Stars as Laboratories for Fundamental Physics - MPP Theory Group

382 Chapter 10
one needs a large mixing angle to achieve a substantial suppression of
the measurable flux. If one includes the possibility of three-flavor oscillations
one can achieve, in principle, a maximum ν e reduction by a
factor 1 3 .
In this picture the ν e flux is reduced by the same factor for all energies,
contrary to what is indicated by the observations. Therefore, one
needs an oscillation length of order the Earth-Sun distance (astronomical
unit 1 AU = 1.496×10 −13 cm) because in this case the oscillation
pattern is not completely smeared out. Then one may measure a different
flux suppression for different energies; the wiggles in the reduction
factor as a function of neutrino energy can be resolved (Fig. 10.17).
For two-flavor oscillations the relationship Eq. (8.15) between neutrino
energy E ν , the mass-square difference ∆m 2 ν, and the oscillation
length l osc can be written as
∆m 2 ν = 1.66×10 −10 eV 2 1 AU
l osc
E ν
10 MeV . (10.23)
Therefore, the neutrino masses are very small if these “long-wavelength
oscillations” are the explanation for the observed flux deficits.
Fig. 10.17. Survival probability of solar ν e ’s for ∆m 2 ν = 0.8×10 −10 eV 2
and sin 2 2θ = 0.8, parameters which reconcile all measurements with the
standard solar flux predictions. The solid line is for the minimum annual
distance to the Sun of 1.471×10 13 cm (January), the dashed line for the
maximum distance 1.521×10 13 cm (July). The right panel is an enlarged
section around the line of beryllium neutrinos. Between January and July,
their flux is suppressed by very different amounts.