A marginally fast-cooling proton-synchrotron model for prompt GRBs
MMNRAS , 1–16 (2021) Preprint 5 February 2021 Compiled using MNRAS L A TEX style file v3.0
A marginally fast-cooling proton-synchrotron model forprompt GRBs
Ioulia Florou (cid:63) , Maria Petropoulou ID † , Apostolos Mastichiadis ID Department of Physics, National and Kapodistrian University of Athens, University Campus Zografos, GR 15783, Greece
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
A small fraction of GRBs with available data down to soft X-rays ( ∼ . keV) have been shown to feature a spectralbreak in the low-energy part ( ∼ γγ pair production affect the broadband photon spectrum. We support our findings with detailednumerical calculations. Strong modification of the photon spectrum below the break energy due to the synchrotronemission of secondary pairs is found, unless the bulk Lorentz factor is very large ( Γ (cid:38) ). Moreover, this scenariopredicts unreasonably high Poynting luminosities because of the strong magnetic fields ( − G) that are necessaryfor the incomplete proton cooling. Our results strongly disfavor marginally fast cooling protons as an explanation ofthe low-energy spectral break in the prompt GRB spectra.
Key words: gamma-ray burst: general– radiation mechanisms: non-thermal – neutrinos
Gamma-ray bursts (GRBs) are extremely energetic explo-sions that release most of their electromagnetic output in γ -rays within a brief period of time, typically lasting from afraction of a second to several hundred seconds. GRB lightcurves are highly variable and consist of several pulses, eachof them having a typical width of 10 ms–1 s. The promptemission is typically observed in the 10 keV–1 MeV energyband (Preece et al. 2000) and its isotropic luminosity can beas high as erg s − , making GRBs the most luminous ob-jects in the sky (for reviews, see Piran 2005; Kumar & Zhang2015).The radiation mechanism behind the GRB prompt emis-sion is still under debate. A standard approach adopted toinvestigate the origin of the prompt emission is spectral anal-ysis. This procedure involves fitting empirical functions to theprompt spectra and comparing them with the expectationsfrom different high-energy radiative processes. In many cases,the prompt GRB spectra can be described by a smoothlyconnected broken power law, known as the “Band-function”(Band et al. 1993). (cid:63) E-mail: ifl[email protected] † E-mail: [email protected]
The fact that the overall spectrum of the GRB promptemission might be non-thermal had led to the suggestion thatthe radiation is dominated by synchrotron emission from apower-law distribution of relativistic electrons (Katz 1994;Sari et al. 1996). However, a major criticism of this model isthe predicted low-energy spectral slope. Many prompt emis-sion spectra are represented at low energies by a power lawwith a hard photon index, namely dN/dε ∝ ε α with α ∼ − .However in the standard synchrotron fast cooling model, thespectrum below the peak is expected to have a softer index( α = − / ), making synchrotron radiation a debated pro-cess for the interpretation of GRB prompt emission. Thisproblem is known as the synchrotron “line of death” (Crideret al. 1997; Preece et al. 2002). Other problems with a syn-chrotron interpretation of the prompt emission are the narrowpeak-energy distribution and the narrow spectral width of theobserved “Band-function” as compared with the synchrotronpeak, making the suggestion of the synchrotron model highlydebatable.Variants of the non-thermal electron emission models havebeen discussed to overcome the “line-of-death” problem. Oneassumption is that the electron synchrotron spectrum is mod-ified, on the low-energy part, by inverse Compton (IC) scat-tering in the Klein Nishina regime (Derishev et al. 2001).Alternatively, IC scattering of slow cooling electrons on the © 2021 The Authors a r X i v : . [ a s t r o - ph . H E ] F e b Florou et al. self-absorbed part of the synchrotron spectrum was suggestedto produce the keV part of the prompt emission spectrum(Panaitescu & Mészáros 2000). It was also proposed that theelectrons should be continuously accelerated over the lifetimeof a source in order to reproduce the hardness of most of theobserved GRB spectra by the synchrotron self-Compton ra-diation (Stern & Poutanen 2004) . Some other models takeinto account the electron distribution in pitch angles (Lloyd-Ronning & Petrosian 2002) and the decay of the magneticfield over a length scale shorter that the comoving widthof the emitting region (Pe’er & Zhang 2006; Uhm & Zhang2014a) in order to overcome the problem of the electron fastcooling and reproduce the prompt emission spectrum. Re-cently, Burgess et al. (2020) performed time-resolved spectralanalysis to the prompt spectra of single-pulse GRBs detectedwith the
Fermi
Gamma-ray Burst Monitor (GBM), whileconsidering a time-dependent synchrotron spectral modelwith an electron distribution ranging from extremely cooledto extremely uncooled. These authors conclude that the elec-tron synchrotron interpretation of the prompt GRB emissionis a feasible option, once time dependence and cooling areproperly included.Recent studies have argued that observations extendingto lower frequencies (optical and soft X-rays) can addressthe questions regarding the non-thermal interpretation ofthe GRB prompt emission, by determining in a more robustway the low-energy photon index and the spectral width ofthe GRB spectrum. Oganesyan et al. (2017) have performedspectral analysis to GRB spectra during the prompt phaseincluding data from the
Neil Gehrels Swift
X-Ray Telescope(
Swift /XRT) and
Fermi
GBM (Ravasio, M. E. et al. 2018;Ravasio et al. 2019). Oganesyan et al. (2018) extended theiranalysis by including data from the
Swift
Burst Alert Tele-scope (BAT), and even more recently (Oganesyan et al. 2019)by including optical observations from the
Swift
UltraVioletand Optical Telescope (UVOT) (Roming et al. 2005) andfrom ground based robotic telescopes (Lipunov et al. 2004;Burd et al. 2005; Klotz et al. 2009; Beskin et al. 2017). Allthese studies have shown that the spectrum below 10 keVdoes not lie on the extrapolation of the low-energy power lawof the “Band function”, but shows a spectral break at arounda few keV. Oganesyan et al. (2019) have adopted another phe-nomenological function to fit the overall prompt GRB spec-tral energy distribution (SED) that consists of three powerlaws joined at two energies, E c , obs and E pk , obs . Below E c , obs the average photon spectral index is found to be α ∼ − / ,between E c , obs and E pk , obs is α ∼ − / , and above E pk , obs the photon index β becomes close to − . or slightly steeper.Relying on these spectral fitting results, Oganesyan et al.(2019) reproduced the GRB spectrum with an electron syn-chrotron model. They showed that synchrotron radiation isstill a viable mechanism for the GRB prompt emission, ifthe electron cooling is not complete, but stops at an electronLorentz factor γ c1 that is comparable to the minimum elec-tron Lorentz factor γ m (cid:29) . This idea of marginally fast cool-ing electrons with γ m /γ c = O (10) is a possible solution to theinconsistency between the expected and the measured pho-ton indices found in previous works. Oganesyan et al. (2019)generated successfully electron synchrotron spectra and from γ c corresponds to the synchrotron energy E c , obs . the observables managed to compute some source parametersas a function of the bulk Lorentz factor, such as the magneticfield of the source, the distance from the central region, theelectron distribution power law index and the number of rel-ativistic emitting electrons.Ghisellini et al. (2020) also examined the idea that syn-chrotron radiation from marginally fast cooling particles isable to reproduce GRB prompt emission. They calculatedthe proper values of source radii and magnetic fields ex-pected in the case of a leptonic scenario, where the radi-ating particles are electrons, and showed that the resultscontradicted the general ideas of GRB phenomenology. Morespecifically, the short variability timescales, often seen in theprompt emission of GRBs, indicate that the emitting regionmust be compact and located at relatively short distances( R γ ∼ − cm) from the central engine. Assumingthat the magnetic field of the jet is decaying with distancefrom the central engine, a compact emitting region shouldalso contain a strong magnetic field. According to the calcu-lations of Ghisellini et al. (2020), the synchrotron radiationfrom marginally fast cooling electrons is able to reproduce theGRB prompt emission, if the radius of the emitting region is R γ (cid:38) cm and the comoving magnetic field strength is B (cid:46) G; both values suggest that the minimum variabil-ity timescale should be much larger than the one observed.As a way out of these inconsistent results, Ghisellini et al.(2020) proposed that the prompt emission originates frommarginally fast synchrotron cooling protons.The aim of this project is to extend the work of Oganesyanet al. (2019) and Ghisellini et al. (2020) by considering ahadronic scenario for the prompt emission. More specifically,our working hypothesis is that synchrotron radiation from apopulation of relativistic protons gives rise to the observedprompt GRB emission. Using the best-fit values obtained byOganesyan et al. (2019) for the low-energy spectral break E c , obs of the prompt GRB spectrum, the flux at that en-ergy, F c (in units of mJy), and the ratio of the two breakenergies, E pk , obs /E c , obs , we estimate analytically the sourceparameters in the proton synchrotron scenario for about twodozens GRBs as a function of the bulk Lorentz factor. Wethen numerically compute the photon spectra while includ-ing all relevant radiative processes besides proton synchrotronradiation. In particular, our numerical calculations take intoaccount γγ pair production processes as well as interactions ofprotons with radiation (photohadronic interactions), namelyphotopair (Bethe-Heitler) pair production and photomesonproduction processes. All these processes are responsible forthe injection of ultra-relativistic pairs in the emitting re-gion. These can efficiently radiate their energy through syn-chrotron radiation, thus shaping the overall GRB spectrum.Another result of the photohadronic interactions is the pro-duction of photons with energies well above the peak syn-chrotron energy. Depending on source parameters, these pho-tons can be attenuated via γγ pair production, thus produc-ing even more pairs and modifying the GRB prompt spec-trum (see also Petropoulou 2014). Our goal is to test the roleof these additional physical processes in shaping the overallGRB spectrum, and identify parameter regimes where theproton synchrotron scenario for the prompt emission is valid.Lastly, we complement our analysis by calculating the accom-panying high-energy neutrino signal expected in this scenariofor all plausible parameter sets. MNRAS , 1–16 (2021)
GRB proton-synchrotron model This work is structured as follows. In Section 2 we derivethe model parameters of the problem. Continuously, in Sec-tion 3 we present the numerical code that we utilize in orderto construct the GRB photon spectra. In Section 4 we calcu-late semi-analytically the neutrino fluxes of all the GRBs ofour sample and discuss the significance of the photohadronicprocesses and of γγ pair production, by showing some analyt-ical results for the parameter values of one specific example.Afterwards we show the numerical results for the same pa-rameter values. We conclude in Section 5 with a summaryand a discussion of our results. Throughout this study weuse H = 69 .
