The Scattering Mean Free Path of Cosmic Ray Particles in Isotropic Damped Plasma Wave Turbulence

A&A 555, A111 (2013) Astronomy DOI: 10.1051/0004-6361/201321191 & c ESO 2013 Astrophysics

The scattering mean free path of cosmic ray particles in isotropic damped plasma wave turbulence

M. Vukcevic1,2

1 Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universität Bochum, 44780 Bochum, Germany e-mail: [email protected] 2 University of Defence, Military Academy, Pavla Jurisica Sturma 33, 11000 Belgrade, Serbia

Received 30 January 2013 / Accepted 3 May 2013

ABSTRACT

An analytical expression for the mean free path of single-charged cosmic ray particles, especially for positrons, is derived in isotropic plasma wave turbulence, where the crucial scattering of cosmic ray particles with small pitch-angle cosines is caused by resonant cyclotron interactions with oblique magnetosonic waves. In this calculation, viscous damping effects are included, which results in broadening of the resonance function. It is demonstrated that including resonance function broadening ensures a finite mean free path for cosmic ray energies, for which previously reported types of turbulence predicted an infinitely large mean free path. Key words. scattering

1. Introduction a decaying-dark-matter origin for the GeV Galactic positron anomaly measured by PAMELA. A Galactic bulge-to-disk ratio of the luminosity of diffuse To quantify scattering mean free paths for MeV interstel- 511 keV positron annihilation radiation is, as measured by lar positrons, an analytic approximation has been used (Teufel INTEGRAL, four times larger than a stellar bulge to disk ratio of & Schlickeiser 2002) for slab-like dynamical turbulence, which the Galactic supernovae (SNe). SNe are thought to be the princi- predict that is a mean free path in MeV energies proportional pal source of the annihilating positrons. This large discrepancy to r1/3, where r is defined as particle rigidity. This model for GeV has started a search for new sources. It has been shown that the particles predicts an infinitely large mean free path. The other measured 511 keV luminosity ratio can be understood well in the slab plasma-wave turbulence model (Schlickeiser et al. 2010), context of a Galactic SN origin when the differential propagation which is more plausible when compared to the dynamical one, of these MeV positrons in the various phases of the interstellar predicts an infinite positron scattering mean free path at MeV medium is taken into consideration (Higdon et al. 2009), since energies, while it has finite values proportional to r1/3 for GeV these relativistic positrons must first slow down to energies less energies. than 10 eV before they can annihilate. Here, we propose a damped plasma wave turbulence model, A large number of potential sources have been proposed over which can assure a finite mean free path for MeV and GeV the years: cosmic ray interaction with the interstellar medium positrons, and compare our result with results of previously (Ramaty et al. 1970), pulsars (Sturrock 1971), radioactive nu- mentioned models. clei produced in SN (Clayton 1973), compact objects (Ramaty A key parameter for the cosmic ray transport is the parallel & Lingelfelter 1979), dark matter (Boehm 2004), and micro- spatial diffusion coeffcient κ = vλ/3, which is conventionally ex- quasars (Guessoum et al. 2006). Some of these theories either pressed in terms of the mean free path λ along the background are problematic or have a wide range of uncertainties. For ex- magnetic field and the particle speed v. In many studies, the ample, the predicted distributions of positrons from radionuclei parallel mean free path also controls the perpendicular spatial synthesized in SN are only marginally compatible with obser- diffusion coefficient κ⊥ = ακk, which is assumed to be propor- vations (Milne et al. 2002). However, the recent mapping of the tional to κk due to the lack of a rigorous theory of perpendic- Galaxy at 511 KeV (Knödlseder et al. 2005) has placed severe ular diffusion. When discussing the ratio of perpendicular and constraints on the possible positron sources. parallel mean free path, it has been published from proton ob- On the other hand, the positron excess in Galactic cosmic servations that 0.02 < λ⊥/λk < 0.083 over the rigidity range ray positrons above 10 GeV has been confirmed by PAMELA of 0.5MV < R < 5GV (Palmer 1982), which assures parallel (Adriani et al. 2009). The most recent confirmation of the mean free path values to be high enough that we can neglect the PAMELA result on positrons in the Galaxy cosmic ray spec- perpendicular component. Hereafter, we only consider parallel trum has just been published (Aguilar et al. 2013). There are mean free path. several astrophysical explanations for possible sources of these The scattering mean free path is the result of resonant positrons, such as dark matter annihilation (Hooper et al. 2009) particle-wave interactions of cosmic ray particles with the tur- or decay (Arvanitaki et al. 2009). According to a recent in- bulent component of cosmic magnetic fields, and thus depends vestigation (Pohl & Eichler 2009), Fermi measurements of the on the nature and geometry of cosmic turbulent magnetic fields high-latitude γ-ray background (Abdo et al. 2009) constrain (Schlickeiser 2002). Many observations of plasma turbulence

