Stars as Laboratories for Fundamental Physics - MPP Theory Group

Supernova Neutrinos 433
ber of ν e ’s in the burst is of order 10 56 , its duration of order 50 ms. Thus,
while it passes it represents a ν e density of about 10 32 cm −3 r7 −2 which
exceeds the local electron density for r > ∼ 10 9 cm. However, the phasespace
distribution of the neutrinos is locally far from isotropic and so
the energy shift involves a factor ⟨1 − cos Θ⟩ where Θ is the angle between
the “test neutrino” and a “background neutrino;” the average is
to be taken over all background neutrinos (Sect. 9.3.2). A typical angle
between two neutrinos moving within the burst at the same location
is given by the angle subtended by the neutrino sphere as viewed from
the relevant radial position, i.e. Θ ≈ R/r with R the radius of the neutrino
sphere. For a large r one thus finds ⟨1 − cos Θ⟩ ≈ Θ 2 ≈ (R/r) 2 .
Then, with R ≈ 10 7 cm the effective neutrino density is approximately
10 32 cm −3 r7 −4 , a value which is always smaller than the electron density
Eq. (11.11). Therefore, in the present context one may ignore neutrinoneutrino
interactions. This will not be the case for the issue of r-process
nucleosynthesis (Sect. 11.4.5).
One may proceed to determine the MSW triangle as in Fig. 8.9
for solar neutrinos, except that there an exponential electron density
profile was used while now Eq. (11.11) pertains. Nötzold (1987) found
a conversion probability in excess of 50% if
∆m 2 ν sin 3 2θ ∼ > 4×10 −9 eV 2 E ν /10 MeV, (11.12)
assuming that ∆m 2 < ν ∼ 3×10 4 eV 2 E ν /10 MeV so that a resonance can
occur outside of the neutrino sphere. The region in the ∆m 2 ν-sin 2 2θplane
(mixing angle θ) with a conversion probability exceeding 50% for
E ν = 20 MeV is shown as a shaded area in Fig. 11.17, together with
the MSW solutions to the solar neutrino problem. For orientation, the
Kamiokande-allowed range for solar neutrinos is also indicated.
The solar small-angle MSW solution would seem to have a small
impact on the prompt ν e burst from a collapsing star. Thus, the first
Kamiokande SN 1987A event may still be interpreted as a prompt ν e ,
and one may well observe nearly the full ν e burst from a future SN.
It is interesting that the MSW triangle for the prompt ν e burst
reaches to relatively large neutrino masses. In the 3−30 eV regime neutrino
masses would play an important cosmological role as dark matter
and for the formation of structure in the universe. Such massive neutrinos
likely would mix with ν e . Unless the mixing angle is very small
the appearance of an unoscillated prompt ν e burst from a stellar collapse
would be in conflict with a cosmological role of massive neutrinos
(Arafune et al. 1987b).