Neutrinos from Choked Jets Accompanied by Type-II Supernovae
Hao-Ning He, Alexander Kusenko, Shigehiro Nagataki, Yi-Zhong Fan, Da-Ming Wei
aa r X i v : . [ a s t r o - ph . H E ] M a r Draft version March 21, 2018
Preprint typeset using L A TEX style emulateapj v. 12/16/11
NEUTRINOS FROM CHOKED JETS ACCOMPANIED BY TYPE-II SUPERNOVAE
Hao-Ning He , Alexander Kusenko , Shigehiro Nagataki , Yi-Zhong Fan , Da-Ming Wei Draft version March 21, 2018
ABSTRACTThe origin of the IceCube neutrinos is still an open question. Upper limits from diffuse gamma-ray observations suggest that the neutrino sources are either distant or hidden from gamma-rayobservations. It is possible that the neutrinos are produced in jets that are formed in the core-collapsing massive stars and fail to break out, the so-called choked jets. We study neutrinos from thejets choked in the hydrogen envelopes of red supergiant stars. Fast photo-meson cooling softens theneutrino spectrum, making it difficult to explain the PeV neutrinos observed by IceCube in a one-component scenario, but a two-component model can explain the spectrum. Furthermore, we predictthat a newly born jet-driven type-II supernova may be observed to be associated with a neutrinoburst detected by IceCube.
Keywords: neutrinos – stars: jets – stars: massive – supergiants – supernovae: general INTRODUCTION
The origin of high-energy neutrinos observed bythe IceCube observatory is still under debate. Thedistribution of the observed neutrino events is con-sistent with being isotropic, suggesting extragalacticsources (Aartsen et al. 2014). Active galactic nu-clei (AGNs; Stecker et al. 1991; Essey et al. 2010,2011; Kalashev et al. 2013; Padovani & Resconi 2014;Kimura et al. 2015; Murase et al. 2016), gamma-raybursts (GRBs; Waxman & Bahcall 1997; Murase & Ioka2013; Liu & Wang 2013; Cholis & Hooper 2013;Murase et al. 2013), starburst/star-forming galax-ies (Loeb & Waxman 2006; He et al. 2013a;Tamborra et al. 2014; Chang & Wang 2014; Liu et al.2014), supernova remnants (Mandelartz & Becker Tjus2015; Xiao et al. 2016; Zirakashvili & Ptuskin 2016) andyoung pulsars (Murase et al. 2009; Fang et al. 2014) areamong possible candidates.The flavor composition of the neutrino flux is consis-tent with the standard scenarios where neutrinos areproduced via pp collision or photo-meson interaction,but disfavor a neutron-decay scenario at 3.6 σ signifi-cance (Aartsen et al. 2015a).In addition to the spatial distribution of neutrinoevents and the flavor composition, the features of theobserved neutrino spectrum are also crucial for iden-tifying the sources. The IceCube Collaboration hasreported a neutrino spectrum that can be best fitby a soft unbroken power-law spectrum with indexof − . ± .
09 for neutrinos with energies above 10TeV. A single power-law neutrino spectrum with in- Key Laboratory of Dark Matter and Space Astronomy,Purple Mountain Observatory, Chinese Academy of Sciences,Nanjing 210008, China Astrophysical Big Bang Laboratory, RIKEN, Wako,Saitama, Japan Department of Physics and Astronomy, University ofCalifornia, Los Angeles, CA 90095-1547, USA Kavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba277-8568, Japan Interdisciplinary Theoretical Science Research Group(iTHES), RIKEN, Saitama 351-0198, Japan Interdisciplinary Theoretical & Mathematical ScienceProgram (iTHEMS), RIKEN, Saitama 351-0198, Japan dex − . σ (Aartsen et al.2015a). An even softer spectrum with index of − . +0 . − . from six years of High Energy Starting Event(HESE) IceCube data at energies above 40 TeV wasreported (IceCube Collaboration et al. 2017). However,cosmic muon neutrinos with energies >
194 TeV fromthe northern hemisphere from six years of IceCube datahave a hard spectral index of − . ± .
13 (Aartsen et al.2016). The difference between the best-fit spectral in-dices in different energy bands can be explained by atwo-component model(He et al. 2013b): a soft compo-nent explaining the lower-energy neutrinos, and a hardcomponent explaining the higher-energy neutrinos. Thismodel can be tested with improved statistics.The flux of the observed all-flavor neu-trinos around 30 TeV is as high as10 − GeVcm − s − sr − (The IceCube Collaboration et al.2015). Meanwhile, the Fermi
Large Area Telescope(LAT) collaboration reported that blazars’ contribu-tions dominate the diffuse extragalactic gamma-raybackground (EGB) above 50 GeV (Ackermann et al.2016), while blazars only contribute to less than 10% ofthe neutrinos observed by the IceCube. This leads tothe puzzling fact that diffuse gamma-ray emission fromneutrino sources is much weaker than one would expectbased on IceCube neutrino observations (Murase et al.2013, 2016). The tension can be avoided if the neutrinosources are distant (Chang et al. 2016), if the neutrinoproduction occurs in pγ interactions along the line ofsight (Essey et al. 2010, 2011; Kalashev et al. 2013),or if the gamma-ray emission in the GeV–TeV band issuppressed, so that the neutrino sources are hidden fromgamma-ray observations. The hidden sources can beAGN cores (Stecker 2005; Murase et al. 2016), chokedjets in tidal disruption events of supermassive blackholes (Wang & Liu 2016) and a pair of choked jets incore-collapse massive stars (M´esz´aros & Waxman 2001;Razzaque et al. 2004; Murase & Ioka 2013; Xiao & Dai2014; Senno et al. 2016).At the end of its life, a massive star can collapse intoa neutron star, a quark star, or a black hole, and it canproduce an energetic jet. The core collapse of massivestars which lost their outer layer of hydrogen and he-lium due to strong winds or mass transfer to a compan-ion (Filippenko 2005) may produce Type Ib/c supernovae(SNe), while the core collapse of massive stars which didnot lose their stellar envelope may produce Type II