Stars as Laboratories for Fundamental Physics - MPP Theory Group

470 Chapter 12
Fig. 12.10. Geometry of the radiative neutrino decay.
The photons which arrive first are the ones from decays near the
source. Then the limit d D ≪ d LMC leads to d γ = d LMC − d D cos θ lab and
t = t D (1 − β cos θ lab ). With cos θ lab = (β + cos θ)/(1 + β cos θ) where θ
is the angle of photon emission in the parent frame one finds
t =
t D
γ 2 (1 + βx) , (12.20)
where γ = E ν /m ν is the neutrino Lorentz factor and x ≡ cos θ. Therefore,
photons detected between t and t + dt result from decays at
t D = γ 2 (1 + βx) t during an interval dt D = γ 2 (1 + βx) dt. (Recall
that t is measured after the first massless neutrinos arrived while t D
is measured after emission at the source.) The number of parent neutrinos
diminishes in time as e −t D/γτ tot
with τ tot the total decay time.
Therefore, the number of photons traversing a spherical shell of radius
d LMC per unit time is
Ṅ γ (t) =
γ(1 + βx)
τ γ
e −γ(1+βx) t/τ tot
, (12.21)
where as before τ γ is the radiative decay time.
Photons produced before the parent has left the envelope of the
progenitor star (radius R env ) cannot be detected at Earth. If this
absorption effect is to be included, Eq. (12.21) will involve a step
function 76 Θ(d D − R env ) with d D = βt D = βγ 2 (1 + βx) t. Put another
way, photons emitted at an angle θ in the rest frame will first
arrive at a time t 0 = R env [βγ 2 (1 + βx)] −1 . The progenitor of SN 1987A
has been unambiguously identified as the blue supergiant Sanduleak
−69 202 (Schramm and Truran 1990). From its surface temperature
(15,000 K), its luminosity (5×10 38 erg/s) and the distance to the LMC
one can infer its radius to be R env ≈ 3×10 12 cm = 100 s. Using this
76 The step function is defined by Θ(z) = 0 for z < 0 and Θ(z) = 1 for z > 0.