Could a plasma in quasi-thermal equilibrium be associated to the "orphan" TeV flares ?

6.4 MB·English

by  Nissim Fraija

Couldaplasmainquasi-thermalequilibriumbeassociatedto
the”orphan”TeVflares?
N. Fraija∗
Instituto de Astronom´ıa, Universidad Nacional Auto´noma de Me´xico,
Circuito Exterior, C.U., A. Postal 70-264, 04510 Me´xico D.F., Me´xico
TeVγ-raydetectionsinflaringstateswithoutactivityinX-raysfromblazarshaveattractedmuchattention
duetotheirregularityofthese”orphan”flares. Althoughthesynchrotronself-Comptonmodelhasbeenvery
successfulinexplainingthespectralenergydistributionandspectralvariabilityofthesesources,ithasnotbeen
abletodescribetheseatypicalflaringevents.Ontheotherhand,anelectron-positronpairplasmaatthebaseof
theAGNjetwasproposedasthemechanismofbulkaccelerationofrelativisticoutflows.Thisplasmainquasi-
themalequilibriumcalledWeinfireballemitsradiationatMeV-peakenergiesservingastargetofaccelerated
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protons. In this work we describe the ”orphan” TeV flares presented in blazars 1ES 1959+650 and Mrk421
1
assuminggeometricalconsiderationsinthejetandevokingtheinteractionsofFermi-acceleratedprotonsand
0
MeV-peak target photons coming from the Wein fireball. After describing successfully these ”orphan” TeV
2
flares,wecorrelatetheTeVγ-ray,neutrinoandUHECRfluxesthroughpγinteractionsandcalculatethenumber

r ofhigh-energyneutrinosandUHECRsexpectedinIceCube/AMANDAandTAexperiment, respectively. In
p
addition, thermalMeVneutrinosproducedmainlythroughelectron-positronannihilationattheWeinfireball
A
willbeabletopropagatethroughit. Byconsideringtwo-(solar,atmosphericandacceleratorparameters)and
three-neutrinomixing,westudytheresonantoscillationsandestimatetheneutrinoflavorratiosaswellasthe
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numberofthermalneutrinosexpectedonEarth.
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∗LucBinette-Fundacio´[email protected]
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I. INTRODUCTION
Flaresobservedinvery-high-energy(VHE)γ-rayswithabsenceofhighactivityinX-rays,areverydifficulttoreconcilewith
thestandardsynchrotronself-Compton(SSC)althoughithasbeenverysuccessfulinexplainingthespectralenergydistribution
(SED)ofblazars[1–3]. AlthoughmostoftheflaringactivitiesoccuralmostsimultaneouslywithTeVγ-rayandX-rayfluxes,
observationsof1ES1959+650[4–6]andMrk421[7,8]haveexhibitedVHEγ-rayflareswithouttheircounterpartsinX-rays,
called”orphan”flares.
Leptonic and hadronic models have been developed to explain orphan flares. A leptonic model based on geometrical consid-
erations about the jet has been explored to reconcile the SSC model [9] whereas hadronic models where accelerated protons
interactwithbothexternalphotonsgeneratedbyelectronsynchrotronradiation[10]andSSCphotonsatthelow-energytail[11]
havebeenperformedtoexplainthisanomalousbehaviorintheseblazars. BasedonthesemodelsHEneutrinoemissionhasbeen
studiedbyReimeretal.[12]andHalzenandHooper[13].Inparticularfortheblazar1ES1959+650,HalzenandHooper(2005)
based on proton-proton (pp) and proton-photon (pγ) interactions estimated the number of events expected in Antarctic Muon
AndNeutrinoDetectorArray(AMANDA).Theyfoundthattheneutrinorateswere1.8(10−3)eventsforpp(pγ)interactions.
Ithasbeenwidelysuggestedthatrelativisticjetsofactivegalacticnuclei(AGN)containelectron-positronpairsproducedfrom
accretiondisks[14–16]. Alsoelectron-positronpairplasmahasbeenproposedasamechanismofbulkaccelerationofrelativis-
ticoutflows. Ingammarayburst(GRB)jets,thisplasma”fireball”formedinsidetheinitialscale∼107 cmismadeofphotons,
asmallamountofbaryonsande± pairsinthermalequilibriumatsomeMeV[17]. However,inAGNjetsthefireballcannotbe
formedbecausethecharacteristicsizeistoolarge(3r ∼ 1014 cm)incomparisonwithGRBjets. Someauthorsfoundthatif
g
thepairplasmaisexpectedtobeopticallythintoabsorptionbutthicktoscattering,the”Weinfireball”couldexist,eventhough
forthesizeandluminosityofAGNs[18–20]. Afterward,simulationswithprotonsinsidethisplasmawereperformedbyAsano
andTakahara[21,22].
AttheinitialstageoftheWeinfireball,thermalneutrinoswillbemainlycreatedbyelectron-positronannihilation(e++e− →
Z → ν +ν¯ ). By considering a small amount of baryons, neutrinos could also be generated by processes of positron cap-
j j
ture on neutrons (e+ +n → p+ν¯), electron capture on protons (e− +p → n+ν ) and nucleon-nucleon bremsstrahlung
e e
(NN → NN +ν +ν¯ )forj = e,ν,τ. TakingintoaccountthatthetemperatureofWeinfireballisrelativistic[19,20],then
j j
neutrinosof1-5MeVcanbeproducedandfractionsofthemwillbeabletogothroughthisplasma. Asknown,theneutrino
properties are modified when they propagate through a thermal medium, and although neutrino cannot couple directly to the
magneticfield,itseffectcanbeexperimentedthroughcouplingtochargedparticlesinthemedium[23]. Theresonanceconver-
sionofneutrinofromoneflavortoanotherduetothemediumeffect,knownasMikheyev-Smirnov-Wolfensteineffect[24],has
beenwidelystudiedintheGRBfireball[25–27].
