Limits on compression of cosmic rays in superno v a remnants
Iurii Sushch
1,2‹and Robert Brose
3‹1CentreforSpaceResearch,North-WestUniversity,2520Potchefstroom,SouthAfrica
2AstronomicalObservatoryofIvanFrankoNationalUniversityofLviv,KyrylaiMethodia8,UA-79005Lviv,Ukraine
3DublinInstituteforAdvancedStudies,AstronomyandAstrophysicsSection,31FitzwilliamPlace,D02XF86Dublin2,Ireland
Accepted2023February19.Received2023January23;inoriginalform2022October12
ABSTRACT
Thespectralshapeofthegamma-rayemissionobservedfordynamicallyoldsupernovaremnantsthatinteractwithmolecular cloudstriggeredanexcitingscenarioofadiabaticcompressionandfartherre-accelerationof Galacticcosmicrays(GCRs)in radiativeshellsoftheremnants,whichwasextensivelydiscussedandappliedtovarioussourcesoverrecentyears.Indeed,the observedgamma-rayspectrumfromanumberofremnantsstronglyresemblestheexpectedspectrumofthegamma-rayemission fromthecompressedpopulationofGCRs.Inthefollowingwediscussthefeasibilityofthisscenarioandshowthatitisvery unlikelythatcompressedGCRscouldproducesufficientamountofgamma-raysandthattheobservedspectralshapeisputting stronglimits onthe allowedcompressionfactors. Further,absence of curvatureinfeaturelesspower-law spectraof evolved supernovaremnantsatradiowavelengthsisstronglydisfavouringthecompressionscenarioforelectronsandhenceforhadrons.
Ourcalculationsshowthatthecontributionofcompressedelectronstotheobservedradiofluxcouldreachatmost∼10 per cent.
Keywords: cosmicrays– ISM:supernovaremnants– gamma-rays:general– radiocontinuum:general– ISM:clouds.
1 I N T R O D U C T I O N
Two decades of observations with the Fermi LAT telescope has revealeda largepopulationof gamma-brightsupernovaremnants (SNRs;Aceroetal.2016).Asignificantfractionofthispopulation constitutedynamicallyoldSNRs,interactingwithdensemolecular clouds,whosegamma-rayemissioncanbeconfidentlyattributedto hadronicprocesses(Giuliani&AGILETeam2011;Ackermannetal.
2013;Jogler&Funk2016;Ambrogietal.2019;deOna˜ Wilhelmi etal.2020).Anotherfeaturethattheseremnantshaveincommon istheshapeoftheirgamma-rayspectrumwhichappearstobesoft indicatingasoftspectrumoftheunderlyingprotonpopulationwith spectralindicesof∼2.4–2.8.Suchasoftspectrumisexpectedfor dynamicallyoldSNRs,e.g.duetothecombinedeffectofthedecrease ofthemaximumenergyandparticleescape(Cellietal.2019;Brose etal.2020)and/orthedecreaseoftheshockcompressionduetothe propagationthroughthehotshockedwindoftheprogenitorstar(Das etal.2022).Theresemblanceoftheunderlyingprotonspectrumwith thespectrumofGalacticcosmicrays(GCRs)triggeredalsoanother excitingscenarioin whichpre-existingGCRscanbecompressed andre-acceleratedsubsequentlyemittinggamma-rayradiation.Such amechanismispossiblewhentheSNRshockisinteractingwitha densematerialeitherinformofalargecloudorsmallverydense clumpsthatleadstoformationoftheradiativeshellbehindtheshock front.ThisscenariowasproposedforanumberofSNRsincluding themostprominenthadronicemittersW44andIC443(Uchiyama etal.2010;Tang&Chevalier2014;Leeetal.2015;Tang&Chevalier 2015;Cardillo,Amato&Blasi2016;Tang2019)closelyfollowing
E-mail:[email protected](IS);[email protected](RB)
theideaproposedbyBlandford&Cowie(1982).However,itwas shownrecentlythatsomeofmodelssufferfromstronginconsistency connectedtotheunrealisticallylargeamountofgasrequiredinthe shelloftheremnant(deOna˜ Wilhelmietal.2020).Indeed,tomatch theobservedflux,itisrequiredthatthestronglycompressedshell whichconsistsofthecrushedcloudmaterialcoversalargefraction oftheSNRvolume,whichresultsinmorematerialintheshellthan theSNRcanpossiblyacquirefromtheambientmediumduringits evolution.Ontheotherhand,itshouldalsobestressed,thatunder theassumptionthatSNRsaresourcesofGCRs,itisexpectedthatthe spectrumofparticlesreleasedbySNRsintothemediumfollowsa powerlawwiththespectralindexof∼2.4andhence,thedownstream spectrumofparticlesacceleratedatshocksofdynamicallyoldSNRs mustbesofterthanthat.
Becausecompression/re-accelerationmodelsarestillwidelydis- cussedandpromotedwithinthecommunitywewouldliketofollow- up and explore furthertheirfeasibility. Inthe following, wewill consideronlycompressionofGCRsasthere-accelerationprocess doesnotstronglyimpactthenormalizationoftheresultinggamma- rayspectrumbutrathershiftsitinenergy.
