STOCHASTIC ACCELERATION OF ELECTRONS AND PROTONS. I. ACCELERATION BY PARALLEL-PROPAGATING WAVES

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1 The Astrohysical Journal, 610: , 2004 July 20 # The American Astronomical Society. All rights reserved. Printed in U.S.A. A STOCHASTIC ACCELERATION OF ELECTRONS AND PROTONS. I. ACCELERATION BY PARALLEL-PROPAGATING WAVES Vahé Petrosian 1 and Siming Liu Center for Sace Science and Astrohysics, Deartment of Physics, Stanford University, Stanford, CA 94305; vahe@astronomy.stanford.edu, liusm@stanford.edu Received 2004 January 27; acceted 2004 March 31 ABSTRACT Stochastic acceleration of electrons and rotons by waves roagating arallel to the large-scale magnetic fields of magnetized lasmas is studied with emhasis on the feasibility of accelerating articles from a thermal background to relativistic energies and with the aim of determining the relative acceleration of the two secies in one source. In general, the stochastic acceleration by these waves results in two distinct comonents in the article distributions, a quasi-thermal and a hard nonthermal, with the nonthermal one being more rominent in hotter lasmas and/or with higher level turbulence. This can exlain many of the observed features of solar flares. Regarding the roton-to-electron ratio, we find that in a ure hydrogen lasma the dominance of the wave-roton interaction by the resonant Alfvén mode reduces the acceleration rate of rotons in the intermediate energy range significantly, while the electron-cyclotron and Whistler waves are very efficient in accelerating electrons from a few kev to MeV energies. The resence of such an acceleration barrier rohibits the roton acceleration under solar flare conditions. This difficulty is alleviated when we include the effects of 4 He in the disersion relation and the daming of the turbulent waves by the thermal background lasma. The additional 4 He cyclotron branch of the turbulent lasma waves suresses the roton acceleration barrier significantly, and the stee turbulence sectrum in the dissiation range makes the nonthermal comonent have a near ower-law shae. The relative acceleration of rotons and electrons is very sensitive to a lasma arameter ¼! e = e,where! e and e are the electron lasma frequency and gyrofrequency, resectively. Protons are referentially accelerated in weakly magnetized lasmas (large ). The formalism develoed here is alicable to the acceleration of other ion secies and to other astrohysical systems. Subject headings: acceleration of articles MHD lasmas Sun: flares turbulence On-line material: color figures 1. INTRODUCTION One of the imortant questions in acceleration of cosmic articles is the fractions of energy that go into acceleration of electrons and rotons (and other ions). In this aer we investigate this question for acceleration by lasma wave turbulence, a second-order Fermi acceleration rocess, which we call stochastic acceleration (SA). The theory of SA has received little attention in high-energy astrohysics excet in solar flares where it has achieved significant successes during the ast few years. The turbulence or lasma waves required for this model are resumably generated during the magnetic reconnection that energizes the flares. The first alication of SA was to the acceleration of rotons and other ions to exlain the observed nuclear gamma-ray lines from solar flares (see, e.g., Ramaty 1979; Miller & Roberts 1995). Combining with the nuclear reaction rates (Ramaty et al. 1975, 1979; see also Kozlovsky et al. 2002) and a magnetic loo model, Hua & Lingenfelter (1987a, 1987b) and Hua et al. (1989) showed that the SA model can rovide natural exlanations for the many observed features in the 1 7 MeV range. Later this model was also investigated in the acceleration of electrons in several studies (Miller & Ramaty 1987; Bech et al. 1990; Miller et al. 1996; Park & Petrosian 1995, 1996), and the first quantitative comarison of redictions of this model with the observed 1 Deartment of Alied Physics, Stanford University, Stanford, CA hard X-ray ( kev) sectra in some solar flares was carried out by Hamilton & Petrosian (1992). With a more detailed modeling, Park et al. (1997) showed that the SA of electrons by some generic turbulent lasma waves can reroduce the many sectral breaks observed over a broad energy range, from tens of kev to 100MeV,intheso-called electron-dominated flares via the bremsstrahlung rocess (Rieger et al. 1998; Petrosian et al. 1994). The strongest evidence suorting the SA model comes from the Yohkoh discovery of imulsive hard X-ray emission from the to of a flaring loo, in addition to reviously known emission from loo footoints (FPs; Masuda et al. 1994; Masuda 1994). The resence of the loo-to (LT) emission requires temorary traing of the accelerated electrons at the to of the loo where the reconnection is taking lace. The turbulence required for SA will naturally accomlish this by reeated scatterings of the articles (Petrosian & Donaghy 1999). More imortantly, an analysis of a larger samle of Yohkoh flares (Petrosian et al. 2002) has shown that the LT emission is a common roerty of all flares, and a reliminary investigation of RHESSI data aears to confirm this icture (Liu et al. 2003). Finally, a third and equally imortant iece of evidence in suort of the SA model comes from the sectra and relative abundances of the flare-accelerated rotons and other ions observed at 1 AU from the Sun (Mazur et al. 1992; Reames et al. 1994; Miller 2003). These several indeendent lines of argument have established the SA as the leading model for solar flares. This may be also true in many

