Twinkling pulsar wind nebulae in the synchrotron cut-off regime and the gamma-ray flares in the Crab Nebula
aa r X i v : . [ a s t r o - ph . H E ] D ec Mon. Not. R. Astron. Soc. , 1– ?? () Printed 17 September 2018 (MN L A TEX style file v2.2)
Twinkling pulsar wind nebulae in the synchrotron cut-offregime and the gamma-ray flares in the Crab Nebula
A.M.Bykov ⋆ , G.G.Pavlov , , A.V.Artemyev ,Yu.A.Uvarov , Ioffe Institute for Physics and Technology, 194021 St.Petersburg, Russia
525 Davey Laboratory, Pennsylvania State University, University Park, PA 16802 State Politechnical University, St. Petersburg, Russia Space Research Institute, Russian Academy of Sciences, Moscow, Russia
17 September 2018
ABSTRACT
Synchrotron radiation of ultra-relativistic particles accelerated in a pulsar wind neb-ula may dominate its spectrum up to γ -ray energies. Because of the short coolingtime of the γ -ray emitting e ± , the γ -ray emission zone is in the immediate vicinity ofthe acceleration site. The particle acceleration likely occurs at the termination shockof the relativistic striped wind, where multiple forced magnetic field reconnectionsprovide strong magnetic fluctuations facilitating Fermi acceleration processes. The ac-celeration mechanisms imply the presence of stochastic magnetic fields in the particleacceleration region, which cause stochastic variability of the synchrotron emission.This variability is particularly strong in the steep γ -ray tail of the spectrum, wheremodest fluctuations of the magnetic field lead to strong flares of spectral flux. In par-ticular, stochastic variations of magnetic field, which may lead to quasi-cyclic γ -rayflares, can be produced by the relativistic cyclotron ion instability at the terminationshock. Our model calculations of the spectral and temporal evolution of synchrotronemission in the spectral cut-off regime demonstrate that the intermittent magneticfield concentrations dominate the γ -ray emission from highest energy electrons andprovide fast, strong variability even for a quasi-steady distribution of radiating parti-cles. The simulated light curves and spectra can explain the very strong γ -ray flaresobserved in the Crab nebula and the lack of strong variations at other wavelengths.The model predicts high polarization in the flare phase, which can be tested withfuture polarimetry observations. Key words: shock waves — turbulence— ISM: supernova remnants—gamma-rays—supernovae: individual (Crab nebula)
Strong flares of a few days duration have been discov-ered recently by the
AGILE and
Fermi γ -ray observatoriesin the Crab nebula at energies above 100 MeV (see e.g.Tavani et al. 2011; Abdo, et al. 2011; Vittorini et al 2011,and references therein). The most striking features of theflares are the extreme amplitude of the photon flux changes,especially at energies above the exponential cut-off energyof the quiescent spectrum, and the fast hour-timescale vari-ability. While the exponential cut-off energy E c of a typicalquiescent γ -ray spectrum is ∼
100 MeV, the cut-off energyof >
500 MeV was found in the April 2011 flare spectrum.This value of E c exceeds the energy ˜ E c ∼ m e c /α ∼ α = e / ¯ h c is the fine-structure constant), con- ⋆ E-mail:[email protected]ffe.ru sidered to be the maximal cut-off energy in the synchrotronmodels of γ -ray emission from the Crab and other pulsarwind nebulae (PWNe) (de Jager et al. 1996; Uzdensky et al.2011; Striani et al. 2011). The value ˜ E c is the synchrotronphoton energy emitted by an electron whose energy is suchthat the synchrotron cooling time is equal to the charac-teristic gyration time ω − g (see e.g. Guilbert et al. 1983;de Jager et al. 1996; Atoyan & Aharonian 1996). The par-ticle gyration time is considered to be the fastest accelera-tion time in a plasma system with frozen-in magnetic fieldof the r.m.s. amplitude h B i / that exceeds the electricfield magnitude. The value of ˜ E c corresponds to the elec-tron Lorentz factor γ m that can be found from the equa-tion ˙ E syn ( γ m ) = ˙ E acc ( γ m ). Both the synchrotron loss rate˙ E syn ( γ ) and the electron acceleration rate ˙ E acc ( γ ) dependon the moments (often just h B i ) of the stochastic magneticfield. c (cid:13) RAS
A.M. Bykov, G.G. Pavlov, A.V. Artemyev, Yu.A. Uvarov
Figure 1.
