4.2 Nuclear cascade source classes
4.2.1 Development of the nuclear cascade in GRBs
The Empty Cascade source class is defined as the case that is optically thin to Aγ interactions of the primary injected isotope and thus also for all lighter isotopes including nucleons, since the cross section scales with the mass number A. This source class requires low radiation density, hence low luminosities or large radii. In Fig. 4.2, we show a benchmark for this class with a luminosity of Lγ = 1049 erg s−1. In the top left panel, the rates of the dominant processes for the injected primary are shown, including the photo-meson production for protons t′−1pγ for comparison. For the sake of simplicity, Bethe-Heitler pair production is included but not shown as it is sub-dominant in all the examples we show. One can clearly see that the source is
Figure 4.2: Benchmark for the Empty Cascade source class with isotropic luminosity Lγ = 1049 erg s−1and pure56Fe injection: Interaction rates (top left, for iron,pγalso included), the nuclear cascade (top right), particle densities in the source (bottom left) and ejected cosmic ray fluence per shell (bottom right, without interactions on EBL and CMB) as a function of the energy in the observer’s frame. The color code in the nuclear cascade shows the fraction of energy stored in each isotope, relative to the total isotropic equivalent energy Eiso,tot (white: fraction smaller than 10−4).
Different curves in the bottom panels correspond to different isotopes according to the legend. The other GRB parameters are R≃108.3 km,ε′γ,br= 1 keV andz= 2.
Taken from [159].
optically thin to photo-hadronic interactions by comparing t′−1dis and t′−1Aγ to t−1dyn, as indicated by the arrows. The maximum energy is dominated by adiabatic losses, such that cosmic rays
Figure 4.3: Same as Fig. 4.2 but for thePopulated Cascadesource class with isotropic luminosity Lγ = 1051 erg s−1. Taken from [159].
rather cool than interact. However, a small fraction will still interact, such that the nuclear cascade (shown in the top right panel) will be populated around the injected isotope and a few nucleons and 4He will be produced. The color shows the integrated energy per isotope relative to the total injection energy, where white boxes represent energy fractions smaller than 10−4.
The (quasi-)steady state particle densities in the source are shown in the lower left panel of Fig. 4.2. Since the source is optically thin, the injected spectrum ∝ E−2 of 56Fe is hardly modified and extends up to the maximum energy. Nucleons and secondary nuclei are suppressed and their maximum energies follow Lorentz factor conservation. The ejected cosmic ray spectra, depicted in the lower right panel as the fluence at Earth including adiabatic losses only, show
Figure 4.4: Same as Fig. 4.2 but for the Optically Thick Case source class with isotropic lumi-nosityLγ = 1053 erg s−1. Taken from [159].
harder spectra∝E−1 because of the direct escape mechanism, see Sec. 3.1.3. However, neutrons are not magnetically confined, so they soften the overall spectrum as they decay and convert to protons on their way to Earth. The characteristics of the Empty Cascade are that the ejected cosmic rays are dominated by the hard spectrum of injected primaries, while only few nucleons are produced, making the composition heavy. Nevertheless, compared to the density in the source, the contribution of neutrons, especially at low energies, is more substantial than expected, due to the additional suppression of the escape mechanism at the highest energies.
In the case of the Populated Cascade, which is obtained for Lγ = 1051 erg s−1, the source is optically thick to Aγ interactions of the injected primary which will disintegrate and populate
the cascade. At the same time, the source is still transparent to photo-hadronic interactions of nucleons, such that proton and neutron fluxes will be hardly affected by these interactions. This source class is encountered for intermediate radiation densities, i.e., intermediate luminosity and radii. The example for this source class is shown in Fig. 4.3. From the interaction rates, it is clear that at the maximum energy, photo-meson production and disintegration are now the limiting processes for the injected 56Fe. However, nucleons (and light nuclei) are still limited by adiabatic cooling. The nuclear cascade is well populated with integrated energies in nucleons and helium between 1% and 10% of the total energy, similar to isotopes close to the injection.
