Radiation pulse propagation

We have now performed many checks of the particle transport in stationary-state situations. In this subsection, we check the dynamics of the radiation field itself. The neutrino transport in the supernova is most dynamic when the shock crosses the neutrinosphere. As soon as the shock has propagated to densities where the opacities are low enough that neutrinos can escape from the hot material behind the shock, immediate electron capture will occur to refill the emptied phase space with new electron neutrinos. A neutrino burst carries an energy of order erg throughout the star towards the distant observer. Although the neutrino burst is not completely interaction free, it is an excellent example for a time-dependent radiation field where pulse propagation can be studied. We do this by plotting the luminosity profiles at selected times after the launch of the neutrino burst. We shift the radial coordinate, , in each time slice by the amount a free massless neutrino would have propagated during the time since bounce, . With an ideal numerical solution to free propagation (and assuming a point source), we would expect congruent pulse shapes from each time slice at the same positions . Figure ()

Figure: Shown is the radial profile of the electron neutrino luminosity at different times when the neutrino burst propagates through the star ( M model). The radial coordinate of each time slice has been shifted to the left by the distance a signal with light speed would have traveled since bounce. This makes the pulse profile roughly stationary in position. The levels on the right hand side in the graph give estimates of the peak height for the next time slice. The evaluation is based on the estimated numerical diffusion from first order upwind differencing in the free streaming regime in the transport equation. The expected peak heights accurately explain the visible decay in the luminosity pulse.

shows the neutrino burst with the described radius adjustment in the evolution of the M progenitor. We first observe, that the position of the pulse is fairly stationary. Therefore, the pulse indeed travels with speed of light through the star. However, we also observe that the shape of the pulse broadens considerably with ongoing time. From the check of the neutrino number luminosity conservation in Fig. (), we already know that the number and energy of the emitted neutrinos is conserved during their propagation to larger radii. The broadening is likely to occur by artificial diffusion in our numerical finite differencing scheme. Indeed, from considerations in Liebendoerfer_Rosswog_Thielemann_02 we remember that first order upwind differencing introduces a numerical diffusivity proportional to the advection speed and zone width,

$$D_{i'}=cdr_{i'}.$$ (125)

We quantify the influence of this effect to the evolution of our neutrino pulse by approximating the diffusion between the time steps with . We evaluate the diffusivity () and the second derivative of the luminosity at the pulse peak in the first time slice. The time span to the next time slice gives us an estimate for the change of the peak luminosity. We subtract from the pulse height and draw a horizontal line on the right hand side in Fig. () at this estimated peak level for the next time slice. The close to perfect agreement with the numerically evolved pulse height leads to the conclusion that the diffusivity () introduced with the low order advection scheme fully accounts for the observed pulse broadening. The resolution study in the next section illustrates that this is the dominant effect.