Light curves of synchrotron emission at 5 keV (dot-dash line), 500 MeV (dashed line), 1 GeV (dotted line) and 2GeV (short-dash line) as a response to an imposed fluctuationwith magnetic field B ( t ) (solid line) simulated in Model I. Thelight curves are normalized to maximal intensity. The backgroundmagnetic field in the emission region was modeled as a stochasticgaussian field of h B (0) i / = 0 . B ( t ) (solid line) is localized in a stripe of a 0 . thickness( ≫ r f ). The maximum of B ( t ) = 1 mG is at ct/ ∆ = 0.6. Thespatial scale ∆ ≈ × cm for the photon energy E ≈ s in theGeV regime. In this work we consider the case when the formationlength, r f = m e c /e h B i / , of incoherent synchrotron ra-diation is much smaller than the typical synchrotron cool-ing and acceleration lengths, while the typical wavelength λ of the fluctuating magnetic field is larger than r f . In thiscase ˙ E syn ( γ m ), ˙ E acc ( γ m ), and E c are determined by the samer.m.s. value of the fluctuating magnetic field, and the bulk ofthe relativistic electron distribution may vary on scales thatare much larger than the gyration radius r g = γr f . Sincethe synchrotron emissivity of a power-law electron distribu-tion with spectral index p is proportional to B ( p +1) / , thelocal emissivity sharply grows with B for large p values. Thismeans that the synchrotron radiation in the cut-off regime(which corresponds to large effective p values) is governedby high statistical moments of the stochastic magnetic fielddistribution, and it is intermittent. The intermittency ef-fect implies that rare strong peaks of the magnetic field dis-tribution dominate the synchrotron emission (Bykov et al.2008, 2009). It is particularly important in the synchrotroncut-off regime, when the typical size of the distribution ofradiating electrons (the synchrotron cooling length) can becomparable with the correlation length of strong magneticfield fluctuations. For instance, this is expected to be thecase in supernova shells, where magnetic fluctuations areproduced by instabilities of anisotropic distributions at themaximal energy of particles accelerated in the source (see,e.g., Bykov et al. 2011, and references therein).Since the source emission in the cut-off regime is domi- Figure 2.
Normalized spectra of synchrotron radiation at twodifferent time moments ct/ ∆ = 0.2 (solid line) and 0.6 (dashedline), which model the quiescent and flare spectra, respectively(see Fig. 1). The dotted curve shows the contribution of the vari-able magnetic field. The power emitted in the GeV flare is about2 × erg s − , for the Crab parameters. nated by just a single (or a few strongest) concentration(s)of the stochastic magnetic field, the light curve of the sourcein this regime reflects the lifetime of the magnetic concentra-tions rather than the electron acceleration/losses timescales.Fast temporal variations will appear even for a quasi-steadyelectron distribution. The light curve and the spectral be-haviour in the synchrotron cut-off regime are determined bystatistical characteristics of the magnetic field, which can bedescribed by the probability distribution function (PDF) ofmagnetic fluctuations, P ( B ).It is worthwhile to note that intermittent magneticfields can be found in quite different circumstances. For in-stance, non-Gaussian distributions of fluctuations, exhibit-ing gradual tails at large field amplitudes, have been foundin the Earth magnetotail after the current disruption asso-ciated with magnetospheric substorms observed by Geotail and
