Nuclearand ParticlePhysics Proceedings NuclearandParticlePhysicsProceedings00(2017)1–8 Cosmic Ray (Stochastic) Acceleration from a Background Plasma V.A.Dogiel1a,b,e, K.S.Chengb, D.O.Chernyshova,b,A.D.Erlykina,c,C.-M.Kod,andA.W.Wolfendalec aP.N.LebedevPhysicalInstitute,Moscow,Russia bDepartmentofPhysics,UniversityofHongKong,HongKong,China 7 cDepartmentofPhysics,DurhamUniversity,Durham,UK 1 dNationalCentralUniversity,ZhongliDist.,TaoyuanCity,Taiwan(R.O.C.) 0 eMoscowInstituteofPhysicsandTechnology(StateUniversity),Dolgoprudny,141707,Russia 2 n a J 9 Abstract 1 WegiveashortreviewofprocessesofstochasticaccelerationintheGalaxy. Wediscuss: howtoestimatecorrectly ] thenumberofacceleratedparticles, andatwhichconditionthestochasticmechanismisabletogeneratepower-law E nonthermal spectra. We present an analysis of stochastic acceleration in the Galactic halo and discuss whether this H mechanismcanberesponsibleforproductionofhighenergyelectronsthere,whichemitgamma-rayandmicrowave . h emissionfromthegiantFermibubbles. Lastly,wediscusswhethertheeffectsofstochasticaccelerationcanexplain p theCRdistributionintheGalacticdisk(CRgradient). - o Keywords: r t s a [ 1. Introduction whereuisthevelocityofmagneticfluctuationsandLis theaveragedistancebetweenfluctuations. 1 v ThetheoryofCRoriginstartedfromthekeypapers Anadvantageofthismodelwasthatthismechanism 1 of Baade and Zwicky [1] and Fermi [2, 3]. In the first generates power-law spectra of accelerated particles, 8 paper the authors assumed that the bulk of CRs ob- justasneededfortheobservedCRpower-lawspectrum. 4 served near Earth are produced by supernovae explo- However,spectraintheFermimodelweremuchharder 5 sionsintheMetagalaxy.Theyexcludedthatsupernovae (∝ E−1inthelimitcase)thantheobservedforCRspec- 0 . in our Galaxy were sources of CRs because, accord- trum. Besides, tooaverytimewasnecessarytoaccel- 1 ing to their estimates, their energy density had to be erateparticlesuptothehighenergiesneeded. 0 too high. An alternative explanation was suggested by 7 In1964GinzburgandSyrovatskii[4]suggestedtheir Fermi. He assumed that CRs were of the Galactic ori- 1 theory of CR origin. Using the observed CR chemical : gin and the whole volume of the Galaxy was a source composition they estimated the CR luminosity, which v i of CRs. CR acceleration is in this case due to regular was about 1040 erg s−1, and concluded that probably X collisionsofchargedparticleswithchaoticallymoving Galactic supernovae are the sources of CRs, because r magneticfluctuations. Theaccelerationisduetothein- only 10% of supernovae shocks energy was needed to a ducedelectricfieldE generatetheGalacticCRflux. Theydevelopedalsothe 1dH diffusionmodelofCRpropagationintheGalaxywhere curlE=− (1) c dt particlescatteringbymagneticfluctuationsintheinter- stellarmediumwasdescribedasspatialdiffusion. excitedbyatime-varyingmagneticfieldH. The next important mile-stone was connected with Asaresulttheenergyofparticles,E,increasesas the papers of Krymskii and Bell [5, 6] who suggested dE (cid:18)u(cid:19)2 E thetheoryofCRaccelerationbysupernovaeshocks. In ∼ =α E (2) dt c L/c 0 principle, this model is a modification of the classical /NuclearandParticlePhysicsProceedings00(2017)1–8 2 model of Fermi, because the important component of The solution of this equation is the equilibrium theaccelerationprocessisthatofparticlesscatteringon Maxwelliandistribution: moving magnetic fluctuations but the new component (cid:114) 2 (cid:18) E (cid:19) of the model is a velocity ”jump” on the shock front f (p)= n exp − (5) that changes the acceleration process drastically. First, M π 0 kT this acceleration generates steeper spectra of particles wheren isthedensityofbackgroundplasmaand E is 0 (∝ E−2)justasneededtoexplaintheobservedCRspec- theparticleenergy. trum. Secondly the rate