Non-linear Cosmic Ray propagation close to the acceleration site

Non-linear Cosmic Ray propagation close to the acceleration site The Hebrew University of Jerusalem, Jerusalem, 91904, Israel E-mail: lara.nava@mail.huji.ac.il Stefano Gabici Astroparticule et Cosmologie (APC), CNRS, Université Paris 7 Denis Dierot, Paris, France E-mail: stefano.gabici@apc.univ-paris7.fr Alexanre Marcowith Laboratoire Univers et particules e Montpellier, Université Montpellier/CNRS, 34095 Montpellier, France E-mail: alexanre.marcowith@umontpellier.fr Giovanni Morlino INFN Gran Sasso Science Institute, viale F. Crispi 7, 67100 L Aquila, Italy E-mail: giovanni.morlino@gssi.infn.it Vlaimir Ptuskin Pushkov Institute of Terrestrial Magnetism, Ionosphere an Raio Wave Propagation of the Russian Acaemy of Sciences, Troitsk, Moscow, 142190 Russia E-mail: vptuskin@izmiran.ru Recent avances on gamma-ray observations from SuperNova Remnants an Molecular Clous offer the possibility to stuy in etail the properties of the propagation of escaping Cosmic Rays (CR). However, a complete theory for CR transport outsie the acceleration site has not been evelope yet. Two physical processes are thought to be relevant to regulate the transport: the growth of waves cause by streaming instability, an possible wave amping mechanisms that reuce the growth of the turbulence. Only a few attempts have been mae so far to incorporate these mechanisms in the theory of CR iffusion. In this work we present recent avances in this subject. In particular, we show results obtaine by solving the couple equations for the iffusion of CRs an the evolution of Alfvén waves. We iscuss the importance of streaming instabilities an wave amping in ifferent ISM phases. The 34th International Cosmic Ray Conference, 30 July- 6 August, 2015 The Hague, The Netherlans Speaker. c Copyright owne by the author(s) uner the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/

1. Introuction We stuy the non-linear iffusion of a population of Cosmic Rays (CRs) after they escape from the acceleration site (for example, a supernova remnant shock). As a zero orer approximation the problem can be escribe by an impulsive injection of CRs of a given energy at a given location in the Galaxy. We stuy the transport of CRs taking into account the growth of Alfvén waves cause by particle-wave interactions, an their back-reaction on the CR iffusion. We also consier possible linear wave amping mechanisms that limit the growth of the waves. The problem is escribe by two ifferential equations that govern the evolution of the CR pressure an of the wave energy ensity as a function of time an istance from the acceleration site. We present numerical solutions of these two couple equations for ifferent values of the physical parameters involve in the problem. 2. The moel In our moel the transport of CRs is assume to be regulate by the resonant scattering off Alfvén waves, i.e. a CR of energy E resonates with waves of wave number k= 1/r L (E) where r L is the particle Larmor raius (see e.g. [1]). The (normalize) energy ensity I(k) of Alfvén waves is efine as: δb 2 8π = B2 0 I(k)ln k, (2.1) 8π where B 0 is the ambient magnetic fiel an δb the amplitue of the magnetic fiel fluctuations. Accoring to quasi linear theory, CRs iffuse along the magnetic fiel lines with a iffusion coefficient equal to [2]: D= 4 c r L(E) 3π I(k) = D B(E), (2.2) I(k) which can be expresse as the ratio between the Bohm iffusion coefficient D B (E) an the energy ensity of resonant waves I(k = 1/r L ). Formally, quasi linear theory is vali for δb/b 0 1. In this limit the iffusion perpenicular to the fiel lines can be neglecte, being suppresse by a factor of(δb k /B 0 ) 4 I(k) 2 with respect to the one parallel to B 0 (see e.g. [3, 4]). This implies that uner the conition that δb/b 0 remains small, the problem is one imensional. We consier the streaming of CR to be the main source of Alfvénic turbulence. For CRs that stream along the irection of