The role of hadronic cascades in GRB models of efficient neutrino production
aa r X i v : . [ a s t r o - ph . H E ] M a y Mon. Not. R. Astron. Soc. , 1– ?? (2014) Printed 22 October 2018 (MN L A TEX style file v2.2)
The role of hadronic cascades in GRB models of efficientneutrino production
Maria Petropoulou , ⋆ Department of Physics and Astronomy, Purdue University, 525 Northwestern Avenue, West Lafayette, IN, 47907, USA NASA Einstein Postdoctoral Fellow
Received.../Accepted...
ABSTRACT
We investigate the effects of hadronic cascades on the gamma-ray burst (GRB) promptemission spectra in scenarios of efficient neutrino production. By assuming a fiducialGRB spectrum and a power-law proton distribution extending to ultra-high energies,we calculate the proton cooling rate and the neutrino emission produced throughphotopion processes. For this, we employ a numerical code that follows the formationof the hadronic cascade by taking into account non-linear feedback effects, such as theevolution of the target photon field itself due to the contribution of secondary particles.We show that in cases of efficient proton cooling and subsequently efficient high-energyneutrino production, the emission from the hadronic cascade distorts and may evendominate the GRB spectrum. Taking this into account, we constrain the allowablevalues of the ratio η p = L p /L γ , where L p and L γ are the isotropic equivalent protonand prompt gamma-ray luminosities. For the highest value of η p that does not leadto the dominance of the cascading emission, we then calculate the maximum neutrinoluminosity from a single burst and show that it ranges between (0 . − . L p and(0 . − . L γ for various parameter sets. We discuss possible implications of otherparameters, such as the magnetic field strength and the shape of the initial gamma-ray spectrum, on our results. Finally, we compare the upper limit on η p derived herewith various studies in the field, and we point out the necessity of a self-consistenttreatment of the hadronic emission in order to avoid erroneously high neutrino fluxesfrom GRB models. Key words: neutrinos – radiation mechanisms: non-thermal – gamma ray burst:general
Gamma-ray bursts (GRBs) are candidate sites of pro-ton acceleration to ultra-high energies (UHE) (Waxman1995; Vietri 1995) and, therefore, potential sources ofhigh-energy (HE) neutrino emission (Paczynski & Xu 1994;Waxman & Bahcall 1997). The problem of GRB neu-trino production has been considered by many authors(Murase 2008; Mannheim et al. 2001; Dermer & Atoyan2003; Guetta et al. 2004; Asano 2005; Rachen & M´esz´aros1998; Baerwald et al. 2013; Reynoso 2014) and it maygain even more interest under the light of the recent Ice-Cube HE neutrino detection (IceCube Collaboration 2013;Aartsen et al. 2014).Another aspect of GRB models for neutrino productionis the formation of hadronic cascades (HC), i.e. cascadesconsisting of relativistic electron-positron pairs, initiated ⋆ E-mail: [email protected] by photopion interactions of UHE protons with the GRBprompt radiation which, in this framework, serves as the tar-get field. The emission produced by such cascades has someinteresting implications as far as the prompt emission is con-cerned. For example, it has been proposed as an alternativeexplanation for the underlying power-law components seenin some bright bursts (e.g. GRB 090902B (Abdo et al. 2009);GRB 080319B (Racusin et al. 2008)), which extend from thehard X-rays up to GeV energies and do not agree with sim-ple extrapolations of the MeV spectrum (Asano et al. 2010).More general studies regarding the emission signatures ofhadronic cascades in the GeV and TeV energy bands andhow these can be used as diagnostic tools of UHECR ac-celeration in GRBs have been made by various authors (e.g.B¨ottcher & Dermer 1998; Asano & Inoue 2007; Asano et al.2009). One common feature of the aforementioned studies isthat the emission produced by the hadronic cascade doesnot dominate over the MeV emission of the burst, whichfurther implies that the proton cooling and neutrino pro- c (cid:13) M. Petropoulou duction through photopion interactions are not very effi-cient. On the other hand, models that focus on the GRBneutrino emission require high efficiency while at the sametime they neglect any effects of the hadronic cascade on theMeV part of the GRB spectrum (e.g. Abbasi et al. 2010;Zhang & Kumar 2013).In the present work we try to bridge the gap betweenthe two approaches by demonstrating, in the most generalway possible, the gradual dominance of the cascading emis-sion over the MeV (and/or GeV) part of the spectrum as theefficiency in neutrino production progressively increases. Forthis, we simulate the GRB prompt emission spectrum as agrey-body photon field, which can be characterized only bytwo parameters, i.e. its effective temperature T and its com-pactness ℓ γ . Using this as a fixed target field, we graduallyincrease the injection luminosity of protons, or equivalently,their compactness, up to that value where the emission fromthe HC begins to affect the gamma-ray spectrum. The domi-nance of the HC sets, therefore, an upper limit on L p and onthe ratio η p = L p /L γ , where L p and L γ are the proton andprompt gamma-ray (isotropic) luminosities, respectively, asmeasured in the observer’s frame. For the maximum value ofthis ratio ( η p , max ) we then calculate the neutrino productionefficiency ( ξ ν ) as well as the efficiency in the injection of sec-ondary pairs and photons ( ξ sec ) into the HC. We show thatthe upper limit of η p is anticorrelated with the ℓ γ , ξ ν and ξ sec . As a next step, we investigate the robustness of our find-ings by repeating the procedure for a higher magnetic fieldstrength and a gamma-ray spectrum that can be describedby the more accurate for GRBs Band function (Band et al.2009). We comment also on the role of the Bethe-Heitlerprocess in the formation of the HC. Finally, we calculate ℓ γ for different parameter sets used in the literature and com-pare the values of ‘proton loading’ used therein, i.e. the ratioof proton to gamma-ray luminosities (or energy densities),with our upper limit η p , max .The present work is structured as follows: in § §