Telescope Array (TA) experiment reported the arrival of 72 ultra-high-energy cosmic rays (UHECRs) above 57 EeV with a
statisticalsignificanceof5.1σ. Theseeventscorrespondtotheperiodfrom2008May11to2013May4. Assumingtheerror
reportedbyTAexperimentinthereconstructeddirections,someUHECRsmightbeassociatedtothepositionofMrk421[28].
In addition, IceCube collaboration reported the detection of 37 extraterrestrial neutrinos at 4σ level above 30 TeV [29, 30],
althoughnoneofthemlocatedinthedirectionofneither1ES1959+650norMrk421,asshowninfig. 1.
BecauseTeVγ-ray”orphan”flaresareverydifficulttoreconcilewithSSCmodel,inthisworkweintroduceahadronicmodel
bymeansofpγinteractionstoexplaintheseatypicalTeVflaresregisteredinblazars1ES1959+650andMrk421. Inthismodel,
weconsidersomegeometricalassumptionsofthejetandtheinteractionsbetweentheMeV-peakphotonscomingfromtheWein
fireballandrelativisticprotonsacceleratedattheemittingregion. Then,wecorrelatetheTeVγ-ray,neutrinoandUHECRfluxes
tocalculatethenumberofHEneutrinoandUHECRevents. Inaddition, westudytheresonanceoscillationsofthermalMeV
neutrinos. The paper is arranged as follows. In Section 2 we show the dynamic model of the radiation coming from Wein
fireballanditsinteractionswiththeprotonsacceleratedattheemittingregion. Insection3westudytheemission,production
andoscillationofneutrinos. InSection4wediscussthemechanismsforacceleratingUHECRsandalsoestimatethenumberof
theseeventsexpectedintheTAexperiment,supposingthattheprotonspectrumisextendeduptoenergiesgreaterthan57EeV.
InSection5wedescribetheTeVorphanflaresoftheblazars1ES1959+650andMrk421andgiveadiscussiononourresults;a
briefsummaryisgiveninsection6. Wehereafteruseprimes(unprimes)todefinethequantitiesinacomoving(observer)frame,
naturalunits(c=(cid:126)=k=1)andredshiftsz(cid:39)0.
II. ORPHANTEVγ-RAYEMISSION
Different hadronic models have been considered to explain TeV γ-ray observations presented in blazars [31–34]. In those
models, SEDs are described in terms of co-accelerated electrons and protons at the emitting region. In this hadronic model,
we describe the TeV γ-ray emission through π0 decay products generated in the interactions of accelerated protons and seed
photonscomingfromtheWeinfireball,asshowninfig. 2.
3
II.1. MeVradiationfromtheWeinfireball
TheWeinfireballconnectsthebaseofthejetwiththeblackhole(BH).Weassumethatattheinitialstate,itisformedbye±
pairswithphotonsinsidetheinitialscaler =2r =4GM,beingGthegravitationalconstantandMtheBHmass. Theinitial
o g
temperaturecanbedefinedthroughmicroscopicprocessesatthebaseofthejet(Comptonscattering,γγ pairproduction,etc).
Atthefirststate,photonsinsidetheWeinfireballareatrelativistictemperature. Theinternalenergystartstobeconvertedinto
kinetic energy and the Wein fireball begins to expand. As a result of this expansion, temperature decreases and bulk Lorentz
factorincreasesatthefirststate. Theinitialopticaldepthis[19,20]
n σ r
τ (cid:39) e,o T o , (1)
o Γ
W,o
(cid:113)
whereσ =6.65×10−25cm2istheThompsoncrosssection,Γ =1/ 1−β2 istheinitialLorentzfactoroftheplasma
T W,o W,o
andn istheinitialelectrondensitywhichisgivenby
e,o
1 1 (cid:18)m (cid:19)(cid:18)r (cid:19)2(cid:18) L (cid:19)
n = p g j , (2)
e,o 4σ G M Γ2 β (cid:104)γ (cid:105) m r L
T N W,o W,o e,o e o Edd
whereL isthetotalluminosityofthejet,L =2πm r /σ istheEddingtonluminosity,(cid:104)γ (cid:105)=K (1/θ )/K (1/θ )−θ
j Edd p g T e,o 3 o 2 o o
is the average Lorentz factor of electron thermal velocity, θ = T /m is the initial temperature normalized to electron mass
o o e
(m ), m is the proton mass and K is the modified Bessel function of integral order. For a steady and spherical flow, the
e p i
conservationequationsofenergyandmomentumcanbewrittenas
1 d
[r2(ρ +P )Γ2 β ]=0, (3)
r2dr T T W W
1 d dP
[r2(ρ +P )Γ2 β2 ]+ T =0. (4)
r2dr T T W W dr
Hereρ isthetotalenergydensityofpairs(ρ =2m n (cid:104)γ (cid:105))andphotons(ρ =3m n θ)andP isthetotalpressureofpairs
T e e e e γ e γ T
(P = 2m n θ) and photons (P = m n θ). The number density of electrons and photons in the Wein equilibrium, and the
e e e γ e γ
numberconservationequationsaregivenby[35]
n K (1/θ)
e = 2 ≡f(θ), (5)
n 2θ2
γ
and
1 d
(r2n Γ β )=n˙ , (6)
r2dr e W W e
1 d
(r2n Γ β )=n˙ , (7)
r2dr γ W W γ