2 A D I A B AT I C C O M P R E S S I O N O F G C R P R OT O N S A N D E L E C T R O N S
Thebasicideaofthecompressionprocessisthattheinteractionof theSNRshockwiththedenseenvironmentresultsintheformation oftheradiativeshellbehindtheshockfront.Thematerialbehindthe shockisadiabaticallycompressedtoveryhighdensitiespotentially boostinganypion-decaygeneratedgamma-rayemissionandalsora- dioemissionbyenhancingthemagnetic-fieldstrength.Theadiabatic compressionofthepre-existingambientGCRsintheradiativeshell
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enhancestheCRspectrumbothenergizingparticlesandincreasing the normalizationofthe spectrum.ThecompressedCR spectrum canthenbeexpressedas(Blandford&Cowie1982;Uchiyamaetal.
2010)
ncomp(p)=ξ2 / 3 nGCR(ξ−1 / 3 p), (1) where nGCR (p) isthe densityof GCRs as a functionof momen- tum and ξ ≡ nshell /(rn0 ) is the adiabaticcompression ratio, with nshell the density of the cooled gas in the shell, n0 the density of the ambient medium (cloud), and r the shock compression ratio. The interactionof compressed GCRswith the compressed cloud material of high density and high magnetic field in the shell can potentially result in substantial gamma-ray and radio emission.
2.1 Galacticprotons
For the proton CRspectrum, weadopt the approximation of the observed proton flux proposed by Bisschoff & Potgieter (2016) imposedwithaspectralhardeningathigherenergies(Adrianietal.
2011;Aguilaretal.2015):
JGCR (E)=0.3719E1.03 β2
E1.21+0.771.21 1+0.771.21
−3 . 18
×
1+ E
335
00..1190240 . 024
GeV−1 cm−2 sr−1 s−1 , (2)
whereEisthekineticenergyofprotoninGeVandβistheproton velocityinc.ThenumberdensityofCRsasafunctionofmomentum isgiventhenby
nGCR (p)=βcnGCR (E)=4π JGCR (E) (3) andthecompressedspectrumcanbefoundusingequation(1).
The level of the resulting gamma-ray emission is completely determinedbythe compressionratioandtheamountofthe target materialintheshell.ForaspecificSNRwithknownsizeanddensity oftheambientmedium(cloud)thissimplifiestojusttwoparameters:
the totalcompressionratioχ ≡ nshell /n0 =ξrand volumefilling factor f= Vshell /VSNR , theratio ofthe volume ofthe shell to the volumeofthe remnant.Itishoweveroftenignoredthatthesetwo parametersareinterdependentandtheirparameterspaceisstrongly constrained.
2.2 Galacticelectrons
TheGalacticelectronspectrum iswelldescribedbyapower law withapower-lawindexofs1 =3.04athigherenergies.However,the spectrumshowsasmoothtransitiontoaspectrumwiths≈1atlower energies(Jaffeetal.2011).Thisspectralshapecanbedescribedbya log-parabolaatlowenergiesthattransitionstoapowerlawathigher energies,
Ne (E)= N 1
E exp −log2(E/σEB)
forE≤EB
N2 E−s forE>EB . (4) A fit to the electron spectra given in Jaffe et al. (2011) for a galactocentric radius of 6.5kpc with expression (4) yields s = 3.04and EB = 5GeV.These spectra are compatible withdirect observationsof theelectron spectra in thelocal ISMby Voyager 1, which alsoshow spectra harder than s = 2.0at low energies (Cummingsetal.2016).
2.3 Constraintimposedbytheavailablecloudmaterial Theradiativeshellbehindtheshockfrontwhereadiabaticcompres- siontakesplaceconsistsofthecloudmaterialoverranbytheshock.
Thisimposesahardupperlimitonthetotalnumberofparticlesin theshellforacurrentgivenvolumeoftheSNR
Nmax =VSNR n0 , (5)
where VSNR=4/3π R3sh , Rsh is the shock radius, and n0 is the particledensityofthecloud.Thisupperlimitfollowsfromasimple consideration thatiftheSNRhasbeenalwaysexpandingintothe cloudthroughoutitsevolutionthatisthemaximumpossiblenumber ofparticlesthatcanbelocatedbehindtheshock.
Now,thetotalnumberofparticlesintheshellcanbeexpressedas Nshell =fVSNR nshell =fVSNR χ n0 . (6)
Fromthisimmediatelyfollowsaconditionontheproductofthe volumefactorandthetotalcompressionratio:
fχ≤1. (7)
It shouldbenotedthatthe aboveconstraintisindependentofthe structure of the medium, size of clouds, their density, and the amount of volume that they occupy. It follows solemnly from a basicconditionthatthecrushedshellcannotpossiblycontainmore particles than available in the ambient medium confined by the currentsizeoftheSNR.Thisconditionisnotfulfillede.g.inCardillo etal.(2016)forW44wherethemodelrequiresthefillingfactoroff= 0.14andthetotalcompressionratioofχ=50,1 whileinUchiyama etal.(2010)themodeloperatesat10–20percentlevelofthishard upperlimit.