3. We continue in § § § In the present study we do not attempt a self-consistent cal-culation of the GRB prompt emission nor we pinpoint theGRB emission itself. Instead, we focus on the formation ofthe hadronic cascade and the self-consistent calculation ofthe neutrino emission after taking into account the modifi-cation of the initial gamma-ray spectrum. We choose, first,to depict the prompt emission spectrum as a grey-body pho-ton field instead of using the typical Band function for tworeasons: the effects of the cascade emission on the primaryphoton field become more evident and only two parameters,namely the photon compactness ℓ γ and the effective tem-perature T , are required for its description. The effects of a Band-shape photon spectrum on our results will be dis-cussed separately in § r from the central engineprotons are accelerated into a power-law distribution withindex p to ultra-high energies (UHE), e.g. E p < eVin the comoving frame, and are subsequently injected at aconstant rate into a spherical region of size r b that movesoutwards from the central engine with Lorentz factor Γ. Thisregion is equivalent to the shell of shocked ejecta in the in-ternal shock scenario, and has a comoving width r b ≃ r/ Γ(for reviews, see Piran 2004; Zhang & M´esz´aros 2004).We further assume that the region contains a magneticfield of strength B (in the comoving frame), which is usuallyrelated to the jet kinetic luminosity L j through the param-eter ǫ B as follows: ǫ B L j = cB Γ r . (1)Protons with gyroradii larger than the size r b cannot beconfined, and thus escape from the region (Hillas 1984). Thissets a maximum Lorentz factor that is given by γ H = eBr b m p c . (2)In principle, the maximum proton energy is given by theminimum of γ H and γ sat , where the latter denotes the satura-tion energy of the acceleration process due to energy losses.For the parameters used throughout the text, we find that γ H . γ sat (see Appendix B). For this, we set γ max = γ H .The total proton injection luminosity L p , which is just afraction of L j , can be used for defining the proton injectioncompactness as: ℓ injp = σ T L p πr b Γ m p c . (3)Although electron acceleration at high energies is also ex-pected to take place, here, in our attempt to minimize thenumber of free parameters, we assume that the injection lu-minosity of primary relativistic electrons is much lower thanthat of protons, making their contribution to the overall pho-ton emission negligible.The following parameters were kept fixed in all ournumerical simulations, unless stated otherwise: Γ = 225, r b = 10 cm, B = 960 G, γ max = 10 . and T = 10 K.From this point on, we will refer to this parameter set as thebenchmark case. For these values, the typical pulse durationin the internal shock scenario would be δt ≈ r b /c Γ = 0 .
15 s,while ǫ B ≃ × − for L j = 3 × erg/s. For a givenpair of r b and Γ, different values of ℓ γ correspond to dif-ferent γ -ray (isotropic) luminosities ( L γ ), since the photoncompactness is defined as ℓ γ = σ T L γ πr b Γ m e c (4)In particular, we chose the following set of ℓ γ val-ues, namely { . , . , . , . , . } , which corresponds toprompt gamma-ray luminosities in the range L γ = 10 − erg/s with a logarithmic step of 0.5. Then, for each The assumption of a spherical region is valid as long as thebeaming angle 1 / Γ is smaller than the opening angle of the jet,which holds during the GRB prompt phase. The choice of a low ǫ B value will be justified later on.c (cid:13) , 1– ?? RB neutrinos and hadronic cascades value of ℓ γ , we performed a series of numerical simulationswhere we increased ℓ injp over its previous value by a fixedlogarithmic step δx .The method outlined above highlights the main differ-ence between our approach and the one usually adopted inthe literature (e.g. Asano & Inoue 2007; Asano et al. 2009).Here, we use as free parameters the gamma-ray and protoninjection compactnesses instead of Γ, r , δt , L γ and L p , sinceonly the former are the intrinsic quantities of the physicalsystem. Note that very different combinations of Γ , r, δt, L γ may lead to the same ℓ γ and to similar derived properties ofthe leptohadronic system, such as the neutrino productionefficiency. In order to study the formation of the hadronic cascadeand its effects on the multiwavelength photon spectrum, weemploy the time-dependent numerical code as presented inDimitrakoudis et al. (2012) – hereafter DMPR12. This fol-lows the evolution of protons, neutrons, secondary pairs,photons and neutrinos by solving the coupled integro-differential equations that describe the various distribu-tions. The coupling of energy losses and injection intro-duces a self-consistency in this approach that allows thestudy of the system at various conditions, e.g. in the pres-ence of non-linear electromagnetic (EM) cascades (see alsoPetropoulou & Mastichiadis 2012 for a relevant discussion).Although details can be found in DMPR12, for the sake ofcompleteness, we summarize in Table 1 the physical pro-cesses that are included in the code.Photohadronic interactions are modelled using theresults of Monte Carlo simulations. In particular, forBethe-Heitler pair production the Monte Carlo resultsby Protheroe & Johnson (1996) were used (see alsoMastichiadis et al. 2005). Photopion interactions were in-corporated in the time-dependent code by using the resultsof the Monte Carlo event generator SOPHIA (M¨ucke et al.2000), which takes into account channels of multipion pro-duction for interactions much above the threshold.Synchrotron radiation of muons was not included in theversion of the code presented in DMPR12 and for the exacttreatment we refer the reader to Dimitrakoudis et al. (2014).As synchrotron cooling of pions is not yet included in thenumerical code, we restricted our analysis to cases where theeffects of pion cooling are minimal.Pairs that cool down to Lorentz factors γ ∼ kT e ≪ m e c (Lightman & Zdziarski1987). For the pair annihilation and photon downscatter-ing processes we followed Coppi & Blandford (1990) andLightman & Zdziarski (1987), respectively (for more detailssee Mastichiadis & Kirk 1995). For completeness, we notethat the limits on η p derived in § -3-2.5-2-1.5-1-0.5 0 0.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 l og ( < t p γ - > / t d - ) log l p l γ =0.07l γ =0.7l γ =7 Figure 1.
Photopion cooling rate (in units of c/r b ) of protonswith γ p = 10 > E th / . kT ≃ , where E th ≃ .
15 GeV,as a function of the proton injection compactness ℓ injp for threevalues of the gamma-ray compactness marked on the plot. Forcomparison reasons, the cooling rate given by eq. (5) is plottedwith grey lines. First, we show that for high enough proton injection com-pactnesses, photons produced through the hadronic cascadecontribute to the target photon field for photopion interac-tions and therefore enhance the respective proton energy lossrate. For this, we compare the analytic expression for the en-ergy loss rate on a grey-body photon field of certain ℓ γ and T with the one derived numerically after taking into accountthe modification of the photon spectrum because of the cas-cade. It can be shown (for more details, see Appendix A)that the fractional energy loss rate of protons with Lorentzfactor γ p > γ th = E th / . kT , where E th ≃ .
