Stars as Laboratories for Fundamental Physics - MPP Theory Group

Radiative Particle Decays 469
weak limit would not apply. This is the case for τ lab ∼ < R env ≈ 100 s
(envelope radius R env of the progenitor star) and so with E ν ≈ 20 MeV
one needs to require τ tot /m ν ∼ > 10 −5 s/eV.
12.4.4 Decay Photons from High-Mass Neutrinos
For “high-mass” neutrinos with m > ν ∼ 40 eV a calculation of the photon
fluence is more involved because of the dispersion of the neutrino burst
and the corresponding delay of the decay photons. The parent neutrino
travels with a velocity β = (1 − m 2 ν/Eν) 2 1/2 ≈ 1 − m 2 ν/2Eν 2 so that it
arrives at Earth with a time delay of about (m 2 ν/2Eν) 2 d LMC relative
to massless ones; here, d LMC = 50 kpc = 5.1×10 12 s is our distance
to the LMC where SN 1987A had occurred. With T ν ≈ 6 MeV for
ν µ or ν τ an average neutrino energy is 3T ν ≈ 20 MeV, spreading out
the arrival times of massive neutrinos over an approximate interval of
10 −2 s (m ν /eV) 2 . As the radiative decay may occur anywhere between
the LMC and here, the arrival times of the decay photons will be spread
out by a similar amount even though the photons themselves travel
with the speed of light. Thus for m < ν ∼ 40 eV all decay photons fall
within about a 10 s time window around the arrival time of the first
ν e ; the results of Sect. 12.4.3 apply to this case. For m < ν ∼ 200 eV
they fall within the 223.2 s interval for which GRS fluence limits exist.
Therefore, one may easily scale the previous limits to this case by using
the 223.2 s fluence limits in Tab. 12.1 instead of the 10 s ones. Of course,
for a given neutrino mass there would be an optimum time window for
which fluence limits could be derived on the basis of the original data.
For larger masses only a certain portion of the photon pulse falls
into the 223.2 s window. In order to calculate this fraction, I follow
Oberauer et al. (1993) and begin with the simple case where all neutrinos
are emitted at the same time with a fixed energy E ν . The radiative
decay occurs at a time t D after emission and thus at a distance
d D = βt D from the source (neutrino velocity β), and the photon is emitted
at an angle θ lab relative to the neutrino momentum (Fig. 12.10).
It has to travel a distance d γ until it arrives here; elementary geometry
yields d γ = [d 2 LMC − d 2 D(1 − cos 2 θ lab )] 1/2 − d D cos θ lab . Therefore,
relative to the first (massless) neutrinos the photons are delayed by
t = t D + d γ − d LMC . This delay has two sources: The parent moves
with a speed less than that of light, and the photon is emitted at an
angle so that a detour is taken from the LMC to us. For ultrarelativistic
parents both effects disappear as the relativistic transformations
squeeze all laboratory emission angles into the forward direction.