g. if CR electrons are stored for long times in the dilute relativistic medium filling the radio cocoons (radio plasma in the following) they are protected against Coulomb losses and thus are able to produce a significant non-thermal Comptonisation signature on the CMB. On the other hand, CR protons and positrons which leak out of radio plasma can interact with the ambient medium, leading to characteristic gamma ray radiation by pion decay and pair annihilation. In order better understand such problems a model for the escape of CR particles from radio galaxy cocoons is presented here. It is assumed that the radio cocoon is poorly magnetically connected to the environment. An extreme case of this kind is an insulating boundary layer of magnetic fields, which can efficiently suppress particle escape. More likely, magnetic field lines are less organised and allow the transport of CR particles from the source interior to the surface region. For such a scenario two transport regimes are analysed: diffusion of particles along inter-phase magnetic flux tubes (leaving the cocoon) and cross field transport of particles in flux tubes touching the cocoon surface. The cross field diffusion is likely the dominate escape path, unless a significant fraction of the surface is magnetically connected to the environment. Major cluster merger should strongly enhance the particle escape by two complementary mechanisms. i) The merger shock waves shred radio cocoons into filamentary structures, allowing the CRs to easily reach the radio plasma boundary due to the changed morphology. ii) Also efficient particle losses can be expected for radio cocoons not compressed in shock waves. There, for a short period after the sudden injection of large scale turbulence, the (anomalous) cross field diffusion can be enhanced by several orders of magnitude. This lasts until the turbulent energy cascade has reached the microscopic scales, which determine the value of the microscopic diffusion coefficients.

Key words. ISM: cosmic rays – diﬀusion – magnetic ﬁelds – galaxies: intergalactic medium – galaxies: active

1. Introduction which also could explain the existence of cluster radio halos (Dennison 1980). If radio plasma contains a significant frac- 1.1. Motivation tion of positrons, they could lead to a detectable annihilation The outflows of radio galaxies fill large regions in the inter- line, if they would be able to leave the radio cocoon and intergalactic space with relativistic, magnetised plasma. The fate act with a dense ambient intra-cluster medium (Furlanetto & of this radio plasma is unclear, since it rapidly becomes unde- Loeb 2002). tectable for radio telescopes due to the radiative energy losses On the other hand, if the relativistic electrons are efficiently of the higher energy electrons, which emit the observable syn- confined for cosmological time-scales in radio cocoons, they chrotron emission at radio bands. Lower energy electrons, and are shielded from severe energy losses by Coulomb interac- any possible present relativistic proton population, may reside tion with the environmental gas. In such a case they would for cosmological times, unless they are able to escape by spatial be long-lived and would therefore be able to produce a non- diffusion. thermal Comptonisation signature in the cosmic microwave The escape of CRs out of radio plasma is of high astro- background (Enßlin & Kaiser 2000; Enßlin & Sunyaev 2002). physical relevance. If CR electrons are able to leak out of the Further, if old, invisible radio plasma with confined relativistic cocoon, they are possible seed particles for the Mpc-sized ra- electrons is dragged into a shock wave of the large-scale struc- dio halos in clusters of galaxies if they are re-accelerated in ture formation, its radio emission is possibly revived and forms the cluster turbulence to radio-observable energies (Giovannini the observed cluster radio relics (Enßlin & Brüggen 2002, and et al. 1993; Brunetti et al. 2001). If radio cocoons release CR references therein). protons into the intra-cluster medium, secondary CR electrons The recent detections of several ghost cavities in galaxy are produced in hadronic interactions with the background gas, clusters (for lists of recent detections e.g. Enßlin & Heinz 2002; Soker et al. 2002), which are often but not always radio- ? e-mail: [email protected] emitting, support the assumption that the environmental gas

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20021827 410 T. A. Enßlin: On the escape of cosmic rays and the bulk of the relativistic plasma stays separated on a fluctuations, which strongly determines the transport coef- timescale of 100 Myr. However, leakage of higher energy par- ficients, is assumed to be a single power law, connecting ticles is not excluded by these observations. the length scales on which field line wandering happens, CR escape from radio plasma can also play a role in our down to CR gyro-radii scales. In any numerical example galaxy. Microquasars are proposed to contribute characteristic a Kolmogorov-like turbulence spectrum is adopted. The spectral features to the galactic CR spectrum, if CRs are able detailed dependency on the turbulence wave types (e.g. to leave the ejected plasma (Heinz & Sunyaev 2002). Michalek & Ostrowski 1997, 1998; Michalek et al. 1999; Chandran 2000b) is put into a phenomenological “fudge factor”. 1.2. A simplified approach – The macroscopic cross field diffusion coefficient (κa)isas- The usual description of CR escape from systems like our sumed to exceed the microscopic one by orders of magni- Galaxy is done in terms of leaky box models, where the CR tude due to magnetic field line wandering. This allows CRs particles trapped in some region are attributed a characteristic that are rapidly diffusing along the field lines to be trans- 1 ported across them (Rechester & Rosenbluth 1978). This escape time or frequency (τ = νescape− ). This escape time is often empirically determined (e.g. for the CR escape from the also ensures that the length scales of the diffusion problem Galaxy by radioactive CR clocks (e.g. Cesarsky 1980)). It is are always shorter than length scales on which a magnetic the goal of this work to provide an estimate of this escape time flux tube loses its identity due to field line wandering. This and its dependency on the geometrical and plasma parameters should justify the use of the flux tube picture. of the system under investigation. The focus of this work is cocoons of radio galaxies, but the presented model may have 1.3. The structure of the paper other applications. In many applications, the physical parameters will not be The structure of this article is the following: in Sect. 2 the theo- sufficiently constrained to allow an accurate calculation of the retical tools for the description of the CR propagation in inho- escape time. Nevertheless, insight into the dependencies on pa- mogeneous media are compiled. The inter-phase CR transport rameters such as turbulence, fraction of the magnetically open is analysed in Sect. 3. Three escape routes are considered: the surface etc. allows statements of the relative CR escape rates in penetration of an isolating boundary layer (Sect. 3.1), the paral- different situations to be made. This should help to formulate lel diffusion along inter-phase magnetic flux tubes (Sect. 3.2), hypotheses about the conditions in which CR escape is effi- and the cross field escape (Sect. 3.3). In Sect. 4 the CR es- cient, which may be tested observationally. Examples of such cape from radio plasma cocoons is investigated and in Sect. 5 tests were given in Sect. 1.1. the main findings of the paper are listed. In Appendix A the The following simplifications are used to compile the CR spatially homogenised transport equation of CRs in a small- escape model: scale inhomogeneous medium is derived. Appendix B contains a glossary of the frequently used symbols and abbreviations. – Convective CR transport, e.g. by plasmoids which detach by reconnection events from the source region, is neglected. – The detailed magnetic structure of the source interior is not 2. Cosmic ray diffusion modelled and is treated as being homogeneous. Only the 2.1. Diffusion in inhomogeneous magnetic fields topological properties of the magnetic flux tubes touching the source surface are modelled. Since the magnetic field topology dominates the mobility of – The magnetic field at the source region surface is virtually CR particles, adapted coordinates are chosen. The length x is split into flux tubes with diameters of the perpendicular au- measured along the (local) mean direction B of a field bundle tocorrelation length of the magnetic field. The transport of with local field strength M = B+δB. Although individual field particles along and across the flux tubes is described as lines may leave the flux tubes, due to magnetic fluctuations δB, adiffusion process. A similar phenomenological descrip- this mean field flux tube is well defined and gives an appropriate tion of CR transport within flux tubes can be found in e.g. local coordinate system. Chandran (2000a), where limits of such a picture are also The analysis is restricted to particles with gyro-radii much discussed. smaller than the length-scale of the magnetic fields. In this – It is assumed that such a surface flux tube bends at some case the cross field diffusion is orders of magnitude slower point into the interior of the source region. This allow than the diffusion along the field lines. Particles can be re- CRs from the interior to enter the tube there and to fol- garded as being confined in a flux tube. Since the diameter low it to the surface. In the case that the magnetic fields are and the field strength of the tube can change as a function more onion-like in structure, the particle transport will be of x, it is convenient to work with the number of particles much more suppressed than predicted with this model. A F per flux tube length dx andmagneticfluxdφB = B dy dz, brief discussion of such a hypothetical situation is given in instead of the usual volume normalisation dV = dx dy dz = Sect. 3.1. dx dφB/B. The particle phase space distribution function is – A simplified description of the diffusion process is F(x, p,µ,t) = dN/(dx dφB dp dµ), and its source density is adopted: the microscopic diffusion coefficients (κ , Q(x, p,µ,t) = dN˙ /(dx dφB dp dµ). p is the momentum of the κ ) are parametrised. The spectrum of magnetic fieldk particle, v its velocity, m is its mass, and c is the speed of light. ⊥ T. A. Enßlin: On the escape of cosmic rays 411

