On the origin of variable gamma-ray emission from the Crab Nebula

0.77 MB·English

by  S. S. Komissarov

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed1February2011 (MNLATEXstylefilev2.2)
On the origin of variable gamma-ray emission from the
Crab Nebula
S. S. Komissarov,1(cid:63) M. Lyutikov1†
1 1Department of Applied Mathematics, The University of Leeds, Leeds, LS2 9GT
1 2Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907-2036, USA
0
2
n Received/Accepted
a
J
1 ABSTRACT
3 The oblique geometry of pulsar wind termination shock ensures that the Doppler
beaminghasastrongimpactontheshockemission.Weillustratethisusingtherecent

] relativistic MHD simulations of the Crab Nebula and analysis of oblique shocks. We
E
also show that the observed size, shape, and distance from the Crab pulsar of the
H Crab Nebula inner knot are consistent with its interpretation as a Doppler-boosted
. emission from the termination shock. If the electrons responsible for the synchrotron
h
gamma-rays are accelerated only at the termination shock then their short life-time
p
ensures that these gamma-rays originate close to the shock and are also strongly
-
o effected by the Doppler beaming. As the result, bulk of the observed synchrotron
r gamma-rays of the CrabNebula around 100 MeVmaycome from its inner knot. This
t
s hypothesis is consistent with the observed optical flux of the inner knot, provided
a
its optical-gamma spectral index is the same as the injection spectral index found
[
in the Kennel & Coroniti model of the nebula spectrum. The observed variability of

2 synchrotron gamma-ray emission on the time scale of wisp production can be caused
v bytheinstabilityoftheterminationshockdiscoveredinrecentnumericalsimulations.
0 Given the small size of the knot, it is also possible that the September 2010 gamma-
0 ray flare of the Crab Nebula also came from the knot, though the actual mechanism
8
remains unclear. The model predicts correlation of the temporal variability of the
1
synchrotron gamma-ray flux in the Fermi and AGILE windows with the variability of
.
1 the un-pulsed optical flux from within 1(cid:48)(cid:48) of the Crab pulsar.
1
0 Key words: ISM: supernova remnants – MHD – shock waves – gamma-rays: theory
1 – radiation mechanisms: non-thermal – relativity – pulsars: individual: Crab
:
v
i
X
r 1 INTRODUCTION celeration at the wind termination shock, the second-order
a
Fermi acceleration in the turbulent plasma of the nebula,
The Crab Nebula has been a source of intriguing discov-
including secondary shocks, and the magnetic reconnection
eries and served as a test bed of astrophysics for decades.
events.
ThisisoneofthebeststudiedobjectsbeyondtheSolarsys-
Compared to the highly filamentary thermal emission,
tem. It has been observed at all wavelengths, from radio to
thenon-thermalemissionisrelativelyfeatureless.Yet,itwas
veryhighenergygamma-rays.Itsnon-thermalemissionbe-
low Eb (cid:39)500MeV is a synchrotron emission of relativistic discovered already in 1920 that fine and dynamic “wisps”
ph are somehow produced in the center of the nebula (Lamp-
electrons in the nebula magnetic field and above Eb it is
ph land1921;Scargle1969).Later,theX-rayobservationsdis-
the inverse Compton emission of the same electrons. The
covered the famous jet-torus structure in the inner nebula
emitting electrons are accelerated up to PeV energies, indi-
(Weisskopfetal.2000;Hesteretal.2002),andthehighreso-
catingthattheaccelerationmechanismisveryefficient.The
lutionopticalobservationswithHubbleSpaceTelescopere-
sourceofenergyistheultra-relativisticmagneticwindfrom
vealedfinesub-arcsecondstructureofthenon-thermalemis-
the pulsar (Rees & Gunn 1974; Kennel & Coroniti 1984a),
sion, including few optical knots (Hester et al. 1995, 2002).
but the actual mechanism of particle acceleration is still a
The synchrotron life-time of electrons emitting in op-
mystery. The main candidates are the diffusive shock ac-
ticsiscomparabletothedynamicaltime-scaleofthenebula,
andthismakesitdifficulttospottheexactlocationsofthe
(cid:63) E-mail:[email protected](SSK) particleaccelerationcites.Ingamma-raysbelowEpbh,where
† E-mail:[email protected] the life-time becomes short, the angular resolution of the
(cid:13)c 0000RAS
2 Komissarov & Lyutikov
telescopes is insufficient to see the nebula structure. How- elongated, with the main axis perpendicular to the jet. Its
ever,theobservationsindicatedvariabilityofthegamma-ray lengthandwidthareψ (cid:39)0(cid:48).(cid:48)50andψ (cid:39)0(cid:48).(cid:48)16respectively
⊥ (cid:107)
emissioninthe1-150MeVrange(Muchetal.1995;deJager (Hester et al. 1995). A number of recent optical and infra-
etal.1996)onthetimescalearoundoneyear.deJageretal. red observations of the inner Crab Nebula focused on the
(1996)proposedthatthisemissioncouldoriginatefromthe knot variability. Melatos et al. (2005) reported no variabil-
variableopticalfeaturesseenwithHSTinthepolarregions ity on the timescales from 1 kilosecond to 48hr. Tziamtzis
oftheinnernebula,inparticulartheso-called“anvil”.Vari- et al. (2009) compared the measurements separated by two
ability on a similar time-scale has been recently discovered andhalfmonthsandfoundnosignificantdifferencetoo.On
in the X-ray emission (Wilson-Hodge et al. 2010). theotherhand,Sollerman(2003)analyzedtheHSTarchive
In September 2010, AGILE collaboration reported a dataanddiscoveredfluxvariationsonthelevelof50%over
three-foldincreaseofthegamma-rayflux(>100MeV)from the period of 6 years1. They also reported possible random
thedirectionoftheCrabNebula(Tavanietal.2011),which displacementsoftheknotonthescaleof0(cid:48).(cid:48)1.Morerecently,
was immediately confirmed by Fermi LAT collaboration, Sandberg & Sollerman (2009) reported twice as higher flux
who reported a six-fold increase of the flux (Abdo et al. fromtheinnerknotin2003comparedtotheearlierobserva-
2011). The flare continued for four days, September 18-22, tions in 2000. Thus, we tentatively conclude that the inner
after which the gamma-ray flux returned to the pre-flare knot does show significant variations of its flux, and pos-
level. Fermi also reported that the pulsed emission of the sibly location, on the time-scale comparable to that of the
Crab pulsar remained unchanged during the flare, suggest- gamma-rays variability reported by de Jager et al. (1996).
