Very high energy emission as a probe of relativistic magnetic reconnection in pulsar winds

0.2 MB·English

by  Iwona Mochol

Mon.Not.R.Astron.Soc.000,1–6(2014) Printed5March2015 (MNLATEXstylefilev2.2)
Very high energy emission as a probe of relativistic
magnetic reconnection in pulsar winds
Iwona Mochol1,2 and J´erˆome P´etri1 ⋆
1 Observatoire Astronomique de Strasbourg, Universit´e de Strasbourg, CNRS, UMR 7550, 11 rue de l’Universit´e, F-67000 Strasbourg, France.
5 2 Institute of Nuclear Physics, Polish Academy of Sciences, ul. E. Radzikowskiego 152, 31-342 Krak´ow, Poland
1
0
2
Accepted .Received;inoriginal

r
a
M
ABSTRACT
The population of gamma-ray pulsars, including Crab observed in the TeV range,
4
and Vela detected above 50 GeV, challenges existing models of pulsed high-energy

emission. Such models should be universally applicable, yet they should account for
]
E spectral differences among the pulsars.
H We show that the gamma-ray emission of Crab and Vela can be explained by
synchrotron radiation from the current sheet of a striped wind, expanding with a
.
h modest Lorentz factor Γ.100 in the Crab case, and Γ .50 in the Vela case. In the
p Crabspectrumanewsynchrotronself-Comptoncomponentisexpectedtobedetected
-
by the upcoming experiment CTA.
o
r Wesuggestthatthegamma-rayspectrumdirectlyprobesthephysicsofrelativistic
t magnetic reconnection in the striped wind. In the most energetic pulsars, like Crab,
s
a with E˙338/2/P−2 &0.002(where E˙ is the spin down power, P is the pulsar period, and
[ X =Xi×10iinCGSunits),reconnectionproceedsintheradiativecoolingregimeand
2 results in a soft power-law distribution of cooling particles; in less powerful pulsars,
v likeVela,particleenergizationis limited bythe currentsheetsize,anda hardparticle
3 spectrum reflects the acceleration mechanism. A strict lower limit on the number
2 density of radiating particles corresponds to emission close to the light cylinder, and,
1 in units of the GJ density, it is &0.5 in the Crab wind, and &0.05 in the Vela wind.
7
0 Key words: acceleration of particles - magnetic fields - magnetic reconnection -
. radiation mechanisms: non-thermal - pulsars: general - gamma-rays:stars.
1
0
5
1
:
v 1 INTRODUCTION ther very close to the light cylinder (Lyubarskii 1996;
i Bai & Spitkovsky 2010; Arka& Dubus 2013), or in the
X It is now well established that the pulsed spectrum of the
far wind zone (P´etri 2012). Inverse Compton scattering
Crabextendsupto400GeV(Aliu et al.2011;Aleksi´c et al.
r of the thermal stellar photons by the instantaneously ac-
a 2012) and very recently newmeasurements of TeV photons
celerated wind has been also considered in this context
have been announced (Zanin et al. 2014). Although this is
(Aharonian, Bogovalov & Khangulyan 2012).
the most extreme example, several other pulsars exhibit
In the standard wind scenario (the “striped wind”
emission above 25 GeV (Ackermannet al. 2013), includ-
model), the bulk speed of the flow is roughly constant and
ing the Vela pulsar with reported detection above 50 GeV
highly relativistic. The entire physics of pulsed emission is
(Leunget al. 2014). The Fermi-LAT data of these objects
related to strong beaming effects and the presence of cur-
have been fitted with a sub-exponential cut-off or a broken
rentsheets(Kirk,Skjæraasen & Gallant 2002)–narrowre-
power-law.