of acceleration is proportional Ifbackgroundparticlesareundertheinfluenceofany tothefirstdegreeofu/c acceleration, (dE/dt) = α E, (see Eqs. (2) and (3)) ac 0 dE u thentherateofenergyvariationsis ∝ E (3) dt c (cid:32) (cid:33) (cid:32) (cid:33) dE dE dE thatmakestheshockaccelerationmuchmoreeffective dt = dt − dt (6) ac C thanthatofFermi. Hereuistheshockvelocity. Since then the theory of shock acceleration in the interstellar where(dE/dt)C istherateofionizationlossesinagas mediumhasbeendevelopedfervently,andforthemod- with the density n, which can be presented as (see e.g. erntheoryofthisprocessreadersarereferredtothetalk [12]) byDamianoCaprioliatthisconference. (cid:32)dE(cid:33) (cid:18) E (cid:19)−3/2 We notice, nevertheless, that the classical stochastic =ν E (7) Fermi acceleration may also be effective in an astro- dt C 0 kT physicalplasma. Stochasticaccelerationmaybeeffec- where tive near shocks of supernovae where magnetic turbu- lence is generated by the Rayleigh-Taylor and Kelvin- ν = 4πn0e4m1p/2Λ (8) Helmholtzinstabilities,seee.g.[7].Anotherexampleis 0 (kT)3/2m e thediscoveryoffreshlyacceleratedCRsintheCygnus is the frequency of Coulomb collisions of thermal par- Superbubble[8]. Itwasassumed,thatCRsareacceler- ticles, Λ is the Coulomb logarithm, m and m are the atedtherebythecollectiveactionofshocksinthisarea. e p restmassesofelectronsandprotons. Thetheoryofstochastic(multi-shock)accelerationbya ThenEq. (6)hastheform supersonicturbulencewasdevelopedin[9,10]. dE (cid:18) E (cid:19)−3/2 =α E−ν E (9) 2. Theory of In-Situ Acceleration from a Back- dt 0 0 kT groundPlasma. TheNumberofAcceleratedPar- FromEq.(9)wecaderivethethresholdenergy,ε thr ticles (cid:32)ν (cid:33)2/3 Oneoftheimportantquestions,whichmodelsofac- ε (n ,α )(cid:39)kT 0 (10) thr 0 0 α celerationshouldsolve,is: howmanyhighenergypar- 0 ticlescanbeproducedbytheseprocesses? Thereareno whichdefinestheenergyrangeofacceleratedparticles other sources of particles for acceleration except those withE >ε . thr fromabackgroundplasmaorhighenergyparticlespre- Thestochasticaccelerationformsapower-lawspec- acceleratedbyothersorces. Westartfromacceleration trumofnonthermalparticles fromabackgroundplasma. The spectrum of background plasma, f , is formed f = Kp−γ (11) M nth byCoulombcollisionsandisdescribedbytheequation Thesimplestwaytoestimatethenumberofacceler- (see[11]) atedparticles(constantK)isjusttomatchthermal(Eq. (cid:34)(cid:32) (cid:33) (cid:35) 1 d p2 dp f +D dfM =0 (4) (5))andnonthermal(Eq. (11))componentsofthetotal p2dp dt M C dp spectrumattheenergyE =ε thatgives C thr where pistheparticlemomentum,(dp/dt)C istherate (cid:114)2 (cid:18) ε (cid:19) of Coulomb losses, D = (dp/dt) mkT/p is the coef- K = n exp − thr (12) C C π 0 kT ficient of momentum diffusion due to Coulomb colli- sions,misthemassoftheacceleratedparticles,kisthe However,asGurevichnoticed(see[13]),theparticle BoltzmannconstantandT istheplasmatemperature. distributionbecomesnon-equilibriumandtime-varying /NuclearandParticlePhysicsProceedings00(2017)1–8 3 inthiscase,andaproperestimateofthenumberofac- clusterandshowedthatforreasonableparameters,this celeratedparticlescanbeobtainedintheframeworkof stochastic acceleration is able to produce enough high equationwhichincludesthetermofstochasticacceler- energyparticlesneededtoexplaintheComaX-rayex- ation (described by another momentum diffusion with cess. the coefficient DF) and the terms describing Coulomb However, in [13] a linear equation of acceleration collisions, which form the Maxwellian distribution of was analysed with a constant temperature of plasma, thermalparticles. Thetotalequationhastheform T = const, which did not take into account a back re- (cid:34)(cid:32) (cid:33) (cid:35) actionofacceleratedparticlesontotheparameterofthe ∂f 1 d dp df − p2 f +(D +D ) =0(13) thermal pool. This was done by Wolfe and Melia[16] ∂t p2dp dt C F dp C andPetrosianandEast[17], whoshowedfromnumeri- that gives the solution for the distribution function of calcalculationsofasystemofnon-linearequationsthat particleswithe.g. D =α p2as this mechanism of in-situ acceleration did not work at F 0 allbecausetheenergysuppliedbytheaccelerationwas    N(p¯)= (cid:114)π2n0exp−(cid:90)p¯ 1+(αud0/uνo)p¯5 ismulmtinegdieaffteelyctaobfsoacrbceedlerbaytiothnewthaesrmapallapsomoal,oavnedrhtehaetirneg- insteadofapower-lawspectraofnon-thermalparticles. 