Attenuation factors for the presentation of interaction rates

Whenever one composes a graph for the discussion of weak interaction rates in the supernova environment, one has to circumvent the inherently large scale differences of the rates at different locations in the star. In a logarithmic presentation, many details are hidden and differences between absorption and emission are difficult to appreciate. One can focus on the neutrinospheres and investigate the rates of interest under the corresponding conditions. However, the definition of the neutrinosphere is only based on the opacity and does not account for large emissivities outside the neutrino sphere. One may miss important sources of the total neutrino luminosity and fail in the explanation of the spectra. Moreover, it is common to average the extremely energy-dependent location where a given optical depth is reached to one single neutrinosphere--an even more problematic concept. In this appendix, we motivate a convenient presentation of interaction rates according to their relevance to the total luminosities. The approach aims to produce intuitively accessible figures with information about where the neutrinos come from, which reactions contribute to the total luminosity, how they locally compare to other reactions, and how the neutrino spectra are formed.

We derive auxiliary attenuation factors in terms of a staggered grid with zone edge indices and zone center indices . We start with a conserved luminosity with respect to spheres around the symmetry center. We may choose the neutrino number luminosity in units of particles per second or the neutrino energy luminosity in ergs per second. Each zone may have a number or energy source, , and a number or energy sink, , in number or ergs per gram and second. The luminosity is then recursively defined by its central value and

$$L_{i+1}=L_{i}+\left( em_{i'}-ab_{i'}\right) da_{i'},$$ (136)

where denotes the rest mass contained in the mass shell . The entering luminosity, , and the source, , are subject to absorption in this shell. We define an attenuation factor, , which accounts for the reduction of these quantities,

$$L_{i+1}=x_{i'}\left( L_{i}+em_{i'}da_{i'}\right) .$$ (137)

From Eq. () and (), we can readily isolate :

$$x_{i'}=\frac{L_{i+1}}{L_{i}+em_{i'}da_{i'}}=\frac{L_{i+1}}{L_{i+1}+ab_{i'}da_{i'}}.$$

In the rare cases of a negative , we set to zero. If we calculate a mean free path, , from all reactions that may act as a sink, the absorption rate can also be derived from the local specific number or energy density, , according to . The attenuation factor can then be expressed by physical quantities that are well accessible in a numerical evolution of radiation hydrodynamics,

$$x_{i'}=\frac{L_{i+1}}{L_{i+1}+\frac{cJ_{i'}}{\lambda _{i'}}da_{i'}}.$$ (138)

On the other hand, we derive from Eq. () by two recursive self-substitutions,

$$L_{i+1}=x_{i'}\left( x_{i'-1}\left( x_{i'-2}\left( L_{i-2}+e... ...'-2}\right) +em_{i'-1}da_{i'-1}\right) +em_{i'}da_{i'}\right) .$$

The continuation to and a rearrangement of the terms leads to

$$L_{n+1}=\sum _{i=1}^{n}\left( \prod _{l=i}^{n}x_{l'}\right) em_{i'}da_{i'}.$$ (139)

This equation describes how the total number or energy luminosity at a radius is composed by contributions of distributed emissivities in the star. The attenuation coefficients

$$\xi _{n+1,i'}=\prod _{l=i}^{n}x_{l'}=\prod _{l=i}^{n}\frac{L_{l+1}}{L_{l+1}+\frac{cJ_{l'}}{\lambda _{l'}}da_{l'}}$$ (140)

suppress irrelevant sources that are subject to large reabsorption. It is instructive to go a step further. We introduce the flux factor and rewrite Eq. () as

$$\xi _{n+1,i'}=\left( 1+\frac{dr_{i'}}{h_{i'}\lambda _{i'}}\right) ^{-1}\xi _{n+1,i'+1}$$

in order to extract the logarithmic derivative of ,

$$\frac{\xi _{n+1,i'+1}-\xi _{n+1,i'}}{\xi _{n+1,i'}}=\frac{dr_{i'}}{h_{i'}\lambda _{i'}}.$$