2 ELECTRON AND PROTON ACCELERATION. I. 551 other astrohysical nonthermal sources. Thus, a more detailed investigation of the SA model and its comarison with observations are now fully warranted. In articular, the SAs of electrons on the one hand and rotons and other ions on the other are investigated searately; a unified treatment and comarison with the total nonthermal radiative signatures of all secies have not been carried out yet. The urose of this investigation is to obtain the relative acceleration of electrons and rotons from the thermal backgrounds of solar flare lasmas with the same sectrum of turbulence. We resent some general results of the model and qualitative comarisons with observations. More detailed comarisons with observations and the acceleration of other ions, such as the anomalous overabundance of the flare-accelerated 3 He, will be addressed in subsequent aers. Secifically, we will address the energy artition between the flare-accelerated electrons and rotons. Observationally, in some flares, or during the earlier imulsive hase of most flares, there is little evidence for gamma-ray lines and therefore roton acceleration. These are called electron-dominated cases. In the majority of solar flares the energy artition favors electrons, but there are a significant fraction of flares where more energy resides in rotons than in electrons in their resective observed energy bands. The ratio of energy of the observed electrons (with greater than 20 kev range) to that of rotons (with greater than 1 MeV range) in solar flares varies aroximately from 0.03 to 100 (see, e.g., a comilation by Miller et al. 1997). In what follows, we use solar flare lasma conditions, but the formalism described here is alicable to other astrohysical sources. In x 2 we describe the general theory of SA and argue that in most cases the Fokker-Planck (F-P) equation can be reduced to the diffusion-convection equation with the corresonding coefficients given by itch-angle averaged combinations of the F-P coefficients. In x 3 we study the resonant interactions of electrons and rotons with arallel-roagating waves in a ure hydrogen lasma and calculate the resultant F-P coefficients and acceleration arameters for interactions with a ower-law turbulence sectrum of the wavenumber. The new and surrising result here is that the roton acceleration is suressed by a barrier in its acceleration rate in the intermediate energy range. This is what is required by observations of electron-dominated cases, but as shown in this aer, this barrier is too strong and makes the acceleration of rotons unaccetably inefficient relative to the electron acceleration, excet for in very weakly magnetized lasmas. In x 4 we oint out that this difficulty can be alleviated by a more comlete descrition of the disersion relation that includes the effects of helium ions and by an inclusion of the effects of the thermal daming of the turbulence at high wavenumbers. The resence of an aroriate amount of fully ionized helium introduces an extra wave branch that lowers the barrier, and the thermal daming steeens the turbulence sectrum toward high wavenumbers, making the acceleration of electrons and rotons more in agreement with observations. The results resented here are summarized in x 5, and their alications to solar flare observations are discussed qualitatively. Some useful aroximate analytical exressions for the interaction rates are resented in Aendices A, B, C, and D. 2. GENERAL THEORY OF STOCHASTIC ACCELERATION In this section we resent the general theory of SA and show that in most astrohysical situations the diffusionconvection equation is adequate to address the article acceleration rocesses Fokker-Planck Equation The study of SA in a magnetized lasma starts from the collisionless Boltzmann-Vlasov equation and the Lorentz force (Schlickeiser 1989). In the quasi-linear aroximation, it can be treated by the F-P equation (e.g., Kennel & ¼ ṗ L f þ S; where the wave-article interaction is arameterized by the F-P coefficients D ij ½i; j2(; )Š. Here f (t; s; ;) is the gyrohase-averaged article distribution and s, v,, and are the satial coordinate along the field lines, the velocity, the itch-angle cosine, and the momentum of the article, resectively. The energy-loss (minus systematic energy gains, if any) rocesses are accounted for by ṗ L,andSis the source function. For weak turbulence (BTB), as is the case for solar flares, the F-P coefficients can be evaluated by assuming that the articles and waves are couled via a resonant rocess. The acceleration of articles at a given energy is then dominated by interactions with certain secific wave modes, e.g., the Alfvén or Whistler waves. For a study of acceleration in a narrow energy band it is usually sufficient to consider waves in a narrow frequency range (Miller & Ramaty 1987). In order to address the energy artition between electrons and ions, however, one has to calculate the article acceleration over the whole energy range. For examle, the Alfvén waves can efficiently accelerate ions but not nonrelativistic electrons. Models with ure Alfvénic turbulence are not adequate to address the energy artition of accelerated articles in solar flares and many other astrohysical lasmas, esecially for the acceleration of articles from a thermal background. For the acceleration of such low-energy articles interactions with turbulent lasma waves extending over a broad range of wavenumbers (and frequencies) must be considered. We consider here interactions with a broad sectrum of lasma waves roagating along a static background magnetic field. The interactions of arallel-roagating waves with electrons are described fully by Dung & Petrosian (1994, hereafter DP94) and Pryadko & Petrosian (1997, hereafter PP97) (see also Steinacker & Miller 1992). We use their formalism and evaluate the relative rates of interaction and acceleration of electrons and rotons in cold but fully ionized lasmas Disersion Relation and Resonance Condition Waves roagating arallel or antiarallel to the large-scale magnetic field in a uniform lasma have two normal modes that are olarized circularly (Sturrock 1994). Because their electric fields are erendicular to their corresonding wavevectors, the waves are also referred to as transverse waves. The disersion relation for these waves in a cold lasma is (see DP94) " # ðkcþ 2 ¼! 2 1 X i! 2 i!!þ ð i Þ ð1þ ; ð2þ

3 552 PETROSIAN & LIU Vol. 610 where! i ¼ (4n i q 2 i =m i) 1=2 and i ¼ (q i B 0 )=(m i c) are, resectively, the lasma frequencies and the nonrelativistic gyrofrequencies of the background articles (with charges q i, masses m i, and number densities n i ). The term B 0 stands for the large-scale magnetic field, c is the seed of light, and! and k are the wave frequency and wavenumber, resectively. One of the imortant arameters characterizing a magnetized lasma is the ratio of the electron lasma frequency to the electron nonrelativistic gyrofrequency: ¼! e = e ¼ 3:2 n e =10 10 cm 3 1=2 ð B0 =100 GÞ 1 ; ð3þ where e ¼ (eb 0 )=(m e c)ande and m e are the elemental charge unit and the electron mass, resectively. The value of is small for a strongly magnetized lasma. A article with a velocity c (Lorentz factor ) and a itchangle cosine interacts most strongly with waves satisfying the resonance condition:! k k ¼ n! i ; where n is for the harmonics of the gyrofrequency (not to be confused with the background article number densities n i ),! and k k are