ingthattheflareoriginatedintheNebula.JodrellBankra- Thesedatasuggesttoconsiderthepossibilitythattheinner
diotimingobservationsoftheCrabpulsarshowednoglitch knot may be a strong source of gamma-rays, both during
during the flare, supporting this conclusion (ATel#: 2889). and between the flares.
Atthesametime,INTEGRALreportednodetectionofthe The synthetic maps of synchrotron emission from the
flareduringSep19inthe20-400keVwindow(ATel#:2856) Crab Nebula, based on numerical relativistic MHD sim-
and Swift/BAT did not see any significant variability dur- ulations, reveal a bright compact feature strikingly simi-
ing the gamma-ray flare in the 14-150 keV range (ATel#: lar in its appearance and location to the inner knot (see
2893). Swift also reported no evidence for active AGN near Figure 2). In Komissarov & Lyubarsky (2004) this feature
the Crab, suggesting that the Crab itself is responsible for was identified with the Doppler-boosted emission from the
theflare(ATel#:2868).ARGO-YBJcollaborationreported high-velocity flow downstream of the oblique termination
asignificantenhancementoftheveryhighenergyemission, shock of the pulsar wind. The more recent simulations of
around 1 TeV, from the Crab nebula during the AGILE- the Crab Nebula, which had a significantly higher resolu-
Fermi flare (ATel#: 2921). However, this has not been con- tion, discovered strong variability of the termination shock,
firmed by VERITAS and MAGIC collaborations (ATel#: involvingdramaticchangesofitsshapeandinclination(Ca-
2967,2968). This discovery seems to have given credit to mus et al. 2009). This discovery suggests that the gamma-
another event, detected in February 2009, which lasted for rayvariabilitymayberelatedtothechangesintheDoppler
approximately 14 days, during which the gamma-ray flux beaming of the post-shock emission, associated with this
increased by a factor of three or four (Tavani et al. 2011; structural variability of the termination shock. Volpi et al.
Abdoetal.2011).OntheSEDplotstheflaresappearasan (2008)havealreadyaddressedtheissueofvariabilityofboth
extensionofthesynchrotroncomponentfurtherouttowards the synchrotron and inverse Compton emission of the Crab
thehigherenergies,upto1GeVfortheSeptemberflareand Nebula in their computer simulations. They reported the
a bit less dramatic for the February flare. strongest variability in the part of the gamma-ray window
The short duration of these flares suggests that their where the synchrotron emission is still dominant over the
source is rather compact. Unfortunately, no high angular inverse Compton emission, and on the time-scale compara-
resolution observations of the nebula were carried out dur- ble to that found by (de Jager et al. 1996). However, they
ing the flares. The Crab Nebula images from Chandra and could not identify the source of variability with any partic-
HST, obtained after the September flare, have not revealed ular feature in their numerical solutions.
anything especially unusual (ATel#: 2882, ATel#: 2903). Here we present new arguments in favour of the inter-
Theydoshowachangeinthestructureofthenebulawisps pretationoftheinnerknotasaDoppler-boostedshockemis-
compared to previous observations, carried out years ear- sion,andinvestigatewhethertheinnerknotcanbeastrong
lier. However, the large length scale of these wisps shows sourceofvariablegamma-rayemission.InSection2weanal-
that they can hardly be a source of the flares. The Chan- yse the observed shape and location of the inner knot and
dra images also show a significant change in the position show that they are consistent with this interpretation. In
of one of the jet knots, which apparently had moved about Section3wearguethatatleastasignificantfractionofthe
3(cid:48)(cid:48) towards the pulsar. This may be more significant as this synchrotron gamma-ray emission of the Crab Nebula orig-
feature is more compact. inates from the inner knot. The key factors are the short
Evenmorepotentsourceoftheflarescouldbethemys- cooling time of electrons and the strong Doppler beaming
terious “inner knot”, discovered only 0(cid:48).(cid:48)65 away from the of the emission originated in the vicinity of the termina-
Crab pulsar along the jet direction. This knot, named as tion shock. In Section 4 we discuss the possible connection
“knot 1” in Hester et al. (1995), is the brightest and most betweentheobservedvariabilityofthegamma-rayemission
compact persistent feature of the Crab Nebula. It is seen
at more or less the same location in many observations,
both with space and ground based telescopes with adap- 1 For some reason the actual observational data have not been
tive optics, which followed its discovery. It is extended and presentedinthispaper.
(cid:13)c 0000RAS,MNRAS000,000–000
Gamma-ray emission from the Crab Nebula 3
thepropermagneticfieldinthisregionisstrongest.Several
factorsareresponsibleforthisresult.Althoughthethemag-
netic field of dissipationless wind behaves as B ∝ sinθ/r
φ
(Michel1973;Bogovalov1999),inthesimulationsthisfunc-
tionwasmultipliedby|1−2θ/π|,inordertoaccountforthe
magneticfielddissipationinthestripedwindzone.Thus,the
wind magnetic field peaked at θ (cid:39) 57◦ instead of θ (cid:39) 90◦.
