Electron acceleration in solar noise storms Prasad Subramanian Indian Institute of Astrophysics, Koramangala, Bangalore- 560034, India e-mail: [email protected] 7 0 0 Abstract 2 Wepresentanup-to-datereviewofthephysicsofelectronaccelerationinsolarnoisestorms. Wede- n a scribe the observedcharacteristics of noise storm emission, emphasizing recent advances in imaging J observations. Webrieflydescribethegeneralmethodologyoftreatingparticleaccelerationproblems 3 and apply it to the specific problem of electron acceleration in noise storms. We dwell on the is- 2 sueoftheefficiencyoftheoverallnoisestormemissionprocessandoutlineopenproblemsinthisarea. 1 1. Introduction: v 7 4 1.1 Motivation: 6 1 Noise storms are the most common form of meter wavelengthradio emission from the solar corona. 0 Thenomenclaturearisesfromhissingsoundsproducedinshort-waveradioreceivers,andwascoined 7 0 aroundthe1930s. Noisestormsaresitesoflong-lastingquasi-continuouselectronaccelerationinthe / solar corona,and we will focus on this aspect here. Electron acceleration (and particle acceleration h in general)is of centralimportance in severalastrophysicalproblems. A thoroughunderstanding of p - this process in the solar corona can therefore be of considerable use in understanding its import in o objects that are farther away and less accessible to observations. r t s a 1.2 Brief History: : v i The recognition that the sun could be a source of intense meter wavelength emission took place X around1942,when the operationsof Britishanti-aircraftradarwere severely affected by such emis- r sions. However, owing to the then-ongoing war, the scientific results were not published until 1946 a (Hey[4]),andthisheraldedthebirthoftherichfieldofsolarradiophysics. Agoodoverviewofmeter wavelengthsolar phenomena can be found in McLean & Labrum[5]. More recent overviewsof solar radio emission that are not confined only to meter wavelength phenomena can be found in Gary & Keller[3] and Bastian & Gary[1]. 1.3 Brief overview of solar meter wavelength emission There are several kinds of emission from the solar corona at meter wavelengths, which have dis- tinctobservationalsignatures. Someexamplesaregiveninfigures1and2. Thesefiguresarecalled dynamic spectra. They are multifrequency records of intensity from the entire sun, and contain no spatial information. Time runs along the x-axis of these figures and the observing frequency is along the y-axis. The observed intensity is represented by a colorscale/grayscale. Different kinds of emission are characterized by different kinds of characteristic signatures on such dynamic spec- tra. In figure 1 for instance, the emission labelled ’type 2’ has a characteristic signature where the bright emission drifts downwards in frequency with time. It is taken to be a signature of elec- trons accelerated at a shock front that is travelling outwards through the solar corona. The kind 1 Figure 1: Examples of types 2 and3 meter wavelength radio emission: Thisis adynamic spectrum from the HiraisoradiospectrographinJapan. Timerunsalongthex-axis,andfrequencyalongthey-axis. Theintensity fromthewholesunisexpressed bythecolorscale. of emission labelled ’type 3’ comprises of bright, almost vertical tracks on the dynamic spectrum; this kind of emission is taken to be a signature of relativistic electrons escaping outwards through the solar corona along open magnetic field lines. On the other hand, the kind of emission depicted in figure 2 is called noise storm (or type 1) emission. It is rather unspectacular, and comprises of a broadband continuum that lasts for several hours to days on which there are superposed several randomlydistributed,shorttimescale(0.1–1second)narrowband,intensebursts. Thiskindofemis- sionis thought to be causedby nonthermal/acceleratedelectrons. We will concentrate here only on these noise storms. The reasons are twofold; on the one hand, they are the most common signature of accelerated electrons at meter wavelengths, and have been very well observed. Owing to their ubiquitous and long-lastingnature,they arealso wellsuited to repeated observations. Onthe other hand, some important aspects of noise storm emission (both with regard to the continuum and bursts) that are still rather poorly understood, despite decades of study. Furthermore, the plasma emission process, (which we will describe later) which is thought to be operational in noise storm emission,isalsocentraltootherkindsofmeterwavelengthemissionsuchastype2,3and5emission. A thorough understanding of the noise storm phenomenon is therefore of considerable utility. 