Prompt Emission of High Energy Photons from Gamma Ray Bursts
Mon. Not. R. Astron. Soc. 000, 1–?? (2007) Printed 30 October 2018 (MN LATEX style file v2.2)
Prompt Emission of High Energy Photons from Gamma Ray
Bursts
Nayantara Gupta⋆ and Bing Zhang†
Department of Physics and Astronomy, University of Nevada Las Vegas, Las Vegas, NV 89154, USA
Accepted 2007; Received 2007; in original form 2007
ABSTRACT
Within the internal shock scenario we consider different mechanisms of high energy
(> 1 MeV) photon production inside a Gamma Ray Burst (GRB) fireball and derive
the expected high energy photon spectra from individual GRBs during the prompt
phase. The photon spectra of leptonic and hadronic origins are compared within dif-
ferent sets of parameter regimes. Our results suggest that the high energy emission
is dominated by the leptonic component if fraction of shock energy carried by elec-
trons is not very small (e.g. ǫ
−3). For very small values of ǫ
the hadronic
emission component could be comparable to or even exceed the leptonic component
in the GeV-TeV regime. However, in this case a much larger energy budget of the
fireball is required to account for the same level of the observed sub-MeV spectrum.
The fireballs are therefore extremely inefficient in radiation. For a canonical fireball
bulk Lorentz factor (e.g. Γ = 400), emissions above ∼ 10 GeV are attenuated by
two-photon pair production processes. For a fireball with an even higher Lorentz fac-
tor, the cutoff energy is higher, and emissions of 10 TeV - PeV due to π0-decay can
also escape from the internal shocks. The flux level is however too low to be detected
by current TeV detectors, and these photons also suffer attenuation by external soft
photons. GLAST LAT can detect prompt emission of bright long GRBs above 100
MeV. For short GRBs, the prompt emission can be only barely detected for nearby
bright ones with relatively “long” durations (e.g. ∼ 1 s). With the observed high
energy spectrum alone, it appears that there is no clean picture to test the leptonic
vs. hadronic origin of the gamma-rays. Such an issue may be however addressed by
collecting both prompt and afterglow data. A moderate-to-high radiative efficiency
would suggest a leptonic origin of high energy photons, while a GRB with an ex-
tremely low radiative efficiency but an extended high energy emission component
would be consistent with (but not a proof for) the hadronic origin.
Key words: Gamma Rays, Gamma Ray Bursts.
1 INTRODUCTION
The study of Gamma Ray Bursts (GRBs) has been one of the most interesting areas in astrophysics in the past few years. Ongoing
observational and theoretical investigations are disclosing the physical origin, characteristics of these objects as well as bringing
new puzzles to us. EGRET detected high energy photons from five GRBs coincident with triggers from the BATSE instrument
(Jones et al. 1996). GRB 940217 was detected by EGRET independent of BATSE trigger, which has extended emission and with
the highest energy photon of 18GeV (Hurley et al. 1994). Gonzalez et al. (2003) discovered a distinct high energy component up to
⋆ nayan@physics.unlv.edu
† bzhang@physics.unlv.edu
c© 2007 RAS
http://arxiv.org/abs/0704.1329v3
2 Nayantara Gupta and Bing Zhang
200 MeV in GRB 941017 that has a different temporal evolution with respect to the low energy component. Although even higher
energy gamma rays/neutrinos have not been firmly detected from GRBs yet, Atkins et al. (2000) have provided tentative evidence
of TeV emission from GRB 970417A. For a long time, GRBs have been identified as potential sources of ultrahigh energy cosmic
rays (Waxman 1995; Vietri 1997). Within the standard fireball picture (e.g. Mészáros 2006), there are about a dozen mechanisms
that can produce GeV-TeV gamma-rays from GRBs (e.g. Zhang 2007). More theoretical and observational efforts are needed to
fully understand high energy emission from GRBs. From the theoretical aspect, it is essential to investigate the relative importance
of various emission components to identify the dominant mechanisms under certain conditions.
The high energy photon spectra expected from GRBs during the prompt and the afterglow phases have been derived
by various groups. In the scenario of external shock model the high energy photon spectra during the early afterglow phase
due to synchrotron and synchrotron self Compton (SSC) emission by shock accelerated relativistic electrons and protons
have been studied (Mészáros et al. 1994; Mészáros & Rees 1994; Panaitescu & Mészáros 1998; Wei & Lu 1998; Totani 1998;
Chiang & Dermer 1999; Dermer et al. 2000a,b; Panaitescu & Kumar 2000; Sari & Esin 2001; Zhang & Mészáros 2001; Fan et al.
2007; Gou & Mészáros 2007). In the case of a strong reverse shock emission component, the SSC emission in the reverse shock
region or the crossing inverse Compton processes between the forward and reverse shock regions are also important (Wang et al.
2001a,b; Pe’er & Waxman 2005). The discovery of X-ray flares in early afterglows in the Swift era (Burrows et al. 2005) also
opens the possibility that scattering of the flaring photons from the external shocks can give strong GeV emission (Wang et al.
2006; Fan & Piran 2006). The effect of cosmic infrared background on high energy delayed γ-rays from GRBs has been also
widely discussed in the literature (Dai & Lu 2002; Stecker 2003; Wang et al. 2004; Razzaque et al. 2004; Casanova et al 2007;
Murase et al. 2007). The most important high energy emission component is believed to be emitted from the prompt phase. Swift
early X-ray afterglow data suggest that the GRB prompt emission is of “internal” origin, unlike the external-origin afterglow emis-
sion (Zhang et al. 2006, cf. Dermer 2007). The most widely discussed internal model of prompt emission is the internal shock model
(Rees & Mészáros 1994). Within the internal shock model the spectrum of high energy photons expected during the prompt phase
has been studied (Pilla & Loeb 1998; Fragile et al. 2004; Bhattacharjee & Gupta 2003; Razzaque et al. 2004; Pe’er & Waxman
2004; Pe’er et al. 2006). The various processes of high energy photon production in the internal shocks are electron synchrotron
emission, SSC of electrons, synchrotron emission of protons, photon production through π0 decay produced in proton photon
(pγ) interactions and radiations by secondary positrons produced from π+ decays. In this paper we consider all these processes
self-consistently with a semi-analytical approach and study the relative importance of each component within the internal shock
scenario. The derived photon spectra are corrected for internal optical depth for pair production, which is energy-dependent and also
depends on various other parameters of GRBs e.g. their variability times, luminosities, the low energy photon spectra inside GRBs,
and photon spectral break energies. If the electrons cool down by synchrotron and SSC emission to trans-relativistic energies, then
they accumulate near a value of Lorentz factor of around unity. The accumulated electrons affect the high energy photon spectrum
by direct-Compton scattering and other processes, which make the spectrum significantly different from the broken power laws
considered in this work, see (Pe’er et al. 2005, 2006) for detailed discussions. In any case, for the values of parameters considered
in the present paper this effect is not significant.
GLAST’s (Gehrels & Michelson 1999) burst monitor (GBM) will detect photons in the energy range of 10keV to 25MeV and
large area telescope (LAT) will detect photons in the energy range of 20MeV and 1000GeV. With a large field of view (> 2 sr
for LAT), GLAST will detect high energy photons from many GRBs and open a new era of studying GRBs in the high energy
regime. This is supplemented by AGILE (Longo et al. 2002), which is designed to observe photons in the energy range of 10-40
keV and 30MeV-50GeV and also has a large field of view. There are several other ground based detectors e.g. Whipple/VERITAS
(Horan et al. 2007), Milagro (Atkins et al. 2004), which have been searching or will search for ∼ TeV photons from GRBs. De-
tections or non-detections of high energy gamma rays from GRBs with space-based and ground-based detectors in the near future
would make major steps in revealing the physical environment, bulk motion, mechanisms of particle acceleration and high energy
photon production, photon densities, etc., of GRBs.