More important, however, is the axial compression of the
nebulabythemagnetichoopstress,whichleadstothetotal
pressure downstream of the termination shock to be signif-
icantly higher at small polar angles. As the result, the arch
shock is pushed closer to the pulsar, leading to a stronger
upstreamandhencedownstreammagneticfield.Second,the
emission from the upper arch shock is Doppler boosted. In-
deed,asonecanseeintheleftpanelofFigure2,theLorentz
factor of the flow downstream of the arch shocks is quite
Figure1.InterpretationoftheinnerknotastheDoppler-boosted
high.
emission from the high velocity flow located downstream of the
The analysis of oblique relativistic MHD shocks given
oblique termination shock. The termination shock is shown by
thesolidline.Thedashedlineisthelineofsight. in Appendix A shows that in the case of ultra-relativistic
cold upstream flow the downstream Lorentz factor is
1 1
aicnadlstihmeunloanti-osntas,tiionnpaarrytischuolcakrtdhyenraomleicosfdthisecovvaerriaebdleinDnoupmpelerr- γ2 = (cid:112)1−χ2sinδ1, (1)
beamorientation.OurconclusionsaregiveninSection5.In whereδ istheanglebetweentheupstreamvelocityandthe
1
AppendixAwepresenttheanalysisofobliqueMHDshocks. shock plane and
ItsresultsallowustodeterminehowhightheLorentzfactor
(cid:112)
1+2σ + 16σ2+16σ +1
downstream of the Crab’s termination shock can be and to χ= 1 1 1 , (2)
confirm the results of numerical simulations. 6(1+σ1)
where σ = B2/4πρ c2 is the magnetization parameter of
1 1 1
the upstream flow. For σ (cid:29)1 this yields
1
2 ORIGIN OF THE INNER KNOT
σ1/2
γ (cid:39) 1 , (3)
The first numerical simulations of the Crab Nebula by 2 sinδ
1
(Komissarov&Lyubarsky2003,2004;DelZannaetal.2004;
and for σ (cid:28)1
1
Bogovalov et al. 2005) had rather low numerical resolution
3 1 1
and imposed reflectional symmetry in the equatorial plane. γ (cid:39) √ (1+ σ ) . (4)
In these simulations the termination shock appeared as a 2 2 2 2 1 sinδ1
more or less stationary complex structure, which in fact in- In the simulations σ varies with the polar angle between
1
cludedseveraldifferentshocks.Thesecomponents,namedas 0 and 0.05, with the volume averaged value (cid:104)σ (cid:105) (cid:39) 0.014,
1
archshocks,rimshocks,andtheMachbeltinKomissarov& and thus the latter limit applies. On can see that for small
Lyubarsky(2004),areshowninFigure1.Thelatestsimula- shock inclination angles the Lorentz factor can indeed be
tionsbyCamusetal.(2009)hadmuchhigherresolutionand quite high. Even higher values are expected for high-sigma
the computational domain included the whole range of the pulsar wind.
polarangle,θ∈(0,π).Inthesesimulations,thestructureof In addition to having high Lorentz factor in the down-
termination shock appeared highly distorted and dynamic, stream flow, the upper arch shock is inclined at the angle
butasonecanseeinFigure2,theseindividualcomponents of ∼ 60◦ to the polar axis, near the axis. Observations of
were still identifiable. the inner Crab Nebula suggest that the angle between the
The right panel of Figure 2 shows the typical distri- line of sight and the symmetry axis of the nebula is also
bution of the synchrotron emissivity as measured in the close to 60◦ (Weisskopf et al. 2000). Thus, the upper arch
pulsar/observer frame, found in the latest numerical sim- shockiswellalignedwiththelineofsight,resultinginstrong
ulations 2. One can see that it is strongly enhanced in the Doppler-boostingofitsemission.Thisisschematicallyillus-
vicinity of the upper arch shock of the termination shock trated in Figure 1. The left panel of Figure 3 shows the
complex.Therearetworeasonsforthisenhancement.First, syntheticopticalsynchrotronimageoftheinnerpartofthe
simulatedPWNatthetimecorrespondingtotheageofthe
Crab Nebula. One can see prominent wisps and a bright
2 The details of these numerical simulations and the radiation
knot located very close to the origin, where the projected
transfercomputationsaredescribedinCamusetal.(2009).Here
image of the pulsar would appear if it was included in the
we only note that the radiation model assumes that relativistic
emissionmodel.Inthesimulations,thereisnoemissionfrom
etelercmtrinonatsedwiatht Ethmeaxpo=wer1PlaewV,eanreergiynjsepcteecdtruamt tNhe(Etee)rm∝inaEte2io.2n, thepulsarwindastheemittingelectronsareinjectedatthe
e
shock, to be more precise at the arch shock and the Mach belt, termination shock only. Thus, all the fine features of the
and then they evolve subject to advection and synchrotron and synthetic synchrotron images, including the inner knot, are
adiabaticenergylosses. produces inside the nebula.
(cid:13)c 0000RAS,MNRAS000,000–000
4 Komissarov & Lyutikov
Figure2. TerminationshockinnumericalsimulationsofCamusetal.(2009).TheleftpanelshowstheflowLorentzfactor.Inthisplot
thewindzonehasbeencutoff(Thethinlineatr∼0.7isanartifactofthisprocedure,whichconvenientlyindicatesthelocationofthe
Machbelt).Therightpanelshowstheobservedsynchrotronemissivityintheopticalrange,log j ,intheplanewhichincludesthe
10 ν,ob
lineofsightandthesymmetryaxis.Theanglebetweenthelineofsightandthesymmetryaxisis60◦.
Figure 3. Synthetic images of the inner Crab nebula in optics. The panel shows the proper synchrotron image, log10Iν, where Iν is
theintensityofradiation.Theanglebetweenthelineofsightandthesymmetryaxisis60◦.Therightpanelshowshowtheimagewould
lookifthetheDopplerbeamingwasnottakenintoaccount.