respectively. Takingintoaccountthemomentumconservationlaw 1dΓ+ 1 dPT =0,eqs. from(3)to(7)andfollowingto
Γdr (ρT+PT) dr
IwamotoandTakahara[19,20],itispossibletowritetheevolutionoftheLorentzfactor,thetemperatureandthetotalnumber
densityofphotonsandpairsas
r
Γ =Γ , (8)
W 0,Wr
0
r
θ =θ 0, (9)
0 r
and
(cid:16)r (cid:17)3
n +2n =3n 0 , (10)
γ e e,0 r
respectively. Eqs. (8),(9)and(10)representtheevolutionoftheWeinfireballintheopticalthickregime. AstheWeinfireball
expands,thephotondensity,opticalthicknessandtemperaturedecrease.Atacertainradius,flowbecomesopticallythin(τ =1),
thenphotonemissionwillberadiatedaway. Definingthisradiusasthephotosphereradius(r=r ),thenfromeqs. (1),(2),(5)
ph
and(10),thephotondensity,radius,temperatureandLorentzfactoratthephotospherecanbewrittenas
θ 3L
n = 0 γ , (11)
γ θ (1+2f(θ ))4πr2 (cid:15)Γ2 β m (cid:104)γ (cid:105)
ph ph ph W,ph W,ph e e,o
4
r (cid:39)τ1/3r , (12)
ph o o
θ =τ−1/3θ , (13)
ph o o
and
Γ =τ1/3Γ , (14)
W,ph o W,o
respectively. HereL =(cid:15)L istheluminosityradiatedatthephotosphereand(cid:15)isaparameter0≤(cid:15)≤1. Onecanseethatn ,
γ j γ
r , θ and Γ only depend on the initial conditions (θ , r , Γ and L /L ). The numerical results exhibit a radiation
ph ph ph o o W,o j Edd
centeredaround5MeV[19]. Hence,theoutputspectrumofsimulatedphotonscouldbedescribedby
(cid:26)
dnγ((cid:15)γ) ∝ ((cid:15)γ)−βl, if (cid:15)γ <(cid:15)γb, (15)
d(cid:15)γ ((cid:15)γb)−βl+βh((cid:15)γ)−βh, if (cid:15)γ ≥(cid:15)γb,
whereβ (cid:39) 2,β (cid:39) 1[36]andthepeakenergyaround(cid:15) (cid:39)5MeV.Alsowecandefinetheopticalthicknesstopaircreation
h l γb
aroundthepeakas[37]
σ (cid:15) r
τ (cid:39) T γb ph n , (16)
γγ 4m Γ γ
e W,ph
wherewehavetakenintoaccountthatthecrosssectionofpairproductionreachesamaximumvalueclosetotheThomsoncross
section. Additionally,itisveryimportanttosaythatintheopticallythickregimen,theangulardistributionofMeVphotonsis
almostisotropic,whereasintheopticallythinregimeithasaskewdistribution. Specificallyatadistancer (cid:29) r ,theoutgoing
g
photonsaredistributedintherange0◦ ≤φ ≤60◦[19].
ph
II.2. Geometricalconsiderationsandassumptions
We consider a spherical emitting region with a uniform particle density and radius r , located at a distance R from the BH
j
and moving at relativistic speed with bulk Lorentz factor Γ , as shown in fig. 2 (above) [38, 39]. Seed photons coming from
j
thephotosphereofthe’Wein’fireballwillinteractwithFermi-acceleratedelectronsandprotonsinjectedintheemittingregion.
Taking into account R (cid:29) r , these photons (with an angular distribution; Iwamoto and Takahara [19]) will arrive and go
ph
throughtheemittingregionfollowingdifferentpathswithanangle(φ );longerpathsaroundthecenterandshorteronesasthe
ph
trajectories get farther away from the center. To estimate the distances of any path, we find the common points of the circle,
(y−y(cid:48))2+(x−x(cid:48))2 = r2,andthestraightline,y−y = m(x−x ),throughtheirintersections(pointsaandb)(seefig.
0 0 j 0 0
2(below)). Asshowninfig. 2(below),weassumey(cid:48) = R−r ,x(cid:48) = 0 = x = y = 0andm = π/2−φ . Solvingthis
0 ph 0 0 0 ph
equationsystem,weobtainthatthedistancebetweentwopoints(aandb)asafunctionofφ canbewrittenas
ph
(cid:115) (cid:2) (cid:3)
4(R−r )2−4(1+tanφ2 ) (R−r )2−r2
d= ph ph ph j . (17)
1+tanφ2
ph
Photonscomingfromthe’Wein’fireballwillbeabletogoornotthroughtheemittingregiondependingontheirpaths(distances)
andthemeanfreepathλ =1/(σ n )insideofit. Forinstance,photosphericphotonsgoingthroughtheemittingregionwill
γ,e T e
be absorbed in it if the paths are longer than the mean free path (d > λ ); otherwise, (d < λ ), they will be transmitted.
γ,e γ,e
Definingtheopticaldepthasafunctionofthesetwoquantities(dandλ )
γ,e
d
τ = (18)
λ
γ,e
(cid:115)(R−r )2−(1+tanφ2 ) (cid:2)(R−r )2−r2(cid:3)
=2σ n ph ph ph j ,
T e 1+tanφ2
ph
itispossibletorelatethisanglewiththeelectronparticledensity. Takingintoaccountthatamediumissaidtobeopticallythick
or opaque when τ > 1 and optically thin or transparent when τ < 1, we write the electron particle density for this transition
(τ =1)as
(cid:115)
1 1+tanφ2
ne = 2σT (R−rph)2−(1+tanφ2ph)ph(cid:2)(R−rph)2−rj2(cid:3). (19)
Thepreviousequationgivesthevaluesofelectrondensityforwhichphotosphericphotonsgoingthroughtheemittingregion
withaparticularanglecouldbetransmittedorabsorbed.