2 ELECTRON SYNCHROTRON RADIATION
We define three reference frames: (i) the comoving frame or the wind rest frame is the rest frame of the outflowing ejecta expanding
with a Lorentz factor Γ with respect to the observer and the central engine; (ii) the source rest frame is attached to the GRB
central engine at a redshift z; and (iii) the observer’s frame is the reference frame of the observer on earth, which is related to the
source rest frame by the redshift correction factor. We denote the quantities measured in the comoving frame with primes. The
shock accelerated relativistic electrons lose energy by synchrotron radiation and SSC in the shock region. Assuming a power law
distribution of fresh electrons accelerated from the internal shocks and considering a continuous injection of electrons during the
propagation of the shocks, the relativistic primary electron number distribution in the comoving frame can be expressed as a broken
power law in energy (Sari et al. 1998)
c© 2007 RAS, MNRAS 000, 1–??
Prompt Emission of High Energy Photons from Gamma Ray Bursts 3
dNe(E
E′e,m < E
e < E
E′e,c < E
in the case of slow cooling, where E′e,m is the minimum injection energy of electrons and E
e,c is the energy of an electron that loses
its energy significantly during the dynamic time scale, known as the cooling energy of the electrons. If the electrons are cooling
fast so that even the electrons with the minimum injection energy have cooled during the dynamical time scale, by considering
continuous injection of electrons from the shock the comoving electron number distribution can be expressed as
dNe(E
E′e,c < E
e < E
E′e,m < E
If the electrons cool down to sub-relativistic energies then they accumulate near electron Lorentz factor γ′e ∼ 1. This effect may
distort the high energy photon spectrum by direct-Compton scattering (Pe’er et al. 2005, 2006), and we focus on the parameter
regime where this effect is not significant. The energies in the source rest frame and the comoving frame are related as Ee ≃ ΓE
where Γ is the average bulk Lorentz factor of the GRB fireball in the prompt phase. The expression for the minimum injection
energy of electrons in the comoving frame is E′e,m = mec
2γ̄′pg(p)
, where g(p) = p−2
for p >> 2 and g(p) ∼ 1/6
for p = 2 (Razzaque & Zhang 2007), mp, me are the masses of proton and electron, respectively, and γ̄′pmpc
2 is the average
internal energy of protons in the comoving frame. We have assumed γ̄′p to be of the order of unity (in principle γ̄′p could be smaller
than unity). The total internal energy is distributed among electrons, protons and the internal magnetic fields within the internal
shocks. The fractions of the total energy carried by electrons, protons and internal magnetic fields are represented by ǫe, ǫp and ǫB ,
respectively, where ǫe + ǫp + ǫB = 1. We have assumed that all the electrons and protons are accelerated in internal shocks. In
reality, the shock accelerated particles may be only a fraction of the total population and additional fractional parameters (ξe, ξp)
may be introduced (e.g. Bykov & Mészáros 1996). In such a case, the following treatments are still generally valid by re-defining
ǫ′e = ǫe/ξe and ǫ
p = ǫp/ξp, while the relation ǫe + ǫp + ǫB = 1 still holds.
The relativistic electrons lose their energy by synchrotron radiation and inverse Compton scattering (Panaitescu & Mészáros
1998; Sari & Esin 2001; Zhang & Mészáros 2001). The comoving cooling break energy in the relativistic electron spectrum can be
derived by comparing the cooling and the dynamical time scales. The comoving cooling time scale t′cool of electrons is a convolution
of the cooling time scales for synchrotron radiation t′syn and for inverse Compton (IC) scattering t
t′cool
t′syn
. (3)
We denote U as the internal energy density of the internal shock, and Ue, UB as the energy densities of electrons and magnetic
fields, respectively. The energy density of the synchrotron radiation is Ue,syn =
ηeǫeU
(Sari & Esin 2001), where the
radiation efficiency of electrons is ηe = [(E
e,c/E
2−p, 1] for slow and fast cooling, respectively, and
Le,IC
Le,syn
Ue,syn
1 + 4ηeǫe/ǫB
denotes the relative importance between the IC and the synchrotron emission components1. Le,IC and Le,syn are the luminosities
of radiations emitted in SSC and synchrotron emission of relativistic electrons respectively. The inverse of the cooling time scale
of electrons can be expressed by the power divided by energy (E′e = meγ
t′cool
σe,Tβ
(UB + Ue,syn) =
σe,Tβ
(1 + Ye) , (5)
where σe,T is Thomson cross-section of electrons, βe
≃ 1 is the dimensionless speed of the relativistic electrons. The comoving
dynamical time scale is t′dyn ≃ Γtv , where Γ is the average Lorentz factor of the GRB, and tv is the variability time in the source
rest frame of the GRB, which denotes the variability time scale of the central engine. Throughout the paper, we assume that electron
synchrotron radiation from the internal shocks is the mechanism that power the prompt gamma-ray emission in the sub-MeV band.
However, for standard parameters within this scenario the cooling time scale of electrons is much shorter than the dynamical time
scale of GRBs. As a result the flux density
dNγ,s(Eγ,s)
below the cooling break energy is proportional to E
γ,s and cannot
explain the harder spectral indices observed in many GRBs (Ghisellini et al. 2000). If the magnetic field created by internal shocks
decays on a length scale much shorter then the comoving width of the plasma, then the resulting synchrotron radiation can explain
1 Strictly speaking, such a treatment is valid for the IC process in the Thomson regime. However, this is also a reasonable approximation if the
peak of the spectral energy distribution of the IC component is in the Thomson regime, which is generally the case for the calculations performed
in this paper.
c© 2007 RAS, MNRAS 000, 1–??
4 Nayantara Gupta and Bing Zhang
some of the broadband GRB spectra observed by Swift (Pe’er & Zhang 2006). In this case the effective dynamical time scale is
shorter by a factor of fc than its actual value. Hence, the ratio of the cooling and the dynamical time scale can be expressed as
t′dyn
t′cool
= fc (6)
at the cooling energy E′e = E
e,c. The expression of the electron cooling energy in the comoving frame can be written as
e,c = γ
e,cmec
= mec
2 3mec
4Γtvσe,T cUǫB(1 + Ye)
= 530keV
tv,−2Γ
2fc,2
Liso,51ǫB,−1(1 + Ye)
. (7)
Here and throughout the text the convention Qx = Q/10
x is adopted in cgs units. In the above expression Liso is the luminosity
corresponding to the energy Eiso carried by all particles and the magnetic fields in the shocks. It is a fraction of the wind (outflow)
luminosity Liso ∼ ηLw, where η is the efficiency of converting the kinetic energy of the wind to the shock internal energy.