In reality,the product fχ shouldbe muchlower than unityas consideredSNRsinteractwithcloudsonlyduringacertainfraction of their evolution and the interaction does not cover the whole shocksurface.Moreover,theradiusoftheSNRexpandingintothe densecloudmediumthroughoutitsevolutionisexpectedtobemuch smallerthanobservedforspecificSNRs.Infact,theradiusofthe SNRatthebeginningoftheradiativeorpressure-driven-snowplough (PDS) stagecanbe expressedas (Cioffi, McKee & Bertschinger 1988)
RPDS =1.2 n0
300cm−3 −3 / 7
ESN 1051erg
2 / 7
ζm−1 / 7 pc, (8)
whereESN isthesupernovaexplosionenergyandζm isthemetallicity factor,ζm=1forsolarabundances.Thisradiuscanbeconsidered asanupperlimitonthedistancethattheSNRwouldexpandintothe cloudatanystageofitsevolution,whichwouldprovideuswitha moreconstraininglimitonthefχproduct.Assumingthatthewhole surfaceoftheSNRshockexpandedintothecloudforthedistance of R,thetotalvolumeofthecloudthatshockinteractedwithcan bewrittenas
Vcloud = 4
3π(R3 sh −(Rsh − R)3 )≈4π Rsh 2 R
1− R Rsh
. (9)
1Note,Cardilloetal.(2016)usesthesurfacefillingfactorintheircalculations andtherelationoffillingfactorfthatisdefinedthesamewayasinthiswork andsurfacefillingfactorξisshownonpage5ofCardilloetal.(2016).f= 0.14correspondstoξ=0.55requiredbytheirmodel.χ=50followsfrom therequireddensityofthecrushedshellof10000cm−3withtheadopted clouddensityof200cm−3.
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Hence,thetotalnumberofparticlesthatcanbeaccumulatedinthe shellis
N=Vcloudn0=4π Rsh 2 R
1− R Rsh
n0. (10)
Combiningthiswithequation(6)onegets fχ=3 R
Rsh
1− R
Rsh
(11) or
fχ≤3RPDS Rsh
1−RPDS Rsh
. (12)
Thelargeradiiofobserved SNRsaredifficult toaccommodate withintheassumptionofanexpansionintoauniformmediumwitha highdensity.Thisproblemcanbeovercomeusingtheassumptionof aclumpymedium,wheredensebutsmallclumpsareresponsiblefor thecreationofthecrushedshellbutaredynamicallyunimportant.
Thisassumptionwouldalsoallowforthehighshockvelocitiesduring interactionwithdenseclumpsevenatlatestagesofevolution.The importantparameterthatcharacterizessuchamediumisthefilling factorφ whichexpresses the fractionofthe volumeoccupiedby theclumps.Itisclearthatthefillingfactorcannotbetoolargeto justifytheassumptionthattheclumpsaredynamicallyunimportant.
Inthe following, weconsideranupperlimit onthe filling factor ofφ≤0.1whichfollowsfromhydrodynamicsimulations(seee.g.
Slavinetal.2017)and stillreflectsaquiteoptimisticscenario. It shouldbenoted,however,thatevenforlowfillingfactorstheSNR shockwillexpandslowerthanin theuniform intercloudmedium duetoconductiveevaporationofclumpsembeddedinthehotgas behind the shock and such evolution would be accompanied by significantthermalX-rayemission(White&Long1991;Slavinetal.
2017).
Forthemostoptimisticscenario,wheretheSNRexpandsintothe clumpymediumthroughoutitswholeevolutionthefillingfactorof clumpsφcanbeexpressedbythemeansofthedefinedabovefilling factorofthecrushedshellfandtotalcompressionratioχas φ= Vclumps
VSNR = Vshell VSNR
Vclumps
Vshell =fχ (13)
andthereforeimpliesacondition
fχ≤0.1 (14)
tojustifytheneglectofclumpsintheshockdynamics.Note,thatthis conditionisnotsatisfiedinUchiyamaetal.(2010)wherebest-fitting modelsrequiretheclumpsfillingfactorofφ∼0.2−0.42 andhence clumpscannotbeconsideredasdynamicallyunimportant.
In the following, we probe the non-thermal emission from compressed CRs in both scenarios, i.e. the expansion of the SNR into one large cloud (condition 12) and the expansion into a clumpy medium (condition 14). In both cases we use the same radius of the SNR ignoring differences in the SNR evolution and focusing on testing the requirement of a suffi- cient amount of cloud material. This would naturally result in a larger age and a lower shock velocity in the one-large-cloud scenario.
2Note,thatφisnotthesameasthefillingfactorfinUchiyamaetal.(2010) whichisalsodefinedastheratioofthevolumeoccupiedbyclumpstothe volumeoftheSNR,butitisassumedthattheSNRinteractswiththeclumpy mediumonlyduringahalfofitsage.
2.4 Constraintsonthetotalcompressionratio
Assuming the magnetic field in the cloud, B0, is turbulent, the magneticfieldinthecompressedshellcanbeexpressedby Bshell =
2χ2+1
3 B0 (15)
or Bshell ≈
2/3χ B0 (16)
forlargeχ.Now,followingtheassumptionthatcompressiondueto radiativecoolingdownstreamislimitedbythemagneticpressureone canestimatethe totalcompressionratioby equatingthemagnetic pressurewith theshock rampressure(Blandford & Cowie1982; Uchiyamaetal.2010):
χ9.4 n0
1cm−3 1 / 2
B0 1μG
−1 vsh 106 cms−1
. (17)
The study on deducing magnetic field strengths in molecular cloudsfromZeemanobservationsbyCrutcheretal.(2010)indicates roughly constant magnetic fields of B0 =10μG in clouds with densities ofn0300cm−3 witha power-lawincreasefordenser clouds.Thegeneralizedempiricalmodelforthemaximummagnetic fieldstrengthinthecloudcanbeexpressedas(Crutcheretal.2010):
B0 =
10μG, n0 ≤300cm−3 10μG n0
300cm−3
0.65
n0 >300cm−3. (18) Adoptingthisresultintoequation(17)impliesthatthecompres- sionratiocouldbeassmallas
χ
16.3 n0
300cm−3
1/2 vsh
106cms−1
n0≤300cm−3 16.3 n 0
300 cm −3
−0 . 15 v sh
10 6cm s −1
n0 >300cm−3 . (19)
Whileforhigh-densitycloudsthisshouldbeconsideredasalower limit,forlow-densitycloudsitbasicallyreflectstheestimateofthe compressionratioasformanycloudsthemagneticfieldisfoundto beatthelevelof∼10μG(Crutcheretal.2010).
