Stars as Laboratories for Fundamental Physics - MPP Theory Group

Supernova Neutrinos 411
Figure 11.6 shows a normalized Maxwell-Boltzmann spectrum and
a Fermi-Dirac spectrum with η = 2, both for the same average energy
⟨E ν ⟩. This choice implies T ν = 1 ⟨E 3 ν⟩ (Maxwell-Boltzmann) and
T ν = 0.832 1 ⟨E 3 ν⟩ (Fermi-Dirac with η = 2). What is shown in each
case is the normalized number spectrum, i.e. dN ν /dE ν ∝ Eν 2 e −Eν/Tν
(Maxwell-Boltzmann) and Eν/(1 2 + e E ν/T ν −η ν
) (Fermi-Dirac). It is apparent
that for a fixed average energy the Fermi-Dirac spectrum is
“pinched,” i.e. it is suppressed at low and high energies relative to the
Maxwell-Boltzmann case.
The most important difference between the two cases is the suppressed
high-energy tail of the pinched spectra which causes a significant
reduction of neutrino absorption rate. This may be important
for neutrino detection in terrestrial detectors as well as for neutrinoinduced
nuclear reactions in the SN mantle and envelope. However,
for the sparse SN 1987A signal the differences would not have been
overly dramatic. Therefore, in view of the many other uncertainties
at predicting the spectrum most practical studies of neutrino emission
and possible detector signals used simple Maxwell-Boltzmann spectra,
or Fermi-Dirac spectra with a vanishing chemical potential which differ
from the former only in minor detail.
11.2.3 Time Evolution of the Neutrino Signal
The schematic time evolution of the (anti)neutrino luminosities of the
different flavors was shown in Fig. 11.3. The prompt ν e burst has relatively
high energies (⟨E νe ⟩ ≈ 15 MeV), but the total energy content of
a few 10 51 erg renders it negligible relative to the integrated luminosity
of the subsequent emission phases.
The average energies and luminosities of the other flavors rise during
the first few 100 ms while the shock stalls, matter is accreted, and the
initially bloated outer region of the protoneutron star contracts. The
temperature of the region near the edge of the lepton number step
rises substantially during this epoch (Fig. 11.2). During the first 0.5 s
somewhere between 10% and 25% of the total binding energy is radiated
away; the remainder follows during the Kelvin-Helmholtz cooling phase
of the settled star.
Detailed parametric studies of the Kelvin-Helmholtz cooling phase
were performed by Burrows (1988), and more recently by Keil and
Janka (1995). In these works the expected SN 1987A detector signal
was studied as a function of the assumed nuclear equation of state
(EOS), the mass of the collapsed core at bounce, the amount of post-