32 km Mpc − s − , Ω M = 0 . , Ω Λ = 0 . (Hin-shaw et al. 2013). We use published values for GRB redshifts,while we adopt z = 2 for GRBs without measured redshift. As we mentioned above, our first goal is to reexamine thegeneral synchrotron model for the GRB prompt emission, fol-lowing the analysis of Oganesyan et al. (2019) and Ghiselliniet al. (2020), in the case of a hadronic scenario and deter-mine the regime in the parameter phase space in which thisscenario is viable. For the analytical calculations presentedin this section, we make the implicit assumption that syn-chrotron radiation dominates the proton energy losses insidethe source and is responsible for the prompt GRB emission.We assume that at a distance R γ from the central engine ofthe GRB, relativistic protons are injected inside a sphericalregion that moves with a bulk Lorentz factor Γ . This sphericalregion can be thought of as the shell of the shocked ejecta inthe internal shock GRB model. It has got a comoving width r b = R γ / Γ and contains a tangled magnetic field of comovingstrength B .Relativistic protons are injected in the emitting region afterbeing accelerated only once into a power law distribution ofspectral index p , starting from a minimum Lorentz factor γ m up to a maximum value γ max . The proton injection rate (perunit volume) can be written as Q p ( γ, t ) = Q γ − p H ( γ − γ m ) H ( γ max − γ ) H ( t ) , (1)where H ( x ) is the Heaviside function and γ max =min( γ H , γ eq ) . Here, γ eq is the Lorentz factor where the syn-chrotron loss timescale equals the acceleration timescale t acc = ηr g /c = ηm p γc/qB with η ≥ , and γ H is the Lorentzfactor of protons with r g = r b (Hillas 1984). Here, q, m p arethe charge and mass of the proton, and c is the speed of light.The injection rate of eq. (1) translates also to an injectionluminosity of relativistic protons in the comoving frame as L p = 4 πr m p c Q (cid:90) γ max γ m γ − p +1 dγ. (2)The proton injection luminosity L p can in turn be used todefine the proton injection compactness, (cid:96) p , as (cid:96) p = L p σ T πr b m p c · (3)Upon entering the source, the relativistic protons interactwith the magnetic field and any soft photons present, creat-ing radiation and secondary particles, through proton syn-chrotron radiation and photohadronic (i.e., photomeson and photopair) interactions respectively. In order to make the freeparameters as few as possible, we assume that the injectionof primary relativistic electrons has a negligible contributionto the photon emission. Because of this, the main target pho-tons for photomeson and photopair processes are the protonsynchrotron photons.According to Oganesyan et al. (2019), for most of the anal-ysed spectra of that work, spectral fits with a synchrotronfunction returned a well constrained cooling energy E c , obs and ratio γ m /γ c . These in turn constrain the peak energy E pk , obs of the prompt GRB spectrum E pk , obs = (cid:18) γ m γ c (cid:19) E c , obs . (4)These quantities in addition to the flux at the cooling energy F c and the variability timescale of the GRB prompt emission t γ, obs , are four observables that can lead us to the source char-acteristics needed to explain the observed spectra as protonsynchrotron radiation.The observed proton synchrotron spectrum can be de-scribed by the following free parameters: the magnetic fieldstrength B , the minimum Lorentz factor of the proton distri-bution γ m , the power-law slope p of the proton distribution,the distance from the central engine R γ (or comoving size ofthe emitting region r b ), the comoving proton luminosity L p ,and the bulk Lorentz factor Γ . The latter parameter cannotbe determined from the four available observables. Hence, weexpress all quantities as a function of the bulk Lorentz factor.In the marginally fast cooling scenario considered here, thepower-law slope p is related to the photon index β above thepeak energy of the prompt photon spectrum, as β = − p/ − .Oganesyan et al. (2019) could not constrain the high-energypart of the spectrum because of the lack of Fermi -GBM data,except for one GRB. Motivated by the results of spectral anal-ysis performed on large GRB samples with empirical models(e.g. Nava et al. (2011), Goldstein et al. (2013), Gruber et al.(2014), we use β = − . throughout this work.We express the three observables ( E pk , obs , E c , obs , and F c ),as a function of the bulk Lorentz factor and the other sourceparameters, that are required to give rise to proton syn-chrotron spectra with the same observational characteristics E pk , obs = m e m p qhBγ πm e c Γ1 + z (5) E c , obs = (cid:18) m p m e (cid:19) πqhm e c (1 + z ) σ B t γ, obs Γ(1 + Y ) (6) t γ, obs = R γ (1 + z ) c Γ , (7)where z and d L are the redshift and luminosity distance, Y = U γ /U B is the Compton parameter, and U γ is the photonenergy density. In our analysis we assume that Y → , sincethe losses via IC scattering are negligible for the parametervalues in this hadronic scenario (for details, see Appendix B).The last observable, F c , is expressed through the bolomet-ric flux F γ as F c = F γ (cid:18) E c , obs h (cid:19) − (cid:32)
34 + 2 (cid:115) E pk , obs E c , obs − p − (cid:115) E pk , obs E c , obs (cid:33) − · MNRAS , 1–16 (2021)
Florou et al. (8)From eq. 6 we find that the comoving magnetic field of thesource is expressed as B = 1Γ / (cid:32) πqhm e c (1 + z ) σ t γ, obs E c , obs (cid:33) / (cid:18) m p m e (cid:19) / (9) (cid:39) × G (cid:18) Γ300 (cid:19) − / (cid:18) t γ, obs (cid:19) − / · (cid:18) E c , obs
10 keV (cid:19) − / (cid:18) z (cid:19) / (10)The minimum proton energy γ m is computed from eq. 4, tak-ing into account eq. 5 γ m = (cid:115)(cid:18) γ m γ c (cid:19) πm e c (1 + z )3 qh m p m e E c , obs B Γ (11) (cid:39) . × (cid:18) ζ (cid:19) (cid:18) Γ300 (cid:19) − / (cid:18) t γ, obs (cid:19) / · (cid:18) E c , obs
10 keV (cid:19) / (cid:18) z (cid:19) / , (12)where ζ ≡ γ m /γ c . The distance from the central engine iscomputed from eq. 7 and reads R γ = c Γ t γ, obs (1 + z ) − (13) = 9 × cm (cid:18) Γ300 (cid:19) (cid:18) t γ, obs (cid:19) (cid:18) z (cid:19) − , (14)Accordingly, the size of the emitting region r b is r b = c Γ t γ, obs (1 + z ) − (15) = 3 × cm (cid:18) Γ300 (cid:19) (cid:18) t γ, obs (cid:19) (cid:18) z (cid:19) − · (16)In order to compute the comoving proton luminosity, we as-sume that the observed luminosity is approximately equal tothe injection luminosity of particles. This hypothesis is validas long as particles are in the marginally fast cooling regime.We therefore may write L p ≈ L γ, obs Γ (cid:39) . × ergs (cid:18) F c (cid:19) (cid:18) E c , obs
10 keV (cid:19) (17) (cid:18)
Γ300 (cid:19) − (cid:20)
34 + 2 (cid:18) ζ (cid:19) − . (cid:18) ζ (cid:19)(cid:21) · where L γ, obs = 4 πd F γ is the bolometric γ -ray luminosity ofthe prompt emission in the observer’s frame, and the bolo-metric flux, F γ , is computed from eq. 8.All the above source parameters are going to be displayedas a function of the bulk Lorentz factor Γ in Sec. 4. Thebest-fit values (and the σ uncertainties) of the four observ-ables, required for the calculation of source parameters, aretaken from a sample of 21 GRBs, shown in Oganesyan et al.(2019) (see Table B1 of Appendix B). For GRBs where anal-ysis is performed in multiple time intervals, we show resultsfor one time interval that corresponds to the brightest fluxstate, unless stated otherwise. In our analysis we do not in-clude GRB 060814 and GRB 090715B because more than oneobservables are not constrained (they have upper or lowerlimits). Other bursts that have upper limits on only one ob-servable have been included in the analysis. In the previous section we provided analytical expressions forcomputing the source parameters in the context of a protonsynchrotron model for the GRB prompt emission. To verifythis assumption, we utilize a time-dependent numerical code(