The luminosity Liso and internal energy U are related as U = Liso/(4πΓ
2c), where ris = Γ
2ctv is the internal shock
radius. The synchrotron spectrum is a multi-segment broken power law (Sari et al. 1998) separated by several breaks, including
the emission frequency from electrons with the minimum injection energy, the cooling break frequency, and the synchrotron self-
absorption frequency (Rybicki & Lightman 1979). In the internal shocks, the magnetic field in the comoving frame can be expressed
as (Zhang & Mészáros 2002)
≃ 4.4× 10
G(ξ1ǫB,−1)
iso,51r
is,13Γ
2 = 1.5× 10
(ξ1ǫB,−1Liso,51)
Γ32tv,−2
where ξ is the compression ratio, which is about 7 for strong shocks. The synchrotron self absorption energy (Essa) in internal
shocks can be expressed as (Li & Song 2004; Fan et al. 2005; cf. Pe’er & Waxman 2004)
Essa ≃ 0.24 keVL
γ,s,51Γ
is,13 B
= 0.69 keVL
iso,51t
v,−2 Γ2
(ξ1ǫB,−1)
1 + Ye
where Lγ,s = Lisoǫeηe/(1 + Ye) is the isotropic gamma-ray luminosity due to synchrotron radiation. The cooling break energy
E′e,c and the minimum injection energy E
e,m of the electrons define two break energies in the synchrotron photon spectrum. The
cooling break energy in the photon spectrum in the source rest frame is
Eγ,c = Γ
E′e,c
)2 eB′c
≃ 1.9 × 10
tv,−2Γ
2fc,2
Liso,51ǫB(1 + Ye)
5 = 2.8eVtv,−2
(Liso,51ǫB,−1)3/2
Γ42fc,2
1 + Ye
Notice that Eγ,c very sensitively depends on Γ and some other parameters so that it could become a large value when parameters
change. For example, for B′ = 104G, Γ = 400, fc = 500, Liso = 10
51erg s−1, tv = 0.01s and ǫB = 0.1 we get Eγ,c ∼ 1.9
MeV. The break energy in the photon spectrum due to the minimum electron injection energy is
Eγ,m = Γ
E′e,m
)2 eB′c
≃ 0.58 MeVΓ2
5 = 8.5MeV
(ξ1ǫB,−1Liso,51)
2tv,−2)
−1 (11)
Assuming Essa < Eγ,m,s < Eγ,c,s the photon energy spectrum from synchrotron radiation of slow-cooling relativistic electrons
is as follows
dNγ,s(Eγ,s)
dEγ,s
γ,s Essa < Eγ,s ≤ Eγ,m,s
4/3+(p−3)/2
γ,m,s E
−(p−3)/2
γ,s Eγ,m,s < Eγ,s ≤ Eγ,c,s
4/3+(p−3)/2
γ,m,s E
γ,c,sE
−(p−2)/2
γ,s Eγ,c,s ≤ Eγ,s
In the case of slow-cooling electrons for very small values of ǫe (e.g. ∼ 10
−3, which is relevant when the hadronic emission
component becomes important), the break in the photon spectrum due to the minimum injection energy of electrons goes below
the synchrotron self absorption energy. The order in the spectral break energies becomes Eγ,m,s < Essa < Eγ,c,s, and the
spectrum is also modified. The spectral indices of the electron synchrotron spectrum for different ordering of the spectral break
energies are derived by Granot & Sari (2002). For Eγ,m,s < Eγ,s < Essa the spectral index of E
dNγ,s(Eγ,s)
dEγ,s
is 7/2, and for
Eγ,s < Eγ,m,s the spectral index is 3. The indices of the spectrum between Essa, Eγ,c,s and above Eγ,c,s remain as −(p− 3)/2
and −(p − 2)/2, respectively. When Essa is greater than both Eγ,m,s and Eγ,c,s their relative ordering becomes unimportant. In
that case the spectral indices of E2γ,s
dNγ,s(Eγ,s)
dEγ,s
are 7/2 between Eγ,m,s and Essa, and −(p− 2)/2 above Essa. Below Eγ,m,s
the index is 3.
For fast-cooling electrons the synchrotron photon energy spectrum for Essa < Eγ,c,s < Eγ,m,s is
dNγ,s(Eγ,s)
dEγ,s
γ,s Essa < Eγ,s ≤ Eγ,c,s
γ,c,sE
γ,s Eγ,c,s < Eγ,s ≤ Eγ,m,s
γ,c,sE
(p−1)/2
γ,m,s E
−(p−2)/2
γ,s Eγ,m,s ≤ Eγ,s
c© 2007 RAS, MNRAS 000, 1–??
Prompt Emission of High Energy Photons from Gamma Ray Bursts 5
When the ordering of break energies in the photon spectrum becomes Eγ,c,s < Essa < Eγ,m,s the photon energy spectrum is
dNγ,s(Eγ,s)
dEγ,s
γ,s Essa < Eγ,s ≤ Eγ,c,s
γ,c,sE
γ,s Eγ,c,s < Eγ,s ≤ Eγ,m,s
γ,c,sE
(p−1)/2
γ,m,s E
−(p−2)/2
γ,s Eγ,m,s ≤ Eγ,s
The total energy emitted in synchrotron radiation by relativistic electrons is Eisoηeǫe/(1+Ye). The normalisation constant for the
synchrotron photon energy spectrum can be calculated from
∫ Eγ,max
Eγ,min
dNγ,s(Eγ,s)
dEγ,s
dEγ,s = Eiso
(1 + Ye)
The maximum electron energy Ee,max can be calculated by equating the acceleration time and the shorter of the dynamical
and cooling time scales of the relativistic electrons. The expression of the acceleration time scale is t′acc = 2πζrL(E
e)/c =
2πζE′e/eB
′c. Here rL(E
e) is the Larmor radius of an electron of energy E
e in a magnetic field B
′, ζ can be expressed as
ζ ∼ βsh
−2y, where βsh is the velocity of the shock in the comoving frame of the unshocked medium and y is the ratio of diffusion
coefficient to the Bohm coefficient Rachen & Mészáros (1998). In ultra-relativistic shocks βsh ≈ 1 and numerical simulations for
both parallel and oblique shocks gives ζ ∼ 1. With
acc = min[t
cool, t
dyn] , (16)
one can derive the maximum comoving electron energy
e,max = min
Liso,51ǫB(1 + Ye)
, 14.3× 10
Γ2tv,−2B
GeV (17)
For electrons, the cooling term (first term in the bracket) always defines the maximum electron energy. The maximum synchrotron
photon energy in the source rest frame can be then derived as
Eγ,max = Γ
E′e,max
)2 eB′c
= 0.48GeV
t2v,−2
Liso,51ǫB,−1(1 + Ye)
= 102GeV
1 + Ye
This is used in eqn.(15) to define the normalization of the spectrum. The result has a very steep dependence on Γ. We also notice that
B′ is not an independent parameter, but can be calculated from other parameters according to eqn.(8). For example, for Γ = 400,
Liso = 10
51erg/s, tv = 0.01s and ǫB , ǫe ∼ 0.1, the magnetic field is of the order of 10
4G and the maximum photon energy
becomes a few hundred GeV.
3 ELECTRON INVERSE COMPTON SCATTERING
The relativistic electrons can be inverse Compton scattered by low energy synchrotron photons inside the GRB fireball and transfer
their energy to high energy photons. Below, we derive the IC photon spectrum using the electron and synchrotron photon spectra.
dNγ,i(Eγ,i)
dEγ,i
dNe(Ee)
dEe ×
dNγ,s(Eγ,s)
dEγ,s
dEγ,s (19)
The electron Lorentz factor (γe
′), IC and synchrotron photon energies (Eγ,i, Eγ,s) are related as Eγ,i ∼ γ
Eγ,s, this can be used
to simplify the above equation. The final expression for the IC photon spectrum considering slow cooling of electrons is
dNγ,i(Eγ,i)
dEγ,i
γ,i Essa,i < Eγ,i ≤ Eγ,m,i
4/3+(p−3)/2
γ,m,i E
−(p−3)/2
γ,i Eγ,m,i < Eγ,i ≤ Eγ,c,i
4/3+(p−3)/2
γ,m,i E
γ,c,iE
−(p−2)/2
γ,i Eγ,c,i < Eγ,i ≤ Eγ,K
4/3+(p−3)/2
γ,m,i E
γ,c,iE
(p−2)/2
γ,K E
−(p−2)
γ,i Eγ,K < Eγ,i
Here Essa,i = γ
Essa, Eγ,m,i = γ
Eγ,m,s, and Eγ,c,i = γ
Eγ,c,s, where γ
e,m = E
e,m/mec
g(p)(mp/me)(ǫe/ǫp), γ
e,c = E
e,c/mec
2 are Lorentz factors corresponding to the minimum injection energy of electrons and
the cooling break energy of electrons. In the case of fast cooling Eγ,m,i > Eγ,c,i and the IC photon spectrum has to be modified
accordingly.
dNγ,i(Eγ,i)
dEγ,i
γ,i Essa,i < Eγ,i ≤ Eγ,c,i
γ,c,iE
γ,i Eγ,c,i < Eγ,i ≤ Eγ,m,i
γ,c,iE
(p−1)/2
γ,m,i E
−(p−2)/2
γ,i Eγ,m,i < Eγ,i ≤ Eγ,K
γ,c,iE
(p−1)/2
γ,m,i E
(p−2)/2
γ,K E
−(p−2)
γ,i , Eγ,K < Eγ,i
c© 2007 RAS, MNRAS 000, 1–??