Article published by EDP Sciences A111, page 1 of 10 A&A 555, A111 (2013) in the solar wind indicate that the turbulent magnetic field is a It is the purpose of this work to involve oblique magene- mixture of slab (i.e., parallel to the ordered background magnetic tosonic waves and quantitatively investigate the influence of field) waves, a dominating 2D component (Bieber et al. 1996) wave damping on the quasilinear scattering mean free paths with a negligible contribution to particle scattering (Shalchi of cosmic rays, especially for positrons with Lorentz factors & Schlickeiser 2004), and obliquely propagating magnetosonic γ < mp/me = 1836. In a plasma in rough equipartition of ki- waves that reveal through electron density fluctuations because netic and magnetic pressure, oblique magnetosonic waves are of their compressive nature. In the static magnetosonic limit, expected to be overdamped by Landau absorption (Fedorenko the scattering rate from the 2D component vanishes (Shalchi & 1992). Thus, the second-order Fermi acceleration using oblique Schlickeiser 2004). fast magnetosonic waves cannot work in a plasma in rough For small amplitude waves, an appreciable interaction be- equipartiton (Achterberg 1979). It is possible to overcome this tween a wave and a particle arises only when the particle is difficulty either by considering that this mechanism works, in moving at nearly the parallel wave phase speed. The interaction the interstellar medium with small β (relatively cold medium) or is resonant and the subsequent physics is close to that of Fermi by including the finite wave cascading that exists in the interstel- acceleration: particles with vk slightly greater than ω/kk suffer a lar medium. Due to the large amplitude of interstellar turbulence, trailing collision with a wave compression and slow down, while there is no doubt that wave cascading exists and will then trans- particles with vk slightly less than ω/kk are stuck by a compres- fer this spectral energy to higher wavenumbers. One property sion and speed up. This process could be called resonant Fermi of the diffusion equation for isotropic turbulence (Eq. (11.3.6), acceleration, but the usual term is transit-time acceleration or Schlickeiser 2002) is that it yields finite positive wave spectral transit-time damping (TTD). The name is given because the res- energy densities at all wavenumbers k, even if the net damping onance condition can be rewritten as λk/vk ≈ t, where t is the rate is negative in some wavenumber intervals. wave period and λk = 2π/kη is the parallel wavelength. That We show then that the inclusion of resonance broadening means a wave and a particle will interact strongly when the par- caused by wave damping in the resonance function guarantees ticle transit time across the wave compression is approximately that the transit-time damping contribution holds at small pitch equal to the period. angle cosines µ ≤ |Va/v|, unlike the case of neglible wave Schlickeiser & Miller (1998) and Schlickeiser & Vainio damping (Vukcevic & Schlickeiser 2007). We expect that TTD (1999) have investigated the quasilinear interactions of charged makes an overwhelming contribution to particle scattering be- particles with Alfven and magnetosonic plasma modes. A cos- cause the cosmic ray particle interacts with the whole wave spec- mic ray particle of given velocity v, Lorentz factor γ = (1 − trum in this interaction. This contrasts gyroresonances that sin- 2 2 −1/2 (v /c )) , pitch angle cosine µ = vk/v, mass m, the speed gle out individual resonant wave numbers (Schlickeiser 2003). of light c, charge qi = e|Zi|Q with Q = sgn(Zi), and gyrofre- However, quantitative analysis of gyroresonance contribution quency Ωc = QΩ with Ω = qiB0/(mcγ) interacts with waves in damped plasma wave turbulence will be studied in a sepa- whose wavenumber k, cosine of propagation angle η = kk/k, and rate, forthcoming investigation. Therefore we only consider the real frequency f obey the resonance condition f (k) = vµkη+nΩc TTD-contribution to particle scattering in the following and as- for entire n ∈ [−∞, ∞]. For slab(η = 1) Alfven waves, only gy- sume n = 0 both in the resonance function and in the calculation roresonant interactions with n = 1 are possible. In contrast, the of the Fokker-Planck coefficients. n = 0 resonance for obliquely propagating fast magnetosonic In Sect. 2, we present the general plasma wave turbulence waves, which do not include gyroresonance interactions (n , 1), and relevant magnetohydrodynamic plasma modes. In Sect. 3, is possible due to their compressive magnetic field component. we calculate the Fokker-Planck coefficient for fast mode waves The n = 0 interactions are referred to as transit-time damp- and discuss relevant domains for application to cosmic ray ing (TTD). Using n = 0 the resonance condition requires the par- positrons. In Sect. 4, we derived the Fokker-Planck coefficient allel cosmic ray velocity vµ = f /(kη) f /k ' VA to be greater than for slow mode waves. The analytical expressions of a mean free the Þfinite phase velocity of magnetosonic waves, which is given path for fast and slow plasma waves and the comparison with dif- by the Alfven speed. TTD interactions therefore only occur for ferent previously reported turbulence models are given in Sect. 5. cosmic ray particles with pitch-angle cosines µ > |VA/v| = . At The summary and conclusion are presented in the last section. these pitch angle cosines, TTD interactions provide most of the particle scattering. Thus, the crucial scattering in small µ is provided for energetic particles by only gyroresonances with slab plasma waves 2. Plasma wave turbulence and oblique magnetosonic waves (Schlickeiser et al. 2010): According to our discussion in the paragraph above and in the 2 two paragraphs after Eq. (1), the relevant mean free path for the 3v Z 1 − µ 3V λ= dµ ' A · (1) positrons with Lorentz factor γ < mp/me = 1836 in small pitch- slab g−ms slab g−ms 8 − Dµµ (µ) + Dµµ (µ) 4[Dµµ (0) + Dµµ (0)] angle cosine becomes

According to Schlickeiser & Miller (1998), the magnetosonic 3VA g−ms slab λ ' · (2) contribution Dµµ (0) ' Dµµ (0) for energetic particles ( 1) 4DTTD(0) is much smaller than the slab contribution for similar turbu- µµ lence power spectra of slab and magnetosonic waves. However, this latter reduction is not justied for cosmic ray positrons with Hereafter, we omit TTD notation and keep only notation for relevant magnetosonic waves. The Fokker-Planck coefficient D Lorentz factors γ < mp/me = 1836. These positrons no LH µµ polarized slab waves to find resonantly interact with, so that is computed by employing the Kubo formula (Kubo 1957), slab Dµµ (0)− > 0. In this case, the contribution from gyroresonant interactions with magnetosonic waves provides small but finite Z ∞ scattering (Schlickeiser et al. 2010). Dµµ = dthµ˙(t)˙µ(0)i. (3) 0 A111, page 2 of 10 M. Vukcevic: The scattering mean free path of cosmic ray particles in isotropic damped plasma wave turbulence