The densities in the source show now a clear depletion of the E−2 spectrum beyond the disintegration threshold, while the peak density of secondaries are comparable to the primary density. The densities of neutrons and protons do not reach this level yet, still the ejected cosmic ray spectrum is already dominated by neutrons. The reason is that the ejected spectra are suppressed by the optical thickness in the escape mechanism. The Larmor radius of the cosmic rays reaches only about 1/30 the size of the region at the maximum energy, so particles which are not contained in this thin layer on the outside of the shell take too long to escape. The effective cosmic ray escape spectrum is therefore relatively soft because of the neutron component, which can be controlled by the luminosity or, more general, by the radiation density.
If we further increase the luminosity, the source will become optically thick even to nucle-ons (and cnucle-onsequently all other isotopes), which defines the Optically Thick Case as shown in Fig. 4.4. This requires extremely high luminosities Lγ = 1053 erg s−1 which drives up the in-teraction rates, such that adiabatic cooling is sub-dominant for all species. The nuclear cascade appears to be less populated off the main diagonal, which is because of the strong disintegration, i.e., intermediate isotopes efficiently interact and most of the energy is dumped into nucleons, which are now occupied at a similar level as the primaries. The densities of heavy nuclei in the source are strongly suppressed beyond the photo-disintegration threshold, while protons and neutrons are populated at a level comparable to the injection. The density of nucleons peaks at the photo-meson threshold as they cascade down in energy due to multiple interactions. The ejected cosmic rays are dominated by neutrons as the heavy nuclei are strongly suppressed be-cause of the small Larmor radii. In fact, the strongAγinteractions make it complicated to reach ultra-high energies at all.
In Fig. 4.5 we show the all-flavor neutrino fluence per shell for the different source classes.
In each panel, the total flux is shown (thin black curve) as well as the decomposition of this total flux according to the contribution of primary nuclei (solid, blue), secondary isotopes (dot-dashed, green) and nucleons ((dot-dashed, red). For the Empty Cascade source class, the dominant contribution to the neutrino flux are photo-hadronic interactions of the injected isotope. On the other hand, for the Populated Cascade, the dominant contribution is given by Aγ interactions
Figure 4.5: All-flavor neutrino fluence per shell for the Empty Cascade (top left), Populated Cascade (top right) and Optically Thick Case (bottom left) source class as a function of the energy in the observer’s frame. The total neutrino fluence (thin, black) is split into the contribution of primaries (solid, blue), secondary isotopes (dot-dashed, green) and nucleons ((dot-dashed, red) to neutrino production. Taken from [159].
off the secondary nuclei. In the Optically Thick Case, photo-meson production of protons and neutrons has the largest contribution. Neutrinos from beta decay are included and visible as a bump in the nucleon contribution at low energies, i.e., it is sub-dominant at the peak. By increasing the luminosity, the neutrino fluence grows quadratically as the photon and the baryon density both scale with luminosity.
As mentioned in Sec. 3.1.2 as well as in Refs. [24] and [177], photo-meson production is rela-tively well understood in the case of nucleons. However, predictions for photo-meson production off nuclei rely typically on a superposition model with an implicit scaling of the cross section (here: ∝A). This implies that the neutrino prediction for the Optically Thick Case is robust,
Figure 4.6: Parameter space scan showing the regions for the different nuclear cascade source classes as a function of luminosity Lγ and collision radius R in the internal shock scenario. The injection composition is pure16O (left) or56Fe (right). We fix Γ = 300 and scale tv ∝ R according to Eq. (3.1), the other parameters are the same as in Fig. 4.2. Black dots in the right panel represent the benchmarks shown in Sec. 4.2.1.
The gray dashed contours indicate the maximum energy log10(Ei,max[GeV]) in the observer’s frame while the gray dotted line indicates the transition from one region to another. Below the photosphere (red solid line), gamma-rays cannot escape from the source due to electron-positron pair production. Taken from [159].
whereas the other classes (where the fluence is however low) could carry large uncertainties which will only be quantified in the future. Improved models as in [177] show that the contribution of nucleons may be also dominating in the Populated Cascade, but in the superposition model these predictions are less reliable because of the poorly understood cross sections.