6 Nayantara Gupta and Bing Zhang
In eqn.(21) the expressions for Essa,i, Eγ,c,i and Eγ,m,i are Essa,i = γ
Essa, Eγ,c,i = γ
Eγ,c,s and Eγ,m,i =
γ′e,m
Eγ,m,s. When EeEγ,s >> Γ
2m2ec
4 the cross section for IC scattering decreases as the scattering enters the Klein Nishina
(KN) regime. A break in the photon spectrum at Eγ,i = Eγ,K appears when the Klein Nishina effect becomes important. We define
a parameter κ =
EeEγ,peak
Γ2m2ec
, where Eγ,peak = max[Eγ,c,s;Eγ,m,s]. The KN regime starts when κ = 1 (e.g. Fragile et al. 2004),
Eγ,K =
Γ2m2ec
Eγ,peak
= 2.5 GeV
Eγ,peak,MeV
In the KN regime the emissivity of electrons decreases by κ2, and the photon energy spectral index simply follows the electron
energy spectral index, i.e. −(p− 2). The IC photon spectrum in eqn.(20) can be normalised as
∫ Eγ,max,i
Eγ,m,i
dNγ,i(Eγ,i)
dEγ,i
dEγ,i = Eiso
ηeǫeYe
1 + Ye
, (23)
where Eγ,max,i = ΓE
e,max due to the KN effect.
4 PROTON SYNCHROTRON RADIATION
Relativistic protons lose energy by synchrotron radiation and photo-pion (π0, π+) production inside GRBs. They interact with the
low energy photons in the GRB environment and pions are produced. There is a threshold energy for this interaction (pγ) to happen,
EpEγ ≥ 0.3GeV
2Γ2, where Ep and Eγ are proton, photon energy in the source rest frame respectively. The π
0s decay to a pair of
high energy photons, while the π+s decay to neutrinos and leptons. The threshold condition therefore suggests that the photon-pion
related high energy spectrum is typically more energetic than the electron IC spectrum. We assume that the proton spectrum in the
internal shocks can be expressed as a power law in proton energy. We consider a proton spectral index similar to electrons for our
present discussion. Since protons are poor emitters, we only consider the scenario of slow-cooling in the comoving proton spectrum
dNp(E
E′p,m < E
p < E
E′p,c < E
where E′p,m is the minimum injection energy of the protons and E
p,c is break energy in the spectrum due to proton cooling. The
minimum injection energy E′p,m = γ̄
2g(p), where g(p) = p−2
for p ≫ 2 and g(p) ∼ 1/6 for p = 2. The cooling break
energy can be derived by comparing the comoving and the cooling time scales. The inverse of the cooling time scale t′cool of a
proton is
t′cool
t′syn
The photo-pion cooling time scale t′π has been derived earlier in the context of estimation of neutrino fluxes from GRBs
(Waxman & Bahcall 1997; Gupta & Zhang 2007). If fπ is the fraction of proton energy going to pion production in the ∆ res-
onance of pγ interactions one has 1/t′π ∼ fπ/t
dyn where the comoving time scale is
2 t′dyn = Γtv. The peak value of pγ
interaction cross section at the ∆ resonance is σpγ = 5 × 10
−28cm2. This is much higher than the Thomson cross section for
protons σp,T =
σe,T , where σe,T = 6.625 × 10
−25cm2. We therefore neglect the IC process of protons. Substituting for
t′syn and t
π in eqn.(25), we get
t′cool
σp,Tβ
where, β′p is dimensionless speed of relativistic protons. We use the general expression for fπ from Gupta & Zhang (2007)
fπ(Ep) = f0
1.34α2−1
)α2−1 Ep < Epb
1.34α1−1
)α1−1 Ep > Epb
where
0.9Liso,51
810Γ42tv,−2Eγ,peak,MeV
1 + Ye
. (28)
2 In this definition, on average protons loose ∼ 20% energy in the time scale of t′π . Although it is not strictly the e-folding timescale usually used
to define cooling, for order-of-magnitude estimates this is good enough.
c© 2007 RAS, MNRAS 000, 1–??
Prompt Emission of High Energy Photons from Gamma Ray Bursts 7
In our present discussion α2 = (p + 2)/2 and α1 = (p + 1)/2. Eγ,peak,MeV is the peak energy in the electron synchrotron
photon spectrum expressed in MeV, and Liso,51 is the GRB luminosity in unit of 10
51 erg s−1, which is the typical value for GRB
luminosities. Epb = 0.3Γ
2/Eγ,peak,GeV GeV is the threshold proton energy for interaction with photons of energy Eγ,peak,GeV .
For typically observed values of GRB parameters one has Epb ∼ 1 PeV. The break energy in the proton spectrum due to proton
cooling can be calculated by comparing the comoving and cooling time scales of protons as discussed in the case of electrons in §2.
We assume β′p ∼ 1 then for Ep < Epb the expression of cooling break energy in the comoving frame is
p,c =
σp,Tβ
2 cUǫB
m2pc4
EpbΓtv
1.34α2−1
α2 + 1
108GeVfc,2
Γ2tv,−2
Liso,51ǫB
6t2v,−2
Epb(PeV)Γ2tv,−2
1.34α2−1
α2 + 1
where fc =
. The synchrotron photon spectrum from relativistic protons is
dNγ,ps(Eγ,ps)
dEγ,ps
−(p−3)/2
γ,ps Eγ,m,ps < Eγ,ps ≤ Eγ,c,ps
γ,c,psE
−(p−2)/2
γ,ps Eγ,c,ps < Eγ,ps
The minimum injection energy in the photon spectrum from proton synchrotron radiation is related to that from electron synchrotron
radiation as (Zhang & Mészáros 2001)
Eγ,m,ps
Eγ,m,s
E′p,m
E′e,m
)2(me
The cooling break energy in the photon spectrum from proton synchrotron radiation is the characteristic synchrotron photon energy
for proton energy E′p,c. To normalize the proton synchrotron spectrum, it is important to find out the relative importance between
proton synchrotron radiation and pγ interactions. Similar to the treatment of electrons, one can define
Lp,pγ
Lp,syn
Ue,syn
Ye . (32)
where, Lp,pγ and Lp,syn are the luminosities of radiations emitted in pγ interactions and synchrotron emission of protons respec-
tively. Notice that protons interact with the synchrotron emission of the electrons, so that Ye enters the problem. Eqn. (32) suggests
that Yp is usually much greater than unity since σpγ ≫ σp,T . As a result, most of the proton energy is lost through pγ interaction
rather than proton synchrotron radiation.
