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Stochastic Fermi acceleration in turbulent fields with non-vanishing wave helicities
Astroparticle Physics 16 (2002) 425±428
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Stochastic Fermi acceleration in turbulent ®elds with non-vanishing wave helicities G. Siemieniec-Oziez bøo Obserwatorium Astronomiczne, Uniwersytet Jagiello nski, Orla 171, 30-244 Krak ow, Poland Received 10 October 2000; received in revised form 29 November 2000
Abstract In the previous paper [Astropart. Phys. 10 (1999) 121] we showed that the opposite helicity circularly polarized Alfven waves of ®nite amplitudes provide conditions to forward±backward asymmetry of particle scattering. In this letter we present an analytic solution of kinematic equation proving the enhancement of stochastic acceleration eciency due to regular (asymmetry) term. The process is controlled by the ratio of the regular and the ordinary diusion term. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Acceleration mechanisms; Fermi acceleration; Cosmic rays; Non-linear waves
1. Introduction The acceleration of charged particles to superthermal energies occurs in a number of astronomical objects harboring a turbulent magnetized plasma in the process of stochastic Fermi acceleration [7]. Such mechanisms were considered, e.g., for providing relativistic electrons in extragalactic radiosources [2±6] or for generation of energetic ions during impulsive solar ¯ares (see, e.g., Refs. [9,12,13,17]). The MHD waves are attractive candidates because they can be produced by either large-scale restructuring of the magnetic ®eld, which presumably occurs during the ¯are release phase, or by a shear in the plasma bulk velocity [15], which would be found in regions of reconnection-driven plasma out¯ows (see, e.g., Refs. E-mail address: [email protected] (G. Siemieniec-OzieÎbøo).
[8,11]). Ostrowski and Siemieniec-Oziez bøo [14] demonstrated that the forward±backward asymmetry of particle scattering (as measured in the scattering center rest frame) at randomly moving scattering centers leads to a ®rst-order regular acceleration term, in addition to the one resulting from the momentum diusion. A physical example of such asymmetric scatterer provides a (®nite amplitude) circularly polarized Alfven wave [16]. The aim of the present paper is to study in¯uence of the forward±backward asymmetry of particle scattering on energetic (relativistic) charged particles acceleration in space ®lled with ®nite amplitude circularly polarized Alfven waves. We do not consider any particular site for application of the present results, the only restriction is provided by the considered simpli®ed structure of magnetic ®eld perturbations. An analytical solution for a simple acceleration model involving a regular term is presented below. We con®rm a substantial
0927-6505/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 6 5 0 5 ( 0 1 ) 0 0 1 2 2 - 0
426
G. Siemieniec-Oziez bøo / Astroparticle Physics 16 (2002) 425±428
increase of acceleration eciency due to the regular acceleration modi®cation. 2. Stochastic particle acceleration in the presence of a regular acceleration term Here we undertake the eect of second-order Fermi acceleration enhanced by the asymmetric scattering term announced in Ref. [16]. We are going to consider the modi®cation to energy distribution function produced by the systematic acceleration contribution discussed in mentioned paper. Even before the detailed calculation one can anticipate that it will lead to an increase of acceleration eciency. A pure second-order Fermi acceleration corresponds to momentum diusion while the asymmetry eect should provide the additionary boosting of momentum distribution to higher energy. In order to prove it, we will analyze a simple example of the process of particle momentum diusion in the presence of a regular acceleration/deceleration term p_ reg const. For simplicity we will also consider a constant momentum diusion D. Evolution of the particle phase space is given by distribution function, f p; t, satisfying the kinetic equation of p; t 1 o of p; t 2 p2 D ot p op op 1 o 2 fp p_ reg f p; tg: 1 p2 op The nature of time evolution is simultaneously diusive and convective in momentum space. The term containing D takes into account the stochastic acceleration in which energy is transferred from turbulence in the background plasma to energetic particles while the p_ reg term describes the scattering asymmetry eect due to e.g., wave helicity. Both coecients D and p_ reg are a priori energy dependent. Although we do not indicate any particular astrophysical circumstance when the both eects are important, we would rather restrict our interest to all these situations where the ordinary Fermi II mechanism loses its eectiveness. This is the case of ultrarelativistic particles. For them, the trans-