Cluster satellites. The energy injection during the sub-storms feeds an energy cascade to small-scale fluctuationswith the corresponding increase of intermittency (see, e.g.,Zimbardo et al. 2010, for a recent review of magnetosphericobservations).In this Letter we show that the model of synchrotronemission in fluctuating magnetic fields, with account for theintermittency in the spectral cut-off regime, can explain thenature of the γ -ray flares observed in the Crab. We demon-strate the effect of intermittency on the GeV regime emissionusing two models. In first model, we simulate the spectra ofaccelerated electrons and positrons in a simple kinetic modelof diffusive Fermi acceleration in the termination shock ofstriped wind with account for synchrotron losses. Then weconstruct the GeV regime flare light curve and spectra by in-tegrating the synchrotron emissivity of spatially inhomoge-neous particle distribution in the shock downstream with im-posed magnetic field variation. Second model demonstratesthe effect of the magnetic field PDF shape on the syn- c (cid:13) RAS, MNRAS , 1– ?? winkling pulsar wind nebulae in the synchrotron cut-off regime and the gamma-ray flares in the Crab Nebula chrotron photon spectra in the cut-off regime. The emissionis produced by Fermi-accelerated pairs in the spectral cut-offregime, in which the acceleration is balanced by synchrotronlosses. The synchrotron origin of the observed gamma-ray flaresassumes the presence of pairs accelerated to PeV energies.The thickness of the γ -radiating region depends on the cool-ing rate in the magnetic field: ∆ ∼ τ syn c ∼ . × · B − / · E − / cm, where B mG is the r.m.s. magnetic fieldin mG, and E GeV is the photon energy in GeV. The sizeof the γ -ray emitting region is very small compared to thesize of the nebula. The thickness of the layer ∆ is estimatedusing the standard electron synchrotron cooling time (e.g.Rybicki & Lightman 1979) of the electron radiating photonsat the peak of the power spectrum of synchrotron emis-sion. The particle acceleration mechanism in PWNe is notfully understood yet, although some basic features and con-straints have been established (see, e.g., Kennel & Coroniti1984; Arons 2008, 2011; Keshet et al. 2009; Kirk et al.2009; Lemoine & Pelletier 2010; Sironi & Spitkovsky 2011;Bykov & Treumann 2011; Uzdensky et al. 2011).Recent models of particle acceleration in PWNe con-sider a relativistic wind in the equatorial plane with toroidalstripes of opposite magnetic field polarity, separated by cur-rent sheets. Sironi & Spitkovsky (2011) modeled particle ac-celeration and magnetic field dissipation at the termina-tion shock of a relativistic striped wind using 2D and 3Dparticle-in-cell (PIC) simulations. They found a complexstructure of the flow in the vicinity of the termination shock.In that model, the shock-driven reconnection in the down-stream transfers the magnetic energy of alternating fields ofthe striped wind to accelerated relativistic pairs. The en-ergy spectra of electrons accelerated in the reconnection re-gion take the form of power laws, N ( γ ) ∝ γ − p , with spec-tral indices p ∼ . γ -ray emitting electrons. This is an important findingas it simultaneously addresses two problems – the termina-tion shock formation in magnetized PWN winds and particleacceleration . Therefore, the combined action of the recon-nection processes and shock acceleration is expected in theintense equatorial pulsar wind. Extensive PIC simulationsin a wide dynamical range are needed to demonstrate thefeasibility of this approach and identify the mechanism ofparticle acceleration to PeV energies and their synchrotronemission. However, to construct the synthetic spectra of the Note that the standard model of diffusive Fermi accelera-tion in a transverse relativistic shock in a non-striped uniformwind encounters problems when applied to PWN terminationshocks, see, e.g., Niemiec et al. (2006); Pelletier et al. (2009);Bykov & Treumann (2011).