SNe.There is no essential difference between the Type Ib/cand II SNe, though their spectra are superficially dif-ferent due to the different properties of their stellar en-velopes. For massive stars with strong winds, which blowout most of the materials of their stellar envelope, i.e.,Wolf-Rayet stars, the jets with large injected energy caneasily break through the star and produce gamma-rayemission, usually observed as a GRB, which is suggestedto be associated with a Type Ib/c SN (Hjorth & Bloom2012). These GRB jets are believed to accelerate pro-tons and produce high-energy neutrinos in interactionsof protons with gamma rays (Waxman & Bahcall 1997).However, the flux of the diffuse neutrinos from GRBs isnot high enough to explain the observed neutrino flux,and no associations between the observed GRBs and neu-trinos have been observed so far (Aartsen et al. 2015b,2017). If the jets are propagating in a thick stellar en-velope or extended material (M´esz´aros & Waxman 2001;Razzaque et al. 2004; Murase & Ioka 2013; Xiao & Dai2014; Senno et al. 2016), they may not be able to breakout through them. The choked jets might be more ubiq-uitous than the break-out ones. In that case, neutrinosand gamma-rays are produced via the interaction of ac-celerated protons and thermal photons in the choked jets.Since the neutrinos and gamma rays are produced insidethe stellar envelope, the source is opaque to gamma-rayphotons but transparent for neutrinos. This can explainthe lack of association between the observed GRBs andIceCube neutrinos, as well as the tension between the dif-fuse gamma-ray and neutrino observations. The detailedfeatures of the observed neutrino spectrum can help usto study the sources hidden in the gamma-ray band, andthe possible association between neutrinos and SNe willprovide us with more information on the progenitor stars.There have been many investigations of choked jetsfrom the progenitor of Type Ib/c SNe, correspondingto the scenario of failed GRBs, with the duration ofthe central engine less than 100 s, which is typical forlong GRBs (M´esz´aros & Waxman 2001; Razzaque et al.2004; Murase & Ioka 2013; Senno et al. 2016). However,for jets choked in the stellar envelope of red/blue su-pergiant stars, associated with Type II SNe, the dura-tion of the central engine could be longer than for longGRBs (Xiao & Dai 2014); the feasibility of the long du-ration is discussed in Section 2. With the same energybudget, the scenario with longer duration and lower lu-minosity will lead to a neutrino spectrum with a lowercutoff energy at the high energy end, as will be discussedin Section 5.Since the rate of Type II SNe is about three timesthat of Type Ib/c SNe (Li et al. 2011), the study of thechoked jets from the progenitors of Type II SNe is alsoimportant. In this paper, we study neutrinos associatedwith Type II SNe, i.e., neutrinos from jets choked in thehydrogen envelope of red supergiant stars (RSGs). Wediscuss the conditions under which jets are choked andprotons are accelerated in Sections 2 and 3, respectively.Then we discuss the interactions between protons andthe target photons in Section 4, and study the features of the produced neutrinos spectrum in Sections 5 - 7.We predict that a newly born jet-driven type II SN maybe observed to be associated with the neutrino bursts inSection 8. Finally, we discuss and summarize in Section9. CHOKED JET DYNAMICS
It is widely believed that, at the end of their lives,a fraction of rapidly rotating massive stars will un-dergo core collapse, form a compact star or a blackhole, and launch a pair of jets. We assume that thecore of an RSG is surrounded by a helium envelopeof size ∼ cm, and a hydrogen envelope of size ∼ × cm with a slowly varying density ρ H ≃ − gcm − ρ H , − (M´esz´aros & Rees 2001). A success-ful jet will propagate in the helium and hydrogen en-velopes, and finally break out through the star. How-ever, if the jet lifetime is shorter than the jet cross-ing time, the jet is stalled before it breaks throughthe star (MacFadyen et al. 2001; M´esz´aros & Rees 2001;M´esz´aros & Waxman 2001).A forward shock and a reverse shock are producedwhen the jet is propagating in the hydrogen envelope.The shocked region between the two shocks is the jethead. The Lorentz factor of the jet head and the un-shocked jet plasma is Γ h and Γ, and the Lorentz factorof the shocked jet plasma in the rest frame of the un-shocked jet plasma is¯Γ = ΓΓ h (1 − ββ h ) , (1)where β = p − / Γ and β h = p − / Γ h . The en-ergy density of the region behind the reverse shock is(Blandford & McKee 1976; Sari & Piran 1995) e r = (4¯Γ + 3)(¯Γ − n j m p c , (2)where n j = L/ (4 π Γ R cm p c ) is the number density ofthe un-shocked jet, with the radius R h ≃ β h ct . Theenergy density of the region behind the forward shock is e f = (4Γ h + 3)(Γ h − ρ H c . (3)Balancing the energy densities of these two regions, onecan derive the Lorentz factor and the radius of the jethead, Γ h and R h .According to Equation (1) in M´esz´aros & Rees(2001), for a larger isotropic luminosity, i.e., L iso > erg s − r . ρ H , − , the velocity of the jet headis relativistic (Razzaque et al. 2004; Ando et al. 2005;Murase & Ioka 2013; Senno et al. 2016). In this paper,we scan the parameter space of the isotropic luminosity L iso and the lifetime t of the jet, and find that, for mostof the parameters we are concerned with, the jet head isnon-relativistic, i.e., Γ h ∼