The pitch-angle variationµ ˙(t) is obtained from the where σ(k) is the magnetic helicity and the function g(k) de- Newton-Lorentz equation: termines different turbulence geometries. This was to illustrate p the result, although this model is not in accord with the known Ω 1 − µ2 " q c 2 i iφ c polarization properties of fast-mode waves at oblique angles. µ˙ = 1 − µ δE|| + √ e δBr + iµ δEr B0 v 2 v c − e−iφ δB − iµ δE . (4) 2.1. Relevant magnetohydrodynamic plasma modes l v l It has been already emphasized that there is no TTD for shear In these equations, we use pitch-angle cosine µ, the particle Alfven waves (Teufel et al. 2003) in the case of negligible damp- speed v, the gyrophase φ, the cosmic ray particle gyrofrequency ing, and the gyroresonant interactions provided by shear Alfven in the background field B and the turbulent fields δB and δE , 0 l,r l,r waves is small compared to the same contribution provided by which are related to the left-handed and right-handed polarized fast magnetosonic waves (Vukcevic & Schlickeiser 2007). As field components. The term δE is parallel component relative to || a consequence, we consider only fast and slow magnetosonic the background magnetic field. waves. The simplest method to calculate Dµµ is the application of perturbation theory (Jokipii 1966). In this case, we have applied a quasilinear approximation for a fluctuating electric and 2.1.1. Fast magnetosonic plasma modes magnetic field, a quasistationary turbulence condition, and the existence of a finite correlation time tc. The last two assump- For the fast plasma modes, we use a simplified dispersion tions guarantee a diffusive behavior of transport (Shalchi & relation, Schlickeiser 2004). The assumption of homogeneous turbulence will imply that the turbulence fields at different wavevectors are ωR ' jkVA, (11) uncorrelated. Next, we define the properties of the plasma turbulence that which is relevant at wavenumbers kc k ξkc, where will be considered. We follow the approach for the electromag- k = ω /c is the inverse ion skin length, and ξ = pc p,i netic turbulence that represents the Fourier transforms of the mp/me = 43. magnetic and electric field fluctuations as a superposition of N In the dispersion equation, forward ( j = 1) and backward individual weakly damped plasma modes of frequencies: ( j = −1) moving fast mode waves are described. The associated ω = ω (k) = ω (k) − iγ (k), (5) electric field and magnetic field polarizations are (Dogan et al. j R, j j 2006) where j = 1, ...N, which can have both the real and imaginary parts with |γ j| |ωR, j|, so that δEL = −δER, δEk = 0, δBL = δBR, δBk , 0. (12) N X h j j i −iω t [B(k, t), E(k, t)] = B (k), E (k) e j . (6) 2.1.2. Slow magnetosonic plasma modes j=1 The dispersion relation for slow magnetosonic waves in low-β Damping of the waves is counted with a positive γ j > 0. plasma reads (Dogan et al. 2006) In the case we consider here, the time integration of the ! Fokker-Planck coefficient Dµµ yields the Lorentzian resonance η2β η4β2 function: ω2 ' k2V2 + (13) R A 1 + β (1 + β)3 Z ∞ −i(kkvk+ωR, j+nΩ)u−γ ju R j(γ j) = du e 0 with η = cos θ and β as the ratio of thermal and magnetic pressure. In the last equation, we neglect the second term in brack- γ j(k) = · (7) 2 2 ets in the first approximation, since it is one order smaller than γ j (k) + [kkvk + ωR, j(k) + nΩ)] the first term. The associated electric field and magnetic field The detailed derivation of the Fokker-Planck coefficient for this polarizations are case (vanishing magnetic helicity and isotropic turbulence) is performed in the PhD Thesis of Vukcevic (2007), which con- δEL = −δER, δEk = 0, δBL = δBR, δBk , 0. (14) trasts the case of negligible damping γ− > 0 (Schlickeiser 2001) in which the use of the δ-function representation Consideration of the dispersion relation in high frequencies could change the result, since we expect nondispersive effects γ lim = πδ(ξ), (8) in that range. The polarization of the waves in that domain could γ−>0 γ2 + ξ2 also affect the transport coefficients through the correlation ten- reduces the resonance function (7) to sharp δ-functions, sors. However, a detailed inspection of the dispersion relation in the high-frequency domain is needed to draw a quantitative j R (γ = 0) = πδ(kkvk + ωR, j + nΩ). (9) conclusion, and it will be discussed in a separate investigation. Next, it is necessary to specify the geometry of the plasma wave turbulence itself through the correlation tensors, which will be 2.2. Damping rate adopted throughout this work in the form (Mattheus & Smith 1981) of The damping of magnetosonic waves is caused by both colli- sionless Landau damping and collisional viscous damping and j " # j gi (k) kαkβ kλ by Joule damping and ion-neutral friction. The dominant contri- P (k) = δαβ − + iσ(k)αβλ , (10) αβ k2 k2 k bution is provided by viscous damping with the rate calculated A111, page 3 of 10 A&A 555, A111 (2013) for plasma parameters of the diffuse intercloud medium (Spanier for kmin < k < kmax. & Schlickeiser 2005) The magnetic energy density in wave component j is given by 1 2 2 h 2 −9 2 i γF = βVAτik sin θ + 5 × 10 cos θ 12 Z ∞ 5 2 2 h 2 −9 2 i 2 j = 2.9 × 10 βVAk sin θ + 5 × 10 cos θ (15) (δB j) = dkg (k) (23) 0 6 in terms of the ion-ion collisional time τi = 3.5 × 10 s. Except at very small propagation angles, the second term in Eq. (15) is which implies negligible, and we infer 2 1−q 1−q 2 q−1 5 2 2 2 2 2 gtot = (q − 1)(δB) / k − kmax w (q − 1) (δB) k (24) γF ' 2.9 × 10 βVAk sin θ = αFk sin θ, (16) min min

1 2 where αF = βV τi. 12 A for kmax kmin. With Eqs. (22), (23), (24) the pitch-angle Fokker-Planck F 3. Fast mode waves coeﬃcient Dµµ reads as

With Eqs. (11) and (16), the resonance function (7) for forward Z k Ω2 max and backward moving fast mode waves becomes DF ' q − δB 2kq−1 − µ2 kk−q µµ 2 ( 1)( ) min 1 d 4B k 2 2 0 min j αFk sin θ R (n) = , (17) Z 1 F 2 2 2 2 2 2 (αFk sin θ) + [kvµ cos θ + jVAk + nΩ)] dη RF (0)J1 (W) 1 + η . (25) −1 which describes both gyroresonant (n , 0) and transit-time damping (n = 0) wave-particle interactions. Now, we must approximate the resonance function (Eq. (18)). In The non-vanishing parallel magnetic field component Bk , 0 doing this, we consider two cases: (see Eq. (12)) of fast mode waves allows TTD interactions with n = 0, so that we procede with n = 0. The resonance a) η < ηc; function (17) becomes b) η > ηc, α k2 sin2 θ R j (0) = F F 2 2 2 2 where = VA/v and ηc = /µ. The parameter ηc divides the (αFk sin θ) + [kvµ cos θ + jVAk] integration domain with respect to η, in which either VA, vµη or 2 F F αF(1 − η ) both values are relevant. By using D (−µ) = D (µ) and the = · (18) µµ µµ 2 2 2 p 2 (αFk(1 − η )) + [vµη + jVA] substitution s = RLk 1 − µ and RL = v/|Ω|, we derive Throughout this work, we consider isotropic turbulence g j(k) = j !2 q+1 Z ∞ g (k). Modifications due to different turbulence geometries are F q−1 δB 2 2 −q Dµµ ' αF(q − 1)(kminRL) 1 − µ √ dss possible and will be the subject of further analysis (in particular 2 B0 kminRL 1−µ anisotropic turbulence).  Z min(1,/µ) ! For energetic cosmic ray particles with v VA, the  q 1 ×  dη 1 − η4 J2 s 1 − η2 pitch-angle Fokker-Planck coefficient then simplifies as  1 α2 (1−η2)2 s2  0 F V2 R2 (1−µ2) + A Ω2 X Z ∞ Z 1 L DF ' 1 − µ2 dk dη R j (0)g j(k)J2(W) 1 + η2 , (19)  µµ 2 F 1 Z 1 !  4B −∞ −1 q 1  0 j=±1 + dη 1 − η4 J2 s 1 − η2  · (26) 1 α2 (1−η2)2 s2  min(1,/µ) F 2  where J1(W) is a Bessel function with the argument W = 2 2 + (vµη) RL(1−µ ) v p 2 p 2 |Ω| · k⊥ 1 − µ = RL · k⊥ 1 − µ that involves the cosmic ray Larmor radius RL = v/|Ω|. We further simplify Eq. (19) by assuming equal intensity of 3.1. High values of µ > forward and backward waves (a vanishing cross helicity of each For large pitch-angles µ > we obtain plasma mode):