0   −exp−(cid:90)∞ 1+(αud0/uνo)p¯5 (14) tteumrLeavotaefrrei,aqCtuihoaentrison.ynTsshhdoeevyscesrthiaoblwi.n[e1gd8a]tchacanetaltlehyresaetrideosanunlatoninndlgitneeeffmaerpcsetyroasf-- 0 √ in-situaccelerationfrombackgroundplasmadepended wherethedimensionlessmomentum p¯ = p/ 2mkT. strongly on its parameters. If the momentum diffusion coefficienthasacut-offatlowenergies,e.g. intheform 1 D(p) = αpζθ(p− p ) the situation depends drastically 0 on the relation between the cut-off momentum, p and √ 0 10-3 theinjectionmomentum, p = 2mε . If p < p inj thr 0 inj the effect of acceleration is similar to [16, 17], i.e. the Np()10-6 plasmaisoverheated. Asurpisinglydifferentresultwas obtainedforthecaseof p > p . Inthiscaseacceler- 0 inj 10-9 ationsubtractsfromthethermalpoolonlyhighenergy particlesoftheMaxwelliandistribution. Asaresultthe 1 10 100 plasmacoolsdown(analoguetoMaxwelldemon),and p the power-law ”tail” is formed by the acceleration. As a restriction of this model we should mention that the larger is the value of p , the smaller is the number Figure1: Illustrativepicturewhichshowsthesolutionofcorrectki- inj neticequation(14)(solidline)andsimplematchingofthermaland of accelerated particles. On the other hand, for high nonthermal(power-law)compon√ents(dottedline). Here p¯ isthedi- enough pinj theMaxwellianspectrummatchesdirectly mensionlessmomentump¯=p/ 2mkT. withthepower-lawtailat p insteadof p asassumed 0 inj inEq. (12). The total spectrum of particles is shown in Fig. 1. The question is what could be the reason for the ac- Two important conclusions follow from this solution. celerationcut-offatrelativelylowmomenta. Thiscould The first one is that in the case of stochastic accelera- be due to absorption of MHD waves by CRs of rela- tiontheMaxwellianandpower-lawcomponentsdonot tivelylowenergy[19]. Inthestationarycasetheequa- matchwitheachotherasassumedinEq. (12). Thereis tionforspectrumofMHD-waves,W(k,t)canbewritten an extended region of a distorted Maxwellian distribu- as[20] tionformedbyCoulombcollisions,whichtriestocom- pensate the flux of particles running-away into the re- gionofacceleration. Secondly,asonecanseefromFig. dΠ(W,k,t) =−2Γ W+Φδ(k−k ), (15) 1, thesimpleestimate(12)underestimatesstronglythe dk cr 0 numberofacceleratedparticles. Dogieletal.[14,15]appliedthismodelofparticleac- where k is the wave-number, Π(W,k,t) decribes the celerationfromabackgroundplasmaforinterpretation non-linearcascadeofwaves,Φisenergyfromexternal of the hard X-ray excess in the spectrum of the Coma sourcesatk=k ,andΓ isthedecrementofabsorption 0 cr /NuclearandParticlePhysicsProceedings00(2017)1–8 4 byCRs,see[21] Analysis of this acceleration for both situations was providedin[22,27]. Inthefirstcase,thekineticequa- πZ2e2V2 (cid:90)∞ dp tion for the distribution function is similar to Eq. (13) Γ (k)= A F(p), (16) cr 2kc2 p but it includes also synchrotron and inverse Compton pres(k) energylossesforelectrons. Thenumberofaccelerated where p (k) = ZeH/ck, F(p)istheCRspectrumand electrons depends strongly on the value of cut-off mo- res mentum p . Itcannotbetoosmallbecauseoftheeffect Histhemagneticfieldstrength. 