ATHE ν A , Dimitrakoudis et al. 2012) that follows the evolu-tion of spatially averaged particle populations inside a homo-geneous spherical emitting region. The numerical approachof the problem gives us also the opportunity to extend thework of Ghisellini et al. (2020) by investigating the contribu-tion of photohadronic interactions and γγ pair production tothe overall photon spectrum and by computing the associatedneutrino flux.The numerical code computes the electromagnetic, neu-trino, and cosmic-ray (i.e., proton and neutron) fluxes emerg-ing from a single radiation zone in the jet under certain as-sumptions. These are summarized below. After being injectedinside a spherical source with a constant injection rate (asin eq. 1 but with a high-energy exponential cutoff insteadof a sharp cutoff at γ max ), relativistic protons interact withthe magnetic field and any soft photons present, creating ra-diation and other secondary particles. Among those, pionsand muons decay almost instantaneously into lighter particles(i.e., pairs, electron and muon neutrinos). However, a fractionof their populations can cool by emitting synchrotron photonsbefore their decay (for more details, see Appendix A). Pionand muon synchrotron cooling can affect both the photon andneutrino spectra (Baerwald et al. 2011; Petropoulou et al.2014b; Tamborra & Ando 2015). Neutrons do not typicallyinteract with soft photons before they escape the source (i.e.,the source is optically thin to neutron-photon interactions),and neutrinos escape the source without any interactions ona light-crossing time.At any given time, there are five stable particle speciesin the emitting region, namely protons, photons, electron-positron pairs, neutrons and neutrinos. The production andloss rates of these five stable particle species are tracked self-consistently with five time-dependent coupled kinetic equa-tions, which can be written in a compact form as ∂n i ∂t + n i t i , esc + L i = Q i . (18)Here, n i is the differential number density of particle species i , t i , esc = r b /c is the respective escape timescale, which isassumed to be energy-independent for all particles, and Q i and L i are the injection (source) and loss (sink) terms, re-spectively. These terms include the following processes: • synchrotron radiation for both electrons and protons • proton-photon pair production (photopair) • proton-photon pion production (photopion) • neutron-photon pion production • pion, kaon, muon and electron synchrotron radiation • synchrotron self-absorption • electron inverse Compton scattering • photon-photon ( γγ ) pair production • electron-positron pair annihilation Neutrons typically escape the source without undergoingneutron-photon interactions, and deposit their energy in the sur-rounding region once they are transformed back to protons.MNRAS , 1–16 (2021)
GRB proton-synchrotron model Summarizing, the numerical code we utilize is the one pre-sented in Mastichiadis et al. (2005); Dimitrakoudis et al.(2012) but augmented in a way to include pion, muon andkaon synchrotron cooling. Details about the implementationof the latter can be found in Petropoulou et al. (2014b).
In this section we adopt the idea that the marginally fastsynchrotron cooling can relax the inconsistency between thesynchrotron radiation and the harder spectra of some GRBs.We adopt the idea of a hadronic scenario (Ghisellini et al.2020), according to which, the proton synchrotron is the dom-inant radiative process in the emitting source responsible forthe prompt GRB emission. Based on this framework, we an-alytically obtain constraints on the physical parameters ofthe source (Sec. 4.1). As a next step, we numerically calcu-late the broadband photon spectra and investigate whethertheir shape is modified when additional physical processesare taken into account (Sec. 4.2).
In order to find some constraints on the physical parametersof the source, such as the magnetic field, the size of the emit-ting region, the proton luminosity, and the minimum protonLorentz factor, we adopt the best-fit values (and uncertain-ties) for several observables from Oganesyan et al. (2019) anduse the analytical expressions presented in Sec. 2. To com-pute the uncertainties on the inferred model parameters, weuse the python package soad . Using the methods of Erdim &Hudaverdi (2019), soad searches for a likelihood function thathas a maximum at the best-fit value of a measured quantitywith asymmetric uncertainties.As an example, we show in Fig. 1, the distributions of E c values for two GRBs of the sample. The E c values ofGRB 110102A follow approximately a normal distribution,while the distribution of GRB 100906A is asymmetric withpositive skewness. The turquoise solid and dashed lines showsthe median value and the 68% range of these distributionsrespectively, while the solid blue line indicates the best-fitvalue, which can also be interpreted as the most probablevalue of the distribution. The median value does not co-incide with the most probable value, only for a few GRBsin the sample (GRB 081008, GRB 100906A, GRB 121123A,GRB 140206, GRB 151021A) that have very asymmetric dis-tributions for one (or more) observables. In what follows, wereport and use the median and 68% uncertainty range for themodel parameters.All model parameters are plotted as a function of the bulkLorentz factor of the outflow for a wide range of plausiblevalues, namely ≤ Γ ≤ . As an illustrative example,we show in Fig. 2 the values of the comoving magnetic field,the minimum proton Lorentz factor and the proton compact-ness as a function of Γ , obtained for GRB 061121 (at redshift z = 1 . (Bloom et al. 2006)). As we have already shownin eqs. 8, 10, and 16, the values of B , γ m and L p decrease as https://github.com/kiyami/soad/ the bulk Lorentz factor increases. We also show the size ofthe spherical source r b as a function of Γ , which is commonfor all the bursts, since it is computed following eq. 14 and16, under the assumption that t γ = 1 s for all the GRBs in-cluded in the sample. Shorter variability timescales translateto smaller source sizes and higher photon number densities.We repeat the same analysis for all GRBs in our sampleand report the values of B, γ m , and L p in Fig. 3 for an in-dicative value of the bulk Lorentz factor Γ = 300 . Resultsfor another choice of Γ can be found using the scaling re-lations B ∝ Γ − / , γ m ∝ Γ − / , L p ∝ Γ − . The parame-ters for individual GRBs for Γ = 300 and 1000 are listedin Table 1. We find that comoving magnetic fields strengthsof the order of (1 − × G are required for explain-ing the spectral break at ∼ keV energies as result of protonsynchrotron cooling. Because of the strong magnetic fields,acceleration of protons to ultra-high energies is not neces-sary for explaining the keV to MeV part of the GRB promptspectrum. More specifically, we find that the required min-imum Lorentz factor of the proton distribution, γ m , rangesbetween . and . . The proton injection luminosity inthe jet comoving frame has a strong dependence on the bulkLorentz factor (see eq. 2). As a result, we find that the pro-ton luminosity in the comoving frame spans a wide range ofvalues, namely erg s − (cid:46) L p (cid:46) erg s − . However,the isotropic proton luminosity in the observer’s frame is al-most independent of Γ and the distribution of L p , obs valuesreflects the distribution of L γ, iso in the GRB sample (see Ta-ble 1). Finally, we find that the size of the comoving regionis cm < r b (cid:46) × cm . These results are consistentwith those of Ghisellini et al. (2020), who assumed a power-law particle distribution of minimum lower Lorentz factor γ m = 10 . , a typical magnetic field B ∼ G , and comov-ing radius of the emitting source r b ∼ cm for a fixedvariability timescale of t var = 1 s. γγ pairproduction in GRB prompt photon spectra Relativistic protons can become targets to their own radia-tion, after interacting with the strong magnetic field of thesource and emitting synchrotron photons. Before we proceedwith fully numerical calculations of the emerging electromag-netic radiation of the GRB prompt phase, we estimate semi-analytically the photohadronic efficiency and γγ opacity ofthe emitting region.Fig. 4 shows the photopion production efficiency f p π = t − π r b /c (see eq. C1) and the photopair production efficiency f pe = t − r b /c (see eq. C2) as a function of the comoving pro-ton energy ε p for Γ = 100 (thick lines) and 1000 (thin lines).Solid red lines show the total efficiency of photohadronic in-teractions, defined as f p γ ≡ f p π + f pe . Proton losses are dom-inated by photopion production, except for Γ = 1000 wherephotopair production becomes more important at the lowestenergies. The dependence of f p γ on the bulk Lorentz factorfor a given proton energy is strong (i.e., ∝ Γ − ) and reflectsthe dependence of the photon number density on Γ (see alsoeq. C4).High-energy photons that are produced by the power-lawdistribution of synchrotron-cooled protons and/or by photo-hadronic interactions may be absorbed by the isotropic syn-chrotron photon field inside the emitting region. Therefore, MNRAS , 1–16 (2021)
Florou et al. E c [keV] D i s t r i b u t i o n GRB 110102A E c [keV] D i s t r i b u t i o n GRB 100906A
Figure 1.