Crab nebula at γ -ray energies (where the synchrotron cool-ing is very important), one needs to account for the radiativereaction force, which is not yet attainable in the PIC simu-lations (Sironi & Spitkovsky 2009; Nishikawa, et al. 2011).Kinetic models can be used to simulate particle ac-celeration due to the repetitive interaction of electronswith magnetic turbulence in the energetic outflows withaccount for the synchrotron cooling. The kinetic modelby Bykov & Meszaros (1996) for particle acceleration byboth relativistic and transrelativistic shocks, accompaniedby broad dynamic spectra of magnetic fluctuations with vi-olent motions of relativistic plasma, predicts a hard brokenpower-law electron distributions with slopes 1 p
2. Inthis model, particles are accelerated by strong magnetic fluc-tuations on timescales comparable to their gyration periodin the r.m.s. magnetic field.In the simulations described below we complement thekinetic model for particle acceleration in the vicinity of thestriped wind termination shock with the synchrotron coolingeffects to account for the spectral cut-off regime. To estimatethe spectra of nonthermal leptons accelerated downstreamof the PWN termination shock by Fermi mechanism, onecan use a Fokker-Planck-type kinetic equation: ∂N∂t = k ( γ ) ∂ N∂z − u ∂N∂z + 1 γ ∂∂γ γ (cid:20) D ( γ ) ∂N∂γ + a ( γ ) N (cid:21) , (1)where z is the coordinate along the shock normal . Thisequation, averaged over the ensemble of strong electromag-netic fluctuations in the vicinity of the wind terminationshock, accounts for diffusion and advection of electrons inphase space due to interactions with the fluctuations. Theterm with the momentum diffusion coefficient D ( γ ) corre-sponds to the stochastic Fermi acceleration, k ( γ ) is the fastparticle spatial diffusion coefficient, u is the flow velocitycomponent along the shock normal, and a ( γ ) is the energyloss rate of an electron due to synchrotron radiation. Model I . To construct the synchrotron emission spectraand flaring light curves in the diffusive shock accelerationmodel, we simulated spatially inhomogeneous acceleratedpair distribution downstream of the termination shock ofthe striped wind using Eq. 1. Short-scale magnetic fluctu-ations are required to be present upstream of the shock toallow an efficient diffusive Fermi acceleration in the trans-verse relativistic shocks (see e.g. Bykov & Treumann 2011,and footnote ). Recently, Sironi & Spitkovsky (2011) havefound in the PIC simulations that the fluctuations can begenerated upstream of the striped wind termination shock.We assume the fluctuations provide the Bohm diffusion with k ( γ ) ≈ cr g ( γ ) /
3. The stochastic Fermi acceleration was ne-glected in the model (i.e., D = 0).To illustrate the intermittency effect in the cut-offregime, we simulated a light curve and spectra for a mag-netic field fluctuation imposed in the GeV photon emittingregion of scale size ∆ = 2 × cm (for a quiescentmagnetic field B = 0.2 mG, downstream of the termina-tion shock). The fluctuation δB ( t ) is localized in a stripeof a 0 . width and has the time dependence shown by Since the scale size ∆ of the PeV electron distribution is muchsmaller than the termination shock radius, the problem can beconsidered as one-dimensional.c (cid:13)
RAS, MNRAS , 1– ?? A.M. Bykov, G.G. Pavlov, A.V. Artemyev, Yu.A. Uvarov the solid line in Figure 1. Such magnetic field variationsmay be produced by the relativistic ion cyclotron instabil-ity at the termination shock. The instability, proposed bySpitkovsky & Arons (2004) to explain the origin of the opti-cal wisps in the Crab nebula, can produce quasi-cyclic γ -rayflares in our model. The scale ∆ , used in our simulations,corresponds to about 0.1 of the magnetic field limit cyclefound by Spitkovsky & Arons (2004) (see their Figure 2).The light curves of the γ -ray (0.5 , 1 GeV and 2 GeV) andX-ray (5 keV) emission show the strong response of the γ -ray emission in the cut-off regime, while the response is verymodest at X-ray energies. Note that the variability time inthe GeV regime is shorter than the imposed field fluctuationand the cooling time of PeV electron distribution. This is be-cause the emission in the cut-off regime is governed by high-order momenta of the magnetic field. The energy loss rate a ( γ ) that determines the electron cooling depends on h B i .Therefore, the stochastic magnetic field realizations in theemission region with the same h B i but different high-ordermomenta (determined by their PDFs) would correspond tothe same particle distributions. However, their photon spec-tra in the cut-off regime are very different. This effect isclearly seen in Figure 3 discussed below. The synchrotronflares in the cut-off regime will appear even in the case ofsteady electron distributions.In Figure 2 we show the simulated spectra of syn-chrotron emission, generated downstream of the termina-tion shock, for η ≈ .
5, corresponding to quiescent regime(solid line) and fluctuation maximum (dashed line). Theyare similar to the quiescent and flare spectra observed inthe Crab nebula by
AGILE and
Fermi (see Tavani et al.2011; Abdo, et al. 2011; Vittorini et al 2011, and referencestherein).
Model II.
