GRBs as ultra-high energy cosmic ray sources: clues from Fermi
aa r X i v : . [ a s t r o - ph . H E ] M a r GRBs as ultra-high energy cosmic raysources: clues from
Fermi
C. Dermer
Naval Research Laboratory, Code 7653, 4555 Overlook Avenue, SWWashington, DC 20375-5352 USA
Abstract
If gamma-ray bursts are sources of ultra-high energy cosmic rays,then radiative signatures of hadronic acceleration are expected inGRB data. Observations with the
Fermi Gamma-ray Space Telescope(Fermi) offer the best means to search for evidence of UHECRs inGRBs through electromagnetic channels. Various issues related toUHECR acceleration in GRBs are reviewed, with a focus on the ques-tion of energetics.
Fermi observations of GRBs provide a new probe of particle acceleration inthe relativistic outflows of GRBs. Some generic features of the high-energybehavior of 10 Large Area Telescope (LAT) GRBs consisting of 8 long-softand 2 short-hard GRBs at the time of this conference have been identified,as summarized by Omodei [1]. These are:1. A delayed onset of the > ∼
100 MeV radiation observed with the LATcompared with the start of the keV/MeV GBM radiation;2. Long-lived LAT emission extending well after the Gamma ray BurstMonitor (GBM) radiation has fallen below background, as known pre-viously for long duration GRBs from EGRET [2];1. Existence of a distinct hard spectral component in addition to acomponent described by the Band function in both long and shortGRBs, confirming the discovery of such a component from jointBATSE/EGRET TASC analyses [3].In this contribution, the possibility that these features can be explainedby UHECRs in GRBs is considered. Because protons and ions are weaklyradiative compared to electrons, even with escaping energies E ≈ eVneeded to explain UHECRs, large amounts of nonthermal hadronic energyare required. We consider energetics arguments to constrain models forUHECRs from GRBs. Bohdan Paczy´nski, in his seminal article on hypernovae [4], proposed an ex-planation for the large apparent isotropic energies of GRBs by appealing tothe enormous energy available in the process whereby the core of a massivestar collapses to form a long GRB. He noted that the ∼
10 M ⊙ core of aType II supernovae carries ≈ × erg of rotational energy, which couldbe tapped through Blandford-Znajek processes if surrounded by a highlymagnetized torus formed during the stellar collapse event. Paczy´nski’s ar-guments suggest that the maximum energy available from core collapse su-pernovae is therefore E max < ∼ erg , (1)with perhaps an order-of-magnitude less in the coalescence events thoughtto form the short hard GRBs.The association of long-duration GRBs with Type Ib/c rather than TypeII supernovae does not alter his energetics argument, but the recognitionthat GRBs are jetted greatly reduces the absolute energy requirements. En-ergy extracted through Blandford-Znajek processes is likely beamed in viewof the rapid rotation of the newly formed black hole. Even so, we can stillapply the energy bound given by eq. (1) to determine the plausibility ofmodels with large energy requirements. The idea that GRBs could accelerate UHECRs was proposed by Waxman[5] and Vietri [6] in 1995, and subject to criticism on energetics grounds[7], though here related to whether GRBs within the GZK radius have theequisite volume- and time-averaged luminosity density. We can reformulatetheir argument in light of new data. The Auger results show that the differ-ential energy density of UHECRs at 10 eV is ≈ × − erg cm − . Thehorizon distance for 10 eV UHECR protons is ≈
50 Mpc (shorter thanthe mean-free-path of ≈
140 Mpc, because this is the distance from whichprotons with measured energy E originally had energy ≈ . E ), so that therequired emissivity to power > ∼ eV UHECRs is ˙ ε CR ≈ c × × − ergcm − /50 Mpc ≈ × erg Mpc − yr − . If UHECRs are protons, then˙ ε CR ≈ ˙ ε erg Mpc − yr − , with ˙ ε ≃
1, is required to power > ∼ eV UHECRs, noting that the exponentially decreasing horizon distance withenergy approximately cancels the ∝ E − decrease in UHECR energy densityover this energy range.With long-duration GRBs taking place about twice per day over the fullsky for BATSE-type detection capabilities [8], and are found at a typicalredshift of unity, implying a local space density for unbeamed sources of ∼ ×
365 yr − ζ/ [4 π (4 Gpc) / ≈ . ζ/ .
1) Gpc − yr − , consistent withthe value of ≈ . − yr − found in more detailed treatments [9]. Here ζ accounts for the smaller star-formation activity occurring at z ≈ E CR inUHECRs, then the local volume- and time-averaged cosmic-ray emissivity is˙ ε CR ≃ × − E CR ( ζ/ .
