Stars as Laboratories for Fundamental Physics - MPP Theory Group

Supernova Neutrinos 423
ter of 53 events! Therefore, any cluster following a high-energy muon
would be very suspicious. Performing a cut on the data for this background
leaves no burst with multiplicity 3 or larger in several data sets
of several hundred days each (Hirata 1991). Therefore, it is extremely
unlikely that the event cluster 10−12 in the Kamiokande data has been
caused by background.
The BST detector has a relatively large background rate. Event
clusters of multiplicity 5 or more within 9 s occur about once per day.
Thus the probability for such a background cluster to fall within a
minute of the IMB and Kamiokande events is about 5×10 −4 .
11.3.3 Analysis of the Pulse
Many authors have studied the distribution of energies and arrival times
of the reported events. Probably the most significant work is that of
Loredo and Lamb (1989, 1995) who performed a maximum-likelihood
analysis, carefully including the detector backgrounds and trigger efficiencies.
Loredo and Lamb also gave detailed references to previous
works, and in some cases offered a critique of the statistical methodology
employed there. Their more extensive 1995 analysis supersedes
certain aspects of the earlier methodology and results.
Because of the small number of neutrinos observed, a relatively
crude parametrization of the time-varying source is enough. Among a
variety of simple single-component emission parametrizations, Loredo
and Lamb (1989, 1995) found that an exponential cooling model was
preferred. It is characterized by a constant radius of the neutrino
sphere, R, and a time-varying effective temperature
T (t) = T 0 e −t/4τ , (11.8)
so that τ is the decay time scale of the luminosity which varies with
the fourth power of the temperature according to the Stefan-Boltzmann
law. It should be noted, however, that numerical cooling calculations
do not yield exponential lightcurves. For example, the model shown in
Fig. 11.7 displays an exponential decline of the effective temperature,
but a power-law decline of the neutrino luminosity. Other calculations
even yield early heating and a constant temperature for some time (see
Burrows 1990b for an overview).
Of course, for the time-integrated spectrum the exponential cooling
law is just another assumption concerning the overall spectral shape.
For example, one easily finds that the average ν e energy of the timeintegrated
spectrum is ⟨E νe ⟩ = 2.36 T 0 if Fermi-Dirac distributions with