!2 q+3 + − 1 F (q − 1) q−1 δB 2 2 g (k) = g (k) = gtot(k), (20) Dµµ(µ > ) ' (kminRL) 1 − µ 2 αF B0 Z ∞ " Z /µ q ! which reads as −q 4 2 2 × √ dss dη 1 − η J1 s 1 − η 2 Z ∞ Z 1 2 Ω X kminRL 1−µ 0 DF ' 1 − µ2 dk dη R j (0)g j (k)J2(W) 1 + η2 . (21) µµ 2 F tot 1 Z 1 ! 8B −∞ −1 1 q 0 j=±1 + + dη 1 − η4 J2 s 1 − η2 R2 (1−µ2)V2 1 22 2 L A /µ To illustrate our results, we adopt a Kolmogorov-type power 1 − η s + 2 αF law dependence on g j(k) above and below some minimum and # 1 maximum wavenumber kmin and kmax, respectively × · (27) R2 (1−µ2)(vµη)2 − 22 2 L −q 1 η s + α2 gtot(k) = gtotk (22) F

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3.2. Small values µ < which can be simpliﬁed if we consider the energetic cosmic ray particles v VA, or namely, the last two terms, to be small This case is important when treating damped waves. For the (order of and 2) and neglected. Then, we obtain the same small pitch-angles µ < , we obtain expression as in the case for fast mode waves:

!2 q+1 δB Z k F q−1 2 2 Ω2 max Dµµ(µ < ) ' αF(q − 1)(kminRL) 1 − µ S 2 q−1 2 −q B Dµµ ' (q − 1)(δB) k 1 − µ dkk 0 B2 min  4 0 kmin ∞ ! Z Z 1 q Z 1 −q  4 2 2 2 2 × √ dss  dη 1 − η J1 s 1 − η × dη RS(0)J (W) 1 + η . (33) 2  1 kminRL 1−µ  0 −1   Next, we have to approximate the resonance function for slow 1  mode waves. As in the fast mode case, there are two cases: ×  · (28) α2 (1−η2)2 s2  F 2  S 2 2 + VA RL(1−µ ) a) η < ηc ;

We have already discussed that inclusion of resonance broad- b) η > ηS, ening due to wave damping in the resonance function guaran- c tees dominance of transit-time damping. The main contribution where of wave damping comes exactly in the region |µ| < that is relevant for deriving the spatial diffusion coefficient and related S β ηc = · (34) mean free path, both of which are given by the average over µ of µ 1 + β the inverse of Dµµ. Therefore, we can further consider only the Note that ηS < ηF. Using DS (−µ) = DS (µ) and the substitution case Dµµ(µ = 0), which simplifies the analysis enormously and c c µµ µµ p 2 reads as s = RLk 1 − µ , we derive 2 2 2 − ! Z ∞ 2 F Ω (q 1)RL q−1 δB −q ! q+1 ' S q−1 δB 2 2 Dµµ(µ = 0) (kminRL) dss D ' α (q − 1)(k R ) 1 − µ 4αF B0 k R µµ F min L min L B0 Z 1 q ! β 1  1, 4 2 2 Z ∞ Z min µ 1+β × dη 1 − η J s 1 − η 2 2 · (29)  1 V R −q  4 0 (1 − η2)2 s2 + A L × RL√ dss  dη 1 − η α2 1−µ2  F kmin  0 ! In the last equation, s = kRL, and each term under integration is q 1 Z 1 dimensionless. × J2 s − η2 1 1 2 + α2 (1−η2) s2 β β F 2 2 min 1, µ 1+β 2 2 + VAη 1+β RL(1−µ )  4. Slow mode waves ! q 1  ×dη 1−η4 J2 s 1−η2 , (35) With Eqs. (13) and (16), the resonance function (7) for slow 1 α2 (1−η2)2 s2  F 2 mode waves becomes 2 2 +(vµη) RL(1−µ ) 2.9×105βV2 k2 sin2θ R j (n) = A · (30) where = VA/v. S 2 q 2 × 5 2 2 2 β 2.9 10 βVAk sin θ + kvµ cos θ+ jVAkη 1+β +nΩ) 4.1. High values of µ > The non-vanishing parallel magnetic field component Bk , 0 (see Eq. (14)) of slow mode waves allows TTD interactions For large pitch-angles µ > we obtain with n = 0, so that we procede with n = 0. The resonance !2 q+3 function (30) becomes S (q − 1) q−1 δB 2 2 Dµµ(µ > ) ' (kminRL) 1 − µ 5 2 2 2 αF B0 j 2.9 × 10 βVAk sin θ Z q RS (0) = 2∞ −q 2 q 2 × RL 1 − µ dss 5 2 2 2 β 2.9 × 10 βV k sin θ + kvµ cos θ + jVAkη kmin A 1+β  Z β ! α sin2 θ  µ(1+β) q = F , (31) × dη 1 − η4 J2 s 1 − η2 q 2  1 2 2 β  0 αFk sin θ + vµ cos θ + jVAη 1+β 1 where α is the same as in the fast mode case. × β F R2 (1−µ2)V2 η2 Following the same procedure for fast mode waves in Sect. 3, − 2 2 2 L A 1+β (1 η ) s + α2 we assume (20), (22), (23), and imply (24). We derive F Z 1 q ! ∞ 4 2 2 2 Z Z 1 + dη 1 − η J1 s 1 − η S Ω 2 X j j 2 β D ' − µ k η R g k J W µ(1+β) µµ 2 1 d d S (0) ( ) 1 ( ) 4B −∞ −1  0 j=±1  s 1   !  ×  · (36)  β β  R2 (1−µ2)(vµη)2  × 1 + η2 1 + µ22η2 − 4µ jη  , (32) 2 2 2 L    (1 − η ) s + 2  1 + β 1 + β αF