Distribution of E c values for two GRBs created using the python package soad and the best-fit values with their uncertaintiesfrom Oganesyan et al. (2019). The solid dark blue and turquoise lines correspond to the most probable value and to the median valuerespectively, while and the turquoise dashed lines indicate the 68% uncertainty range.
200 400 600 800 10006.56.66.76.86.9 o g B [ G ]
200 400 600 800 10004.24.34.44.54.64.7 o g m
200 400 600 800 100041424344 o g L p [ e r g s ]
200 400 600 800 100012.212.412.612.813.0 o g r b [ c m ] Figure 2.
Inferred parameters of the proton synchrotron model for GRB 061121 using observables from the time interval with the highestphoton flux. From top left and in clockwise order we show the magnetic field B , the minimum proton Lorentz factor γ m , the sourceradius r b , and the proton injection luminosity L p as a function of the bulk Lorentz factor Γ . Shaded regions indicate the σ uncertainties,whenever relevant. All quantities are measured in the comoving frame of the outflow.MNRAS000
Inferred parameters of the proton synchrotron model for GRB 061121 using observables from the time interval with the highestphoton flux. From top left and in clockwise order we show the magnetic field B , the minimum proton Lorentz factor γ m , the sourceradius r b , and the proton injection luminosity L p as a function of the bulk Lorentz factor Γ . Shaded regions indicate the σ uncertainties,whenever relevant. All quantities are measured in the comoving frame of the outflow.MNRAS000 , 1–16 (2021) GRB proton-synchrotron model Table 1.
Median values and 68% uncertainty ranges (in logarithmic scale) of parameters in the proton synchrotron model for the GRBsanalyzed by Oganesyan et al. (2019) for
Γ = 300 . Values enclosed in parentheses are obtained for
Γ = 1000 .Source B (G) γ m L p (erg s − ) L p , obs a (erg s − )GRB 060510B . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 061121 . +0 . − . ( . +0 . − . ) > . ( > . ) > . ( > . ) > . GRB 070616 . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 080928 . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 081008 . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 100906A . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 110102A . +0 . − . ( . +0 . − . ) > . ( > . ) > . ( > . ) > . GRB 110119A . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 110205A . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 111103B . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 111123A . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 121123A . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 121217A . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 130514A . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 130907A . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 140108A . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 140206A . +0 . − . ( . +0 . − . ) > . ( > . ) > . ( > . ) > . GRB 140512A . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . GRB 151021A . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . ( . +0 . − . ) . +0 . − . a The proton luminosity in the observer’s frame is independent of Γ .For GRBs with multiple time intervals analyzed, we report values for the interval of peak flux. In three cases,the minimum Lorentz factor and the proton luminosity are quoted as lower limits, because one of the observables( γ m /γ c ) that is required in order to compute them, is given as a lower limit. it is important to estimate the photon opacity of the source.We compute the optical depth τ γγ by using eq. C6 for theparameter values of GRB 061121 and overplot the solutionin Fig. 4 for two values of Γ (solid black lines). For most val-ues of Γ , we find that the source is optically thick to γγ pairproduction ( τ γγ >> ) for a wide range of comoving photonenergies. Only for Γ = 1000 (thin black line) we find τ γγ (cid:46) for almost all photon energies. Therefore, we conclude that,for parameters that correspond to lower bulk Lorentz fac-tors, γγ photon absorption is unavoidable and the producedelectron-positron pairs might affect the overall spectrum ofthe GRB prompt emission. GRBs are candidates of proton acceleration to ultra-high en-ergies and, therefore, potential sources of high-energy neu-trino emission (Vietri 1995; Waxman & Bahcall 1997b, 1998).After investigating the role of the photohadronic interactionsin the prompt emission spectrum, we calculate analyticallythe neutrino flux and energies, given the parameter values wehave already computed. The all-flavour neutrino flux can beapproximately computed as ε ν, obs F ε ν, obs ≈
38 Γ f p π ( ε p ) r cε p U p ( ε p ) d L (19) where ε ν, obs ≈ η p π Γ ε p (1 + z ) , η p π = 1 / is the fractionof proton energy that is transferred to each neutrino pro-duced, and U p ( ε p ) is the differential proton energy densityat steady state. This can be written as U p ( ε p ) = ε p n p ( ε p ) ,where n p ( ε p ) = n p ( γ ) /m p c . The steady-state proton distri-bution n p ( γ ) is given by (Inoue & Takahara 1996) n p ( γ ) = Q γ c t p , esc γ e − γ c γ (cid:90) γ max max[ γ m ,γ ] e γ c γ γ − p d γ (20)where Q is computed through eq. 2. The above equationdescribes essentially a broken power law, i.e., n p ( γ ) ∝ γ − p for γ c < γ < γ m and n p ( γ ) ∝ γ − p − for γ > γ m .We apply eq. 19 to each GRB of the sample using the pa-rameters that correspond to Γ = 300 (see Table 1 below).The resulting all-flavour neutrino fluxes are shown in Fig. 5.Most of the curves show a triple broken power-law shapethat arises from the multiplication of the proton distribu-tion function with the double broken power-law function ofphotopion efficiency (eq. C1). The shape of the latter is de-termined by the proton-synchrotron photon number density(see eq. C4). A closer look at these results and the observablesfor each burst shows that as the ratio γ m /γ c decreases andapproaches unity, the first branch of the neutrino spectrumdisappears. The neutrino break energy between the first andthe second branch of the neutrino energy spectrum occurstypically when protons of Lorenz factor equal to γ m interactwith photons of energy greater than E pk , obs , and corresponds MNRAS , 1–16 (2021)
Florou et al. og B [G] N u m b e r o f G R B s o f t h e s a m p l e =300 og m N u m b e r o f G R B s o f t h e s a m p l e =300 og L p [erg s ] N u m b e r o f G R B s o f t h e s a m p l e =300 Figure 3.
Logarithmic histograms of the inferred
B, γ m , and L p values for our GRB sample and Γ = 300 . Results for another choiceof Γ can be obtained using the scaling relations B ∝ Γ − / , γ m ∝ Γ − / , L p ∝ Γ − . The values of individual GRBs are listedin Table 1. og , og p [eV] o g f p , o g f p e , o g =100=1000 f pe f p f p Figure 4.