B 0 (that we consier aligne along the coorinate z) the growth rate of Alfvén waves Γ CR is proportional to the prouct between the Alfvén spee V A an the graient of the pressure of resonant CRs, an can be efine as (see e.g. [3, 5]): 2Γ CR I = V A z. (2.3) The sign inicates that only waves traveling in the irection of the streaming are excite. For imensional consistency, the pressure of CRs P CR has to be normalize to the magnetic energy ensity B 2 0 /8π. On scales smaller than the magnetic fiel coherence length, a flux tube characterize by a constant magnetic fiel strength B 0 an irecte along the z axis can be consiere. Two couple equations (for the evolution of CRs an waves along the flux tube) must be solve: the iffusion 2

coefficient of CRs of energy E (Eq. 2.2), inee, epens on the energy ensity of resonant waves I(k), an in turn the growth rate of these waves (Eq. 2.3) epens on the graient of the pressure of resonant CRs. The two couple equations then rea: t +V A z = z ( DB I ) z (2.4) I t +V I A z = V A z 2Γ I+ Q (2.5) where the left sie in both expressions is the time erivative compute along the characteristic of excite waves: t = t +V A z. (2.6) The avective terms V A / z an V A I/ z are neglecte in the following since we verifie that they play little role in the situation uner examination. The last two terms in Eq. 2.5 account for possible mechanisms of wave amping, operating at a rate Γ, an for the injection Q of turbulence from an external source (i.e. other than CR streaming). In this work we consier only linear amping mechanisms, for which the amping rate is inepenent of space an time. The term representing the external source of turbulence can be set to Q = 2Γ I 0, so that when streaming instability is not relevant the level of the backgroun turbulence is at a constant level I = I 0. As alreay pointe out by [6], the approach escribe above ecouples the process of acceleration of particles, which operates, for example, at a SNR shock, from the particle escape from the acceleration site. Though such a separation might seem artificial, the problem efine above can still be useful to escribe the escape of particles form a ea accelerator, in which the acceleration mechanism oes not operate any more, or operates at a much reuce efficiency [6]. This situation woul probably apply to the case of ol SNRs. On the other han, [7] suggeste that Equations. 2.4 an 2.5 coul be also use to escribe an intermeiate phase of CR propagation in which the CRs have left the source but are not yet completely mixe with the CR backgroun. For the case of a SNR shock, the equations above coul thus be applie to those CRs characterize by a iffusion length large enough to ecouple them from the shock region. Typically, this happens uring the Seov phase to the highest energy particles confine at the SNR shock when the iffusion length D B /u s graually increases with time up to a value larger than some fraction χ of the SNR shock raius R s, where u s is the shock spee an χ 0.05...0.1 [8, 9]. In both scenarios, the initial conitions for Eqs 2.4 an 2.5 can be set as follows: P CR = P 0 CR an I 1 for z<a (2.7) P CR = 0 an I = I 0 1 for z>a (2.8) where a is the spatial scale of the region fille by CRs at the time of their escape from the source. Following [6], we introuce the quantity Π, efine as: where: Π = V A D B Φ CR (2.9) Φ CR z P CR = PCR 0 a. (2.10) 0 3