µ = px/p is the cosine of the pitch angle between B (or x) With the help of a quasi-linear approximation the evolution and p. The particle distribution is assumed to be rotationally equation for f can be derived from Eq. (1) following the calcu- symmetric with respect to B, therefore the azimuthal angle has lation steps described in Schlickeiser (1989a): been integrated out. The particles entering or leaving the flux ∂ f ∂ ∂ ∂(Bf) f tube are included in the source term Q andinthelossterm + (˙pf) = κ + q. (5) F/τ in the Fokker-Planck equation for F: ∂t ∂p ∂x k B ∂x ! − τ − This equation describes the diffusive transport of particles ∂F ∂ ∂ ∂ + (vµ F) + (˙pF) + (˙µ F) = along the flux tube. In the case that small-scale variations of the ∂t ∂x ∂p ∂µ coefficients in this equations exist, effective large-scale coeffi- ∂ ∂ F cients can be estimated, as shown in Appendix A. The diffusion Dµµ F + Q. (1) coefficient is ∂µ ∂µ − τ 2 2 1 1 µ2 v (p) − Dµµ is the Fokker-Planck pitch angle diffusion coefficient, κ (x, p) = dµ (6) k 8 Z 1 Dµµ(x, p,µ) which can in principle be calculated from quasi-linear theory − of plasma wave-particle interaction for a given spectrum of and the pitch angle averaged momentum losses are plasma waves (e.g. Jokipii 1966, 1967; Kulsrud & Pearce 1969; 1 1 Hasselmann & Wibberenz 1970; Skilling 1975b; Schlickeiser p˙(x, p) = dµ p˙(x, p,µ). (7) 2002, and references in the latter). Further Fokker-Planck dif- 2 Z 1 fusion coefficients, which describe momentum changes of the − In order to demonstrate that Eq. (5) is the proper transport equa- particles, can be found in in the literature (e.g. Schlickeiser tion the volume density of particles with momentum p is intro- 1989a,b). Here, momentum diffusion is ignored due its small duced: g(x, p, t) = dN/(dV dp) = B(x) f (x, p, t). Equation (5) impact on spatial transport processes, and only continuous en- then transforms into ergy or momentum losses are included byp ˙. ∂g ∂ ∂ κ ∂g g The continuous pitch angle changes can be calculated from + (˙p g) = B k + Bq. (8) the adiabatic invariants of a particle with charge Z moving ∂t ∂p ∂x B ∂x! − τ in spatially slowly varying, and temporally constant mag- This equation is consistent with Eq. (2) of Cesarsky & Völk netic fields. Specifically, these are the linear momentum p 2 (1978). Diffusion of g vanishes for ∂g/∂x = 0. This means that and the magnetic flux within a gyro-radius φB(rg) = π rg B, diffusion tries to approach a state where the space density of where r (x, p,µ) = (1 µ2)1/2 pc/(ZeB(x)) is the gyro-radius. g − particles with identical momentum is constant, as it should do. One gets Due to variations in the diameter of the flux tube this can result in a non-constant f as a function of position. v ∂B µ˙ = 1 µ2 (2) 2 B ∂x − − · 2.2. Diffusion coefficients Klepach & Ptuskin (1995) and Chandran (2000a) investigate The parallel diffusion coefficient depends on the pitch an- the weak scattering regime, in which the mean free path of a gle diffusion coefficient according to Eq. (6). By defining the CR is large compared to typical length-scales of the inhomo- particle-wave scattering frequency νµ geneous magnetic fields and CR density scale-length. Here, a 1 2 2 more conventional standpoint is adopted, by assuming that the 1 3 (1 µ ) νµ− = dµ − (9) particle distribution is rapidly isotropised by plasma wave in- 8 Z 1 Dµµ(x, p,µ) teractions1. Therefore the anisotropic part of the distribution − function is small compared to the isotropic one. The pitch an- one can write the parallel diffusion coefficient as gle integrated distribution and injection densities are defined by κ (x, p) κ (x, p) = Bohm (10) k ε(x, p) · dN 1 f (x, p, t) = = dµ F(x, p,µ,t), (3) Here, ε(xp) is the ratio of scattering frequency νµ to the dx dφB dp Z 1 − gyro-frequency Ω(x, p) = v(p)/rg(x, p, 0), and κBohm(x, p) = dN˙ 1 v(p) rg(x, p, 0)/3 = v(p) pc/(3 ZeB(x)) is the Bohm diffusion q(x, p, t) = = dµ Q(x, p,µ,t). (4) coefficient. νµ can be regarded as the decay time of the paral- dx dφB dp Z 1 − lel particle velocity autocorrelations, which in the following is 1 This is supported by the observation that the pitch angle diffu- assumed to be identical to the decay time of the perpendicular sion coefficient is the fastest of all the Fokker-Planck coefficients (e.g. velocity autocorrelations. This latter assumption allows us to Schlickeiser 2002). The relatively fast pitch angle scattering should write the microscopic perpendicular diffusion coefficient as help to maintain a nearly isotropic CR distribution – as observed in our ε own galaxy. A criterion to test if pitch angle scattering is indeed suf- κ = κBohm (11) ⊥ 1 + ε2 ficient to establish an isotropic pitch-angle distribution – as assumed throughout this paper – is given later. (Bieber & Matthaeus 1997). 412 T. A. Enßlin: On the escape of cosmic rays