1, allowing us to perform ananalytical calculation as follows.In the non-relativistic case, the velocity of the jet headat radius r is approximated to be v h ≃ (cid:18) L iso (4Γ h + 3) πr cρ H (cid:19) / (4)and the corresponding propagating time is t prop = Z drv h ≃ R h v h . (5)Combining Equations (4) and (5), one can derive theradius and the velocity of the jet head at the end of thejet lifetime t analytically, i.e., R h = 9 . × cm L / , t / ρ − / , − , (6)and v h = 0 . c L / , t − / ρ − / , − . (7)The above results are consistent with the Lorentz factorof the jet head being Γ h ≃ L iso , t − . × ρ H , − . (8)According to Equation (6), the jet crossing time, i.e.,the time that the jet takes to break through the stellarenvelope of radius of R , is t cros = 1 . × s R . L − / , ρ / , − (9)If the jet lifetime is smaller than the jet crossing time,i.e., t < t cros , (10)the jet cannot break out through the stellar envelope. Inthis case, the jet is choked and this object will not beobserved as a shock breakout phenomenon or a GRB.Rather, it will become a jet-driven type II SN. We willdiscuss this point in Section 8.According to observations, the duration of long GRBsis about <
100 s, while very long GRBs and ultra-longGRBs with durations of ∼ s and ∼ s have beenobserved (Greiner et al. 2015), and a few GRBs can reach ∼ s, with the central engine unclear (Levan 2015).The majority of the durations of the choked jet eventsare plausibly different from those of the successful jetevents, i.e., the observed GRBs. If we assume the jetis powered by the rotation energy of the central magne-tar (Usov 1992; Metzger et al. 2011; Mazzali et al. 2014;Greiner et al. 2015; Gompertz & Fruchter 2017) with amass of 1 . M ⊙ and a radius of 10 km, the upper limitof the jet lifetime is about the spin-down duration of themagnetar, which is (Ostriker & Gunn 1969) t sd = 2 . × s P , − B − , , (11)where P i is the initial spin period, and B m is the strengthof magnetic field of the magnetar. If we assume the jetis powered by the accretion of the star material ontothe central black hole, the upper limit of the jet life-time is the free-fall time of the star material, whichis (Kippenhahn & Weigert 1994) t fb = (cid:18) π R GM c (cid:19) = 1 . × s R / . (cid:18) M c M ⊙ (cid:19) − , (12)where G is the gravitational constant, and M c is themass of the central black hole. Therefore, in this paper,we assume the jet lifetime to be a free parameter with avalue in the range of ∼ − s. PARTICLE ACCELERATION CONSTRAINT IN THEINTERNAL SHOCK
Due to the inhomogeneities of the jet, an internalshock can be produced beneath the jet head at radius R IS . R h , by a rapid shell catching up with and merg-ing into the slow shell. The relative Lorentz factor be-tween the two shells is Γ rel ≃ (Γ r / Γ + Γ / Γ r ) / ∼ r and Γ are the Lorentz factors of the rapidand merged shells, respectively. Cosmic rays are accel-erated efficiently in the internal shock, if the comovingsize of the upstream flow l u = R IS / ΓΓ rel is much smallerthan the mean free path length of the thermal photonsin the upstream flow, l dec = 1 / ( n u σ T ) (Murase & Ioka2013), where the number density of particles in the up-stream flow is n u = n p , IS Γ rel , and n p , IS = L iso πR Γ m p c =2 . × cm − Γ − L / , t − ρ / , − is the proton num-ber density in the un-shocked material. This accelera-tion constraint of l u < l dec requires that (Murase & Ioka2013; Senno et al. 2016), τ = n p , IS σ T ( R IS / Γ) < min[Γ , . C − Γ ] , (13)with C = 1 + 2 ln Γ is a coefficient accounting for pairproduction. When the internal shock approaches the jethead, i.e., R IS ∼ R h , the criterion of efficient accelerationof protons can be rewritten as τ = 0 .
13 Γ − L / , t − / ρ / , − < min[Γ , . C − Γ ] , (14)Then the accelerated protons interact with the photonsescaping from the jet head.If the constraint shown in Equation (14) is satisfied,the internal shock is able to accelerate protons, with anacceleration timescale of t p , acc ( ǫ p ) = φ ǫ p q e Bc = 0 .
14 s φ ǫ − / B, − ǫ p , L − / , Γ t / ρ − / , − , (15)where φ is the number of the gyro-radii required to e -fold the particle energy, q e and ǫ p are the charge and theenergy of the proton, and B = 9 . × G ǫ / B, − Γ − L / , t − / ρ / , − , (16)is the strength of the magnetic field in the internal shockwith ǫ B as the fraction of the magnetic field energy. THE TARGET PHOTONS
In the shocked jet head region, the Thomson opticaldepth of the shocked plasma is τ h = n h σ T R h = 52 Γ − L / , t − / ρ / , − f a , (17)where n h = (4¯Γ + 3) n j is the number density of electronsin the shocked jet head region, and f a = ∼ ≫ h ≃
1. If τ h > , (18)the electrons in the reverse shock are heated and lose alltheir energy ǫ e e r into a thermal radiation with a temper-ature of k B T h = 99 eV ǫ / , − L / , t − / ρ / , − f c , (19)where k B is the Boltzmann constant, and f c = (cid:16) (4¯Γ+3)(¯Γ − (cid:17) / ∼ ≫ h ≃
1. The numberdensity of thermal photons in the shocked jet is n γ, h = 19 π ( hc ) ( k B T h ) = 3 . × cm − ǫ / , − L / , t − / ρ / , − f (20)When the internal shock approaches the jet head re-gion, the thermal photons will escape into the internalshock with a fraction of f esc = 1 /τ h (Murase & Ioka2013), since it’s optically thin in the internal shock.There, the energy of the thermal photons from the jethead is boosted by a factor of ¯Γ ∼ Γ if Γ ≫ Γ h andΓ h ≃
1. Then the number density of thermal photons inthe internal shock is n γ, IS = ¯Γ n γ, h f esc = 6 . × cm − ǫ / , − Γ L − / , t − / ρ / , − f f − (21)and the peak energy of the thermal photons in the inter-nal shock frame is boosted to be ǫ γ, IS = ¯Γ(2 . k B T h ) = 2 . × eV ǫ / , − Γ L / , t − / ρ / , − f c . (22)On the other hand, the electrons are accelerated in theinternal shock. The acceleration timescale for electronswith Lorentz factor of γ e is t e , acc ( γ e ) = φ e γ e m e c q e Bc = 2 . × − s φ e , γ e , . ǫ − / B, − Γ L − / , t / ρ − / , − , (23)and the cooling timescale via synchrotron emis-sion (Sari et al. 1998) is t e , c ( γ e ) = γ e m e c σ T cγ B π = 0 .