0 Thederivedcoefficientofthemomentumdiffusionis ofplasmaoverheating, andtoolargebecausethenum- ber of accelerated particle is smaller than needed for (see[22]) theobservedgamma-rayfluxfromthebubbles. In[22] D(p)=α (p)J (ξ) (17) it was shown that stochastic acceleration from a back- 0 2 ground is able to explain the observed emission from whereα (p)isapower-lawfunctionandJ istheBessel 0 2 theFermibubblesbutforanexceptionallynarrowrange function,whereξisacomplicatedfunctionofp(seefor oftheaccelerationparameters,whichmakesthismodel detailsofcalculationsAppendixin[22]). Thediffusion doubtful. coefficienthasacut-off D(p) = 0at J = 0,thatcorre- 2 The analysis of the second case [27] showed that it spondstoξ = 5.14or p0 = 0.2mcforparametersofthe is more effective for production of electrons than the Galactichalo. accelerationfromabackgroundplasma,becauseinthe caseofSNRelectronre-accelerationtheirenergyshould 3. Stochastic Acceleration in the Galactic Halo. be increased by three orders of magnitude only while ModelsoftheFermiBubbles foraccelerationfrombackgroundpoolelectronsareac- celerated from their temperatures (about several keV). Recent Fermi-LAT observations found new sources Thekineticequationinthiscasehasamorecomplicated of CRs in the Galaxy whose origin is enigmatic. First formthan(13)becauseitincludesalsotermsofparticle of all, we mention mysterious giant gamma-ray fea- propagation, turesinthecentralpartoftheGalaxy(FermiBubbles) elongated perpendicular to the Galactic plane[23, 24]. −∇·(cid:2)κ(r,z,p)∇f −u(r,z)f(cid:3)+ (cid:34)(cid:32) (cid:33) (cid:35) Severalmodelsweresuggestedtoexplaintheoriginof 1 ∂ dp ∇·u ∂f p2 − p f −D(r,z,p) = the bubbles which include phenomenological assump- p2∂p dt 3 ∂p tionsabouttheprocessesofparticleaccelerationthere. Q(p,r)δ(z), (18) Thus,Chengetal. [25]assumedthatgamma-rayemis- sion is generated by high energy electrons accelerated where r is the galactocentric radius, z is the altitude inthehalobygiantshocksresultingfromtidaldisrup- abovetheGalacticplane, p = E/cisthemomentumof tion of stars captured by the central black hole. Al- electrons, u is the velocity of the Galactic wind, κ and ternatively, Mertsch and Sarkar[26] assumed that this D are the spatial and momentum (stochastic accelera- emissionisproducedbyelectronsin-situacceleratedby tion)diffusioncoefficients,c(dp/dt) = dE/dtdescribes MHD-turbulence behind the shock. In this paper the the rate of electron energy losses, and Q describes the authorstriedtoreproducespectralcharacteristicsofthe spatial distribution of cosmic ray (CR) sources in the emission from the Fermi bubbles but did not estimate Galacticplane(z=0)andtheirinjectionspectrum. whether this mechanism could provide enough elec- The problem of the second model is that the spec- trons needed for the observed nonthermal fluxes from trum of re-accelerated electrons is too steep to repro- thebubbles. ducethemicrowaveemissionfromthebubblesasmea- As we mentioned above there are no other evident sured by Planck [28]. Thus, both models of stochastic sources ofelectrons foracceleration exceptthose from accelerationofelectronsinthebubbleshaveproblems, the background plasma or those injected by supernova andinthatsensethemodelofelectronaccelerationby remnants. In the first case electrons are injected with shocks [25] seems to be more attractive. We do not energies close to their background temperature in the discuss here the hadronic model of Fermi bubbles (see halo(i.e. aboutkeV).Inthesecondcase,onlyelectrons e.g. [29, 30, 31, 33]), whose problems were presented withenergies E ≤ 1GeVcanreachthealtitudesofthe in[34] Fermi bubbles. In both case further re-acceleration up Ontheotherhand,thestochasticaccelerationofpro- toenergiesabout1012eVisneededtogenerategamma- tonsintheFermibubblesmayexplaintheoriginofCRs raysfromtheFBs. with energies above the ”knee” (> 1015 eV) as shown /NuclearandParticlePhysicsProceedings00(2017)1–8 