Apart from the diffusive shock accelerationa variety of other particle acceleration mechanisms can beimportant in the region. The coalescence of magnetic is-lands, particle reflection by magnetic islands (both firstand second order Fermi-type processes), are expected tobe in action, as it occurs in the Earth magnetosphere (e.g.Drake et al. 2006; Zelenyi et al. 2010; Sironi & Spitkovsky2011; Uzdensky et al. 2011; Daughton et al. 2011). Whenthe source of strong turbulence is quasi-steady on timescaleslonger than ω − g ( γ ), a simple analytical treatment of theproblem is also possible. For the case of fast stochastic Fermiacceleration (comparable to the particle gyration period) byan energetic plasma outflow with strong magnetic turbu-lence, the momentum diffusion coefficient in Eq.1 takes theform D ( γ ) = γ η ω g ( γ ), where η < ∼ ω g ( γ ) = e h B i / / ( m e cγ ) and a ( γ ) = 4 r h B i γ / (9 m e c )(where r = e /m e c ) depend on the same ensemble-averaged value h B i .The asymptotical shape of the particle spectrum in thecut-off regime is N ( γ ) ∝ γ − p exp (cid:20) − Z dγ a ( γ ) /D ( γ ) (cid:21) . (2)It is important that, while the index p depends on theturbulence spectrum and system geometry, the exponen-tial cut-off in the particle spectrum is rather universal: N ( γ ) ∝ γ − p exp[ − ( γ/γ ) ], where γ = 9 eη/ (2 r h B i / ),i.e., γ ≈ × η h B i − / . The synchrotron emissivity ǫ ( ω, B ) in a local magnetic field B , which is assumed to beuniform on spatial scales larger than r f = m e c /eB , is givenby the equation ǫ ( ω, B, z ) = √ Be πmc Z dγ γ N ( z, γ ) R ( ω/ω c ) , (3)where ω c = 3 eBγ / m e c is the characteristic frequency ofsynchrotron radiation. Approximate analytic expressions forthe function R ( x ) were derived by Crusius & Schlickeiser(1986) and Zirakashvili & Aharonian (2007). The spectrumof synchrotron emission from the downstream region filledwith strong magnetic field fluctuations can be expressed as J ( ω ) = Z dB dz ǫ ( ω, B, z ) P ( B ) . (4)To illustrate the effect of the magnetic field fluctuationsin the cut-off regime, where high-order statistical momentsdominate the integral in Equation (4), we used the PDF ofmagnetosonic type fluctuations, which corresponds to thewisp structures seen in the polarized optical images pre-sented by Hester (2008). The PDF has the form P ( B ) = C n exp( − b n / Θ n ), with n = 1 and 2, and b = | B − B | /B .Here C n is the normalization constant, and Θ n is the di-mensionless width of the distribution. The simulated syn-chrotron spectra are presented in Figure 3 for the gaussian( n = 2), exponential ( n = 1), and non-fluctuating magneticfield distributions. The characteristic frequency ω for theLorentz factor γ is given by ¯ hω = 27 η m e c / (4 α ) ≈ η MeV. The synchrotron curves in Figure 3 are simulated forΘ n = 1 (i.e., for the case of strong fluctuations). The resultsillustrate a strong effect of the PDF shape on the spectralbehaviour in the cut-off regime (even for a fixed rms field h B i ), in contrast to a very modest effect in the power-lawregime at ω < ω . Thus, Model II demonstrates that a re-construction of the PDF tail of magnetic fluctuations in thesynchrotron emission region (with the same h B i ) would re-sult in a strong change of the synchrotron photon spectrum,similar to that observed in the GeV flares in the Crab neb-ula. In Model I, γ -ray photons are radiated downstream of thePWN termination shock by ultrarelativistic electrons. Theparticles are accelerated by some kind of Fermi mechanismin the vicinity of the termination shock in the striped windwith reconnecting magnetic fields. Electrons in the sharplydecreasing high-energy tail of the distribution function radi-ate synchrotron γ -ray photons in fluctuating magnetic fields,which results in flaring behaviour in the cut-off regime (i.e.,at the high-energy end of the synchrotron spectrum).The compressed magnetic field structures, movingthrough the very narrow γ -ray emitting layer just down-stream of the termination shock, result in γ -ray flareswith the strong spectral variations shown in Figure 2. Thetimescale of the γ -ray photon variability in the flare is afraction of the time ∆ /c ∼ × s in the cut-off regimebecause of the effect of high-order statistical moments. Atthe same time, the variability at energies below the cut-off energies is slower, and of much smaller amplitudes, asit is clearly seen from comparison of the light curves at c (cid:13) RAS, MNRAS , 1– ?? winkling pulsar wind nebulae in the synchrotron cut-off regime and the gamma-ray flares in the Crab Nebula Figure 3.