The dimensionless fractional energy loss rate of rela-tivistic protons due to photopion production (dashed orange lines),photopair production (dotted cyan lines) and the sum of the twoprocesses (solid red lines) as a function of the comoving protonenergy. The opacity of the emitting region to γγ pair productionis also plotted as a function of the comoving photon energy (solidblack lines). Thick and thin lines show results for Γ = 100 and1000, respectively. Displayed results are for GRB .
10 12 14 16 18 og v, obs [eV] o g v , o b s F v , o b s ( v , o b s ) [ e r g s c m ] GRB 060510GRB 061121GRB 07616GRB 080929GRB 081008GRB 100906AGRB 110102 GRB 110119GRB 110205GRB 111103BGRB 111123AGRB 121123AGRB 121217AGRB 130514A GRB 130907AGRB 140108GRB 140206GRB 140512GRB 151021GRB 07616
Figure 5.
Semi-analytical results for the all-flavour neutrino en-ergy flux spectra in the proton synchrotron model for the GRBsample of Oganesyan et al. (2019), assuming
Γ = 300 . Results areshown for the best-fit values of observables from the interval ofpeak γ -ray flux. Diamonds mark the characteristic energy abovewhich pions cool due to synchrotron radiation before they decay,leading to a steeper neutrino spectrum.MNRAS000
Γ = 300 . Results areshown for the best-fit values of observables from the interval ofpeak γ -ray flux. Diamonds mark the characteristic energy abovewhich pions cool due to synchrotron radiation before they decay,leading to a steeper neutrino spectrum.MNRAS000 , 1–16 (2021) GRB proton-synchrotron model og F pk, obs [erg sec cm ] o g F p k v , o b s [ e r g s e c c m ] Figure 6.
A scatter plot that shows the peak of the predictedall-flavour neutrino energy flux for the GRBs of the sample, as afunction of the ratio of the photon to neutrino peak fluxes, assum-ing that
Γ = 300 (purple circles). The coloured solid line has aslope of two and is plotted in order to guide the eye. to the peak energy of the neutrino spectrum. This ranges be-tween TeV and 1 PeV for
Γ = 300 , and scales as ∝ Γ .Because of the strong magnetic fields ( − G) involvedin the proton-synchrotron model, the cooling timescale of pi-ons and muons can be faster than their decay timescale (seeAppendix A), thus leading to a steepening of the neutrinospectrum (e.g., Waxman & Bahcall 1997a; Abbasi et al. 2010;Zhang & Kumar 2013). If F ε v , obs ∝ ε χ v , obs , we approximatelyaccount for the cooling effects by describing the neutrino fluxas F ε v , obs ∝ ε χ − , obs above an energy that corresponds to thatof a pion with equal synchrotron cooling and decay timescales(this is indicated with a diamond in the figure). While thisis a rough approximation (for a detailed analytical treatmentof cooling effects, see e.g., He et al. 2012; Tamborra & Ando2015; Pitik et al. 2021), it is sufficient for the purposes of thisstudy. More details on the neutrino flux predicted by thismodel will be presented elsewhere (Pitik et al. 2021).We next compare the peak energy fluxes of the all-flavourneutrino and photon spectra, which we denote respectivelyas F (pk) ν, obs and F (pk) γ, obs . Our results are presented in Fig. 6 for Γ = 300 . Here, the error bars indicate the 68% range of valuesof the respective distributions (see also Fig. 1). The predictedpeak fluxes of a few GRBs (see 4.1) have large uncertaintiesbecause at least one of their observables (i.e., E c , F c , γ m /γ c )has large asymmetric error bars. The neutrino peak flux ismany orders of magnitude below the peak γ -ray flux, re-flecting the low photopion production efficiency that is ex-pected for conditions that enhance the synchrotron cooling ef-ficiency of protons (see also Fig. 4). Fig. 6 also shows that formost GRBs the predicted fluxes scale as F (pk) ν, obs ∝ (cid:16) F (pk) γ, obs (cid:17) .Such quadratic relation is expected in the proton-synchrotronmodel, since both the photopion efficiency and the proton en-ergy density scale linearly with the photon number density(for details, see Appendix C). Table 2.
Parameter values used in the numerical proton syn-chrotron model of GRB 061121. Γ B (G) r b (cm) γ m L p (erg s − )300 . × . × . × . × . × . × . × . × Note. – The parameter values are inferred for the first timeinterval used in the time-resolved analysis of Oganesyan et al.(2019). Other parameters used are γ max = 10 and p = 2 . . Utilizing the numerical code presented in Sec. 3 and the pa-rameter values obtained in Sec. 2, we compute the GRBprompt photon and neutrino spectra in the proton syn-chrotron scenario after taking into account all relevant ra-diative processes. In particular, we demonstrate how thephotohadronic interactions, γγ pair production, and syn-chrotron cooling affect the overall photon spectrum of theGRB prompt emission. We also take into account synchrotroncooling of kaons, pions, and muons, as this may be importantfor certain parameter values (see Appendix A). As an illus-trative example we use GRB 061121. Model parameters areinferred by the observables for the first time interval (63.99-68.83 s) analyzed by Oganesyan et al. (2019) (see Table B1and Figure 4 therein), and are summarized in Table 2 for twoindicative values of the Lorentz factor.The broadband photon and all-flavour neutrino spectra arepresented in Figure 7. For comparison, we also show the pureproton synchrotron spectrum (dotted lines) that provides agood description of the prompt emission from a few eV to ∼ keV energies. When all processes are taken into ac-count we find a modification of the spectra at ε obs < E c , obs (cid:39) keV, which becomes stronger for lower Γ values. This ex-tra emission can be attributed to ultra-relativistic pairs in-jected in the source mainly via γγ pair production that havecooled down to γ ∼ due to synchrotron radiation in thevery strong magnetic field of the emitting region. Becauseof the complete electron cooling down to trans-relativisticenergies (i.e., . (cid:46) γ (cid:46) ), cyclotron-synchrotron effects be-come important at energies ε ∗ , obs (cid:46)
22 eV ( B/ G)(Γ / ) .While the numerical code computes correctly the total powerlost by a trans-relativistic electron (see e.g., eq. 2 in Ghis-ellini et al. 1998), it does not use the appropriate spectrumfor the single-particle cyclotron-synchrotron emissivity, whichapproximately scales as j ν ∝ const instead of ν / (Beck-ert & Duschl 1997; Ghisellini et al. 1998). As a result, theshape of the low-energy part of the synchrotron spectrum( ε obs (cid:46) ε ∗ , obs ) is not described accurately. Because of thislimitation of our numerical treatment we cannot excludethe proton synchrotron interpretation for Γ = 1000 , wherethe secondary electron emission is not very luminous andcauses deviations from the data below a few eV. However,for
Γ = 300 , the secondary synchrotron emission overshootsthe
Swift /XRT flux, i.e., at energies above ε ∗ , obs (cid:39) eV.Given that the main source of secondary pairs comes from the This is still a phenomenological approximation. The true spec-trum is more complicated, as it is composed of a series of harmonics(see e.g., Marcowith & Malzac 2003). MNRAS , 1–16 (2021) Florou et al. −12 −10 −8 −6 −4 ε obs (eV)10 −12 −10 −8 −6 −4 ε ob s F ε ob s ( e r g c m − s − ) γ (all processes) γ (p syn) ν (all processes) ν (no K, π , µ cooling) Γ =1000 [63.99−68.83 s] Swift/UVOTSwift/XRTSwift/BAT −12 −10 −8 −6 −4 ε obs (eV)10 −12 −10 −8 −6 −4 ε ob s F ε ob s ( e r g c m − s − ) γ (all processes) γ (no γ γ ) γ (p syn) ν (all processes) ν (no K, π , µ cooling) Γ =300 [63.99−68.83 s] Swift/UVOTSwift/XRTSwift/BAT
Figure 7.
