Stars as Laboratories for Fundamental Physics - MPP Theory Group

464 Chapter 12
Fig. 12.6. Expected fluence of photons from the decay ν → ν ′ γ of low-mass
SN neutrinos according to Eq. (12.15) with m ν /τ γ = 10 −15 eV/s and the
indicated neutrino temperatures. The shaded bands for each temperature
are for the range −1 ≤ α ≤ +1 with the harder edge corresponding to
α = +1. Also shown are the upper limits from the GRS channels taken from
the 10 s column of Tab. 12.1.
Because details of the spectral form are not known it is easiest to
use a Boltzmann distribution as a generic case,
Φ ν (E ν ) = E2 ν e −E ν/T ν
2T 3 ν
. (12.14)
Then one finds explicitly 74
F ′ γ(E γ ) = F ν
m ν
τ γ
d LMC
2T 2 ν
[
(1 − α) e −ε + 2α ε E 1 (ε) ] , (12.15)
where ε ≡ E γ /T ν . In Fig. 12.6 this spectrum is shown for T ν = 4 and
8 MeV with m ν /τ γ = 10 −15 eV/s. The envelopes of the shaded bands
in Fig. 12.6 correspond to α = ±1 where for each temperature the
“harder” edge corresponds to α = +1.
The observational constraints give a limiting γ fluence for certain
energy bands. Thus one needs the expected fluence for a given energy
74 The exponential integral function is defined as E n (x) = ∫ ∞
1
dt e −x t /t n . Note
that E n (∞) = 0 while for n > 1 E n (0) = (n − 1) −1 .