5
II.3. Pγinteractions
Onceemitted,theMeV-peakphotonsalsointeractwithprotonsacceleratedattheemittingregion.Assumingabaryoncontent
inthisregion[32,40–43],acceleratedprotonslosetheirenergiesbyelectromagneticandhadronicinteractions. Electromagnetic
interactionsuchasprotonsynchrotronradiationandinverseComptonwillnotbeconsideredhere,weonlyassumethatprotons
willbecooleddownbypγ interactions. Theopticaldepthofthisprocessis
r n σ
τ (cid:39) j γ pγ, (20)
pγ Γ
j
where the photon density (n ) is given by eq. (11) and σ is the cross section for pγ interactions. The photopion process
γ pγ
pγ →Nπhasathresholdphotonenergy(cid:15) =m +m2/2m wheretheneutralandchargedpionmassarem =135MeV
th π π p π0
andm =139.6MeV,respectively[44]. Thetwomaincontributionstothetotalphotopioncrosssectionatlowenergiescome
π0
fromtheresonanceproductionanddirectproduction.
ResonanceProduction(∆+). Thephotopionproductionisgiventhroughtheresonances∆+(1.232),∆+(1.700),∆+(1.905)
and∆+(1.950)wherethemassm andLorentzianwidthΓ forallresonancesarereportedbyMu¨ckeetal.[45]. Theπ0
∆+ ∆+
thresholdis(cid:39)145MeV,andonlythepγ −→∆+ −→pπ0iscinematicallyallowed.
Direct Production. This channel exhibits the non resonant contribution to direct two-body channels consisting of outgoing
chargedpions. Thischannelincludesthereactionpγ →nπ+,pγ →∆++π−,andpγ →∆π0. Fortheπ+ threshold,thedirect
pionchannelpγ →nπ+isdominantforanenergybetween0.150GeVand0.25GeV.
Othercontributionstothetotalphotopioncrosssection(multipionproductionanddiffraction)areonlyimportantathighener-
gies.
II.3.1. π0decayproducts
Neutral pion decays into two photons, π0 → γγ, and each photon carries less than 15% of the proton’s energy (cid:15) (cid:39)
γ
ξ /2E =0.15E . Theπ0coolingtimescaleforthisprocessis[46,47]
π0 p p
1 (cid:90) (cid:90) dn
t(cid:48)−1 = d(cid:15)σ ((cid:15))ξ (cid:15) dxx−2 γ((cid:15) =x), (21)
π0 2Γ2 π π0 d(cid:15) γ
p γ
where Γ is the proton Lorentz factor, σ ((cid:15) ) (cid:39) σ is the cross section of pion production. The target photon spectrum
p π γ pγ
dn /d(cid:15) isgivenbyeq. (15)whichisnormalizedthrough(cid:82)∞ (cid:15) (dn /d(cid:15) )d(cid:15) =U (cid:39)(cid:15) n /2Γ2 . Thethresholdforpγ
γ γ 0 γ γ γ γ γ γb γ W,ph
interactioniscomputedthroughtheminimumprotonenergy,whichcanbewrittenas
(m2 +2m m )
E =Γ Γ π0 p π0
p,min j W,ph 2(cid:15) (1−cosφ)
γb
Γ Γ (cid:15)−1
(cid:39)0.14TeV j W,ph γb,MeV , (22)
(1−cosφ)
whereφistheangleofthisinteraction. Thephotopionproductionefficiencyfπ0 = tt(cid:48)(cid:48)d canbedefinedthroughthedynamical
π0
(t(cid:48) (cid:39) r /Γ )andphotopiontimescales[48,49]
d j j
fπ0 = tt(cid:48)π(cid:48)d0 (cid:39) rp24hΓnjγΓσWpγ,pξhπr0j∆(cid:15)(cid:48)p(cid:15)e(cid:48)paekak ×(cid:40)(cid:16)1(cid:15)π0(cid:15),0γ,c(cid:17)−1(cid:16)(cid:15)π(cid:15)00,γ(cid:17) (cid:15)(cid:15)ππ00,,γγ,c<<(cid:15)π(cid:15)0π,0γ,,γc,
wherethebreakphotopionenergyisgivenby
(cid:15) (cid:39)0.25Γ Γ ξ (m2 −m2)(cid:15) −1, (23)
π0,γ,c j W,ph π0 ∆+ p γb
(cid:15)(cid:48) and∆(cid:15)(cid:48) correspondtotheenergyandthewidtharoundtheresonanceoftotalcrosssection,respectively. Takinginto
peak peak
accounttheprotonspectrumasasimplepowerlaw
(cid:18)dN(cid:19) (cid:18) E (cid:19)−α
=A p , (24)
dE p GeV
p
thephotopionproductionefficiency(f ),andf E (dN/dE) dE = (cid:15) (dN/d(cid:15)) d(cid:15) ,thenwecanwritethepho-
π0 π0 p p p π0,γ π0,γ π0,γ
topionspectrumas
(cid:18)(cid:15)2 dN(cid:19) =A ×(cid:16)(cid:15)π0(cid:15),0γ,c(cid:17)−1(cid:16)(cid:15)π(cid:15)00,γ(cid:17)−α+3 (cid:15)π0,γ <(cid:15)π0,γ,c
d(cid:15) π0,γ p,γ (cid:16)(cid:15)π(cid:15)00,γ(cid:17)−α+2 (cid:15)π0,γ,c <(cid:15)π0,γ,
6
withtheproportionalityconstantgivenby
(cid:16) (cid:17)1−α
3r2 n (cid:15)2σ 2 ∆(cid:15)(cid:48)
A (cid:39) ph γ 0 pγ ξπ0 peak A . (25)
p,γ 4Γ Γ r (cid:15)(cid:48) p
j W,ph j peak
Fromeqs. (24)and(25),theprotonluminosityL (cid:39)4πD2F =4πD2E2(cid:0)dN(cid:1) canbewrittenas
p z p z p dE p
(cid:16) (cid:17)α−1
L (cid:39) 8πΓjΓW,phrjDz2 ξπ20 (cid:15)(cid:48)peak A (cid:18) Ep (cid:19)2−α, (26)
p (α−2)r2 n σ ∆(cid:15)(cid:48) p,γ GeV
ph γ pγ peak
whereA isobtainedthroughthephotopionspectrum(eq. 25).