28 s γ − , . ǫ − B, − Γ L − / , t ρ − / , − , (24)where φ e is the number of the gyro-radii required to e -fold the electron energy. Comparing the accelerationtimescale with the cooling timescale above, one can de-rive the maximum Lorentz factor of electrons acceleratedin the internal shock, which is γ e , max = 3 . × φ − / , ǫ − / B, − Γ / L − / , t / ρ − / , − (25)We assume a fraction of ǫ ′ e of the shock energy goesinto the electrons, and the spectral index of the accel-erated electrons is p = 2 .
2, then the minimum Lorentzfactor of the accelerated electrons in the internal shockis (Sari et al. 1998) γ e , min = ǫ ′ e (2 − p )(1 − p ) m p m e = 31 ǫ ′ e , − . (26)So the cooling timescale for electrons with Lorentz factor γ e , min is t e , c ( γ e , min ) = 0 .
30 s ǫ ′ e − , − ǫ − B, − Γ L − / , t ρ − / , − ,which is much smaller than the dynamic timescale of theshock t dyn = R IS / (Γ c ) = 30 s Γ − L / , t / ρ − / , − . (27) In addition, from Equation (4), one has t e , c ( γ e ) ∝ γ − .Therefore, electrons with Lorentz factor γ e > γ e , min arecooled even faster via synchrotron radiation. Finally,most of the electrons are cooled to the low-energy endwith Lorentz factor γ e < γ e , min . The energy of the cor-responding photons is smaller than (Sari et al. 1998) ν syne , min = γ , min q e B πm e c = 0 .
10 eV ǫ / B, − Γ − L / , t − / ρ / , − , (28)which is too low to reach the threshold of photo-mesoninteraction and interact with the accelerated protons inthe internal shock (the maximum energy of the acceler-ated protons is calculated in Section 5.2.). Therefore, thesynchrotron photons of electrons in the internal shock asthe target photons of photo-meson interaction can be ig-nored. FEATURES OF THE NEUTRINO SPECTRUM
As discussed above, our model is valid only if someconditions are met, namely the jet is choked, protons areaccelerated to high energy efficiently, and thermal pho-tons are produced in the jet head and propagate into theinternal shock region. Then the accelerated protons in-teract with the photons from the choked jet head andproduce pions. A charged pion with the characteris-tic energy ǫ π ∼ . ǫ p , decays into a neutrino and amuon with characteristic energy ǫ µ ∼ . ǫ p via theprocess π + ( π − ) → µ + ( µ − ) + ν µ (¯ ν µ ). The producedmuon further decays into one lepton and two neutrinos: µ + ( µ − ) → e + ( e − ) + ν e (¯ ν e ) + ¯ ν µ ( ν µ ). The average energyof the neutrinos is ǫ ν ∼ . ǫ p . The produced neutrinospectrum has two features: one is a cutoff at low energydue to the threshold of the photo-meson interaction, andthe other is a cutoff at high energy due to the photo-meson cooling of protons, or the synchrotron cooling ofpions and muons. Below we calculate the two character-istic energies of the neutrino spectrum. The low-energy cutoff
For a proton energy in the range0 . / (2 ǫ γ, IS ) ≤ ǫ p ≤ . / (2 ǫ γ, IS ) , (29)the ∆-resonance dominates the photo-meson interaction,with cross section σ ∆ = 5 × − cm . Assuming theinelasticity is f in , p∆ = 0 . t ∆ = 1 σ ∆ n γ, IS cf in , p∆ ≃ .
056 s ǫ − / , − L / , Γ − t / ρ − / , − f a f − , (30)which is much smaller than the dynamic timescale of theshock t dyn as in Equation (4), meaning that the photo-meson interaction efficiency is as high as 100%. Thetimescale of the protons in the internal shock losing en- We also estimated the density of photons produced by elec-trons at the high-energy tail γ e , min < γ e < γ e , max , and found outthat those ∼ keV synchrotron photons are not dominant over thethermal photons from jet head. t(s) L i s o ( e r g s − ) erg10 erg10 erg10 erg eV10 eV10 eV ε obsν,th t(s) L i s o ( e r g s − ) erg10 erg10 erg10 erg eV10 eV10 eV ε obsν,th Figure 1.
Constrained luminosity-duration ( L iso − t ) space withfixed Lorentz factor Γ = 10 (upper panel) and Γ = 100 (bottompanel), respectively. We fix the outer radius and the density of thehydrogen envelope as R H = 3 × cm and ρ H = 10 − g cm − ,with ǫ e = 0 . ǫ B = 0 .
01. The parameter space to the left of theblack solid line corresponds to the choked jet situation, that be-low the red solid line satisfies the acceleration constraint, and thatabove the blue solid line corresponds to the jet head optically thicksituation. The color contours denote the values of the low-energycutoff of the neutrino spectrum, which is only valid in the param-eter space enclosed by the blue, black, and red solid lines. Theyellow dotted, dashed, dash–dotted and solid lines correspond tothe isotropic injected energy of E iso = 10 erg , erg , erg,and 10 erg, respectively. ergy via the pp collision is t pp = 1 σ pp n p , IS cf in , pp = 1 . × s L − iso , t ρ − H , − Γ , (31)where the cross section of the pp collision is approxi-mated to be σ pp ≃ × − cm , and the inelasticity isassumed to be f in , pp = 0 .
2. The pp collision can be ne-glected since the timescale of the energy loss t pp is muchlonger than the dynamic timescale t dyn .For ǫ p < . / (2 ǫ γ, IS ), the cross section of thephoto-meson interaction decreases rapidly, leading toa cutoff at the low-energy end of the neutrino spec- trum (M¨ucke et al. 1999).Therefore, the threshold of the photo-meson interac-tions in the internal shock frame corresponds to the pro-ton energy ǫ p , th = 0 . / (2 ǫ γ, IS )= 3 . × eV ǫ − / e, − Γ − L − / , t / ρ − / , − f − . (32)Thus, a lower-energy cutoff of the observed neutrinospectrum appears due to the threshold of the photo-meson interaction at ǫ obs ν, th = 0 . ǫ p , th Γ= 1 . × eV ǫ − / e, − L − / , t / ρ − / , − f − . (33)We plot the value of ǫ obs ν, th as a function of the isotropicluminosity L iso and the lifetime t of the jet using colorcontours in Figure 1. The parameter space to the left ofthe black solid line satisfies the choking condition as dis-cussed in Section 2. The parameter space below the redsolid curve satisfies the internal shock acceleration con-straint as discussed in Section 3. The parameter spaceabove the blue solid curve satisfies the condition of pro-ducing thermal photons in the jet head as discussed inSection 4. The color contours are valid for parameterswhich satisfy these constraints, i.e., in the region enclosedby the red, blue, and black solid lines. Comparing thetwo panels, the choking condition does not change fordifferent Lorentz factors, but the shock acceleration con-straint is tighter for a smaller Lorentz factor. Thus, fora smaller Lorentz factor Γ = 10, only jets with a lowerluminosity and a longer lifetime can accelerate protonsefficiently in the internal shock, and the low-energy cut-off is at around a few tens to a few hundreds of TeV forlocal sources. For a larger Lorentz factor Γ = 100, thelow energy cutoff of the observed neutrino spectrum isabout 1 −
