Stars as Laboratories for Fundamental Physics - MPP Theory Group

Radiative Particle Decays 471
value, t 0 is shown in Fig. 12.11 as a function of the neutrino velocity
β for several values of x = cos θ. The absorption effect is negligible
except for nonrelativistic neutrinos or for backward emission which, in
the laboratory frame, corresponds to very soft photon energies.
Fig. 12.11. Arrival time t 0 of first decay photons from a parent neutrino
with velocity β, taking the envelope radius of the source to be R env = 100 s.
The curves are marked with the respective values of x = cos θ, the direction
of photon emission in the neutrino rest frame.
In the parent frame the photons (energy ω) follow a normalized
distribution f(ω, x) which yields d 3 N γ (t, ω, x) = Ṅγ(t) f(ω, x) dt dω dx
with Ṅγ(t) from Eq. (12.21). The usual relativistic transformations
lead to a laboratory photon energy of E γ = γ(1 + βx) ω. Transforming
from dωdx to dωdE γ and integrating over the unobserved rest-frame
energy ω yields
d 2 N γ
dE γ dt = 1 ∫
E ω− γ f(ω, x)
dω
τ γ βγ ω + ω 2
( )
βγEγ t
e −Eγt/ωτtot Θ − ω ,
R env
(12.22)
where ω ± = E γ [γ(1 ± β)] −1 and x = (E γ /γω − 1)/β. The photon flux
at Earth is obtained by multiplication with the neutrino fluence F ν of
Eq. (12.12) and integration over a suitable spectrum Φ ν (E ν ) of neutrino
energies.
If the neutrinos are sufficiently long-lived (the exact meaning of
this is quantified below) the exponential can be ignored. If one also
ignores the absorption effect by the progenitor star (R env = 0), the