15 GeV, is t − γ ≃ × − ℓ γ Θ t − , (5)where Θ = kT /m e c and t d = r b /c . As expected (see e.g.WB97, Aharonian 2000), the above expression does not de-pend on the proton injection compactness. The character-istic loss rate , as derived by the simulations, where thefeedback on the target field is taken into account in a self-consisent way, is plotted against ℓ injp in Fig. 1 for three valuesof ℓ γ marked on the plot. In all cases, the energy loss rateis constant and in agreement with the analytic expression(5), shown with grey lines in Fig. 1, only up to a certainvalue of ℓ injp . Its subsequent rapid increase is a sign of themodification of the initial photon spectrum because of thehadronic cascade. Hence, the analytic estimates of the frac-tion of energy lost by proton to pions can be considered onlyas a lower limit. The loss rate is calculated by h t − γ i = R dγ p γ p n p ( γ p ) ˙ P p γ / R dγ p γ p n p ( γ p ), where n p is the steady-state proton distribution.c (cid:13) , 1– ?? M. Petropoulou
Table 1.
Physical processes that act as injection (source) and loss terms in the kinetic equations of each species.protons neutrons pions muons relativistic pairs photons neutrinosInjection external pγ pγ pγ pγ neutral pion decay pγpγ a BH b pair production proton synchrotron β - decay β -decay γγ pair production electron synchrotronmuon synchrotroninverse ComptonLoss pγ pγ decay decay synchrotron γγ pair production escapeBH β -decay synchrotron inverse Compton synchrotron self-absorptionsynchrotron escape annihilation Compton downscatteringescape escape escape a photopion process b Bethe-Heitler process
46 47 48 49 50 51 52 53 0 5 10 15 20 l og L γ , ν ( e r g s - ) log ε (eV) photons ν e + ν µ Figure 2.
Observed photon and neutrino spectra obtained for ℓ γ = 0 . ℓ p = 10 − . (bottom curve) up to ℓ p = 10 − . (upper curve)with logarithmic increaments of 0.4. The last spectra before thedominance of the hadronic cascade are shown with thick lines. The modification of the photon spectrum due to the emis-sion from the secondaries produced in the hadronic cascadeis demonstrated in Figs. 2 and 3, where we plot the mul-tiwavelength photon spectra (black lines) in the observer’sframe for two indicative values of the gamma-ray compact-ness, namely ℓ γ = 0 . r b (see §
2) these correspond to observed gamma-ray luminosi-ties L γ = 10 and 10 erg/s, respectively. The neutrino( ν e + ν µ ) spectra (grey lines) are also plotted, for compar-ison reasons. In both figures, each spectrum is obtained byincreasing ℓ injp (or equivalently L p ) over its previous valuewith a logarithmic step δx – for the exact values see labelsof the respective figures.Above a certain value of ℓ injp , which depends on the valueof ℓ γ , the emission from the hadronic cascade begins to ei-ther ‘cover’ the prompt emission spectrum (see top spectrain Figs. 2 and 3) or peak in the sub-GeV energy band with
47 48 49 50 51 52 53 0 5 10 15 20 l og L γ , ν ( e r g s - ) log ε (eV) photons ν e + ν µ Figure 3.
Observed photon and neutrino spectra obtained for ℓ γ = 7 and values of the proton injection compactness startingfrom ℓ p = 10 − . (bottom curve) up to ℓ p = 10 − . (upper curve)with logarithmic increaments of 0.4. Thick lines have the samemeaning as in Fig. 1. a luminosity approximately equal to that of the MeV emis-sion (see second spectrum from the top in Fig. 2). The lat-ter is ruled out by the current status in observations (e.g.Dermer 2010). For illustration reasons, the spectra obtainedjust before the dominance of the HC are shown with thicklines in Figs. 2 and 3. From this point on, we will charac-terize cases that modify the initial MeV emission or have L (0 . − & . − . L (0 . − commonly as ‘HC dom-inant’ cases. We note that if a stricter upper limit on theGeV luminosity was used this would also be reflected at theupper limit imposed on the proton to gamma-ray luminosityratio discussed in the following section.For a fixed ℓ γ , higher secondary emission and neutrinoluminosities can be obtained by increasing the proton injec-tion compactness. This is demonstrated in Fig. 4, where weplot the observed luminosities of protons, muon and electronneutrinos, photons and (0.1-1) MeV gamma-ray photons, forthe case shown in Fig. 2. For low enough values of L p thebolometric photon luminosity is actual equal to the lumi- c (cid:13) , 1– ?? RB neutrinos and hadronic cascades
49 49.5 50 50.5 51 51.5 52 52.5 53 51 51.5 52 52.5 53 l og L i ( e r g / s ) log L p (erg/s)HC dominanceL ν L γ ,tot L (0.1-1) MeV Figure 4.
Log-log plot of the total neutrino luminosity (solidline), the bolometric photon luminosity (dashed line) and the (0.1-1) MeV gamma-ray luminosity (dotted line) as a function of theproton injection luminosity L p for ℓ γ = 0 .
7. The grey coloredarea is obtained for L p values that lead to the dominance of theHC emission. nosity emitted in 0.1-1 Mev energy band. However, for L p above a certain value, which for the particular example is ≃ . × erg/s, we obtain L tot γ > L (0 . − . This indi-cates that the photon component of the hadronic cascade isno longer negligible. Moreover, we find that L ν ∝ L qp , where q = 1 for low enough L p but q = 1 . L ν and L p is one more indication of the spectral modifica-tion due to the HC. Interestingly enough, the ratio of theneutrino to the bolometric photon luminosity becomes max-imum only for proton luminosities leading to HC dominantcases (grey colored area in Fig. 4). Still, for the last casebefore the dominance of the HC, we find that the photonand neutrino components are energetically equivalent with L ν ≃ . L tot γ (see also Table 2). For each value of ℓ γ we can derive a maximum value ofthe proton compactness above which the hadronic cascadesignificantly alters the GRB photon spectrum, as previouslydescribed. This can also be translated to a maximum valueof the ratio η p = L p /L γ , which in terms of compactnessesis written as η p = ℓ injp ( m p /m e ) /ℓ γ . For these maximumvalues, we then derive (i) the ratio η ν = L ν /L γ , (ii) the ratio ζ ν = L ν /L tot γ , (iii) the production efficiency in neutrinos( ξ ν ), and (iv) the production efficiency in secondaries thatcontribute to the HC ( ξ sec ), where the efficiencies are definedas ξ ν, sec = L ν, sec /L p = ℓ ν, sec ( m e /m p ) /ℓ injp . Our results aresummarized in Table 2 and a few things worth mentioningfollow: An alternative definition for η p would be η p = L p /L tot γ , wherethe superscript ‘tot’ denotes the bolometric photon luminosity.In the present work, however, we adopt the definition with thegamma-ray luminosity, as this appears mostly in the literature. l og η p , m a x log l γ Figure 5.
