TeV-PeV Neutrinos from Low-Power Gamma-Ray Burst Jets inside Stars
aa r X i v : . [ a s t r o - ph . H E ] S e p TeV–PeV Neutrinos from Low-Power Gamma-Ray Burst Jets inside Stars
Kohta Murase and Kunihito Ioka
2, 3 Hubble Fellow – Institute for Advanced Study, Princeton, New Jersey 08540, USA Theory Center, Institute of Particle and Nuclear Studies, KEK, Tsukuba 305-0801, Japan Department of Particles and Nuclear Physics, the GraduateUniversity for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan (Dated: 17 September 2013)We study high-energy neutrino production in collimated jets inside progenitors of gamma-raybursts (GRBs) and supernovae, considering both collimation and internal shocks. We obtain simple,useful constraints, using the often overlooked point that shock acceleration of particles is ineffectiveat radiation-mediated shocks. Classical GRBs may be too powerful to produce high-energy neutrinosinside stars, which is consistent with IceCube nondetections. We find that ultralong GRBs avoidsuch constraints and detecting the TeV signal will support giant progenitors. Predictions for low-power GRB classes including low-luminosity GRBs can be consistent with the astrophysical neutrinobackground that IceCube may detect, with a spectral steepening around PeV. The models can betested with future GRB monitors.
PACS numbers: 95.85.Ry, 97.60.Bw, 98.70.Rz
Long gamma-ray bursts (GRBs) are believed to origi-nate from relativistic jets launched at the death of mas-sive stars. Associations with core-collapse supernovae(CCSNe) have provided strong evidence for the GRB-CCSN relationship [1]. But, there remain many impor-tant questions. What makes the GRB-CCSN connec-tion? How universal is it? What is the central engineand progenitor of GRBs? How are jets launched and ac-celerated? Observationally, it is not easy to probe physicsinside a star with photons until the jet breaks out and thephotons leave the system. This is always the case if thejet is “chocked” rather than “successful” [2]; that is, thejet stalls inside the star, where the electromagnetic signalis unobservable. Such failed GRBs may be much morecommon than GRBs (whose true rate is ∼ − of thatof all CCSNe), and CCSNe driven by mildly relativisticjets may make up a few present of all CCSNe [3–5].Recent observations suggest interesting diversity inthe GRB population. “Low-power GRBs” such as low-luminosity (LL) GRBs [3, 6, 7] and ultralong (UL)GRBs [8, 9] have longer durations ( ∼ –10 s) com-pared to that of classical long GRBs, suggesting differentGRB classes and larger progenitors. While they werelargely missed in previous observations, they are impor-tant for the total energy budget and the GRB-CCSNconnection.Neutrinos and gravitational waves (GWs) can presentspecial opportunities to address the above issues. In par-ticular, IceCube is powerful enough to see high-energy(HE) neutrinos at & Q x = Q/ x in CGS unitwith cosmological parameters of H = 71 km s − Mpc − ,Ω m = 0 .
3, and Ω Λ = 0 . Jet propagation in a star.—
To make a GRB, a jetmust penetrate the progenitor successfully. The jet dy-namics is governed by the jet head, cocoon and collima-tion [22]. The jet is decelerated by a reverse shock whilea forward shock is formed in the stellar envelope. The jethead, which is the shocked region between the two shocks,is controlled by the ram pressure balance between thereverse-shocked jet and forward-shocked envelope [2, 23].This shocked region is so hot to expand sideways to forma cocoon. For a given initial opening angle θ j , if the abso-lute jet luminosity L j is low enough and/or the ambientdensity ̺ a is high enough, the hydrodynamic jet is colli-mated by the cocoon pressure via collimation shocks (seeFig. 1). At time t , the (collimation-)shocked jet becomescylindrical through the collimation shock at [22] r cs ≈ . × cm t / L / j , ( θ j / . − / ̺ − / a, , (1)beyond which the cylindrical, collimated flow has a con-stant Lorentz factor (with Γ cj ≈ θ − j ) because of the flux FIG. 1: The schematic picture of a collimated GRB jet insidea progenitor. CR acceleration and HE neutrino productionmay happen at collimation and internal shocks. The pictureof the radiation-mediated shock is also shown. conservation. The subsequent jet head position r h is r h ≈ . × cm t / L / j , ( θ j / . − / ̺ − / a, . (2)Even if the jet achieves Γ ≫ Γ cj in the star, Γ cj ≈ θ j / . − implies that the collimated jet is radiationdominated. The jet breakout time t bo is determined by r h ( t bo ) = R ∗ , where R ∗ is the progenitor radius.The progenitor of long GRBs has been widely believedto be a star without an envelope, such as Wolf-Rayet(WR) stars with R ∗ ∼ . R ⊙ [24]. Let us approximatethe density profile to be ̺ a = (3 − α ) M ∗ ( r/R ∗ ) − α / (4 πR ∗ )( α ∼ . M ∗ is the progenitor mass [25].Then, taking α = 2 .