Predicted photon (solid blue line) and all-flavour neutrino (dashed blue line) energy spectra of GRB 061121 in the proton-synchrotron prompt emission model for
Γ = 1000 (left panel) and
Γ = 300 (right panel). The proton synchrotron spectrum (dotted blueline) and the neutrino spectrum in the absence of meson and muon synchrotron cooling (dashed-dotted red line) are overplotted. In theright panel, the photon spectrum without γγ pair production is also shown for comparison (solid light blue line). In both panels, promptobservations by Swift /UVOT,
Swift /XRT and
Swift /BAT (adopted from Oganesyan et al. (2019)) are overplotted with symbols (see insetlegend). attenuation of proton synchrotron photons above the peak(compare light blue and blue solid lines in the right panel),the secondary emission could not have been avoided even ifphotohadronic interactions were not taken into account.In both cases, the neutrino energy spectrum (dashed lines)peaks at PeV energies, but the neutrino peak flux is manyorders of magnitude lower than the peak γ -ray flux (see alsoFig. 6). Because the photopion production efficiency is a sen-sitive function of the Lorentz factor (see also Fig. 4), a de-crease of Γ by a factor of 3 leads to an increase of ∼ orders ofmagnitude in the neutrino flux. Still, neutrino fluxes compa-rable to the peak γ -ray fluxes can be excluded in this scenariobecause of the strong modification of the prompt emissionspectrum by secondaries expected for Γ (cid:46) (with the ex-act value depending on the properties of individual GRBs).We also show the effects of meson and muon cooling on theall-flavour neutrino spectra (compare dashed blue and dash-dotted red lines). Above the peak energy of the neutrino spec-trum, the flux decreases because pions and muons cool beforethey decay (see also Baerwald et al. 2011; Petropoulou 2014;Tamborra & Ando 2015).A comparison of the numerical and semi-analytical resultsfor the neutrino emission is presented in Fig. 8. For complete-ness, we also show the respective proton synchrotron spectra,with the analytical one given by eq. C5. For this illustrativeexample, we used the parameter values of GRB 061121. Inthe absence of meson and muon cooling, the peak flux andpeak energy of the numerical and semi-analytical neutrinospectra are in good agreement (compare solid blue and dash-dotted lines). The shape of the numerical neutrino spectrumabove the peak energy is smoother than the one found semi- −14 −12 −10 −8 −6 −4 −2 GRB 061121 ε obs (eV)10 −14 −12 −10 −8 −6 −4 −2 ε ob s F ε ob s ( e r g c m − s − ) γ : p−syn (num.) ν : no K, π , µ cooling (num.) ν : K, π , µ cooling (num.) γ : p−syn (an.) ν : no π cooling (an.) ν : π cooling (an.) Figure 8.
Comparison of numerical and analytical results for theproton-synchrotron spectrum and all-flavour neutrino spectra ofGRB 061121 for
Γ = 300 . For a description of the curves, see insetlegend. analytically, which is related to the smoothness and curvatureof the target photon spectra (compare solid dark blue anddashed lines). Pion cooling leads to a steepening of the neu-
MNRAS , 1–16 (2021)
GRB proton-synchrotron model trino spectrum above a characteristic energy, which is seenin both numerical and analytical spectra (compare solid lightblue and dotted lines). However, the peak neutrino flux inthe numerical solution is ∼ times lower than the one of thesemi-analytical solution. This drop is not seen in the latterspectrum because our analytical approach does not take intoaccount the decrease in the number of produced muons (andconsequent muon neutrinos) caused by the pion cooling (fora more accurate analytical treatment, see He et al. 2012). Inthis regard, the semi-analytically derived neutrino fluxes (seeSec. 4.3.1) are optimistic. In this paper we have investigated whether a synchrotronmodel of relativistic protons that do not cool completelycould fit observations of prompt GRB spectra. We have alsoexamined how important are additional physical processes,such as photohadronic interactions and photon-photon pairproduction, in the overall shape of the GRB spectrum andwhat is the expected neutrino flux, assuming parameters thatlead to proton-synchrotron radiative dominance. We havesupported our analytical calculations by utilizing a time-dependent numerical code, which computes the GRB promptemission spectra taking into account all relevant physical pro-cesses without making approximations about cross sections,inelasticities, and production spectra of secondary particles.Using the best-fit values from Oganesyan et al. (2019) forfour observables of GRB prompt spectra, we have shown thatin case of a marginally cooled proton synchrotron model theemitting region is compact and strongly magnetised. Thissuggests that the dominant form of energy carried by the jethas to be electromagnetic. We therefore compute the Poynt-ing luminosity of the jet, which is defined as: L B , j = cR γ ( B Γ) ∝ Γ / t / γ, obs E − / , obs (1 + z ) − / (21)and the jet luminosity due to the relativistic proton popula-tion, which is written as: L p , j = cR γ Γ m p c (cid:90) γ max γ c γn p ( γ )d γ (22)where n p is given by eq. (20). For Γ = 300 , the medianvalues for the GRB sample read L B , j ≈ erg s − and L p , j ≈ erg s − . Given that the Poynting luminosity hasonly a weak dependence on the observables (see eq. 21) wecan conclude that extreme jet luminosities cannot be avoidedin the marginally cooled proton-synchrotron model for GRBprompt emission.Proton-synchrotron models of γ -ray emission from jetsof active galactic nuclei (AGN) have been widely discussed(Aharonian 2000; Mücke & Protheroe 2001). Several studies(e.g., Sikora et al. 2009; Cerruti et al. 2015; Petropoulou &Dimitrakoudis 2015; Zdziarski & Böttcher 2015; Petropoulouet al. 2016; Liodakis & Petropoulou 2020) have pointedout the excessive energetic requirements of the proton syn-chrotron model needed to counterbalance the inefficiency ofthe proton synchrotron radiation. Moreover, it was shownthat photohadronic processes are suppressed, leading to neu-trino fluxes from individual AGN that lie well below the Ice-Cube sensitivity threshold (e.g., Dimitrakoudis et al. 2014;Keivani et al. 2018; Gao et al. 2019; Liodakis & Petropoulou 2020). While these results are similar to the findings of thiswork, there are some important differences between the twoproton synchrotron models. The magnetic field strength inthe GRB model considered here is much stronger than thevalues used in AGN models ( B ∼ − G), as in thelatter proton synchrotron cooling is not a requirement. Thisfact also leads to the predominance of the magnetic lumi-nosity over the proton luminosity of the jet in contrast toAGN models where the opposite is noticed. Moreover, thetypical energy of protons radiating at the peak of the GRBphoton spectrum is several orders of magnitude lower thanthe one in AGN models. This is related to differences in thecomoving magnetic field strengths ( O (10 ) G in GRBs ver-sus O (10 ) G in AGN), peak energies of the γ -ray spectra(sub-MeV in GRBs versus GeV in AGN), and bulk Lorentzfactors ( O (10 ) G in GRBs versus O (10) G in AGN). Finally,in the GRB proton synchrotron models the emitted neutrinosare of PeV energies (with the peak determined by the pionand muon synchrotron cooling), while AGN models predictneutrinos in the EeV energy band.Most of the hadronic models assume that protons carrythe majority of the energy and only a small amount is radi-ated away mainly by proton synchrotron radiation. However,there is an intriguing intrinsic property of hadronic systems,the so-called hadronic supercriticality, during which photo-hadronic processes, like photopair and photopion, dominateinside the source and essentially drain the proton energy andtransfer it to secondaries, thus increasing the photon effi-ciency to high values (Kirk & Mastichiadis 1992; Mastichiadiset al. 2005; Petropoulou & Mastichiadis 2012). Different vari-ants of hadronic supercriticalities have been applied to GRBs(Kazanas et al. 2002; Petropoulou et al. 2014a; Petropoulou &Mastichiadis 2018). In particular, Petropoulou et al. (2014a)have shown that the proton synchrotron radiation producesVHE γ -rays that are quenched, producing as a result electron-positron pairs, while the MeV emission is the outcome ofComptonization of photons by cooled pairs. Moreover, a thor-ough analysis of the parameters of hadronic supercriticalitythat could be implemented to GRB prompt emission has beendiscussed in Mastichiadis et al. (2020). Both studies used verydifferent source parameters compared to those of the proton-synchrotron model. Apart from assuming a much smallerminimum proton Lorentz factor ( γ m = 1 ), these works uselower values of magnetic field strength ( B ∼ G) insidethe spherical volume and lower values of proton injection lu-minosity L p per radius of the emitting volume r b , stated ascompactness (cid:96) p , in order to push the system to the onset ofsupercriticality and reproduce a GRB.Long-duration GRBs are thought to be associated with thecollapse of the core of massive stars (Woosley 1993; Hjorthet al. 2003; Stanek et al. 2003; Woosley & Bloom 2006). Theburst is proposed to be powered by the rotation of a rapidlyrotating strongly magnetized neutron star (i.e., magnetar)(e.g., Usov 1992; Thompson et al. 2004; Metzger et al. 2011)or by the accretion on to a