Theshockvelocity isstronglydependenton the densityofthe cloud.Fortheconstantdensityoftheambientmediumof1cm−3an SNRreachestheradiativestageofevolutionwhenitsshockvelocity is afew hundredkms−1 (see e.g. Cioffi etal. 1988). Theshock velocityinthedensecloudcouldbeconsiderablysmallerthanthat.
Generally,theshockvelocityinthecloudcanbeapproximatedby (McKee&Cowie1975)
vsh∼vsh n0
n0, (20)
wherevshandn0aretheshockvelocityandthedensityofthemedium beforethe interaction.Combiningthis withequation(17) onecan concludethatthetotalcompressionratioχdoesnotstronglydepend onthedensityofthecloud.Moreover,itshoulddecreasefordensities largerthan∼300cm−3 duetotheincreaseofthemagneticfield.The shockvelocityinthecloudfortypicalvaluesshouldbearound∼10–
100kms−1 resultingincompressionratioofχ ∼10–100.Onthe otherhand,theshockvelocityhastobelargerthanthespeedofsound inthecloud(typicallyabout10kms−1)atleastbyafactoroftwoto ensurethattheshellisnotdisruptedandmergedwiththemedium.
Thismeansthatthecompressionratioshouldalsobeatleastafew tens.
Another constraint on the compression ratiofollows from the observed spectralshapeofthegamma-ray emissionforthesedy- namicallyoldinteractingSNRs,forwhichthecompressionscenario couldbeapplicable.Mostofthemexhibitapeakintheenergyfluxat
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Figure1. Normalizedspectralenergydistributionofthegamma-rayemis- sionproducedbythecompressedGCRprotonsfordifferentvaluesofthe compressionratioχ.Curvesareplottedforχintherangefrom10to100with thestepof10.Gamma-rayemissioniscalculatedasdescribedinSection3.
࣠1GeVwithmaybeoneoutlier,G349.7+0.2,whichseemstohave apeakataround2GeV(seefig.1inTang2019).Suchalowpeak canbereproducedonlyformoderatevaluesofthecompressionratio asthe adiabaticcompressionalsoenergizesparticles,and fortoo highvaluesofχthepeakinthegamma-rayspectrumwouldmoveto higherenergies.InFig.1,weshowthenormalizedspectralenergy distributionoftheresultinggamma-rayradiationfordifferentvalues ofχ.Theenergyofthepeakis0.7GeVforχ=10,anditshiftsto 1GeValreadyforχ=30.Forχ=100thepeakislocatedat1.5 GeVwithaquitesteepdecreasebelowthatenergy.Thissuggeststhat fromtheobservationalpointofviewthecompressionratiocannotbe largerthanafewtens.Note,thatthissimpleanalysisdoesnottake intoaccountadditionalre-accelerationofGCRs,whichwouldbring theconstraintonχtoevenlowervalues.
Asonecanseethevalueofthecompressionratioisconstrained onbothsidesthroughdifferentconsiderationtoafewtens,which stronglylimitstheparameterspaceandleaveslittleroomtomaneu- ver.
3 G A M M A - R AY E M I S S I O N
Tocalculatethegamma-rayemission,weusethepost-processing radiationroutineoftheRATPaCcode(Telezhinsky,Dwarkadas&
Pohl2012,2013;Brose,Telezhinsky&Pohl2016;Sushch,Brose&
Pohl2018) designedfor numericalsimulations of particle accel- eration in SNRs. The module to calculate gamma-ray radiation frompiondecaysreliesonMonteCarloeventgenerators,namely
DPMJET-III(Roesler,Engel&Ranft2001)andUrQMD(Bassetal.
1998;Bleicheretal.1999),forthecalculation ofinelasticcross- sections and differential production rates of secondary particles producedinnucleicollisions(Bhattetal.2020).Astargetparticles forhadronicinteractionswetakeintoaccountbothHandHewitha typicalcompositionH:He=10:1.