To unerstan the physical meaning of Π, consier the initial setup of the problem, in which CRs are localize in a region of size a. The CR pressure within a is PCR 0, an then Φ CR = PCR 0 a. The initial iffusion coefficient outsie the region of size a is equal to D B /I 0. The problem of transport is characterise by three relevant timescales: i) the growth time τ g (V A /I 0 / z) 1 ai 0 /V A PCR 0, ii) the time it takes the CR clou to sprea ue to iffusion τ i f f a 2 /D a 2 I 0 /D B, an iii) the amping time τ = 1/2Γ. In orer to have a significant growth of waves ue to CR streaming, the timescale for growth must be shorter than the two other timescales: τ g < min(τ i f f,τ ). In terms of the parameter Π this conitions reas: Π>max(1,τ i f f /τ ). It is evient, then, that the parameter Π regulates the effectiveness of CR inuce growth of waves. In the absence of a amping term for waves, Π > 1 is a necessary conition for streaming instability to be relevant, while in the presence of efficient wave amping, a more stringent conition on Π applies. For Π max(1,τ i f f /τ ) CRs play no role in the generation of Alfvén waves, an Equation 2.4 can be solve analytically: P CR = I0 I 0 z 2 πd B t Φ CR e 4D B t (2.11) Equation 2.11 is referre to as test particle solution. When wave growth cannot be neglecte, the solution eviates from this analytic test-particle solution. In the extreme scenario Π >> max(1,τ i f f /τ ) waves grow so quickly that the the iffusive term in Eq. 2.4 becomes negligible when compare to avection term V A / z, an the avection term cannot be neglecte anymore. This escribes a situation in which CRs are "locke" to waves an move with them at a velocity equal to V A [10]. An ientical result was foun by [11] in a stuy of the escape of MeV CRs from sources, an also suggeste by [12]. 3. The metho In orer to solve Eqns. 2.4 an 2.5 it is convenient to perform the following change of coorinates [6]: s z ( ) VA τ t (3.1) a a an re-normalise the parameters as follow: ( ) VA a P CR P CR D B ( ) VA a W I D B ( ) a Γ Γ (3.2) V A In these notations (an after ropping the avection terms) the equations become: = ( ) 1 τ s W s (3.3) W τ = 2Γ (W W 0 ) s (3.4) Their solution (i.e., P an W as a function of the variables τ an s) epens only on three parameters: the initial values PCR 0, W 0, an Γ 0. Note that: i) P0 CR = Π, ii) W 0 = V A a/d 0, an iii) Γ = Γ 0 (since we consier here only a linear amping constant in both time an space). In these notations, the conition for effective growth of waves is Π>max(1,Γ W 0 ). 4

Figure 1: Left: ion-neutral amping rate (Γ IN, soli curves), Farmer & Golreich amping rate (ΓFG, ashe curves) an ratio between Alfvén spee an initial size of the source (V A /a, ot-ashe curves), as a function of the CR energy E. Different colours refer to two ifferent ISM phases: warm ionise meium (WIM, re), an warm neutral meium (WNM, blue). Right: parameter Π as a function of the CR energy for a WIM an WNM (soli lines). In absence of efficient wave amping, values of Π<1 correspon to a test particle solutions, since the wave growth rate is inefficient. Conversely, for Π larger that Π max 10 the growth rate is large an the iffusion coefficient becomes smaller than the Bohm iffusion coefficient D B. In presence of efficient wave amping, both these limits are shifte to higher values, epening on the value of Γ. We present examples of numerical solutions to the couple equations 3.3 an 3.4 for values of the parameters in the ranges: Π = [2 20], W 0 = [10 5 10 2 ], an Γ = [0 100]. In orer to unerstan why we chose these values, an to which physical conitions they correspon, it is necessary to specify the values of V A,a,D 0, D B, an the amping mechanisms. These quantities in general epen on the properties of the ISM, an on the particle energy. We consier two ifferent ISM phases: a warm neutral meium (WNM) an a warm ionise meium (WIM). The typical values of ensity, temperature, ionisation fraction, an magnetic fiel strength for these two phases are taken from [13]. In these ISM phases, the two most relevant linear amping mechanisms are the ion-neutral friction (Γ IN ) an the amping by backgroun MHD turbulence suggeste by Farmer & Golreich [14] (Γ FG ). The value of ΓFG as a function of the particle energy is shown in Fig. 1 (left panel, long-ashe lines), for both ISM phases. The ion-neutral amping rate is estimate by numerically solving the equations in [15]. The results are shown in Fig. 1 (left panel, soli lines), for both ISM phases: at low energies Γ IN = ν IN/2 (where ν IN is the ion-neutral collision