5, we obtain r cs ≈ . × cm t / L / , ( θ j / . / ( M ∗ / M ⊙ ) − / R / ∗ , and r h ≈ . × cm t / L / , ( θ j / . − / ( M ∗ / M ⊙ ) − / R / ∗ , [22], where L = 4 L j /θ j is the isotropictotal jet luminosity. The GRB jet is successful if t bo ≈
17 s L − / , ( θ j / . / ( M ∗ / M ⊙ ) / R / ∗ , isshorter than the jet duration t dur . With t dur ∼
30 s, wetypically expect r cs ∼ cm for classical GRBs [26].The comoving proton density in the collimated jetis n cj ≈ L / (4 πr Γ cj ηm p c ) = L/ (4 πr Γ cj Γ m p c ) ≃ . × cm − L r − , Γ − (5 / Γ cj ). Here, L = (Γ /η ) L , L is the isotropic kinetic luminosity, and η is the maxi-mum Lorentz factor. The density in the precollimatedjet at the collimation or internal shock radius r s is n j ≈ L/ (4 πr s Γ m p c ) ≃ . × cm − L r − s, Γ − ,which is lower than n cj due to Γ ≫ Γ cj . This quantity isrelevant in discussions below. Note that inhomogeneitiesin the jet lead to internal shocks, where the Lorentz fac-tor can be higher (Γ r ) and lower (Γ s ) than Γ ≈ √ Γ r Γ s . Radiation constraints.—
Efficient CR acceleration atinternal shocks and the jet head has been suggested,since plasma time scales are typically shorter than anyelastic or inelastic collision time scale [12–14]. How-ever, in the context of HE neutrinos from GRBs, it hasoften been overlooked that shocks deep inside a starmay be radiation mediated [27]. At such shocks, pho-tons produced in the downstream diffuse into the up-stream and interact with electrons (plus pairs). Thenthe upstream proton flow should be decelerated by pho-tons via coupling between thermal electrons and pro- tons [28]. As a result (see Fig. 1), one no longer ex-pects a strong shock jump (although a weak subshockmay exist [29]), unlike the usual collisionless shock, andthe shock width is determined by the deceleration scale l dec ≈ ( n u σ T y ± ) − ≃ . × cm n − u, y − ± when thecomoving size of the upstream flow l u is longer than l dec .Here n u is the upstream proton density, and y ± ( ≥
1) isthe possible effect of pairs entrained or produced by theshock [30].In the conventional shock acceleration, CRs are in-jected at quasithermal energies [31]. The Larmor ra-dius of CRs with ∼ Γ m p c is r uL ∼ Γ m p c / ( eB ) ≃ . × − cm ǫ − / B L − / , r s, Γ Γ , where B is themagnetic field, Γ rel is the relative Lorentz factor and ǫ B ≡ L B /L [32]. If the velocity jump of the flow is smallover r uL , the CR acceleration is inefficient. For l dec ≪ l u ,since significant deceleration occurs over ∼ l dec , includ-ing the immediate upstream [28, 29], CRs with r uL ≪ l dec do not feel the strong compression and the shock accel-eration will be suppressed [27, 33, 34]. CRs are expectedwhen photons readily escape from the system and theshock becomes radiation unmediated, which occurs when l u . l dec [30, 36]. Regarding this as a reasonably neces-sary condition for the CR acceleration, we have τ uT = n u σ T l u . min[1 , . C − Γ rel ] , (3)where C = 1 + 2 ln Γ is the possible effect by pair pro-duction [29], although it may be small when photons startto escape. Since the detailed pair-production effect is un-certain, τ uT . L r cs , Γ − . . × − min[1 , . C − Γ rel ] , (4)where n u = n j , l u ≈ r cs / Γ, and Γ rel ≈ (Γ / Γ cj + Γ cj / Γ) / n cj ≫ n j and Γ cj ≪ Γ.We can also apply Eq. (3) to internal shocks in theprecollimated jet, which have been considered in theliterature [12, 13]. Internal shocks may occur above r is ≈ s cδt ≃ . × cm Γ s, . δt − , and the relativeLorentz factor between the rapid and merged shells isΓ rel ≈ (Γ r / Γ + Γ / Γ r ) /