the wave frequency and the arallel comonent of the wavevector in units of e and e /c, resectively (we use these units in the following discussion unless secified otherwise and in our case k k ¼ k and n ¼ 1), and! i ¼ q i m e =em i are the article gyrofrequencies in units of e ;! e ¼ 1 for electrons and! ¼ m e =m for rotons, where m is the roton mass (for more details see also DP94). One notes that low-energy articles mostly resonate with waves with high wavenumbers and only relativistic articles interact with large-scale waves with low frequencies. The resonant wave-article interaction can transfer energy between the turbulence and articles with the details deending on the article distribution and the sectrum and olarization of the turbulence Fokker-Planck Coefficients The evaluation of the F-P coefficients requires a knowledge of the sectrum of the turbulence. Following revious studies (DP94; PP97), we first assume a ower-law distribution of unolarized turbulent lasma waves. For unolarized turbulence, the amlitudes of the waves only deend on k. ThenwehaveE(k) ¼ (q 1)E tot k q 1 min k q for k > k min (i.e., a large-scale cutoff ), where the turbulence sectral index q > 1. For a given turbulence energy density E tot, k min, resumably larger than the inverse of the size of the acceleration region, determines the maximum energy that the accelerated articles can reach and the characteristic timescale of the interaction. The general features of this situation have been exlored in the aers cited above. For the sake of comleteness, we briefly summarize the key results here. The F-P coefficients can be written as ð D ab ¼ 2 1Þ i 2 X N j¼1 ð4þ x >< j for ab ¼ ; k j x j 1 x j for ab ¼ ; >: 2 x 2 j for ab ¼ ; ð5þ where k j q k j ¼ g k j ; x j ¼! j =k j : ð6þ The sum over j is for the resonant interactions discussed in the revious section. The characteristic interaction timescale for each of the charged article secies is i ¼ =! 2 i with that for electrons given by (see DP94) 1 ¼ 2 E tot e B 2 0 =8 (q 1)k q 1 min : In general, the F-P coefficients have comlicated deendence on the turbulence sectral index q, the lasma arameter, and the energy and itch angle of the articles. The exact solution of the full F-P equation is a difficult task. Fortunately, under certain conditions considerable simlifications are ossible. These conditions are defined by the relative values of the three F-P coefficients. The itch-angle change rate of the articles is roortional to D, while the momentum or energy change rate is roortional to D / 2. As evident from equation (5), the behavior of D / is intermediate between the two Diffusion-Convvection Equation The relative values of the F-P coefficients determine the tye of aroximations that can be used for solving the F-P equation. We now show that for most conditions reasonable aroximations lead to the well-known transort equation (eq. [10]). In order to justify these aroximations, it is convenient to define two ratios of the coefficients: R 1 (; ) ¼ D 2 ; ð8þ D R 2 (; ) ¼ D : ð9þ D We show in the following sections for most energies and itch angles both R 1 and jr 2 jt1, which means that D 3 D = 2. Under these conditions the articles are scattered frequently before being significantly accelerated and the accelerated article distribution is nearly isotroic. Then the itch-angle averaged article distribution function F(s; t; ) ¼ 0:5 R 1 1 d f (; s; t; ) satisfies the well-known diffusionconvection equation (see, e.g., Kirk et al. 1988; DP94; PP97). In this study we are interested in the relative acceleration of electrons and rotons that is not sensitive to the detailed geometry or the inhomogeneities of the source. Therefore, we can assume a homogeneous and finite (size L) sourceor alternatively confine our discussion to satially integrated sectra. In this case we can treat the satial diffusion or advection of the articles by an energy-deendent escae term. Then the above-mentioned equation is @E 2 ð Ė L A N N T esc þ Q; ð7þ ð10þ where E ¼ ( 1)m i c 2 is the article kinetic energy, N(t; E ) de ¼ 4 2 d R L 0 F(s; t; ) ds, Ė L describes the net systematic energy loss, and Q(t; E) ¼ 0:5 R 1 1 d R L 0 S(s;;t; E) ds is the total injection flux of articles into the acceleration region. The term D EE describing the diffusion in energy is

4 No. 1, 2004 ELECTRON AND PROTON ACCELERATION. I. 553 related to D and defines the acceleration time, and T esc is related to the scattering time sc : T esc ¼ L2 =v 2 sc ; sc ¼ 1 2 ac ¼ E2 D EE ; Z 1 D EE ¼ E2 2 ð1 2Þ 2 d TL=v; ð11þ D 1 Z 1 1 d D R 1 R 2 2 : ð12þ Note that equation (10) describes the energy diffusion with two terms, D EE and the direct acceleration rate: A(E) ¼ 1 d 2 D EE 2 ¼ dd EE de de þ D EE 2 2 : ð13þ E 1 þ 1 There are several imortant features in the diffusion coefficients that we emhasize here: 1. The first is that in the extremely relativistic limit the diffusion coefficients (and their ratios) for rotons and electrons are identical and assume asymtotic values such that both of the ratios are much less than 1. Therefore, equations (10), (11), and (12) are valid. (Strictly seaking, this is not true for very strongly magnetized lasmas 1=2 where one gets R 1 jr 2 j1; see eq. [5].) 2. The second is that at low energies, as ointed out by PP97, R 1 and R 2 2 are not necessarily less than 1, esecially for lasmas with low values of. In the extreme case of R 1 3 jr 2 j 3 1, three of the four diffusion terms in equation (1) can be ignored. Again, if we assume a finite homogeneous region or integrate over a finite inhomogeneous source, the resultant equation becomes similar to equation (10). Now because of the lower rate of itch-angle scatterings, the escae time may be equal to the transit time T esc L=(v), the other transort coefficients D EE and Ė L (and consequently the accelerated article sectra) may deend on the itch angle, and the assumtion of isotroy may not be valid. However, as can be seen in the next section (Figs. 5 and 6), these coefficients change slowly with, excet for some negligibly small ranges of, so that the exected anisotroy is small. In addition, at lower energies Coulomb scatterings become increasingly imortant and can make the article distribution isotroic. In many cases, esecially for lasmas not comletely dominated by the magnetic field (i.e., for 1), one can neglect the small exected anisotroy and integrate the equation over the itch angle, in which case the transort equation becomes identical to equation (10) excet now T esc ¼ L= ffiffiffi 2 vtsc 1=D ; ð14þ 2 2 ac ¼ R 1 1 d D () ; ð15þ where angle brackets denote averaging over the itch angle. 