A111, page 5 of 10 A&A 555, A111 (2013)

4.2. Low values of µ < For kminRL 1: This case is treated in detail in Appendix A, where we derive This case is important for treating damped waves for the same reason as discussed for fast mode waves (Sect. 3). For the small 2 10−14 G(T 1) = T −(q+2), (42) pitch-angles µ < , we obtain 5 q !2 2 δB q+1 0F 15αF q B0 14 3 DS (µ < ) ' α (q − 1)(k R )q−1 1 − µ2 2 λ (T 1) = 10 T . (43) µµ F min L VA q − 1 δB B0 Z q 0 ∼ 3 2∞ −q At relativistic rigidities, we find that λ T . As we consider × RL 1−µ ds s certain positron energies 1–100 GeV, this case is not relevant. kmin   For kminRL 1: ! Z 1 q 1  This case is treated in detail in Appendix A, where we derive × dη 1−η4 J2 s 1−η2  , (37)  1 α2 (1−η2)2 s2 β   0 F V2 η2  1 1 1−q R2 (1−µ2) + A 1+β G(T 1) = T , (44) L 3 q − 1 or for µ = 0, we read 9α B 2 λF0(T 1) = F 0 · (45) 2 − 2 !2 Z VA δB S Ω (q 1)RL q−1 δB ∞ −q Dµµ(µ = 0) ' (kminRL) RL dss In this energy limit, the mean free path is constant with respect 4αF B0 kmin   to T, which is relevant for considered positron energies.   Z 1 q 1   dη (1 − η4)J2(s 1 − η2)  · (38)  1 2 β 2   V η2 R  5.2. Slow mode waves  0 − 2 2 2 A 1+β L  (1 η ) s + α2 F In this subsection, we calculate the mean free path which is con- In the last equation, s = kRL. nected with the spatial diffusion coefficient through 3κS 3v Z 1 (1 − µ2)2 λS µ · = = d S (46) 5. Cosmic ray mean free path v 4 0 Dµµ 5.1. Fast mode waves For the case for which we are interested, we can write In this section, we calculate the mean free path, which is con- 3κS 3v 1 Z 3 V λS0 = = dµ = A nected with the spatial diffusion coefficient through S v 4 Dµµ(µ = 0) 0 4 Dµµ(µ = 0) Z 1 2 2 2 3κ 3v (1 − µ ) αF 1−q B0 1 λ = = dµ · (39) = 3 (kminRL) , (47) v 4 0 Dµµ (q − 1)VA δB G For the case in which we are interested, we can write where Z ∞ Z 1 q ! Z −q 4 2 3κ 3v 1 3 VA G = dss dη 1 − η J s 1 − η2 λ0F = = dµ = 1 F kminRL 0 v 4 Dµµ(µ = 0) 0 4 Dµµ(µ = 0) 1 α B 2 1 × · (48) F 1−q 0 V2 η2 β R2 = 3 (kminRL) , (40) − 2 2 2 A 1+β L (q − 1)VA δB G (1 η ) s + 2 αF where We consider two limits: kminRL 1, and kminRL 1, where k R = T = E and is normalized with respect to E . Z ∞ Z 1 q ! min L c −q 4 2 2 For k R 1: G = dss dη 1 − η J1 s 1 − η min L kminRL 0 This case is treated in detail in Appendix B, where we derive 1 √ × · (41) 2 |b/(b − 1)| + | log(1 − b)| V2 R2 −(q+2) 2 2 2 A L G(T 1) = T , (49) (1 − η ) s + 2 π qp2 αF 2 0S 3αF q B0 Now, we consider two limits: kminRL 1, and kminRL 1, λ (T 1)= √ − where kminRL = T = E and is normalized with respect to 2VA (q 1) δB kc 2 2 Ec = Tc = mec . Expressing kmin = Lmax/2π in the terms πp kmin × T 3, (50) of the longest wavelength of isotropic fast mode waves, kc = |b/(b − 1)| + | log(1 − b)| Ω0,e/vA = ωp,e/c and for following plasma conditions (vA = where T = k R and b 1. 33.5 km , B0 = 1, Beck 2007), T = 10.7n1/2( Lmax ) × 109 MV). min L s δB c e 10 pc At relativistic rigidities, we find that λ0S ∼ T 3. For these values, the particle mean free path is measured by For kminRL 1: 9αF B0 2 λ1 = ( ) . We have used interplanetary plasma conditions VA δB This case is treated in detail in Appendix B, where we derive that have been used in both models (Schlickeiser et al. 2010; 1 1 Teufel & Schlickeiser 2002) for comparison. We note that for G(T 1) = T 1−q, (51) 3 q − 1 interstellar medium plasma conditions, λ1 is one or two orders 2 of magnitude larger than for interplanetary conditions, which is 0S 9αF B0 also a reasonable value. λ (T 1) = · (52) VA δB A111, page 6 of 10 M. Vukcevic: The scattering mean free path of cosmic ray particles in isotropic damped plasma wave turbulence

Schlickeiser, Lazar, Vukcevic (2010) In this energy domain, the mean free path is constant with re- Teufel, Schlickeiser (2002) spect to T, which is important for cosmic ray positrons with Damped plasma wave turbulence 103 Lorentz factor γ < mp/me = 1836. Although the mean free path for slow magnetosonic waves has the same expression as in the