It is interesting that both in the synthetic and the real size but that is about it. Several factors contribute to this
optical images of the Crab Nebula the termination shock effect. Firstly, the inner cavity filled with the pulsar wind
is not clearly identifiable3. The inner wisps give away its
ring,calledthe“innerring”,aroundthepulsar.Thisringisoften
3 TheChandraX-rayimageoftheinnerCrabNebularisoverall identified with the termination shock. The fact that the ring is
similar to the HST image (Hester et al. 2002). There are how- relatively symmetric, in contrast with the optical image, is hard
eversomenoticeabledifferences.Inparticular,theX-rayimageis to explain in our model and indicates that some other factors
muchknottierandsomeoftheseX-rayknotsarrangeinasortof havetobeincluded.
(cid:13)c 0000RAS,MNRAS000,000–000
Gamma-ray emission from the Crab Nebula 5
Figure 4. Basicgeometricparametersoftheinnerknot.
is small compared to the size of the nebula, making the compact and that it appears at the base of the Crab jet
central brightness reduction rather weak. Secondly, the re- are more suggestive of some jet feature rather than the ex-
gion of enhanced proper emissivity around the termination tendedterminationshock.However,theknotshapeandsize
shockformsageometricallythindistortedshell.Becauseof are nicely explained in the shock model. Denote as ψ the
p
this, the observed emission is strongly enhanced in places observedangulardistanceoftheknotfromthepulsar,asd
n
where the line of sight is tangent to the shell surface. This the linear distance to the nebula, and as d the linear dis-
k
leads to the appearance of several bright rings on synthetic tancebetweenthepulsarandthepointontheshocksurface
maps where the Doppler beaming is not included (see the where the line of sight is tangent to the shock. The angle
right panel of Figure 3). The Doppler beaming leads to in- betweentheshocksurfaceandtheupstreamvelocityvector
creasedemissivityinthepartofsucharingwhereitsplasma denote as δ (see Fig.4). Then in the small angle approxi-
1
flows towards the observer, and decreased emissivity where mation
itflowsaway,turningtheringintoanarc(seetheleftpanel
δ =ψ (d /d ). (5)
of Figure 3). Because of the non-spherical shape of the ter- 1 p n k
mination shock, the jet base is located much closer to the Using Eq.1 we can now find the Lorentz factor of the post-
pulsarcomparedtotheMachbeltradius,makingthewrong shock flow
impressionthatthejetoriginatesfromtheinsideoftheter-
f
mination shock and suggesting that it is produced by the γ2 = δσ (6)
pulsar. 1
and the beaming angle
Comparing the images presented in Figure 3 one can
see how some features brighten up and others get dimmer φ = 1 = δ1, (7)
because of the Doppler beaming. In particularly, the bright D γ2 fσ
knot in the center of the left image is certainly Doppler-
wheref =(1−χ(σ ))−1/2.Thisallowsustofindthetrans-
σ 1
boosted.Komissarov&Lyubarsky(2004)proposedthatthis
verse angular size of the knot
syntheticknotisacounterpartoftheCrab’sinnerknot.In-
deed, like the inner knot it is positioned on the jet-side of ψ = dkφD = 1 ψ . (8)
the nebula at the base of the jet (only in projection) and it ⊥ dn fσ p
iselongatedinthedirectionperpendiculartothejet(Hester For σ ≤1, one has f (cid:39)1 and, thus,
1 σ
etal.1995).Othersyntheticjetknots,whichareseeninFig-
ure3,havemoreorlessthesamebrightnessinboththeseim- ψ⊥ (cid:39)ψp, (9)
ages,indicatingthattheDopplerbeamingisnotthatimpor-
whichisinexcellentagreementwiththeobservationsofthe
tant.Theyindeedoriginateatthebaseofthepolarjets.The
inner knot. The same argument shows that HST knot 2,
jetLorentzfactorisvariablebutonaverageitisratherlow,
which has similar size but is located much further out from
γ ∼ 1.5. Combined with the large viewing angle, 60◦, this
j thepulsar,cannotbetheDoppler-boostedpartofthetermi-
explain why the Doppler effect is rather weak. Phenomeno-
nationshock,unlessthepulsarwindmagnetizationσ (cid:29)1.
1
logically, these jet knots are created via unsteady inhomo-
The other size of the inner knot, ψ , is determined by
geneous axial magnetic pinch, which is responsible for the (cid:107)
thethicknessofthepost-shockplasmaflowandtheshocklo-
jet formation. The flow towards the polar axis, which feeds
calcurvature.Theflowthicknessatthedistanced fromthe
k
thejet,ishighlyinhomogeneouswithpatchesofstrongand
pulsarcanbeestimatedasδ d ,whereδ isthedownstream
2 k 2
weakmagneticfield,resultinginstrongspatialvariationsof
anglebetweenthevelocityvectorandtheshockplane.Since
the magnetic hoop stress and highly variable jet dynamics.