port coecient D ceases to grow with energy and becomes constant [10]. For the particles with v c, almost all the MHD waves participate in acceleration. Thus D does not depend on energy any more. In particular, in high energy regime, the consideration of regular term may become important in improving the acceleration eciency. However one has to remember that this process is limited by acceleration ± wave damping equilibrium condition. Thus it is sucient to analyze the transport equation in energy limited domain. The value of pmax has to be understood as a momentum at which the turbulence damping on CR particles occurs and the energy equipartition settles. The energy dependence of p_ reg is unknown. One may however speculate that it should be rather weak. The asymmetry eect due to circular polarization of waves should not essentially depend on magnetic ®eld perturbation spectrum. For simplicity also p_ reg can be assumed as energy constant. Above we have tried to justify that our restriction to the case of momentum independent transport coecients may not be completely unrealistic. We also treat the turbulence spectrum as a static up to energy when its damping occurs. The coef®cients which do not depend neither on energy nor on time, we denote by D A and p_ reg B. The kinetic equation is given now in the form of o2 f of 2A 2B A 2 f: 2 B ot op op p p According to the above discussion one should ®nd the time-dependent solution of Eq. (2) in the interval p0 pmax . Let us assume an impulse injection of monoenergetic particles with momentum p0 at t 0. f p; 0
1 d p p2
p0 :
3
The value of pmax gives the energy threshold above which the turbulence damping occurs. To simplify the Eq. (2) we replace the distribution function f p; t by f p; t
ea p p0 g p; t; p
4
where a B=2A. This quantity serves as a measure of the relative strength of the regular and the
G. Siemieniec-Oziez bøo / Astroparticle Physics 16 (2002) 425±428
stochastic acceleration. Now, Eq. (2) can bee transformed to the form 1 og o2 g 2a 2 a : g 5 A ot op2 p After applying the standard separability condition we look for the solution in the form 1 X 2 Cn e kn t vn p: 6 g p; t n1
The phase-space density g p; t can be expressed as an in®nite sum of functions vn p with expansion coecient Cn . Each function vn p satis®es an equation below corresponding, to the appropriate eigenvalue 2a d2 v 2 2 v k 0: 7 a n dp2 p The above has a form of the Schr odinger equation for the particle wave function in the Coulomb potential. Then, its solution is well known and can be expressed in terms of the Coulomb wave function F0 g; p [1]. From two linearly independent solutions we choose the one which is regular at p 0. Therefore one can write 1 X a a k2n t g p; t C n e F0 ; jn p0 F0 ; jn p ; jn jn n1 8 where the eigenvalues jn : j2n
k2n A
a2
have to be calculated from the boundary condition g pmax ; t 0. It can be expressed with the Coulomb function as a F0 ; jn pmax 0: 9 jn The coecients Cn are given by orthogonality condition for eigenfunctions F0 , i.e. 2 Z pmax a F0 ; jn p dp 10 Cn jn 0 and a full solution of the transport Eq. (1) is represented, up to a constant factor, by
f p; t /
1 X Cn e
427
ea p p0 a k2n t F0 ; jn p0 jn p n1 a F0 ; jn p : jn
11
One can see immediately that there does not exist a steady-state solution. The most important property of the above solution can be derived from its asymptotic form for large t. Then the in®nite but rapidly converging series can be approximated by its leading term expression. Comparing to strictly diusion behavior in the case when distribution function peak is located p p0 , we have for solution (11), a peak transfer toward higher energy. Its position is determined by the value of a parameter i.e. the ratio of regular to statistical acceleration terms (Fig. 1). One obtains the correct correspondence to the purely diusive case, i.e. when a goes to zero, the solution (11) becomes the well known solution of the diusion equation. The above solution may also be applied when stochastic acceleration problem includes losses. In this case the value of a is negative.
3. Conclusions and ®nal remarks In any astrophysical medium ®lled with tenuous magnetized plasma with propagating Alfven waves allowing for an asymmetric scattering of particles (in non-linear regime), one can expect the enhancement of stochastic acceleration eciency. The conditions leading to such process can occur for example, when the particles interact with circularly polarized Alfven waves having the opposite helicities (e.g., R L ). Then in kinetic equation, in addition to the diusive acceleration a positive regular acceleration term can arise. This term is responsible, which is clearly seen in Fig. 1, for energy distribution shift toward the higher energy. This transfer measure may be substantial wherever the regular term contribution becomes signi®cant, in particular for increasing amplitude and velocity of waves. Possibility of acting regular acceleration processes in turbulent MHD media is of interest in all situations, where the second-order Fermi acceleration is considered to energize cosmic rays.
428
G. Siemieniec-Oziez bøo / Astroparticle Physics 16 (2002) 425±428
Fig. 1. Shows the particle distribution function. The peak position increases with the value of a. The p-distribution with negative a is indicated by the dashed line.