3. It is easy to see that one can combine the above two sets of exressions (eqs. [11] [15]) for the acceleration rates (or timescales) and the escae times at the nonrelativistic and extremely relativistic cases as T esc ¼ L ffiffi 2 L ffiffi 1 þ ; ac ¼ E2 ; ð16þ 2 v v sc D EE and D EE ¼ E2 2 Z 1 1 d D R 1 if R 1 3 j R 2 j3 1; R 1 R 2 2 if R 1 ; jr 2 jt1 ð17þ The first exression in equation (17) is valid at low values of E and and the second at higher energies and in weakly magnetized lasmas. However, it turns out that at extremely relativistic energies and in weakly magnetized lasmas ( >1), indeendent of other conditions, R 2 2 TR 1 and the first exression can be used. These exressions and equation (10) then describe the roblem adequately for most uroses in highenergy astrohysics, in articular for solar flares, the focus of this aer. 4. Finally, in certain cases, esecially in the intermediate energy range, the quantity R 1 R 2 2 aearing in equations (12) and (17) can be small. The acceleration rate can be reduced dramatically when both R 1 and jr 2 j are much less than 1 and R 1 R 2 2. From the definitions of these ratios and exressions for the F-P coefficients (eqs. [8], [9], and [5]) it is clear that if there were only one resonant interaction, one would have R 1 ¼ R 2 2 and there would be no acceleration. Thus, strictly seaking, the use of equation (10) with interactions involving only one wave mode (say the Alfvén) is incorrect. However, as we show in x 3.1, there are always at least two resonant interactions in unolarized turbulence, in which case R 1 6¼ R 2 2 so that the acceleration rate is finite. But if one of the interactions is much stronger than the others, R 1 R 2 2 can be small. In the next section we show some examles where this is true (Fig. 5) and that this haens at the intermediate values of energy (Fig. 6). The acceleration rate is then reduced greatly. The much lower acceleration rate at the intermediate energies comared to the higher rates in the nonrelativistic and extremely relativistic limits introduces an acceleration barrier. As we shall see, in the intermediate energy range the behaviors of rotons and electrons are quite different and a much stronger acceleration barrier aears for rotons Loss Rate To determine the distributions of the accelerated rotons and electrons by solving equation (10) with the above formalism, in addition to the transort coefficients D EE, A, and T esc, we need to secify the loss term Ė L. For electrons the loss rocesses are dominated by Coulomb collisions at low energies and by synchrotron losses at high energies: Ė Le ¼ 4r0 2 m ec 3 n e ln = þ B =9m e c 2 ; ð18þ where r 0 ¼ 2:8 ; cm is the classical electron radius and ln ¼ 20 is a reasonable value in our case (See Leach 1984). The ion losses in a fully ionized lasma are mainly due to Coulomb collisions with the background electrons and rotons (Post 1956; Ginzburg & Syrovatskii 1964). For electron-ion collisions, we have Ė Li ¼ 2r0 2 m ec 3 q i 2 n e 8 e 2 ffiffi 2 2 Te 3 ln 3 1=2 1 ln m2 e c2 4 >< r ; 0 n e f 2 m 2 e ln c2 2 2r 0 n e f 2 ln m em i c 2 >: 4r 0 n e f 2 for < Te; for 1 3 > Te ; for m i m e 3 3 1; for 3 m i m e ; ð19þ

5 554 PETROSIAN & LIU Vol. 610 where Te ¼ (3k B T e =m e c 2 ) 1=2 is the mean thermal velocity of the background electrons in units of c and k B is the Boltzmann constant. For roton-ion collisions, which are imortant for ions with even lower energies, we have (Sitzer 1956) Ė Li ¼ 4r0 2 m ec 3 n ðq i =eþ 2 m e =m 1 ln : ð20þ These loss rocesses dominate at different energies, and we can define a loss time loss ¼ E=Ė L Steady State Solution and Normalization We use the imulsive hase conditions of solar flares for our demonstration. In this case, we can assume that the system is in a steady state because the relevant timescales are shorter than the dynamical time (the flare duration). We also assume the resence of a constant sectrum of turbulence. We are interested in the acceleration from a thermal background lasma; therefore, a thermal distribution is assumed for the source term Q. As described above, equation (10) may not be valid at low (kev) energies where R However, for solar flare conditions and in the kev energy range, Coulomb scatterings become imortant (D Coul 3 D wave ; see Hamilton & Petrosian 1992). In this case R 1 T1 and the article distribution will be nearly isotroic at all energies. We therefore calculate the acceleration rate with the second exression of equation (17) and solve equation (10) to get the distributions of the accelerated articles over all energies. To areciate the relevant hysical rocesses, one can comare the acceleration time with the escae and the loss time. We are mostly interested in the energy range above the energy of the injected articles. Thus, the source term is not as imortant in shaing the sectrum as the other terms. In the energy band where the escae and loss terms are negligible, from the flux conservation in the energy sace, one can show that AN d(d EE N)=dE ¼ const. On the other hand, when the acceleration terms are negligible, no acceleration occurs. When the escae time becomes much shorter than the acceleration time and both of them are much shorter than the loss time, articles escae before being accelerated. This results in a shar cutoff in the article distribution at the energy where T esc E=A(E) E 2 =D EE. When the escae time is long and the loss time is much shorter than the acceleration time, one would then exect a quasi-thermal distribution for the Coulomb collisional losses (Hamilton & Petrosian 1992) and a shar high-energy cutoff for the synchrotron losses (Park et al. 1997). Power-law distributions can be roduced only in energy ranges where the loss term is small and the acceleration and escae times have similar energy deendence. The normalization of the steady state article distributions is determined by their rates of acceleration, escae, and injection. The injection rates deend on the geometries of the reconnection and the turbulent acceleration site and on ossible contributions of the charged articles to reverse currents that must exist when a net charge current leaves the acceleration site. A more detailed time-deendent treatment is required to determine the relative normalization. This is beyond the scoe of the aer and will be dealt with in the future. Here we concentrate on the relative shaes of the electron and roton sectra in the LT and FP sources. We assume that the injection flux R QdE ¼ 1s 1 