the distance of the shock from the pulsar across the line of
The non-linear sausage-mode of the magnetic pinch insta-
sight is δ d , this yields ψ = (δ /δ )ψ . Using Eq.A19 to
bility could be another contributor to the jet variability. 1 k (cid:107) 2 1 p
evaluate δ /δ and Eq.9 we then find that if ψ is fully de-
2 1 (cid:107)
The fact that the inner knot of the Crab Nebula is so termined by the thickness of the post-shock flow then
(cid:13)c 0000RAS,MNRAS000,000–000
6 Komissarov & Lyutikov
1
ψ (cid:39) ψ , (10) theterminationshock.Anumberofdifferentideashavebeen
(cid:107) 3 ⊥ put forward but the issue is far from settled (Lyubarsky
in excellent agreement with the observations again. & Kirk 2001; Kirk & Skjæraasen 2003; Lyubarsky 2003b,a;
Theshockcurvaturewouldleadtoafinitelinearwidth Arons 2008). An alternative model, where the flow remains
of the knot in projection on the plane of the sky even if Poynting-dominated even inside the nebula, has been put
the postshock flow was infinitely thin. This width can be forward recently (Lyutikov 2010). As the result, it is not
estimatedasR (1−cos(φ /2)),whereR isthelocalshock clearastowhatmodelofthepulsarwindshouldbeusedin
c D c
curvature radius. For the small angles this yields setting the inflow boundary condition in the MHD simula-
tions of PWN. Moreover, so far only two-dimensional sim-
(R /d )
ψ (cid:39) c k ψ . (11) ulation have been carried out, which leaves unexplored the
(cid:107) 8γ f ⊥
2 σ effectsofnon-axisymmetricinstabilitiesonthenebulastruc-
Thus, unless (R /d )>8γ f , we have ψ <ψ , in agree- ture and dynamics (Begelman 1998).
c k 2 σ (cid:107) ⊥
ment with the observations. Numerical simulations show
that normally R /d <10.
c k
Thetransverseangularsizeoftheknotcouldhavebeen How and where the emitting particles are accelerated
used to infer the Lorentz factor of the post-shock flow if isalsodebated.Inthesimulations,itwasassumedthatthe
we knew the distance d between the knot and the pulsar. synchrotronelectronsareacceleratedonlyatthearchshock
k
Indeed, combining Eqs. 6 and 5 one finds that and the Mach belt, but the acceleration can also occur at
the rim shocks and in the turbulent interior of the nebula.
d 1
γ (cid:39) k .
2 d ψ
n ⊥
Sinceweonlyknowthatd cannotexceedtheradiusofthe Giventheseuncertainties,itisnotsurprisingthatthere
k
termination shock, this equation only allows us to find the arestillsomesignificantquantitativedifferencesbetweenthe
upper limit theory and the observations. For example, the observations
reveal comparable isotropic luminosities of the inner knot
ψ
γ2 < ψts (cid:39)20, and the brightest wisps (Hester et al. 1995). In contrast, in
⊥ the synthetic synchrotron images the knot is several times
where ψ (cid:39) 10(cid:48)(cid:48) is the angular size of termination shock brighter. This is illustrated in Figure 2, which shows the
ts
inferred from the observations (Hester et al. 2002). intensity of radiation using logarithmic scaling. One possi-
If the inner knot is indeed a part of the termination ble reason for this is the excessive axial compression of the
shock then its spectrum can be used to infer the proper- nebula via the hoop stress of the azimuthal magnetic field,
tiesoftheparticleaccelerationattheshock.Severalgroups caused by the condition of axisymmetry. In fact, the syn-
have carried out optical and near-infrared observations of theticopticalimagesoftheCrabNebulashowstrongglobal
theknotinrecentyears.Unfortunately,theirresultsdonot enhancement of the surface brightness along the symmetry
quiteagree.Sandberg&Sollerman(2009)reportedtheopti- axis, which is not present in the images of the Crab Neb-
calspectralindexα=1.3,assumingthatI ∝ν−α.Onthe ula. The observed large scale kink of the Crab jet indicates
ν
otherhand,thedatapresentedintheirFig.2suggestamuch thatsomekindofkink-modeinstabilitysignificantlyreduces
flatternear-infraredspectrum,withα∼0.3.Itisdifficultto the degree of symmetry in the polar regions of the nebula.
seehowthesynchrotronmechanismcanaccommodatesuch This is important as, in addition to the anisotropic power
a large variation of spectral index within only one decade distribution in the pulsar wind, the overall geometry of the
offrequency.Perhaps,thesemeasurementssufferfromlarge termination shock is also influenced by the pressure distri-
systematic errors. The proximity of the knot to the pulsar bution inside the nebula. The enhanced pressure near the
could be one of the complications. According to the data axispushesthecuspofthearchshockfurtherdowntowards
obtainedbyMelatosetal.(2005)thenear-infraredspectral thepulsar,makingtheterminationshocklesssphericalcom-
index of the inner knot is α = 0.78. Unfortunately, the ac- pared to what it would be in the case of uniform pressure
curacy of this measurement is not given, whereas for other distributioninsidethenebula.Thisleadstohighermagnetic
featurestheerrorisgivenanditisabout±0.13.Theproxim- fieldandnumberdensityofemittingparticles,andhencein-
itytothepulsarsuggeststhatfortheinnerknottheerroris creased volume emissivity near the shockcusp. In addition,
higher. Finally, Tziamtzis et al. (2009) give α=0.63±0.02 theshockgeometrydeterminesthevelocitydistributionand
for the optical emission of the inner knot. The very small hence the Doppler beaming.
error indicates that this is the most accurate measurement
todate.Thisresultisinexcellentagreementwiththevalue
oftheinjectionspectralindexinferredbyKennel&Coroniti In three dimensional simulations of the Crab Nebula,
(1984)viamodelfittingoftheintegralspectrumoftheCrab thealmostperfectalignmentofthearchshockwiththeline
Nebula. ofsighmaynolongerbefound.Lesssquashedalongthepo-
Although, the MHD model in general, and the recent lardirection,theterminationshockwouldhavelowerproper
numerical MHD simulations in particular, have enjoyed a emissivitynearthecuspregion.TheDoppler-boostingofthe
lot of success in explaining the properties of the Crab Neb- knotemissionwillbereducedaswell,andnotonlybecause
ula,aswellasotherPWN,itisbynomeansproblemsfree. of the less perfect alignment of its Doppler beam with the
Theso-calledσ-problemisitsmainweakness.Itisnotclear lineofsight.Moresphericalshapeoftheterminationshock
how exactly the pulsar wind turns from being Poynting- wouldalsoleadtohighershockinclinationanglesandlower
dominatednearthepulsartokinetic-energy-dominatednear Lorentz factor of the downstream flow.
