Stars as Laboratories for Fundamental Physics - MPP Theory Group

Radiative Particle Decays 473
ignored (t env = 0) the spectrum is shown in Fig. 12.12 for the anisotropy
parameters α = 0, ±1. Note the difference to the triangular shape of
Fig. 12.1 for a stationary source. High-energy photons are now enhanced
because lower-energy ones correspond to larger emission angles
in the parent frame and so they take a larger “detour” from the source
to us (Fig. 12.10). Hence, their flux is spread out over a larger time
interval even though photons of all energies begin to arrive at the same
time if t env = 0.
Fig. 12.12. Photon spectrum from the decay of a short burst of relativistic
neutrinos, energy E ν , according to Eq. (12.24) taking p ν = E ν and e −t/τ∗ = 1
(relativistic and long-lived parent), and ignoring absorption effects by the
progenitor (t env = 0).
In order to compare with the GRS fluence limits one needs to integrate
the expected flux between t = 0 and t = t GRS = 223.2 s.
The Θ function in Eq. (12.24) is accounted for by using t env as a lower
limit of integration. Integrating also over the neutrino source spectrum
F ν Φ ν (E ν ) yields
where
F ′ γ = F ν
t GRS
m ν τ γ
∫ ∞
E min
dE ν Φ ν
2E γ
p ν
[
1 + α 2E ]
γ − E ν
I, (12.27)
p ν
I ≡ e−t env/τ ∗
− e −t GRS/τ ∗
t GRS /τ ∗
. (12.28)
For sufficiently long-lived parents (τ ∗ ≫ t GRS ) the exponentials can
be expanded and I = (t GRS − t env )/t GRS . The lower limit of integration
is set by the condition Eq. (12.26) and by the requirement that
t env < t GRS , i.e. that I > 0. This condition may be expressed as
p ν > (m 2 ν/2E γ ) (R env /t GRS ).