Energetics of gamma-ray bursts

From the structure of a GRB’s light curve, the variability time scale t_v can be deduced, which is one of the main parameters for the model. Within the internal shock scenario, we can relate the size of the region to this time variability by assuming that the engine emits shells which collide at a radius R according to Eq. (3.1). Note that by applying the internal shock model, we place ourselves in the prompt emission scenario in the following, which is assumed to be the main emission channel for most GRBs since the photon densities are much higher than in external shocks with the circumburst medium. The width of the shock is given by ∆d^′ =R/2Γ with the shock Lorentz factor Γ, which is typically between 100 and 1000 for conventional GRBs. This shock or shell width defines the characteristic time scale of the system, i.e., the dynamical time scale t_dyn= ∆d^′/c, which corresponds to the variability time scale boosted back into the SRF.

In the static burst approximation or one-zone model, we assume that in total N such collisions happen which are all alike in the SRF. The number of collisions is given by N ≃ T₉₀/t_v, i.e., the ratio of the duration of the burst T₉₀, during which 90% of the emission is observed, and the variability time scale. This averaged picture of a GRB yields reliable results in most cases, but we will discuss the impact of dynamical models for GRBs, which includes multiple, different emission zones, too.

The target photon spectrum assumed for GRBs follows observations and can typically be described as a broken power law

with the photon break energyε^′_γ,br and two spectral indicesαandβ. In the following, we choose ε^′_γ,br= 1 keV,α= 1.0 andβ= 2.0, unless noted otherwise. The minimum and maximum photon energies, ε^′_γ,min and ε^′_γ,max, are chosen as small and large enough such that nuclei will always find interaction partners for disintegration at the GDR. However, we studied the impact of the

photon field on the neutrino and cosmic ray fluxes in [159] and [177], showing that there is a small effect, but we will not consider this in the following. The factor C_γ^′ is the normalization of the photon spectrum, which will be obtained from matching the photon density in the SRF

u^′_γ=

∫︂

ε^′N_γ^′(ε^′)dε^′ = L_γ∆d^′/c

Γ²V_iso = L_γ

4πcΓ²R² (4.2)

with measured quantities. The observed isotropic equivalent luminosity Lγ has to be boosted back into the SRF by dividing with Γ²and converted to an energy by the characteristic time scale of the collision t_dyn. The energy density is then obtained by dividing by the isotropic volume Viso= 4πR²∆d^′. The integration limits are determined by the Fermi-GBM energy range.

In the following, Eq. (4.2) will become important as it clearly shows how the radiation density scales with the parameters of the model: It scales linearly with the luminosity ∝ Lγ and is strongly dependent on the radius ∝R⁻², which directly comes from the volume of the radiation zone. In the same way it scales ∝Γ⁻², which is however fixed in our calculations to a typical value for conventional GRBs, i.e., Γ = 300. Note that there is a degeneracy between collision radius R, Lorentz factor Γ and variability time scale tv on account of the internal shock model Eq. (3.1). Therefore, varying the collision radius implicitly varies t_v. It is possible to fix one of the other parameters as well, vary Γ instead and compute the remaining one in the internal shock scenario.

A pure or mixed composition of nuclei is injected into the radiation zone with a spectrum following the expected power law of Fermi shock acceleration

Q^′_i(E_i^′) =C_i^′·

with a spectral index k≃2.0 and an exponential cut-off function with p= 2.0 for every species i. The cut-off function will have a small impact on the neutrino spectra shown in the following, while the cosmic ray fit is more sensitive to it. The maximum energy E_i,max^′ is obtained by balancing the energy gain with the energy loss processes, i.e.,

t^′−1_acc(E_i,max^′ ) =t^′−1_dyn+t^′−1_syn(E_i,max^′ ) +t^′−1_e+e⁻(E_i,max^′ ) +t^′−1_Aγ(E_i,max^′ ) , (4.4) where the rate for photo-hadronic interactions includes both, photo-meson production and photo-disintegration. For the computation of these rates, we calculate the magnetic field by

u^′_B = B^′2

where we typically assume that the fraction of energy in the magnetic fieldϵBis in equipartition with the fraction of energy in gamma-raysϵ_e,i.e.,ϵ_B=ϵ_e. With typical magnetic field strengths B ∼10²−10⁵ G and sizesR∼10⁷−10¹²km, GRBs fulfill the Hillas criterionE_max= ΓZeBR, which uses the argument that the size of UHECR sources has to be at least the Larmor radius of the accelerated particles [217].

The normalization factorC_i^′ is determined by normalizing the total energy per isotope

∫︂ 10E^′_i,max 0

E_i^′Q^′_i(E_i^′)dE_i^′ =ξA,i·u^′_γ· c

∆d^′ (4.6)

to the energy density in photons times a baryonic loading factor ξ_A,i, which is assumed to be 10 as this is a typical value for GRBs [13]. It will, however, be adjusted a posteriori by the cosmic ray fit, which is no problem as the model is linear in baryonic loading. The additional factor c/∆d^′ originates from the conversion from density to flux according to Eq. (3.5).

Based on the primary injection spectrum, we can give a simple estimate for the predicted neutrino fluence E_ν²ϕ_ν by using the relation

E_ν²ϕ_ν = 1

4f_pγE_p²dN_p

dE_p/(4πd²_L(z)) , (4.7) where E_p²dN_p/dE_p denotes the primary injection spectrum in [GeV] in the source, integrated over the volume. The factor fpγ =t_dyn/tpγ is called pion production efficiency, as it describes the ratio of the photo-hadronic interaction rate to the dynamical time scale. Here, it is defined for protons, but there are scaling relations for nuclei as well [22]. Assuming the ∆-resonance approximation, it is possible to derive an analytical expression for the pion production efficiency

f_pγ ≃ L^′_γ

ε^′_γ,brΓ²R = L^′_γ

ε^′_γ,brΓ⁴t_v , (4.8)

similar to our radiation density in Eq. (4.2).¹ Neutrinos produced by pion decay get about 25%

of the energy of the pion, thus the factor 1/4 in Eq. (4.7). Dividing by 4πd²_L(z) boosts the spectrum in the observer’s frame by distributing the energy on a sphere with the luminosity distance dL(z) as radius.

In fact, by similar considerations it is possible to place an upper bound on the diffuse neutrino flux from astrophysical sources, independent from the model and whether those sources are detected or not. From Sec. 3.2.1, we know that the energy injection rate of cosmic rays is a few

×10⁴⁴ erg Mpc⁻³ yr⁻¹ for a cosmologically distributed source. The present-day diffuse flux of

1The difference between the pion production efficiency and the photon density in Eq. (4.2) is a factor ∆d^′/c originating from the definition offpγ relative to the dynamical time scale.

muon neutrinos (combined νµ and ̄νµ) can then be estimated via E_ν²ϕ_ν = 1

4f_pγξ_zt_H c

4πE²_CRdN_CR

dE_CR ≈a few×10⁻⁸ξ_zGeV cm⁻²s⁻¹sr⁻¹ . (4.9) With respect to Eq. (4.7), this equation includes not only the energy in cosmic rays per source and primary, but the injection rate of the whole population E_CR² dN_CR/dE_CR. The factort_H ≈ 10¹⁰yr is the Hubble time andξ_zis a factor of order unity taking into account possible unobserved contributions to UHECRs from high redshift sources. This bound is named after Waxman and Bahcall who first derived it [218]. It cannot be exceeded by any source, however for GRBs the stacking limits due to the non-observation of correlated neutrinos are stronger by now.