frequency), while at high energies Γ IN E 2. The level of coupling between ions an neutrals affects also the Alfvén spee, that is in the range 10 6 10 7 cm/s. Since it is useful for our calculations, we show in Fig. 1 the value of the ratio V a /a, as a function of energy (ash-otte lines). In orer to estimate a we consier a R esc (E), where R esc (E) is the escape raius an epens on the CR energy. We moel its epenence on the energy following [16]: we foun that its value ranges between a few pc (for the highest energy particles) an 20 pc (for GeV particles). Since we are intereste in following the CR iffusion from the scale a where they are release up to a istance z 10 3 pc, this roughly correspons to solve the normalise equations in a range of s from 1 to several hunres. Moreover, we want to stuy the CR pressure at ifferent times t in the range 10 3 yr to 10 5 yr. Consiering the value of V a /a as a function of energy isplaye in Fig. 1, this 5

correspons to τ = 10 2 1. In orer to ientify the relevant values of the parameters Π,W 0,Γ we procee as follow: Γ the amping rate Γ = max(γ IN,ΓFG ) ranges between 10 9 an 10 12 s 1 (Fig. 1, left panel), epening on the particle energy an ISM phase. This correspons to 1<Γ 10 4. Π the values of Π as a function of the CR energy, for the two ifferent ISM is shown in Fig. 1 (right panel, soli lines). W 0 for the backgroun iffusion D 0 we can refer to the average Galactic value D 0 = D ISM = 10 28 (E/10GeV) 0.5 cm 2 /s. Since V A a ranges between 10 27 cm 2 /s an 10 25 cm 2 /s (at low an high energies, respectively), the range of relevant values for W 0 is 10 5 < W 0 < 10 1. 4. Results an Discussion We numerically solve Eqs. 3.3 an 3.4 an show the results for P(s,τ) vs. s (soli lines in Fig. 2) at three ifferent normalise times τ = 10 2,10 1,1 (corresponing to ifferent colours in the figure). The test particle solution is also shown for comparison (otte lines). Each row in the figure correspons to results obtaine by varying only one parameter at the time. The upper row shows the impact of the parameter Π for fixe values of Γ = 10 an W 0 = 10 4. Since for Π < 1 the numerical solution coincies with the test particle solution, we show our results for values of Π>1. When Π=2, the solution starts to eviate from the test particle case: ue to the growth of wave by streaming instability the iffusion is slower, the CR pressure in the vicinity of the accelerator is larger, an the iffusion length is smaller. The effect is more an more evient for increasing values of Π (upper row, mile an right panels). The mile row shows the effect of wave amping, for fixe Π = 10 an W 0 = 10 4. Since Π >> 1, when wave amping is negligible (Γ =0, left panel) the iffusion is suppresse as compare to the test particle case. Larger values of Γ limit the growth of Alfvén waves an the solution approaches the test particle solution. Note that Γ =10 (mile row, central panel) correspons to a amping mechanism that becomes important at τ > 0.1. For this reason the solution at τ = 0.01 an τ = 0.1 are unaffecte by the increase amping rate, an only the solution at late time τ = 1 is moifie, an approaches the test particle solution. The right panel shows the case of a even faster amping mechanism, Γ =100, whose effect starts to be relevant at times corresponing to τ > 0.01, explaining why the solution at intermeiate an late times approaches the test particle solution, while the solution at early time is unaffecte by wave amping. The bottom row shows calculations performe at Π=5 an Γ =10, with W 0 varying from 10 5 to 10 2. A large W 0 (left panel) correspons to slow iffusion an small iffusion length, both in the test particle case an in the numerical solution. The two solutions iffer at early an intermeiate times because the growth rate is faster than the iffusion rate, an suppresses the iffusion. At later times instea the growth rate ecreases an the numerical solution oes not appreciably iffer from the test particle case. A change in the initial value of the backgroun turbulence has the same effect on both the analytical an numerical solution: the iffusion is faster for ecreasing values of W 0 (see bottom row, from left to right), an the CRs can reach larger istances. The eviation from a 6

