Mon.Not.R.Astron.Soc.000,1–9(2008) Printed27January2009 (MNLATEXstylefilev2.2) Some properties of synchrotron radio and inverse-Compton gamma-ray images of supernova remnants 9 O. Petruk1,2,4, V. Beshley2, F. Bocchino3,4, S. Orlando3,4 0 0 1Institute for Applied Problems inMechanics and Mathematics, Naukova St. 3-b, 79060 Lviv, Ukraine 2 2Astronomical Observatory,National University,Kyrylaand Methodia St. 8, 79008 Lviv, Ukraine 3INAF - Osservatorio Astronomico di Palermo “G.S.Vaiana”, Piazza del Parlamento 1, 90134 Palermo, Italy n 4Consorzio COMETA, viaSanta Sofia 64, 95123 Catania, Italy a J 7 2 Accepted ....Received...;inoriginalform... ] E ABSTRACT H The synchrotron radio maps of supernova remnants (SNRs) in uniform interstellar . medium and interstellarmagnetic field (ISMF) areanalysed,allowingdifferent ‘sensi- h tivity’ of injection efficiency to the shock obliquity. The very-high energy γ-ray maps p duetoinverseComptonprocessarealsosynthesized.Thepropertiesofimagesinthese - o different wavelength bands are compared, with particulr emphasis on the location of r the bright limbs in bilateral SNRs. Recent H.E.S.S. observations of SN 1006 show t s that the radio and IC γ-ray limbs coincide, and we found that this may happen if: i) a injection is isotropic but the variation of the maximum energy of electrons is rather [ quicktocompensatefordifferencesinmagneticfield;ii)obliquitydependenceofinjec- 1 tion (either quasi-parallel or quasi-perpendicular) and the electron maximum energy v isstrongenoughtodominatemagneticfieldvariation.Inthelattercase,the obliquity 8 dependences of the injection and the maximum energy should not be opposite. We 5 argue that the position of the limbs alone and even their coincidence in radio, X-rays 2 andγ-rays,asitis discoveredbyH.E.S.S. inSN1006,cannotbe conclusiveaboutthe 4 dependenceoftheelectroninjectionefficiency,thecompression/amplificationofISMF . 1 and the electron maximum energy on the obliquity angle. 0 Key words: ISM:supernovaremnants–shockwaves–ISM:cosmicrays–radiation 9 0 mechanisms: non-thermal – accelerationof particles : v i X r 1 INTRODUCTION realy correlate if the VHE γ-radiation originates from elec- a trons? What should be the limitations for theory once ob- The observation of the supernova remnants (SNRs) in served patterns are really quite similar, especially in sym- very-high energy (VHE) γ-rays by H.E.S.S. and MAGIC metricalbilateralSNRs,likeinSN1006(H.E.S.S.Sourceof experiments is an important step toward understanding theMonth, August 2008). the nature of the Galactic cosmic rays and kinematics Another key issue for particle kinetics is the 3-D mor- of charged particles and magnetic field in vicinity of the phology of bilateral SNRs in general and SN 1006 partic- strong nonrelativistic shocks. However, the spectral analy- ularly. Is it polar-cap or barrel-like? The answer of this sis of multi-wavelenght data allows both for leptonic and questionisstronglyrelatedtothemodelofinjection(quasi- hadronic origin of VHE γ-ray emission (e.g. RX J1713.7- parallel in the former and isotropic or quasi-perpendicular 3946: Berezhko & V¨olk (2006),Aharonian et al. (2007)).In inthelattercase),givingthereforeanimportanthintforac- this context, the broad-band fitting of the spectrum of the celeration theory. The properties of brightness distribution nonthermal emission from SNRs is one of the hot topics in may bethemost conclusive issue in this task (e.g. criterion present studies of SNRs. At the same time, another very ofRothenfluget al.(2004),azimuthalprofilescomparisonin important source of scientific information, the distribution Petruk et al. (2009)). ofthesurface brightness,isnot in great demand.Thereare just some discussions emphasyzing that observed correla- An experimental investigation of SNR images have to tions of brightness in radio, X-raysand γ-rays may be con- be complemented with theoretical modelling of SNR maps sidered to favor electrons to be responsible for VHE emis- in different energy domains. Radio and X-ray synchrotron sioninRXJ1713.7-3946,VelaJr.andsomeotherSNRs(e.g. images in the uniform interstellar medium (ISM) and the Aharonian et al.