Stars as Laboratories for Fundamental Physics - MPP Theory Group

Supernova Neutrinos 441
be taken to be the same at 3×10 51 erg s −1 each, and R = 11 km. For
these conditions the estimated contributions to ∆V from electrons and
neutrinos are shown in Fig. 11.22 as a function of radius.
As long as no swap has occurred the neutrinos do not play a major
role because the contribution of ν µ cancels exactly against ν µ . Further,
the difference between ν e and ν e is smaller than the electron contribution,
and it has the same sign whence it simply causes a slightly larger
effective matter density.
However, on resonance a relatively large number of ν e ’s exchange
flavor with ν µ ’s so that the neutrino contribution changes sign. Moreover,
on resonance the vacuum ∆m 2 ν by definition cancels against the
medium-induced contribution. Therefore, “switching on” the neutrino
term shifts the resonance position for given vacuum mixing parameters.
Also, in a self-consistent treatment one needs to consider the full non-
Fig. 11.22. “Mass splitting” between ν e and ν µ (equivalently ν τ ) of momentum
p ν = 10 MeV caused by the regular medium (the electrons), and
by different neutrino flavors according to Eq. (11.17). For the electrons the
density profile of Fig. 11.20 was used with Y e = 0.5; in a real SN core Y e is
much lower near the neutrino sphere so that the thick line would increase
less steeply toward the neutrino sphere than shown here. For the neutrinos,
a luminosity in each degree of freedom of 3×10 51 erg s −1 was assumed, the
average energies were taken to be ⟨E νe ⟩ = 11 MeV, ⟨E νe ⟩ = 16 MeV, and
⟨E νµ ⟩ = ⟨E νµ ⟩ = 25 MeV, and the neutrino-sphere radius is 11 km. The
signs of the contributions relative to electrons are: + for ν e and ν µ , − for
ν e and ν µ . The contribution of ν µ cancels exactly against ν µ unless a swap
ν e ↔ ν µ has taken place.