Surface spectral density of synchrotron radiationin the cut-off regime (in arbitrary units) simulated for differentprobability distribution functions of fluctuating magnetic field inthe source (Model II). The characteristic frequency is defined by¯ hω ≈ η MeV. (Tennant et al. 2011; Caraveo et al.2010). The frequency of the γ -ray flares is determined inour model by the frequency of occurrence of extreme valuesof magnetic fluctuations. The strong magnetic field compres-sions could be associated with the variable wisps observedwith the quasi-periodic cycle of a year timescale (e.g. Scargle1969; Hester et al. 2002; Bietenholz et al. 2004). To modelthe dynamics of the wisps, Spitkovsky & Arons (2004) simu-lated outward-propagating magnetosonic waves downstreamof the shock generated by relativistic cyclotron instabil-ity of gyrating ions. The process achieves a limit cycle, inwhich the waves are launched with periodicity on the or-der of the ion Larmor time (a year timescale). Compres-sions in the magnetic field and pair density, associated withthese waves, can reproduce the behaviour of the wisp andring features. The high-resolution relativistic MHD mod-els by Camus et al. (2009) have revealed a highly variablestructure of the pulsar wind termination shock with quasi-periodic behaviour within the periods of about 2 years, andMHD turbulence on scales shorter than 1 year.The synchrotron emission in the γ -ray regime is domi-nated by infrequent quasi-coherent structures, and thereforeit should be highly polarized, that can be tested with futuretime resolved polarimetry observations.The strong γ -ray variability predicted by our model isconsistent with that observed in the Crab nebula. No simul-taneous strong flares at other wavelengths is expected in ourmodel. The γ -ray flares can occur even for a steady electron Note that we are discussing variations in the flux integratedover the downstream region (over z in our 1D model). The lackof strong variations in the flux does not contradict local variationsin the synchrotron intensity, such as the moving wisps. distribution function with the maximal energies not exceed-ing the limit of particle acceleration by the Fermi mechanismwith strong magnetic fluctuations at the extended wind ter-mination shock. This is the main difference from the modelby Yuan et al. (2011), which explains the flares by varyingmaximal electron energies in isolated knots that must trans-fer a substantial power (above 10 erg s − ) to the observedGeV photons. The variability of the maximal energies of theelectrons would result in simultaneous GeV and 100 TeVregime flares since the electrons producing the synchrotronGeV photons radiate also 100 TeV regime photons by inverseCompton scattering. Also, Bednarek & Idec (2011) pointedout that the TeV photons produced by inverse Comptonscattering of soft radiation by the variable distribution ofaccelerated electrons should be variable on timescales simi-lar to those observed at GeV energies by AGILE and
Fermi-LAT .Recently, Komissarov & Lyutikov (2011) proposed amodel in which the Crab flares originate in the inner knot(within about 1 ′′ from the Crab pulsar) with strong Dopplerbeaming effects. In our model the region of GeV photonemission is about 10 ′′ away from the pulsar. The Doppler-boosted synchrotron emission from a corrugated shock, pro-posed by Lyutikov et al. (2011), would be accompanied byflares in the cut-off regime of the inverse Compton TeV pho-tons. This is different from the prediction of our model, inwhich the strong variability of GeV synchrotron emission isdue to magnetic field variability, strongly amplified in thecut-off regime, while the amplitude of TeV emission varia-tion is expected to be much less prominent.We thank the referee J.Arons for constructive com-ments. A.M.B, G.G.P and Y.A.U were supported in part bythe Russian government grant 11.G34.31.0001 to the Saint-Petersburg State Politechnical University, and also by theRAS Programs and by the RFBR grant 11-02-12082-ofi-m-2011. The numerical simulations were performed at JSCCRAS and the SC at Ioffe Institute. The research by A.M.Bwas supported in part by the National Science Founda-tion under Grant PHY05-51164 and by ISSI (Bern). G.G.Pwas supported in part by NASA grants NNX09AC84G andNNX09AC81G, and NSF grant AST09-08611. REFERENCES
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