1.4 Organization of paper: The rest of this paper is organized as follows: we describe further details of the observed charac- teristics of noise storm emission in 2. We pay particular attention to multifrequency observations § of noise storms and to recent imaging observations that have the potential to significantly impinge on some long-unsolvedtheoretical issues. We next turn our attention to the physics of the emission process via which the observed noise storm radiation is produced in 3. We emphasize the role of § 2 Figure 2: Exampleof type1 emission: rest of thecaption same as thatforfigure1. acceleratedelectronswhicharethe startingpointofthe overallemissionprocess. The mainfocus of this paper is the manner in which electrons are accelerated in a quasi-continuous manner to power noise storm emission at meter wavelengths. Accordingly, we review the development of standard theoretical treatments of particle acceleration in 4. We first elaborate on the physical scenarios § whereparticlesareaccelerated,suchasreconnectionregionsandshocks. Wethentracethedevelop- ment of particle transportequations that treat diffusion in velocity/momentum space and yield the Fermi acceleration scenario. We then turn to the specific problem of electron acceleration in noise storm sources in the solar corona in 5. We estimate the power input to the accelerated electrons § and compare it with the power observed in the observed radiation, deriving an efficiency for the overall process. In 6 we outline the applicability of such an efficiency estimate to other kinds of § radio emission from the solar corona, and its role in furthering our understanding of solar coronal transients. We also include a brief discussion of open problems in this interesting area. 2. Observations of noise storm emission 2.1 Multifrequency observations: The most basic identification of type1/noise storm emission, as mentioned earlier, is from dynamic spectra (figure 1). Type 1 emission is characterized by a long-lived (few hours to days), wideband (δf/f 100%) continuum, together with several intense, randomly interspersed short-lived (0.1– ∼ 1s), narrowband (δf/f 1) bursts. Early reviews on noise storms can be found in Wild et al.[8], ≪ Kundu[9] and Kruger[10]. An extensive review of noise storm observations is given by Elgaroy[2]. Noisestormsaretypicallyobservedfrom50–500MHz,andarebrightestaround100-200MHz. The detailed multifrequency characteristics of noise storms have not been studied very well; to the best of our knowledge, the only such study after the early pioneering study of Smerd[7] have been those 3 of Thejappa & Kundu[13], Kundu & Gopalswamy[12] and Sundaram & Subramanian[6]. While Smerd’s[7] early results showed that noise storms tended to be brightest around 100 MHz, with in- tensities tapering off on either side of this (approximate) frequency, Sundaram & Subramanian’s[6] more recent, relatively sophisticated studies were confined only to the 50–80 MHz frequency range, and showed that noise storm intensities clearly rose as a function of frequency in this range. We will have occasion to comment on the interesting implications of such multifrequency observations in 6. § 2.2 Imaging observations of noise storms: EarlyimagingobservationsofnoisestormsweremadewiththeCulgooraradioheliographinAustralia (e.g.,Dulk & Nelson[11]). Lateron,noise stormswereimagedwiththe ClarkLakeRadioheliograph in the USA (e.g., Kundu & Gopalswamy[12]; Thejappa & Kundu[13]). Around the same time, the Nancay radioheliograph (NRH) in France also carried out extensive noise storm observations (e.g., Kerdraon & Mercier[14]; Malik & Mercier[15]). Occasional noise storm observations are also car- ried out with the Very Large Array (VLA) in the USA (Willson, Kile & Rothberg[16]; Willson[17]; Habbal, Ellman & Gonzalez[18]; Habbal et al.