10 TeV.
The high-energy cutoff
For protons with energies ǫ p > . / (2 ǫ γ, IS ), i.e., ǫ p > . × eV ǫ − / e, − Γ − L − / , t / ρ − / , − f − , (34)the other resonances and multi-pion channel willdominate the photo-meson interaction, with the av-erage cross section approximated to be ∼ × − cm (M¨ucke et al. 1999). Assuming the inelas-ticity is 0.6 (Atoyan & Dermer 2001), the correspondingtimescale of multi-pion production is t pγ ≃ .
093 s ǫ − / , − L / , Γ − t / ρ − / , − f a f − . (35)When t pγ ≥ t p , acc ( ǫ p ), the protons cannot be acceleratedto an energy as high as ǫ p before cooling. Comparingthe photo-meson cooling timescale (Equation (35)) andthe acceleration timescale (Equation (15)), one can de-rive the maximum energy of accelerated protons due to Since t ∆ ≪ t dyn and t p γ ≪ t dyn , the source is considered tobe “calorimetric”, i.e., the protons lose all their energy and cannotescape from the internal shock, thus we do not have to consider thepp collision between the protons accelerated in the internal shockand hydrogen in the stellar envelope. the photo-meson cooling as ǫ p , c ≃ . × eV × φ − ǫ − / , − ǫ / B, − L / , Γ − t − / ρ / , − f a f − . (36)Consequently, the observed neutrino spectrum has a cut-off at energy ǫ obs ν, c = 0 . ǫ p , c Γ ∼ . × eV × φ − ǫ − / , − ǫ / B, − L / , Γ − t − / ρ / , − f a f − . (37)Moreover, the synchrotron cooling of pions and muonswill suppress the flux of neutrinos when the synchrotroncooling timescales of pions and muons are shorter thantheir lifetimes. Adopting the approximations that theenergy of the produced pions is ǫ π = 0 . ǫ p and the energyof the produced muons is ǫ µ = 0 . ǫ p , the lifetimes ofpions and muons can be written as functions of ǫ p , i.e., τ π = 0 .
038 s ǫ p , and τ µ = 3 .
15 s ǫ p , . The synchrotroncooling timescales of pions and muons are τ π, syn = 1 . × s ǫ − , ǫ − B, − Γ L − / , t ρ − / , − (38)and τ µ, syn = 59 s ǫ − , ǫ − B, − Γ L − / , t ρ − / , − (39)Comparing the synchrotron cooling timescale with thelifetime of the leptons, we find that, during the photo-meson interactions, muons with the energy larger than ǫ µ, sup cool first before decaying, where the critical energyof muons in the internal shock is ǫ µ, sup = 6 . × eV ǫ − / B, − Γ L − / , t / ρ − / , − (40)Then the observed neutrino spectrum is suppressed atenergies above ǫ obs ν, sup ≃ . × eV ǫ − / B, − Γ L − / , t / ρ − / , − (41)due to the synchrotron cooling of the muons.Muons with energy larger than ǫ µ, sup will produce syn-chrotron photons with the energy larger than the char-acteristic energy of E µ, syn = γ µ, sup q e B πm µ c = 1 . × eV ǫ − / B, − Γ L − / , t / ρ − / , − (42)in the internal shock, where γ µ, sup = ǫ µ, sup / ( m µ c ) isthe Lorentz factor of a muon with the energy of ǫ µ, sup .Therefore, the energy of the synchrotron photons is muchhigher than that of the thermal photons from the jethead, leading to the production of . GeV neutrinos viathe ∆-resonance of photo-meson interaction, which is be-yond our scope.On the other hand, a low luminosity and a largeLorentz factor result in a slow acceleration according toEquation (15), but a fast photo-meson cooling accord-ing to Equations (30) and (35). For the extreme case t ∆ ≤ t p , acc ( ǫ p , th ), the protons are quickly cooled via thephoto-meson interaction as long as they are acceleratedto be above the threshold energy. Then the spectrum isno longer an extended power law spectrum, but a peakedspectrum with peak energy around ǫ p , th . We estimate the overall maximum energy of the pro-duced neutrinos by comparing the three energies calcu-lated above, which are ǫ obs ν, c , ǫ obs ν, sup , and ǫ obs ν, th . The overallmaximum energy of the produced neutrinos is ǫ obs ν, max ≃ max( ǫ obs ν, th , min( ǫ obs ν, c , ǫ obs ν, sup )) . (43)The fast cooling case where ǫ obs ν, max = ǫ obs ν, th can be realizedfor a larger Lorentz factor and lower luminosity. We plotthe value of ǫ obs ν, max in Figure 2.From Figure 2, one can see that the parameter spaceis roughly divided into two regions: the red region cor-responds to a maximum energy of the spectrum ≥ eV, which is hereafter called the “hard phase”, while thedark blue region corresponds to a maximum energy of thespectrum ≤ eV, which is called the “soft phase”, andthe light blue region corresponds to a maximum energyof the spectrum in between 10 and 10 eV, which iscalled the “intermediate phase”.As shown in the upper panel in Figure 2, for a lowLorentz factor, Γ = 10, it is unlikely that the spectrumis hard enough to explain PeV neutrinos; this is due tothe photo-meson cooling. On the other hand, as in thebottom panel in Fig 2, for the case with a large Lorentzfactor, Γ = 100, a high luminosity of the jet is requiredto explain PeV neutrinos.In the next section, we adopt a few parameter sets fordifferent phases with the same isotropic injection energy E iso = L iso t = 10 erg, to study the neutrino spectrafrom individual sources for different phases. NEUTRINOS FROM INDIVIDUAL SOURCES