gions in the wind, which are embedded in the cold, highly
Such emission has been modelled as inverse Comp-
ton scattering of magnetospheric photons by the e± magnetized stripes of opposite magnetic polarity (Coroniti
1990). Radiating particles are thought to be energized dur-
pairs created and accelerated in the magnetosphere
ing reconnection in the current layer and they are thought
(Lyutikov 2013). As an alternative, wind models have
tohaverelativisticthermaldistributionthatsatisfiesaHar-
been proposed, which attribute the gamma-ray emis-
ris equilibrium (P´etri & Kirk 2005). However, recent sim-
sion to the synchrotron radiation of energized plasma ei-
ulations of relativistic reconnection in plasmas with high
magnetization, σ 1 (where σ is the ratio of Poynting
≫
⋆ E-mail: [email protected]; flux to bulk kinetic-energy flux), suggest that the ener-
[email protected] gized particles are accelerated to a power-law distribution
(cid:13)c 2014RAS
2 I. Mochol and J. P´etri
(Sironi & Spitkovsky 2014; Werneret al. 2014). Moreover, period, and r =c/ω thelight-cylinder radius. The current
L
it has been argued that in highly magnetized, relativistic sheetthicknessisparametrised byλ∆,whereλ=2πβr is
L
systems shearing of magnetic field lines leads to formation thestriped wind wavelength and ∆ [0;0.5].
∈
of nearly force-free current layers rather than hot plasma The ordered field exists mostly in stripes, vanishing in
sheets, and therefore a force-free equilibrium is a more ap- the middle of a current sheet. We assume, however, that
propriatemodelthanaHarrisequilibrium(Guo et al.2014, the current sheet is magnetized such that there is an ad-
and references therein). ditional field component, generated in the course of recon-
In pulsar electrodynamics, the formation of a current nection,whichcanbeeithershearingorrandom,andwhich
layer can start close to the neutron star surface, along the wecall “turbulent”.Therearetworeasonsforinvokingthis
lastopenfieldlines,andextendfaroutsidethelight-cylinder component:(1)anonvanishingsynchrotronemissivityinthe
intothestriped wind region. Wedescribe acurrentlayeras middle of the current sheet (where the ordered component
approximately force-free up to the light cylinder, beyond vanishes), (2) the turbulent nature of the field seems to be
which reconnection sets in and proceeds over many wave- requiredtoexplainthedecreaseinlinearpolarizationdegree
lengths in the wind as it propagates. Since reconnection of theCrab optical pulse (P´etri 2013).
results in energy dissipation and acceleration of particles, Weassumethatonlyafractionε oftheavailablemag-
d
the energy density in the particle content increases and, netic energy is dissipated during reconnection and most
together with the reconnecting field in a sheet, it main- of the field survives as a turbulent field. The simplest
tains the pressure balance against the fields outside the model for this turbulent component proves to be B′ =
tur
layer.ThisapproachcanbecomparedwiththeFIDOmodel b B′ /cosh(ψ /∆),whereb =(1 ε )1/2.Thischoiceen-
0 max s 0 − d
(force-free inside – dissipative outside the light cylinder) of sures that in thecase of no dissipation (b =1) aturbulent
0
Kalapotharakos, Harding & Kazanas(2014),who,however, field provides a pressure balance between a current sheet
attribute the energy dissipation to a changing plasma con- and the field outside. The energy dissipation and particle
ductivity. They have demonstrated that the model is able energization may be interpreted as a change in the fraction
to reproducethe variety of Fermi-LATpulsar lightcurves. of plasma and field contributions to the pressure balance
We focus on the spectra of gamma-ray pulsars. In our (Harrison & Neukirch 2009), which, in the wind comoving
model the spectral shape is an imprint of the acceleration frame, takesthe form
regime, which can be determined either by the radiative
B′2 B′2 B′2
cooling of particles or by the size of the acceleration region max = +p+ tur (2)
8π 8π 8π
(current sheet thickness). We show that the wind scenario
can be distinguished from other competing models by ob- The pressure of ultrarelativistic particles can be approxi-
servationsofitsuniquesignature:synchrotronself-Compton matedbyp u′/3,whereu′ =mc2 γn′(γ)dγistheenergy
(SSC)emissionextendinguptotensofTeVinthespectrum density and≈n′(γ) is thedistributionRfunction.
of the Crab pulsar.
In section 2, we describe the basic picture of our wind
model, including the magnetic configuration and the parti- 2.2 Particle distribution function
cle distribution function. Next, as typical examples of two
acceleration regimes we investigate the emission properties Sofarthecurrentsheetinastripedwindhasbeenmodelled
of the winds from the Crab pulsar and the Vela pulsar, as by assuming a thermal particle distribution with a temper-
explained in section 3. Conclusions are drawn in section 4. ature determined by the synchrotron cooling (P´etri 2012).