2, which may lead to the upstreamdensity in the rapid shell ∼ n j / Γ rel . Using l u ≈ r is / Γ r ∼ l/ Γ rel , we get τ T = n j σ T l . min[Γ , . C − Γ ] or L r is , Γ − . . × − min[Γ , . , . C − Γ , . ] . (5)As shown in Fig. 3, unless Γ & , it seems difficult toexpect CRs and HE neutrinos for high-power jets insideWR-like progenitors (where r is . r cs ∼ cm). Notethat although the constraint is relevant for shocks deep !!" + , - . (cid:1) / +,-.0 123-456/3 $! $$" !" FIG. 2: Lower limits on Γ for given L , above which CRsand HE neutrinos can be expected from the collimation shock(without mediation by radiation) at r cs . Thick (thin) curvesrepresent cases without (with) the possible pair effect withthe approximation of C ≃
10. Typical parameters of classicalGRBs and UL GRBs are depicted. (cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:4)(cid:2)(cid:3)(cid:5)(cid:5)(cid:2)(cid:3)(cid:6)(cid:6)(cid:2)(cid:3)(cid:7)(cid:7)(cid:8) (cid:7)(cid:9) (cid:7)(cid:10) (cid:3)(cid:1) (cid:3)(cid:4) (cid:3)(cid:5) (cid:3)(cid:6) (cid:11) (cid:12) (cid:13) (cid:14) (cid:1) (cid:15) (cid:11)(cid:12)(cid:13)(cid:14)(cid:16) (cid:17)(cid:18)(cid:19)(cid:13)(cid:20)(cid:21)(cid:22)(cid:15)(cid:19) (cid:23)(cid:21) (cid:24)(cid:4)(cid:1) (cid:4)(cid:1) (cid:25)(cid:26)(cid:19) (cid:23)(cid:21) (cid:24)(cid:4)(cid:1) (cid:4)(cid:4)(cid:2)(cid:3) (cid:25)(cid:26) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:8)(cid:12)(cid:13)(cid:14)(cid:15)(cid:12)(cid:16)(cid:17)(cid:17)(cid:18)(cid:19)(cid:20)(cid:14)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:8)(cid:26)(cid:12)(cid:14)(cid:15)
FIG. 3: The same as Fig. 2, but for the internal shock (withΓ rel = 3) at r is . The IceCube upper limit on the slow-jetmodel for SN 2008D [21] is also shown for t dur = 100 s. inside the stars we here consider, CRs may be expectedaround the photosphere τ T ∼ ∼ E j = 2 L j t dur model dependently [21], and its upper limitis shown in Fig. 3 after converting E j to L . We may ex-pect HE neutrinos from sufficiently low-power jets with L . . –10 erg s − for WR-like progenitors. Ultralong GRBs.—
As seen above, efficient CR accel-eration may not occur in high-power jets inside WR-likeprogenitors. However, the situation is different for less-power GRBs such as UL GRBs [8, 9] and LL GRBs [3].In particular, UL GRBs are as energetic and possiblycommon as classical GRBs [8]. Their lower-luminosity L γ ∼ –10 erg s − and longer duration t dur ∼ ssuggest bigger progenitors like blue supergiants (BSGs) with R ∗ ∼ –10 cm [9, 40].Assuming a stellar envelope with ̺ a ( r ) =10 g cm − ̺ BSG r − [41], with Eqs. (1) and (2),we obtain r cs ≃ . × cm t L / , . ( θ j / . ̺ − / , r h ≃ . × cm t L / , . ( θ j / . − / ̺ − / , and t bo ≃ L − / , . ( θ j / . / ̺ / R ∗ , (comparable to t dur ). We may typically expect the collimation shock at r cs ∼ . cm.Interestingly, thanks