[19]). Lately, there have been attempts at combining visibilities from the Giant Metrewave Radio Telescope (GMRT) in Pune, India with those from the NRH to obtain images of noise storm sources in the solar corona. This approach combines several complementaryadvantagesofferedbythesolar-dedicatedNRHandtheGRMT.Ityieldsthehighest dynamic range images of the solar corona at meter wavelengths (Mercier et al.[20]). Figures 3 and 4 show a couple of examples of results obtained by using this technique. This technique allows Figure3: 17secondsnapshotofthesolarcoronaat327MHzaround09:04UTonAug272002. Theresolution ′′ of this image is 49 and the rms dynamic range is 283. The bright noise storm emission in the southwestern quadrantis clearly evident, as are thetwo weak sources and the interveningdiffuseemission near disk center. Thisimage was madebycombiningvisibilitiesfromtheNRHandtheGMRT (Mercier etal.[20]). 4 Figure4: 2secondsnapshotofthesolarcoronaat236MHzaround10:36UTonApr062006. Theresolution ′′ is29 and therms dynamicrange 447. Thisimage was madebycombiningvisibilitiesfrom theNRHandthe GMRT(Mercier etal.[20]). dynamic ranges of 300-500 to be achieved with integration times of a few seconds. Such dynamic ranges were previously obtained only with integration times of around 3–4 hours. High dynamic ranges allow simultaneous imaging of bright and dim features, as is evident from figure 3. This feature is rather important in several situations, for bright noise storms are often accompanied by relativelydim, large-scalefeatures suchas coronalmass ejections (e.g., Bastianet al.[21]; Habbal et al.[19];Willson[22];Willson[23]). Thehighresolutionaffordedbythistechniquecanbeimportantin resolving the important question of angular broadening due to coronalturbulence (e.g., Bastian[24] and references therein). While there are already some hints that theoretical estimates of angular broadening might be somewhat exaggerated (Zlobec et al.[25]), this technique has the potential to set a definitive lower limit on the smallest observable source size in the solar corona. It can also set a firm upper limit on the brightness temperature of noise storms, which can have important implications for the plasma physics involved in the emission process (Robinson[26]; Kerdraon[27]). 3. Physics of noise storm emission process: overview As mentioned earlier, noise storms are sites of long-lasting, quasi-continuous electron acceleration in the non-flaring corona (e.g., Raulin & Klein[29]; Klein[28] and references therein). They have been observed to occur in conjunction with other transient events such as coronal mass ejections and soft X-ray brightenings. In general, noise storms seem to be associated with emergence of new material in the corona and/or magnetic restructuring of some kind (e.g., Kerdraonet al.[30]; Bent- ley et al.[31]). Such rearrangements of magnetic fields are envisaged to take place via the process of magnetic reconnection (e.g., Priest & Forbes[32]; Biskamp[33]), which results in the release of magnetic energy and consequent heating/acceleration of particles. Magnetic reconnection is also 5 thoughttoresponsibleforaccelerationofparticlesinviolenteruptiveeventsinthesolarcoronasuch as flares. Emerging magnetic flux into the solar corona can also drive an ensemble of weak shocks, which can accelerate electrons (Spicer, Benz & Huba[34]). We will return to the subject of electron acceleration in the next section. Fornow,wenotethatmostoftheoreticaltreatmentsofnoisestormemissionsimplyassumethe presence of nonthermal electrons and proceed from there. If this nonthermal electron distribution has an anisotropy in velocity and/or physical space, it will emit an intense population of Langmuir wavesthroughamechanismanalogoustoCherenkovemission(e.g.,Robinson[26];Mikhailovskii[35]; Melrose[36,37,38]and references therein). The anisotropy can be viewed as a source of free energy that is givento the populationof Langmuirwaves. The Langmuir waves(alsoreferredto as plasma waves) comprise bodily oscillations of the plasma. These waves coalesce with a suitable population of low frequency waves such as ion-sound waves or lower hybrid waves (Wentzel[39]) in order to produce the observed electromagnetic (radio frequency) emission. This overall picture is usually called the “plasma emission” hypothesis, and is usually invoked for high brightness temperature emission that necessitates a coherent emission mechanism. Itmaybe notedthatallthetheoreticalrequirements,startingfromthatofelectronacceleration and including the plasma emission process, are much more severe for the very intense, short-lived (0.1–1sec)type1burststhantheyareforthelong-lived(fewhourstodays)backgroundcontinuum. Detailedmeasurements(Malik&Mercier[15];Kruckeretal.