However its importance in real astrophysical situations is a matter of speculation. Perhaps it may contribute to ions spectra modi®cation in solar ¯ares. Also one cannot exclude its role during the formation of primordial, helical magnetic ®elds. This would be of great interest especially in the case when magnetic ®eld generation is associated with gravitational structure formation. Although the structure formation is usually accompanied by complex cosmic shock structure generation, the presence of dierent helicity MHD waves provides the required asymmetry and large amplitudes and thus a supplementary process of particle acceleration. In consequence it may account for non-thermal activity evidenced near the cosmic structures like galaxy clusters, ®laments and sheets. Acknowledgements The work was supported by the KBN grant 2 P03B 112 17.
References [1] M. Abramovitz, I. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Wiley-Interscience, New York, 1964. [2] A. Achterberg, A&A 76 (1979) 276. [3] R.D. Blandford, M.I. Rees, Physica Scripta 17 (1978) 3543. [4] B.I. Burn, A&A 45 (1975) 435. [5] D.S. De Young, ARAA 14 (1976) 447. [6] J.A. Eilek, ApJ 230 (1979) 373. [7] E. Fermi, Phys. Rev. 75 (1949) 1169. [8] T.G. Forbes, in: R. Ramty, N. Mandzhavidze, X.-M. Hua (Eds.), High-Energy Solar Physics, AIP, New York, 1996, p. 275. [9] R.J. Hamilton, V. Petrosian, ApJ 398 (1992) 350. [10] S.A. Kaplan, V.N. Tsytovich, A.S. Chikhachev, Radio®zyka XIV (2) (1971) 204. [11] T.N. LaRosa, R.L. Moore, ApJ 418 (1993) 912. [12] J.A. Miller, R. Ramaty, Solar Phys. 326 (1987) 195. [13] J.A. Miller, D.A. Roberts, ApJ 452 (1996) 912. [14] M. Ostrowski, G. Siemieniec-Oziez bøo, Astropart. Phys. 6 (1997) 271. [15] D.A. Roberts, M.L. Goldstein, W.H. Mattheaus, S. Ghosh, JGR 97 (1992) 17115. [16] G. Siemieniec-Oziez bøo, G. Michaøek, M. Ostrowski, Astropart. Phys. 10 (1999) 121. [17] J. Steinecker, J.A. Miller, ApJ 393 (1992) 764.
www.elsevier.com/locate/astropart
Stochastic Fermi acceleration in turbulent ®elds with non-vanishing wave helicities G. Siemieniec-Oziez bøo Obserwatorium Astronomiczne, Uniwersytet Jagiello nski, Orla 171, 30-244 Krak ow, Poland Received 10 October 2000; received in revised form 29 November 2000
Abstract In the previous paper [Astropart. Phys. 10 (1999) 121] we showed that the opposite helicity circularly polarized Alfven waves of ®nite amplitudes provide conditions to forward±backward asymmetry of particle scattering. In this letter we present an analytic solution of kinematic equation proving the enhancement of stochastic acceleration eciency due to regular (asymmetry) term. The process is controlled by the ratio of the regular and the ordinary diusion term. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Acceleration mechanisms; Fermi acceleration; Cosmic rays; Non-linear waves
1. Introduction The acceleration of charged particles to superthermal energies occurs in a number of astronomical objects harboring a turbulent magnetized plasma in the process of stochastic Fermi acceleration [7]. Such mechanisms were considered, e.g., for providing relativistic electrons in extragalactic radiosources [2±6] or for generation of energetic ions during impulsive solar ¯ares (see, e.g., Refs. [9,12,13,17]). The MHD waves are attractive candidates because they can be produced by either large-scale restructuring of the magnetic ®eld, which presumably occurs during the ¯are release phase, or by a shear in the plasma bulk velocity [15], which would be found in regions of reconnection-driven plasma out¯ows (see, e.g., Refs. E-mail address: [email protected] (G. Siemieniec-OzieÎbøo).
