Stars as Laboratories for Fundamental Physics - MPP Theory Group

484 Chapter 12
12.5 Galactic Supernovae and ν τ → ν e e + e −
12.5.1 Bounds on the Positron Flux
A core-collapse SN produces about 3×10 57 ν τ ’s which may subsequently
decay into ν e e + e − . What is the long-term fate of all these positrons If
the ν τ ’s decay mostly outside of the galaxy it is well possible that the
positrons will linger in intergalactic space “forever.” Those positrons
produced within the galaxy, however, will be trapped by the magnetic
fields (typical strength a few µG) which render the galactic disk a magnetic
“bottle” for charged particles. The interstellar electron density is
on the order 1 cm −3 , leading to a positron lifetime against annihilation
of order 10 5 years. Moreover, elastic e + e − scattering (Bhabha scattering)
is very efficient at slowing down relativistic positrons because of
the perfect mass match which allows for an efficient energy exchange
in collisions. Therefore, most annihilations occur at rest, producing a
sharp γ-ray feature at 511 keV.
The galactic SN rate is a few per century while the decay positrons
annihilate on a much longer time scale. Therefore, the galactic disk
should contain a stationary positron population with a density determined
by the galactic SN rate and the ν τ lifetime. A comparison with
the measured photon flux at E γ = 511 keV of about 5×10 −3 cm −2 s −1
then leads to a very restrictive limit on the e + e − decay channel (Dar,
Goodman, and Nussinov 1987).
In detail these authors used a galactic rate of two core-collapse SN
per century, an e + lifetime against annihilation of 10 5 yr, and a typical
distance of the decays from Earth of 10 kpc. If most neutrinos decay
within the galactic disk, these assumptions lead to an expected photon
flux of 400 cm −2 s −1 . Thus, it is enough that one in 10 5 neutrinos decays
within the galactic disk to outshine the measured flux.
This estimate is corroborated by the more recent work of Skibo, Ramaty,
and Leventhal (1992) who devised detailed models of the positron
distribution in the galaxy in order to account for the 511 keV diffuse
galactic line feature measured in the direction away from the galactic
center. They found a total stationary positron annihilation rate in
the galaxy of 0.6−3 × 10 43 s −1 , where the precise coefficient depends on
model assumptions. With two core-collapse SN per century the average
galactic ν τ production rate is 2×10 48 s −1 . Again, it is enough if one ν τ
in 10 5 injects an e + into the galaxy to account for the observations.
If all positrons produced within about 1 kpc = 3×10 21 cm from the
source (the scale height of the galactic disk) were magnetically trapped,