tokens: The overall diffuse intensity is comparable within both models. Figure 6. Fraction of successful jets in the (, ζ SN ) plane for the simple GRB model. Only a small fraction of all CCSNe can harbor a successful jet. 4. Neutrino Diffuse Emission in the Advanced GRB Model In this section, we take a somewhat more realistic model of the physics within a jet and allow Γ to vary across the jet angle from some maximum value Γ max at the center of the jet ( θ = 0) out to Γ = 1: the advanced GRB model. In this model, we do not rely on θ j = 1/Γ anymore. Because we anticipate that multiple shocks will be accelerating the protons, the resulting Γ distribution is a von-Mises–Fisher distribution,where the concentration κ ≈ 1/ σ 2 for σ small, with σ the usual standard deviation. We take motivated by random walks of the accelerated particles within the jet. In this model, we define the volume of the jet (Equation ( 3 )) with where is defined by Γ( θ max ) = 1, orWe note that for κ = Γ max , θ max has a maximum at Γ max = e . For a representative jet with Γ max = 300 this corresponds to θ max = 11°. The component of the GRB rate R ( z , Γ) introduced in Section 3 depending on Γ is assumed to follow a distribution similarly defined as in the previous section but this time this is a function of Γ max : . The constraint in Equation ( 30 ) to reproduce the observed HL-GRB rate becomes,The maximum value of is 1 and the minimum value of (the maximum value of θ ) is when Γ = 200, which is .Similarly to Equation ( 29 ), for the advanced GRB model we havewhere the beaming angle factor on each side has canceled. The diffuse neutrino intensity isThe integral goes over θ ∈ [0, θ max ], and we again require that the jet can successfully accelerate protons. The fraction of successful jets for this model is shown in the top panel of Figure 7 . One can see that the fraction of CCSNe harboring successful jets is larger in this model with respect to the simple GRB model, given the different scaling laws intrinsic to the models. Figure 7. Top panel: fraction of successful jets in the (, ζ SN ) plane for the advanced GRB model. Bottom panel: fraction of jets that are choked for the advanced GRB model. Below erg, 50% of the jets are choked; below erg, 90% of the jets are choked. The total diffuse neutrino intensity for our canonical GRB and ζ SN = 0.1 for all successful jets is plotted in the bottom panel of Figure 5 for the advanced GRB model to be compared with the one from the simple GRB model. Both models produce comparable intensities. We now turn to the issue of electromagnetically choked jets. Depending on the jet properties, we expect that there will be some successful jets that are choked or invisible. In the simple model, all of the jets are bright if they successfully form. On the other hand, among all successful jets, the fraction of choked jets for the advanced model is shown in the bottom panel of Figure 7 . Below erg, we find that 50% of the jets are choked;
ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula O B-Model I-Model I-Model O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O B-Model I-Model O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O