black hole created after the corecollapse (e.g., Woosley 1993; MacFadyen & Woosley 1999;MacFadyen et al. 2001). In the first scenario, the magnetarwind is considered to be the source of the outflow respon-sible for the relativistic GRB jet (Metzger et al. 2011). Inthe simplest magnetar model, where the jet is solely pow-ered by the solid-body spin-down energy of the magnetar,the jet luminosity L j ∝ L SD ∝ P − B , where P is the ini- MNRAS , 1–16 (2021) Florou et al. tial spin period and B is the surface magnetic field. For B = 10 − G and P ∼ ms, L j ∼ − ergs − . If the central engine is a black hole, accretion in com-bination to the rotation energy of the black hole can powerthe GRBs (e.g., Mészáros & Rees 1997). The in-falling stellarmaterial can drag in the large-scale magnetic flux through theprogenitor star and the jet can be powered via the Blandford-Znajek process (Blandford & Znajek 1977). In this scenario,the jet power (equivalent to the power of the central engine)is determined by the magnetic flux through the black-holehorizon Φ BH , namely L j ≡ L BZ ∝ a BH Φ M − , where a BH and M BH are the BH spin and mass, respectively. As long asthe accretion rate ˙ M is high enough as to sustain the mag-netic flux Φ BH on the BH, the jet power is independent of ˙ M and approximately constant (Tchekhovskoy & Giannios2015). Assuming that the collapsing core of the star formsa black hole with M BH = 4 M (cid:12) and B BH ∼ G at thehorizon, the resulting jet power L j ∼ erg s − . Based onthe jet luminosities expected in collapsars, the jet luminos-ity predicted by the proton-synchrotron model (see eq. 21) isprohibitively high.Apart from the unreasonably high jet luminosity, anotherproblem of the marginally fast-cooling proton synchrotronmodel for the GRB prompt emission is the modification of thelow-energy part of the photon spectrum by secondary pairsand the disagreement with the data, especially for low valuesof Γ . While the proton synchrotron spectrum (see blue dottedlines in Fig. 7) fits perfectly the observations, other physicalprocesses that cannot be neglected (e.g., proton-photon in-teractions and γγ absorption) reshape the photon spectrumin a way that it becomes incompatible with the soft X-rayand optical observations (see blue solid lines in Fig. 7). Be-cause of the strong inverse dependence of the γγ opacity andp γ efficiency on Γ (see Fig. 4 and Appendix B), the contri-bution of secondaries to the spectrum of GRB 061121 canbe suppressed only for Γ (cid:38) . While the exact value of Γ will vary among different GRBs, it will be of the same or-der. Thus, the cascade emission from secondaries can set verystrong lower limits on the acceptable values of Γ . These lim-its may be inconsistent with the bulk Lorentz factors inferredwith other methods, e.g., afterglow light curves (Ghirlandaet al. 2018), high-energy spectral modeling (Tang et al. 2015)and others (Ghirlanda et al. 2011; Nava et al. 2016).Based on the above discussion, it becomes clear that themarginally fast-cooling proton synchrotron model for theGRB prompt emission cannot explain the observations pre-sented in Oganesyan et al. (2019). The leptonic synchrotroninterpretation, which was put forward by Oganesyan et al.(2019), does not face the problems of the proton-synchrotronmodel. However, it was disfavoured by Ghisellini et al. (2020)because the values of magnetic field and radii needed to re-produce the GRB prompt spectrum with a low Compton ra-tio suggest a variability timescale much longer than the oneobserved. Nonetheless, a leptonic synchrotron explanation ofthe GRB spectra may be viable when studied with a time-dependent approach. One proposed solution is to include aspatially decaying magnetic field within an expanding emit-ting region. The effects of a decaying magnetic field on theelectron and photon spectra have been studied by Uhm &Zhang (2014b). These authors showed that the spectral indexcan become harder than in the case of cooling in a constantmagnetic field strength, and the low-energy spectrum below the injection frequency may appear more curved. Anotherproposed idea is the assumption of an electro-synchrotron,internal collision induced magnetic reconnection and turbu-lence (ICMART) model (Zhang & Yan 2011). In this scenario,the injection rate of electrons on the synchrotron radiationspectrum increases rapidly with time leading to the harden-ing of the corresponding radiation spectrum (Liu et al. 2020).Gill et al. (2020) investigated the steady-state photon spec-tra produced by electrons that are either accelerated via mag-netic reconnection into a power law or heated via magnetohy-drodynamic instabilities in a strongly magnetized relativisticoutflow. In their scenario, power-law electrons cool mainly bysynchrotron radiation, while the almost monoenergetic elec-trons from heating cool by Comptonization on thermal peakphotons. In both cases, the photon spectra show a low-energybreak at ∼ keV energies that may be consistent with GRB ob-servations (Oganesyan et al. 2017, 2019). In conclusion, thelow-energy break seen in the prompt spectra of certain GRBsremains a puzzle. Note added:
Neutrino predictions of the proton-synchrotron model of GRB prompt emission are alsopresented in the independent work of Pitik et al. (2021). Ourpaper investigates in detail the plausibility of the proton-synchrotron interpretation by analytical and numericalmeans.
ACKNOWLEDGEMENTS
We would like to thank Dr. Gor Oganesyan for providingthe observational data of GRB 061121 and Dr. GeorgiosVasilopoulos for discussions on the python uncertainty rou-tine. I.F. acknowledges that this research is co-financed byGreece and the European Union (European Social Fund-ESF) through the Operational Programme “Human Re-sources Development, Education and Lifelong Learning” inthe context of the project “Strengthening Human ResourcesResearch Potential via Doctorate Research” (MIS-5000432),implemented by the State Scholarships Foundation (IKY).
DATA AVAILABILITY
The GRB sample of Oganesyan et al. (2019) was used withthe best-fit values of observables taken from Table B.1 therein(under the column “synchrotron model”). Flux points shownin Fig. 7 were provided to us by Dr. Oganesyan upon request.Uncertainties have been computed using the python packagethat is available from https://github.com/kiyami/soad/ .Numerical calculations of photon and neutrino spectra wereperformed making use of the non-public code
ATHE ν A (Dimi-trakoudis et al. 2012). Results shown in Figs. 7 and 8 and anyother accompanying outputs of the code are available uponrequest to the corresponding authors. REFERENCES
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APPENDIX A: PION & MUON SYNCHROTRONCOOLING
The charged byproducts of photohadronic interactions,namely pions and muons, cannot be always consideredto decay instantaneously after their production. If theirsynchrotron cooling timescale is shorter that their decaytimescale, these particles can cool via synchrotron radiationbefore they decay. They can therefore contribute to the over-all GRB photon spectrum, while suppressing the neutrinoflux above a certain energy. In this section, we investigatewhether pion and muon synchrotron cooling are relevant forthe source parameters we use in Sec. 4.In the case of pions, we calculate the minimum pionLorentz factor above which the synchrotron cooling timescaleis shorter than the decay timescale: γ π > (cid:115) πm e c σ T B τ π (cid:18) m π m e (cid:19) (A1)where τ π = 2 . × − s the charged pion mean lifetime inthe lab frame and m π its rest mass. The kinematics of thisresonance decay predicts a nucleon inelasticity (Mücke et al.1999) κ = γ π m π c γm p c = 0 . . (A2)Therefore the minimum energy of relativistic protons thatwould produce pions which emit synchrotron radiation beforethey decay is given by the following relation: γ > γ π, synp ≡ (cid:18) m π m e (cid:19) / m e m p (cid:114) πm e c σ T τ π B − κ − (A3)where the magnetic field B is given by eq. 10.In Fig. A1 we plot the minimum values of the protonLorentz factor required to produce pions that cool via syn-chrotron before they decay, γ π, synp , as a function of Γ (solidgrey line). In the same figure we also show the minimumLorentz factor of the proton distribution, γ m , calculatedthrough eq. 12 as a function of Γ (solid light blue line). Thisplot is created using the observables of GRB 061121. Thecoloured bands indicate the uncertainties on each parameter.These have been calculated following the method of Erdim& Hudaverdi (2019), as described in Sec. 2. We find thatthe two regions do not intersect for all values of Γ consid-ered. Thus, only a fraction of the proton distribution (with γ php ≤ γ ≤ γ max ) can lead to the production of pions thatwill cool via synchrotron before they decay. How much pioncooling will affect the observed photon and neutrino spectrawill also depend on the shape of the proton energy distribu-tion (e.g., power-law index).We follow the same procedure to compute the minimumproton energy that would produce muons which radiate syn-chrotron photons before they decay. The muon energy, abovewhich its synchrotron cooling timescale is smaller than itsdecay timescale, is given by the relation: γ µ > (cid:115) πm e c σ T B τ µ (cid:18) m µ m e (cid:19) (A4)
200 400 600 800 10004.24.44.64.85.05.2 o g m , o g , s y n p Figure A1.