[40])showthattheburstsaremuchmore compact than the continuum source, and that they tend to move randomly within the continuum source. Type1 bursts couldbe manifestations of“nanoflares”,whicharesmallelementalreleasesof energyviasmall-scalereconnectionsandaresuspectedtoberesponsibleforheatingtheambientsolar corona(Mercier&Trottet[41]). Thephysicaloriginofsuchburstscouldbestatisticalnonequilibrium fluctuations in the coronal plasma, and there are few theories that attempt to treat the continuum and bursts in a self-consistent manner (e.g., Thejappa[42]). 4. Particle acceleration: 4.1 Physical situations for particle acceleration: reconnection sites and shocks Whilethereareseveralinterestingquestionstobe answeredwithregardtonoisestormemission,we will focus here only on the aspect of electron acceleration, which is the starting point for the entire process. Wefirstbrieflydescribethephysicalsituationswhereparticlesaretypicallyacceleratedand follow it up with typical theoretical treatments. It may be emphasized that particle acceleration in general is ubiquitous in astrophysics, and is responsible for several interesting phenomena, ranging from ultra-high energy cosmic rays to radio and high energy emission from extragalactic jets to several situations in the solar corona and the earth’s bowshock and magnetotail. Even in the solar corona,particle accelerationis responsible for observationsassociatedwith diverse phenomena such as nonthermalradio bursts, flaresand coronalmass ejections (see, for e.g., Aschwanden[43]). In the solarcorona,particles are thought to be acceleratedprimarily due to magnetic reconnection(Priest & Forbes[32]; Biskamp[33]) and at shocks (e.g., O’C Drury[44]; Quenby & Meli[45]; Malkov & O’C Drury[46]; Jones & Ellison[47]). Reconnectioncanbebroadlythoughtofasaprocessbywhichstressedmagneticfieldsrearrange their topology in order to relax to a lower energy configuration. The excess energy is partially expended in accelerating particles. From a microscopic standpoint, the process of reconnection involves two oppositely directed magnetic field lines coming very close to each other and eventually changing their connectivities. When oppositely directed magnetic fields come close to each other, there will be a large magnetic field gradient in the reconnection region, necessitating the presence 6 of a strong electric field in the plane perpendicular to the one which contains the magnetic fields. This electric field is capable of accelerating particles (e.g., Onofri, Isliker & Vlahos[48]; Turkmani et al.[49]; Arzner et al.[50]). However, this is a complex problem involving treatments in both the magnetohydrodynamicandkineticregimes,anditisprobablyfairtosaythatweareonlybeginning to develop an understanding of this process. Direct electric field acceleration apart, there can also be substantial turbulence in the vicinity of the reconnection regions; specifically, the reconnection outflows are typically turbulent. Particles can resonate with part of the turbulent wave spectrum and gain energy via wave-particle interactions. Thesolarcoronaisforeverinastateofflux,andmagneticfluxisconstantlybeingadded/removed orbeingmovedaround. Thereisthusplentyofscopeforreconnectiontooccurandreorganizefields onseveralscales,bothsmallandlarge. Assuch,reconnectionisheldresponsibleforseveraltransient phenomena in the solarcoronasuch as the initiationof flaresand coronalmassejections. Before we leave this discussion, it may be noted that the wide scope of magnetic reconnection and its crucial importance has spurred laboratory studies to investigate this phenomenon in detail (e.g., Yamada et al.