However,simulations suggest thatduringreconnection par-
ticles are energized to a broader power-law distribution.
Power law indices s 1 are attributed to the acceleration
2 THE MODEL ≈
at the primary X-point, whereas indices up to s 3 result
≈
2.1 Magnetic field structure from acceleration inmanyX-points(Jaroschek et al.2004).
Thesespectralfeaturesdependalsoontheplasmamagneti-
Thewinddescription isbasedon thestripedwind modelof
zationσ (Guoet al.2014;Werner et al.2014),suchthatfor
P´etri (2013), valid beyond the light cylinder. In this model
thehighestvaluesofmagnetizationthespectralindextends
thewind(fluid)expandswithaconstantrelativisticLorentz
factor Γ=(1 β2)−1/2 in the radial direction. In thewind toone,s→1.Radiativecoolinghasnotbeenprobedinsim-
− ulations,butwearguethatinthiscaseacooleddistribution
comovingframetheelectricfieldvanishes.Weformulatethe
may be observed if the acceleration and cooling timescales
relevant physics in this frame and all the quantities in this
are shorter than the timescale of particle injection into the
framearedenotedwithaprime,whilenonprimedquantities
acceleration process.
should beunderstood as expressed in thelab frame.
Wesuggest thatdependingonthepulsarperiod P and
Inthefluidcomovingframethemagnitudeofthemag-
spin down power E˙, particle acceleration at reconnection
netic field is
sites in thewind proceeds in two different regimes.
B′2 =β2B2rL2 sin2θ +β2rL2 tanh2(ψ /∆) In the first – cooling regime – the particle spectrum is
Lr2 (cid:18) Γ2 r2(cid:19) s modelled as a power law with an exponential cut off:
=Bm′2axtanh2(ψs/∆) (1) n′(γ)=n′γ−se−γ/γr′ad (3)
0
where ψ = cosθcosχ+sinθsinχcos[φ ω(t r/βc)] de-
s
scribesthelocation ofthecurrentsheeto−fthew−ind,with χ γ′ istheparticleLorentzfactor,forwhichthesynchrotron
rad
beingtheanglebetweenmagneticandrotationalaxes(obliq- cooling timescale in the total magnetic field B′ = (B′2+
tot
uity),ω=2π/P being the pulsar frequency,P – thepulsar B′2 )1/2 is equal to the acceleration timescale in the recon-
tur
(cid:13)c 2014RAS,MNRAS000,1–6
VHE photons from reconnection in pulsar winds 3
nection electric field: instance,fortheCrabpulsarE˙338/2/P−2 ≈3andfortheVela
γr′ad =(cid:18)σ6πBe′τ (cid:19)1/2 (4) dpiufflsearernEt˙33s8/c2e/nPa−ri2os≈. 0.002. Theyaretypicalexamplesof two
T tot Synchrotron emission of an accelerated particle can be
3.4 105Γ3/2P1/2ε1/2 E˙−1/4 1+sin2χ 1/4 observed if its gyroradius is larger than the electron iner-
≈ × 1 −2 d,−2 38 (cid:18)Γ2rˆ−2+sin2θ(cid:19) tial length, a scale on which the reconnection takes place
1 1
(5) (Jaroschek et al. 2004; Treumann & Baumjohann 2013).
This defines the low energy cut-off γ′ in the particle dis-
w(Shpeirtekovwsekyu2s0e06B)Land≈th(e2nπo/tPa)t(ioE˙n/cX3)1=/2X(1i×+1s0ini2inχ)C−G1/S2 tgreinbeurtailonfo,rsmucthhtehcautti-noffboisthgievqe.n(b3y) aγnc′cd=eBq.′/(6(8)πγN>′mγcc′2.)I1n/2a,
units. We have also assumed that the reconnection electric
where theparticle density is
field has a form E′ τB′, where τ . 1 is a constant re-
≈ ∞
connection rate. We estimate it by comparing the recon- ′ ′
N = n(γ)dγ (9)
ntiemcteiotn′extpim∼escradlises/t(′reεcdΓ∼c)Γr(LG/i(aτncn)iowsit2h01t3h)e,wwihnedreexǫpdan≪sion1 and the parameter n′ in eZqγ.c′(3) and eq. (6) is determined
determines the fraction of the available magnetic energy by the pressure balan0ce, eq. (2). Calculation of γ′ can be
that is dissipated within the distance r . Thus, the re- c
diss simplified by approximating particle distributions, eq. (3)
connection rate can be related to the dissipation distance
andeq.(6),byapowerlawwithasharphighenergycut-off
τ ≈ Γ2εdrL/rdiss (the dissipation distance in units of the at γ′ or γ′, respectively.