The proton synchrotron photon spectrum can be normalised as
∫ Eγ,max,ps
Eγ,m,ps
Eγ,ps
dNγ,ps(Eγ,ps)
dEγ,ps
dEγ,ps = Eiso
1 + Yp
, (33)
where ηp =
E′p,c/E
. The maximum proton synchrotron photon energy is derived by Eγ,max,ps =
E′p,max
, where E′p,max is again defined by comparing the comoving acceleration time with the shorter of the co-
moving dynamical and cooling times scales
p,max = min
Liso,51ǫB(1 + Yp)
, 1.4 × 10
Γ2tv,−2B
TeV . (34)
p,max = min
ǫB,−1Liso,51
)1/2 Γ32tv,−2
1 + Yp
(ξ1ǫB,−1Liso,51)
TeV . (35)
5 π0 DECAY
The relativistic protons interact with the low energy photons and photo-pions (π0,π+) are produced as a result. The probabilities
of π0 and π+ production are 1/3 and 2/3, respectively. Pions subsequently decay, i.e. π0 → γγ and π+ → µ+νµ → νµν̄µνee
As the cross section for the γγ interactions is much higher than the peak value of pγ interaction cross section, above the threshold
energy of pair production γγ interactions are expected to dominate over pγ interactions. If the photon energy is 2mec
∼ 1 MeV in
the comoving frame, then in the source rest frame it is of the order of a few hundred MeV as the Lorentz factors are typically of the
order of few hundred for canonical GRBs. For example, for Γ = 400 the photons of energy 400 MeV can produce photo-pions by
interaction with protons of minimum energy Ep ∼ 120 TeV. The π
0 typically carries 20% of the proton’s energy and the photons
produced in π0 decay share its energy equally. Hence, the minimum energy of the photons produced from π0 decay is expected to
c© 2007 RAS, MNRAS 000, 1–??
8 Nayantara Gupta and Bing Zhang
be ∼ 10%Ep ∼ 12 TeV. The photon spectrum produced from π
0 decay has been derived below using the proton spectrum defined
in eqn.(24) and assuming the fraction fπ/3 of protons’ energy goes to π
dNγ,π0(Eγ,π0)
dEγ,π0
fπ(Eγ,π0)
Eγ,π0 ≤ Eγ,π0,c
Eγ,π0 > Eγ,π0,c
where, Eγ,π0,c = 0.1Ep,c. For the expression for fπ , see eqn.(27), which contains a break energy. The break energy in the photon
spectrum contained within fπ is Eγ,π0,b = 0.03Γ
2/ǫbr,GeV GeV assuming 10% of the proton’s energy goes to the photon produced
via π0 decay. ǫbr is the break energy in the low energy photon spectrum (in the scenario of slowly cooling electrons it is the cooling
break energy in the photon spectrum and for fast cooling electrons it is the photon energy corresponding to the minimum injection
energy of electrons). The photon flux can be normalised in the following way
γ,π0,max
γ,π0,min
Eγ,π0
dNγ,π0(Eγ,π0)
dEγ,π0
dEγ,π0 =
ǫpηpYp
1 + Yp
where Eγ,π0,min = 30Γ GeV and Eγ,π0,max = 0.1Ep,max. Although high energy photons (∼ TeV) are absorbed by lower
energy photons and e+e− pairs are produced, at extreme energies the pair production cross section decreases with increasing energy
(Razzaque et al. 2004). Hence, ultrahigh energy photons can escape from the internal shocks for suitable parameters depending on
the values of their various parameters and the low energy photon spectra.
6 SYNCHROTRON RADIATION OF POSITRONS PRODUCED IN π+ DECAY
The shock accelerated protons may interact with the low energy photons to produce π+s along with π0s as discussed in the
previous section. The π+s subsequently decay to muons and neutrinos. The energetic muons decay to positrons and neutrinos
(pγ → π+ → µ+νµ → e
+νµν̄µνe). The charged pions, muons and the positrons are expected to lose energy through synchrotron
radiation and IC inside the shock region. As the Thomson cross section for positrons is much larger than pions or muons, they
are expected to emit much more radiation compared to the heavier charged particles. On the other hand, since these positrons are
very energetic, most IC processes happen in the Klein Nishina regime. We therefore neglect the contribution of the positron IC
processes. The positron synchrotron spectrum produced in pγ interactions can be derived in the following way. The fraction of the
protons’ energy tranferred to pions is denoted by fπ (eqn[27]). If we assume that the final state leptons share the pion’s energy
equally then one fourth of the pion’s energy goes to the positron. The energy of the positron spectrum
at the energy Ee+
can be expressed using the proton spectrum defined in eqn.(24)
dN(Ee+)
fπ(Ee+)
Ee+ ≤ Ee+,c
Ee+ > Ee+,c
where, Ee+,c is the cooling break energy in the positron specrum and
fπ(Ee+) = f0
1.34α2−1
)α2−1 Ee+ < Ee+b
1.34α1−1
)α1−1 Ee+ > Ee+b
where f0 has been defined in eqn.(28), Ee+b = 0.05Epb , Epb = 0.3 GeVΓ
2/ǫbr,GeV , and ǫbr,GeV is the break energy in the
photon spectrum as defined earlier. The positron spectrum in eqn.(38) can be normalised using the total energy carried by the
positrons,
e+,max
e+,min
dN(Ee+)(Ee+)
dEe+ =
ǫpηpYpEiso
1 + Yp
The maximum and minimum positron energies are Ee+,max = 0.05Ep,max and Ee+,min = 15Γ GeV (which is ∼ 6TeV for
Γ = 400). The synchrotron photon spectrum from the positrons can be subsequently derived using the same treatment for primary
electrons as discussed in §2. The IC emission is in the KN regime and therefore not important. Also, photons having energies
above a few hundred GeV are annihilated by lower energy photons as discussed in the following section. The relativistic muons
produced in π+ decay lose energy by synchrotron radiation. We compare the decay and synchrotron energy loss time scales of the
high energy muons. The maximum energies of positrons can be calculated in this way. If the muons decay before losing energy
significantly high energy positrons are produced carrying approximately 5% of the initial proton’s energy. On the otherhand if
the muons lose energy before they decay lower energy positrons are produced. These positrons radiate energy and produce lower
energy photons. The muons initially carry approximately 10% of the relativistic protons’ energy hence, we expect the low energy
photon flux produced by cooling of positrons is lower than that produced by relativistic electrons if ǫe and ǫp are comparable.
c© 2007 RAS, MNRAS 000, 1–??
Prompt Emission of High Energy Photons from Gamma Ray Bursts 9
7 INTERNAL PAIR-PRODUCTION OPTICAL DEPTHS OF HIGH ENERGY PHOTONS
Inside GRBs high energy photons interact with low energy photons to produce electron-positron pairs (e.g. Baring & Harding 1997;
Lithwick & Sari 2001). The optical depth depends on the values of various parameters of the GRB fireball. We follow the approach
discussed in Bhattacharjee & Gupta (2003) to derive internal optical depths of GRBs in detail. For two photons (a high energy
photon γh and a low energy photon γl), the pair production cross section depends on the energies of the photons and the angle
between their directions of propagation. The cross section is (Berestetskii et al. 1982)
σγhγl (E
, θ) =
σT (1− β
(3− β
1 + β′
1− β′
(2− β
where σT is the Thomson cross section, and β
′ = [1 − (E′γl,th/E
)]1/2 is the center of mass dimensionless speed of the pair
produced. The threshold energy of pair production with a high energy photon of energy E′γh is
γl,th
2(mec
E′γh(1− cosθ)
For the photons with energy higher than the threshold energy, the pair production cross section decreases with increasing photon
energy (Jauch & Rohrlich 1955; Razzaque et al. 2004). In the present work we calculate internal optical depths in different energy
regimes using the cross sections with different energy dependences. The mean free path for γh γl interactions lγhγl can be calculated
using the low energy photon spectrum.
