Stars as Laboratories for Fundamental Physics - MPP Theory Group

410 Chapter 11
Apart from the highly nontrivial problem of neutrino transport,
the expected signal depends on a variety of physical assumptions such
as the nuclear equation of state which determines the stellar equilibrium
configuration and the amount of energy that is liberated, on the
properties of the progenitor star (notably the iron core mass), on the
duration of the accretion phase while the shock stalls, on the treatment
of convection during the first few 100 ms, and others.
In the Monte Carlo integrations of Janka and Hillebrandt (1989a,b)
the neutrino spectra at a given time are found to be reasonably well
described by the Fermi-Dirac shape
dL ν
dE ν
∝
E 3 ν
1 + e E ν/T ν −η ν
, (11.5)
where T ν is an effective neutrino temperature and η ν an effective degeneracy
parameter. This ansatz allows one to fit the overall luminosity by
a global normalization factor as well as the energy moments ⟨E ν ⟩ and
⟨Eν⟩; 2 finer details of the spectrum are probably not warranted anyway.
Throughout the emission process, η νe decreases from about 5 to 3, η νe
from about 2.5 to 2, and η νµ,τ ,ν µ,τ
from 2 to 0. This effective degeneracy
parameter is the same for ν µ,τ and ν µ,τ , in contrast with a real chemical
potential which changes sign between particles and antiparticles.
Fig. 11.6. Normalized neutrino spectral distribution according to a Maxwell-
Boltzmann distribution and a Fermi-Dirac distribution with an effective degeneracy
parameter η = 2, typical for the Monte Carlo transport calculations
of Janka and Hillebrandt (1989a,b). The temperatures are T ν = 1 3 ⟨E ν⟩
(Maxwell-Boltzmann) and T ν = 0.832 1 3 ⟨E ν⟩ (Fermi-Dirac with η = 2).