[51]). The other agentthatis typically invokedforparticle accelerationis a shock. Roughly speaking, a shock is formed when a pressure/temperature/density disturbance propagates through a medium ata speed that is greaterthanthe characteristicspeed for the propagationofsmall pressuredistur- bances in that medium. For an unmagnetized medium this means that shocks are typically formed when a disturbance travels at supersonic speeds. For magnetized media, the situation is somewhat more complicated, since there are several characteristic speeds to consider: the Alfven speed and the slow and fast magnetosonic speeds, for instance. The shocking agent can be a localized energy release, such as that in a flare, which causes what is referred to as a “blast wave” shock, akin to that in a supernova explosion. It canalso be a piston, like a coronalmass ejection. The shock itself canbeviewedasapropagatingdiscontinuity(inallphysicalquantitiessuchastemperature,density and pressure). There is usually some form of turbulence present at the shock front which enables particles to diffuse back and forth across it. A particle that diffuses from the upstream side of the shock onto the opposite (i.e., downstream) side will collide with scattering centers moving with the shock and gain energy. If it manages to diffuse back upstream and then back downstream, it will gain more energy in a second collision. This is a rough description of what is usually referred to as diffusive shock acceleration (e.g., O’C Drury[44]; Quenby & Meli[45]; Malkov & O’C Drury[46]; Jones& Ellison[47]). As with reconnection,shock accelerationis a mechanismthat is applicable for a wide variety of astrophysicalphenomena,not to mention severalobservationalfeatures in connec- tionwiththesolarcorona. Inthecontextofnoisestorms,electronsarepresumedtoacceleratedbya seriesofweakshocksdrivenbytheemergenceofnewmagneticfluxintothecoronafrombeneaththe photosphere (Spicer, Benz & Huba[34]). Emerging magnetic flux can also cause repeated episodes of small-scale reconnection, which can be accompanied by electron acceleration. 4.2 Mathematical treatment of particle acceleration: Having discussed the physical situations where particles can be accelerated, we now turn our at- tention to theoretical methods of treating this process. We start with the collisionless Boltzmann equation Df ∂f ∂x∂f ∂p∂f = + + =0. (1) Dt ∂t ∂t ∂x ∂t ∂p The quantity x represents a spatial coordinate and p represents a momentum coordinate and the 7 distribution function f is normalized as follows to yield the number density n: n= dpf(p) (2) Z Forthesakeofsimplicity,wehavewrittentheBoltzmannequationwithonlyonespaceandmomen- tum dimension; it can easily be generalized to include three dimensions in each of these quantities. The intepretation of equation (1) is simple: it says that, in the absence of collisions, the number of particles in the x-p phase space is conserved. In other words, the shape of the elemental volume dxdp itselfcanbe distortedwithtime,butaslongastherearenocollisions,thenumberofparticles inthis volumewillstay constant. Itis worthnoting thatthe well-knownthermalMaxwelliandistri- butionisanequilibriumsolutionofthecollisionlessBoltzmannequation. Inotherwords,anyinitial particle distribution will eventually relax to a Maxwellian, given enough time. However, this is a highly idealized equation; in reality, there will be several kinds of collisions that can move particles in and out of the dxdp volume. These can be represented on the right hand side of the Botlzmann equation as Df ∂f = C +C = , (3) Dt − out in ∂t (cid:12) (cid:12)c (cid:12) where Cin represents the flux of particles entering and Cout t(cid:12)he flux of particles leaving the phase space as a result of collisions. It may be noted that these collisions need not actually be physical collisions; they can be any process that scatter particles in an elastic or inelastic manner. For instance, they couldrepresentintermittent episodes of accelerationthat a particle mightexperience inmovingthroughreconnectionregionsorinteractionswithpartofaturbulentwavespectrumwith which the particle resonates. If the cumulative effect of severalsmalldeflection collisions dominates overlargedeflectioncollisions,itturnsoutthatthecollisionterm(eq.3)canbeconvenientlywritten in what is called a Fokker-Planck form. A rigorousderivation of the Fokker-Planckequationcan be