p,γ
II.3.2. π±decayproducts
Muons and positrons/electrons are produced through charged pion decay products (π± → µ± +ν /ν¯ → e± +ν /ν¯ +
µ µ µ µ
E(cid:48)
ν¯ /ν +ν /ν¯ ). Althoughmuon’slifetimet(cid:48) = µ+ τ isveryshortwithτ =2.2µsandm = 105.7MeV,muons
µ µ e e µ+,dec mµ µ+ µ+ µ
couldberapidlyacceleratedforashortperiodoftimeinthepresenceofamagneticfield(B(cid:48))andradiatephotonsbysynchrotron
emission(cid:15)(cid:48) = 3πqeB(cid:48) E(cid:48)2 [31,34,40,50]. Aftermuonsdecay,positrons/electronscouldradiatephotons(cid:15)(cid:48) = 3πqeB(cid:48) E(cid:48)2 at
γ 8m3 µ γ 8m3 e
µ e
thesameplace. Therefore, muonsandpositrons/electronscooldowninaccordancewiththecoolingtimescalecharacteristic,
t(cid:48) = 6πm4/(σ m2B(cid:48)2E(cid:48))uptoamaximumaccelerationtimescalegivenbyt(cid:48) = 16E(cid:48)/(3q B(cid:48)). Hereq isthe
syn,i i T e i syn,max i e e
elementarychargeandthesubindexiisforµande. Hence,thebreakandmaximumphotonenergiesintheobservedframeare
m5
(cid:15) = i(cid:15) ,
γ,c m5 γ,c−e
me
(cid:15) = i(cid:15) , (27)
γ,max m γ,max−e
e
where we have taken into account the Lorentz factor ratios γ = m2/m2γ . Assuming that Fermi-accelerated muons and
i i e e
electrons/positrons within a volume (cid:39) 4πr3/3 with energies γ m and maximum radiation powers P (cid:39) dE/dt are
j i i ν,max,i (cid:15)γ(γi)
well described by broken power laws N (γ ): γ−α for γ < γ and γ γ−(α+1) for γ ≤ γ < γ , then the observed
i i i i i,b i,b i i,b i i,max
synchrotronspectrumcanbewrittenas[51,52]
(cid:16) (cid:17)−1/2(cid:16) (cid:17)−(α−3)/2
(cid:18)(cid:15)2 dN(cid:19) =A × (cid:15)(cid:15)γ0,c (cid:15)(cid:15)γ0 (cid:15)γ <(cid:15)γ,c,
d(cid:15) syn,γ syn,γ−i (cid:16)(cid:15)(cid:15)γ0(cid:17)−(α−2)/2 (cid:15)γ,c <(cid:15)γ <(cid:15)γ,max,
with
4σ m4
A = T i r3D−2Γ (cid:15) B(cid:48)N . (28)
syn,γ−i 27π2q m3 j z j γ,c−e i
e e
HereN isthenumberofradiatingmuonsand/orpositrons/electrons.
i
III. NEUTRINOEMISSION
Althoughinthecurrentmodeltherearemultipleplaceswhereneutrinoswithdifferentenergiescouldbegenerated,onlywe
aregoingtoconsiderthethermalneutrinoscreatedattheinitialstageoftheWeinfireballandtheHEneutrinosgeneratedbypγ
interactionsattheemittingregion(seefig. 2(above)).
III.1. ThermalNeutrinos
At the initial stage of the Wein fireball, thermal neutrinos are created by electron-positron annihilation (e+ +e− → Z →
ν +ν¯ ). Byconsideringasmallamountofbaryons,neutrinoscouldbegeneratedbyprocessesofpositroncaptureonneutrons
j j
(e++n→p+ν¯),electroncaptureonprotons(e−+p→n+ν )andnucleon-nucleonbremsstrahlung(NN →NN+ν +ν¯ )
e e j j
7
forj =e,ν,τ. Electronantineutrinoscanbedetectedindirectly(inwaterCherenkovdetector)throughtheinteractionswiththe
positronscreatedbytheinverseneutrondecayprocesses(ν¯ +p→n+e+). Theexpectedeventratecanbeestimatedby
e
(cid:90) (cid:18)dN(cid:19)
N =T ρ N V σν¯ep(E ) dE , (29)
ev N A w cc ν dE ν
Eν¯e ν
where N = 6.022 × 1023 g−1 is the Avogadro’s number, ρ = 2/18gcm−3 is the nucleons density in water [53], V
A N w
is the volume of the detector, σν¯ep (cid:39) 9 × 10−44E2 /MeV2cm2 is the cross section [54, 55], T is the observation time
cc ν¯e
and (dN/dE) is the neutrino spectrum. Taking into account the relation between the luminosity L and neutrino flux F ,
ν ν ν
L =4πD2F (<E >)=4πD2(E2dN/dE) ,thenthenumberofeventsexpectedwillbe
ν z ν z ν
(cid:18) (cid:19)
T dN
N (cid:39) V N ρ σν¯ep <E >2
ev <E > w A N cc ν¯e dE
ν¯e ν¯e
T
(cid:39) V N ρ σν¯epL . (30)
4πD2 <E > w A N cc ν¯e
z ν¯e
Here we have averaged over the electron antineutrino energy. After thermal neutrinos are produced, they oscillate firstly in
matter(duetoWeinplasma)andsecondlyinvacuumontheirpathtoEarth.