To calculate the neutrino spectrum distribution, weadopt the analytical description of the energy distribu-tion of neutrinos produced in the photo-meson interac-tion from Kelner & Aharonian (2008). We also considerthe effect of photo-meson cooling on the proton accelera-tion and the suppressions due to the synchrotron coolingof the secondary pions and muons, as discussed in Section5.2.According to Figure 2, we choose one parameter set of L iso = 3 . × erg s − and t = 3 . × s, which is inthe soft phase for Γ = 100, while is in the intermediatephase for Γ = 10. We also choose the parameter set of L iso = 1 . × erg s − and t = 1 . × s, which is inthe hard phase for Γ = 100, while is not valid for Γ = 10since it violates the acceleration constraint. The spec-tra of the neutrino fluence from individual sources at adistance of 1 Gpc are plotted in Figure 3. We can seethree different types of spectra. The dashed line shows asoft spectrum, which can only contribute to the 10 − DIFFUSE NEUTRINOS
To calculate the diffuse neutrino spectrum, we assumethat the rate of choked jets R cj ( z ) at redshift z followsthe star formation rate ρ sf ( z ), i.e., R cj ( z ) = A cj ρ sf ( z ) , (44) t(s) L i s o ( e r g s − ) erg10 erg10 erg10 erg eV10 eV10 eV10 eV10 eV ε obsν,max t(s) L i s o ( e r g s − ) erg10 erg10 erg10 erg eV10 eV10 eV10 eV10 eV ε obsν,max Figure 2.
Color contours denoting the estimated maximum en-ergy of the observed neutrinos ǫ obs ν, max , for Lorentz factor Γ = 10(upper panel) and Γ = 100 (bottom panel), respectively. The solidlines denote the same constraints as in Figure 1. where the star formation rate is (Madau & Dickinson2014) ρ sf ( z ) = 0 . M ⊙ yr − Mpc − (1 + z ) . z ) / . . , (45)and A cj is a free normalization coefficient in units of M ⊙− .Here we adopt a constant luminosity instead of a lu-minosity function for the choked jets, because in ourmodel only sources with luminosities satisfying thoseconstraints as in Figure 1 can produce neutrinos, andtheir luminosity distribution would not follow a knownluminosity function. The diffuse neutrino flux is calcu-lated via integrating the neutrino spectrum over the red- E ν (GeV)10 −6 −5 −4 −3 −2 −1 E ν d N ν / d E ν ( G e V c m − ) L iso = 3.3 × 10 ergs −1 , t = 3.3 × 10 s, Γ = 10 (Intermediate)L i o = 3.3 × 10 erg −1 , t = 3.3 × 10 , Γ = 100 (Soft)L i o = 1.0 × 10 erg −1 , t = 1.0 × 10 , Γ = 100 (Hard) Figure 3.
Fluence of neutrinos produced from individual sourcesat a distance of 1 Gpc for three parameter sets. The other param-eters are fixed as ǫ B = 0 . ǫ e = 0 . p = 2 . R H = 3 × cm,and ρ H = 10 − g cm − . The dashed, dash–dotted, and dottedlines denote the soft, intermediate, and hard phases, respectively. shift from 0 to 8, i.e., (Murase 2007) ǫ obs ν dN ν dǫ obs ν ( ǫ obs ν ) dǫ obs ν = c πH Z ǫ ν dN ν dǫ ν ((1 + z ) ǫ ν ) dǫ ν × Ω4 π A cj ρ sf ( z ) dz (1 + z ) p Ω Λ + Ω M (1 + z ) (46)where the cosmological parameters are adopted as H =70 kms − Mpc − ,Ω M = 0 .
3, and Ω λ = 0 . ǫ ν dN ν dǫ ν ( ǫ ν ) dǫ ν is the neutrino flux at the source frame, and Ω =2 π (1 − cos θ ) is the solid angle of a jet with the openangle assumed to be θ = 0 . z = 1 −
3, since the star formation ratedensity peaks around those redshifts. However, there arestill many uncertainties on the total star formation ratedensity at high redshifts due to problems with selectionsof high-redshift galaxies and accurate measurements oftheir star formation rates (e.g., Wang et al. 2016). Itis possible that there are more massive stars at high red-shift, for example, PoP III stars, which may contributemore to the neutrinos.We calculate the neutrino spectra with three fixed pa-rameter sets, implying three fixed spectral shapes. Thenormalization parameter A cj is the only free parame-ter; then the flux of the spectra in Figure 4 are nor-malized via the parameter A cj . By fitting the IceCubecombined data (Aartsen et al. 2015a) and the IceCubesix year HESE data (IceCube Collaboration et al. 2017)for all flavor neutrinos, we get the best values of A cj ,which are listed in Table 1. The three spectra, com-pared with the combined and six year data for all fla-vor neutrinos, are listed in Figure 4. Then the corre-sponding constrained local rate of the choked jet events R cj ( z = 0) can be calculated via Equation (44), and thevalues are also listed in Table 1. The jet’s kinetic energy, L iso t Γ A cj R cj ( z = 0) N S ( N ν µ > N S ( N ν µ > N S ( N ν µ > − s M − ⊙ Gpc − yr − yr − yr − yr − Soft Phase 3 . × . ×
100 1 . × − . × . × . ×
10 3 . × − . × . × . ×
100 1 . × − . × Table 1
Isotropic luminosity L iso , Lifetime t , and Lorentz Factor Γ of the Jet, Constrained Normalization Parameter A cj , and Constrained LocalRate of the Choked jets R cj for the Single-component Fitting in Figure 4. In the last three columns, N S ( N ν µ > N S ( N ν µ > N S ( N ν µ >