In Fig. 2, weshow upperlimit curveswith solidlines for the gamma-rayspectralluminosityintheuniformcloudscenario(top panel)andclumpymediumscenario(bottompanel)forthreevalues of the compressionratioindicated with differentcolours. Weset the radiusofthe shockto 10 pc.Thedensityofthe cloudin the uniformcloudscenarioissetto300cm−3 whichreflectstheturning pointabovewhichobservationallymagneticfieldincloudsstartsto
Figure2. Thesolidcurvesshowupperlimitsonthespectralgamma-ray luminosityresultedfromthecompressionofGCRprotonsindynamically oldSNRsinteractingwithauniformcloud(toppanel;satisfiesthecondition inequation12)andclumpymedium(bottompanel;satisfiesthecondition in equation14) fordifferentvaluesofthecompression ratio.Theshock radiusissetto10pcforbothscenarios.Thedensityoftheuniformcloud issetto300cm−3whilethedensityofclumpsissetto1000cm−3.Inboth scenariosthedensityoftheintercloudmediumissetto1cm−3whichroughly results in theaverage densityof∼100cm−3 for both cases. Thedash- dottedlinesillustratemodelsconstructedspecificallyfortheIC443SNR underassumptionoftheuniformdensemedium(toppanel)andcomplex clumpymedium(bottompanel;seetheAppendixfordetaileddescription).
Thedashedanddottedlinesinthebottompanelshowindividualcomponents ofthesecondmodel.TheredmarkersshowtheobservedspectrumoftheIC 443SNRwithFermiLAT(circles;Ackermannetal.2013),MAGIC(upward triangles;Albertetal.2007),andVERITAS(downwardtriangles;Acciari etal.2009).TheFermiLATdatafortheW44SNR(Ackermannetal.2013) isshownwithbluesquares.
increasewiththedensity.Intheclumpymediumscenariothedensity ofclumpsissetto1000cm−3 .Inbothscenariosthedensityofthe intercloudmediumisfixedat1cm−3 ,i.e.theaveragedensityisabout 100cm−3forbothcases.Thevolumefillingfactorofthe crushed shellfissettotheupperlimitvalueasfollowsfromequation(12) for theuniform cloud scenarioand equation(14) for theclumpy medium. The volume filling factor of clumps in the medium is setto φ=0.1.Thedatapointsshowthespectraofthe twomost famousandstudiedhadronicSNRs,IC443(FermiLAT(redcircles;
Ackermannetal.2013),MAGIC(redupwardtriangles;Albertetal.
2007),andVERITAS(reddownwardtriangles;Acciarietal.2009)),
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andW44(FermiLAT(bluesquares;Ackermannetal.2013)).IC 443istheleastluminousin gamma-raysamongotheragedSNRs while W44 falls roughly in the middle of the distribution based onthegamma-rayluminosity(Tang2019).IC 443alsoexhibitsa slightlyhigherpeakenergy,closeto1GeV,thatallowsustovary χ up to30. ForotherSNRs exceptG349.7+0.2the peak energy islower thanthat,constraining thecompression ratioevenmore.
G349.7+0.2, on the other hand, exhibits a veryhigh luminosity whichwould behardtoreachevenforhighervaluesofχ.Itcan beseeninthetoppanelofFig.2thatforthescenarioofthelarge uniformcloudcoveringthewholesurfaceoftheforwardshockthe derivedupperlimitongamma-rayemissionfromthecompression ofGCRprotonscanbarelyreachthegamma-rayluminosityofIC 443andfallsalmostanorderofmagnitudebelowthe gamma-ray luminosityof W44forthe most optimisticcase. The situation is betterfor the clumpy medium scenario(bottom panel of Fig. 2) where derivedupper limits do not exclude the GCRsmodel for IC 443 but still fall below the observed gamma-ray luminosity ofW44.For SNRs moreluminous thanW44 the contribution of compressed GCR should be negligible. Moreover, any potential additional re-acceleration of CRs would shift the peak towards higherenergiesevenmorelimitingtherangeofallowedvaluesof χ.Therefore,thefailureoftheadiabaticcompressioninproviding enough gamma-ray flux around the peak energy, means that re- accelerationmodelsbasedonthecompressedCRspectrumwillfail too.
Forcomparisonwealsoshowtwomodelsconstructedspecifically fortheIC 443SNR whichfollowobservationalpropertiesofthe remnant.Thefirstmodelassumesuniformdensemediawithdifferent densitiesontwosidesoftheremnant(Fig.2,toppanel)andsecond modelconsidersaclumpymediumintheNorthernhemisphereof theremnant and additionalinteractionwithaverydensetoroidal molecularcloud.ThemodelsarefurtherdescribedintheAppendix.
Bothmodelsoptimisticallyassumethecompressionratioofχ=30 andfallbelowrespectiveupperlimits.The‘uniformdensemedia’
modelfailstoexplaintheobservedIC443fluxwhilethe‘clumpy medium’modelcanroughlyfitthedataindicatingthatindeedthere mightbesomecontributionfromcompressedGCRsinthecaseof IC443.However,itshouldbestressedhereagainthatfluxesinboth modelsarederivedunderoptimisticconditionsandcanbeconsidered asupperlimitsofthegamma-rayemissionfromcompressedGCRs inIC443,whichisprobablythebestcandidateforsuchamodel givenitslowgamma-rayluminosityandaverydenseandcomplex medium.
4 R A D I O E M I S S I O N
Thecompressed Galactic electronstogether withthe compressed magneticfieldsin theSNRshellwillboost theradiosynchrotron emission from these objects. In fact, it is hard to explain how CR protons shall not be accelerated by the SNR shock, so that compression is the main driver of the gamma-ray emis- sion, while electrons get freshly accelerated. This is especially constraining since radio emitting electrons and protons respon- sible for the gamma-ray emission have approximately the same energy.