Log-log plot of η p , max as a function of ℓ γ . The actualresults of our simulations are shown with points while the lines arethe result of interpolation. Results shown with filled circles/solidline are obtained for η p , max while open circles/dashed line cor-respond to η p = η p , max δx . The grey colored area denotes theuncertainty of the derived η p , max . • the maximum value of η p decreases as the photon com-pactness becomes larger. This is exemplified in Fig. 5. Anextrapolation of our results to even larger values of ℓ γ (seee.g. Murase 2008; Zhang & Kumar 2013) would imply that L p ≈ (1 − L γ – see also § • there is an anticorrelation between the maximum al-lowable value of η p and ξ ν ( ζ ν ) that reflects the fact thatthe upper limit imposed on ℓ injp by the HC, becomes morestringent as ℓ γ increases. • the maximum ratio ζ ν increases as the gamma-ray com-pactness becomes larger and may as high as 60% for ℓ γ & • we find that η p , max ∝ /ξ sec , max , which can be under-stood as follows the photon compactness from the cascade isdefined as ℓ sec = ξ sec ℓ injp ( m p /m e ) while ℓ injp = η p ℓ γ ( m e /m p ).Roughly speaking, the secondary photon emission will startaffecting the overall photon spectrum, if ℓ sec ≈ αℓ γ where α is a numerical factor that contains all the details aboutthe hadronic cascade and depends on a series of parame-ters, such as the magnetic field strength and the spectralshape of the secondary emission itself. Our results show,however, that α varies less than a factor of 2 among the fiveparameter sets presented in Table 2, and we can, therefore,consider it a constant. By combining the above we find that η p , max ≈ α/ξ sec , max ∝ /ξ sec , max . In the previous section we demonstrated the effects of theHC on the photon and neutrino spectra but more impor-tantly we showed that the HC imposes an upper limit on theratio L p /L γ . Here, in an attempt to test the robustness ofthese results, we discuss in detail the effects of the magneticfield strength, of the initial spectral shape in gamma-rays,and of a larger value of Γ. c (cid:13) , 1– ?? M. Petropoulou
Table 2.
Maximum values of various ratios-efficiencies derived for five values of the gamma-ray compactness. ℓ γ ℓ tot γ log ℓ injp , max η p , max η ν, max ζ ν, max ξ sec , max (%) ξ ν, max (%)0.07 0.2 − . . . − . − . . . . − . − . . .
46 47 48 49 50 51 52 53 0 5 10 15 20 l og L γ , ν ( e r g s - ) log ε (eV) photons -- B=960 G ν e + ν µ --B=960 Gphotons -- B=1.5e4 G ν e + ν µ -- B=1.5e4 G Figure 6.
Comparison of photon and neutrino spectra for ℓ γ =0 . B = 960 Gand B = 1 . × G, shown with solid and dashed lines, re-spectively. The proton injection compactness ranges from 10 − . (bottom) to 10 − . (top). Thick lines have the same meaning asin Fig. 1. In general, stronger magnetic fields enhance the cascadeemission in two ways: • through synchrotron radiation of secondary pairs,which leaves an imprint mainly at energies below 1 MeVin the comoving frame because of the strong photon atten-uation that affects higher energy photons. • through synchrotron cooling of muons and pions, whichstarts playing a role for high enough magnetic fields. Thisintroduces more photons into the system, since part of theenergy that would have been transfered to the neutrino com-ponent goes now into the photon component.The effects of the magnetic field on the photon and neutrinoemission are exemplified in Fig. 6. For the same proton in-jection compactness, we find that the flux of the HC emis-sion increases, while the neutrino spectra become harder.The change of the neutrino spectral shape is non-trivial (seealso Asano & Meszaros 2014) and it requires a self-consisenttreatment, as it strongly depends on the shape of the tar-get photon field. Finally, we find that the value of η p , max is not significantly affected by the choice of the magneticfield, while the values of ξ sec and ξ ν increase at most by afactor of two – see Table 3. Although synchrotron coolingof pions is not relevant for the benchmark case, it startsplaying a role for the higher magnetic field strength con-sidered here. In particular, we find that E π, c ≃ E π, max /
10, where E π, max ≃ . γ max m p c = 8 × eV and E π, c is thetypical energy of a pion that cools due to synchrotron ra-diation before it decays. Since the numerical code does notaccount for the synchrotron losses of pions (see also § L ν ≃ . × erg/s. This would be translatedto a slight increase of the the bolometric luminosity, i.e. L tot γ ∼ . × erg/s instead of 4 × erg/s that is thevalue we obtain when we neglect pion synchrotron losses.Thus, the limits derived for η p are robust, while the valuesof ζ ν, max and η ν, max listed in Table 3 should be lower by afactor of a few.It has been already pointed out by various authors (e.g.WB97, Asano 2005), that the photopion processes is thedominant energy loss mechanism of high-energy protons insources with large photon compactnesses. This is also illus-trated in Fig. 7, where we plot the ratio p i of the proton en-ergy loss rate due to a process i (synchrotron, Bethe-Heitlerand photopion) to the total energy loss rate. The photopionprocess dominates indeed, over all other processes for thewhole range of η p values. This is also the case for the othervalues of ℓ γ .The contribution of the Bethe-Heitler process to thehadronic cascade, however, is not always negligible. Themain reason for this is that only a part of the energy lostby the protons through photopion interactions contributesto the hadronic cascade itself; the rest goes to the neutrinoand high-energy neutron components. This is exemplified inFig. 8, where we plot q BH and q π as a function of η p for ℓ γ = 0 . q BH and q π are the energy injection rates definedas q BH = ˙ E BH / ˙ E t ot and q π = ( ˙ E π ± → e ± pγ + ˙ E π → γγpγ ) / ˙ E tot ,where ˙ E k denotes the energy injection rate of the process k into secondary pairs and/or photons. The dependance ofthe ratios q BH and q p γ on η p is non-trivial and it is mainlydetermined by the characteristics of the target photon field,e.g. spectral shape and compactness, which highlights onceagain the role of the HC. From a quantitative point of view,we find that for indermediate values of η p the contributionof the Bethe-Heitler process to the HC may exceed 50% and30% for the high and low magnetic field values, respectively. c (cid:13) , 1– ?? RB neutrinos and hadronic cascades -3.5-3-2.5-2-1.5-1-0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 l og p i log η p l γ =0.7synchrotronBethe-Heitlerphotopion Figure 7.
