Stars as Laboratories for Fundamental Physics - MPP Theory Group

Supernova Neutrinos 409
must then be multiplied with a factor between 4 and 6 to obtain an
estimate of E b .
Even though the emission of neutrinos is a quasithermal process
their energies are not set at the “neutrino sphere” which is defined
to be the approximate shell from where they can escape without substantial
further diffusion. Of course, even the notion of a neutrino
sphere is a crude concept because of the Eν 2 dependence of the scattering
cross section on nonrelativistic nucleons which implies that there is
a separate neutrino sphere for each energy group. The scattering with
nucleons does not allow for much energy transfer apart from recoil effects.
What is relevant for determining the neutrino energies is their
“energy sphere” where they last exchanged energy by the scattering on
electrons, by pair processes, and by charged-current absorption. Naturally,
this region lies interior to the neutrino sphere—see the shaded
areas in Fig. 11.1 as opposed to the dotted line which represents the
neutrino sphere.
The concept of an “energy sphere” (where neutrinos last exchanged
energy with the medium) and of a “transport sphere” (beyond which
they can stream off without further scattering) helps to explain the
apparent paradox that the ν µ spectrum is, say, twice as hard as that
of ν e , yet the same amount of energy is radiated. Both fluxes originate
from the same radius of about 15 km so that the Stefan-Boltzmann
law (L ∝ R 2 T 4 ) would seem to indicate that the ν µ flux should carry
16 times as much energy. However, the place to which the Stefan-
Boltzmann law should be applied is the energy sphere, yet the neutrinos
cannot escape from there because the flow is impeded by neutral-current
scattering on an overburden of nucleons. One may crudely think of the
energy sphere being covered with a skin that does not allow the radiation
to stream off except through some holes. Thus the effectively
radiating surface is smaller than 4πR 2 . (For a more technical elaboration
of this argument see Janka 1995a.)
Evidently, neutrino transport is a rather complicated problem, especially
in the transition region between diffusion and free escape. The
most accurate numerical way to implement it would be a Monte Carlo
integration of the Boltzmann collision equation (Janka and Hillebrandt
1989a,b). In practice, this is not possible because of the constraints
imposed by the limited speed of present-day computers so that a variety
of approximation methods are used to solve this problem. While
there is broad agreement on the general features of the expected neutrino
signal, there remain differences between the predicted spectra and
lightcurves of different authors.