The scattering frequency νµ depends on details of the un- that the quasi-linear approximation is not fully applicable in derlying plasma turbulence on scales comparable to the gyro- this case). radius of the particle. There are two contributions important In order that the propagation of CRs is diffusive the pitch to this small-scale turbulence: the first is the Kolmogorov angle distribution should be sufficiently isotropic. This is given (or Kraichnan) cascade of large-scale turbulent kinetic energy if the scattering length lscatt(p) = v/νµ is small compared to to smaller length-scale, and the second is turbulence induced by any CR density scale-length ( f /(∂ f /∂x)). Within the above CR streaming (Wentzel 1968, 1969; Skilling 1975a). Since the parametrisation we get focus of this work is on the transport of poorly connected re- 1 γ gions, the amount of CR streaming is expected to be low. In the lB rg − lscatt(p) = following numerical examples, only external Kolmogorov-like ε0 δ0 lB ! 1 2 1 turbulence is assumed. In other cases, the theory of this article 3 3 3 10− kpc pc 3 lB ZB− can still be applied if the appropriate diffusion coefficients are = (14) ε δ GeV kpc! µG ! · used. 0 0 A simplified parameterisation of the scattering frequency is This is sufficiently small in most of our cases to allow the dif- adopted here. It is assumed that fusive approximation to be used. The resulting diffusion coefficients for CR particles are then 2 2 ε(p) = ε0 δB (rg(p))/B , (12) 1 γ v lB pc − where ε is a fudge factor, which allows one to trace and correct κ = 0 k 3 ε δ ZeBl ! the error made in this simplification, and 0 0 B 1 2 1 2 3 3 cm v pc 3 lB ZB− 2 2 γ 28 δB (l) = δ0 B (l/lB) , (13) = 3.2 10 (15) × s c ε0 δ0 GeV kpc! µG ! γ+1 is the power in the magnetic fluctuations on scale l v lB ε0 δ0 pc γ / κ = (and smaller). In Kolmogorov-like turbulence = 2 3, which ⊥ 3 ZeBlB ! will be used in numerical examples. The reference scale lB 2 5 2 5 cm vε δ pc 3 l − 3 ZB− 3 is the largest length scale which contains significant magnetic = 3.5 1016 0 0 B (16) power (compared to the power-law given by Eq. (13)), so that × s c GeV kpc! µG ! · 2 2 δB (lB) = δB is the total magnetic fluctuation power. This h i Note that these diffusion coefficients are valid under the as- length scale is also of the order of the coherence length of sumption that the turbulence dissipation length is smaller than the fields. Throughout this paper (with the exception of the the gyro-radius. If this is not the case, a much higher κ and a enhanced anomalous diffusion discussed in Sect. 2.3) it is as- much lower κ would result. k sumed that a single power-law describes the magnetic fluctua- Cross field⊥ diffusion is strongly inhibited. The small motion spectrum from the scales important for field line wandering bility of particles perpendicular to the field lines can be am- down to the CR gyro-radii. plified by rapid parallel diffusion along diverging field lines The parameter δ0 and ε0 play an important role for many (Rechester & Rosenbluth 1978). A particle’s microscopic dis- of the addressed questions and their expected values should placement from its original field line of the order of the gyro- briefly be discussed here, although our knowledge of these radius by microscopic cross-field diffusion grows exponentially quantities is still very limited. δ0 is roughly speaking the ra- while it follows its new field line. This can be described in tio of the magnetic power on the largest length-scale, on which terms of a Liapunov length λL, which depends on the behaviour the power-law spectrum of the inertia range of the turbulence of the magnetic field autocorrelation function at small displace- is valid, to the total magnetic energy density. For a sharply ments, since the old and new field lines will be strongly cor- peaked, single power-law magnetic spectrum it is δ 1. For a 0 ∼ related. As soon the particle is sufficiently far from its original broken power-law spectrum, or a very broad maximum above field line, it can be regarded as de-correlated from it. During the the inertia-range length-scales, we expect δ 1. ε gives 0 0 time needed for the particle to de-correlate from its present field the efficiency of CR scattering per magnetic power on length- line, the particle is tied to it and follows its stochastic wander- scales of the order of the particle gyro-radius, thus the power on ing while diffusing along it. The stochastic field line wandering scales which can resonate with the particle gyro-orbit. For an can be described as a diffusion process, with a diffusion coeffi- Alfvén wave spectrum with slab-like geometry (waves-vectors cient DB which gives the perpendicular displacement (squared) are mainly parallel to the magnetic main direction) one ex- per unit length travelled along the field line. Thus, during a de- pects ε 1.2 However, the nature of MHD turbulence might 0 ∼ correlation time the particle is doing perpendicular steps of the be anisotropic on small scales in the sense that mainly wave- length given by the field line wandering times the typical dif- modes with wave-vectors perpendicular to the main field are fusion length along the field line (during this time). Since this populated (Sridhar & Goldreich 1994; Goldreich & Sridhar leads to a stochastic displacement, the combined propagation 1997). In that case a strong reduction of the scattering fre- can be described as a macroscopic diffusion process, which is quency can be expected (ε 1) (a quasi-linear estimate is 0 called anomalous diffusions. This anomalous diffusion allows provided by Chandran 2000b, however it is also noted there the CRs to use the usually more powerful large-scale magnetic 2 As can be found by an order of magnitude estimations of the pitch fluctuations for their transverse propagation. A mathematical angle scattering frequencies as given by e.g. Schlickeiser (2002). description of this can be found in Duffy et al. (1995), which T. A. Enßlin: On the escape of cosmic rays 413 should be consulted for details. In their formalism the anoma- definitions one gets lous diffusion depends strongly on the parameter Λ2 λ2 κ κ 2 a = 1 + ⊥2 (20) δB (lB) λ ⊥ ln Λ l ! Λ= k , (17) ⊥ 2 2 1 γ √2 B ε(rg) λ v lB δ0 λ pc − ⊥ k 2 (21) ≈ 6 ε0 ln Λ l ZeBlB ! where λ , λ are the correlation length of the field fluctuations ⊥ k ⊥ 2 1 2 1 along and perpendicular to the main field direction (usually 2 vδ λ 3 3 27 cm 0 pc 3 lB ZB− 1.2 10 k lB λ λ ). Λ is therefore mainly the ratio of the mag- 2 ≈ k ≥ ⊥ ≈ × s c ε0 l GeV kpc! µG ! · netic turbulence energy density on the largest scales (lB)tothat ⊥ on the small gyro-radius scale. For the assumed magnetic tur- The ratio of the anomalous to parallel diffusion coefficients bulence spectrum (Eq. (13)) one gets δ2 λ2 0 2 2 2 κ /κ k . δ λ /l γ a = 2 0 037 0 (22) δ0 λ λ pc − k 2lnΛ l ≈ k ⊥ Λ= k = k (18) ⊥ √2 ε(rg) λ √2 ε0 λ ZeBlB ! · ⊥ ⊥ is nearly independent of the particle momentum, and it is independent of the fudge factor ε . The numerical result is in The anomalous cross field diffusion coefficient is 0 good agreement with detailed Monte-Carlo simulation of par- 2 DB κ ticle diffusion in slab-geometry and homogeneous turbulence κa = κ + k (19) ⊥ λL ln Λ by Giacalone & Jokipii (1999). These authors find a ratio of κa/κ = 0.02...0.04 for a turbulence strength comparable with for time-scales longer than the particle-field de-correlation time δ =k 1 (see their Fig. 3) and a scaling which seems to be only 2 2 0 (Duffy et al. 1995). DB = δB (lB) λ /(4 B ) is the field line slightly weaker than δ2 (see their Fig. 4). k 2 2 2 0 wandering diffusion coefficient, and λL = B l /(λ δB (lB)) is ∝ the Liapunov length-scale3. This is the amplification⊥ k length of (the expectation value of the square of) a small displacement 2.3. Enhanced anomalous diffusion of a particle from its initial field line while travelling along its Under certain circumstances the anomalous cross field diffu- new field line. The characteristic perpendicular length-scale l sion is extremely efficient. The anomalous diffusion coefficient of the size of chaotic, non-trivially twisted magnetic field fluc-⊥ κa depends quadratically on Λ (Eq. (20)), which is mainly the tuations is expected by us to be of the order of lB, although 4 ratio of large-scale to small scale turbulence (Eq. (18)). Λ is arguments exist that it might be much smaller . With these therefore independent of the level of the turbulence, as long 3 Strictly speaking, only the perpendicular component of the field as the spectral shape of the turbulence energy spectrum is not 2 fluctuations δB (lB) enter the equations for DB and λL.However,the changed. Increasing the turbulence energy density of a system ⊥ difference between the perpendicular and total fluctuation levels are by a factor XT increases κa also by XT, due to the dependence not very large. Since also the diffusion coefficients depend on the 1 of κ XT (or κ XT− ) on the small scale turbulence. perpendicular magnetic fluctuations, one can interpret δB2 as the per- If⊥ ∝ the slopek of∝ the turbulence is changed, drastic changes pendicular fluctuations right from the beginning and get a consistent in κa can result. If e.g. large-scale kinetic energy is suddenly formalism. injected into a system, it may need a period on the order of 4 2 Pertubatively (to lowest order in δB ) one can show that l is given the eddy turnover time τ = l /v before the turbulent cas- by the second Taylor coefficient of the magnetic field autocorrelation⊥ T T T cade has also raised the small-scale turbulence level. During function perpendicular to the main field direction: δB(x) δB(x + 2 1 2 2 4 h · this period the microscopic diffusion coefficients κ and κ stay ξ ) = δB (lB)(1 2 ξ /l + O(ξ )). More strictly speaking, l seems k ⊥ ⊥ i − ⊥ ⊥ ⊥ ⊥ 2 to be the Taylor length of the along the mean field direction integrated unchanged, but κa is increased by a factor of XT. and by λ normalised autocorrelation function (of the chaotic part of k the fluctuation spectrum). However, if the autocorrelation function is a 3. Cosmic ray escape direct product of a parallel and perpendicular profile, then this is identical with the simplified definition given here. Otherwise it can be ex- 3.1. Boundary layer penetration pected to be close to that. However, it has been pointed out by Narayan & Medvedev (2001) that if the chaotic (topologically non-trivially If a boundary layer of insulating magnetic fields (without any twisted) magnetic fluctuations extend from the turbulence energy in- flux leaving the boundary layer regions) separates the interior jection range down to the smallest turbulence scale, the anomalous from the surrounding of the source region the escape of cosmic diffusion seems to become very efficient. If all the magnetic power ray particles is strongly inhibited. Although radio polarization is in such chaotic modes the effective cross field diffusion coefficient observations of radio cocoons indicate that magnetic fields are seems to be close to the order of the parallel diffusion coefficient. In well aligned with the cocoon surface (e.g. Laing 1980, 1981; the case that the magnetic turbulence in radio cocoons is completely of Spangler et al. 1984), it is far from obvious if this implies an chaotic this nature our results would be changed in a way that particle insulating boundary layer. The origin of such a hypothetical escape from radio cocoons is always very fast. Thus the observational tests of particle escape mentioned in the introduction may confirm or largest scales we do expect strong non-trivial topologies, produced by refute such a picture. In the following it is assumed that the small- the turbulence driving forces. Their strength should be limited (and scale turbulence is dominated by travelling Alfvénic waves, which do therefore possibly confined from the smaller-scales) by reconnection not disturb the topological properties of the main field. Only on the events. Therefore we expect l lB. ⊥ ∼ 414 T. A. Enßlin: On the escape of cosmic rays layer may be due to mixing of radio plasma with the ambient gas in a thin surface mixing layer, which is then dynamically decoupled from the turbulent interior due to its much higher inertia. The expected shear between the interior and the mixing layer may amplify and align magnetic fields into such an insulating boundary layer. As a simplified model we assume the source region to be spherical with radius rs, the boundary layer to have a thickness db. The layer, which extends from radius ri = rs db to rs,is assumed to be filled with magnetic fields which are− mainly tan- gential to the source surface at rs. Therefore the radial diffusion coefficient is that of the macroscopic cross field diffusion κa. In a quasi-stationary situation (slow particle escape) the space density of particles is given by x s x l 1 1 source region transition zone loss region r− rs− g(r, p) = g0(p) 1 − 1 (23) ri− rs− − Fig. 1. Geometry of the inter-phase region. The left (x < xs) and right regions (x < x) are the source and loss regions of the particle. A within the boundary layer (ri < r < rs), where g0(p)isthe l bundle of field lines connect these regions through a transition zone particle spectrum inside the layer (r < ri). From this the number of CRs within the source region is given by (xs < x < xl). 2 π N(p) = dV g(r, p) = g (p) r r (r + r ), (24) given by the diameter of the flux tube. The typical time a par- s 3 0 s i s i Z ticle requires to enter or leave a flux tube with flux φ = π r2 B 2 and the CR escape rate through the source surface area As is and radius r λ /2isgivenbyτ = r /(4 κa) = φ/(4 π B κa). given by In the loss region≈ ⊥q = 0. In the transition zone κ and B might be functions of the position, but no particle lossesk or sources 4 πκag0(p) N˙ (p) = As κa g(r, p) r=r = . (25) are assumed there. Furthermore, any energy loss processes are ∇ | s − r 1 r 1 i− − s− ignored. These conditions lead to the following parameters: The escape frequency follows to be q(x, p) = H(x x) f (p)/τ (p), (27) s − s s N˙ (p) 6 κa(p) 3 κa(p) τ(x, p) = H(x x) τ (p) + H(x x ) τ (p), (28) ν p s s l l layer( ) = = 2 2 = (26) − − − N(p) rs r rs db (1 db/(2 rs)) p x, p , − i − ˙( ) = 0 (29) 2 4 2 3 1 1 3 3 B(x) = H(xs x) Bs + H(x xl) Bl 4.0 vδ0 λ lB kpc pc 3 ZB− k − − 2 +H(x xs) H(xl x) Bt(x), and, (30) ≈ Gyr c ε0 rs db l GeV µG ! · − − ⊥ κ (x, p) = H(xs x) κs(p) + H(x xl) κl(p), It should be noted that enhanced anomalous diffusion due to k − − +H(x xs) H(xl x) κt(x, p). (31) a sudden raise of the large-scale turbulence level by a factor XT − − increases the particle escape rate through a boundary layer by H(x) is the Heaviside step function. Since the momentum is 2 a factor XT during the period of not fully developed turbulence, conserved (p ˙(x, p) = 0), Eq. (5) can be solved independently and XT afterwards (see Sect. 2.3). for all momenta. The explicit dependency on p is dropped in the following. The solution of Eq. (5) with the parameters given 3.2. Flux tube escape route above, smooth field lines (dB/dx = 0atxs and xl)andthe boundary conditions f ( ) = fs, f ( ) = 0is If some magnetic flux φB leaks from the source to the loss re- −∞ ∞ gion, the CR can escape following the field lines. A possible fs ( fs fa)exp((x xs)/ls);x

