A treatise on elementary algebra by by James Hamblin Smith

By by James Hamblin Smith

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We need to make an important point about these spikes. Each spike in fact go all the way up to infinity. The finite height of a spike in the figure is an artifact of the finite step size taken in plotting it. In reality, of course, a cross-section cannot be infinite. In the present case, the crosssection remains finite due to the fact that the initial particles cannot be exactly monochromatic due to the uncertainty relation. For an example if we concentrate on the initial neutrino alone then there must be a spread in its energy, which can be represented by a probability distribution Φ(Ω), defined by dΩ Φ(Ω) = 1 .

84) Keeping the central value of neutrino energy as 10 MeV as before, we have drawn these plots for two different values of the spread, as mentioned in the caption. For the smaller 28 value of the spread in particular, the graph looks very similar to that drawn in Fig. 1, but the difference is that now the height of the spikes denote the actual enhancement, and is not an artifact of the plotting procedure. For the higher value of the energy spread, we see that the spikes have smoothened out. 5 Asymmetric emission of neutrinos from a proto-neutron star From what has been discussed so far we know that the inverse beta decay cross-section is direction sensitive through its dependence on the angle θ, the angle between the neutrino 3-momentum and the magnetic field direction.

They found the cross-section and went on to derive expressions for the neutrino opacity. From the final expressions, one can only recognize the terms linear in B. The opacity was calculated also by Chandra, Goyal and Goswami [50]. Like the previous authors, they also considered the contribution to the opacity from other reactions like neutrino-nucleon elastic scattering. 4 Consequences of neutrino energy spread The enhancement factor in Fig. 1 shows some spikes. They appear at values of the magnetic field for which the denominator of Eq.

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