1) Mpc − yr − . Equating this with the emissivityrequired to power the UHECRs implies that each GRB must release E CR ≈ × ˙ ε ( ζ/ .
1) erg in UHECRs. This can be compared with the typicalelectromagnetic energy release per GRB of 4 π (4 Gpc) − F − erg cm − ≈ × F − erg, where F = 10 − F − erg cm − is the average BATSE long-duration GRB fluence. Thus the energy released in UHECRs has to be > ∼ × the energy measured from electromagnetic processes, independentof beaming.The need for large baryon loads in GRB blast waves is confirmed bydetailed fits to the UHECR energy spectrum, which imply a factor ≈
10 – 100times more energy in UHECRs than observed in electromagnetic radiation[11]. An explanation for the rapid declines in
Swift
X-ray light curves interms of escaping UHECR neutrons from photohadronic production alsorequires highly baryon-loaded GRB outflows [12]. A δ -function approximation for the γγ opacity constraint, given the detec-tion of a γ -ray photon with energy m e c ǫ and variability time t v , implies ainimum bulk Lorentz factorΓ min ≈ " σ T d L (1 + z ) f ˆ ǫ ǫ t v m e c / , ˆ ǫ = 2Γ (1 + z ) ǫ . (2)where f ǫ is the νF ν flux at photon energy m e c ǫ . Thus Γ min ≈ f − ǫ (3 GeV) /t v (s)] / , where the νF ν flux is 10 − f − erg cm − s − ,using values corresponding to time bin b for GRB 080916C [13]. This GRB,at redshift z ∼ = 4 . ± .
15 and luminosity distance d L = 1 . × cm, hada total 10 keV – 10 GeV energy fluence F = 2 . × − erg cm − , implyinga total γ -ray energy release E γ,iso ∼ = 8 . × erg.Provided the target photon number spectrum has an index softer than −
1, eq. (2) gives a result within ∼
10% of a numerical integration overspectral parameters. Issues in the evaluation of the uncertainty, ∆Γ min , inthe value of Γ min include (1) uncertainties in redshift and νF ν spectral flux;(2) the definition of variability time t v ; (3) the cospatial assumption that thetarget photons are made in the same region as the high-energy photon; (4)the assumed geometry and dynamical state [14] of the emission region, givingthe escape probability of a high-energy photon; (5) statistical fluctuationsfor the detection of a high-energy photon that furthermore depend on theactual high-energy photon spectrum.Writing the GRB blast wave Lorentz factor Γ = q Γ min , a considerationof these various issues show that q > ∼ . The total apparent energy release of GRB 080916C is E iso = 10 E erg,with E > ∼
1. The corresponding deceleration length is r dec = 1 . × E / /n Γ / cm, where 10 Γ is the coasting Lorentz factor and n (cm − )is the external medium density. The implied radius for internal shock emis-sion is r ∼ = Γ ct v / (1 + z ) ∼ = 6 × Γ t v (s) cm. The unexpectedly largeemission radius, close to the deceleration radius when q ≈
2, has led a num-ber of researchers to argue that the LAT radiation originates from leptonicsynchrotron radiation at an external forward shock [15, 16]. Another pos-sibility is that the delayed LAT emission results from upscattered cocoonradiation [17]. γ rays from UHECRs in GRB blast waves We can make some simple estimates to deduce the total energy needed to ob-tain bright hadronic γ -ray emission from GRBs through proton synchrotronand photopion processes [18, 19, 20]. Here we consider only UHECR protonacceleration, using parameters appropriate to GRB 080916C.The internal photon energy density u ′ γ ∼ = d L Φ /R Γ c , where Φ =10 − Φ − erg cm − s − is the measured energy flux, R ∼ = Γ c ˆ t/ (1 + z ) is theshock radius, and ˆ t is a fiducial timescale (corresponding to t v for internalshocks, or the GRB duration for an external shock). Writing the magnetic-field energy density u ′ B = ǫ B u ′ γ , where u ′ B = B / π , then the magneticfield in the emission region of GRB 080916C is B (kG) ∼ = 