(cid:13)c 0000RAS,MNRAS000,000–000
Gamma-ray emission from the Crab Nebula 7
Downstream of the Mach belt the magnetic field and
the number density of the emitting particles is significantly
lower compared to the arch shock, and for this reason the
synchrotron emissivity is low as well (see Figure 2). Thus,
it is the wind plasma which has just passed through the
arch shock which is likely to be the main emitter of the
synchrotron gamma-rays. This plasma flows with relativis-
ticspeedandissubjecttostrongDopplerbeaming(seeFig-
ure2).Thisresultsinboostingoftheemissionfromthepart
oftheflowwherethevelocityvectorisclosetolineofsight,
the inner knot region, and dimming of the emission from
other parts, where the viewing angle exceeds 2/γ . Thus, a
2
significantfraction,ifnotmost,oftheobservedsynchrotron
gamma-rayemissionoftheCrabNebulamayoriginatefrom
itsinnerknot.Infact,quickinspectionofournumericalso-
lutions shows that at 100 MeV the inner knot is essentially
theonlyfeatureinthesky(seeFigure5).However,giventhe
uncertainties of the numerical model, one cannot exclude a
contribution from few brightest wisps.
In order to test this idea against the observations one
can compare the observed flux from the Crab’s inner knot
inopticswiththeobservedfluxfromthewholeoftheCrab
Figure 5. Synthetic synchrotron map of the Crab Nebula at
Nebula at 100 MeV. Given the small light crossing time of
100MeV.Theimageshowslog10Iν inarbitraryunits.
the knot compared to the synchrotron cooling time even at
100MeV, its synchrotron electrons must still have the en-
3 WHY THE INNER KNOT CAN BE A ergy spectrum which is very close to the one produced by
STRONG SOURCE OF GAMMA-RAY the shock acceleration mechanism. At ν = 3.76×1014Hz
EMISSION the de-reddened flux from the inner knot is F (cid:39) 1.6 ×
ν
10−27ergs−1cm−2Hz−1 (Tziamtzisetal.2009).At100MeV
Suppose that the termination shock is the main accelera-
the observed flux is F (cid:39) 1.7 × 10−32ergs−1cm−2Hz−1
tion site of gamma-ray emitting electrons. The synchrotron ν
(Abdo et al. 2010). The corresponding two point spectral
cooling timescale is
index is α (cid:39) 0.64. This is indeed the value of the injection
(cid:18) B (cid:19)−3/2(cid:18) E (cid:19)−1/2 spectral index inferred by Kennel & Coroniti (1984), and
tcool (cid:39)3.7D1/2 103G 100phM,oebV days, measured in optics by Tziamtzis et al. (2009)!
Thereexistsanupperlimitontheenergyofsynchrotron
whereEph,ob =DEphistheobservedenergyofphotonsemit- photons,whichisindependentonthedetailsoftheaccelera-
ted at the energy Eph in the fluid frame of the downstream tionmechanism(Lyutikov2010).Iftheacceleratingelectric
plasma, fieldE isafractionη≤1ofthemagneticfieldthentherate
1 of energy gain can be estimated as
D=
γ(1−βcosφ) dE
e =eEc=ηeBc. (12)
is the Doppler factor, B is the magnetic field strength as dt
measured in the fluid frame, and φ is the angle between The corresponding acceleration time scale τ =
acc
the line of sight and the velocity vector of the plasma E /(dE /dt) = (ηω )−1, where ω = ceB/E is the rela-
e e B B e
bulkmotion.FittingoftheCrabNebulaspectrumwiththe tivistic Larmor frequency. The energy loss rate due to syn-
synchro-ComptonmodelyieldsthetypicalB ∼100−200µG chrotron emission
(Horns & Aharonian 2004; Abdo et al. 2010). However, in- dE
e =−c B2E2, (13)
dividual bright features can have stronger magnetic field. dt 2 e
In particular, Hester et al. (1995) give the equipartition
where c = 4e4/9m4c7 and we also assumed effective pitch
B (cid:39)2.5×10−3G for the inner knot. Moreover, the numeri- 2
anglescattering,growswiththeelectronenergy.Thebalance
calsimulationsshowthatnearthearchshockthemagnetic
ofenergygainsandlossesyieldsthemaximumenergy,which
field can be significantly higher, up to ten times, than the
can be reached by the accelerated electrons
volumeaveraged4.Thus,evenfortheDopplerfactorashigh
as D=10, the cooling length scale of electrons emitting at Emax =(ηec/c B)1/2. (14)
e 2
E ∼ 100MeV is likely to be small compared to the ter-
ph
mination shock radius, which is about (cid:39)10(cid:48)(cid:48) or (cid:39)120light The characteristic energy of the synchrotron photons emit-
ted by the electron of energy Emax in the magnetic field of
days (Hester et al. 2002) in linear scale. Thus, the gamma- e
strength B,
rayemittingregionmustbelocatedveryclosetotheshock.