cm 2 for both electrons and rotons (see also x 5). In the steady state this is equal to the flux of the escaing articles Nesc tot ¼ R 1 0 N LT (E )=T esc (E ) de. Since the escaing articles lose most of their energy at the FPs, instead of N esc (E ) ¼ N LT (E )=T esc (E ) we show the effective article distribution for a thick target (comlete cooling) FP source, which is related to the corresonding LT distribution N LT via (Petrosian & Donaghy 1999) N FP (E ) ¼ 1 Z 1 N LT ðe 0 Þ Ė L T esc ðe 0 Þ de0 : ð21þ E 3. APPLICATION IN COLD HYDROGEN PLASMAS In this section we describe the relative acceleration of electrons and rotons in cold, fully ionized, ure hydrogen lasmas. This is an aroximation because all astrohysical lasmas contain some helium and traces of heavy elements. Ignoring the effects of helium (trace elements will, in general, have no influence on the following discussion) and adoting a turbulence sectrum of a single ower law of the wavenumber simlify the mathematics and allow us to demonstrate the differences between the acceleration rates of electrons and rotons more clearly. Moreover, in some low-temerature lasmas, most of the helium may be neutral and not be involved in the SA rocesses. The results resented here are a good aroximation. Pure hydrogen lasmas can also be realized in terrestrial exeriments to test the theory. The formalism can also be easily generalized to the case of electronositron lasmas and to more comlicated situations. In the next section we resent our results for lasmas including about 8% by number of helium and for turbulence with a more realistic sectrum Disersion Relation and Resonant Interactions In a ure hydrogen lasma, equation (2) reduces to (PP97) k 2! 2 ¼ 1 2 (1 þ ) (! 1)(! þ ) ; ð22þ and the Alfvén velocity in units of c is given by A ¼ 1=2 =. (For e air-dominated lasmas ¼ 1). The left anels of Figure 1 deict the normal modes of these waves, which comose four distinct branches. From to to bottom in the to left anel, we have the electromagnetic wave branch (EM; long-dashed line), electron-cyclotron branch (EC; dot-dashed line), roton-cyclotron branch (PC; dotted line), and a second electromagnetic wave branch (EM 0 ; shortdashed line). The bottom left anel is an enlargement of the region near the origin. The ositive and negative frequencies mean that the waves are right- and left-handed olarized, resectively, where the olarization is defined relative to the large-scale magnetic field (Schlickeiser 2002). The right anels of Figure 1 deict the grou velocities g ¼ d!=dk of these waves. One may note that the signs of the hase velocity h ¼!=k and the grou velocity of a secific wave mode are always the same. In the left anels of Figure 1, the two solid straight lines deict equation (4) for an electron (uer line) andaroton (lower line) with ¼ 0:5 and ¼ 0:25. The intersections of these lines with the wave branches satisfy the resonance condition. The electron interacts resonantly at the indicated oint with the EC branch and at another oint with the PC branch at a high negative wavenumber that lies outside the figure. The roton, on the other hand, resonates with not only one PC wave but also three EC waves (only two of which are seen in the bottom left anel of the figure). As we show below, the fact that certain rotons can resonate with more than one

6 No. 1, 2004 ELECTRON AND PROTON ACCELERATION. I. 555 Fig. 1. Left: Disersion relation of arallel-roagating waves in a cold ure hydrogen lasma with ¼ 0:5. The bottom left anel is an enlargement of the region around the origin. The lines, from to to bottom, describe the EM (long-dashed line), EC (dot-dashed line), PC (dotted line), and EM 0 (short-dashed line) waves. The uer and lower solid lines give, resectively, the resonance conditions for electrons and rotons with v ¼ 0:5c ( ¼ v=c) and ¼ 0:25. Resonant interactions occur at the oints where these lines cross the lines that deict the waves. Right: Same as the left anels, but for the grou velocity g ¼ d!=dk vs. the wavenumber k. The line tye remains the same for each wave branch. Negative grou velocities mean that the energy fluxes of the waves are in the direction antiarallel to the large-scale magnetic field. [See the electronic edition of the Journal for a color version of this figure.] EC wave has significant imlications for the overall roton acceleration rocess Critical Anggles and Critical Velocities In general one exects four resonant oints. However, for a given article velocity or energy, at critical angles, wherethe grou velocities of the waves are equal to the arallel comonent of the article velocity, the number of resonant oints can change from four to two or vice versa. Figure 2 shows the velocity deendence of the critical angles for electrons (to) and rotons (bottom) in lasmas with ¼ 0:5 (left) and 0.1 (right). (The results for electrons are the same as those given by PP97.) Both articles have at least two resonant interactions (one with the PC and one with the EC branch excet for ¼ 0 where electrons interact with two EC waves and rotons interact with two PC waves). Electrons with a large can have two additional resonances with the EM branch, and those with a small have two additional resonances with the EC branch. The two regions with four resonances grow with decreasing and shrink as increases. For larger values of the interaction is weaker because for large ranges of velocities and itch angles electrons interact with only two waves (e.g., the interactions with the EM branch disaear for >1; see Figs. 2 and 3). But as aroaches zero, the region with two wave interactions diminishes and the two lines for the critical angles merge into one, satisfying the relation cr ¼ ( 1)= for! 0: ð23þ In this case there are always four resonances and the total interaction is strong at all energies. Protons have a similar, but slightly more comlicated, behavior. As increases, one obtains interactions with 1EC+3PC, 1EC+1PC, 3EC+1PC, and back to 1EC+1PC waves. With the decrease of, the uer two regions diminish, while the lower ortions increase in size. Protons can also be accelerated by the EM 0 waves, but this only occurs in more highly magnetized lasmas (< 1=2 =2 0:012) as comared with the interactions of electrons with the EM branch. At such low values of a region with four interactions (1EC+1PC+2EM 0 ) aears in the uer ortion of the - lane and its lower boundary eventually merges with the lower line for cr as aroaches zero. Just like electrons, the critical angle is given by equation (23). In this limit, articles are basically