foundinplasmaphysicstextbookssuchasSturrock[52]andMontgomery&Tidman[53]. TheFokker- Planckequationcanalsobeheuristicallymotivatedbyconsideringasituationwherestochasticforces acting on a typical particle cause its momentum vector to execute a random walk as a function of time and diffuse in momentum space. In addition to this, the particle would also experience a drag/frictional force while moving through a dilute medium. The Fokker-Planck equation for such a situation can then be written as ∂f ∂ 1 ∂2 = Bf + Cf , (4) ∂t (cid:12) −∂p(cid:18) (cid:19) 2∂p2(cid:18) (cid:19) (cid:12)c (cid:12) (cid:12) wherethefirsttermontherighthandsideisthefrictionaltermandthesecondrepresentsdiffusionin momentumspace. The solutionofequation(4)withonlythe firsttermofthe righthandsidewould beaGaussianinmomentumspacewhosemeankeepsdecreasingwithtime; hencethe nomenclature ’drag’ term. On the other hand, if only the second term on the right hand side were retained, the solution would be a Gaussian whose variance (in momentum space) increases as a function of time, while its amplitude decreases so as to conserve the area under the curve. This shows why it is called the diffusion term. Although the frictional term as written in equation (4) represents energy loss, it could well representenergy gain if B is negative; this represents a process called first order Fermi acceleration, andoccurs in situations where the scattering centers move systematically, imparting energy to the particle. The import of the second (diffusion) term on the right hand side ofequation (4)is somewhatless obvious;however,if it is recognizedthat the averagemomentum of aparticleisthefirstmomentofthe distributionfunction(the zerothmomentisthenumberdensity, aspereq2),itcanbeunderstoodthatparticlesdiffusing towardshighermomentaarepreferentially 8 weighted, so that the mean momentum actually increases (see appendix of Subramanian, Becker & Kazanas[54]). ThiseffectisreferredtoassecondorderFermiacceleration. Fermioriginallyconceived these acceleration processes (especially the second order process) from a kinematic point of view, as a consequence of a test particle bouncing off stochastically moving clouds. A treatment of Fermi acceleration from this viewpoint can be found in Longair[55]. In addition to these effects, there can also be particles escaping from and also injected into the system. The full transport equation including these terms can be written as (Becker, Le & Dermer[56]): ∂f 1 ∂ ∂f f = p2 A(p)f D(p) +S(p,t), (5) ∂t (cid:12) −p2∂p(cid:18) (cid:20) − ∂p(cid:21)(cid:19)− t (p) (cid:12)c esc (cid:12) where the first term on(cid:12)the right hand side is and equivalent representation of the right hand side of equation (4). The second term on the right hand side represents particle escape from the system with a mean timescale t while the last term on the right hand side represents particle injection esc into the system. This completes our cursory overview of the mathematical formulation of particle acceleration. In addition to what we have described, it may be mentioned that there can be additional terms on the right hand side of equation 5), such as diffusion in physical space and losses of various kinds (e.g., Becker[57]). There are a variety of numerical and analytical techniques that are employed to solve such particle transport equations. Much effort is devoted to calculating various parameters of the equationsuch as the diffusion coefficient in momentum space, escape timescale, etc., for a given physical situation. For an extensive treatment of such transport equations and various methods of solution, see Schlickeiser[58]. 