27 mhc3
Emax =c B(Emax)2 = η =236ηMeV, (15)
ph 1 e 16π e2
4 Thisisincontrastwiththeone-dimensionalMHDmodelwhere
themagnetic fieldincreases withthedistance fromthetermina- where c1 = 3eh/4πm3c5, does not depend on the magnetic
tionshock(Kennel&Coroniti1984a) fieldstrength.Thisistheutmostupperlimit,whichmaybe
(cid:13)c 0000RAS,MNRAS000,000–000
8 Komissarov & Lyutikov
impossibletoreachinpractice.Forexample,deJageretal. variability of the gamma-ray emission from the Crab Neb-
(1996) give an almost ten times smaller value for Emax for ula. Figure 6 shows the intra-year variability of the inner
ph
the shock acceleration. knot at E = 100 MeV based on the results of these sim-
ph
Infact,thesynchrotroncomponentoftheCrabNebula ulations. The epoch corresponds to the present age of the
spectrum becomes very steep above 10MeV, and can be Crab Nebula. Within this particular period the total flux
fitted with the function changeswerelimitedby(cid:39)27%.(Unfortunately,mostofthe
simulation data is now lost and we cannot comment on the
F ∝ν−αexp(−hν/Ec ), (16)
ν ph statisticalsignificanceofthisresult.)Thedataalsoindicate
wherethecut-offenergyEc (cid:39)100MeV(Abdoetal.2010). noticeablechangesintheknotappearanceandsmallchanges
ph
ThisissurprisinglyclosetoourvalueofEmax.However,the in its location.
ph
above limit applies only in the frame of emitting plasma. If The mechanism of this shock variability is not very
the plasma is moving with relativistic bulk speed relative clear. It seems to be related to the unsteady axial pinch,
to the observer then it has to be multiplied by the Doppler which is behind the origin of the Crab jet in this model.
factorinordertoobtainedthecorrespondingobservedpho- As we have mentioned already, the magnetic field in the
ton energy. For γ (cid:29) 1 the maximum value of the Doppler backflowatthebaseofthejetishighlyinhomogeneousand
factor is D (cid:39)2γ, and thus even for the rather moderate this results in strong spatial and temporal fluctuations of
max
postshockvalueofγ (cid:39)5thesynchrotroncutoffenergycan the magnetic hoop stress, and hence the axial pressure. As
2
beincreasedbyafactoroften.Thishasbeenusedtoargue the result, the arch shock dives towards the pulsar at times
that the observed synchrotron emission of the Crab Nebula when the pressure is high, and moves further out when it
withE (cid:38)100MeVoriginatesinrelativisticflow(Lyutikov is low. Another factor is the presence of strong vortices in
ph
2010). Here, this argument can be refined to support the the backflow, which can appear all the way along the arch-
inner knot as the source of this gamma-ray emission be- shock.Intheireyesthepressureislowerandontheoutside
cause we are now almost certain that its emission is indeed it is higher. Moreover, there are significant fluctuations of
Doppler-boosted. the ram pressure inside the simulated nebula as well.
Itseemsreasonabletoexpectthegyroradiusofelectrons Inthesimulations,thewispsareassociatedwithregions
acceleratedattheterminationshocktobebelowtheradius ofhighmagneticfieldintheunsteadyoutflowfromtheter-
oftheterminationshock.AneffectiveDopplerboostingmay mination shock. New wisps are produced approximately on
alsorequirethegyroradiustobebelowthetransversesizeof the light crossing time of the termination shock, which was
thefastflowdownstreamofthearch-shock.Onlyinthiscase around 10 months in the simulations and which is around
onecanfirmlyconcludethattheelectronsareadvectedwith 3-4 months for the Crab Nebula. Strong variations of the
the flow. For the electrons emitting synchrotron photons of shockstructureoccurredonthesimilartimescale.Thus,the
theenergyE inthecomovingframe,thegyroradiusradius variability of gamma-ray emission on the time-scale around
ph
is several months may well have this origin. The variations of
gamma-rayfluxmaybeattributedtochangesoftheproper
(cid:18) E (cid:19)1/2
r = ph . (17) emissivity of the inner knot, associated with changes of the
L c1e2B3 magnetic field strength and the number density of emitting
particles,butalsotochangesinthedirectionoftheDoppler
Ournumericalsimulationsshowthatthemagneticfieldmea-
beam (see Figure 7).
sured just downstream of the arch shock is significantly
higherthanthevolumeaveragedone,uptoaboutoneorder Assuming the power law spectral distribution for the
of magnitude. This indicates that B =10−3G may well be emissivity in the comoving frame, jν ∝ ν−α, the observed
typicalforthisregion.Thenthetypicalgyroradiusradiusof emissivity is
the electrons is
r (cid:39) 1.6 (cid:18) Eph,ob (cid:19)1/2(cid:18) B (cid:19)−3/2 light days, (18) jν,ob =D2+αjν
L D1/2 100MeV 10−3G
whichissignificantlylessthantheterminationshockradius
(Lind&Blandford1985).Whentheanglebetweentheflow
and even below the size l (cid:39)2 light days of the inner knot,
(cid:107) direction and the line of sight decreases from φ = 1/γ to
which is an observational indicator of the thickness of the
φ=0 the Doppler factor increases from γ to 2γ. Thus, the
fast post-shock flow (see the discussion leading to Eq.10).
difference in the inclination angle of the arch shock at the
Given this result, we conclude that the observed emission
location of inner knot ∆φ = 1/γ can bring about the dif-
up to E =1GeV can be the Doppler-boosted emission
ph,ob ference in the boosting factor up to 22+α. Above 100MeV,
produced by the electrons accelerated at the termination
where the observed synchrotron spectrum can be approx-
shock.
imated by a power law with α ∼ 3 (Abdo et al. 2010),
thiscorrespondstoa30-foldfluxvariation.Atlowerphoton
energies, where the knot spectral index is expected to be
α(cid:39)0.6,thecorrespondingfluxvariationisfivetimeslower.