This is a finite difference representation of the equation

$$\frac{d\ln \xi }{dr}=\frac{1}{h\lambda }$$

with the solution . It is the familiar attenuation , if the radius-dependent flux factors are properly accounted for in the evaluation of the optical depth, . However, we evaluate the attenuation coefficients according to Eq. () where the finite differencing is consistent with the conservation laws in our implementation of the Boltzmann equation. As a consistency check, we compare in Fig. ()

Figure: The thin lines show the luminosity profiles in the evolution of the M progenitor star at ms after bounce. The solid line represents the electron neutrino luminosity, the dashed line the electron antineutrino luminosity, and the dash-dotted line the - and -neutrino luminosity. The thick lines show the luminosity profiles evaluated according to Eq. (), i.e. based on attenuated local emissivities. The reconstruction obviously misses the observer corrections in regions with low interaction rates. Otherwise, it reproduces the original luminosities sufficiently well to be accurate in the analysis of the formation of the neutrino spectra.

the original luminosities from the simulation with the luminosities reconstructed according to Eq. ().

We can use the attenuation coefficients in many convenient ways. A graph showing

$$g_{n+1}\left( r_{i'}\right) =\xi _{n+1,i'}em_{i'}\frac{da_{i'}}{dr_{i'}}$$

as a function of radius, , visualizes the contribution of each region in the star towards the total luminosity (represented by the area under the graph) at radius . In a comparison of different neutrino species, the graphs illustrate the decoupling at different radii. If the gravitational well is not too deep, the neutrino energy is approximately a constant of motion and the neutrino in different energy groups can be treated like different species. Moreover, instead of considering total emissivities and opacities, one can disentangle them into different reactions which we will enumerate with a superscript . A figure with graphs showing

$$g_{n+1}^{\ell }(r_{i'})=\xi _{n+1,i'}em_{i'}^{\ell }\frac{da_{i'}}{dr_{i'}}$$ (141)

as a function of radius would visualize the contribution of reaction to the total luminosity at radius by the enclosed area under the line . The graph is automatically scaled such that the most important reactions for the total luminosity are presented most prominently. If a reaction conserves the analyzed quantity we are free to include or not include it in the evaluation of the mean free path in Eq. () and the emissivities in Eq. (). Scattering, for example, does not change the neutrino number. If we include scattering as a reaction in the analysis of the number luminosity, the attenuation coefficients would indicate the probability to escape from a given location without any further scattering. If we do not include scattering, the attenuation coefficients indicate the (in many cases larger) probability to escape from a given location without being absorbed on the way out. The number luminosity is the same in both cases because scattering conserves the number of propagating neutrinos. If we work with the energy luminosity instead of the number luminosity, we have to include neutrino-electron scattering because this reaction affects the energy of escaping neutrinos after their production. We simply decompose a neutrino-electron scattering reaction into a neutrino absorption at the incoming neutrino energy and a neutrino production at the outgoing neutrino energy. These terms are then included in the opacities and emissivities in Eqs. () and (). In this case, the attenuation coefficients indicate the probability to escape from a given location without an energy-changing reaction. We found this choice to be the most interesting for the analysis of the formation of the neutrino spectra. The omitted isoenergetic scattering reactions are reflected in the transport spheres which are easily displayed as a complement. Finally, we mention the potential use of the attenuation coefficients for statistical evaluations, e.g. for the average radius of neutrino emission,

$$\left\langle r\right\rangle =\frac{\sum _{i=1}^{n}r_{i'}\xi ... ...'}em_{i'}da_{i'}}{\sum _{i=1}^{n}\xi _{n+1,i'}em_{i'}da_{i'}}.$$ (142)

In contrast to the classical definition of the neutrinosphere, the quantity accounts for regions with high emissivities in transparent regimes. Many other statistical informations at the origin of the neutrino emission may be obtained analogously.