light cylinder radius, rˆ = rdiss/rL, is a parameter in our rUadsingesql.(9)andtherelationN′(r)=κN r2/Γr2,
model, and we obtain it by fitting of the computed syn- GJ,LC L
where N = ωB /2πec is the Goldreich-Julian par-
chrotronfluxtothepulsarspectrum,seeSect.3.2).Forusu- GJ,LC L
ticle number density at the light cylinder, one can con-
ally invokedreconnectionrates(τ 0.01 0.2) thefulldis-
∼ − strainthenumberdensityparameterκofradiatingparticles
sipation (εd ≈ 1) would proceed over many wavelengths in κ (ecPrˆ/B′) n′(γ)dγ. Note that this expression deter-
thewind,possiblyacceleratingit(Lyubarsky& Kirk2001). ≈
minesonlyalowRerlimitonthemultiplicity,becausethelow
Theassumptionεd ≪1allows ustoneglect thewindaccel- energy cut-off γ′ is an estimation of the lowest energy of
eration. c
radiating particles, whose emission is observed.
Inthesecondregime,accelerationofparticlesislimited
by the current sheet size, thus they escape the acceleration
region before they reach the cooling regime. This case has
been investigated by Werneret al. (2014), who observe in 3 PULSED EMISSION
theirsimulationsformationofapower-lawparticledistribu-
3.1 The gamma-ray spectra
tion with a super-exponentialcut-off
Synchrotron flux produced by a particle distribution n′(γ)
n′(γ)=n′0γ−se−γ2/γs′2l, (6) is given by
dwsiimhuesurlebaetγcios′olnmsisehstaheveqeuLnaoolrtteonbteazenffraaicncttiotioria,nlifzoζerdowfwhthiitcehhsaypsaftorertmcicel-esfirzgeeey.rcoTorhnae-- ǫFǫsyn =ZV dV 4Dπd32√3eh3Bt′otǫ′Zγc∞′ dγ′n(γ′)R(z) (10)
figuration,whichisofinteresthere,andtheydonotinclude where z = (2ǫ′B )/(3B′ γ′2) and B = m2c3/e¯h = 4.4
radiativelosses,neverthelessweexpectthatwhentheescape 1013 G is the quacrntumctroittical field, cr ×
effects are dominant, a generic type of particle distribution ∞
(6)isformed,whichweadopthere.Inourcase,however,the R(z)=z dξK (ξ) 1.78z0.297e−z. (11)
Z 5/3 ≈
current sheet contains disordered fields, and locally the ac- z
celeration lengthscale is comparable to theparticle confine- ǫandǫ′ isthephotonenergynormalizedtotheelectronrest
ment scale, which we estimate as being roughly thecurrent mass in the lab and in the comoving frame, respectively; d
sheet thickness∆λ=∆2πrLβ ∆2πrL,thus is the distance between the observer and the pulsar, and D
≈
γs′l = Γζ(∆2mπrcL2)eBt′ot (7) i(s20th05e)D).oIpnptrleordufaccitnogryo=faγw′/iγnr′dad(faonrdduetsainilgstsheeepPa´erttriicl&esKpeirck-
trum (3), we obtain
≈2×108ζ−1∆−1rˆ1−1E˙318/2(cid:18)Γ211rˆ1−+2s+ins2inχ2θ(cid:19)1/2 (8) ǫFsyn x1.3 ∞ dye−(s+0.6)lny−y−x/y2, (12)