2 . √ ǫ B Φ − / Γ ˆ t (s).The Hillas criterion whereby the Larmor radius r ′ L = m p c γ ′ p /eB < ∆ R ′ ∼ = R/ Γ, where ∆ R ′ is the comoving shell width, implies that the escap-ing UHECR proton Lorentz factor γ p ∼ = Γ γ ′ p < ∼ ceB Γ ˆ t/ (1 + z ) m p c . ForGRB 080916C, this relation implies that UHECRs can be accelerated to γ p < ∼ (2 d L e/ Γ m p c ) p πǫ B Φ /c ∼ = 4 × √ ǫ B Φ − / Γ , independent of time(note that B ∝ t − , with the divergence at t → E p < ∼ × B (kG)Γ ˆ t ( s ) eV.A further restriction on the maximum proton energy is obtained by bal-ancing the acceleration rate, given by c/ ( φr ′ L ), with φ ≫
1, with the syn-chrotron loss rate. This gives γ sat,p ∼ = 2 × Γ / p ( φ/ B (kG), compa-rable to the value obtained above from the Hillas condition for the chosenparameters. The proton synchrotron energy loss timescale, as measured by an observer,is t syn ∼ = 3 m e c (1 + z ) / [4Γ µ cσ T u ′ B γ ′ p ], where µ ≡ m e /m p . The typicalphoton energy (in m e c units) of the measured proton synchrotron emissionis ǫ p,syn ∼ = Γ µBγ ′ p / [(1 + z ) B cr ], where B cr = 4 . × G is the criticalmagnetic field. From this, one obtains the jet power associated with themagnetic field, given by L B ∼ = R c Γ B ∼ = 2 × Γ / t / syn (s) E γ (100 MeV) / erg s − (3)[19]. The absolute energy requirements are E abs ∼ = L B t syn f b / (1 + z ) ∼ =1 . × Γ / f b t / syn (10 s) /E γ (100 MeV) / erg, where f b is a beaming fac-tor. In the scenario of Ref. [18], the delayed onset corresponds to the timeor protons to accumulate and cool such that they are radiating most of theproton-synchrotron photons near energy E γ Such large energies disfavor this interpretation for the LAT radia-tion. To salvage the proton synchrotron model [18], a narrow jet open-ing angle of order 1 ◦ along with a value of q ∼ = 0 . E abs ∼ =4 × (Γ / . / ( f b / − ) t / syn (10 s) /E (100 MeV) / erg, within accept-able ranges. Another way to avoid the large energy losses is to suppose thatthe UHECR protons are accelerated to their allowed maximum energy andradiate proton synchrotron photons that cascade to energies < ∼
100 MeV.This possibility seemed unlikely given that the spectrum of GRB 080916C isconsistent with a single Band function [13]. The detection of distinct com-ponents in GRB spectra suggests that a cascading interpretation be morecarefully considered [20, 21].
Proton-photon interactions making secondary pions, γ rays, and neu-trinos represents another likely channel for making a γ -ray componentidentifiable in the Fermi data. The efficiency for extracting the energyof a proton with escaping energy E p from photohadronic processes canbe written as η pγ ( E p ) = t dyn /t pγ ( E p ) ∼ = ( R/ Γ c ) t − pγ ( E p ), where t dyn isthe dynamical time scale, the rate for photohadronic energy losses is t − pγ ( E p ) ∼ = c ( K pγ σ pγ ) R ∞ ǫ ′ thr dǫ ′ n ′ ( ǫ ′ ), and the comoving photon spectrum n ′ ( ǫ ′ ) ∼ = d L f ǫ / ( m e c ǫ ′ R Γ ), with ǫ ′ ∼ = (1 + z ) ǫ/ Γ. Here K pγ σ pγ ∼ = 70 µ babove threshold photon energy defined by γ ′ ǫ ′ ∼ = 400 [11]. Defining protonswith energy E pkp which interact at threshold with photons with energy ǫ pk at the peak f ǫ pk of the νF ν spectrum, then E pkp ∼ = 400 m p c Γ / [(1 + z ) ǫ pk ∼ =7 × Γ /ǫ pk eV. One obtains η pγ ( E pkp ) = K pγ σ pγ d L f ǫ pk Γ m e c t v (1 − b ) ǫ pk ∼ = 0 . f − Γ t v (s) ǫ pk (4)[5, 22, 23]. Here b ( <