5 by Cheng et al. in [35]. We presented, however, these self-generated turbulence. In their model they investi- results in details at the last San-Vito conference which gatedasystemofequationsforCRpropagation(aone- werepublishedin[36]. dimensional version of Eq. (18) for propagation in the directionperpendiculartotheGalacticplane)andequa- tionsforMHD-waveexcitationbytheCRstreamingin- 4. Stochastic Acceleration in the Galaxy and the stabilityandtheirdamping. Inthismodelofnon-linear problemsOfCRGradient CRtransportthegradientandthespectralshapeofCRs The main questions of the theory of CR origin atdifferentgalactocentricradiiwerereproduced. are where CRs are generated and how they propa- If we return to models of CR acceleration we no- gate through the Galaxy. Necessary information can tice that at some conditions the injection energy of be obtained from investigations of the diffuse Galac- stochastic acceleration is a function of plasma temper- tic gamma-ray emission. First investigations of this ature. This may also be true for shock wave accelera- emission showed that the derived distribution of CRs tion because in some models of particle injection into in the Galactic disk was flatter than the radial distribu- shocks the temperature of the ambient ISM has rele- tion of their potential sources: SNRs and pulsars (see vance[48,49,50]. Itisknownthatthereisaradialin- e.g. [37, 38]). The first attempts to interpret this dif- creaseoftemperatureinHIIregionsfromabout6600K ference were performed in terms of CR propagation in atR = 2kpctoabout10000KatR = 14kpc[51]. Re- the Galactic halo. In [39, 40, 41] it was assumed that cently,Erlykinetal. [52]examinedhowmanyparticles an effective mixture of CRs in the Galactic halo due can be acclerated from background plasma depending toCRscatteringdiffusionmadetheirdistributioninthe onitstemperature. Galaxymoreorlessuniform. However,numericalcal- For simple estimates of the fraction of background culations showed that even in the most favorable case particles accelerated by Fermi mechanism we take Eq. of an extended halo the diffusion is unable to remove (14), and for the shock acceleration from background thesignatureoftheobservationallyinferredSNRsource plasmawetaketheequationfrom[48],whichis distribution. The problem is even more aggravated for (cid:32) (cid:33) ∂ ∂f 1du 1 ∂ thesharperSNRdistributionofGreen[42]. u(x)f −D =− (p2f) (19) ∂x ∂x 3dx p2∂p Recent analysis of the Fermi-LAT gamma-ray data [43,44]ingeneralconfirmedaflatterCRdistributionin where the coordinate x is perpendicular to the shock the outer part of the Galaxy, although showed a sharp front, Disthecoefficientofparticlespatialdiffusion,u drop of CR density near the GC. Interpretation of the istheparticlefluidvelocityandthevelocityjumpatthe last result is beyond the scope of our analysis. If con- shock (x = 0) is u−/u+ = 4 for strong shocks with the firmed, it may be due to specific processes in the GC. Machnumber M >> 1. Fromthisequationthenumber Interpretation of the flat CR distribution in the outer ofacceleratedparticlescanbeestimatedas Galaxy can be obtained in two different ways. One of (cid:32) (cid:33) n p m themwassuggestedby[39,45]whoassumedthatthisis nth (cid:39) T e δ1/3× aneffectof”unseen”SNRsatlargegalactocentricradii. n0 pthr mp [s4e6rTv]headenoCdtRhRerdeciwsctahryiibawuetaitosanslu.ingg[t4eh7set]eGdwabhlayocBtiincrteedirtipsscrkehtwiendetredthrtmeestoaoblf-. exp−δ1/21+ 21ln(cid:32)mmei(cid:33)231δ (20) convectivetransport(galacticwind). In[46]theauthors where m and m are the electron and proton mass re- e p concludedfromanalysesofasystemofhydrodynamic spectively,m isthemassoftheacceleratedparticles, i and kinetic equations that the wind velocity is propor- (cid:112) tional to the CR pressure PCR which in turn is propor- pT = 2kTmi (cid:114) tionaltothedensityofCRsources QCR. Fromanalyti- p = p mpδ1/3 calandnumericalcalculationsofthethree-dimensional thr T m e equationsofCRpropagationsimilartoEq. (18),where Dν¯ the wind velocity is a function of coordinates and the δ= u2 sourcedensity,u(r,z,Q),theyshowedthatCRsescaped 2πn e4m2 faster from regions of higher source density. Just this ν¯ = 0 pΛ effectexplainsinthemodelflatterCRdistribution. mep¯3 (cid:114) In[47]theauthorsanalysedanon-linearmodelofCR 2kT p¯ =m propagation,inwhichtheirtransportisdeterminedbya p m e /NuclearandParticlePhysicsProceedings00(2017)1–8 6 Coulombcollisionstrytocompensateafluxofpar- 1.×10-2 ticles running-away from a background pool into 5.×10-3 the acceleration region. Therefore, a very broad n01.×10-3 transfer region is formed by Coulomb collisions n/nth between thermal and non-thermal components of the total spectrum. In order to estimate the num- 1.×10-4 ber of accelerated particles one should analyse a kinetic equation which includes both the term of 6000 7000 8000 9000 10000 11000 12000 Coulomb collisions, which forms the Maxwellian T(K) distribution of thermal particles, and the term of stochastic acceleration, which forms a nonlinear Figure2:Illustrativepictureofthefractionofacceleratedparticlesfor powerlowspectrumofnon-thermalparticles. stochastic(solidline)andshockwave(dottedline)accelerationasa functionoftemperatureTofbackgroundplasma. • Ifthestochasticaccelerationinteractswithallpar- ticles of the Maxwellian spectrum, then it does Asonecanseeinbothcasestheinjectionefficiencyin- not form power-law ”tails” of accelerated parti- creaseswiththetemperatureT. InFig. 2wepresented cles. The energy supplied by acceleration is ab- the fraction of accelerated particles as it follows from sorbed immediately by the thermal pool through Eqs. (14)and(20). ionization losses of accelerated particles. The re- Asitfollowsfromtheradiodata,thedensityofSNRs sulting effect of stochastic acceleration is plasma dropsby30timesfromtheradiusR = 2kpctoR = 14 overheating. kpc[42],whilethedensityofCRsdropsby2timesonly fortheseradii. However,asweseeinFig. 2,themodel • If the stochastic acceleration interacts with a high efficiencyofparticleaccelerationrisesin20-100times energy fraction of the Maxwellian distribution forthesedistancesfromtheGCbecauseofthetempera- only, then power-law spectra of particles are gen- turevariations. Thus,theefficiencyofaccelerationmay erated,andthenumberofacceleratedparticlesde- compensatepartiallythedropofsupernovadensityand pends strongly on the position of acceleration the with this effect the observed CR density can be repro- low momentum cutoff of the stochastic accelera- duced in the ”temperature” model. Erlykin et al. [52] tionprocess. derivedfromtheMaxwell-Boltzmandistributionatdif- ferent temperatures the necessary value of ε , which thr • Weanalysedwhetherthestochasticaccelerationin shouldbeabout2.5eVfortheobservedCRdenstyvari- the Galactic halo can produce there enough num- ations. Thisanalysisgivesareasonablecoincidenceof ber of high energy electrons needed to explain the”temperature”modelwithvariationsofCRdensity the gamma-ray and microwave emission from the intheGalacticdiskalthoughthereareanumberofun- enigmaticFermibubblesintheGalacticcentralre- certaintieswhichrequirefurtherinvestigations. gion. Weanalysedtwocasesofelectronaccelera- We notice, however, that this estimates presented in tion:a)in-situaccelerationofelectronsfromback- Fig. 2, are mainly illustrative and give qualitative im- ground plasma, and b) re-acceleration in the halo pressionabouttheinjectionprocesses. Thus,shocksin ofelectronsgeneratedbySNRsinthedisk,which theGalaxyarecollisionless,andparticleinjectionisde- reachthealtitudesoftheFermibubbleedges. We terminedbyinteractionswithmagneticfluctuations(not showed that there are problems in both cases, but byCoulombcollisions, seethetalkofDamianoCapri- the needed number of electrons can be provided oli). underspecificconditions. 