WeusetheGalacticelectronspectrumasparametrizedinequation (4) and compressed according to equation (1). We then use the emissionroutines of the RATPaC code (Telezhinskyet al.2012, 2013; Brose et al. 2016; Sushch et al. 2018) to calculate the synchrotron emission that canbe expected from the compressed shell.
4.1 Spectralbreak
The spectral turnover of the Galactic electron spectrum below EB =5GeVleadstoaspectralbreakinthesynchrotronemission fromthecompressedelectrons.Theradiobreak-frequencyνB canbe calculatedaccordingto
νB =16MHz √
2/3χ B0 μG
ξ1/3EB GeV
. (21)
Asaresult,theradiospectrashowsaturnoverfromaradiospectral index of α = 0 to α = 1 belowνB=40GHz forχ = 30 and B0 =10μG.Foralowercompressionratioofχ=10,thetransition happensbelowνB =9GHz.
Thefact that particularlyevolved SNRs showremarkably fea- tureless radio spectra from low to high energies (e.g. Castelletti etal.2007) comprehendsstrongevidenceagainst acompression- onlyoriginoftheradioemission.
4.2 Radioflux
Wecalculatedthe radiofluxfromcompressedelectronsusingthe parameters from Section 3. Here, we also consider two distinct scenarios, one assuming only a homogeneous medium and one assumingthepresenceofadditionaldenseclumps.Forthemagnetic fieldwefollowtheupperlimitsgivenbyCrutcheretal.(2010)given inequation(18).Note,thatusageofequation(12)forsynchrotron radiationdoesnotnecessarilyprovideanupperlimitontheemission astheemissiondoesnotdependonthenumbertargetparticlesin thiscase.
Fig.3showsacomparisonoftheexpectedradioluminositywith the luminosities of W44 and IC 443.It is evident that the radio luminosityfromthecompressedelectronsiswellbelowtheobserved radioluminosityforbothremnantsandbothconsideredscenarios, contributing at most on alevel of≈10 per cent.The dedicated modelsforIC443showslightlyhigherfluxescomparedtothegeneric modelsonaccountofthe compressedfieldinthe clumpsandthe densecloud(seealsoAppendixA).Itshouldbenotedhere,thatthe magnetic-fieldscalingderivedbyCrutcheretal.(2010)constitutes anupper limit onthe field in the cloudand clumpsand so does consequentlyourpredictedradioemission.
Theunderpredictionoftheradiofluxfromcompressedelectrons stronglyindicatesthatadditionalelectronsneedtobeacceleratedin theSNRs.Theseadditionalelectronscaneitherbeacceleratedbefore the SNR–cloudinteractionorcontinuously duringtheinteraction.
These freshly accelerated electrons would need to reach at least energiesofafewGeV.Inthiscase, however,alsoprotons should beacceleratedtoatleastthoseenergiesandwouldcontributetothe gamma-rayemission.
Whereas the compression of Galactic electrons falls short in providingenoughradioluminosityin general,thesituation might be different forother radiosources. Recently, it was shownthat non-thermalradioemissionforthestellarbow-shockBD43canbe well explained by the compression of Galactic electrons in this system (Moutzouri et al. 2022). However, in these systems, the shockvelocityisneverexceedingafewtensofkms−1,whileSNRs experience considerablyhigher shock velocitiesduring the initial phasesoftheirevolution.Here,asalreadynotedinSection3,the freshaccelerationofelectronsandprotonsintheinitialphasesofthe remnantsevolutioncanprovideenoughgamma-rayandradioflux, whiletheinteractionwiththedensecloudenhancesparticleescape andshapestheobservedsoftspectra(Cellietal.2019;Broseetal.
2020;Brose,Pohl&Sushch2021).
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Figure 3. The solid coloured curves show upper limits on the radio luminosityfroma populationofcompressedelectronsin andynamically oldandinteractingSNR.Theupperpanelshowstheemissionforanambient densityof300cm−3andthelowerpanelfora1cm−3mediumand1000cm−3 clumpswithavolume-fillingfactorofφ=0.1.Theshock-radiusissetto 10pcinbothcases.Theredcirclesandbluesquaresindicatetheobserved radioluminositiesforW44andIC443,respectively(Egronetal.2017;Loru etal.2019).TheblacklinesillustratededicatedmodelsforIC443(seethe Appendix)includingdistinctcomponentsasdescribedinthelegend.
5 PA RT I C L E C O N F I N E M E N T
Aconsequenceofconditioninequation(7)is,thatthethicknessof thecompressedshellDshell canbewrittenas
Dshell=Rsh
1− 3
1− 1
χ
. (22)
Thislengthscalecanbecomparedtothemeanfreepathofparticles inordertoinvestigateiftheycanbeconfinedbycompressioninthe shelloriftheyescapethecompression-processdiffusely.Themean freepathLisroughlygivenby
L= D( cE) ≈A·1019 E
10GeV
1/3 √ 2/3χ B0 3μG
−1/ 3
cm, (23)
where A is a numerical factor describing the suppressionof the diffusionaroundtheSNR.AvalueofA=1impliestheGalactic diffusioncoefficient, while A = 10−2 is suppressed diffusion as observed around gamma-ray pulsars (Abeysekara et al.2017) or asderivedforSNRs(Fujita,Ohira&Takahara2010;Broseetal.
2021).Equations(22)and(23)canbecombinedtoobtaintheenergy uptowhichparticlescanbeconfined,
E=1
Rsh A·1019 cm
3√ 2/3χ B0
3μG 1− 3
1− 1
χ 3
TeV.