3) denote the expected amount of sources from which more than 1, 2, and 3 muon neutrinos can be detected by IceCube peryear, respectively. . × erg, which we adopt in this paper, is typical fora SN, and the constrained local choked jet rate is about1 . × − . × Gpc − yr − , as seen in Table 1. Forcomparisons, the required event rate is larger than therate of observed successful GRBs, where the rate of high-luminosity long GRBs (with isotropic luminosity above10 erg s − ) is 0 . +0 . − . Gpc − yr − , and the rate of low-luminosity long GRBs (with isotropic luminosity above5 × erg s − ) is 164 +98 − Gpc − yr − (Sun et al. 2015).However, the required event rate is only about 1% − ∼ Gpc − yr − (Atteia2013), which is consistent with our assumption.Since the angular resolution of observing muon neu-trinos is better than that of observing the other flavors,considering the beam correction, we calculate the rates ofobserving more than 1, 2, and 3 muon neutrinos for thethree parameter sets based on the constrained source ratefor the single-component fitting, and list them in the lastthree columns of Table 1. Here we assume the ratio be-tween the three flavors is approximated to 1 : 1 : 1 due totheir oscillations (Learned & Pakvasa 1995; Athar et al.2000), although the real flavor ratio is slightly differentfrom 1 : 1 : 1 considering their oscillations in the stellarenvelope (Sahu & Zhang 2010). As we can see from Ta-ble 1, IceCube can observe about four triplets of muonneutrinos during 10 years of operation.As seen in Figure 4, the hard phase (dotted line) pro-duces a hard spectrum with a cutoff at ∼ PeV, whilethe soft phase (dashed line) produces a soft spectrumwith a cutoff at lower energy, but cannot produce PeVneutrinos. If we assume that both the hard and thesoft phases contribute to the neutrino spectrum, one canget a soft spectrum with a cutoff at ∼ PeV. The two-component spectra, assuming that the flux ratio of thehard phase to the soft phase at 10 TeV is 0.2, are plot-ted in the upper panel of Figure 5. The existence of thecutoff at ∼ PeV is evidence of this model. If the en-ergy of the cutoff is above a few to 10 PeV, one needs anadditional component to explain the ≥ PeV neutrinos.Neutrinos from AGN cores (Stecker 2005) or from dis-tant blazars(Kalashev et al. 2013) may contribute at the ≥ PeV energy, and neutrinos from the choked jets con-tribute at the lower energy. This possibility is plotted inthe bottom panel of Figure 5. OBSERVATIONAL PREDICTIONS
The choked jet in our model might not result in GRBs,rather, it might result in a type-II (especially type-IIP,Arcavi 2017) SN, since the total jet energy exceeds thegravitational binding energy of the progenitor star. Thisis consistent with the observations that only a small frac- E ν (GeV)10 −9 −8 −7 −6 E ν d N ν / d E ν ( G e V c m − s − s r − ) L iso = 3.3 × 10 ergs −1 , t = 3.3 × 10 s, Γ = 10L iso = 3.3 × 10 ergs −1 , t = 3.3 × 10 s, Γ = 100L iso = 1.0 × 10 ergs −1 , t = 1.0 × 10 s, Γ = 100IC2015IC2017 Figure 4.
Diffuse neutrino spectra for three parameter sets asin Figure 3. The light gray squares denote the data from the Ice-Cube combined analysis (IC2015, Aartsen et al. 2015a), and theblack diamonds denote the IceCube six year HESE data (IC2017,IceCube Collaboration et al. 2017). tion of SNe are associated with GRBs.After the central engine stops, the jet tail catches upwith the jet head after δt = R h /βc , where βc ≃ c isthe velocity of the jet tail. Then the jet head startsto decelerate and propagate like a jet-driven SN ex-plosion (Nagataki 2000). Since the typical radial ve-locity of an SN shock is about v sh ∼ cm s − (Wongwathanarat et al. 2015), the time for the jet headto break out through the outer stellar envelope, withlength of ( R − R h ), is evaluated as δt ≃ ( R − R h ) /v sh ,where R and R h are the radii of the star and the jethead. Meanwhile, it takes δt = ( R − R h ) /c for neu-trinos to reach the stellar surface. Therefore, the ob-served time delay of photons from the beginning of theSN explosion to neutrinos is δt = δt + δt − δt = R (1 /v sh − /c ) − R h (1 /v sh − /c ) = 3 . × s R . − . × s L / , t / ρ − / , − . We note here, since the lightcurve of the SN explosion takes a few days to a few tensof days to reach the peak flux, we may not be able to de-tect photons at the very beginning of the SN explosion.Fortunately, the light curve of Type II SNe usually ex-tends from around a hundred days to a few hundred days,we have a large time window to observe them. Once weobserve an SN spatially associated with a muon neutrinotriplet, we can trace back to the explosion time accordingto the observed light curve, and then measure the timedifference between the neutrino burst and the explosion. E ν (GeV)10 −9 −8 −7 −6 E ν d N ν / d E ν ( G e V c m − s − s r − ) L iso = 3.3 × 10 ergs −1 , t = 3.3 × 10 s, Γ = 100L iso = 1.0 × 10 ergs −1 , t = 1.0 × 10 s, Γ = 100IC2015IC201710 E ν (GeV)10 −9 −8 −7 −6 E ν d N ν / d E ν ( G e V c m − s − s r − ) L iso = 1.0 × 10 ergs −1 , t = 1.0 × 10 s, Γ = 100AGN (Kalashev et al. 2013)IC2015IC2017 Figure 5.