(2006),Plaga(2008)).However,shouldthe uniform interstellar magnetic field (ISMF) are modeled by patterns of surface brightness in radio, X-rays and γ-rays Reynolds (1998). The role of gradient of ISM density and 2 Petruk O. et al. ISMF strength on radio morphology of SNRs are stud- line of ied by Orlando et al. (2007). These papers bases on the sight classical MHD and assumes unmodified shocks. Studies on Z nonthermal images of SNRs with non-linear acceleration theory undergo development (Leeet al. 2008). The pro- files of the synchrotron brightness in such SNRs are sub- jectofinvestigationinEllison & Cassam-Chena¨ı(2005)and Cassam-Chena¨ı et al. (2005). In the present paper, we present for the first time the Θ inverse-Compton γ-ray images of SNRs in uniform ISM Bo o n and ISMF produced on the basis of the model of Reynolds Y (1998).Inadditiontothismodel,weallowfordifferent‘sen- sitivity’ of injection efficiency to the shock obliquity like it projection isapparentinnumericalresultsofEllison et al.(1995).The plane synthesized maps are compared with the radio ones. Some consequenciesfororigin ofVHEemission ofSNRsandelec- X tron injection scenario are drawn. Figure 1. Geometry of the task. The obliquity angle Θo, the 2 MODEL aspectangleφo andtheazimuthangleϕareshown.ISMFBo is chosentobeparalleltotheX0Zplane. We consider SNR in uniform ISM and uniform ISMF. At the shock, the energy spectrum of electrons is taken as N(E) = KE−sexp(−E/Emax), Emax is the maximum en- vary with obliquity angle with different ‘sensitivity’ which ergyofelectrons,s=2isusedthroughoutofthispaper.We is given by theparameter Θ : K followReynolds(1998)incalculation oftheevolutionofthe magnetic field and relativistic electrons (see details also in ς(Θo)=ςkexp − Θo/ΘK 2 (1) Petruk (2006), Petruk & Beshley (2008)). The compression factorforISMFσB increasesfromunityatparallelshockto where ςk is the(cid:16)effi(cid:0)ciency(cid:1)fo(cid:17)r the parallel shock. This ex- 4atperpendicularone.Thefiducialenergyatparallelshock, pression restores approximately the results of Ellison et al. which is responsible for the ‘sensitivity’ of relativistic elec- (1995) with ΘK = π/9÷π/4. The classic quasi-parallel in- trons to the radiative losses (Reynolds 1998) and which is jection may be approximated with ΘK = π/6. Isotropical usedinICimagesissettoEmax.Thesynchrotronlossesare injection assumes ΘK = ∞, but the values ΘK ≥ 2π pro- considered as the dominant channel for the radiative losses ducesalmostthesameimagesasΘK =∞becausetherange of relativistic electrons. We assume that K is constant in for obliquity angle is 0≤Θo ≤π/2. time; eventualevolution of K affectstheradial thicknessof Weconsider also quasi-perpendicularinjection: rims and does not modify the main features of the surface brightness pattern (Reynolds1998). ς(Θo)=ςkexp − (Θo−π/2)/ΘK 2 . (2) ElectronsemittingICphotonshaveenergiesE ∼Emax. Inthemos(cid:16)tca(cid:0)sespresentedher(cid:1)e,(cid:17)E isassumedtobe LikeK,E isassumedtobeconstantintime.Itspossible max max constant over SNR surface; this choice allows us to clearly variation in time does not change the pattern of IC bright- see the role of other parameters. Reynolds (1998) consid- nessandleadstoeffectssimilartothoseoriginatingfromthe eredloss-limited,time-limitedandescape-limitedmodelsfor timedependenceofK.Namely,featuresinICimageshaveto E . In all cases, except of the loss-limited one with the beradiallythickerifE decreaseswithtime(i.e.increases max max level of turbulence comparable with the Bohm limit, E with the shock velocity): since E was larger at previous max max shouldgrowwithincreaseofΘ (Reynolds1998).Wemodel times, there are more electrons in the SNR interior able to o the role of possible increase of E with obliquity with a emitICphotonsatthepresenttime.IfE increaseswith max max simple parameterization time(i.e.decreaseswiththeshockvelocity)thenmaximain brightness are expected to beradially thinner. E (Θ )=E exp − (Θ −π/2)/Θ 2 (3) Reynolds (1998) considered three models for injection: max o maxk o E quasi-parallel, isotropic and quasi-perpendicular. The pat- where Θ is a paramet(cid:16)er,(cid:0)E the max(cid:1)im(cid:17)um energy at tern of the radio surface brightness distribution in the E maxk parallel shock. This formula, with different values of Θ , is E case of the quasi-perpendicular injection is quite similar to able to restore approximately different cases considered by the isotropic injection case, though with different contrasts Reynolds(1998). (Fulbright & Reynolds 1990; Orlando et al. 2007). The nu- The surface brightness is calculated integrating emis- merical calculations of Ellison et al. (1995) show that the sivitiesalongthelineofsightwithin SNR.Thesynchrotron obliquitydependenceoftheinjection efficiencyς (afraction emissivity at some radio frequency is q ∝ KB(s+1)/2, sych ofacceleretedelectrons)maybeeitherflatterorsteeperthan B is the strength of magnetic field. The γ-ray emissivity of in the classic quasi-parallel case (ς ∝ cos2Θ where Θ is o o electrons dueto inverse Compton process is calculated as the obliquity angle, the angle between the ISMF and the ∞ normal to the shock, Fig. 1). In order to be more general q (ε)= N(E)p (E,ε)dE (4) than Reynolds (1998), we allow the injection efficiency to ic ic Z0 Synchrotron and IC images of SNRs 3 1.0 a _Q=K /p12 b Q_=K /p6 c _Q=K /p4 0.5 R0.0 r/ -0.5 B -1.0 1.0 Q_=K /p2 Q_=K p _Q=K 2p d e f 0.5 R0.0 r/ -0.5 -1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 r/R r/R r/R Figure 2. Radio images of SNR for an aspect angle φo = 90o and different ΘK: π/12 (a), π/6 (b), π/4 (c), π/2 (d), π (e), 2π (f). Ambientmagneticfieldisorientedalongthehorizontalaxis.Hereafter,theincrementinbrightnessis∆S=0.1Smax. where ε is the photon energy. The spectral distribution p 3 RESULTS ic of radiation power of a ”single” electron in a black-body 3.1 Synchrotron radio images photon field with temperatureT is We stress that all figures in the present paper have been 2e4ǫ computed using complete MHD model. p (γ,ε)= cγ−2I (η ,η ) (5) ic π¯h3c2 ic c o Let us define an aspect angle φo as an angle between interstellar magnetic field and the line of sight (Fig. 1). It where γ is Lorenz factor of electron, ǫc =kT, is shown that the azimuthal variation of the radio surface brightnessS atagivenradiusofprojection̺,inSNRwhich ̺ ǫ ε ε2 isnotcentrallybrightened,ismostlydeterminedbythevari- η = c , η = , (6) c (mec2)2 o 4γmec2(γmec2−ε) ations of the magnetic field compression (and/or amplifica- tion)σ andtheelectroninjectionefficiencyς (Petruk et al. B m ,e,c,¯h,k havetheirtypicalmeaning.I (η ,η )may be 2009): e ic c o approximated as (Petruk 2008) S (ϕ)∝ς Θ (ϕ,φ ) σ Θ (ϕ,φ ) (s+1)/2 (9) ̺ o,eff o B o,eff o π2 5 η 1/2 whereϕis(cid:0)theazimutha(cid:1)lang(cid:0)le.Theeffect(cid:1)iveobliquityangle Iic(ηc,ηo) ≈ 6 ηc exp −4 ηo Θo,eff is related to ϕ and φo as " (cid:18)0.c7(cid:19) # (7) cosΘo,eff(ϕ,φo)=cosϕsinφo, (10) 5 η 2η +2ηoexp −7 ηo exp −3ηo . here,theazimuthangleϕismeasuredfrom thedirectionof " (cid:18) c(cid:19) #! (cid:20) c(cid:21) ISMFin theplane of the sky (Fig. 1). Fig. 2 shows how Θ affects a radio image of SNR. K This approximation is quite accurate, it represents Iic in Complete MHD simulations are in agreement with the ap- any regime, from Thomson to extreme Klein-Nishina. The proximate formula (9). First, we note that smooth increase maximum of spectral distribution pic(ε) for electrons with of ΘK results in transition from the 3-D polar-cap model of energy E is at (Petruk 2008) SNR to the 3-D barrel-like one.ThisisalsovisibleonFig.3 where ISMF is directed toward observer. Namely, increase ε (E)≈ EΓc , Γ = 4ǫcE . (8) of ΘK change the visual morphology from centrally-bright max 1+Γc c (mec2)2 to shell-like. There are three names for a class of SNRs which AllICimagesinthepresentpaper(exceptofthatonFig.10) have two opposite limbs in the literature: ‘barrel-shaped’ are calculated for the initial photon field with T = 2.75 K (Kesteven & Caswell1987),‘bipolar’(Fulbright & Reynolds and for the γ-ray photon energy ε=0.1ε (E ) that is 1990)and‘bilateral’(Gaensler1998).Theywereintroduced max max for example ε=0.3TeV for E =30TeV. on the base of 2-D visual morphology. It is interesting that max 4 Petruk O. et al. 4 ISMFmayhavedifferentorientationversusobserverinvar- 6 iousSNRs.Ifquasi-parallel injection isnotarareexception then the polar-cap SNRs should be projected in a different 3 way and we may expect to observe not only ‘bipolar’ SNRs 5 (Fig.4c,d)butalso SNRswithoneortworadioeyeswithin u. a. 2 thermalX-rayrim(Fig.4a,b).Fulbright & Reynolds(1990) S, developed statistically this thought and showed that the 4 quasi-parallel injection model would be unlikely, but again, 1 we would need a complete study to verify this statement2. Statistical arguments of Fulbright & Reynolds (1990) may 3 2 be affected by the fact that centrally-bright radio SNRs 1 0 (lines1-2onFig.3)areexpectedtobefainterthanbilateral 0 0.2 0.4 0.6 0.8 1 x/R orcircular SNRswith thesamecharacteristics (lines4-6on Figure 3. Profiles of the radio surface brightness for an aspect Fig. 3):it could bethat most of thecentrally-peaked SNRs angleφo=0o (theradialprofileofbrightnessisthesameforany may not beobservable. azimuth).ΘKisπ/12(line1),π/6(line2),π/4(line3),π/2(line 4),π (line5),2π (line6). 3.2 IC γ-ray images the first two names reflects de facto the two different con- Let us consider first the case when the maximum energy ceptions of SNRsin 3-D. of electrons is constant over SNR surface; this allows us to Fig. 2 also shows that an assumption about orienta- clearly see the role of the injection efficiency and magentic tionofISMFleadstolimitationofpossibleinjectionmodel. field variations. AmbientmagneticfieldinallimagesonFig.2isalonghori- Synthesized IC γ-ray images of SNRs are presented on zontalaxis. Thus,if one consider the polar-cap scenario for Fig. 5, for different aspect angles. These images assume al- bilateral SNR (ISMF is along axis which crosses two limbs) mostisotropicinjection(Θ =2π)andshouldbecompared K then one should consider the injection model which strongly with radio maps on the lower panel of Fig. 4. The compo- depends on the obliquity (Θ ≤ π/6, Fig. 2a,b). Instead, if K nent of ISMF which is perpendicular to the line of sight is the barrel is the preferable model (ISMF is parallel to the alonghorizontalaxisonallimages.Animportantdifference symmetry axis between two limbs) then the injection effi- is prominent from these two figures. Namely, the two lobes ciency should be almost independent of obliquity (Θ ≥ π, o developwithincreasingofφ inbothradioandγ-rays.How- o Fig. 2e,f), or prefer quasiperpendicular shocks. ever, their location in respect to ISMF is opposite. The line Gaensler(1998)measuredtheangleψbetweenthesym- conecting two maxima in radio is perpendicular to ISMF metry axis in 17 ‘clearly’ bilateral SNRs and the Galactic while it is parallel to ISMF on IC images (cf. Fig 5d and plane.AxesaremoreorlessalignedwiththeGalacticplane Fig 4h). in 12SNRs(ψ<30o),2SNRshaveψ≈45o and 3SNRsis The reason of this effect is the following. For assumed almost perpendicular(ψ>60o).Ifweassume that ISMFis isotropic injection, the azimuthal variation of the radio parallel totheplaneof Galaxy thenmost of bilateral SNRs brightness is determined only by the dependence σ on B shouldbe3-Dbarrelsprefferingthusisotropic(orquasiper- obliquity (the azimuth angle equals to the obliquity angle pendicular) injection. forφ =π/2).ElectronsemittingVHEγ-rayshaveenergies o An interesting feature appears on images for ΘK = E ∼ E and experience substantial radiative losses (this max π/4÷π/2(Fig.2c,d).Namely,SNRhas‘quadrilateral’mor- effect is negligible for radio emitting electrons). Magnetic phology.Withincreasingofobliquity,theinjectionefficiency field does not appear directly in the formulae for IC emis- decreaseswhilethecompressionfactorofISMFicreases.The sion,butitaffectsthedownstreamdistributionofrelativistic variationofinjectionς(Θ )dominatesσ (Θ )forΘ ≤π/6. o B o K electronsemittingICγ-rays.Thelargerpost-shockmagnetic If Θ ≥π (injection is almost isotropic) then σ (Θ ) plays K B o fieldthelargerradiativelosses.Thedownstreamdistribution the main role in azimuthal variation of the radio surface ofIC-emittingelectronsisthereforesteeperwheremagnetic brightness. In the intermediate range of Θ , the signifi- K field is stronger. This leads to lower IC brightness in SNR cance of the two variations are comparable leading there- regions with larger magnetic field (while radio brightness foretoazimuthalmigrationofthebrightnessmaximainthe increases therebecause of proportionality to B3/2). modelled images. There is no ‘quadrilateral’ SNR reported In VHE γ-ray image of SN 1006 recently reported by in the literature. If there is no such SNR at all, the range H.E.S.S. collaboration (H.E.S.S. Source of the Month, Au- Θ ≃π/4÷π/2maybeexcluded.However,westressthata K gust 2008), thetwo maxima coincide in location with limbs complete statistical studyofthemorphology of radioSNRs in radio and nonthermal X-rays. This fact, in view of the would be needed to