Plot of the contribution of synchrotron (solid lines),photopion (dotted lines) and Bethe-Heitler (dashed lines) energyloss rates to the total energy loss rate of protons as a functionof the ratio η p for ℓ γ = 0 . B = 960 G (black lines) and B = 1 . × G (greylines). Here, log η p > . q B H , q π log η p l γ =0.7 BH (B=960 G)photopionBH (B=1.5e4 G)photopion Figure 8.
Rate of the energy injection into the hadronic cascade(pairs and photons) by the Bethe-Heitler ( q BH ) and the photopion( q π ) processes as a function of the ratio η p , for ℓ γ = 0 . B = 960 G (black lines) and B = 1 . × G (grey lines). Here, log η p > . As stated in §
2, we adopted a fiducial photon spectrum todescribe the GRB prompt emission. We chose, in particu-lar, a grey-body photon field that can be described only bytwo parameters and can give us insight on the effects of thehadronic cascade mainly because of its narrow spectral en-ergy distribution. In this section we investigate the implica-tions of the above choice on the results presented so far, suchas the maximum ratio of proton to gamma-ray luminosity.The gamma-ray spectrum of the prompt emission can
Table 3.
Same as in Table 2 but for B = 1 . × G. ℓ γ η p , max η ν, max ζ ν, max ξ sec , max (%) ξ ν, max (%)0.07 66 1 0.5 7 30.7 16 . . be successfully modelled, at least in most cases , by theso-called ‘Band function’ (Band et al. 1993, 2009). For thepurposes of the present work, it is sufficient to assume a‘Band-like’ photon spectrum and repeat the proceedure of § n γ ∝ x − α , for x min < x < x br and n γ ∝ x − β , for x br < x < x max , where α = 1 . β = 2 . x min = 3 × − , x br = 3 × − , x max = 0 . x is the photon energyas measured in the comoving frame, in units of m e c . Thegamma-ray photon compactness can be also written as ℓ γ = σ T r b u γ m e c , (6)where u γ = L γ πcr Γ (7)is the energy density of the photon field in the comovingframe.A comparison of the multiwavelength photon spectraand neutrino spectra obtained for ℓ γ = 0 . η p , max = 10 .
4, i.e. smaller by a factor of 1.6 than the onederived in the case of a grey-body photon spectrum. Thissmall decrease is found also for other values of the initialphoton compactness (see Table 4), and it can be under-stood as follows: in the case of a Band-like photon field,the same gamma-ray luminosity or compactness as before isnow distributed over a wider energy range. This increases,therefore, the possible combinations of proton and photonenergies with γ p x that lies above the threshold for Bethe-Heitler pair production and/or pion production, and slightlyenhances the emission from the secondaries. Thus, for thesame magnetic field strength, the marginal curve η p , max ( ℓ γ )derived for a grey-body photon spectrum (see Fig. 5) shouldbe shifted towards smaller values by a factor of ∼ . • L p ∼ L γ , L ν ∼ . L γ and L ν ∼ . L tot γ , for ℓ γ ≪ • L p ∼ L γ , L ν ∼ . L γ and L ν ∼ . L tot γ , for ℓ γ ≫ ℓ γ = 30 and 70 correspond to observed gamma-ray lumi-nosities 4 × erg/s and 10 erg/s, respectively. Thus, GRB spectra that cannot be adequately fitted by the Bandmodel, either show evidence of some black-body component (e.g.Axelsson et al. 2012) or require an additional power-law compo-nent (e.g. Guiriec et al. 2010).c (cid:13) , 1– ?? M. Petropoulou
46 47 48 49 50 51 52 53 0 5 10 15 20 l og L γ , ν ( e r g s - ) log ε (eV) photons -- GB ν e + ν µ --GBphotons -- BF ν e + ν µ -- BF Figure 9.
Comparison of photon and neutrino spectra for ℓ γ =0 . − (bottom curve) and 10 − . (topcurve) with logarithmic increaments of 0.4. Table 4.
Same as in Table 2 but for a ‘Band-like’ gamma-rayspectrum and two additional values of ℓ γ . ℓ γ η p , max η ν, max ζ ν, max ξ sec , max (%) ξ ν, max (%)0.07 41 . . . . . the derived values for the various efficiencies are representa-tive for typical and bright GRBs (e.g. Wanderman & Piran2010). Even if η p , max ∼ − L ν /L γ & η p and ζ ν , wefind that the neutrino spectra become harder (see grey linesin Fig. 9). This, however, should not be considered as a gen-eral result, since simulations with Band-like photon spectraand higher ℓ γ resulted in flat (in νF ν units) neutrino spectra,i.e. similar to those in Fig. 3. Our results indicate that for in-dermediate photon compactnesses, e.g. ℓ γ ∼
1, the shape ofthe neutrino spectra is sensitive to various parameters, suchas the magnetic field strength (see Fig. 6) and the shape ofthe initial gamma-ray spectrum, and may deviate from theexpected power-law form ǫ − ν . This requires further investi-gation which, however, is outside the scope of the presentwork. For the benchmark case we chose a moderate value of theLorentz factor (see § > a priori , here we examine the implicationsof a larger Γ on our results. The choice of a larger value of Γhas a twofold effect: the peak energy of the GRB spectrum
46 47 48 49 50 51 52 53 0 5 10 15 l og L γ , ν ( e r g s - ) log ε (eV) Γ =225 Γ =708 Figure 10.