exchanging energy with the Poynting fluxes of the electromagnetic waves. These behaviors can also be seen in Figure 3, where instead of cr we lot what one may call the critical velocities as a function of for two values of. Note that in the roton anel with ¼ 1:0 there is a small region with <0:012 where there are four resonances, including two with the forwardmoving left-handed olarized electromagnetic waves from the EM 0 branch. Protons will not resonate with the electromagnetic waves for larger values of. In general, we have similar atterns of transition between different regions caused by the electromagnetic branches in the - sace, excet that the transitions for rotons occur at a value of that is lower than that for electrons by a factor of 1/2. The main difference in the behaviors of electrons and rotons resides in their four resonant interactions with the PC and EC branches. Protons have two such regions where they resonate with 1EC+3PC or 3EC+1PC, while electrons only have one with 3EC+ 1PC ; electrons never interact with more than one PC wave. This is where the above scaling symmetry of between rotons and electrons is broken. Low-energy aroximations. Because the acceleration of articles at low energies is of articular interest, we resent

7 556 PETROSIAN & LIU Vol. 610 Fig. 2. Left: Velocity deendence of the critical angles in a lasma with ¼ 0:5 for electrons (to) and rotons (bottom). At small itch angles, i.e.,! 1, electrons can resonate with high-frequency electromagnetic waves of the EM branch (region labeled 2EM+1EC+1PC ), while energetic rotons mainly interact with the Whistler and Alfvén waves (region labeled 3EC+1PC ). Right: Same as the left anels, but for ¼ 0:1 where interactions of rotons with the Whistler waves start at a higher energy. See text for details. here some aroximate analytic relations, which are derived in Aendices A, B, C, and D. The first is for the roton critical velocity line dividing the region with two and four resonances (i.e., the middle line in the bottom left anel of Fig. 3). As can be seen in the following sections, at a given the acceleration rate (eq. [17]) increases dramatically once rotons attain the critical velocity or energy and enter the region with four resonant interactions. The itch-angle averaged acceleration rate also increases sharly above this energy. It will be useful to have a formula to estimate this critical velocity. We find the following aroximate exression for this transition: cr ¼ 0:06 0:0012 þ 2 ; or E cr ¼ 1 2 m c 2 cr 2 ¼ 1:7 MeV 2 0:0012 þ 2 ; ð24þ Fig. 3. Same as Fig. 2, but deicting the deendence of the critical velocities on the lasma arameter. Combining this with Fig. 2, one can tell the wave branches resonsible for the critical velocities. [See the electronic edition of the Journal for a color version of the left anels of Fig. 3.]

8 No. 1, 2004 ELECTRON AND PROTON ACCELERATION. I. 557 Fig. 4. Comarison of the analytical exressions (eq. [25]; solid lines) with the exact values of cr (dotted and dashed lines; in the low energy range) due to resonant interactions with the EC (left, for electrons) or PC (right, for rotons; note that the region to the left of E 2 ¼ 40 kev has an exanded scale) branch. The thin dashed (barely visible near the left axis and for ¼ 1:0) and dotted (for ¼ 0:1) lines in the right anel give the critical angles for waves obeying the Alfvénic disersion relation! ¼ k jj A, which clearly give incorrect descritions for the acceleration of low-energy rotons. Low-energy electrons do not interact with the Alfvén waves. which is shown by the dotted line in the bottom left anel of Figure 3 and agrees within 0.2% with the exact result for >0:05. The second aroximation is for the critical angles of rotons below the critical energy (velocity) and low-energy electrons, most of which interact only with two waves with one dominating over the other. When this haens, the acceleration rate for the articles can be very small (see x 2.3). Only articles with very large itch angles ( 0) have four resonances and significant contributions to the itch-angle averaged acceleration rate. The regions for this lie in the small areas below the lowest lines in Figure 2, which are barely visible for the roton and ¼ 0:5 case. As shown in Aendices A and B using the aroximations of equations (A1) and (A2) for the disersion relations, we can derive analytic exressions for the critical angles, which in the nonrelativistic limit give cr / 2 / E=mc 2. Emirically, we find that the following simle aroximate exressions, as shown in Figure 4, agree with the exact results to better than 10% in the indicated energy ranges: cr ¼ 1 3:5 ( 1=2 E=m c 2 for rotons; E < E cr ; E=m e c 2 for electrons; E < 60 kev: ð25þ Here it should be emhasized that the commonly used aroximation of accelerating rotons by the Alfvén waves with the disersion relation! ¼ k j j A for j! j <, which is valid at relativistic energies (Barbosa 1979; Schlickeiser 1989; Miller & Roberts 1995), is invalid at low energies. This can be seen from the bottom left anel of Figure 1, which shows clearly that the disersion relation of the waves in resonance with low-energy rotons deviates far from the simle Alfvénic form. For the simle Alfvénic disersion relation, nonrelativistic rotons resonate with both a forward- and a backwardmoving wave only if jj < cr ¼ A ( 1)=. (Particles only interacting with one wave cannot be accelerated; see x 2.4.) This critical angle is indicated by the thin dashed (barely visible near the vertical axis) and dotted lines in the right anel of Figure 4 for ¼ 1:0 and 0.1, resectively, which clearly overestimate by several orders of magnitude the fractions of low-energy rotons that can be accelerated. The inefficiency of roton acceleration at intermediate energies, combined with the increase of the interaction rate above E cr, gives rise to the acceleration barrier to be described in x 3.4. In most of the article acceleration models, an injection rocess of high-energy articles is ostulated as an inut. If the injected articles have an energy above E cr,itmaybe aroriate to use the Alfvénic disersion relation to describe the waves. If the energy of the injected articles is low, as is the case under study here, one must use the exact disersion relation to calculate the acceleration of low-energy articles. Although most of the turbulence energy is carried by waves with low wavenumbers, the acceleration of low-energy articles is determined by waves with high wavenumbers (see eq. [4]), which can constrain the overall acceleration efficiency. As discussed above, the Alfvénic disersion relation for the waves will overestimate the acceleration efficiency of low-energy rotons significantly Fokker-Planck Coefficients For a given ower-law sectrum of the turbulence, it is straightforward to calculate the F-P coefficients with equation (5). Figure 5 shows variation of these coefficients with at some reresentative energies, and Figure 6 shows the variation of their inverses (i.e., times) with energy at different values of, here ¼ 0:5 andq ¼ 1:6. The discontinuous jums occur at the critical values of cr and cr described in x 3.2. The