III.1.1. Neutrinoeffectivepotential
As known, neutrino properties get modified when they propagate in a heat bath. A massless neutrino acquires an effective
mass and undergoes an effective potential in the background. Because electron neutrino (ν ) interacts with electrons via both
e
neutralandchargedcurrents(CC),andmuon/tau(ν /ν )neutrinosinteractonlyviatheneutralcurrent(NC),ν experimentsa
µ τ e
differenteffectivepotentialincomparison with ν and ν . Thiswould induceacoherenteffectinwhichmaximalconversion
µ τ
ofν intoν (ν )takesplaceevenforasmallintrinsicmixingangle[24]. Ontheotherhand,althoughneutrinocannotcouple
e µ τ
directlytothemagneticfield, itseffectwhichisentangledwiththemattercanbeundergonebymeansofcouplingtocharged
particlesinthemedium. Recently,Fraija[27]derivedtheneutrinoself-energyandeffectivepotentialuptoorderm−4atstrong,
W
moderateandweakmagneticfieldapproximationasafunctionoftemperature,chemicalpotential(µ)andneutrinoenergy(E )
ν
for moving neutrinos along the magnetic field. In this approach, we will use the neutrino effective potential in the weak field
approximation,whichisgivenby
V = √2GF m3e(cid:34)(cid:88)∞ (−1)lsinhα(cid:40)(cid:18)2+ m2e (cid:18)3+4 Eν2 (cid:19)(cid:19)×(cid:18)K0(σl) +2K1(σl)(cid:19)−2(cid:18)1+ m2e (cid:19) B K (σ)(cid:41)
eff π2 l m2 m2 σ σ2 m2 B 1 l
l=0 W W l l W c
−4 m2e Eν (cid:88)∞ (−1)lcoshα(cid:40)(cid:18) 2 − B (cid:19)K (σ)+(cid:18)1+ 4 (cid:19)K1(σl)(cid:41)(cid:35).
m2 m l σ2 4B 0 l σ2 σ
W e l=0 l c l l
(31)
Takingintoaccounttheconditionthattheplasmahasequalnumberofelectronsandpositrons(N −N¯ =0),thentheneutrino
e e
effectivepotentialisreducedto
V =−4√2GF m4eEν (cid:88)∞ (−1)l(cid:40)(cid:18) 2 − B (cid:19)K (σ)+(cid:18)1+ 4 (cid:19)K1(σl)(cid:41),
eff π2m2 σ2 4B 0 l σ2 σ
W l=0 l c l l
(32)
√
where K is once again the modified Bessel function of integral order i, G = 2g2/8m2 is the Fermi coupling constant,
i F W
B =m2/eisthecriticalmagneticfield,m istheW-bosonmass,α =(l+1)µ/(θ m )andσ =(l+1)/θ .
c e W l o e l o
III.1.2. Resonantoscillations
Whenneutrinooscillationsoccurinmatter,aresonancecouldtakeplacethatwoulddramaticallyenhancetheflavormixing
and lead to a maximal conversion from one neutrino flavor to another. This resonance depends on the effective potential and
neutrino oscillation parameters. The equations that determine the neutrino evolution in matter for two and three flavors are
relatedasfollows[27,56].
8
III.1.2.1. Two-NeutrinoMixing. Theevolutionequationforneutrinosthatpropagateinthemedium(ν ↔ ν )isgiven
e µ,τ
by[57]
i(cid:18)ν˙e(cid:19)=(cid:32)Veff − δ2mEν2 cos2ψ δ4mEν2 sin2ψ(cid:33)(cid:18)νe(cid:19), (33)
ν˙µ δm2 sin2ψ 0 νµ
4Eν
whereδm2 isthemassdifference,ψ istheneutrinomixingangleandV istheneutrinoeffectivepotential(eqs. 31and32).
eff
Theoscillationlengthfortheneutrinoisgivenby
4πE
l = ν , (34)
osc (cid:113)
δm2 cos22ψ(1− 2EνVeff )2+sin22ψ
δm2cos2ψ
andtheconversionprobabilityby
δm4sin22ψ (cid:18)ωt(cid:19)
P (t)= sin2 , (35)
νe→νµ(ντ) 4ω2E2 2
ν
(cid:112)
withω= (V −(δm2/2E )cos2ψ)2+(δm4/4E2)sin22ψ. Takingintoaccounttheresonancecondition
eff ν ν
δm2
V =5×10−7eV eV cos2ψ, (36)
eff E
ν,MeV
thentheresonancelength(l )canbewrittenas
res
4πE
l = ν . (37)
res δm2sin2ψ
Thebestfitvaluesofthetwoneutrinomixingare:SolarNeutrinos:δm2 =(5.6+1.9)×10−5eV2andtan2ψ =0.427+0.033[58],
−1.4 −0.029
Atmospheric Neutrinos: δm2 = (2.1+0.9)×10−3eV2 and sin22ψ = 1.0+0.00 [59] and Accelerator Neutrinos: δm2 ≈
−0.4 −0.07
0.5eV2andsin22ψ ∼0.0049[60,61].