Photon (black lines) and neutrino (grey lines) spectrafor Γ = 225 and Γ = 708 shown with solid and dashed lines,respectively. Other parameters used are: r b = 10 cm, B =960 G, L γ = 10 erg/s, L p = 6 . L γ and γ max = 4 × . as measured in the comoving frame decreases and the pho-ton compactness drops significantly, since ℓ γ ∝ r − Γ − . Weshowed that the maximum baryon loading given by η p , max increases as the emission region becomes less compact ingamma-rays (see Fig. 5), whereas the efficiency in the pro-duction of secondaries, such as pairs and neutrinos, drops(see e.g. Table 4).As an indicative example, we choose the case with ℓ γ = 7 and a three times larger Γ than before, i.e. Γ = 708.The photon compactness now becomes 0 .
07 and the peakenergy of the Band spectrum decreases by a factor of three,i.e. x br = 10 − . Following the same steps as in § η p , max = 42. The various efficiencies are η ν, max = 0 . ζ ν, max = 0 . ξ sec , max = 0 . ξ ν, max = 0 .
02 and should becompared with the values listed in Table 4 for ℓ γ = 7.To illustrate better the above, we compare the photonand neutrino spectra between two cases for Γ = 225 and708. The rest of the parameters, which are kept fixed, are: r b = 10 cm, B = 960 G, L γ = 10 erg/s, L p = 6 . L γ and γ max = 4 × . The photon and neutrino spectra are shownin Fig. 10. Note that the distortion of the gamma-ray spec-trum becomes evident only for the first case (black lines),since η p = η p , max = 6 . ∼
100 MeVand in the neutrino emission is the photon compactness,which is decisive for both the intrinsic optical depth for γγ absorption and the efficiency of photopion production (seealso Asano et al. 2009). Finally, the lower peak photon en-ergy in the comoving frame proves to be not as importantas the lower value of ℓ γ . In this section we attempt a comparison of our results re-garding the maximum value of η p imposed by the hadroniccascade with other works in the literature. Our results aresummarized in Fig. 11, where the solid line denotes the max-imum value η p , max as a function of the gamma-ray compact-ness, which divides the η p − ℓ γ into two regions. Above the c (cid:13) , 1– ?? RB neutrinos and hadronic cascades l og η p , m a x log l γ Guetta et al. (2004) -- Table 1Guetta et al. (2004) -- Table 3Zhang Kumar (2013) -- Fig.1Asano et al. (2009) -- Fig.1Asano et al. (2009) -- Fig.2Asano et al. (2009) -- Fig.3Asano Meszaros (2014) -- Fig.2
Figure 11.
The upper limit η p , max plotted as a function of ℓ γ (solid line) along with indicative values derived from studies thatfocus on the diffuse GRB neutrino emission (grey symbols) andon the HC emission (black symbols). The emission from the HCdistorts the GRB spectrum for parameter values drawn from thegrey-colored region. curve (grey colored region) the effects of the HC on the ini-tial gamma-ray spectrum become prominent, whereas thecombination of η p and ℓ γ values drawn from the region be-low the curve does not lead to strong secondary emission.The steps for deriving η p , max for different values of ℓ γ are the same as those described in § α, β and characteristic energies x min , x br and x max as describedin § .The magnetic field is taken to be B = 960 G. We note alsothat the limiting curve in Fig. 11 may be shifted for differentvalues of the spectral indices or/and of the break energy (seealso § η p = L p /L γ used here is equiv-alent to: (i) the parameter f − γ/p in Zhang & Kumar (2013),(ii) the ratio ǫ p /ǫ e in Asano et al. (2009), (iii) the ratio oftotal energies in protons and gamma-rays in Guetta et al.(2004), and (iv) the parameter f p in Asano & Meszaros(2014). For the derivation of ℓ γ (see eq. (6)) we com-bined various parameters given in the aforementioned works,namely Γ, r , δt , and L γ with eq. (7). We note that r = f v c Γ δt , where f v is a numerical factor that ranges be-tween 1-2 among the various studies. For the compilationof Fig. 11, we used the value of f v as given in each work.Let us focus first on the black colored symbols. Most The gamma-ray spectrum in Asano et al. (2009) is the resultof synchrotron radiation of relativistic electrons, which, however,may be described by a Band function. of the points that lie above our upper limit correspond tocases with significant modifications of the gamma-ray spec-trum because of the HC in agreement with the respectivestudies. For example, the first two curves from top of Fig. 3in Asano et al. (2009) are indicative cases of proton-cascadedominated spectra. Using the same values as in Asano et al.(2009), we find that the gamma-ray compactness is ℓ γ = 0 . L γ and r b , the plane η p − ℓ γ proves to be a robust tool fordistinguishing between cases with dominant hadronic cas-cade or not. For the former, the neutrino and secondaryproduction efficiencies may exceed the maximum values de-rived in previous sections (see also Fig. 4). GeV bright GRBsdetected by Fermi -LAT are a good example of bursts thatdeviate from the ‘typical’ ones (e.g. Racusin et al. 2008;Abdo et al. 2009). Assuming that their high-energy emissionis a result of the hadronic initiated cascade (e.g. Asano et al.2010), these bursts might fall into the grey-colored region ofFig. 11 depending on the relative ratio of the GeV/MeVfluxes: bright bursts with comparable fluxes in the two en-ergy bands would be also a good candidate source of brightneutrino emission.The grey colored symbols lie, in general, at the rightpart of the η p − ℓ γ plane, since high gamma-ray compact-nesses are required for efficient neutrino production. Inter-estingly, most of these points lie below but close (within afactor of two) to our limiting curve. For example, if a similaranalysis to the one by Guetta et al. (2004) was performed,but for a higher ratio of proton to gamma-ray luminosities,e.g. η p >
3, the distortion of the pre-assumed