to lower powers and larger shockradii, the Thomson optical depth is low even in a star( τ T ≈ . L . r − s, . Γ − ), allowing the CR accelerationand HE neutrino production, as indicated in Figs. 2 and3. The jet may be sufficiently accelerated by r cs [42],while should slow down to Γ cj after the collimation.Once CRs are accelerated inside a star, the CR power islost to meson production via the pγ reaction with targetphotons or the pp reaction with target nucleons, lead-ing to precursor or orphan neutrinos. We consider twopossibilities: HE neutrinos from CRs accelerated at thecollimation shock (CS) and HE neutrinos from CRs ac-celerated at the internal shock in the precollimated jet(IS).In the CS scenario, CRs are conveyed in the collimatedjet with Γ cj and completely depleted during the advec-tion for R adv /c ≈ min[ R ∗ , r h ( t dur )] /c . Using the photontemperature kT cj ≈ .
70 keV L / , . r − / , . (Γ cj / − / in the collimated jet and ˆ σ pγ ≈ . × − cm , weobtain the high pγ efficiency f cj pγ ≈ n cj γ ˆ σ pγ ( R adv / Γ cj ) ≃ . × L / , . r − / , . (Γ cj / − / R adv , ≫
1. The pp efficiency is also high since f cj pp ≈ n cj ˆ σ pp ( R adv / Γ cj ) ≃ L . r − , . Γ − (Γ cj / − R adv , ≫
1. CRs are de-pleted essentially in the entire energy range, so the sys-tem is “calorimetric” and HE neutrinos are unavoid-able. Since the formation of collimation shocks is alsoquite common for relativistic jets inside stars, HE neu-trinos from UL GRBs can be used as signatures ofjets in big progenitors. Note that due to copious tar-get photons, the maximum energy in the accelerationzone ε Mp is limited by the pγ reaction. By comparingthe acceleration time t acc ≈ ε p / ( eBc ) to the pγ cool-ing time t pγ ≈ / ( n cj γ ˆ σ pγ c ), we obtain ε Mp ≃ . × GeV B . L − / , . r / , . (Γ cj / / .In the IS scenario, during the dynamical time, CRsmainly interact with photons escaping back from thecollimated jet. Using the photon density n jγ ≈ (Γ / cj )( f esc n cj γ ) [where f esc ∼ ( n cj σ T r cs / Γ cj ) − is theescape fraction], which is boosted by Γ rel ∼ Γ / cj ,we have f jpγ ≈ (Γ / cj )( f esc n cj γ )ˆ σ pγ ( r is / Γ) ≫
1, so HECRs with f jpγ & f jpγ ∼ pγ threshold energy, ε th p ≃ . × GeV L − / , . r / , . Γ − (Γ cj / / . On theother hand, since n j is small, the pp efficiency is too lowto be relevant. As in the CS scenario, ε Mp is limited bythe pγ process, leading to ε Mp ≈ eB/ ( n jγ ˆ σ pγ ). (cid:1)(cid:2)(cid:3)(cid:1)(cid:2)(cid:2)(cid:1)(cid:2)(cid:4)(cid:1)(cid:5)(cid:1)(cid:6)(cid:1)(cid:7)(cid:1)(cid:8)(cid:1)(cid:9) (cid:10) (cid:11) (cid:9) (cid:8) (cid:7) (cid:6) (cid:12) (cid:13) (cid:14) (cid:15) (cid:16) (cid:1) (cid:3) (cid:2) (cid:1) (cid:17) (cid:18) (cid:19) (cid:20) (cid:21) (cid:22) (cid:1) (cid:3) (cid:23) (cid:1) (cid:2) (cid:23) (cid:24) (cid:1) (cid:2) (cid:25)(cid:26) (cid:12)(cid:13)(cid:14)(cid:15)(cid:16) (cid:1) (cid:17)(cid:18)(cid:19)(cid:20)(cid:25)(cid:26)(cid:27)(cid:13)(cid:28)(cid:1)(cid:27)(cid:29)(cid:22)(cid:30)(cid:31)(cid:13)(cid:23)(cid:30) ! (cid:18)" (cid:1)(cid:2)(cid:3)(cid:4) (cid:5)(cid:6)(cid:7)(cid:3)(cid:6)(cid:8)(cid:9)(cid:10)(cid:6)(cid:11)(cid:12)(cid:6)(cid:13)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:8)(cid:18) FIG. 4: The cumulative neutrino backgrounds from UL GRBsand LL GRBs. For UL GRBs, we use r s = 10 . cm, Γ cj = 5, kT cj ≃ .