(24) Foracompressionratioofχ=30andA=1,equation(24)yields aenergyof3.4GeV.Ifthediffusioncoefficientissuppressedbya factorof10,theenergy-limitreaches≈3TeV,enoughtopotentially compressCRsatthegamma-rayemittingenergies.However,while the reductionof thediffusion coefficientisnormallyprovidedby escapinghigh-energyCRs,nosuchescapingparticlesareexpected inthecompressionscenario.
Additionally, equation (24) indicates that higher compression ratiosyieldalowerconfinementenergyasthedecreaseintheshell- thicknesswinsovertheincreaseinthemagneticfield.This,again, makesscenariosofveryhighcompressionfactorsunlikely.
6 C O N C L U S I O N S
The amount of cloud material that canbe possibly swept up by the SNR shock as well as constraints on the compression ratio strongly limitsthe potential gamma-ray luminosity generated by compressedGCRprotons.Weshowthatthismodelisnotplausible formostknownSNRsandcanonlymarginallyexplainthegamma- rayemissionfromtheleastluminoushadronic-dominatedSNRIC 443,buteveninthiscasethemodelisintensionwiththeobserved radioemission.Ingeneral,thisscenariocouldbefeasibleonlyinvery specificcases.Moreover,thefeasibilityofthemodelcouldbedirectly checkedthroughradioobservationsasGCRelectronsshouldalsobe compressedandleaveacharacteristiccurvatureinthesynchrotron spectrum at radiowavelengthscorresponding to the break in the spectrum ofGCRelectrons.Suchacurvature isnotobserved for evolvedSNRswhichexhibitfeaturelesspower-lawspectraacrossa widerangeoffrequencies.Todecisivelyclarifythisobservationsat
∼100GHzwouldbenecessary.Wefind,however,thatsynchrotron radiationfromcompressedelectronswouldbeverylowcontributing atmost∼10 percenttotheobservedradioluminosity.Theneedfor a largenumberof freshlyaccelerated electronsmakes it unlikely that protons get not freshly accelerated and likewise exceeding the contribution of the compressed CRs. Finally, we show that theconfinement ofhigh-energyprotonscouldbeaprobleminthe compressedshellwhichfartherstronglydisfavoursthisscenario.
AC K N OW L E D G E M E N T S
RBacknowledgesfundingfromtheIrishResearchCouncilunder the Government of Ireland Postdoctoral Fellowship program. IS acknowledgessupportbytheNationalResearchFoundationofSouth Africa(GrantNumber132276).
DATA AVA I L A B I L I T Y
Thedataunderlyingthisarticlecanbesharedonreasonablerequest tothecorrespondingauthor.
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A P P E N D I X A : M O D E L S F O R T H E I C 4 4 3 S N R WeconstructtwomodelsspecificallyfortheIC443SNRbasedon itsmorphologyandsize.Forbothmodelsweutilizethetwo-shell
ideaadoptedfromTroja,Bocchino&Reale(2006)andreferences therein.WeassumethattheSNRshellcanbeapproximatedbytwo hemispheresofradiiRA=7pcandRB=11pc.Forthefirstmodel weassumeuniformdensemediainbothhemisphereswhileforthe secondmodelweconsideracomplexclumpyenvironment.
A1Uniformdensemedia
Inthismodelthedensitiesofmediainbothhemispheresareassumed to beconstant.TheageofIC 443isveryuncertainrangingfrom
∼1000yrimpliedfromX-rayobservations(Petreetal.1988;Wang et al. 1992) to ∼30 kyr suggested by the proper motion of the compactobjectpotentiallyassociatedwiththeSNR.Weestimatethe highestpossiblevaluesfordensitiesnA=50cm−3andnB=9cm−3 following analytic approximations from Cioffi et al. (1988) and adopting the upperlimit on the discussedrangeof possibleages fortheremnantoftage =30kyr.Theseestimatesareconsistentboth withanalytictreatment(Chevalier1999)andnumericsimulationsof theexpansionofIC443intoamolecularcloud(Zhang&Chevalier 2019).Wealsoassumethatbothshellsareradiativeandtheswept-up materialaswellasCRsarecompressedwiththecompressionratio ofχ =30.Theexpectedgamma-rayluminosityfromcompressed GCRsfailstoexplaintheobservedgamma-rayemission(toppanel inFig.2).Note,inthismodelwecalculatethegamma-rayluminosity fromthewholeSNR,althoughthegamma-rayemissionisdetected onlyfromthehemisphereA.
To calculate the radio emission, we assume a magnetic field strength we follow equation (18) for the upstream field of both hemispheres. The field is then compressed in the downstream, resultingin ahigher fieldstrength.Again,the chosenvalueforB hastobeconsideredasanupperlimitandsoisthederivedradioflux displayedinthetoppanelofFig. 3.
A2Complexclumpyenvironment
IC443interactswithaverycomplexenvironmentthatconsistsof molecular and atomicclouds(seee.g. Rho et al.2001;Su etal.