5. Electron acceleration in noise storms: We now focus on the specific problem of electron acceleration in solar corona. We confine our attention primarily to type I noise storm continua, rather than the sporadic type I bursts, because we are interested in examining the basic energetics of the electron acceleration processes responsi- ble for producing the quasi-continuous radio emission. Most theories of type I phenomena invoke nonthermal electrons as a crucial ingredient in producing the observed radiation. However, little attention has been focused on this problem in the previous literature. The majority of the theories simplyassumethatnonthermalelectronsarepresent,andfocusmostoftheirattentiononexamining thewave-waveinteractionprocessesthroughwhichobservableradioemissionisultimatelyproduced. Althoughthereiscurrentlynotheoreticalconsensusregardingthefundamentalmechanismpowering type I phenomena, there is no question that nonthermal electrons are responsible for the observed emission. In view of the considerable uncertainty surrounding the precise physical mechanism that results in the formation of the nonthermal portion of the electron distribution, we follow the work of Subramanian & Becker[59,60] in characterizing the acceleration in terms of a generic, stochastic (second-order) Fermi process such as the one described in 4. We first note that the total electron § number density, n , including both thermal and nonthermal particles, is related to the momentum e distribution f via n (cm 3) = ∞p2fdp , (6) e − Z 0 where p is the electronmomentum. This is somewhat different from the normalizationexpressed in equation (2). The associated total electron energy density is given by U (ergcm 3) = ∞ǫp2fdp= 1 ∞ p4fdp , (7) e − Z 2m Z 0 e 0 9 where m is the electron mass and ǫ = p2/(2m ) is the electron kinetic energy. We are mainly e e interestedinthe nonthermalelectrons,whicharepickedupfromthe thermalpopulationandsubse- quentlyacceleratedtohighenergies. Inthesituationwheretheaccelerationtimescaleissmallerthan anyrelevantloss timescales,the high-energytailofthe electrondistributionfunction is governedby a rather simple transportequationthat describes the diffusion of electrons in momentum space due to collisions with magnetic scattering centers. The validity of this assumption is examined in detail in Subramanian & Becker[59]. The time evolution of the Green’s function for this process, f , is G described by ∂f 1 ∂ ∂f N˙ δ(p p ) f G = p2 G + 0 − 0 G , (8) ∂t p2 ∂p(cid:18) D ∂p (cid:19) p2 − τ 0 where is the (as yet unspecified) diffusion coefficient in momentum space and τ is the mean D residence time for electrons in the accelerationregion. The source term in equation(8) corresponds tothe injection intothe accelerationregionofN˙ particlesper unitvolumeper unittime, eachwith 0 momentump . Althoughwe do notexplicitly include lossesdue to the emissionofLangmuir/upper 0 hybrid waves by the accelerated electrons, it is expected that these waves will be generated as a natural consequence of the spatial anisotropy of the electron distribution (e.g., Thejappa[42]). Note that in writing equation (8), we have ignored spatial diffusion so as to avoid unnecessary mathematical complexity. Although the specific form for as a function of p depends on the spectrum of the turbulent D waves that accelerates the electrons (Smith[61]), it is possible to make some fairly broad general- izations that help to simplify the analysis. In particular, we point out that a number of authors have independently suggestedthat p2. Examples include the treatment of particle acceleration D ∝ by large-scale compressible magnetohydrodynamical (MHD) turbulence (Ptuskin[62]; Chandran & Maron[63]); analysis of the acceleration of electrons by cascading fast-mode waves in flares (Miller, LaRosa & Moore[64]); and the energization of electrons due to lower hybrid turbulence (Luo, Wei & Feng[65]). Hence we shall write =D p2, (9) 0 D where D is a constant with the units of inverse time. The resulting Green’s function for the 0 nonthermal electron distribution is found to be (Subramanian & Becker[59]; Subramanian, Becker & Kazanas[54]) (p/p0)α1 , p p0 , ≤ f (p,p )=A  (10) G 0 0 (p/p0)α2 , p p0 , ≥ where p is the electron momentum, p is themomentum of the injected mono-energetic electrons, 0 and the exponents α and α are given by 1 2 1/2 1/2 3 9 1 3 9 1 α + + , α + . (11) 1 2 ≡−2 (cid:18)4 D τ(cid:19) ≡−2 −(cid:18)4 D τ(cid:19) 0 0 The quantity τ in these expressions represents the mean residence time for electrons in the acceler- ation region, and the normalization parameter A is computed using 0 N˙0 9 1 −1/2 A + , (12) 0 ≡ 2D p3 (cid:18)4 D τ(cid:19) 0 0 0 where the constant N˙ denotes the number of electrons injected per unit time per unit volume into 0 the acceleration region. The value of the total electron number density associated with the Green’s 10