70 keV, Γ = 100 and L = 10 erg s − . The CR en-ergy generation rate is set to ξ acc E iso γ ρ = 10 erg Gpc − yr − ,with f cho = 1 (thick) and f cho = 10 (thin). For compar-ison, predictions for prompt emission from LL GRBs (with ρ = 500 Gpc − yr − and ξ acc = 10) are taken from Ref. [6]for Γ = 10 (thick) and Γ = 5 (thin). For redshift evolu-tion, the GRB3 model is assumed [44]. The atmosphericbackground [47] is also shown. Note that IceCube suggests E ν Φ ν ∼ a few × − GeV cm − s − sr − [11], which is com-patible with the original Waxman-Bahcall bound [45]. We calculate neutrino spectra, using the numericalcode developed in Refs. [6, 38, 44], where pγ/pp reac-tions and relevant cooling processes are considered in de-tail. Note that we consistently evaluate ε Mp by compar-ing t acc with all relevant competing time scales. We get ε Mp ∼ . GeV and ε Mp ∼ . GeV in the CS and ISscenarios, respectively. Then, we calculate depletion ofCRs and neutrino spectra, assuming a CR spectrum of ε − p e − ε p /ε Mp . The parameters are shown in Fig. 4. Weassume ǫ B = 1 in the IS scenario, while L cj B = 10 − L inthe CS scenario since the collimated jet is radiation dom-inated and its magnetic luminosity would be smaller thanthe kinetic luminosity, but key results are not sensitivewhen the meson synchrotron cooling is subdominant (cf.Ref. [13]).The expected number of neutrino events from a burstat z = 0 . ∼
1, so aggregating many burstsis important. Alhough it is hard for current satellitesto find many low-power GRBs, we can in principle testthe scenarios by stacking neutrino signals from &
100 ULGRBs at z ∼
1, which are detectable by all-sky monitorswith sensitivities better than
Swift .To demonstrate their neutrino spectra and contribu-tions, we numerically calculate the total ENB [44], whichis consistent with the following analytical formula [6, 45]: E ν Φ ν ∼ c πH f sup min[1 , f pγ ] E p dN iso p dE p ρf z f cho (6) ∼ × − GeV cm − s − sr − ( f cho ξ acc / f sup × min[1 , f pγ ]( E iso γ ρ/ erg Gpc − yr − )( f z / , where f z is the evolution factor [45], f sup is the suppres-sion factor due to the meson and muon cooling [38], ξ acc is the CR loading parameter [6], and f cho is the fractionof failed GRBs compared to successful GRBs. Here, ρ isthe local rate that is ∼ − yr − for GRBs and ULGRBs [8] (but see Ref. [9]) while ∼ –10 Gpc − yr − for LL GRBs [3].Results are shown in Fig. 4, where we see that theENB flux from successful UL GRB jets inside stars maybe ∼ − GeV cm − s − sr − . If failed UL GRBs are &
10 times more common, ∼ − GeV cm − s − sr − may even be achieved. Although the uncertainty in ρ islarge, contributions from LL GRBs [6, 7, 30] and/or failedUL GRBs can be compatible to the ENB that IceCubemay start to observe [11]. The spectral steepening is alsoexpected. In particular, in the IS scenario, the mesonradiative cooling or the cutoff from the proton maximumenergy can lead to a break around PeV. In addition, forchoked jets in BSGs, the cutoff at & r h & × cm.In the CS scenario, strong meson cooling leads to a breakat .
10 TeV, so we mainly expect multi-TeV neutrinos.
Summary and discussion.—
We derived general con-straints on HE neutrino production in GRB jets insidestars, based on the point that the shock acceleration is in-efficient at radiation-mediated shocks. They are comple-mentary to observational upper limits, and current non-detections of precursor (orphan) neutrinos from GRBs(CCSNe) are consistent with theoretical expectations.Our work is encouraging and useful for the literatureon the GRB-CCSN connection [15], joint searches withGWs [16], and neutrino mixing [17].We showed that more favorable conditions for HE neu-trino production are satisfied in low-power GRBs such asUL GRBs especially if they originate from bigger pro-genitors like BSGs. The formation of collimation shocksis naturally expected, so TeV neutrinos are useful as asmoking gun of jet physics that cannot be probed withphotons, and will also support the idea of BSG-like pro-genitors. We stress the importance of stacking such lessluminous transients with next-generation all-sky moni-tors like SVOM, Lobster, WF-MAXI and HiZ-Gundam.Internal shocks in a precollimated jet could extend theENB to PeV energies, which may give an important con-tribution if failed UL GRBs are &
10 times more com-mon. Note that the neutrino production site consideredin this work is different from the prompt emission site.Since low-power GRBs may be largely missed, even iftheir successful jets give ∼ − GeV cm − s − sr − ,the results may not contradict with nondetections of“prompt” neutrinos from classical GRBs, which placed . − GeV cm − s − sr − [19]. LL GRBs can give ∼ − GeV cm − s − sr − , as predicted in Refs. [6, 7, 30].They are distinct from classical GRBs and they may bemore baryon rich [46]. Since the uncertainty in ρ is large,revealing these transients, which have been largely missedso far, is important to test the models. Acknowledgments.—
K. M. thanks Omer Bromberg,Boaz Katz, Peter M´esz´aros, Tsvi Piran and Eli Wax-man for useful discussions and acknowledges the CCAPPworkshop, Revealing Deaths of Massive Stars with GeV-TeV Neutrinos. This work is supported by NASAthrough Hubble Fellowship, Grant No. 51310.01 awardedby the STScI, which is operated by the Association ofUniversities for Research in Astronomy, Inc., for NASA,under Contract No. NAS 5-26555 (K. M.) and theGrants-in-Aid for Scientific Research No. 24103006, No.24000004, No. 22244030 of MEXT and JSPS (K. I.). [1] M. Modjaz, Astron. Nachr. , 434 (2011); J. Hjorth,Phil. Trans. R. Soc. A , 20120275 (2013).[2] P. M´esz´aros and E. Waxman, Phys. Rev. Lett. et al. , Nature (London) , 1014(2006); K. Toma, K. Ioka, T. Sakamoto, and T. Naka-mura, Astrophys. J. , 1420 (2007); E. Liang et al. ,Astrophys. J. , 1111 (2007).[4] A. M. Soderberg et al. , Nature (London) , 513 (2010);N. Smith et al. , Mon. Not. R. Astron. Soc. , 1135(2012).[5] P. 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