5. Conclusion • We discussed whether the CR gradient in the Galactic disk can be explained in terms of the model of stochastic acceleration. We showed that Wegiveashortreviewofprocessstochasticaccelera- ifthenumberofacceleratedparticlesisafunction tionintheGalaxy. Theconclusionsareitemizedbelow: ofthetemperatureofbackgroundplasmathenthis • Inthecaseofstochasticaccelerationfromaback- effectmayexplaintheobservedradialvariationof ground plasma the distribution is nonequilibrium. theCRdensityintheGalacticdisk. /NuclearandParticlePhysicsProceedings00(2017)1–8 7 Acknowledegments onInterstellarTurbulenceandCosmic-RayTransport,ApJ642 (2006)902 V.A.D. and D.O.C. acknowledge a partial support [20] C.A.Norman,A.Ferrara,TheTurbulentInterstellarMedium: GeneralizingtoaScale-dependentPhaseContinuum,ApJ467 from the MOST-RFBR grant 15-52-52004 and the (1996)280 RFBR grant 15-02-02358. K.S.C. is supported by the [21] V.S.Berezinskii,S.V.Bulanov,V.A.Dogiel,V.L.Ginzburg, GRFGrantsoftheGovernmentoftheHongKongSAR V.S.Ptuskin,AstrophysicsofCosmicRays,ed.V.L.Ginzburg, (Norht-Holland,Amsterdam),1990 under HKU 17310916. C.M.K. is supported in part [22] K.S.Cheng,D.O.Chernyshov,V.A.Dogiel,C.M.Ko,Multi- by the Taiwan Ministry of Science and Technology wavelengthEmissionfromtheFermiBubbles.I.StochasticAc- Grants MOST 104-2923-M-008-001-MY3 and MOST celerationfromBackgroundPlasma,ApJ790(2014)23 105-2112-M-008-011-MY3. A.D.E. and A.W.W. are [23] G.Dobler,D.P.Finkbeiner,I.Cholisetal.,TheFermiHaze: A Gamma-ray Counterpart to the Microwave Haze, ApJ 717 grateful to the Kohn Foundation for the financial sup- (2010)825 port(GrantRF040081). [24] M.Su,T.R.Slatyer,D.P.Finkbeiner,GiantGamma-rayBub- blesfromFermi-LAT:ActiveGalacticNucleusActivityorBipo- larGalacticWind?,ApJ724(2010)1044 References [25] K.S.Cheng,D.O.Chernyshov,V.A.Dogiel,C.-M.Ko,W.-H. Ip, FermiBubblesasaResultofStarCaptureintheGalactic Center,ApJ731,(2011)L17 [1] W.Baade,andF.Zwicky,RemarksonSuper-NovaeandCosmic [26] P. Mertsch, S. Sarkar, Fermi Gamma-Ray “Bubbles” from Rays,Phys.Rev.46(1934)76 StochasticAccelerationofElectrons,PhRvL107(2011)1101 [2] E.Fermi,OntheOriginoftheCosmicRadiation,Phys.Rev.75 [27] K.S.Cheng,D.O.Chernyshov,V.A.Dogiel,C.-M.Ko,Multi- (1949)1169 wavelength Emission from the Fermi Bubble. III. Stochastic [3] E.Fermi, GalacticMagneticFieldsandtheOriginofCosmic (Fermi) Re-acceleration of Relativistic Electrons Emitted by Radiation,ApJ119(1954)1 SNRs,ApJ804(2015)135 [4] V.L.Ginzburg,S.I.Syrovatskii,TheOriginofCosmicRays, [28] P. Ade et al.(Planck Collaboration), Planck intermediate re- NewYork:Macmillan,1964 sults.IX.DetectionoftheGalactichazewithPlanck,A&A554 [5] G. F. Krymskii, A regular mechanism for the acceleration of (2013)139 chargedparticlesonthefrontofashockwave,SovietPhysics- [29] R.M. Crocker, F. Aharonian, Fermi Bubbles: Giant, Doklady,234(1977)1306 Multibillion-Year-Old Reservoirs of Galactic Center Cosmic [6] A.R.Bell,Theaccelerationofcosmicraysinshockfronts.I, Rays,Phys.Rev.Lett,106(2011)101102 MNRAS182(1978)147 [30] Y. Fujita, Y. Ohira, R. Yamazaki, The Fermi Bubbles as a [7] C. Yang, S. Liu, Energy Partition between Energetic Elec- Scaled-upVersionofSupernovaRemnants,ApJ7752013L20 tronsandTurbulentMagneticFieldinSupernovaRemnantRX [31] S.Thoudam,FermiBubbleγ-RaysasaResultofDiffusiveIn- J1713.7-394,ApJ773(2013)138 jectionofGalacticCosmicRays,ApJ778(2013)L20 [8] M. Ackermann, M. Ajello, A. Allafort et al., A Cocoon of [32] R. M. Crocker, G. V. Bicknell, E. Carretti et al.,Steady-state Freshly Accelerated Cosmic Rays Detected by Fermi in the HadronicGamma-RayEmissionfrom100-Myr-OldFermiBub- CygnusSuperbubble,Science334(2011)1103 bles,ApJ791(2014)L20 [9] A.M.Bykov,G.D.Fleishman,Onnon-thermalparticlegener- [33] R.