2 case for fast waves for energies less than Tc, the low energy limit 10 here is caused by plasma-β (Appendix B). However, it is always less than T and does not violate previous conditions. 1 c 10 The turbulence model considered in this paper ensures that all energy particles below the Tc value are scattered by the 100 proposed resonant interaction. Positron Mean Free Path Considering dynamical magnetic slab turbulence and random sweeping slab turbulence (Teufel & Schlickeiser 2002), 10-1 101 102 103 104 105 106 107 there is a sharp cutoff of the turbulence power spectrum at kmin. The consequence is that the mean free path for all particles with Particle Rigidity (Energy) [MV] energies higher than 1010 eV with typical parameters rapidly Fig. 1. The positron mean free path. The black line represents the mean grows and becomes larger than the size of ambient interstellar free path obtained using the slab turbulence model and gyroresonant in- medium. For the slab plasma waves model (Schlickeiser et al. teractions, which exhibit a cutoff at particle energies less than 2×108 eV. 2010) at rigidities less than 2 × 108 eV, the positron mean free The red line is the mean free path obtained using the dynamical magnetic slab turbulence, or the random sweeping slab turbulence model, path becomes infinitely large. It is because these positrons find 10 no LH polarized slab waves with which to resonantly interact. In which exhibit a sharp cutoff at particle energies of 10 eV. The blue line the turbulence model proposed in this paper, the positron mean is the mean free path obtained using isotropic magnetosonic damped 5 plasma wave turbulence, which remains constant for particle energies free path does not depend on energy, in domain from 10 eV to from 105 eV to 1012 eV. All mean free paths are normalized with re- 1012 eV. spect to the λ1 value. The comparison of our result with previously reported turbulence models are given in Fig. 1, where the mean free path inclusion of nonlinear effects to the slab model would have the 13 is in units of λ1 = 0.2β AU (AU = 1.5 × 10 cm). To inter- cutoff at the same wavenumbers as in the random sweeping lin- pret the MeV positrons propagation and explain their diffusion, ear model. However, it would be useful to compare timescales of the model of the dynamical turbulence can be applied. However, nonlinear effects and damping effects within the same turbulence this model fails to explain transport mechanisms for energetic model, in order to make a general conclusion on the importance positrons with energies of 100 GeV. In contrast, the slab gyrores- of each of them. onat plasma wave turbulence model is able to hold on energetic GeV positrons but fails in explaining the transport properties 6. Summary and conclusion of MeV positrons. The isotropic damped magnetosonic plasma wave turbulence model, which is proposed in this paper, seems We have investigated the implications of isotropically distributed to be more plausible, since it covers energies in both domains. interstellar damped plasma waves on the scattering mean free This model could be used in modeling turbulence that scatters path on the cosmic ray positrons with a Lorentz factor γ < positrons, resolving the problems stated by the results of either mp/me = 1836. We show that inclusion of resonance broaden- the INTEGRAL or PAMELA experiments. ing due to wave damping in the resonance function guarantees The inclusion of viscous damping implies either steep tur- that dominance of transit-time damping also holds for cosmic bulence spectra (q > 2.4) for isotropic turbulence with fixed ray particles at small pitch angle cosines µ ≤ |Va/v|, unlike the values of kmax and kmin, or the existence of high wavenum- case of negligible wave damping. ber cutoff for isotropic turbulence with a spectral index q = For small rigidities, or consequently low energies T < Tc, 5/3 (Spanier & Schlickeiser 2005). To justify the use of kmax the mean free path is constant with respect to energy. The mean and kmin in our model, it is necessary to assume turbulence free path at high energies T > Tc approaches a much steeper with a spectral index that is not less than 2.4. This value is dependence, namely λ ∼ T 3 for both fast and slow waves. higher than the Kolmogorov value of 5/3 (Kolmogorov 1941) It is difficult to draw any general conclusion on the ratio or Kraichan-Iroshnikov value of 3/2 (Kraichnan 1965), but both of fast and slow modes mean free path, since there is an in- theories consider an inertial range, where damping effects have tegral dependence on the pitch angle and the plasma-beta pa- been neglected. Interstellar medium has strong damping fea- rameter are mixed in the latter case. For cold plasma, the mean tures, which does not mean that there is no inertial range. It could free path of slow mode merges to the mean free path of the fast be that there is an intermediate regime between the inertial and one, as expected. However, this turbulence model is able to en- dissipation range. Thus, the request for steep turbulence spectra sure scattering of the cosmic ray positrons with Lorentz factor is not in contradiction to neither Kolmogorov theory, or our cal- γ < mp/me = 1836 via resonant interaction. culation, in which there is only a restriction on q to be greater than 1. Appendix A: Evaluation of the function G for fast The broadening of the resonance function could also be in- sured by nonlinear treatment of the specific turbulence model. mode waves Nonlinear theory (second order theory of Shalchi 2005) applied The task is to calculate the function G (41): to the magnetostatic slab model (Shalchi et al. 2009) would lead Z ∞ Z 1 q ! to a broadened resonance function that is similar to the one ob- −q 4 2 2 G = dss dη (1 − η )J1 s 1 − η tained for the random sweeping model, which is linear (Teufel kminRL 0 & Schlickeiser 2002). We have already mentioned that this lin- 1 × · (A.1) ear model could be used to explain MeV positrons but fails for V2 R2 2 2 2 A L GeV positrons. Thus, we expect, at least mathematically, that (1 − η ) s + 2 αF A111, page 7 of 10 A&A 555, A111 (2013)

V2 R2 A L 14 − First, it is possible to evaluate the term 2 ∼ 10 for used where we substitute m = 1 η. αF plasma parameters (Sect. 5). We then evaluate G in two energy Z ∞ Z 1 − T 2 2xT 2m 1 10 14 limits. I xx−q+2 m m T −1, 3 = d d 4 14 == 4 1 0 10 12 q (A.9) Case G(kminRL 1) For energies T 1 we substitute s = xT. Then, (A.1) reads as where we have used the same substitution as in the previous case 14 2 2 and where 10 4m x . We note that I2 I3. Z ∞ Z 1 q ! G = dxx−qT −(1+q) dη (1 − η4)J2 xT 1 − η2 1 1 Z ∞ Z 1 (1−η4) 1 5 10−14 1 1 0 I xx−(q+1) η · Z ∞ 1 = d d p 14 = 1 πT 1 0 1−η2 10 16 q T × T −(1+q) xx−q 2 2 2 14 = d (1 − η ) x + 10 1 (A.10) Z 1− 1   2xT  1 1 ×  η − η4  Combining all these three integrals, we obtain  d (1 ) p 2 2 2 14  0  2 (1 − η ) x + 10 πxT 1 − η −14 2 10 −(q+2) Z 1 1 G(T 1) = T . (A.11) + dη 1 − η4 x2T 2 1 − η2 5 q 1 4 1− 2xT ! 1 Case G(k R 1) × , (A.2) min L 22 2 14 1 − η x + 10 For energies T 1, we use the approximation for Bessel where we use the approximations of Bessel functions for large function (A.5). Then, (A.1) reads as and small arguments (Abramowitz & Stegun 1972): 2 2 1 Z 1 Z 1 1−η s r ! G(T 1) = dss−q dη 1−η4 2 (2ν + 1)π 2 V2 R2 ≈ − 4 kminRL 0 2 2 A L Jν(z 1) cos νz , (A.3) 1−η s + 2 πνz 4 αF 1 Z 1 Z 1 M2 implying = dss−q dη 1 + η2 − !! 4 4 1 q T 0 2 2 Z 1 Z 1 J1 (z 1) = p 1 − sin 2xT 1 − η πxT 1 − η2 × dss−q dη 1 + η2 1 T 0 ' p (A.4) 1 πxT 1 − η2 × = I5 − I4, (A.12) 1 − η22 s2 + M2 (1/ξ sin ξ/ξ) for the argument, z = xT 1, and 2 2 2 VARL ν where M = 2 . We then evaluate each integral: (z/2) αF Jν(z 1) ≈ , (A.5) Γ(ν + 1) Z 1 Z 1 1 −q 2 1 1−q implying I5 = dss dη 1 + η = T − 1 4 T 0 3(q − 1) 1 1 J2(z 1) = x2T 2 1 − η2 (A.6) ' T 1−q, (A.13) 1 4 3(q − 1) for the argument z = xT 1. We then obtain since T 1, and q > 1. 2 Z ∞ Z 1 (1+q) T −q+2 4 2 Z 1 Z 1 T G(T 1) = dxx dη 1 − η M −q 2 1 1 I ss η η 4 1 1− 4= d d (1 + ) 2 2 2 2 2xT 4 T 0 (1 − η ) s + M (1 − η2) Z 1 1 −q (1 − η2)2 x2 + 1014 = dss I6, (A.14) 4 T 1 Z ∞ Z 1 (1 − η4) 1 xx−(q+1) η where + d d p 2 2 2 14 πT 1 0 1 − η2 (1 − η ) x + 10 Z 1 Z ∞ Z 1 4 2 1 1 (1 − η ) 1 I6 = dη 1 + η · (A.15) − xx−(q+1) η 22 2 2 d d p 2 2 2 14 = 0 1 − η s /M + 1 πT 1 2 (1 − η ) x + 10 1 1− 2xT 1 − η I3 + I1 − I2. (A.7) The exact solution of integral I6 reads as ! Next, we simplify further, namely keeping only terms with re-  (−1)3/4   − 1/4  spect to 1 − η of the lowest order, and we evaluate each integral: (−2i + n) arctan (√1) i(2i + n) arctan √   −i+n −i n  − 1/4  +  Z ∞ Z 1 ( 1)  √ + √  2xT   1 −(q+1) 4m 1  −i + n −i + n  I2 = dxx dm √   πT 4m2 x2 + 1014 1 0 2m √ , −14 2 2 10 − 5 2n 1 + n = T 2 , (A.8) 3 π(q + 3/2) (A.16) A111, page 8 of 10 M. Vukcevic: The scattering mean free path of cosmic ray particles in isotropic damped plasma wave turbulence