4 NATURE OF THE GAMMA-RAY
Moreover,thecoolingtimeoftheelectrons,producingsuch
VARIABILITY
photons, becomes significantly larger – it is already several
Thestrongvariabilityoftheterminationshockdiscoveredin years for the electrons emitting at 1MeV. After traveling
thenumericalsimulationsbyCamusetal.(2009)andassoci- for such a long time the emitting plasma enters the remote
atedwiththewispproductioncouldbebehindtheobserved parts of the nebula, where it inevitably decelerates and its
(cid:13)c 0000RAS,MNRAS000,000–000
Gamma-ray emission from the Crab Nebula 9
Figure6.Variabilityofthegamma-rayemissionfromtheinnerknotincomputersimulations(Camusetal.2009).Thefivecolourplots
showtheimagesoftheinnerknotatEph=100MeV(Iν inlinearscale).Theyareseparatedby0.15year,thetimeincreasingfromleft
torightandfromtoptobottom.TheplotintherightbottomcornershowsthecorrespondingtotalfluxvariationatE =100MeV.
ph
emission is no longer subject to strong Doppler beaming5. time-scale of the order of few days is possible, at least in
The contribution of this unbeamed emission may explain principle. The fact that the flare spectrum extends beyond
theobservationsbydeJageretal.(1996),whonoticedthat thehighestcharacteristicenergyallowedforthesynchrotron
during these observations the flux in the 1-30 MeV energy emission can still be explained by the Doppler effect. Fermi
range was increasing, whereas in the 30-150 MeV range it reportedthespectralindexoftheflaringcomponentα∼1.5.
was decreasing. Most likely, the total flux from the nebula Suchasteepspectrumisexpectedbecauseoftheproximity
at 1-30 MeV is dominated by few recently produced wisps. of the exponential cut-off. The fact that flares are not seen
At100keVtheobservedtotalfluxfromtheCrabNeb- at both lower and higher energies can be explained in the
ula is (cid:39) 4×10−28ergs−1cm−2Hz−1 (Horns & Aharonian same fashion as for the long timescale variability.
2004), whereas the expected flux from the inner knot is
only (cid:39) 1.3×10−30ergs−1cm−2Hz−1. The last estimate is The most difficult task is to explain not only the short
obtainedfromthepowerlawF ∝ν−0.64,normalisedusing duration of the flares but also the fact that they are quite
ν
the observed optical flux of the inner knot (Tziamtzis et al. rare.Ifindeedtheyoriginatedfromtheinnerknotthenthis
2009). Thus, even a 10-fold increase of the X-ray emission had to be associated with some rather peculiar events. For
from the knot would produce a variation below 5% in the example,theycouldbeproducedwhensomeexplosiveevent
total flux. For the similar reason, the knot variability could atthe baseofCrab’sjet drives ashock,whichthencollides
hardlybeseenin10GeV-10TeVrange,whichisdominated withtheterminationshockneartheknotlocation,withboth
bytheInverseComptonemissionofoldelectronsoccupying shocksbeingalmostparalleltoeachother.Theshocksinter-
the whole volume of the nebula (de Jager et al. 1996). section point could move towards the observer with speed
Finally, few words have to be said on the mysterious very close to the speed of light, potentially resulting in a
gamma-rayflaresfromtheCrabNebula.Theobservedlinear shortburstofemissionassociatedwiththispoint.Themag-
sizes of the inner knot are l = ψ d ∼ 6 light days and neticreconnectioncouldbebehindsuchexplosionsbutthis
⊥ ⊥ n
l = ψ d ∼ 2 light days. This implies that the variability wouldprobablyrequireasignificantlyhighermagnetization
(cid:107) (cid:107) n
compared to what is assumed in the current MHD models.
So far, the emission from simulated PWN was computed
5 TheobserveddecelerationoftheCrabwispsisaclearconfir- simplybyintegratingtheinstantaneousemissivityalongthe
mationofsuchevolution(Hesteretal.2002). lineofsight.Therelativisticretardationeffectwasnottaken
(cid:13)c 0000RAS,MNRAS000,000–000
10 Komissarov & Lyutikov
(v) ThesmallsizeoftheinnerknotoftheCrabNebula,2-
6lightdays,showthattherecentlydiscoveredshortgamma-
ray flares can also originate from the knot. However, the
exactmechanismbehindsuchshortandrareeventsremains
unclear.
Themostcriticalpredictionofourmodel,whichallows
arelativelysimpletestwithcurrentlyavailabletelescopes,is
that the un-pulsed synchrotron gamma-ray emission of the
Crab Nebula in the Fermi and AGILE windows originates
fromwithinonearcsecondofthepulsaritself.Althoughthe
angularresolutionofgamma-raytelescopesisnotevenclose
to one arcsecond, the test can be based on comparing the
gamma-ray light curve with the one obtained in optics for
the inner knot. Within one arcsecond the inner knot is the
Figure7. Variabilityoftheterminationshockasareasonbehind
the variability of high energy emission from the Crab Nebula. dominant feature, apart from the pulsar itself. A potential
The solidand dotted linesshow the termination shock withtwo problemofthistestisthecloseproximityoftheknottothe
extreme locations of its polar cusp. The dash-dotted lines show pulsar, which makes image based separation of their fluxes
thecorrespondingdirectionsoftheDopplerbeam. rather tricky, even for HST and the ground-based instru-
mentswithadaptiveoptics.However,thisdoesnotseemto
be needed as the pulsed emission from the Crab pulsar is
into account. This practice has to be abandoned in future
very stable and its un-pulsed emission is not expected to
studies as it filters out the short time-scale variability asso-
bevariabletoo.Hence,oneonlyneedstomeasurethetotal
ciated with the relativistic motion along the line of sight. flux from within ∼ 1(cid:48)(cid:48) of the pulsar and subtract from it
the phase averaged flux of the pulsed emission. The optical
variabilityisexpectedtobestrong,asthetypicalfluxfrom
5 CONCLUSIONS thetheinnerknotisalreadyabout6−10%ofthefluxfrom
the pulsar flux. The light curves in optics and gamma rays
(i) Downstream of oblique termination shocks of pulsar
should correlate.
winds,theLorentzfactorofbulkmotioncanberatherhigh
even for low-sigma winds, up to γ =5 for reasonably small
inclination angles. This is shown using the analytical solu-
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