9 558 PETROSIAN & LIU Vol. 610 Fig. 5. Pitch-angle deendence of the F-P coefficients (absolute values) for rotons of two different energies (8 MeV, left; 1GeV,middle) and 200 kev electrons (right). A, B, C, and D stand for (D D 2 =D )= 2, D / 2, jd j=, andd, resectively. The lasma arameter ¼ 0:5 and the turbulence sectral index q ¼ 1:6. Note that different coefficients are scaled differently and that, for illustration, in the middle anel the region to the left of ¼ 0:2 is exanded. variations of the two ratios defined in equations (8) and (9) with energy at different itch angles are shown in Figure 7. These results justify the discussion in x Barrier in the Proton Acceleration In the revious section we showed that the itch-angle averaged acceleration rate is one of the dominating factors in the article acceleration rocesses. The relative acceleration of rotons and electrons therefore deends on the contrast of their acceleration times. Figure 2 shows that at low energies articles with > cr only resonate with one PC and one EC wave: the EC wave dominates the PC wave for electrons, while the reverse is true for rotons. These articles have significant contributions to the itch-angle averaged acceleration rate for cr T1 (eq. [25]). Because the difference between the wavenumbers of the two waves interacting with rotons is much larger than that for electrons, the resonant interaction is more strongly dominated by one of the resonant waves for rotons than it is for electrons. The factor 1 R 2 2 =R 1 for rotons can therefore be several orders of magnitude smaller than that for electrons at a given energy and itch angle (Fig. 6). Consequently, in the intermediate energies where R 1 R 2 2 1, the itch-angle averaged acceleration time for rotons has a more rominent increase than that

10 No. 1, 2004 ELECTRON AND PROTON ACCELERATION. I. 559 Fig. 6. Energy deendence of the scattering, 1=D ¼ D 1, and acceleration times, 1=B ¼ 2 =D or 1=A ¼ 2 =(D D 2 =D ), in units of at different itch angles for electrons (left) and rotons (right). The lasma arameter ¼ 0:5 and the turbulence sectral index q ¼ 1:6. Note that A ¼ B for ¼ 0. At very low energies D 1 > 2 =D and the acceleration time ac ¼ 2 =D. At intermediate and high energies ac ¼ 2 =(D D 2 =D ). [See the electronic edition of the Journal for a color version of this figure.] for electrons. At still higher energies, articles with four resonances dominate the acceleration rate because R 2 2 TR 1 for the interactions. For both electrons and rotons, the new resonant wave modes come from the EC branch. In the relativistic limit, the acceleration is dominated by resonances with the Alfvén waves, and the interaction rates for electrons and rotons become comarable. Figure 8 shows the itch-angle averaged acceleration (thick lines) and scattering (thin dashed lines) times in units of for electrons (lower lines) and rotons (uer lines) in lasmas with ¼ 0:5 (left) and 0.1 (right). The turbulence sectral index q ¼ 1:6 here. The acceleration times for both cases in equation (17) are lotted with the invalid segments shown by dotted lines. We see that the itch-angle averaged Fig. 7. Ratios R 1 (left) andr 2 2 (right) as functions of energy for several different itch angles for electrons (to) and rotons (bottom). In general, R 1 and jr 2 j exceed unity at low energies and (and small values of ; see PP97). For rotons, at certain ranges of E and, R 2 2 R 1T1, and the acceleration rate, D (R 1 R 2 2 ), can be very small (see eq. [12]).

11 560 PETROSIAN & LIU Vol. 610 Fig. 8. Pitch-angle averaged acceleration (thick lines) and scattering (thin dashed lines) times in units of in lasmas with ¼ 0:5(left )and0.1(right ). Here the turbulence sectral index q ¼ 1:6. The acceleration times defined for both cases of eq. (17) are lotted with the corresonding invalid segments indicated by the dotted lines. The uer two lines are for rotons and the lower two are for electrons. The circles indicate the oints of transitions between low and high energies where R 1 jr 2 j1. The transitions of the electron acceleration times are quite smooth. The thin solid lines show schematically the transitions of the acceleration time for rotons. The acceleration barrier (as indicated by the hatched areas) in the roton acceleration times is rominent. The shar dro of the electron acceleration time with the decrease of energy for ¼ 0:1 is due to interactions with the EM branch (PP97). [See the electronic edition of the Journal for a color version of this figure.] acceleration times are much shorter than the corresonding scattering times for kev articles. The article distributions can be anisotroic at these energies unless there are other scattering rocesses (e.g., Coulomb collisions). In the highenergy range, the scattering time is always shorter than the corresonding acceleration time when A < 1. The transitions where R 1 jr 2 j1 (as indicated by the circles in the figure) occur between 10 2 and 10 3 kev, increase with the decrease of, and deend on the turbulence sectral index q as well. There is clearly an acceleration barrier (as indicated by the shaded area) in the roton acceleration time. The thin solid line shows schematically the acceleration time of rotons in the transition region. The shar increase of the roton acceleration time at lower energies is caused by their low acceleration efficiency when the scattering rate already overtakes the acceleration rate as discussed above. The shar dro of the roton acceleration time at a higher energy is due to interactions with the Whistler waves. Because rotons with small itch angles ( 1) interact with the Whistler waves at the lowest energy and the interaction is very efficient, this energy corresonds to the critical energy E cr identified in equation (24). These characteristics are not true for electrons. In x 3.2 we have shown that a small fraction of articles with < cr can resonate with four waves and jr 2 jtr 1 for the interactions (Fig. 5). Comared with rotons, more electrons can be accelerated this way (eq. [25]). Because the acceleration of electrons with two resonances is very inefficient, the acceleration of this small fraction of electrons already dominates the electron acceleration rocesses where the scattering rate becomes comarable with the acceleration rate. At even higher energies, there are no extra wave modes that can enhance the electron acceleration rocesses. Consequently, electrons have a smooth acceleration time rofile. Exressions for estimating the difference between electron and roton acceleration time are derived in Aendices A, B, C, and D. Briefly, because articles with < cr have significant contributions to the acceleration in the low-energy region (T1), one can estimate the itch-angle averaged acceleration times with the aroximate exressions for the critical itch angles (eq. [25]): ac ¼ 2 Z qþ2 E m e c 2 d D R 1 R 2 2 (1 q)=2 1 1 for electrons; 5=2 for rotons; ð26þ which is consistent with the numerical result within a factor of 2. In the relativistic limit ( 3 1), articles interact with the Alfvén waves and we find 2 ac q(q þ 2) e 2 q E 2 q ¼ 4 q i m e c 2 : ð27þ The difference between these two timescales at the critical energy (eq. [24]) gives an estimate of the height of the acceleration barrier Alication to Solar Flare Conditions Figure 9 shows a model of electron (thick lines) and roton (thin lines) acceleration in a strongly magnetized lasma. The LT size L ¼ 10 9 cm, and the temeratures of the injected electrons and rotons are the same, k B T ¼ 1keV.Themagnetic field and gas density are 400 G and 4:0 ; 10 9 cm 3, resectively, i.e., ¼ 0:5 (see eq. [3]). The relevant timescales are shown in the left anel, where we have defined the

12 No. 1, 2004 ELECTRON AND PROTON ACCELERATION. I. 561 Fig. 9. Left: Timescales for rotons (thin lines) and electrons (thick lines) in a strongly magnetized lasma ( ¼ 0:5) with a stee turbulence sectrum (q ¼ 3). The direct acceleration times a ¼ E=A, which are related to the diffusion acceleration times ac, are shown by the dotted lines. The solid lines are for loss, and the dashed lines are for T esc. Right: Corresonding distributions NE 2 of the accelerated electrons (thick lines) and rotons (thin lines). Here the dotted lines give the thin target LT article distributions, while the dashed lines indicate the effective thick target article distributions at the FPs. The solid line gives the injected thermal article distribution with arbitrary normalization. One can see that, because of the resence of the strong acceleration barrier (exceeding the range of the grah), rotons are basically not accelerated. [See the electronic edition of the Journal for a color version of this figure.] direct acceleration time a ¼ E=A, which is related to ac (eqs. [12] and [13]). The corresonding accelerated article distributions are shown in the right anel. The dotted and dashed lines show the LT and FP sectra, resectively. We note that for the above conditions the electrons can be accelerated to a few hundred kev while the roton acceleration is suressed as a result of the acceleration barrier. The electron distribution steeens with the increase of energy because the escae time becomes shorter than the acceleration time (T esc < a ). At low energies where Coulomb collisions dominate, the LT electrons have a quasi-thermal distribution. The solid line gives the thermal distribution of the injected articles with arbitrary normalization. Because of the absence of acceleration at low energies, the steady state roton distributions are almost identical with the injected roton distribution. To roduce a near ower-law electron distribution, as suggested by solar flare observations, the escae time must be comarable with the acceleration time in the relevant energy band. Because the escae time of the nonrelativistic electrons always decreases with the increase of energy, we adot a turbulence sectral index of q ¼ 3 in the model. The lasma time ¼ 1 s. Then we have the ratio of the turbulent wave energy density to the magnetic field energy density 8E tot =B 2 0 ¼ 4:4 ; kmin 2. To ensure that this ratio is much less than 1 so that the quasi-linear aroximation for the F-P treatment stays valid, one needs k min > 10 5 or an injection length of the turbulent waves of less than 10 6 cm (note that k min is in units of e =c 0:24 cm 1 here). Otherwise, the turbulence sectrum must flatten at low k so that there is less energy content in long-wavelength waves. In xx 3.2 and 3.4 we showed that the roton acceleration barrier moves toward lower energies with the increase of. Hence, in very weakly magnetized lasmas, this barrier can be close to the thermal energy of the injected articles and thus has little effect on the acceleration of rotons. Protons can be accelerated efficiently in the case because their loss time is long. Figure 10 shows such a model, where B ¼ 100 G and n ¼ cm 3. The size of the LT and the injected article temeratures remain the same as those in the revious model. Because the turbulence sectrum is flat (q ¼ 2),wehavea retty hard accelerated roton distribution below 1 MeV. Above this energy, there is a cutoff due to the dominance of the escae term over the acceleration terms. The accelerated electron distribution has a cutoff at less than 100 kev, which is also due to the quick escae of electrons with higher energies from the acceleration site. At a few kev, both electron and roton distributions are quasi-thermal because of the dominance of Coulomb collisions. The above results show that electrons can be accelerated to very high energies by arallel-roagating turbulent waves in ure hydrogen lasmas, but the resence of the acceleration barrier in the intermediate energy range makes the acceleration of rotons very inefficient. Only in very weakly magnetized lasmas where the barrier is close to the background article energy does the acceleration of rotons become efficient. The required value of the lasma arameter is above 10, which is much larger than that believed to be the case for solar flares. However, most astrohysical lasmas including solar flares are not made of ure hydrogen. They contain significant numbers of 4 He. These articles modify the disersion relation used above. Abundances of elements heavier than He are too small to have a significant effect, but 4 He with an abundance (by number) of about 8% can have imortant effects. To roduce a near ower-law distribution of the accelerated articles, the index of the turbulence sectrum must be larger