γhγlθ
, θ) =
γl,th
dnγl(E
dE′γl
σγhγl (E
, θ) (43)
d(cosθ)(1− cosθ)l
γhγlθ
, θ) (44)
where
(E′γl
dE′γl
is the specific number density of low energy photons inside the GRB. The low energy photon spectrum is ob-
servationally known, as revealed by gamma-ray detectors such as BATSE and Swift. Theoretically, it corresponds to the electron
synchrotron component as discussed in §2, which is a broken power law spectrum separated by the synchrotron self absorption
break, the minimum injection break and the cooling break. The low energy photon flux is related to the observed luminosity
through
∫ E′γl,max
E′γl,ssa
dnγl (E
dE′γl
= Uγ =
Lγ,iso
4πcris2Γ2
where Lγ,iso is the isotropic γ-ray luminosity. We have taken it to be equal to the luminosity of the synchrotron photons emitted by
electrons: Lγ,iso = Le,syn =
ǫeηeLiso
. In eqn.(44) we have three variables: angle θ and photon energies E′γl , E
. To simplify
the integration in eqn.(44) we transform the integral with a new variable following Gould & Schreder (1967)
E′γlE
(1− cosθ)
2(mec2)2
E′γl,th
= s0Θ (46)
with s0 =
(mec2)2
, and Θ = 1
(1− cosθ). As β′ = (1− 1/s)1/2, the pair production cross section can be expressed as a function
of the new variable s. It is then possible to write eqn.(44) as
−2 dnγl (E
dE′γl
Q[s0(E
)] (47)
where
Q[s0(E
∫ s0(E
sσ(s)ds , (48)
and σ(s) = 16
σγhγl
. For moderate values of s we use σ(s) ≃ 1 and for s >> 1 it can be approximated as σ(s) ≃ ln(s)/s.
The expressions for Q[s0(E
)] are (s20 − 1)/2 and s0(ln s0 − 1), respectively, in the two cases. Substituting for Q[s0(E
)] in
eqn.(47) we derive the final expression for l−1γhγl (E
). The internal optical depth τint(E
) is the ratio of comoving time scale
and the mean time between two pair production interactions.
c© 2007 RAS, MNRAS 000, 1–??
10 Nayantara Gupta and Bing Zhang
τint(E
) (49)
The final photon energy spectrum to be observed on Earth from nearby GRBs (neglecting further attenuations with the infrared
background and cosmic microwave background) can be obtained by correcting the original flux for the internal optical depth and
the redshift z of the source
dNγ,ob(Eγ,ob)
dEγ,ob
4πd2z(1 + z)
dNγ(Eγ)
exp(−τint(Eγ)) , (50)
where
ΩΛ +Ωm(1 + z′)3
is the comoving distance of the source, H0 = 71km s
−1 Mpc−1 is the Hubble constant, and ΩΛ = 0.73 and Ωm = 0.27 are
adopted in our calculations.
8 PHOTON SPECTRUM FROM SECONDARY ELECTRONS AND POSITRONS
The secondary pairs carry a significant fraction of energy in the primary spectrum, and this energy is re-radiated and converted
to photons. A more realistic treatment should consider a photon-pair cascade process, which requires numerical calculations
(Pe’er & Waxman 2004; Pe’er et al. 2006). Here instead we estimate the emission from the secondary pairs. We first calculate
the photon energy spectra generated by different physical processes as discussed earlier. The photon spectra are then corrected
for internal optical depths and subsequently the total energies carried by these photons are calculated by integrating the corrected
photon energy spectra over photon energies. If we subtract the total energies carried by these high energy photons from their intial
energies before including the effects of internal optical depths, we get the energies of the secondary e− and e+ produced in γγ in-
teractions. These pairs are expected to have spectral indices similar to the high energy photons. With the knowledge of their spectral
indices and the total energies carried by them the synchrotron photon spectra radiated by these secondary leptons are calculated.
For the parameters adopted in this paper, it turns out that the emission contribution from the secondaries is below the emission level
of the primaries, and hence, does not significantly modify the observed the spectrum. We therefore do not include this component
in Figs.1-5, but caution that such a feedback process could be potentially important for the parameter regimes with high opacity.
We refer to Pe’er & Waxman (2004) and Pe’er et al. (2006) for more detailed treatments of such cases.
9 SYNTHESIZED SPECTRA AND DETECTABILITY
Using the procedure delineated above, we have calculated the broad-band emission spectrum from internal shocks for a wide range
of parameter regimes. In particular we focus on the various high energy emission components discussed above and their relative
significance. Our results are presented in Fig.1-5. In each set of calculations we have presented the internal optical depth after the
final photon energy spectrum. For particles accelerated by ultra-relativistic shocks the spectral index is expected to be about 2.26
Lemoine & Pelletier (2003). Afterglow modeling suggests a larger scatter of p values for relativistic shocks, but p = 2.3 is close the
mean value of the data (Panaitescu & Kumar 2002). In all our calculations, the spectral indices of relativistic electrons and protons
are both assumed as p = 2.3.
Figures 1-4 are the calculations for a typical long GRB with duration T90 = 20 s at redshift z = 1 (10 s in the source
rest frame). Since we do not know the physical condition of the internal shocks from the first principles, we vary the parameter
regime in a wide range. In each set of calculations, we design the parameters to make the electron synchrotron emission peaking
at the sub-MeV range (∼ 0.36 MeV, 0.13 MeV, 0.6 MeV, 0.25 MeV for Figs.1-4, respectively), as suggested by the data. The
global energetics of the GRB is also adjusted so that the gamma-ray luminosity in the sub-MeV range is about 1051 ergs s−1 as
suggested by the observations. The variability time scale for these calculations is taken as tv = 0.01 s. The bulk Lorentz factor
is adopted as Γ = 400 in Fig.1-3, as suggested by the recent early optical afterglow observations (Molinari et al. 2006). In order
to check how Γ affects the spectra, we also calculate the case of Γ = 1000 for the parameter set of Fig.1, which is presented in
Fig.4. In all the figures, the different components of the photon energy spectrum from a GRB for both electrons (e) and protons
(p) are displayed with different line styles/colours. The observed energy fluxes E2γ,ob
dNγ,ob(Eγ,ob)
Eγ,ob
in unit of ergs/cm2sec are
plotted against the observed photon energy Eγ,ob(eV ). The green long dashed curves represent the synchrotron emission from the
relativistic electrons. The short dashed curves (blue) represent the IC spectrum from energetic electrons; the dash-dotted curves
(light blue) represents the synchrotron emission of the relativistic protons; the triple short dashed curves (orange) represent for the
synchrotron emission of the relativistic positrons produced in π+ decays; the ultrahigh energy emission component from π0 decays
c© 2007 RAS, MNRAS 000, 1–??
Prompt Emission of High Energy Photons from Gamma Ray Bursts 11
is shown by the double short dashed curves (black) in the extremely high energy regime. The thin black solid lines represent the
synthesized spectra of various components without including the effect of pair production attenuation. Depending on parameters,
the pair opacity becomes important in the GeV - TeV range. The thick black solid lines represent the final photon spectrum after
including the internal optical depths. In order to check whether the predicted high energy components are detectable by GLAST,
we also plot an indicative GLAST sensitivity threshold in the 100 MeV - 100 GeV energy range. The GLAST sensitivity estimate
is based on the criterion of detecting at least a few photons in the band based on the average effective area and photon incoming
zenith angle of LAT. Background is negligible for GRB detections. This gives a rough fluence threshold of ∼ 2× 10−7 erg cm−2
(B. Dingus, 2007, personal communication). The flux thresholds adopted in all the figures are therefore derived from the observed
durations. For T90 = 20 s, this gives a flux threshold of ∼ 10
−8 erg cm−2 s−1. The sensitivity of VERITAS to photon above
energy 200GeV has also been shown in our figures with pink dotted line. It is 2 × 10−8erg cm−2 s−1 (D. Horan, 2007, personal
communication).