Herewehavetakenintoaccount thatin thelightcurvesthe ǫ ∝ Zγc′/γr′ad
widthofthepulsesnormalizedtotheperiodofthepulsar,if where x = (2ǫ′B )/(3B′ γ′2 ). Asymptotic behaviour of
cr tot rad
assumedtoreflecttheratioofthesheetthicknesstothewind this integral can be calculated by the steepest descent
wavelength, implies a parameter ∆ 0.1 (P´etri & Kirk method.Wefindan extremumy oftheintegrandfrom the
≈ 0
2005).Wekeepitthesameforeachpulsar.However,aphys- conditionofthevanishingofitsfirstderivative.Intheexpo-
ically motivated definition of the current sheet thickness is nentialtailγ′ γ′ ,thesolutioncanbefoundanalytically
basedontheregimeinwhichparticleaccelerationproceeds. y (2x)1/3. N≫extr,adwe expand the integrand to the second
0
It depends solely on the combination of two pulsar observ- ord≈eraroundy .Assumingγ′ γ′ ,weintegratearesult-
ables: its spin down power and its period, since γr′ad and ing Gaussian fu0nction. An ascy≪mptoratdics takes theform
γcos′lrdbiencgomtoeocuormfiptasradbilsecuwssheednbE˙e33l8o/2w/PΓ−2 ∼20..50,0rˆ2 (wh3e)r.eFaocr- ǫFsyn x1.3−(s+0.6)/3e−1.9x1/3 (13)
1 ≈ 1 ≈ ǫ ∝
(cid:13)c 2014RAS,MNRAS000,1–6
4 I. Mochol and J. P´etri
10−8
10−9
εd, s, Γ, rˆ
0.026, 2.2, 21, 60
0.01, 2.4, 19, 21
10−10 00..0011,, 22..21,, 2234,, 3467 10−9
1] 3e-7,2.2, 82, 2 1]
− −
−2s 10−11 x0.37CeTxpA(−se1n.s9ixt1iv/i3ty) −2s
m m 10−10
c c
g g
er 10−12 er
[ [
ǫ ǫ
F F
ǫ ǫ 10−11 εd, s, Γ, rˆ
10−13 0.00.2061,, 11..22,, 1178,, 2102
1e-4, 1.2, 40, 2
x0.85exp(−2x1/2)
10−14 10−12
10−2 10−1 100 101 102 103 104 105 10−1 100 101 102
ǫ[GeV] ǫ[GeV]
Figure 1. Several best fits to the Crab spectrum. Thick gray Figure 2.SeveralbestfitstothesynchrotronspectrumofVela.
lineshowsthepredictedCTAsensitivity(deOn˜a-Wilhelmietal. Forahardparticleindex,theSSCcomponentisweakerbyseveral
2012), blackdashed lineisthe asymptote (13), plotted withs= ordersofmagnitudeanditisnotshownintheplot.Blackdashed
2.2.DashedlinesshowSSCcomponents,whilesolidlinesarethe lineistheasymptote(14),plottedwiths=1.2.Blackpointsare
total (synchrotron + SSC) spectrum. Black points are the data thedatafromFermi-LAT(Leungetal.2014)
from Kuiperetal. (2001), Abdoetal. (2010) and Aleksi´cetal.
(2014).
thattheemissionpeaksarenormalizedto1).Thisapproach
is justified by the fact that the model spectra change very
Note that this is very similar to the spectrum exp(x0.35) weakly with the pulsar phase (due to a weak angle depen-
∝
obtained byArka& Dubus(2013) by a fit. denceoftheDopplerfactorwhentheemission isboostedto
In a similar way we calculate an asymptote of a spec- theobserver frame).