III.1.2.2. Three-NeutrinoMixing. Inthethree-flavorframework,theevolutionequationoftheneutrinosysteminthematter
canbewrittenas
d(cid:126)ν
i =H(cid:126)ν, (38)
dt
where the state vector is (cid:126)ν ≡ (ν ,ν ,ν )T, the effective Hamiltonian is H = U ·Hd ·U† +diag(V ,0,0) with Hd =
e µ τ 0 eff 0
1 diag(−δm2 ,0,δm2 ), the neutrino effective potential V is defined by eqs. (31) and (32) and U is the three neutrino
2Eν 21 32 eff
mixingmatrix[62–65]. Theoscillationlengthfortheneutrinoisgivenby
4πE /δm2
l = ν 32 . (39)
osc (cid:113)
cos22ψ13(1− δm2322EcνoVse2ψ13)2+sin22ψ13
Theresonanceconditionandresonancelengthare
δm2
V −5×10−7 32,eV cos2ψ =0, (40)
eff E 13
ν,MeV
and
4πE /δm2
l = ν 32, (41)
res sin2ψ
13
respectively.Inadditiontotheresonancecondition,thedynamicsofthistransitionfromoneflavortoanothermustbedetermined
byadiabaticconversionwhichisgivenby
κres ≡ π2 (cid:18)δ2mE232 sin2ψ13(cid:19)2(cid:12)(cid:12)(cid:12)(cid:12)dVderff(cid:12)(cid:12)(cid:12)(cid:12)−1 ≥1.
ν
π2 (cid:18)δ2mE232 sin2ψ13(cid:19)2 (cid:18)θoθ2ro(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)∂V∂eθff(cid:12)(cid:12)(cid:12)(cid:12)−1 ≥1.
ν
(42)
Combining solar, atmospheric, reactor and accelerator parameters, the best fit values of the three neutrino mixing are,
for sin2 <0.053:∆m2 =(7.41+0.21)×10−5eV2andtan2ψ =0.446+0.030and,forsin2 <0.04:∆m2 =(2.1+0.5)×10−3eV2
13 21 −0.19 12 −0.029 13 23 −0.2
andsin2ψ =0.50+0.083,[58,66].
23 −0.093
9
III.1.2.3. NeutrinoflavorratioexpectedonEarth. Theprobabilityforaneutrinotooscillatefromaflavorstateαtoanother
flavorstateβ initspath(fromthesourcetoEarth)is
(cid:32) (cid:33)
P =δ −4(cid:88) U U U U sin2 δm2ijL , (43)
να→νβ αβ αi βi αj βi 4E
ν
j>i
whereU aretheelementsofthethree-neutrinomixingmatrix[62–65]. Usingthesetofoscillationparameters[58,66], the
ij
mixingmatrixcanbewrittenas
(cid:32) 0.817 0.545 0.191(cid:33)
U = −0.505 0.513 0.694 . (44)
0.280 −0.663 0.694
Additionally, averagingthetermsin∼ 0.5[67]fordistanceslongerthanthesolarsystem, theneutrinoflavorvectoratsource
(ν ,ν ,ν ) andEarth(ν ,ν ,ν ) arerelatedthroughtheprobabilitymatrixgivenby
e µ τ source e µ τ Earth
(cid:32)ν (cid:33) (cid:32)0.534 0.266 0.200(cid:33)(cid:32)ν (cid:33)
e e
ν = 0.266 0.367 0.368 ν . (45)
µ µ
ν 0.200 0.368 0.432 ν
τ Earth τ source
III.2. High-energyneutrinos
HE neutrinos are created in pγ interactions through nπ+ channel. The charged pion decays into leptons and neutrinos,
π± → e± +ν /ν¯ +ν¯ /ν +ν /ν¯ . AssumingthatTeVflarescanbedescribedasπ0 decayproducts, wecanestimatethe
µ µ µ µ e e
numberofneutrinosassociatedtotheseflares[68–70]. Forpγ interactions,theneutrinoflux,dNν/dEν = AνEν−αν,isrelated
withthephotopionfluxby[see,e.g.13,71,andreferencetherein]
(cid:90) (cid:18)dN(cid:19) 1(cid:90) (cid:18)dN(cid:19)
E dE = (cid:15) d(cid:15) . (46)
dE ν ν 4 d(cid:15) π0,γ π0,γ
ν π0,γ
Assuming that the spectral indices of neutrino and photopion spectra are similar α (cid:39) α [72], taking into account that each
ν
neutrinocarries∼5%oftheprotonenergy(E (cid:39)1/20E )[73]andalsofromeq. (25),wecanwritethenormalizationfactors
ν p
ofHEneutrinoandphotopionas
1
A (cid:39) A (10ξ )−α+2 TeV−2, (47)
ν 4 p,γ π0
withA givenbyEq. (25). Therefore,wecouldinferthenumberofeventsexpectedthrough
p,γ
(cid:90) ∞ (cid:18)dN(cid:19)
N ≈Tρ N V σ (E ) dE , (48)
ev ice A i νN ν dE ν
Eth ν
whereE isthethresholdenergy,T correspondstotheobservationtimeoftheflare[13],σ (E )=6.78×10−35(E /TeV)0.363
th νN ν ν
cm2isthechargedcurrentcrosssection[74],ρ =0.9gcm−3isthedensityoftheiceandV istheeffectivevolumeofdetector,
ice i
thentheexpectednumberofneutrinosinferredfromthisflareis
Tρ N V (cid:18)E (cid:19)β
N ≈ ice A i A (6.78×10−35cm2) ν,th TeV, (49)
ev α−1.363 ν TeV
withthepowerindexβ =−α+1.363andA givenbyeq.(47).