Two-component spectra. In the upper panel, a softphase with the parameter set of L iso = 3 . × ergs − , t =3 . × s, and Γ = 100 is adopted as the dashed line, and ahard phase with the parameter set of L iso = 1 . × ergs − , t = 1 . × s, and Γ = 100 is adopted as the dotted line. Inthe bottom panel, an intermediate phase with the parameter set of L iso = 1 . × erg s − , t = 1 . × s, and Γ = 100 is adoptedas the dashed line, and a PeV neutrino component from distantblazars (Kalashev et al. 2013) is adopted for the dotted line. Theother parameters are the same as in Figure 4. The solid line is thesum of the dashed line and the dotted line. As listed in Table 1, about four triplets of muon neu-trinos are expected to be observed during 10 years of Ice-Cube’s observations. The detections on new-born type-II SNe following the observation of muon neutrino sin-glets, doublets, and especially triplets, will be strongevidence for the dominant contribution of the chokedjet neutrino. Moreover, the jet-induced feature, i.e.,the substantially asymmetric and predominantly bi-polarexplosion(Wheeler et al. 2002), will provide us with moreevidence for the existence of the choked jet. Any fu-ture observed time difference between the muon neutrinotriplet and the photon emission will help one to constrainthe model.We calculate the mean number of observed muon neu-trino events from a source at distance of D via N ν µ ( D ) = R F ( E ν µ , D ) A eff ( E ν µ ) dE ν µ , where F ( E ν µ , D ) is the flu- ence of neutrinos produced from an individual source atdistance of D , and A eff ( E ν µ ) is the effective area of Ice-Cube for muon neutrinos. Then we can derive the meanupper limit of the distance of sources from which a tripletof muon neutrinos can be observed by IceCube, via thecriterion of N ν µ ( D ) ≥
3. The mean upper limit of thesource distance is constrained to be 81 Mpc, 135 Mpc,and 188 Mpc, for the soft, intermediate and hard phasesshown in Table 1, respectively, which are within the cur-rent detection radius of core-collapse SNe (Taylor et al.2014).For an extreme high isotropic energy E iso = 10 erg,corresponding to a jet energy of 10 erg for θ = 0 .
2, asseen in the bottom panel of Figure 2, only a small pa-rameter space is consistent with our model, and the asso-ciated SN might be a type II superluminous SN (SLSN).If one adopts parameters Γ = 100, L iso = 10 erg s − ,and t = 10 s, more than three muon neutrinos on aver-age can be observed by IceCube if the source is locatedwithin ∼ . z ≤ .
05) is within the current detection radius of anSLSN.The new-born type-II SN/SLSN associated with thetriplet of muon neutrinos might be observed by opti-cal/infrared and X-ray observatories. The three instru-ments adopted in the IceCube Optical Follow-up pro-gram and X-ray Follow-up program (Kowalski & Mohr2007; Abbasi et al. 2012; Aartsen et al. 2015c), namelythe Robotic Optical Transient Search Experiment(ROTSE; Akerlof et al. (2003)), the Palomar TransientFactory (PTF; Law et al. (2009); Rau et al. (2009)), andthe
Swift satellite (Gehrels et al. 2004), as well as theSubaru Hyper-Suprime-Cam with a field of view (FOV)of 1 . ◦ will be suitable telescopes for the follow-up obser-vations, since the median angular resolution of the muontrack events is around 1 ◦ (IceCube Collaboration et al.2016). The Large Synoptic Survey Telescope (LSST)with a 3 . ◦ FOV (Angeli et al. 2014) as well as thePan-STARRS1 (PS1) telescope with a 3 . ◦ FOV, willprovide a useful archive to search for detections retro-spectively (Aartsen et al. 2015c). CONCLUSIONS
Choked jet sources produce neutrinos without accom-panying gamma rays, thus avoiding a conflict betweenthe diffuse gamma-ray observations and neutrino obser-vations. The choked jets might be very common at theend of the massive stars’ lives. In this paper, we haveconsidered the properties of the neutrino spectrum froma choked jet in a red supergiant star, which is associ-ated with type II SNe/SLSNe. The parameter spaceconsistent with observations is roughly divided into thesoft and hard phases. The soft phase corresponds toa soft part of the spectrum with a cutoff at around afew tens of TeV. This component does not contribute toPeV neutrinos. The hard phase corresponds to a hardspectrum with a cutoff at ∼ PeV. For the choked jetmodel, it is difficult to explain the soft feature of thespectrum and PeV neutrinos using a single component.However, a two-component spectrum can explain the Ice-Cube data. In addition to a soft spectrum contributingto the low-energy neutrinos, a second component, suchas a hard spectrum, neutrinos from AGN cores (Stecker2005) or neutrinos from distant blazars (Kalashev et al.02013) contributing to the PeV neutrinos, is needed. Weplot one-component spectra and two-component spectrain Figures 4, and 5, respectively. The contribution from afast photo-meson cooling case softens the neutrino spec-trum, which is consistent with the currently observed softspectrum. Another predicted feature is the low-energycutoff at ∼
20 TeV, as discussed in Section 5.1. A possi-ble cutoff around PeV can be used to test the origin ofthe second component.Neutrino oscillations can affect the observed flavorcomposition depending on whether the neutrinos gothrough a stellar envelope (Sahu & Zhang 2010). Hence,the future precise observations on the neutrino flavor ra-tio can be another way to test the choked jet model.Furthermore, our model predicts newly born jet-driventype-II SNe/SLSNe associated with the production ofneutrinos. The detection of such SNe associated withan observation of a muon neutrino singlet, doublet, ortriplet will be a strong evidence in favor of the chokedjet model.We thank Ruo-Yu Liu, Hirotaki Ito, Ji-An Jiang, Ko-hta Murase and Peter M´esz´aros for the useful comments.This work is supported in part by the Mitsubishi Founda-tion, a RIKEN pioneering project Interdisciplinary The-oretical Science (iTHES) and Interdisciplinary Theoreti-cal & Mathematical Science Program (iTHEMS). H.N.H.is supported by National Natural Science of China un-der grant 11303098, and the Special Postdoctoral Re-searchers (SPDR) Program in RIKEN. The work ofA.K. is supported by the U.S. Department of EnergyGrant No. de-sc0009937 and by the World Premier In-ternational Research Center Initiative (WPI), MEXT,Japan. S.N. is supported by JSPS (Japan Societyfor the Promotion of Science): No.25610056, 26287056,MEXT (Ministry of Education, Culture, Sports, Scienceand Technology): No.26105521, and Mitsubishi Foun-dation in FY2017. Y.Z.F. is supported by 973 Pro-gram of China under grant 2013CB837000. D.M.W. andY.Z.F. are supported by 973 Programme of China (No.2014CB845800), by NSFC under grants 11525313 (theNational Natural Fund for Distinguished Young Schol-ars), 11273063, 11433009 and 11763003.REFERENCES
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