trum produced by theparticle distribution (6). In this case In Fig. 1 we present several fits to the Crab data from
theexponent has a form f(y)= (s+0.6)lny y2 x/y2 Fermi-LATandMAGIC.Thereisnouniquesolutionaslong
− − −
and its derivative has a maximum at y0 x1/4. The flux as the measurements in the TeV range are missing. Com-
≈
has thefollowing asymptotic behaviour: puted spectra result from an interplay between all four pa-
ǫFsyn x1.3−(s+0.6)/4e−2x1/2 (14) rametersofthemodel,whichmustbefoundsimultaneously
ǫ ∝ in order to give a reasonable fit. The green, blue and or-
The asymptotes (13) and (14), together with fitted spectra ange curves show the spectra with s=2.2 and different ε ,
d
(seenextsection)areshowninFig.1andFig.2,respectively. for which we have found the best-fit Γ and rˆ. In general,
smaller ε lead to smaller rˆand larger Γ, thus, by increas-
d
ing Γ and decreasing ǫ we are able to fit the data with
3.2 Constraining the model parameters d
the model of emission close to the pulsar, at rˆ=2 (orange
There are four free parameters of the model: the Lorentz curve). This fit gives an estimate of the maximum possible
factorofthewindΓ,thedissipation distanceinunitsofthe Lorentzfactor of thewind,on theorderof Γ 100, similar
∼
lightcylinderrˆ,theparticleindexsandthedissipationeffi- tothevalueassumedinpreviousstudies(Arka& Kirk2012;
ciency ε . We constrain them by matching the synchrotron Mochol & Kirk 2013). Note that for the best fits with the
d
flux(10) tothe phase-averaged spectra. same value of s the brightness of the SSC component does
To compute the fluxes in our model we integrate the not change significantly, but the maximum energy of SSC
wind emissivity over 3D volume (see eq. (10)) using spec- photons does. It is estimated by the Klein-Nishina limit,
tral methods (Press, Flannery & Teukolsky 1986). Accord- boosted to theobserver frame:
ing to this approach, the integrand is expanded in a basis ǫ 2Γγ′ mc2 3.6Γ5/2P1/2ε1/2 E˙−1/4TeV. (15)
of orthogonal polynomials and integrated term byterm. As ≈ rad ≈ 1 −2 d,−2 38
bases we choose the same special functions as proposed by Red, blue and pink curves show the fits for ε = 0.01
d
P´etri (2013), i.e., Chebyshev polynomials in coordinate r, anddifferents,forwhich,asbefore, wehavealsofound the
Chebyshev polynomials in cosθ, and a Fourier series in φ. best-fitΓ andrˆ.Inthiscase, thelargeriss,thesmaller are
This choices allow us to use a fast cosine transform to cal- alsorˆandΓ.ThebrightnessoftheSSCcomponentchanges
culatethecoefficientsofexpansioninthepolynomialbases, significantly, being greater for larger s. With the currently
andtoapply directlytheClenshaw-Curtis quadraturerules available VHE data the upper limit on the particle index
for computing integrals (Press et al. 1986). can be estimated to be s 2.4 (red curve). In general, soft
≈
The model spectrum is calculated at the peak of the indicess&2arerequiredtoreproducethegamma-raybump
lightcurve, and we obtain the phase averaged flux using intheCrabspectrum.Thelowenergycut-offintheparticle
F = F I, where F is the flux at the peak maximum, distribution is estimated to beroughly γ′ 50.
av 0 0 c∼
and I is the integral area under the lightcurve (assuming A fit to the Vela spectrum requires a much harder in-
(cid:13)c 2014RAS,MNRAS000,1–6
VHE photons from reconnection in pulsar winds 5
Crab Vela Crab Vela ACKNOWLEDGMENTS
εd 0.01 0.01 εd=3×10−7 εd=10−5 Wearegrateful toJohn Kirk andtotheanonymousreferee
Γ 23 18 82 40 forhelpfulcommentsonthemanuscript.Wealso thankAl-
rˆ 36 12 2 2
ice Harding for drawing our attention to the newest results
τ 0.18 0.26 0.001 0.01
ofMAGIC.ThisworkhasbeensupportedbytheFrenchNa-
κ 7×104 14 0.5 0.05
tionalResearchAgency(ANR)throughthegrantNo.ANR-
Table 1.Reconnectionrateτ andthenumberdensityκofradi- 13-JS05-0003-01 (project EMPERE). We also benefited
atingparticlesobtainedfortwoexemplaryfitstothespectra. from the computational facilities available at Equip@Meso
(Universit´edeStrasbourg).
dex s 1.2 (Fig. 2). In this case the SSC component is
≈
weaker by several orders of magnitude (not shown in the
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Zanin, R., for the MAGIC Collaboration, 2014, 5th Int.
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(cid:13)c 2014RAS,MNRAS000,1–6