ν
IV. ULTRA-HIGH-ENERGYCOSMICRAYS
IthasbeensuggestedthatTeVγ-rayobservationsfromlow-redshiftsourcescouldbegoodcandidatesforstudyingUHECRs[75]. Also
special features in these γ-ray observations coming from blazars favor acceleration of UHECRs in blazars [76]. In the current model we
considerthattheprotonspectrumisextendedupto∼ 1020 eVenergiesandbasedonthisassumption, wecalculatethenumberofevents
expectedinTAexperiment.
10
IV.1. HillasCondition
By considering that the BH has the power to accelerate particles up to UHEs by means of Fermi processes, protons accelerated in the
emittingregionarelimitedbytheHillascondition[77]. AlthoughthisrequirementisanecessaryconditionandaccelerationofUHECRsin
AGNjets[75,76,78],itisfarfromtrivial(seee.g.,Lemoine&Waxman2009foramoredetailedenergeticslimit[79]). TheHillascriterion
saysthatthemaximumprotonenergyachievedis
E =eB(cid:48)r Γ , (50)
p,max j j
whereB(cid:48)isthestrengthofthemagneticfield.Alternatively,duringflaringintervalsforwhichtheapparentluminositycanachieveL∗ ≈1047
ergs−1andfromtheequipartitionmagneticfield(cid:15) ,themaximumenergyofUHECRscanbederivedandwrittenas[43,80]
B
(cid:112)
E ≈1.0×1021 e (cid:15)BL∗/1047ergs−1 eV, (51)
max Φ Γ
j
whereΦ(cid:39)1istheaccelerationefficiencyfactor.
IV.2. Deflections
ThemagneticfieldsintheUniverseplayimportantrolesbecauseUHECRsaredeflectedbythem.UHECRstravelingfromsourcetoEarth
arerandomlydeviatedbygalactic(B )andextragalactic(B )magneticfields.Byconsideringaquasi-constantandhomogeneousmagnetic
G EG
fields,thedeflectionangleduetotheB is
G
ψ (cid:39)3.8◦(cid:18) Ep,th (cid:19)−1(cid:90) LG| dl × BG |, (52)
G 57EeV kpc 4µG
0
andduetoB canbewrittenas[81]
EG
(cid:18) E (cid:19)−1(cid:18)B (cid:19) (cid:18) L (cid:19)1/2 (cid:18) l (cid:19)1/2
ψ (cid:39)4◦ p,th EG EG c , (53)
EG 57EeV 1nG 100Mpc 1Mpc
whereL correspondstothedistanceofourGalaxy(20kpc)andl isthecoherencelength. Duetothestrengthofextragalactic(B (cid:39)1
G c EG
nG)andgalactic(B (cid:39)4µG)magneticfields,UHECRsaredeflected(ψ (cid:39)4◦andψ (cid:39)3.8◦)betweenthetruedirectiontothesource,
G EG G
andtheobservedarrivaldirection,respectively.EvaluationofthesedeflectionangleslinksthetransientUHECRsourceswiththehigh-energy
neutrinoandγ-rayemission.Regardingeqs.(52)and(53),itisreasonabletoassociateUHECRslyingwithin5◦ofasource.
IV.3. Expectednumberofevents
TA experiment located in Millard Country, Utah, is made of a scintillator surface detector (SD) array and three fluorescence detector
(FD)stations[82]. Withanareaof∼700m2, itwasdesignedtostudyUHECRswithenergiesabove57EeV.Toestimatethenumberof
UHECRsassociatedto”orphan”flares, wetakeintoaccounttheTAexposure, whichforapointsourceisgivenbyΞt ω(δ )/Ω, where
op s
Ξt =(5)7×102km2yr,t isthetotaloperationaltime(from2008May11and2013May4),ω(δ )isanexposurecorrectionfactorfor
op op s
thedeclinationofthesource[83]andΩ(cid:39)π.TheexpectednumberofUHECRsaboveanenergyE yields
p,th
N =F (TAExpos.)× N , (54)
UHECR r p
whereF isthefractionofpropagatingcosmicraysthatsurvivesoveradistance>D [84]andN iscalculatedfromtheprotonspectrum
r z p
extendeduptoenergieshigherthanE (eq.24).Theexpectednumbercanbewrittenas
p,th
Ξt ω(δ )(α−2)
N =F op s L , (55)
UHECR r 4π(α−1)Ωd2E p
z p,th
whereL isobtainedfromtheTeVγ-rayobservationsoftheflaringactivities(eq.26).
p
V. APPLICATIONTO1ES1959+650ANDMRK421
FollowingIwamotoandTakahara[19,20], wehaveconsideredthedynamicsofaplasmamadeofe± pairsandphotons, andgenerated
bytheWeinequilibrium[35]. WehaveobtainedtheevolutionequationsofthebulkLorentzfactor(eq. 8),temperature(eq. 9)anddensity
ofphotonsandpairs(eq. 10). Inparticular, wehavecomputedtheradius(eq. 12), temperature(eq. 13), Lorentzfactor(eq. 14)andthe
photondensity(eq. 11)atthephotosphereasafunctionoftheinitialconditions(r ,Γ ,θ andL /L ). Byconsideringthevaluesof
o W,o o j Edd
initialradiusr =2r andLorentzfactorΓ =(3/2)1/2,weplot(seefig. 3)theinitialopticaldepth(left-handfigureabove)andphoton
o g W,o
density(right-handfigureabove)asafunctionofL /L andL ,respectively,andtheLorentzfactor(left-handfigurebelow)andradius
j Edd γ