2 2 2 where n = M /s . We can evaluate I6 by (1/ξ sin ξ/ξ) for the argument, z = xT 1, and 1 (z/2)ν I6 ≈ , (A.17) J (z 1) ≈ , (B.5) 2n3 ν Γ(ν + 1) which for I4 gives implying

1 1 − I ≈ − T (4 q) · 2 1 2 2 2 4 3 1 (A.18) J (z 1) = x T 1 − η (B.6) 8M 4 − q 1 4

To compare I4 and I5 and calculate G(T 1), we have to con- for the argument z = xT 1. We then obtain sider two cases, which are (1 < q < 4) and steep (4 < q < 6) 2 Z ∞ Z 1 turbulence spectrum. (1+q) T −q+2 4 1 1 q−4 T G(T 1) = dxx dη 1 − η For (1 < q < 4) the integral I4 ∼ 4−q , while I4 ∼ q−4 T for 4 1 1 1− 2xT 3 (4 < q < 6). However, M exists in both cases in the denomi- (1 − η2) 1 Z ∞ nator of I , which implies that I I . According to Eq. (A.12) × xx−(q+1) 4 4 5 2 2 2 2 2 + d then: (1 − η ) x + p η πT 1 Z 1 4 Z ∞ (1 − η ) 1 1 −(q+1) 1 1 1−q dη p − dxx G(T 1) = T . (A.19) 2 (1 − η2)2 x2 + p2η2 πT 3 q − 1 0 1 − η 1 Z 1 (1 − η4) 1 η d p 2 2 2 2 2 1 2 (1 − η ) x + p η Appendix B: Evaluation of the function G 1− 2xT 1 − η for slow mode waves = I3 + I1 − I2. (B.7) The task is to calculate the function G (48), Next, we evaluate each integral in turn as in the case of fast mode waves. Z ∞ Z 1 q ! −q 4 2 2 G = dss dη 1 − η J s 1 − η Z ∞ Z 1 1 2xT m kminRL 0 1 −(q+1) 4 1 I2= dxx dm √ 1 πT 4m2 x2 + p2(1 − m)2 · 1 0 2m β (B.1) Z ∞ V2 η2 R2 1 − 22 2 A 1+β L −(q+1) 0 1 η s + α2 = dxx I2(m, x), (B.8) F πT 1 and evaluate it in two energy limits. where we substitute m = 1 − η.

Z ∞ Z 1 T 2 2xT 8m2 Case G(kminRL 1) I = dxx−q+2 dm 3 4 p2(1 − m)2 + 4m2 x2 For energies T 1 we substitute s = xT. Then, (B.1) reads as 1 0 T 2 Z ∞ = dxx−q+2I0 (m, x), (B.9) Z ∞ Z 1 q ! 3 −q −(1+q) 4 2 2 4 1 G = dxx T dη 1 − η J1 xT 1 − η 1 0 where we have used the same substitution as in the previous case. 1 × Z ∞ Z 1 2 1 4m 1 1 − η2 x2 + p2η2 I xx−(q+1) m √ 1= d d 2 2 2 2  1 πT 1 0 p (1 − m) + 4m x Z ∞ Z 1− 2xT 2m −(1+q) −q  4 1 = T dxx  dη (1 − η ) 1 Z ∞  p 2 −(q+1) 0 1 0 πxT 1 − η = dxx I1(m, x). (B.10) πT 1 1 × (1−η2)2 x2 + p2η2 To compare functions under integration with respect to m, we Z 1  write out the following integrals 4 1 2 2 2 1  + dη 1−η x T 1−η  , 1 1 22 2 2 2  Z 2xT 1− 4 1−η x + p η 0 4m 1 2xT I m, x m √ 2( )= d 2 2 2 2 (B.11) (B.2) 0 2m 4m x + p (1 − m) √ where p2 = (V2 R2 β)/(α2(1 + β)). We have used Z 1 A L F 2 2 2xT = dm f , (B.12) r ! p2 1 2 (2ν + 1)π 0 Jν(z 1) ≈ cos νz − , (B.3) πνz 4 Z 1 Z 1 2xT 8m2 8 2xT I0 m, x m m f , 3( )= d 2 2 2 2 = 2 d 2 implying 0 p (1 − m) + 4m x p 0 !! (B.13) 1 q J2(z 1) = 1 − sin 2xT 1 − η2 √ 1 p Z 1 Z 1 πxT 1 − η2 4m 1 2 2 I0 m √ m f , 1(m,x)= d 2 2 2 2 = 2 d 1 1 0 2m p (1 − m) + 4m x p 0 ' p (B.4) πxT 1 − η2 (B.14) A111, page 9 of 10 A&A 555, A111 (2013)