The minimum Lorentz factor that protons must havein order to produce pions that radiate synchrotron photons beforethey decay, γ π, synp , plotted as a function of Γ (solid grey line).Overplotted is also the miminum Lorentz factor of the proton dis-tribution, γ m (solid violet line), calculated through eq. 12. Shadedregions indicate the uncertainties on both parameter values. Theresults are obtained for GRB 061121. where τ µ = 2 . × − s the muon mean lifetime in thelab frame and m µ its rest mass. We assume that a muoncarries half of the parent pion energy. Therefore the pionsthat produce such muons have Lorentz factors γ π ≈ m π m µ γ µ .Furthermore, if we take into account eq. A2, we find that theprotons that produce such pions have energies: γ > γ µ, synp ≡ . (cid:115) πm e c σ T τ µ (cid:18) m µ m e (cid:19) B − (A5)Figure A2 displays γ µ, synp (light green shaded region) and γ m (purple shaded region) as a function of Γ for GRB 061121.The shaded regions have the same meaning as in Fig A1. Weconclude that, for all Γ , protons with γ (cid:38) γ m may producemuons via photopion interaction that will emit synchrotronradiation before they decay. Thus, muon synchrotron coolingis expected to affect the photon and neutrino spectra. APPENDIX B: THE Y COMPTON PARAMETER
Protons can interact with their own synchrotron photons toproduce secondary particles via photohadronic processes, asdescribed in the main text. Another process that could makeprotons lose energy while interacting with their own syn-chrotron photons is inverse Compton (IC) scattering. Thisis equivalent to the electron synchrotron self-Compton pro-cess.The proton cooling timescale due to IC scatterings in theThomson regime is written as t p , ic = 3 m e c cσ T U ph γ (cid:18) m p m e (cid:19) , (B1)where γ is the proton Lorentz factor and U ph is the syn-chrotron photon energy density. This can be computed MNRAS000
Protons can interact with their own synchrotron photons toproduce secondary particles via photohadronic processes, asdescribed in the main text. Another process that could makeprotons lose energy while interacting with their own syn-chrotron photons is inverse Compton (IC) scattering. Thisis equivalent to the electron synchrotron self-Compton pro-cess.The proton cooling timescale due to IC scatterings in theThomson regime is written as t p , ic = 3 m e c cσ T U ph γ (cid:18) m p m e (cid:19) , (B1)where γ is the proton Lorentz factor and U ph is the syn-chrotron photon energy density. This can be computed MNRAS000 , 1–16 (2021)
GRB proton-synchrotron model
200 400 600 800 10004.04.24.44.6 o g m , o g , s y n p Figure A2.
Same as in Fig. A1, but for the minimum Lorentzfactor that protons must have to produce muons that will cool viasynchrotron before they decay (light green shaded region). through the observed bolometric flux F γ (see eq. 8) as (Der-mer et al. 2007) U ph = F γ πr b d c Γ V b , (B2)where V b is the comoving source volume. The proton cool-ing timescale due to synchrotron and Compton processes canthen be written as t p = 3 m e c cσ T U B (1 + Y ) γ (cid:18) m p m e (cid:19) = t p , syn Y , (B3)where U B = B / π is the magnetic energy density, t p , syn isthe proton cooling timescale due to synchrotron radiation,and Y = U ph /U B is the Compton parameter.Using the source parameters that we have derived for ourGRB sample, namely B , F c and r b , we compute the pho-ton and magnetic energy densities and the Y parameter as afunction of Γ . An indicative example is shown in Fig. B1 forGRB . The solid line corresponds to the mean value of U ph /U B while the shaded region indicates the uncertainties.We find that IC scattering on proton synchrotron photonscan be safely ignored as a cooling process for the protons inthe source for all Γ values. Similar results apply to all GRBsin our sample. APPENDIX C: THE ESTIMATION OFPHOTOMESON EFFICIENCY AND γγ OPACITY
Assuming that the photon distribution in the outflow’s restframe is isotropic the fractional energy loss rate of a protonwith energy (cid:15) p due to pion production is written as (Waxman& Bahcall 1997b) t − π ( γ ) = c γ (cid:90) ∞ d x x − n γ ( x ) (cid:90) γx(cid:15) th d (cid:15) (cid:15)σ p π ( (cid:15) ) ξ p π ( (cid:15) ) , (C1)where (cid:15) th (cid:39) is the threshold energy for production of a ∆ + (1232) resonance, σ p π ( (cid:15) ) (cid:39) . mb for (cid:15) th ≤ (cid:15) ≤ is
200 400 600 800 1000876543 o g U p h / U B Figure B1.
The Y Compton parameter, defined as the energydensity ratio of proton synchrotron photons and magnetic fieldsin the emitting region, as a function of the bulk Lorentz factor.Results are shown for GRB 061121. the cross section for pion production for a photon with energy ε in the proton rest frame (in m e c units), and ξ p π ( (cid:15) ) (cid:39) . is the average fraction of energy lost by the proton perinteraction (Dermer & Menon 2009). Moreover, n γ ( x ) is thecomoving differential photon number density (here, this is thenumber density of synchrotron photons), and x = ε/m e c isthe dimensionless photon energy.In order to compute the proton energy loss rate for theBethe Heitler process on the isotropic synchrotron photonfield, we follow the equation given by (Blumenthal 1970) t − ( γ ) = 38 πγ σ T ca m e m p (cid:90) ∞ d (cid:15) n γ (cid:18) (cid:15) γ (cid:19) φ ( (cid:15) ) (cid:15) (C2)where a is the fine structure constant, (cid:15) = 2 γx is the dimen-sionless photon energy in the proton rest frame also used ineq. C1, and φ ( (cid:15) ) is a function defined by a double integral,as shown in Chodorowski et al. (1992) (see eqs. 3.13-3.17therein).In both cases, we compute the synchrotron photon numberdensity in the comoving frame ˜ n γ ( ε ) = 3 d F ε obs δ r cε (C3) n γ ( x ) = 3 d F ε obs Γ r cx (C4)where δ ≈ Γ , ε obs = Γ ε/ (1 + z ) , x = ε/m e c , ˜ n γ ( ε ) dε = n γ ( x ) d ( x ) , and F ε obs is the photon flux per unit energy inthe observer’s frame and is written as F ε obs = h − F c (cid:16) ε obs E c , obs (cid:17) / , ε obs < E c , obs h − F c (cid:16) ε obs E c , obs (cid:17) − / , E c , obs < ε obs < E pk , obs h − F c (cid:16) ε obs E pk , obs (cid:17) − p/ (cid:16) E pk , obs E c , obs (cid:17) − / , ε obs > E pk , obs . MNRAS , 1–16 (2021) Florou et al. (C5)Here, F c is the flux at the cooling break frequency ν c , obs = h − E c , obs (in units of mJy).The same radiation field that serves as a target for pho-tomeson processes is also a source for γγ opacity. The pho-toabsorption optical depth for a γ -ray photon of energy x γ (in units of m e c ), produced through photopion or photopairprocesses, in the isotropic synchrotron radiation field n γ ( x ) is: τ γγ = r b π (cid:90) ∞ d x n γ ( x ) (cid:90) σ γγ ( x, x γ ) (1 − cos θ ) dΩ (C6)where σ γγ is the pair-production cross section, which isapproximated by a step function approximation (Coppi &Blandford 1990). σ γγ (cid:39) . σ T H ( x γ x (1 − cos θ ) − x γ x (1 − cos θ ) (C7) This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS000