Some properties of synchrotron radio and inverse-Compton gamma-ray images of supernova remnants

0.64 MB·English

by  O. Petruk

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
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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