P CR P CR P CR Figure 2: Normalize CR pressure as a function of the variable s, which represents the istance from the source z normalise to the size of the source a. In each row the results are shown by varying only one of the three parameters (whose value is reporte above each panel) an keeping the other two fixe to the values reporte in the shae area. The soli curves show the results from the numerical computation, while otte curves represent the test particle solution, where the role of streaming instabilities in enhancing the turbulence is neglecte. Different colours refer to ifferent normalise times τ = 10 2,10 1,1. 7

test particle solution however remains unchange, since both the growth rate an the iffusion rate scale as W 0. This stuy outlines the importance of consiering, in the physics of CR transport, both the growth of waves ue to streaming instability an the role of mechanisms that amp the turbulence. When amping is neglecte, even relatively small values of the integrate CR pressure Π = 10 (corresponing, for example, to particles of a few TeV, in a WIM - see Fig. 1) in a environment with small backgroun turbulence W 0 = 10 4 (relevant for TeV particles when the backgroun iffusion coefficient D 0 is similar to the average Galactic iffusion coefficient) lea to the conclusion that iffusion is strongly suppresse by the efficient growth of waves, an particles cannot travel large istances, even at late times. This situation is shown in Fig. 2 (mile row, left panel) where the numerical computation is performe in absence of a wave amping term: the iffusion is strongly suppresse also at τ = 1, corresponing to ol remnants (t age 10 5 yr). References [1] D. G. Wentzel, Cosmic-ray propagation in the Galaxy - Collective effects, ARA&A 12 (1974) 71. [2] A. R. Bell, The acceleration of cosmic rays in shock fronts. I, MNRAS 182 (1978) 147. [3] L. Drury, An introuction to the theory of iffusive shock acceleration of energetic particles in tenuous plasmas, Rep. Prog. Phys. 46 (1983) 973. [4] F. Casse, M. Lemoine, G. Pelletier, Transport of cosmic rays in chaotic magnetic fiels, Phys. Rev. D 65 (2002) 3002. [5] J. Skilling, Cosmic ray streaming. III - Self-consistent solutions, MNRAS 173 (1975) 255. [6] M. A. Malkov, P. H. Diamon, R. Z. Sageev, F. A. Aharonian, I. V. Moskalenko, Analytic Solution for Self-regulate Collective Escape of Cosmic Rays from Their Acceleration Sites, ApJ 768 (2013) 73. [7] V. S. Ptuskin, V. N. Zirakashvili, A. A. Plesser, Non-linear iffusion of cosmic rays, Av. Space Res. 42 (2008) 486. [8] V. S. Ptuskin, V. N. Zirakashvili, On the spectrum of high-energy cosmic rays prouce by supernova remnants in the presence of strong cosmic-ray streaming instability an wave issipation, A&A 429 (2005) 755. [9] S. Gabici, Cosmic ray escape from supernova remnants, MmSAI 82 (2011) 760. [10] J. Skilling, Cosmic Rays in the Galaxy: Convection or Diffusion?, MNRAS 170 (1971) 265. [11] C. J. Cesarsky, Interstellar propagation of low energy cosmic rays, ICRC 2 (1975) 634. [12] T. W. Hartquist, G. E. Morfill, Cosmic ray iffusion at energies of 1 MeV to 105 GeV, Astrophys. Space Sci. 216 (1994) 223. [13] P. Jean, W. Gillar, A. Marcowith, K. Ferriere, Positron transport in the interstellar meium, A&A 508 (2009) 1099. [14] A. J. Farmer, P. Golreich, Wave Damping by Magnetohyroynamic Turbulence an Its Effect on Cosmic-Ray Propagation in the Interstellar Meium, ApJ 604 (2004) 671. [15] E. G. Zweibel, J. M. Shull, Confinement of cosmic rays in molecular clous, ApJ 259 (1982) 859. [16] P. Cristofari, S. Gabici, S. Casanova, R. Terrier, E. Parizot, Acceleration of cosmic rays an gamma-ray emission from supernova remnants in the Galaxy, MNRAS 434 (2013) 2748. 8