definitly asses the lack of quadrilateral ‘limb-inverse’ property, could be considered as argument SNRs1. against the leptonic origin of γ-ray emission in SN 1006 The visual morphology of SNRis different for different (if injection is isotropic). However, these IC images are ob- aspect angles. Fig. 4 shows SNR images for quasi-parallel tainedunderassumptionthatE doesnotvaryoverSNR max injection with Θ = π/12 (upper panel) and for isotropic K injection (Θ = 2π, lower panel). We may expect that K 2 G311.5-0.3andG337.2-0.7couldbeexamplesofSNRswithtwo 1 G338.3-0.0couldbeanexampleofquadrilateralSNR radio’eyes’ Synchrotron and IC images of SNRs 5 1.0 a _Q=K/p12 b _Q=Kp/12 c _Q=K/p12 d _Q=K/p12 0.5 R0.0 r/ -0.5 of=0 f=o30 =of60 of=90 -1.0 1.0 e Q_=K2p f Q_=K2p g Q_=K2p h _Q=K2p 0.5 R0.0 r/ -0.5 of=0 f=o30 =of60 of=90 -1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 r/R r/R r/R r/R Figure4.RadioimagesofSNRfordifferentaspectanglesφo:0o(a,e),30o(b,f),60o(c,g),90o(d,h).ΘK=π/12(upperpanel),ΘK=2π (lowerpanel).Componentoftheambientmagneticfieldwhichisperpendiculartothelineofsight,isorientedalongthehorizontalaxis. 1.0 a b c d 0.5 R0.0 r/ -0.5 -1.0 =of0 of=30 f=o60 =of90 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 r/R r/R r/R r/R Figure 5. IC γ-rayimages of SNR.Isotropic injection, Emax is constant over SNR surface. Aspect angles φo: 0o (a), 30o (b), 60o (c), 90o (d).Componentoftheambientmagneticfieldwhichisperpendiculartothelineofsight,isorientedalongthehorizontal axis. surface. If E is high enough at regions with large mag- pendenceς(Θ )isstrongenough(Fig.8b,d).Intherangeof max o netic field (at perpendicular shock), then the ‘limb-inverse’ intermediateΘ ,thequadrilateralmorphologyappearsalso K effectmaybelessprominentorevenmightnotbeimportant inmodelsofICγ-rays(Fig.8c),asanintermediatemorphol- (see below). ogy between those on Fig. 5d and Fig. 8d.(The contrast of Incaseifinjectionstronglyprefersparallelshocks(limbs maxima in the image of quadrilateral SNR is so small that in SN 1006 are polar caps), the dependence ς(Θ ) might thisfeature may probably not be observable.) o dominateσB(Θo).ThemaximaofbrightnessinradioandIC Notethatthequasi-perpendicularinjectionmodelleads γ-raysarethereforelocatedatthesameregionsofSNRpro- to radio images similar to those in the isotropic injection jection(Fig.6,tobecomparedwithFig.4a,d),inagreement case,cf.Fig.8a,bandFig.2f(seealsoOrlandoet al.(2007)), with theChandra and H.E.S.S. observations of SN 1006. because magnetic field and injection efficiency increase at The role of intermediate values Θ for injection which perpendicular shocks both leading to larger synchrotron K prefers parallel shock, Eq. (1), on profiles of IC brightness emission. In contrast, there is a lack of IC radiating elec- is shown on Fig. 7. Increase of the sencitivity of injection trons around perpendicular shocks which may or may not totheobliquity leads toradially thinnerand more contrast (depending on ΘK in (2)) compensate it. Thus IC images features. involving the quasi-perpendicular injection may radically If injection prefers perpendicular shock, Eq. (2), its in- differ from those with isotropic injection, cf. Fig. 8d and Fig. 5d. crease in the regions of larger magnetic field may compen- sate the lack of γ-ray emitting electrons. In that case, the Theobliquityvariationoftheelectronmaximumenergy position of limbs coincide in radio and IC γ-rays if the de- is an additional factor affecting the IC γ-ray brightness in 6 Petruk O. et al. 1.0 a Q_=K/p4 b _Q=K/p4 fo=0 2 fo=p/2 0.5 r/R0.0 a.u. S, 1 -0.5 =of0 =of90 -1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 r/R r/R 0 0 0.2 0.4 0.6 0.8 1 Figure 6. IC γ-ray images of SNR. Quasi-parallel injection (1) x/R withΘK=π/4,Emax(Θo)=const.Aspectanglesφo:0o(a),90o (b).Inthelatter,ISMFisalongthehorizontalaxis. Figure 7. Profiles of the IC surface brightness along X-axis for the aspect angle φo = 0o (the radial profile of brightness is the sameforanyazimuth;tobecomparedwithFig.3)andφo=90o (ISMF is along the horizontal axis). Dependence of injection is SNRs. Actually, Rothenfluget al. (2004) have shown that the cut-off frequency increases at radio limbs of SN 1006 given by(1) withΘK (from below): π/12, π/6, π/4, π/2, π,2π, ∞.Emax isconstant overSNRsurface. thatmay (partially) beduetolarger E there.Therefore