The statistical distribution of energies among particles responsible for long Gamma Ray Burst (GRB) emission is analyzed in light of recent results of the Fermi Observatory. The allsky flux, $F_{\gamma}$, recorded by the Gamma Ray Burst Monitor (GBM) is shown, despite its larger energy range, to be not significantly larger than that reported by the Burst and Transient Explorer (BATSE), suggesting a relatively small flux in the 3 - 30 MeV energy range. The present-day energy input rate in $\gamma$-rays recorded by the GBM from long GRB is found, assuming star-formation rates in the literature, to be $\dot W(0)=0.5 F_{\gamma} H/c = 5 \times 10^{42}\ \rm{erg/Mpc^3 yr}$. The Large Area Telescope (LAT) fluence, when observed, is about 5-10\% per decade of the total, in good agreement with the predictions of saturated, non-linear shock acceleration. The high-energy component of long GRBs, as measured by Fermi, is found to contain only $\sim 10^{-2.5}$ of the energy needed to produce ultrahigh-energy cosmic rays (UHECR) above 4 Eev, assuming the latter to be extragalactic, when various numerical factors are carefully included, if the cosmic ray source spectrum has a spectral index of -2. The observed $\gamma$-ray fraction of the required UHECR energy is even smaller if the source spectrum is softer than $E^{-2}$. The AMANDA II limits rule out such a GRB origin for UHECR if much more than $10^{-2}$ of the cosmic ray energy goes into neutrinos that are within, and simultaneous with, the $\gamma$-ray beam. It is suggested that "orphan" neutrinos out of the $\gamma$-ray beam might be identifiable via orphan afterglow { or other wide angle signatures of GRB in lieu of coincidence with prompt $\gamma$-rays}, and it is recommended that feasible single neutrino trigger criteria be established to search for such coincidences.
The high energy range of the cosmic ray (CR) spectrum is broken into three parts: The knee-to-ankle segment extends roughly from 10 16 to 4 × 10 18 eV, and the ultrahigh energy CR (UHECR) above the ankle can be further segmented into those below the GZK cutoff at about 4 × 10 19 eV, and those above it. Those above the ankle are believed to be extragalactic, showing both a flattening of slope in the observed flux, and little anisotropy below the GZK cutoff, and some anisotropy towards the local supercluster above it. Both UHECR components require particle acceleration well beyond the energies at which γ-rays themselves are detected. The cosmic rays (CR) beyond the ankle have been attributed to active galactic nuclei, but this is not the case for CR in the range 10 15 -10 18 eV, whose origin remains a mystery. In this energy range, CR are probably Galactic in origin and do not fill intergalactic space as the CR above the ankle do.
Gamma-ray bursts (GRB) have been considered by some authors (e.g. Levinson and Eichler 1993, Milgrom & Usov 1995, Waxman 1995, Vietri 1995) as sources of UHECR. Levinson and Eichler (1993), while not taking a position on whether GRBs could account for UHECRs above the ankle (despite giving this possibility serious consideration), suggested tha suggested that those just below are due to Galactic GRB. Doubts that GRB could supply the highest-energy cosmic rays include the total energetics, discussed in this paper, adiabatic losses, which lower the maximum energy should the acceleration be in a compact region, and Galactic isotropy, which is discussed in a forthcoming paper. Later authors (Milgrom & Usov 1995, Vietri 1995, Waxman 1995) focused on the possible connection between GRB and cosmic rays above the GZK cutoff. GRBs have also been considered as sources of UHE neutrinos (e.g. Eichler 1994, Waxman & Bahcall 1997, Eichler and Levinson, 1999). Levinson and Eichler, The detection of UHECR and/or UHE neutrinos in association with GRBs would provide valuable information.
In recent years there have been new and improved data. In particular, the LAT detector on the Fermi observatory provides information about the energy in non-thermal tails of particle distributions, and the Gamma Ray Burst Monitor (GBM), which can measure energies up to 30 MeV, provides a more reliable measurement of the GRB bolometric luminosities. AMANDA II has been operational for over 1000 days and has set limits on the neutrino output of GRB. This paper takes another look at GRB and UHECR energetics in the context of previous suggestions. In the steady state, the total demands of local power per unit cosmic volume on sources of UHECR above the ankle, ẆCR , can be written in terms of the measured allsky UHECR flux in any specified energy range, F CR , as
assuming the UHECR fill intergalactic space. Here τ CR is the cosmic-ray energy loss time due to interactions with the microwave background. The quantity C B is the bolometric correction assuming the source produces a spectrum typical of familiar shock acceleration.
The ratio ẆCR ẆGRB is thus given by a product of four separate dimensionless numbers. Below, we determine the values of F CR F GRB , F GRB H/c ẆGRB (0) , (τ CR H), and C B . We find that for cosmic rays above the ankle, F CR F GRB is 4, F GRB H/c ẆGRB (0) is about 2 for popular estimates of cosmic star formation rates, (τ CR H) -1 is between 5 and 8 for CR above the ankle, and more for CR above the GZK cutoff, and C B is between 4 and 20, depending on assumptions about the acceleration. Altogether, the value of ẆCR ẆGRB is of order 10 2.5 to 10 3 , despite the fact that each of the four factors may be modest.
The average cosmic energy density in γ-rays from GRB is a directly measurable quantity, independent of distracting uncertainties in the beaming angle, average energy, or rate of GRB. For a sample of hundreds of GRB or more, and spectral coverage from 8 keV on upward, as we now have, we can measure the total allsky flux in GRB merely by summing over all events within a given time interval. Most of the photon energy in GRB detected by Fermi is in the GBM range. We have considered all long (T 90 > 2 seconds) bursts detected by the Fermi Gamma-Ray Burst Monitor from August 2008 until February 2010 (see Table 2 of Guetta et al. 2010). The fluences have been collected from the literature (mainly GCN circulars) and usually are given in the 8 keV-1 MeV energy range. There are ∼ 20 GRBs with fluences given in the 50-300 keV energy range. For GRB 080916C, GRB 090902B and GRB 100116A the fluences are given up to 10 MeV. There exists the possibility of some additional unreported fluence outside the quoted range, but we presume the quoted range is selected so that the unreported fluence is relatively small. We have summed the reported GRB fluences and found the sum to be 3.0 × 10 -3 erg/cm 2 . Short bursts, which we do not include here, contributed another several percent. The GBM field of view (FOV) is ro
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