[8,11]). Ostrowski and Siemieniec-Oziez bøo [14] demonstrated that the forward±backward asymmetry of particle scattering (as measured in the scattering center rest frame) at randomly moving scattering centers leads to a ®rst-order regular acceleration term, in addition to the one resulting from the momentum diusion. A physical example of such asymmetric scatterer provides a (®nite amplitude) circularly polarized Alfven wave [16]. The aim of the present paper is to study in¯uence of the forward±backward asymmetry of particle scattering on energetic (relativistic) charged particles acceleration in space ®lled with ®nite amplitude circularly polarized Alfven waves. We do not consider any particular site for application of the present results, the only restriction is provided by the considered simpli®ed structure of magnetic ®eld perturbations. An analytical solution for a simple acceleration model involving a regular term is presented below. We con®rm a substantial
0927-6505/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 6 5 0 5 ( 0 1 ) 0 0 1 2 2 - 0
426
G. Siemieniec-Oziez bøo / Astroparticle Physics 16 (2002) 425±428
increase of acceleration eciency due to the regular acceleration modi®cation. 2. Stochastic particle acceleration in the presence of a regular acceleration term Here we undertake the eect of second-order Fermi acceleration enhanced by the asymmetric scattering term announced in Ref. [16]. We are going to consider the modi®cation to energy distribution function produced by the systematic acceleration contribution discussed in mentioned paper. Even before the detailed calculation one can anticipate that it will lead to an increase of acceleration eciency. A pure second-order Fermi acceleration corresponds to momentum diusion while the asymmetry eect should provide the additionary boosting of momentum distribution to higher energy. In order to prove it, we will analyze a simple example of the process of particle momentum diusion in the presence of a regular acceleration/deceleration term p_ reg const. For simplicity we will also consider a constant momentum diusion D. Evolution of the particle phase space is given by distribution function, f p; t, satisfying the kinetic equation of p; t 1 o of p; t 2 p2 D ot p op op 1 o 2 fp p_ reg f p; tg: 1 p2 op The nature of time evolution is simultaneously diusive and convective in momentum space. The term containing D takes into account the stochastic acceleration in which energy is transferred from turbulence in the background plasma to energetic particles while the p_ reg term describes the scattering asymmetry eect due to e.g., wave helicity. Both coecients D and p_ reg are a priori energy dependent. Although we do not indicate any particular astrophysical circumstance when the both eects are important, we would rather restrict our interest to all these situations where the ordinary Fermi II mechanism loses its eectiveness. This is the case of ultrarelativistic particles. For them, the trans-
port coecient D ceases to grow with energy and becomes constant [10]. For the particles with v c, almost all the MHD waves participate in acceleration. Thus D does not depend on energy any more. In particular, in high energy regime, the consideration of regular term may become important in improving the acceleration eciency. However one has to remember that this process is limited by acceleration ± wave damping equilibrium condition. Thus it is sucient to analyze the transport equation in energy limited domain. The value of pmax has to be understood as a momentum at which the turbulence damping on CR particles occurs and the energy equipartition settles. The energy dependence of p_ reg is unknown. One may however speculate that it should be rather weak. The asymmetry eect due to circular polarization of waves should not essentially depend on magnetic ®eld perturbation spectrum. For simplicity also p_ reg can be assumed as energy constant. Above we have tried to justify that our restriction to the case of momentum independent transport coecients may not be completely unrealistic. We also treat the turbulence spectrum as a static up to energy when its damping occurs. The coef®cients which do not depend neither on energy nor on time, we denote by D A and p_ reg B. The kinetic equation is given now in the form of o2 f of 2A 2B A 2 f: 2 B ot op op p p According to the above discussion one should ®nd the time-dependent solution of Eq. (2) in the interval p0 pmax . Let us assume an impulse injection of monoenergetic particles with momentum p0 at t 0. f p; 0
1 d p p2
p0 :
3
The value of pmax gives the energy threshold above which the turbulence damping occurs. To simplify the Eq. (2) we replace the distribution function f p; t by f p; t
ea p p0 g p; t; p
4
where a B=2A. This quantity serves as a measure of the relative strength of the regular and the
G. Siemieniec-Oziez bøo / Astroparticle Physics 16 (2002) 425±428
stochastic acceleration. Now, Eq. (2) can bee transformed to the form 1 og o2 g 2a 2 a : g 5 A ot op2 p After applying the standard separability condition we look for the solution in the form 1 X 2 Cn e kn t vn p: 6 g p; t n1
The phase-space density g p; t can be expressed as an in®nite sum of functions vn p with expansion coecient Cn . Each function vn p satis®es an equation below corresponding, to the appropriate eigenvalue 2a d2 v 2 2 v k 0: 7 a n dp2 p The above has a form of the Schr odinger equation for the particle wave function in the Coulomb potential. Then, its solution is well known and can be expressed in terms of the Coulomb wave function F0 g; p [1]. From two linearly independent solutions we choose the one which is regular at p 0. Therefore one can write 1 X a a k2n t g p; t C n e F0 ; jn p0 F0 ; jn p ; jn jn n1 8 where the eigenvalues jn : j2n
k2n A
a2
have to be calculated from the boundary condition g pmax ; t 0. It can be expressed with the Coulomb function as a F0 ; jn pmax 0: 9 jn The coecients Cn are given by orthogonality condition for eigenfunctions F0 , i.e. 2 Z pmax a F0 ; jn p dp 10 Cn jn 0 and a full solution of the transport Eq. (1) is represented, up to a constant factor, by
f p; t /
1 X Cn e
427
ea p p0 a k2n t F0 ; jn p0 jn p n1 a F0 ; jn p : jn
11
One can see immediately that there does not exist a steady-state solution. The most important property of the above solution can be derived from its asymptotic form for large t. Then the in®nite but rapidly converging series can be approximated by its leading term expression. Comparing to strictly diusion behavior in the case when distribution function peak is located p p0 , we have for solution (11), a peak transfer toward higher energy. Its position is determined by the value of a parameter i.e. the ratio of regular to statistical acceleration terms (Fig. 1). One obtains the correct correspondence to the purely diusive case, i.e. when a goes to zero, the solution (11) becomes the well known solution of the diusion equation. The above solution may also be applied when stochastic acceleration problem includes losses. In this case the value of a is negative.