m m2 2 2 2 where f1 = (1−m)2 and f2 = (1−m)2 . We have approximated the de- and w = s /p . Here, we compare functions in integrals I5 with 2 2 4x2 respect to η, namely f1 = (1 + η ), and in I6 namely f2 = (1 + nominator as (1−m) , as long as 2 1 holds (which is ensured 2 p η2) η . by consideration of T 1). Analyzing f1 and f2 in given inter- ((1−η2)2w2)+η2 2 vals of integration we deduce that I2 < I3 < I1. However, this We ﬁnd that I5 I6 for w 1 over the interval [0, 1]. For −14 2 2 −14 case is not of particular interest, so we make a rough estimation 10 < w < 1, we still have I5 > I4. Only at w 6 10 , on G. I5 < I6 (low energy value is caused by this limitation; namely, 0 Integral I1 diverges for m = 1, so we integrate to m = b < 1. we consider T 1). However, this treatment is valid as long as Combining all, we obtain β T > 1+β , which is always less than 1. √ As long as I dominates, we obtain 2 |b/(b − 1)| + | log (1 − b)| 5 G(T 1) = T −(q+2). (B.15) 2 1 1 π qp G(T 1) = T 1−q. (B.20) 3 q − 1

Case G(kminRL 1) References For energies T 1, we use the approximation for Bessel function (B.5). Then, (B.1) reads as Abdo, A. A., Ackermann, M., Ajello, M., et al. 2009, Phys. Rew. Lett. 102, 1101 Abramowitz, M., & Stegun, I. A. 1972, Handbook of Mathematical functions 1 Z 1 Z 1 1966 G(T 1) = dss−q dη 1 − η4 Achterberg, A. 1979, A&A, 76, 276 4 kminRL 0 Adriani, O., Barbarino, G. C., Bazilevskaya, G. A., et al. 2009, Nature, 458, 607 (1 − η2)s2 Agular, M., et al. 2013, Phys. Rew. Lett., 110, 141102 × Arvanitaki, A., Dimopoulos, S., Dubovsky, S., et al. 2009, Phys. Rew. D, 80, V2 R2 η2β 2 2 2 A L 55011 (1 − η ) s + 2 αF(1+β) Beck, R. 2007, EAS Pub. Ser., 23, 19 Bieber, J. W., Wanner, W., & Matthaeus, W. M. 1996, J. Geophys. Res., 101, 1 Z 1 Z 1 p2 = dss−q dη (1 + η2) − 2511 4 T 0 4 Boehm, C., Hooper, D., Silk, J., et al. 2004, Phys. Rev. Lett., 92, 1301 Z 1 Z 1 Clayton, D. D. 1973, Nature Phys. Sci., 244, 137 −q 2 Dogan, A., Spanier, F., Vainio, R., & Schlickeiser, R. 2006, J. Plasma Phys., 72, dss dη 1 + η 419 T 0 η Fedorenko, V. N. 1992, Sov. Sci. Rev. E, 8, 4 × Guessoum, N., Jean, P., & Prantzos, N. 2006, A&A, 457, 753 (1 − η2)2 s2 + p2η2 Higdon, J. C., Lingenfelter, R. E., & Rothschild, R. E. 2009, ApJ, 698, 350 = I − I , (B.16) Hooper, D., Stebbins, A., & Zurek, K. N. 2009, PRD, 79, 103513 5 4 Jokippi, J. R. 1966, ApJ 146, 480 V2 R2 β Knodlseder,¨ J., Jean, P., Lonjou, V., et al. 2005, A&A, 441, 513 p2 A L where = α2 (1+β) . We then evaluate each integral. Kolmogorov, A. N. 1941, Dokl. Akad. Nauk. SSSR, 30, 301 F Kraichnan, R. 1965, Phys. Fluids, 8, 1835 Z 1 Z 1 Kubo, R. 1957, J. Phys. Soc. Jpn. 12, 570 1 −q 2 1 1−q Matthaeus, W. H., & Smith, C. W. 1981, Phys. Rev. A 24, 2135 I5 = dss dη (1 + η ) = (T − 1) 4 T 0 3(q − 1) Milne, P. A., Kurfess, J. D., Kinzer, R. L., & Leising, M. D. 2002, New Astron. 1 Rev., 46, 553 ' T 1−q, (B.17) Palmer, I. D. 1982, Rew. Geophys. Space Phys., 20, 335 3(q − 1) Pohl, M., & Eichler, D. 2010, ApJ, 712, L53 Ramaty, R., & Lingenfelter, R. E. 1979, Nature, 278, 127 since T 1. Ramaty, R., Stecker, F. W., & Misra, D. 1970, J. Geophys. Res., 75, 1141 Schlickeiser, R. 2002, Cosmic Ray Astrophysics (Berlin: Springer) To estimate I4, we write Schlickeiser, R. 2003, Lect. Not. Phys., 612, 230 p2 Z 1 Z 1 η2 Schlickeiser, R., & Miller, J. A. 1998, ApJ 492, 352 I ss−q η η2 Schlickeiser, R., & Vainio, R. 1999, Astrophys. Space Sci., 264, 457 4= d d 1 + 2 2 2 2 2 4 T 0 (1 − η ) s + p η Schlickeiser, R., Lazar, M., & Vukcevic, M. 2010, ApJ, 719, 1497 Z 1 Shalchi, A. 2005, Phys. Plasmas, 12, 052905 1 −q Shalchi, A., & Schlickeiser, R. 2004, A&A, 420, 821 = dss I6, (B.18) ˇ 4 T Shalchi, A., Skoda, T., Tautz, R. C., & Schlickeiser, R. 2009, Phys. Rev. D, 80, 023012 where Spanier, F., & Schlickeiser, R. 2005, A&A, 436, 1 Sturrock, P. A. 1971, ApJ, 164, 529 Z 1 η2 Teufel, A., & Schlickeiser, R. 2002, A&A, 393, 703 I η η2 6 = d (1 + ) 2 2 2 2 (B.19) Teufel, A., Lerche, I., & Schlickeiser, R. 2003, A&A 397, 776 0 ((1 − η ) w ) + η Vukcevic, M., & Schlickeiser, R. 2007, A&A, 467, 15

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