Figure 1 is a standard “slow-cooling” leptonic-dominant case. The shock equipartition parameters are ǫe = 0.4 and ǫB = 0.2.
The isotropic shock luminosity is Liso = 10
52 erg s−1. The slow cooling factor fc = 2500 is adopted, which suggests that the
post-shock magnetic field decays on a length scale shorter than the comoving scale (Pe’er & Zhang 2006). The thick black line
shown on the right side around 1015eV is the π0 component after including the effect of absorption due to pair production, indicating
the reduction of pair opacity at high energies (Fig.1b, see also Razzaque et al. 2004). In this figure the break energies in the photon
energy spectrum appear in the order of Essa < Eγ,m < Eγ,c in the electron synchrotron and IC spectral components. The spectral
index of the photon energy spectrum is 4/3 between Essa and Eγ,m, −(p− 3)/2 between Eγ,m and Eγ,c, and −(p− 2)/2 above
Eγ,c. Since ǫe is large, the leptonic components are many orders of magnitude stronger than the hadronic components. The value
of Yp is much larger than 1, so that the proton synchrotron component is below the components due to π
0 decay and positron
synchrotron radiation.
We vary the values of the equipartition parameters (ǫe, ǫB , ǫp) and study the variations in the photon energy fluxes generated
by various processes. The emission level of the electron IC spectral component decreases with decreasing ǫe (fixing ǫB) since Ye
is decreasing. Moreover, as we decrease ǫe the minimum injection energy of electrons Eγ,m also decreases. In the slow cooling
regimes, it is Eγ,c that defines the peak energy in the electron synchrotron spectrum, which could be adjusted to the sub-MeV range
by adopting a suitable fc value. The change of Eγ,m therefore mainly affects the calculated internal optical depth.
By lowering ǫe, we check the parameter regime where the hadronic component becomes comparable. Since eletrons are much
more efficient emitters than protons, the parameter regime for the hadronic component to be comparable to the leptonic component
in the high energy regime is ǫe/ǫp ∼ me/mp < 10
−3.3 A similar conclusion has been drawn for the external shocks (Zhang &
Mészáros 2001). In Fig.2, with ǫe = 10
−3, ǫB = 0.05 and ǫp = 0.849. In order to adjust Eγ,c to the sub-MeV range, fc = 50000
is needed. In order to match the observed MeV emission flux by electron synchrotron, a large energy budget is needed due to a
small ǫe: Eiso = 10
56 ergs and Liso = 10
55 erg s−1. Such a large energy budget has been suggested before (Totani 1998), but
afterglow observations and modeling in the pre-Swift era have generally disfavored such a possibility (Panaitescu & Kumar 2002).
In the Swift era, however, a large afterglow kinetic energy for some GRBs is not ruled out. For example, the bright afterglow
of GRB 061007 demands a huge kinetic energy if the afterglow is produced by isotropic external shocks (Mundell et al. 2007;
Schady et al. 2007). Modeling some X-ray afterglows below the cooling frequency requires a low ǫB and/or a large afterglow
kinetic energy at least for some GRBs (Zhang et al. 2007). We therefore still consider such a possibility. In Fig.2, the break energy
in the photon energy spectrum due to the minimum injection energy of electrons is below the synchrotron self absorption energy.
The break energies appear in the order of Eγ,m < Essa < Eγ,c in the synchrotron and IC electron spectra. The spectral index of
the photon energy flux is 7/2 between Eγ,m and Essa, −(p− 3)/2 between Essa and Eγ,c, and −(p− 2)/2 above Eγ,c. We can
see that in the TeV energy regime beyond the maximum electron synchrotron energy, the positron synchrotron emission from π+
decay becomes dominant. Moreover, when ǫe is small, Ye is small, hence Yp becomes small. In this case the proton synchrotron
component becomes comparable to the spectral components due to synchrotron radiation of the secondary positrons and π0 decays.
The internal optical depth is plotted in Fig.2b, which peaks at a higher energy than that in Fig.1b.
If the post shock magnetic field does not decay within a short distance (fc = 1), internal shocks are in the standard fast-cooling
regime. We calculate such a case in Fig.3. The shock parameters are ǫe = 0.6, ǫB = 0.2, Liso = 10
52 erg s−1, Eiso = 10
53 erg.
In this case the break energies appear as in the order of EC < Essa < Em. The photon energy spectral indices are 13/8, 1/2 and
−(p− 2)/2, respectively, in the three energy regimes.
The pair opacity depends on the bulk Lorentz factor. When Γ is large enough, the ultra-high energy photons would have lower
3 Proton energy loss and their contribution to high energy photon emission in the early afterglow phase has been studied earlier by Pe’er & Waxman
(2005). Our results for the prompt emission phase are generally consistent with them. In order for the proton synchrotron component to be significant,
even smaller ǫe (than 10−3) is demanded. Considering that photon-pion emission is more efficient than proton synchrotron emission, the condition
ǫe/ǫp ∼ me/mp < 10
−3 can allow the hadronic components to be comparable to (but not dominant over) the leptonic components.
c© 2007 RAS, MNRAS 000, 1–??
12 Nayantara Gupta and Bing Zhang
Figure 1. A leptonic-component-dominated slow cooling spectrum. (a) The different components of the photon energy spectrum from the internal
shocks for the following parameters in the slow-cooling regime: Eiso = 10
53erg, Liso = 10
52erg/s, tv = 0.01s and fc = 2500. The thick solid
black curve represents the final spectrum after including the effect of internal optical depths. The thin solid black curve represents the synthesized
spectrum before including the effect of internal opical depths. The long dashed (green) curve is the electron synchrotron component; the short dashed
(blue) curve is the electron IC component; the double short dashed (black) curve on the right side is for π0 decay component; the triple short dashed
(orange) line represents the synchrotron radiation produced by positrons generated in π+ decays; the dash-dotted (light blue) line represents the
proton synchrotron component. The tiny red horizontal line between 108 and 1011eV represents GLAST’s threshold. The pink dotted horizontal
line above 2 × 1011eV represents the sensitivity of VERITAS experiment (b) Internal optical depths plotted against energy for the parameters
adopted in (a).
internal optical depth and may escape from the internal shocks (Razzaque et al. 2004). To test this, in Fig.4, we re-calculate with the
parameter set for Fig.1, but increase Γ to 1000. The slow-cooling parameter fc is adjusted to 50 to maintain the sub-MeV energy
peak. The results indeed suggest that the attenuation of the high energy photons is weaker.
The observational breakthough in 2005 suggests that at least some short GRBs are low-fluence, nearby events that have a
distinct progenitor than long GRBs (Gehrels et al. 2005; Bloom et al. 2006; Fox et al. 2005; Villasenor et al. 2005; Barthelmy et al.