2014).Ustamujicetal.(2021)presentedadetailed3Dhydrodynamic modelforIC 443whichaccountsforinteractionwiththe atomic cloudintheNorthernhemisphereAandwiththetoroidalmolecular cloudthat encirclestheremnantbetweentwohemispheres andis responsibleforthebrightsouthernridge.Thebestfitoftheirmodel tothemultiwavelengthobservationaldataisobtainedforthedensity of the atomiccloudto beabout300cm−3 andthe densityof the molecularcloudtobeabout3000cm−3 whichisingoodagreement with numbers obtained by Rho et al. (2001) from near-infrared observations, 10–1000 and ∼104 cm−3 , respectively. Initiallythe remnantisassumedtopropagateintotheintercloudmediumwith the density of 0.2cm−3 and then atsome point starts to interact with clouds in the northern hemispheres A, while the Southern hemisphere Bcontinuestoevolvein the intercloudmedium.The ageoftheremnantisfoundtobearound8kyr.Althoughitisclear thatalargefractionoftheshocksurfaceinteractswithverydense cloudsatdifferenttimes,itisalsoevidentfrom3Dhydrodynamic simulationsthattheshockdoesnotpropagatestronglyintothecloud, butrathergetscontainedbythecloud(Ustamujicetal.2021).Hence, theamountofthecloudmaterialinthecompressedshellbehindthe shockshouldberathersmallandtheexpectedgamma-rayemission fromcompressedGCRsshouldbenegligible.
Toconstructoursecondmodel,weroughlyfollowthesetupof Ustamujicetal.(2021)butmodifyitintroducingclumpinesstomake thescenarioofCRcompressionmoreoptimistic.Weassumethatthe
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NorthernhemisphereAthroughoutitsevolutionwaspropagatinginto theclumpymediumwiththedensityofclumpsnc =1000cm−3 and densityoftheintercloudmediumofnic=10cm−3 .Thisdescription nicelyfollowsmeasurementsbyRhoetal.(2001).Theexpansion of theshock in the clumpyenvironment canbedescribedby the analyticsolution proposed by White & Long (1991) which is in goodagreementwithnumericsimulations(Slavinetal.2017).This solutionimpliesthattheshockradiuscanbeestimatedbysimple scalingoftheSedov–Taylorradius
R=RST K
1.528 1 / 5
(A1)
withtheparameterKwhichdependsonthecloud-to-intercloudmass ratioCandevaporationtime-scaleτ=te v ap /tage .Numericsimulations bySlavinetal.(2017)fortheclouddensityof25cm−3 implyτ ≈ 5roughlyconstantthroughoutevolution.Giventhatthe te v ap ∝nc (Slavinetal.2017),nc=1000cm−3 yieldsτ ∼200.Wealsoset C=10thatcorrespondstothefillingfactorofφ= 0.091forthe assumeddensities
φ= C
n c
nic +C. (A2)
ForC=10andτ=200theWhite&Long(1991)solutionbasically convergestotheSedov–Taylorsolutionandtheshockshouldreach RA =7pcintage =7400yr.Thisageimpliesthatthedensityofthe mediuminthehemisphereBis1.4cm−3 .
Forthemolecularcloudweassumethedensityofnm =104 cm−3 and thevelocity ofthe shockinteractingwiththe cloudofvm = 30kms−1 inagreementwithRhoetal.(2001).Morerecentwork byCosentinoetal.(2022)reportssimilar measurementsbasedon ESO–AROPublicSpectroscopicSurveySHRECwiththeshocked gasdensityof≥105 cm−3 ,afactorof>10higherthanthepre-shock density,andtheshockvelocityof∼23kms−1 .Further,toremainon theoptimisticside,weassumethatthedistancetheshockpropagates intothecloudis
Rm =vm tage =0.2pc, (A3)
andthatthecloudcoversω=0.1ofthesurfaceofthehemisphereA.
Correspondingvolumesofthematerialintheclumpymediumand inthemolecularcloudthattheshockinteractedwithcanbewritten as
Vc =(1−ω)2
3π RA 3 (A4)
Vm =ω2π RA2 Rm (A5)
andthetotalvolumeofthecrushedshellthus
Vshell =χ(Vc +Vm ), (A6)
wherethe compressionratioisassumedto bethesameand χ = 30. The hemisphere B is completelyignored for the gamma-ray emissionasthecontributionisnegligibleandbecausenosignificant gamma-ray emission isdetectedfrom that region.Thecomputed total gamma-ray luminosity as well as luminosities of separate componentsareshowninthebottompanelofFig.2.Itcanbeseen thatthismodelcanroughlyexplaintheobservedluminosityfromIC 443,butitshouldbestressedagainthatthemodelwaspurposefully constructedtobetoooptimistic.
Fortheradioemission,weconsiderbothhemispheresasthelarger volumeofhemisphereByieldstoasignificantamountofemissionin faceofthecomparablefield-strengthinbothhemispheres.Inaddition tothecontributionfromtheintercloudmediumofbothhemispheres, withafield-strengthofB=10μG,weassumethatthefieldreaches thelimitsgiveninCrutcheretal.(2010)fortheclumpsofhemisphere Bandthetoroidalcloud(seeequation18).
TheradioemissionfromthehemispheresAandBaswellasfrom thecloudisshowninthebottompanelofFig.3.Asaconsequence of thecompressedfieldin theclumpsofhemisphereA,theradio emission from the clumps is comparableto the emission of the interclump mediumfromthathemisphere.However,thestrongest emissionhastobeexpectedfromhemisphereBin thatmodelon accountofthehighertotalnumberofelectronsinthatregion.
ThispaperhasbeentypesetfromaTEX/LATEXfilepreparedbytheauthor.
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