max E is expected to be largest in this SNR at the perpen- max dicular shock (at equatorial belt) if injection is isotropic or usedforSNRwhichiscentrally-brightinγ-raysandisvalid quasi-perpendicularorattheparallelshock(atpolarcaps)if for ̺/R larger than ≃0.9. injectionisquasi-parallel.Inthelattercase,thecalculations ofReynolds(1998) suggest thattheonlypossible modelfor Emax in SN 1006 should be loss-limited one in the Bohm 4 CONCLUSIONS limit. In thepresent paper, we analyse thesynchrotron radio and The role of E increasing with obliquity, Eq. (3), is max the inverse-Compton γ-ray images of Sedov SNRs synthe- shown on Fig. 9. The ‘limb-inverse’ property may not be sizedonthebaseoftheReynolds(1998)model.Ellison et al. important and the limbs may coincide in radio, X-rays and (1995) have shown that the dependence of efficiency of in- IC γ-rays even for the isotropic injection if the maximum jection ς on obliquity angle Θ may differ from commonly energy is large enough at perpendicular shocks to provide o used expression in quasi-parellel case. We therefore param- energetical electrons in despite of radiative losses (Fig. 9b, eterise the dependence ς(Θ ) as it is given by Eq. (1). It cf. with Fig. 4h and Fig. 5d). Note also that the limbs are o is shown that the variation of the parameter Θ provide thicker in this case, because of the more effective radiative K smooth transition from polar-cap (Θ ≤π/6) to barrel-like losses atperpendicularshock(duetolargerISMFcompres- K (Θ ≥ π) models of SNR and that assumed orientation of sion), comparing to limbs if they are at parallel shock. K ISMF should be related to a certain injection model. Some ThedependenceofE onΘ mayalsocausesplitting max o constraints on injection models which follow from morpho- androtationofIClimbsincaseofthequasi-parallelinjection logical considerationsarepointedout.Theazimuthalvaria- (Fig. 9d, cf. with Fig. 6b) or the quasi-perpendicular one. tion of radio brightness ismostly duetovariations of ς and There is a possibility for quadrilateral SNRs to appear in σ , in agreement with theapproximate formula (9). γ-raysduetotheinterplaybetweendependencesE (Θ ), B max o Theoretical γ-ray images of SNR due to the inverse ς(Θ ) and σ (Θ ) (Fig. 9a,d). o B o Compton effect are reported for the first time. We analyse All above IC images are calculated for the photon en- properties of these images and compare them with corre- ergy ε = 0.1ε (E ). The pattern of the γ-ray surface max max sponding radio maps of SNRs. The azimuthal variation of brightness remain almost the same with increasing of the IC brightness is mostly determined by variations of ς, σ photon energy, though regions of maximum brightness be- B andE , in agreement with theapproximate formula (11) come radially thinner and also contrasts change (Fig. 10). max derived in the Appendix. Thisisbecauseelectronswhich contributemost ofemission In case if E is constant over the SNR surface, we at larger photon energy experience higher radiative losses max found an opposite behaviour of azimuthal variation of sur- andthereforethedownstreamdistributionoftheseelectrons facebrightnessinradioandICγ-rays,incaseifinjection is are steeper. isotropicandtheaspectangleislargerthan≃60o.Namely, Totheend,themainpropertiesofICsurfacebrightness thelinecrossingthetwolimbsinradioareperpendicularto maysimplybederivedfromtheapproximateanalyticalfor- the ISMF while they are parallel in IC γ-rays. In particu- mula for the azimuthal variation of IC surface brightness lar, bright radio limbs correspond to dark IC areas, in dis- S (ϕ;φ ,ε) of the adiabatic SNR in uniform ISM and uni- ̺ o agreementwithX-rayandH.E.S.S.observationsofSN1006. form ISMF (Appendix): This happens because IC image is affected by large radia- S (ϕ)∝ς(Θ )exp −Em̺¯−1−5σB(Θo,eff)2Em/2Ef,k (11) tivelosses ofemittingelectronsbehindperpendicularshock ̺ o,eff E F(Θ ) whilethelargermagneticfieldincreasestheradiobrightness (cid:18) max,k o,eff (cid:19) there.VariationofE overSNRsurfacemay(tosomeex- max where E ∝ε1/2, Eq. (A8), ̺¯=̺/R≤1, ̺ is the distance tent)hidethiseffect.Themaximumenergyshould increase m fromthecenterofSNRprojection.Thisformulamaynotbe with obliquity in this case. Synchrotron and IC images of SNRs 7 1.0 a _Q=K /p2 b _Q=Kp/6 1.0 a _Q=K 2p b _Q=K2p 0.5 0.5 r/R0.0 r/R0.0 -0.5 -0.5 -1.0 