Stars as Laboratories for Fundamental Physics - MPP Theory Group

436 Chapter 11
two detectors. In this case the contours refer to an average value ⟨p⟩.
In the shaded area (B) the contours are not independent of the detailed
matter distribution in the stellar envelope.
In Fig. 11.18 the large-angle MSW and the vacuum oscillation solutions
of the solar neutrino problem are indicated. If either one of them
is correct the measurable ν e spectrum in a detector is a substantial mixture
of different-flavor source spectra. Smirnov, Spergel, and Bahcall
(1994) argued on the basis of a joint analysis between the SN 1987A
signals at the Kamiokande and IMB detectors that p < 0.17−0.27 at
the 95% CL, depending on the assumed primary neutrino spectra. If
p were any larger, the expected spectra would be much harder than
has been observed. This analysis excludes the vacuum oscillation solution
to the solar neutrino problem. Kernan and Krauss (1995) arrive
at the opposite conclusion that all mixing angles are permitted by the
SN 1987A signal, and that sin 2 2θ = 0.45 is actually a favored value.
11.4.4 Shock Revival
The “swap fraction” of the ν e with the more energetic ν µ or ν τ spectrum
discussed in the previous section is always very small unless the
mixing angle is very large because of the assumed normal mass hierarchy
which prevents resonant conversions. By the same token a swap
of the ν e with the ν µ or ν τ spectrum will be resonant for certain mixing
parameters and so it can be almost complete even for very small
mixing angles. If the resonance is located between the neutrino sphere
and the stalling shock wave after bounce, but before the final explosion,
the shock would be helped to rejuvenate because the higher-energy
ν µ,τ ’s are more efficient at transferring energy once they have converted
into ν e ’s (Fuller et al. 1992). Of course, approximately equal luminosities
in all flavors have been assumed.
A typical density profile for a SN model 0.15 s after bounce is shown
in Fig. 11.19; the step at a radius of about 400 km is due to the shock
front. A resonance occurs if ∆m 2 ν/2E ν = √ 2G F n e . On the right scale
of the plot the quantity m res ≡ ( √ 2G F n e 2E ν ) 1/2 is shown for E ν =
10 MeV and Y e = 0.5. A resonance occurs inside of the shock wave
only for neutrino masses in the cosmologically interesting regime of
order 10 eV and above. Fuller et al. (1992) have performed a detailed
numerical calculation of the additional heating effect for (∆m 2 ν) 1/2 =
40 eV; they found a 60% increase of the energy of the shock wave.
The conversion probability between neutrinos is large if the adiabaticity
parameter γ defined in Eq. (8.39) far exceeds unity. Accord-