100 100 f(x) f(x) g(x)

10 10

1 g(x) 1 B(x) K(x) 0.1 0.1 K(x)

0.01 B(x) 0.01 -10 -5 Xs 0 Xl 5 10 -10 -5 Xs 0 Xl 5 10

Fig. 2. Profiles of f (x), g(x) = B(x) f (x), B(x), and κ (x). The pa- Fig. 3. As Fig. 2, but Bs = 0.01, Bl = 0.3. rameters are f = 100, B = 0.3, B = 0.01, x = k 2, x = 2, s s l s − l ls = 3, ll = 3. The field in the transition zone is parametrised by 2 2 Bt(x) = [Bl (Bl Bs)exp( √Bs/Bl (x xs) /(xl x) )] + Bs (Bs − − 2 − 2 − − − 2/3 − Strongly magnetised loss region: Bl)exp( √Bl/Bs (x xl) /(xs x) )/2. Further κ (x) = 0.01 B− (x) is assumed.− − − k η B κ η B κ φ κ ν s s l = s s l = B l (42) k ≈ Ls Bl rτl Ls ll Bl Bl Vs ll · An example density profile is shown in Fig. 2 for strong If two regions 1 and 2 with volumes V1 and V2 are connected by fields in the source and weak field in the loss region, and in some magnetic flux φB, then the ratio of the particle exchange Fig. 3 for the reversed geometry. frequencies is equal to the inverse volume ratio, as it is required If one introduces the spatial average within the transition xl by detailed balance: zone by A(x) = dxA(x)/lt,wherelt = xl xs, one can h i xs − write R ν1 2 η1 B1/L1 V2 → = = (43) ¯ ¯ ν2 1 η2 B2/L2 V1 · I = Bt lt/κ¯t = Bt τt/κ¯t , where (37) → B¯ = B , κ¯ = B¯ /pB/κ , and τ = l2/κ¯ . (38) t h i t t h ti t t t In the following it is assumed that the source field region has If B changes by a large factor in the transition zone one expects the strongest magnetic fields and therefore Eq. (41) is used. If a number of M flux-tubes with diameter λ each leave the max(Bs, Bl) > B¯ t min(Bs, Bl). Due to the expected anti- k ⊥ source region the total magnetic flux leaving the source is φB = correlation of B and κ also max(κs,κl) κ¯t > min(κs,κl)is 2 2 k M π B λ /4. Applying the identity M = 4 ηs As/(πλ ) one likely. k ⊥ k ⊥ The total particle loss of the source into the loss region is finds that the particle loss frequency is then given by given by 4 ηs √κa κ f (x) f φ B ν k (44) ˙ ∞ s B s k ≈ λ Ls Ns = dx = (39) ⊥ − k τl √ ¯ √ √ · Zxl Bs τs/κs + Bt τt/κ¯t + Bl τl/κl 1 γ 4 ηs lB λ v pc − If several loss channels exist with similar properties (τs, τl) = k (45) 3 √2lnΛ ε0 l λ Ls ZeBlB ! then φB in Eq. (39) can be replaced by the sum of the abso- ⊥ ⊥ lute fluxes of these channels. In this case the flux leaving the / 1 2 3 3 0.081 v lB λ /(λ l Ls) pc/(ZB) source region is φB = As ηs Bs,whereηs is the fraction of the η k ⊥ ⊥ ≈ Myr s c ε 4/3 GeV/µG ! · source surface (As) penetrated by inter-phase magnetic flux. 0 kpc− After defining the characteristic length-scale of the source by L = V /A ,whereV is the source volume, and noticing that It should be noted that enhanced anomalous diffusion due to a s s s s X N = V g = V f B one can write the typical particle escape sudden raise of the large-scale turbulence level by a factor T s s s s s s X frequency ν as increases the flux tube particle escape rate by a factor T only k during the period of not fully developed turbulence, but not N˙s ηs Bs/Ls afterwards (see Sect. 2.3). ν = k = (40) k − Ns Bs √τs/κs + B¯ t √τt/κ¯t + Bl √τl/κl · If the CRs escape rapidly from the source region, they can excite plasma waves which again scatter the CRs. This limits This formula is simplified if one of the terms in the denomi- the escape velocity to approximately the Alfvén- or thermal nator dominates. Two cases correspond to typical astrophysical velocity, or whichever is larger (Tademaru 1969; Holman et al. situations: 1979). For the high energy part of the CR spectrum the lim- Strongly magnetised source region: iting velocity can be even larger, since the small number den- η κ η κ φ κ sity of high energy CRs in typical power-law distributions does ν s s = s s = B s (41) k ≈ Ls rτs Ls ls Bs Vs ls · not lead to an efficient excitation of the scattering waves 416 T. A. Enßlin: On the escape of cosmic rays