-Z.Yang,F.Aharonian,R.Crocker,TheFermibubblesrevis- ationinsuperbubbles,MNRAS255(1992)269 ited,A&A567A(2014)19 [10] A.M.Bykov,I.N.Toptygin,AModelofParticleAcceleration [34] K.S.Cheng,D.O.Chernyshov,V.A.Dogiel,C.-M.Ko,Multi- toHighEnergiesbyMultipleSupernovaExplosionsinOBAs- wavelength Emission from the Fermi Bubble. II. Secondary sociations,Astron.Lett.,27(2001)625 Electrons and the Hadronic Model of the Bubble, ApJ 799 [11] E. M. Lifshitz, L. P. Pitaevskii, Physical Kinetics, (Oxford: (2015)112 PergamonPress),1981 [35] K.S.Cheng,D.O.Chernyshov,V.A.Dogiel,C.-M.Ko,The [12] V.L.Ginzburg,Theoreticalphysicsandastrophysics,Oxford, FermiBubbleasaSourceofCosmicRaysintheEnergyRange Pergamon Press, (International Series in Natural Philosophy. >1015eV,ApJ746(2012)116 Volume99),1979 [36] D.O.Chernyshov,K.S.Cheng,V.A.Dogiel,C.-M.Ko,Fermi [13] A.V.Gurevich,Ontheamountofacceleratedparticlesinanion- bubblesasasourceofcosmicraysabove>1015eV,NuPhS256 izedgasundervariousacceleratingmechanisms,SovietPhys.J. (2014)179 Exp.Theor.Phys11(1960)1150 [37] A.W.Strong,J.B.G.M.Bloemen,T.M.Dame,etal.,Thera- [14] V.A.Dogiel,OnthethermaloriginofthehardX-rayemission dialdistributionofgalacticgammarays.IV-Thewholegalaxy, fromtheComacluster,A&A357(2000)66 As&A207(1988)1 [15] V.A.Dogiel,S.Colafrancesco,C.-M.Koetal.,In-situacceler- [38] A.W.Strong,J.R.Mattox,GradientmodelanalysisofEGRET ationofsubrelativisticelectronsintheComahaloandthehalo’s influence on the Sunyaev-Zeldovich effect, A&A 461 (2007) diffuseGalacticgamma-rayemission,A&A.308(1996)L21 [39] V. A. Dogiel, A. V. Uryson, Cosmic ray spectra and galactic 433 diffusegamma-rayemission,A&A197(1988)335 [16] B.Wolfe,F.Melia,ApJ638(2006)125 [40] J.B.G.M.Bloemen,V.A.Dogiel,V.L.Dorman,V.S.Ptuskin, [17] V.Petrosian,W.E.East,HeatingandAccelerationofIntraclus- Galacticdiffusionandwindmodelsofcosmic-raytransport.I- terMediumElectronsbyTurbulence,ApJ682(2008)175 InsightfromCRcompositionstudiesandgamma-rayobserva- [18] D.O.Chernyshov,V.A.Dogiel,C.M.Ko,StochasticParticle tions,A&A.267(1993)372 AccelerationandtheProblemofBackgroundPlasmaOverheat- [41] A.D.Erlykin,A.W.Wolfendale,Cosmicrayanisotropiesto5 ing,Astrophys.J.759(2012)113 PeV,JCAP04(2013)006 [19] V.S.Ptuskin,I.V.Moskalenko,F.C.Jonesetal.,Dissipation [42] D.A.Green,Constraintsonthedistributionofsupernovarem- ofMagnetohydrodynamicWavesonEnergeticParticles:Impact /NuclearandParticlePhysicsProceedings00(2017)1–8 8 nantswithGalactocentricradius,MNRAS454(2015)1517 [43] F. Acero, M. Ackermann, M. Ajello et al., Development of theModelofGalacticInterstellarEmissionforStandardPoint- Source Analysis of Fermi Large Area Telescope Data, ApJS, 224(2016)8 [44] R.-z.Yang, F.Aharonian, Evoli, C.Radialdistributionofthe diffuseγ-rayemissivityintheGalacticdisk,Phys.Rev.D,93 (2016)l23007 [45] A.W.Strong,I.V.Moskalenko,O.Reimer,DiffuseContinuum GammaRaysfromtheGalaxy,ApJ537(2000)763 [46] D.Breitschwerdt,V.A.Dogiel,H.J.Vo¨lk,Thegradientofdif- fusegamma-rayemissionintheGalaxy,A&A385(2002)216 [47] Recchia,S.,Blasi,P.,&Morlino,G.,Ontheradialdistribution ofGalacticcosmicrays,MNRAS462(2016)L88 [48] S.V.Bulanov,V.A.Dogiel,TheInfluenceofEnergyLosseson theAccelerationofCosmic-RayParticlesataShockFront,Sov. Astr.Lett.5(1979)278 [49] E.G. Berezhko, V.K. Elshin, L.T. Ksenofontov, Cosmic ray acceleration in supernova remnants, J.Exp.Theor.Phys., 82(1) (1996)1 [50] H. Kang, T. W. Jones, U. D. J. Gieseler, Numerical Studies ofCosmic-RayInjectionandAcceleration, Astrophys.J., 579 (2002)337 [51] C.Quiresa,R.T.Rood,T.M.Baniaetal.,Theelectrontemper- aturegradientintheGalacticDisk,ApJ653(2006)1226 [52] A.D.Erlykin,A.W.Wolfendale,V.A.Dogiel,Theradialgra- dientofcosmicrayintensityintheGalaxy,Adv.SpaceRes.57 (2016)519