2005; Berger et al. 2005). To check the prospect of detecting short GRB prompt emission with high energy detectors such as GLAST,
we perform a calculation for the parameters of a short GRB in Fig.5. Due to their short durations, short GRB detections are favorable
for high luminosity and relatively “long durations”. We therefore take an optimistic set of parameters with Liso = 10
51 erg s−1,
T90 = 1 s, and z = 0.1. Other parameters include: Γ = 800, tv = 1 ms, ǫe = 0.4, ǫB = 0.2, ǫp = 0.4, fc = 50. The photon
flux from synchrotron radiation of electrons peaks at 0.1MeV. Fig.5a suggests that the high energy component of such a burst is
barely detectable by GLAST. The internal optical depth of this set of parameters does not grow to very large values (maximum 10),
so that the attenuation signature is not significant in Fig.5a. The dip around several 1013 eV corresponds to the optical depth peak,
above which the attenuated flux starts to rise. The abrupt drop at several 1014 eV corresponds to the disappearance of the electron
IC component at high energies.
c© 2007 RAS, MNRAS 000, 1–??
Prompt Emission of High Energy Photons from Gamma Ray Bursts 13
Figure 2. A slow-cooling spectrum with significant hadronic contribution (a) The spectra of various components. Parameters: ǫe = 10−3,
ǫB = 0.05, ǫp = 0.849, tv = 0.01s, fc = 50000, Eiso = 10
56erg and Liso = 1055erg/s. Same line styles have been used as in Fig.1. (b) The
corresponding internal optical depths.
10 CONCLUSIONS AND DISCUSSION
We have calculated the broad-band spectrum of GRBs from internal shocks for a wide range of parameter regimes. We did not
take into account the external attenuation of TeV photons by the infrared radiation background and that of the PeV photons by the
cosmic microwave background. These external processes would further attenuate our calculated spectrum in high energy regimes,
and reprocess the energy to delayed diffuse emission (Dai & Lu 2002; Stecker 2003; Wang et al. 2004; Razzaque et al. 2004;
Casanova et al 2007; Murase et al. 2007). Such processes are not relevant for most of the calculations presented, however, since the
internal attenuation already cuts the observed spectrum below TeV. They are however important for high Lorentz factor cases in
which more high energy photons are leaked out of the internal shock region. The external attenuation is also prominant for high
energy emission from the external reverse/forward shocks and the external IC processes related to X-ray flares. These processes
have been extensively discussed in other papers (referenced in Introduction) and they are not discussed in this paper. For nearby
GRBs (e.g. z < 0.3), TeV emission is transparent. It is possible that ground-based Cherenkov detectors such as VERITAS, Milagro
would detect TeV gamma-rays from nearby energetic GRBs.
In previous treatments of hadronic components from internal shocks (Fragile et al. 2004; Bhattacharjee & Gupta 2003), the
shock accelerated protons are assumed to carry mp/me times more energy than electrons. This effectively fixed ǫe ∼ me/mp,
which is not justified from the first principle. In this paper we have taken all the equipartition parameters ǫe, ǫp and ǫB as free
parameters, and explore the relative importance of various components in different parameter regimes. The dominant hadronic
c© 2007 RAS, MNRAS 000, 1–??
14 Nayantara Gupta and Bing Zhang
Figure 3. A leptonic-component-dominated fast-cooling spectrum. (a) The spectra of various components. Parameters: ǫe = 0.6, ǫB = 0.2,
ǫp = 0.2, tv = 0.01s, fc = 1, Eiso = 1053erg and Liso = 1052erg/s. Same line styles have been used as in Fig.1. (b) The corresponding
internal optical depths.
component emission becomes interesting only when ǫe is extremely small. Given the same observed level of sub-MeV spectrum,
the total energy budget of the GRB needs to be very large. Inspecting the calculated spectra for different parameter sets (Figs.1-4),
one finds that there is no clean picture to test the leptonic vs. hadronic origin of the gamma-rays. Such an issue may be however
addressed by collecting both prompt and afterglow data. A moderate-to-high radiative efficiency would suggest a leptonic origin
of high energy photons, while a GRB with an extremely low radiative efficiency but an extended high energy emission component
would be consistent with (but not a proof for) the hadronic origin.
The prompt emission produced by leptons including the effect of pair production has been discussed by Pe’er & Waxman
(2004); Pe’er et al. (2006). They calculated the emergent photon spectra for GRBs located at z = 1. The lower cut-off energy in
the photon flux produced by leptons is determined by the synchrotron self absorption energy, the minimum injection energy or the
cooling energy depending on the values of the various GRB parameters. Our leptonic-component-dominated cases are consistent
with their results, although we do not explore cases with very high compactness. If the electrons cool down to trans-realtivistic
energies then their high energy spectrum significantly deviates from broken power law (Pe’er et al. 2006, 2005). For our choice
of values of the GRB parameters this effect is not important. Razzaque et al. (2004) estimated the internal optical depth for pair
production and showed that at PeV energies the optical depth decreases with increasing photon energies. We have rederived the
optical depths for values of GRB parameters. The results are generally consistent with (Razzaque et al. 2004) except that the growth
of optical depth with increasing energy is more gradual before the optical depth peak. This is a result of including the whole low
c© 2007 RAS, MNRAS 000, 1–??
Prompt Emission of High Energy Photons from Gamma Ray Bursts 15
Figure 4. The case of a higher Lorentz factor. (a) The spectra of various components. Parameters: Γ = 1000 and fc = 50. All the other parameters
are the same as in Fig.1. (b) The corresponding internal optical depths.
energy photon spectrum (rather than the threshold energy photons) for calculating the pair production optical depth. The optical
depths depend on the cross section of γγ interactions, the low energy photon spectra, the various break energies in those spectra,
luminosities, variability times and the GRB Lorentz factors. A change in values of any of these parameters may affect the values
of the optical depths at various energies. For high bulk Lorentz factors, the π0 component may appear in the final spectra due to
the reduced optical depths around PeV energy. However, these ultra-high energy photons will be immediately absorbed in the GRB
neighborhood by cosmic microwave photons (Stecker 2003). The reradiated energy by the e+e− pairs would nonetheless contribute
to the diffuse high energy γ-ray background (Casanova et al 2007).
Upcoming γ-ray detectors have a good chance of detecting prompt emission from GRBs and reveal their physical nature
during the prompt phase. Detection of the hadronic components is difficult but it would be possible to infer the dominance of these
components by a coordinated broadband observational campaign if they are indeed important. More generally, detection or non
detection of high energy photons in the prompt phase would constrain the values of various GRB parameters. In particular, the pair
attenuation feature would help to constrain the bulk Lorentz factor of the fireball. Compared with EGRET, GLAST has a 10 times
larger collecting area and a larger field of view. It is expected that GLAST LAT would detect high energy emission from a large
number of bursts (mostly long GRBs and some bright, relatively “long” short GRBs), which will open a new era of studying GRBs
in the GeV-TeV regime. On the other hand, it is difficult for VERITAS to detect prompt high energy gamma-rays even under the
most optimistic conditions. High energy emissions from the external shock at the early afterglow phase for nearby GRBs may be
c© 2007 RAS, MNRAS 000, 1–??
16 Nayantara Gupta and Bing Zhang
Figure 5. An energetic short GRB. (a) The spectra of various components. Parameters: ǫe = 0.4, ǫB = 0.2, ǫp = 0.4, tv = 0.001s, Γ = 800,
fc = 50, Eiso = 1051erg and Liso = 1051erg/s. Same line styles have been used as in Fig.1. (b) The corresponding internal optical depths.
the better targets for VERITAS and other TeV detectors.
We thank Brenda Dingus, Deirdre Horan, Enwei Liang, Peter Mészáros, Jay Norris, Asaf Pe’er, Soeb Razzaque, and Dave
Thompson for useful discussion/comments and/or technical support. We also thank the anonymous referee for a detailed report
with important suggestions and comments. This work is supported by NASA under grants NNG05GC22G, NNG06GH62G and
NNX07AJ66G.
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c© 2007 RAS, MNRAS 000, 1–??
	Introduction
	0 Decay
	Synchrotron Radiation of Positrons Produced in + decay