0) is the νF ν index above ǫ pk , so that f ǫ = f ǫ pk ( ǫ/ǫ pk ) b when ǫ > ǫ pk , and η pγ ( E p ) ∼ = η pγ ( E pkp )( E p /E pkp ) − b . Likewise in the asymp-totic limit ǫ ≪ ǫ pk , η pγ ( E p ) ∼ = η pγ ( E pkp )( E p /E pkp ) − a , where a is the νF ν index at energies below the peak energy (provided a > E − p spectrum is assumed, thenthe efficiency is reduced in proportion to the number of decades of weaklyradiating low-energy protons. This might not be too severe if the lowestnergy protons have escaping energy E p ≈ Γ m p c ≈ Γ PeV. Furthermore,nonlinear shock acceleration in colliding shells, and second-order processesin the shocked fluid can give harder cosmic-ray spectra, though consistencywhen fitting the measured UHECR spectrum would constrain the assumedaccelerated cosmic-ray spectrum.The final expression in eq. (4), specific to parameters of GRB 080916C,shows that ∼ E p ≈ E pkp , with more than half of this energyreleased in the form of leptons and photons which generates an electro-magnetic cascade emerging in the form of γ rays when the system becomesoptically thin to γγ processes. At E p ≫ E pkp , a larger fraction is extracted.Consequently, the energy of baryons must be ≈
10 to 100 × greater than thenonthermal lepton content if a comparable amount of electromagnetic radia-tion is to be emitted from hadronic as leptonic processes. Note the sensitiveΓ-dependence of the photohadronic efficiency, η pγ ( E pkp ) ∝ Γ − , by compari-son with τ γγ ( ǫ ) ∝ ∼ Γ − . The implications for neutrino and γ -ray productionare considered in [24]. In this contribution, we have sketched the energy requirements for GRBs tobe sources of UHECRs, and for electromagnetic signatures of ultrarelativis-tic hadrons to be found in the
Fermi data. The large amounts of energyneeded has been noted many times in the past, whether from proton syn-chrotron [25, 26] or photohadronic processes. Although the large energyrequirements make uncomfortable demands on γ -ray emission models, aninternal consistency is found insofar as the baryon load in long GRBs mustbe large given their relative rarity within the GZK radius, and that enor-mous energies are available from the rotational and accretion energy in thenewly forming black holes. Future Fermi observations and the possibility ofdetecting PeV neutrinos from GRBs with IceCube could establish whetherGRBs are the sources of UHECRs.
Acknowledgments:
I would like to thank Guido Chincarini and BingZhang for their kind invitation to the Venice Shocking Universe conference,Soeb Razzaque for discussions, and Justin Finke for a careful reading of themanuscript. This work is supported by the Office of Naval Research. eferences [1] OMODEI N., these proceedings[2] HURLEY K. ET AL.,
Nature , (1994) 652.[3] GONZ ´ALEZ M. ET AL., Nature , (2003) 749.[4] PACZY ´NSKI B., Astrophys. J. Lett. , (1998) L45.[5] WAXMAN E., Phys. Rev. Lett. , (1995) 386.[6] VIETRI M., Astrophys. J. Lett. , (1995) 883.[7] STECKER F., Astropart. Phys. , (2000) 207.[8] BAND D., Astrophys. J. , (2002) 806.[9] GUETTA D., PIRAN T., WAXMAN E., Astrophys. J. , (2005) 412.[10] Y ¨UKSEL H., KISTLER M., BEACOM J., HOPKINS A., Astrophys.J. Lett. , (2008) L5.[11] DERMER C. AND ATOYAN A., New J. Phys. , (2006) 122.[12] DERMER C., Astrophys. J. , (2007) 384.[13] ABDO A. ET AL., Science , (2009) 1688.[14] GRANOT J., COHEN-TANUGI J., DO COUTO E SILVA E., Astro-phys. J. , (2008) 92.[15] KUMAR P. AND BARNIOL-DURAN, R., MNRAS , (2009) L75.[16] GHISELLINI G., GHIRLANDA G., NAVA L., CELOTTI, A., MNRAS ,in press (2009) arXiv:0910.2459.[17] TOMA K., WU X.-F., M´ESZ ´AROS P.,
Astrophys. J. , (2009) 1404.[18] RAZZAQUE S., DERMER C., FINKE J., 2009, Open Astronomy J. ,in press (2010) arXiv:0908.0513.[19] WANG X.-Y., LI Z., DAI Z.-G., M´ESZ ´AROS P.,
Astrophys. J. , (2009) L98.[20] ASANO K., GUIRIEC S., M´ESZ ´AROS P., Astrophys. J. , (2009)L191.21] ASANO K., these proceedings[22] DERMER C., Astrophys. J. , (2002) 65.[23] MURASE K., Phys. Rev. Lett. , (2009) 081102.[24] RAZZAQUE S., M´ESZ ´AROS P., ZHANG B., Astrophys. J. , (2004)1072.[25] TOTANI T., Astrophys. J. , (1998) L81.[26] ZHANG B. AND M´ESZ ´AROS, P., Astrophys. J. ,559