(e.g. Felice & Kulsrud 2001). However, an important quantity is the CR escape velocity at the surface of the source region: 1 γ κs 4 v lB λ pc − vCR = = k ls 3 ε √2lnΛ λ l ZeBlB ! 0 ⊥ ⊥ 5/3 1 km v l λ pc/(ZB) 3 B k 80 2/3 (46) ≈ s ε0 c λ l kpc GeV/µG ! · ⊥ ⊥ As long as this velocity is below the limiting speed, which for radio galaxy cocoons should be the environmental sound speed, CR self-conﬁnement due to wave excitation should be negligible.

xx 3.3. Cross field escape route 12 source regionloss region source region In order to model the cross field escape frequency an individual flux tube of diameter λ is investigate first, which touches the Fig. 4. Geometry of a flux tube which is part of the source boundary. ⊥ inter phase surface on a length-scale λ (see Fig. 4). The parti- The left (x < x1) and right regions (x2 < x) are the source and loss k cle distribution function along this tube is governed by Eq. (5) regions of the particles. In between, escape into the other phase is if one adopts the following parameters: possible. q(x, p) = (1 σ (x)) fs(p)/τs(p), (47) − ⊥ This leads to perpendicular escape frequency of τ(x, p) = τs(p), (48) ˙ p˙(x, p) = 0, (49) M N σ y ν = ⊥ ⊥ = ν ,0 (1 ηs) 1 ⊥ (y + e− 1) , (56) ⊥ N ⊥ y B(x) = Bs (50) − s − " − − #

κ (x, p) = κs(p). with k As λ 4 κa σ (x) is the fraction of the flux tubes surface which is in con- ⊥ ⊥ ν ,0 = = (57) tact with the other phase. σ (x) = 0forx < x or x > x . ⊥ 4 Vs τs Ls λ 1 2 ⊥ ⊥ 2 1 Again, the notation of the momentum dependence is dropped 3 2 δ0 lB λ v pc k for convenience. The solution can be found with the help of the = 2 (58) 3 ε0 ln Λ l λ Ls ZeBlB ! method of Green’s functions and is given by ⊥ ⊥ 2/3 2 2 1 l λ / l λ L 3 x 0.016 δ0 v B ( s) pc/(ZB) 1 2 k ⊥ ⊥ 4/3 f (x) = fs 1 dx0 σ (x0)exp( x x0 /ls) . (51) ≈ Myr ε0 c kpc GeV/µG ! · ⊥ − " − 2 ls Zx1 −| − | # In the following it is assumed that σ (x) = const within the The reverse process, in which particles enter the strongly mag- ⊥ netised region, has a frequency which can be calculated from length λ = x2 x1. In this case k − the escape frequency (Eq. (56)) and the condition of detailed σ x x1 x2 x balance (Eq. (43)). f (x) = fs 1 ⊥ h − + h − , with " − 2 ls ! ls !!# It should be noted that enhanced anomalous diffusion due z to a sudden raise of the large-scale turbulence level by a factor h(z) = sgn(z) 1 e−| | , (52) − X increases the cross field particle escape rate by a factor X2 T T which is shown in Fig. 5. The particle escape rate (into the other during the period of not fully developed turbulence, and XT phase) of this filament is then afterwards (see Sect. 2.3).

fs λ σ σ y N˙ = k ⊥ 1 ⊥ (y + e− 1) , (53) − ⊥ τs " − y − # 3.4. The best escape route where The most efficient escape route can be found by comparing the 2 escape frequencies for the flux tube and the cross field escape λ λ λ 4 δ0 y = k = k = k (54) route: ls √κs τs l λ √2lnΛ· ⊥ ⊥ ν κa 1 ηs σ y Usually y 1, otherwise the situation resembles more the iso- ⊥ = − 1 ⊥ (y + e− 1) (59) ν r κ ηs " − y − # lating boundary layer case discussed in Sect. 3.1. The number k k δ0 λ M of flux tubes which touch that part of the surface of the k ⊥ (60) source region, which is not intersected by inter phase magnetic ≈ 5 l ηs · ⊥ flux, is During the injection phase of turbulence (a few τT)thera- As (1 ηs) tio ν /ν is increased by an additional factor of XT over M = − (55) ⊥ k ⊥ πλ λ σ · the estimates given in Eq. (60). Under certain conditions ⊥ k ⊥ T. A. Enßlin: On the escape of cosmic rays 417

of CR electrons, positrons or protons from radio cocoons into 1 10 the centre of galaxy clusters. 3.0 Further, during the buoyant rise of a radio cocoon through 1.0 a cluster atmosphere, the cocoon expands adiabatically. If all magnetic length-scales scale linearly with the cocoon size Ls, 0.3 0.5 and the magnetic ﬁeld strength decreases adiabatically as B 0.1 2 ∝ Ls− then all escape frequencies discussed here (νlayer, ν , ν )de- 2/3 k ⊥ crease according to ν Ls− . This implies that CRs are better conﬁned in a larger cocoon,∝ or that CR escape is most rapidly at an early stage of the cocoon’s buoyant voyage through the intra-cluster medium. X1 X2