radio radio -1.0 _Q=Ep/2 IC _QE=/p4 IC -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0-1.0 -0.5 0.0 0.5 1.0 r/R r/R r/R r/R 1.0 c _Q=K /p2 d _Q=Kp/6 1.0 c _Q=K /p4 d _Q=K /p4 0.5 0.5 r/R0.0 r/R0.0 -0.5 -0.5 -1.0 IC IC -1.0 _Q=E/p2 IC _Q=Ep/4 IC -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 r/R r/R r/R r/R Figure 8. Radio (a,b) and IC γ-ray images (c,d) of SNR for Figure 9. IC γ-ray images of SNR for φo = 90o and Emax φo=90o.Quasi-perpendicularinjection(2)withΘK=π/2(a,c) increasing with obliquity, Eq. (3) with ΘE = π/2 (a,c) and andΘK=π/6(b,d)(to becomparedwithFig.4dandFig.5d). ΘE = π/4 (b,d). Isotropic injection (a,b), to be compared with Emax isconstantoverSNRsurface. Fig.5d;quasi-parallelinjectionwithΘK=π/4(c,d),tobecom- paredwithFig.6b. In case of the polar-cap model of SNR (quasi-parallel injection),themaximainsurfacebrightnessareexpectedto 1.0 coincideinradioandICγ-rays(inagreementwithH.E.S.S. observationofSN1006),unlessincreaseofE withobliq- max 0.5 uitywillbeverystrong,whichisunlikelyincaseofSN1006 becausethecut-offfrequencyislargeratlimbswhichareat parallel shock in this injection model. r/R0.0 Limbs may also coincide in case of the quasi- perpendicular injection, if the lack of electrons (due to ra- -0.5 diative losses) in the regions of large magnetic field is com- pensated by the strong enough increase of ς and/or E max with Θo. -1.0 Isotropic compression/amplification of ISMF on the -1.0 -0.5 0.0 0.5 1.0 r/R shock(i.e. independentof theshockobliquity),likeitcould beunderhighlyeffectiveacceleration,mayalsoberesponsi- Figure 10. The same as Fig. 5d, for 10 times larger photon bleforthesameposition oflimbsinradioandinICγ-rays, energy,ε=εmax(Emax). for the quasi-parallel or quasi-perpendicular injection sce- narios. In this case the dependence of E (Θ ) have to max o follow variation ς(Θ ), namely, to be largest (smallest) at Absence of quadrilateral SNRs in IC γ-rays, if revealed ob- o parallelshockforquasi-parallel (quasi-perpendicular)injec- servationally, may results in limitations on ΘK and ΘE. tion, otherwise the morphology of SNR in IC γ-rays may The detailed characterictics of features on IC image differ from theradio one. (e.g. thickness of rim) depend on the photon energy. They We conclude that the location the γ-ray limbs ver- are radially thinnerat larger photon energies, as expected. sus radio and X-ray ones, recently discovered by H.E.S.S. in SN 1006, cannot be conclusive about the actual de- pendence of the electron injection efficiency, the compres- ACKNOWLEDGMENTS sion/amplification of ISMF and the electron maximum en- ergyontheobliquityangleinthisSNR.Detailedfeaturesof OP acknowledge Osservatorio Astronomico di Palermo for theSNR maps in different wavebands should be considered hospitality.The work of OPwas partially supportedby the for this purpose. program ’Kosmomikrofizyka’ of National Academy of Sci- The interplaybetween dependencesς(Θ ),σ (Θ ) and ences (Ukraine). FB, SO and OP acknowledge Consorzio o B o E (Θ ) may cause the quadrilateral morphology in SNR COMETA under the PI2S2 Project, a project co-funded max o models, due to splitting of maxima in surface brightness. bytheItalian Ministry of UniversityandResearch (MIUR) 8 Petruk O. et al. within the Piano Operativo Nazionale ‘’Ricerca Scientifica, N(E,Θ )∝ς(Θ )K¯(a¯)E−sexp − Ea¯−ψ(E,Θo) (A4) SviluppoTecnologico, Alta Formazione’ (PON 2000-2006). o o E F(Θ ) (cid:18) max,k o (cid:19) where REFERENCES 5σ (Θ )2E ψ(E,Θ )=1+ B o (A5) Aharonian, F. et al., 2006, A&A449, 223 o 2E f,k Aharonian, F. et al., 2007, A&A464, 235 Berezhko, E. G., & V¨olk,H. J. 2006, A&A451, 981 and the obliquity variation of themaximum energy of elec- Cassam-Chena¨ı G., Decourchelle A., Ballet J., Ellison D. trons is given by Emax=Emax,kF(Θo). C., 2005, A&A443, 955 Electrons with Lorentz factor γ emit most of their IC Ellison D. C., Baring M. G., Jones F. C., 1995 ApJ, 453, radiation in photons with energy εm. Let us use the ’delta- 873 function approximation’ (Petruk 2008): Ellison D.& Cassam-Chena¨ı G., 2005, ApJ, 632, 920 ∞ Gaensler B. M., 1998, ApJ,493, 781 Fulbright M. S., & ReynoldsS. P., 1990, ApJ, 357, 591, pic(γ,ε)≈pm(γ)δ(ε−εm), pm(γ)= pic(γ,ε)dε. (A6) Kesteven M. J. & Caswell J. L., 1987, A&A,183, 118 Z 0 Lee S.