Stars as Laboratories for Fundamental Physics - MPP Theory Group

476 Chapter 12
Fig. 12.15. Nonrelativistic correction of the expected photon fluence for
the GRS channels of Tab. 12.1 according to Eq. (12.32) with R env = 100 s,
t GRS = 223.2 s, and T ν = 6 MeV.
So far the nonrelativistic corrections for a neutrino mass in the
10 MeV mass range have been ignored. As an example I consider the
photon fluence of Eq. (12.27) in the limit of a large τ ∗ where the exponentials
in Eq. (12.28) can be expanded. The Boltzmann spectrum
for nonrelativistic neutrinos should include an extra factor β = p ν /E ν .
Then for α = 0 the fluence is 78
with
∫
F γ(E ′ t ∞
GRS E ν e −E ν/T ν
γ ) = F ν dE ν
m ν τ γ E min
T 3 ν
⎧ ⎡ ( ) ⎤
⎪⎨
2
E min = max
⎪⎩ m mν R env
ν
⎣1 + ⎦
2E γ t GRS
(
1/2
)
E γ − m2 ν R env
, (12.32)
2p ν t GRS
,
(
E γ + m2 ν
4E γ
) ⎫ ⎪ ⎬
⎪ ⎭
. (12.33)
Relative to the massless case, the integral expression is suppressed if
m ν ∼ > T ν . A straightforward numerical integration then yields the suppression
of the expected fluence for each GRS channel as shown in
78 Strictly speaking, the fluence of massive neutrinos must be calculated by determining
their neutrino sphere which is different from the massless case. This
problem was recently tackled by Sigl and Turner (1995) by solving the Boltzmann
collision equation by means of an approximation method known from calculations
of particle freeze-out in the early universe. However, because only masses of up
to 24 MeV are presently considered, a precise treatment of the neutrino spectrum
changes the resulting limits only by a small amount.