Fig. 5. Profile of f (x)forσ = 0.5, λ = 2 and different ls as labelled in the graph. ⊥ k 4.2. Suddenly injected turbulence If a cluster merger event suddenly injects large-scale turbulence the radio plasma can become transparent even for low energy (δ0 < 5 ηs < XT δ0) the dominant escape mechanism can switch CR particles. The merger might produce turbulent flows with temporarily from flux tube escape to cross field escape during 1 the onset of turbulence. velocities of vT 1000 km s− on a scale of lT 100 kpc which increase the turbulent≈ magnetic energy density≈ on large-scales by a factor of XT = 10 from initially δ0 = 0.01 for example. For 4. CR escape from radio plasma roughly an eddy turnover time τT = lT/vT 100 Myr the small scale turbulence is not increased in regions≈ far from shock 4.1. A quiet environment waves. During only a tenth of this period (10 Myr, which is of The old radio plasma cocoon of a small radio galaxy in the the order the turbulent cascades needs to transfer energy from central IGM of a galaxy cluster might be of approximately the scales of the radio cocoon (10 kpc) down to the CR gyro- spherical shape. Here the following parameters are assumed: radius length scales) an enhanced anomalous cross field escape should allow roughly 30%/ε0 of the 10 GeV CR particles ini- B = 10 µG, lB = λ = 3 kpc, l = λ = 1 kpc, and a cocoon diameter of 50 kpc.k The turbulence⊥ inside⊥ old radio plasma is tially confined in the radio cocoon to escape. A similar num- likely the turbulence induced by the environment, and there- ber of the CRs would escape in the remaining 90 Myr of fully fore assumed to be Kolmogorov-like. The diffusion coefficients developed turbulence. Significant losses would occur on even 28 2 1 shorter timescales in the case that the pitch angle scattering ef- of 10 GeV CR particles are then κ = 7 10 cm s− /(ε0 δ0), 16 2 1 k × 28 2 1 ficiency is low (ε0 1). The total loss of particles escaping κ = 1.7 10 cm s− ε0 δ0,andκa = 1.9 10 cm s− δ0/ε0. ⊥ × × along inter phase flux tubes is roughly 6% (ηs/0.01)/ε0 1 If the radio cocoon has a boundary layer of thickness db = 2 5 kpc, which is isolating the interior from the exterior due to the during 0.1 τT, and likely negligible if only a few kpc of the cocoons surface is opened (η 0.01). If the cocoon would have lack of any magnetic flux exchange, then the escape frequency s 1 an isolating boundary layer of thickness 5 kpc, then only 1.7% of these CRs is νlayer = 1.7Gyr− δ0/ε0.Thisisextremelyslow of the CR particles would be released during 0.1 τT. for a low level of magnetic turbulence (δ0 < 0.01). If such a isolating layer does not exist, CR particles can reach the surface much easier and leave by cross field diffu- 4.3. Merger shock waves sion (over a short distance) or by following open field lines. Since only reconnection events of the fossil fields with the Cluster mergers happen frequently (Mohr et al. 1995; Jones & much weaker environmental fields could have opened the Forman 1999; Schuecker et al. 2001) so that shock waves ap- magnetic topology, ηs 1 is assumed due to the possible pear often in clusters (Quilis et al. 1998; Miniati et al. 2000), rarity of such events. Under such conditions the escape fre- which should raise the turbulence level on all length scales dur- 1 quencies of 10 GeV particles are ν 57 Gyr− ηs/ε0,and ing their passage. Therefore the strongly enhanced anomalous 1 k ≈ ν 31 Gyr− δ0/ε0. The CR streaming velocity along open diffusivity discussed above is not expected to appear in this ⊥ ≈ 1 flux tubes is 460 km s− /ε0 and therefore likely much below case. However, the turbulence level should increase substan- the cluster sound speed, so that CR self-confinement is not im- tially, leading to some enhancement of the particle escape rate, portant for these particles. For a low level of magnetic turbu- and the typical lengthscale Ls the particles have to travel before lence (δ0 < 0.01) and a mostly closed field topology (ηs < 0.01) leaving the source also decreases substantially. Both effects in the escape of 10 GeV particles requires several Gyr. If one con- combination should lead to efficient particle escape. siders further that such radio plasma is buoyant and therefore The escape length Ls = Vs/As decreases due to two ef- leaves the cluster centre within a few 100 Myr (Churazov et al. fects: first, the radio plasma volume shrinks by a factor C = 3/4 2001; Enßlin & Heinz 2002), the fraction of CRs injected into V2/V1 = (P1/P2) for radio plasma with a relativistic equa- a cluster core is expected to be small under such circumstances. tion of state which is adiabatically (due to the high internal This can impose constraints on several of the theoretical con- sound speed) compressed in an environmental shock wave with siderations listed in Sect. 1.1, which rely on an efficient escape pressure jump P2/P1. Second, the shock wave should disrupt 418 T. A. Enßlin: On the escape of cosmic rays