3. Conclusions and ®nal remarks In any astrophysical medium ®lled with tenuous magnetized plasma with propagating Alfven waves allowing for an asymmetric scattering of particles (in non-linear regime), one can expect the enhancement of stochastic acceleration eciency. The conditions leading to such process can occur for example, when the particles interact with circularly polarized Alfven waves having the opposite helicities (e.g., R L ). Then in kinetic equation, in addition to the diusive acceleration a positive regular acceleration term can arise. This term is responsible, which is clearly seen in Fig. 1, for energy distribution shift toward the higher energy. This transfer measure may be substantial wherever the regular term contribution becomes signi®cant, in particular for increasing amplitude and velocity of waves. Possibility of acting regular acceleration processes in turbulent MHD media is of interest in all situations, where the second-order Fermi acceleration is considered to energize cosmic rays.
428
G. Siemieniec-Oziez bøo / Astroparticle Physics 16 (2002) 425±428
Fig. 1. Shows the particle distribution function. The peak position increases with the value of a. The p-distribution with negative a is indicated by the dashed line.
However its importance in real astrophysical situations is a matter of speculation. Perhaps it may contribute to ions spectra modi®cation in solar ¯ares. Also one cannot exclude its role during the formation of primordial, helical magnetic ®elds. This would be of great interest especially in the case when magnetic ®eld generation is associated with gravitational structure formation. Although the structure formation is usually accompanied by complex cosmic shock structure generation, the presence of dierent helicity MHD waves provides the required asymmetry and large amplitudes and thus a supplementary process of particle acceleration. In consequence it may account for non-thermal activity evidenced near the cosmic structures like galaxy clusters, ®laments and sheets. Acknowledgements The work was supported by the KBN grant 2 P03B 112 17.
References [1] M. Abramovitz, I. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Wiley-Interscience, New York, 1964. [2] A. Achterberg, A&A 76 (1979) 276. [3] R.D. Blandford, M.I. Rees, Physica Scripta 17 (1978) 3543. [4] B.I. Burn, A&A 45 (1975) 435. [5] D.S. De Young, ARAA 14 (1976) 447. [6] J.A. Eilek, ApJ 230 (1979) 373. [7] E. Fermi, Phys. Rev. 75 (1949) 1169. [8] T.G. Forbes, in: R. Ramty, N. Mandzhavidze, X.-M. Hua (Eds.), High-Energy Solar Physics, AIP, New York, 1996, p. 275. [9] R.J. Hamilton, V. Petrosian, ApJ 398 (1992) 350. [10] S.A. Kaplan, V.N. Tsytovich, A.S. Chikhachev, Radio®zyka XIV (2) (1971) 204. [11] T.N. LaRosa, R.L. Moore, ApJ 418 (1993) 912. [12] J.A. Miller, R. Ramaty, Solar Phys. 326 (1987) 195. [13] J.A. Miller, D.A. Roberts, ApJ 452 (1996) 912. [14] M. Ostrowski, G. Siemieniec-Oziez bøo, Astropart. Phys. 6 (1997) 271. [15] D.A. Roberts, M.L. Goldstein, W.H. Mattheaus, S. Ghosh, JGR 97 (1992) 17115. [16] G. Siemieniec-Oziez bøo, G. Michaøek, M. Ostrowski, Astropart. Phys. 10 (1999) 121. [17] J. Steinecker, J.A. Miller, ApJ 393 (1992) 764.