Stars as Laboratories for Fundamental Physics - MPP Theory Group

440 Chapter 11
b) Impact of Neutrino-Neutrino Interactions
The discussion so far has been relatively simplistic because the role of
neutrino-neutrino interactions has been ignored. During the hot-bubble
phase the neutrino refractive index caused by other neutrinos is not necessarily
negligible (Pantaleone 1995; Qian and Fuller 1995). In terms of
the neutrino luminosity the number flux and thus the density of neutrinos
of a given species at a radius r is given by n ν = L ν ⟨E ν ⟩ −1 (4πr 2 ) −1 .
Moreover, as in the discussion in Sect. 11.4.2 one must include an average
of the factor (1 − cos Θ) to account for the anisotropy of the neutrino
phase space distribution (angle Θ between test and background
neutrino). At a given distance r the neutron star (neutrino sphere
radius R) subtends an angle given by sin Θ R = R/r. For a radially
moving test neutrino one finds (for a more rigorous treatment see e.g.
Qian and Fuller 1995)
∫ 1
/ ∫ 1
⟨1 − cos Θ⟩ = (1 − cos Θ) d cos Θ d cos Θ
cos Θ R cos Θ R
[
]
1 −
√1 − (R/r) 2 , (11.15)
= 1 2
which for large r approaches 1 4 (R/r)2 . Therefore, the effective neutrino
density n ν ⟨1 − cos Θ⟩ varies as r −4 at large distances.
The electrons and positrons cause a refractive energy shift between
ν e and, say, ν µ of ∆V = √ 2G F (n e − − n e +) while the effect of the
neutrinos is
∆V ≈ √ 2G F (n νe − n νe − n νµ + n νµ )⟨1 − cos Θ⟩
√ ( 2GF ⟨1 − cos Θ⟩ Lνe
≈
4πr 2 ⟨E νe ⟩ − L ν e
⟨E νe ⟩ − L ν µ
⟨E νµ ⟩ + L )
ν µ
.
⟨E νµ ⟩
(11.16)
The tau-flavored neutrino contribution has not been included because
it cancels exactly between ν τ and ν τ . The same is true for the muflavored
terms before oscillations have taken place. For a test-neutrino
of momentum p ν one finds numerically
2p ν ∆V ≈ 420 eV 2 (10 km/r) 2 ⟨1 − cos Θ⟩ ×
(
p ν Lνe
×
10 51 erg s −1 ⟨E νe ⟩ − L ν e
⟨E νe ⟩ − L ν µ
⟨E νµ ⟩ + L )
ν µ
.
⟨E νµ ⟩
(11.17)
A few seconds after bounce typical values might be ⟨E νe ⟩ = 11 MeV,
⟨E νe ⟩ = 16 MeV, and ⟨E νµ ⟩ = ⟨E νµ ⟩ = 25 MeV, the luminosities can