GRB spectrumcould not be avoided. Thus, in the regime of large gamma-ray compactnesses the calculation of the neutrino emissionnecessitates the treatment of the secondary photon emissionas well.Note that the above discussion does not necessarily referto bright GRBs. Because of the strong dependance of thegamma-ray compactness to the Lorentz factor, namely ℓ γ ∝ L γ / Γ δt , a typical burst with L γ = 3 × erg/s mayhave ten times larger ℓ γ than a bright GRB with L γ =10 erg/s, if its bulk Lorentz factor is by a factor of threesmaller. It is noteworthy that the same conclusion is reachedusing different argumentation by Asano & Meszaros (2014).Although L γ is relatively constrained and it lies in the range10 − erg/s, there is freedom in the choice of Γ and δt ,which makes impossible the definition of an observationallytypical ℓ γ . Thus, studying the effects of the hadronic cascadeusing as a free parameter the gamma-ray compactness coversa wide range of GRBs and may facilitate future parameterstudies of GRB neutrino emission. We investigated the effects of hadronic cascades on thegamma-ray burst (GRB) prompt emission spectra in scenar-ios of efficient neutrino production. For this, we employeda generic method where we approximated the prompt GRBemission by either a grey-body or a Band-function spectrum,while we used as free parameters the proton injection and theprompt gamma-ray compactnesses. Using a numerical code c (cid:13) , 1– ?? M. Petropoulou that follows the evolution of the proton, photon, neutrinoand pair distributions both in energy and time, we calcu-lated the steady-state photon and neutrino emission spectrataking into account in a self-consisent way the formationof the hadronic cascade. We showed that for each value ofthe gamma-ray compactness one can set an upper limit tothe ratio of the proton to gamma-ray luminosity by usingthe fact that the emission from the hadronic cascade cannotalways be neglected. On the contrary, it is an important in-gredient of GRB hadronic models and it may significantlyaffect the overall photon and neutrino emission.The recent PeV neutrino detection by the IceCubeCollaboration (IceCube Collaboration 2013) is beginning toplace stonger constraints on the parameter values used invarious GRB models (Zhang & Kumar 2013). Using a de-tailed example (see Fig. 4) we pointed out that param-eter sets that lead to high neutrino luminosities ( η ν = L ν /L γ ≫
1) also result in modified gamma-ray photon spec-tra due to the emission from the hadronic cascade. On theother hand, we showed that the ratio η ν, max , which is de-rived for parameter values just before the dominance of thehadronic cascade, lies in the range 0 . − .
4. Higher val-ues are obtained for stronger magnetic fields (see Table 3)and for harder/softer GRB spectra below/above the peak,i.e. for spectra more similar to a grey-body photon emis-sion (see Tables 2 and 4). Using this upper limit on theratio L ν /L γ from a single burst, we can make a rough esti-mate of the neutrino energy flux as follows. First, we use¯ η ν = 0 . L γ = 10 erg/s we adopt E γ = 4 × erg as the typ-ical energy emitted in gamma-rays (Ghirlanda et al. 2012;Kakuwa et al. 2012). For the GRB rate at the local universewe use R GRB ≃ − yr − (Wanderman & Piran 2010).We find that the energy production rate of gamma-rays perunit volume is ˙ U γ ≃ × E γ, . erg Mpc − yr − . Then,the present day all-flavour neutrino energy density is givenby U ν ≈ ¯ η ν ˙ U γ T , where T ≃ yr is the Hubble time.Finally, the neutrino flux is given by E ν Φ ν ≈ c π U ν ≈ − ¯ η ν . E γ, . GeVcm − s − sr − . (8)The above rough estimate is lower only by a factor of ∼ Fermi triggered GRBs, however, starts to chal-lenge this assumption (e.g. Abbasi et al. 2012; He et al.2012; Liu & Wang 2013). Even if future observations workagainst this scenario, models where low-luminosity GRBs( ∼ erg/s) are the main contributors to the diffuse PeVemission cannot be ruled out, as long as the statistics of theirpopulation remains undetermined (see e.g. Cholis & Hooper2013; Murase & Ioka 2013 for relevant discussion). Since theanalysis presented here is not based on the absolute valueof the GRB luminosity but on parameters that characterizethe intrinsic properties of the emission region, one can re- peat the above calculation using the appropriate values for E γ , R GRB and ¯ η ν .Our results also indicate that a higher value for theneutrino flux is possible if η ν, max & .
3, which further impliesthat if the observed neutrino signal has indeed a GRB originthen typical GRB sources should be strongly magnetizedwith B above a few kG (see § B = 10 G, r = 10 cmand Γ=300 the spectrum of muons produced via pion decayhas a cutoff at ∼
10 PeV (e.g. He et al. 2012).The shape of the neutrino spectra is strongly affectedby the photon spectrum that serves as a target field forphotopion interactions and, hence, for neutrino production.Although the emitted neutrino spectra are expected to beflat in νF ν units whenever the cooling of a γ − proton dis-tribution is efficient and the target spectra are ‘Band-like’,this is not always the case. Even in the simplest scenariowhere the only available target photon field is the Band-likegamma-ray emission, the neutrino spectral shape also de-pends on the particular pion production channel, e.g. ∆ + resonance or multipion production channel, that is respon-sible for the neutrino emission at the specific energy band(Baerwald et al. 2011). To make things even more compli-cated, photons produced by neutral pion decay or emittedby secondary relativistic pairs through synchrotron and in-verse Compton scattering contribute to the overall multi-wavelength emission and may serve as targets for photopioninteractions, too. The spectral shape of the hadronic cascadeemission is, therefore, an important factor for the determi-nation of the final neutrino spectral shape.Furthermore, our analysis showed that the investigationof the available parameter space in GRB models may be sig-nificantly simplified if this is performed using as basic vari-ables the proton and gamma-ray compactnesses instead ofthe typical GRB model parameters, such as the bulk Lorentzfactor and the observed isotropic gamma-ray luminosity. Af-ter all, it is the compactness of the emitting region in termsof relativistic protons and photons that mainly determinesthe efficiencies of the neutrino and secondary particle pro-duction.Finally, we demonstrated through indicative examplesthe role of the Bethe-Heitler process in the formation of thehadronic cascade. This is not always negligible, as its contri-bution to the injection of secondary pairs into the hadroniccascade may exceed 30-50% for ℓ γ ∼ . −
1, while strongermagnetic fields tend to enhance the role of the Bethe-Heilterprocess. However, we find that its contribution is signifi-cantly suppressed for ℓ γ ≫ c (cid:13) , 1– ?? RB neutrinos and hadronic cascades our results suggest that GRBs can account for the currentneutrino flux level only if there is a substantial contributionof the hadronic emission to the overall GRB photon spectra. ACKNOWLEDGEMENTS
I would like to thank Dr. S. Dimitrakoudis for provid-ing the numerical code and Professors A. Mastichiadis andD. Giannios for fruitful discussions and comments on themanuscript. I would also like to thank the anonymousreferee for his/her comments that helped to improve themanuscript. Support for this work was provided by NASAthrough Einstein Postdoctoral Fellowship grant number PF140113 awarded by the Chandra X-ray Center, which is op-erated by the Smithsonian Astrophysical Observatory forNASA under contract NAS8-03060.