-H., Kamae T., Ellison D.C., 2008, ApJ,686, 325 OrlandoS., BocchinoF., RealeF., Peres G.,& PetrukO., IntheThomsonlimit,whichisvalidforSNRsinmostcases, 2007, A&A,470, 927, εm(γ) ≈ 4kTγ2 (Petruk 2008) and pm(γ) = (4/3)cσTωγ2 Petruk O., 2006 A&A,460, 375 (Schlickeiser2002),T andωarethetemperatureandtheen- Petruk O., 2008, astro-ph/0807.1969 ergydensityof initialblack-bodyphotons,σT istheThom- Petruk O., Beshley V., 2008, KPCB, 24, 159 son cross-section. Petruk O., Dubner G., Castelletti G., Iakubovskyi D., Substitution (4) with (A6) yields Kirsch M., Miceli M., Orlando S., Telezhinsky I., 2009, MNRAS,accepted q = cσTωmec2ε1/2N(E ) (A7) Plaga R.,2008, New Astronomy,13, 73 ic 12ǫ3/2 m c ReynoldsS. P., 1998, ApJ,493, 375 Rothenflug R., Ballet J., Dubner G., Giacani E., De- where courchelle A., & Ferrando P., 2004, A&A,425, 121 m c2ε1/2 Schlickeiser R. Cosmic Ray Astrophysics(Springer, 2002) E = e (A8) m 2(kT)1/2 istheenergyofelectronswhichgivemaximumcontribution APPENDIX A: APPROXIMATE ANALYTICAL to ICemission at photons with energy ε. FORMULA FOR THE AZIMUTHAL Let us consider the azimuthal profile of the IC γ-ray VARIATION OF THE IC γ-RAY SURFACE brightness S̺ at a given radius ̺ from the centre of the BRIGHTNESS IN SEDOV SNR SNRprojection. The obliquity angle Θ is different for each radial sec- o AnapproximateformulaforazimuthalvariationoftheICγ- tor of 3-D object. It is determined, for any position within raysurfacebrightnessallowsonetoavoiddetailednumerical SNR, by the set (ϕ,r¯/̺¯,φ ). Integration along the line of o simulationsandmaybeusefulifapproximateestimationfor sight gathers information from differentradial sectors, with the variation is reasonable. It gives deeper insight in the different obliquities. Let us determine the ‘effective’ obliq- main factors determining the azimuthal behavior of the IC uity angle by therelation surface brightness in SNRs. Let the energy of relativistic electrons is E in a given Θ (ϕ,φ )=Θ (ϕ,1,φ ). (A9) o,eff o o o fluid element at present time. Their energy was E at the i time this element was shocked. These two energies are re- Actually, Θ for a given azimuth equals to the obliquity o,eff lated as angle for a sector with thesame azimuth lying in theplane of the sky (i.e. in the plane being perpendicular to the line E =EE E (A1) i ad rad ofsightandcontainingthecenterofSNR).Θ variesaround o where Ead accounts for theadiabatic losses and Erad for the Θo,eff during integration along the line of sight. The closer radiativelosses.Thereareapproximationsvalidclosetothe ̺ to the edge of SNR projection the smaller the range for shock (Petruk& Beshley 2008): variation of Θ and more accurate is our approximation. o Ead ≈a¯, Erad ≈a¯5σB2E/2Ef,k (A2) fromTthheescuernftaecreabnrdigahttnaezsismoufthSNϕRisprojection at distance ̺ where a¯ = a/R, a is Lagrangian coordinate of the fluid el- ement, Ef,k is the fiducial energy for parallel shock. The S(̺¯,ϕ)=2 1 q (a¯) r¯r¯a¯da¯ . (A10) ic downstream evolution of K in a Sedov SNRis r¯2−̺¯2 Za¯(̺¯) K ∝ς(Θo)K¯(a¯). (A3) where r¯ is the derivatpive of r¯(a¯) in respect to a¯. The az- a¯ With the approximations (A2), the distribution N(E) may imuthalvariation oftheICbrightnessfor fixed̺isapprox- bewritten in themodel of Reynolds(1998) as imately Synchrotron and IC images of SNRs 9 E ̺¯−ψ(Em,Θo,eff) S ∝ς(Θ )exp − m ̺ o,eff E F(Θ ) max,k o,eff ! (A11) 1 r¯r¯ da¯ E (a¯−ψ−̺¯−ψ) × a¯ exp − m Za¯(̺¯) r¯2−̺¯2 (cid:18) Emax,kF (cid:19) If ̺¯→1 thenpa¯(̺¯)→1. Thus, the exponent in the integral is roughly unity because a¯(̺¯) ≤ a¯ ≤ 1 and a¯(̺¯) ≤ ̺¯≤ 1. The integral in (A11) is therefore roughly the same for any azimuthal angle ϕ. The azimuthalvariation of theICγ-ray brightness S (ϕ;φ ,ε) is thusdetermined mostly just by ̺ o S (ϕ)∝ς(Θ )exp −Em̺¯−1−5σB(Θo,eff)2Em/2Ef,k (A12) ̺ o,eff E F(Θ ) (cid:18) max,k o,eff (cid:19) with E given by Eq. (A8), i.e. S depends in this approx- m ̺ imation on the temperature T of the seed black-body pho- tons and the energy ε of observed γ-photons. The relation between the azimuthal angle ϕ, the obliquity angle Θ o,eff and the aspect angle φ is as simple as o cosΘ (ϕ,φ )=cosϕsinφ (A13) o,eff o o fortheazimuthangleϕmeasuredfromthedirectionofISMF in theplane of thesky. Theapproximation(A12)maybeusedfor̺¯largerthan ≃0.9. LikeEq.(9),Eq.(A12)does notgivecorrect profiles in thecase ofcentrally-bright SNRs,i.e. when Θ ≤π/4in K (1) and φ <30o. o