1/3 the radio plasma into filamentary or torus-like morphologies, E.g. for Kolmogorov turbulence it is ν / p ,andfor 1/2 k ⊥ ∝ as seen in numerical simulations (Enßlin & Brüggen 2002) Kraichnan turbulence it is ν / p . and in high resolution radio maps of some cluster radio relics – In the case where the pitchk ⊥ angle∝ scattering frequency is (Slee et al. 2001). The numerical simulations show that a spher- strongly reduced (Chandran 2000b) due to the possible ical radio cocoon transforms into a torus with major radius anisotropic nature of weak magneto-hydrodynamic turbu- equal to the radius of the original sphere (the torus diameter lence (Sridhar & Goldreich 1994; Goldreich & Sridhar is roughly that of the former sphere). Using the dimensions of 1997), this model can still be applicable as long as the parti- the torus given by this observation and the above given volume cle transport is diffusive. In such a case one expects ε0 1. shrinking factor C one finds that the characteristic length-scale – If the chaotic (topologically non-trivially twisted) part Ls = Vs/As decreases due to shock compression and change of of the magnetic fluctuations extends down in the power- the morphology by a factor spectrum to the smallest length-scales, the anomalous cross 1 field diffusion may be much larger than estimated here L 3 C 4 C − 1 s,2 = 1 + √C. (61) (Narayan & Medvedev 2001) leading to a very rapid CR L r 4 π  r3 π2  ≈ 2 escape from radio cocoons. If this is indeed the case is not s,1     yet clear, but may be testable by observations of escaped Thus, for a typical merger shock wave with Mach number of particles5. the order 3, and pressure jump roughly 10 the characteristic – In contrast to a statement by Chandran (2000a) it is shown length scale decreases by a factor of nearly 6. If at the same here that the macroscopic diffusive transport of particles time the level of turbulence increases by an order of magnitude, along a flux bundle with small scale inhomogeneities in the the cross field diffusion escape rate is enlarged by nearly two field strength is reduced by the presence of strong field bot- orders of magnitude. Therefore also for radio plasma passing tlenecks (Appendix A). through a shock wave efficient CR particle escape is expected. – A sudden large-scale turbulence injection can lead temporarily to the enhanced anomalous diffusion regime. This 5. Discussion increases both escape frequencies, but the cross field escape frequency by a larger factor. A model has been presented describing the escape of CR parti- – Cluster merger events can lead to strong particle losses cles from regions which are poorly magnetically connected to from radio plasma. Shock waves disrupt the relativistic the environment. The main application of this theoryis to co- plasma into smaller pieces from which CRs can escape coons of radio galaxies. more easily. It should be noted that cluster merger waves In order to apply this model quantitatively to astrophysi- and turbulence themself are able to accelerate CRs, which cal systems, several poorly known parameters have to be de- can mask (and/or energise) the CRs escaping from radio termined, like the geometry of the source (V , A , L , η , σ ), s s s s s plasma. the field fluctuation length-scales (λ , λ , l , lB), the nature and k ⊥ ⊥ – Merger can also lead to efficient particle escape from radio the level of the turbulence (δ ), and the coupling strength be- 0 plasma not directly affected by shock waves. The enhanced tween small-scale magnetic turbulence and the gyro-motion of anomalous diffusion due to the large-scale turbulence in- charged particles (ε ). Several of these parameters are closely 0 jected by the merger could also lead to efficient particle related and can roughly be estimated (e.g. the characteristic tur- escape. bulence length-scales). Other parameters will remain unknown until a truly detailed knowledge of radio plasma is available. Given the large astrophysical importance of the problem of cos- However, even without detailed knowledge of these param- mic ray escape from radio plasma cocoons, further studies of eters, insight into the qualitative behaviour of CR transport be- this problem – theoretically and observationally – would be of tween different phases is provided: great value. The presented work provides a starting point for – An isolating magnetic layer at the surface of the CR confin- such investigations. ing region can efficiently suppress CR escape compared to the case in which flux tubes from the source interior touch Acknowledgements. I thank Benjamin Chandran, Eugene Churazov, (or even leave) the source surface. Sebastian Heinz, Philipp P. Kronberg, and Kandu Subramanian for – CR escape from a mostly magnetically confined region al- discussions and comments on the manuscript. This work was done in ways requires parallel and cross field particle transport. the framework of the EC Research and Training Network The Physics – The escape frequency of cosmic rays travelling along inter- of the Intergalactic Medium. phase flux tubes is independent of the level of turbulence (Eqs. (44)–(45)). Appendix A: Homogenisation of the diffusion – The cross field escape frequency increases linearly with in- equation creasing turbulence level (Eqs. (56)–(58)). – The parallel and perpendicular escape frequencies are It is instructive to derive the macroscopic transport equation in mainly determined by the properties of the strong field the case that the spatial fluctuations in the parameters of Eq. (5) region. 5 E.g. annihilation of escaped positrons, decaying pions produced – All escape frequencies considered here have an identical by relativistic protons, or even simply a low level diffuse synchrotron scaling with the CR momentum, for any turbulence spectra. halo around a bright radio cocoon due to escaping electrons. T. A. Enßlin: On the escape of cosmic rays 419 can be described by a large-scale x0 and a much smaller scale ηs: fraction of As occupied by inter-phase magnetic flux ξ,sothatx = x0 +ξ/ and is the ratio of the small to the large- f : isotropic particle phase-space density per magnetic flux scale. A quantity A(x) = A(x0,ξ) is averaged over the smaller (Eq. (3)) ξ scale by A (x0) = limξ dξ0 A(x0,ξ0)/ξ. fs: average density of particles ( f ) within the source h i →∞ 0 The evolution of the distributionR function can be expressed γ: spectral index of magnetic turbulence spectrum as an asymptotic series in the small parameter .Tolowestor- g: isotropic particle phase space density per volume der in this method of homogenisation gives κ : parallel diffusion coefficient (Eqs. (6), (10) and (15)) κk : perpendicular diffusion coefficient (Eqs. (11) and (16)) ∂ f¯ ∂ ∂ ∂(B¯ f¯) f¯ ⊥ + (p¯˙ f¯) = κ¯ + q¯, (A.1) κa: anomalous cross field diffusion coefficient (Eqs. (19) ∂t ∂p ∂x k B¯ ∂x ! − τ¯ and (20)) Λ: Eqs. (17) and (18) where the prime was dropped (x x). The details of this 0 λ : parallel correlation length of the field fluctuations kind of calculation can be found in→ Holmes (1995). The ho- λk : perpendicular correlation length of the field fluctuations, mogenised parameters are given by ⊥ typical flux tube diameter f¯ = f (A.2) lB: magnetic turbulence reference scale (Eq. (13)) 1 h i 1 l : δB(x) δB(x+ξ ) = δB2(l )(1 ξ2 /l2 +O(ξ4 )) if only B¯ = 1/B − (A.3) B 2 h i ⊥ theh chaotic· (topologically⊥ i non-trivial− twisted)⊥ ⊥ field⊥ fluctua- p¯˙ = B¯ p˙/B (A.4) h i tions are regarded κ¯ = B¯/ B/κ (A.5) l (l ): characteristic distance a CR travels before leaving a flux k k s l 1h i τ¯ = B¯ − / 1/(τ B) (A.6) tube in the CR source (loss) region (Eq. (33)) h i q¯ = q (A.7) Ls = Vs/As: characteristic length of source h i· lscatt: CR pitch-angle scattering length lscatt = v/νµ In the case of vanishing small-scale fluctuations of all coeffi- M : number of inter phase flux tubes cients, their homogenised averages are, of course, unchanged m:k particle mass by the homogenisation and Eq. (A.1) is identical to Eq. (5). M : number of flux tubes forming the boundary In the case of small scale variations, the homogenised loss µ:⊥ cosine of the CR pitch angle coefficients p¯˙ andτ ¯ are weighted towards the weak field re- Ns: number of CR in source region gions, where the flux tube has a larger diameter and therefore ν : CR escape frequency along flux tubes (Eqs. (40) and (44)) k most of the particles reside. The effective large-scale diffusion νµ: particle-wave pitch-angle scattering frequency coefficientκ ¯ is reduced compared to the typical small scale ν : perpendicular escape frequency k ⊥ coefficient κ , due to the harmonic weighting in Eq. (A.5). φB: magnetic flux h ki The weight of these regions is further increased due to the p: particle momentum expected anti-correlation of field strength and diffusion coeffi- q: isotropic particle phase-space injection rate per magnetic cient. However, even in the case that κ does not change on the flux (Eq. (4)) k micro-scale, its global value is changed due to field strength rg: particle gyro-radius fluctuations on the micro-scale:κ ¯ = κ /( 1/B B ) κ ri: inner radius of (spherical) boundary layer k k h ih i ≤ k (equality only if B(x,ξ) = B(x)). If e.g. 10% of the micro-scale rs: source radius in spherical approximation has a 100 times enhanced field strength, the averaged diffusion σ : fraction of flux tube surface in contact with loss region coefficient is a factor of 10 smaller than the microscopic one, in τ⊥: particle escape time from a flux tube the case of a spatially constant κ , otherwise much more. This τs: flux tube escape time in source region k demonstrates that magnetic traps modify the particle transport τT: eddy turn-over time even in the diffusive regime, in contrast to an opposite state- v: particle velocity ment by Chandran (2000a). 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