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APPENDIX A: PROTON ENERGY LOSS RATEDUE TO PHOTOPION INTERACTIONS WITHA GREY-BODY PHOTON FIELD
The differential number density of a grey-body photon fieldis given by n γ ( x ) = n x e x/ Θ − , (9)where x and Θ are the photon energy and the effective tem-perature in units of m e c , while the normalization n is re-lated to the photon compactness ℓ γ as n = 15 ℓ γ π Θ σ T r b , (10)where r b is the size of the emission region and ℓ γ does notviolate the black-body limit. The fractional energy loss rateof a proton with energy E p = γ p m p c due to photopioninteractions with the above photon field is then given by(see e.g. WB97): t − γ ≡ − E p dE p dt | p γ = (11)= c γ Z ∞ ǫ th d ǫ ǫσ p γ ( ǫ ) ξ p γ ( ǫ ) Z ∞ ǫ/ γ p d xx − n γ ( x ) , (12)where ǫ is the photon energy (in m e c units) as seen inthe rest frame of the proton, ǫ th is the threshold energyof the interaction in m e c units, i.e. ǫ th ≃ σ p γ and c (cid:13) , 1– ?? M. Petropoulou ξ p γ are the cross section and the inelasticity of the inter-action, respectively. Here, we approximate the cross sec-tion by a step function, i.e. σ p γ ≈ σ H ( ǫ − ǫ th ) with σ = 5 × − cm − ≃ (5 / × − ) σ T . Although the in-elasticity may vary between ∼ . ξ p γ ≈ . x , eq. (12) simplifies into t − γ = − c Θ n σ ¯ ξ p γ γ Z ∞ ǫ th d ǫ ǫ ln (cid:16) − e − ǫ/ γ p Θ (cid:17) . (13)The function that appears in the above integral, i.e. f ( ǫ ) = − ǫ ln (cid:16) − e − ǫ/ γ p Θ (cid:17) , is a steep function of ǫ that shows asharp peak at approximately ǫ p ≃ . γ p Θ. Hence, its in-tegral I can be approximated by I appr ≈ ǫ p f ( ǫ p ), where f ( ǫ p ) ≈ . ǫ p . This is illustrated in Fig. 12, where thehatched area that corresponds to the value of I is approxi-mated by the area below the δ -function centered at ǫ p . Wenote here that the sharpness of the peak at ǫ ≃ ǫ p is not ev-ident because of the logarithmic scale. If we compare I appr with the value I obtained after numerical integration of theintegral, we find that their ratio f cor = I/I appr can be mod-eled as f cor ≃ . (cid:18) ǫ p ǫ th (cid:19) , (14)for ǫ p > ǫ th . In what follows, we will incorporate this‘correction’ factor into our analytical expression. Thus, thefractional energy loss rate for protons with Lorentz factor γ p > ǫ th / .
4Θ is given by t − γ ≈ × − f cor ℓ γ π t d Θ , (15)where we have used eq. (10) and t d = r b /c . When normal-ized with respect to t d , we find τ − γ ≈ . × − f cor ℓ γ Θ , (16)which, besides the correction factor, depends only on ℓ γ andΘ. APPENDIX B: MAXIMUM PROTON ENERGY
The maximum energy of protons is typically calculated bybalancing the acceleration and energy loss rates. An inde-pendent constraint comes from the so-called Hillas criterion(Hillas 1984) according to which, the gyroradius of the mostenergetic particles should not exceed the size of the emissionregion. Here we compare the different upper limits on themaximum proton energy.In the simplest scenario protons are accelerated at arate which is inverse proportional to their energy, namely t − ( γ ) = κeBc/γm p c , where κ is an efficiency factor. Forsimplicity, we assume κ = 1. As long as proton accelera-tion takes place at small Thomson optical depths, the mainenergy loss processes that act competitive to the acceler-ation process are synchrotron radiation and photopion in-teractions. Note that for larger optical depths pp-collisions l og f( ε ) log εε th Figure 12.
Function f = − ǫ ln (cid:0) − e − ǫ/ǫ (cid:1) for ǫ = 10 (blackline) and the δ -function approximation (grey line). The hatchedarea denotes I while the white colored area below the delta func-tion corresponds to I appr . consist another important energy loss mechanism for pro-tons as well as another neutrino production channel (e.g.Murase 2008).The synchrotron proton loss timescale is given by t syn =6 πm p cχ /σ T B γ , where χ = m p /m e . By equating t acc and t syn and by using eq. (1) and r b ≈ c Γ δt we find the saturationLorentz factor because of synchrotron losses γ syn ≃ × Γ / δt / − L − / , ǫ − / , − . (17)For the benchmark case of Γ = 225, δt ≈ .
15 s, ǫ B = 2 . × − and L j = 3 × erg/s, we find γ syn ≃ × > γ H .Even for B = 1 . × G (see § γ syn > γ H holds.For large values of the photon compactness, proton cool-ing due to photopion production may overcome synchrotronlosses. The respective loss rate for protons having energiesabove the threshold is constant and is given by eq. (5); forthe derivation, see Appendix A. Setting t acc = t p γ we findthe saturation Lorentz factor to be γ p γ ≃ × Γ δt − ǫ / , − L / , L − γ, E br , obs . z ) , (18)where we replaced the effective temperature Θ in eq. (5)by the peak energy of the photon spectrum E br , obs . For thebenchmark parameter values we find γ p γ ≈ × (1 + z ) /L γ, > γ H .Summarizing, for the present analysis it is safe to as-sume that γ max = γ H . c (cid:13) , 1–, 1–