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production by ultra high-energy protons accelerated in the burst [204].
The fraction of proton energy going into À-production is calculated as before following
Eq. 47,
"R2
fÀ C" xp’!À , (65)
»p³
L³(E³,min) 1 Ã" xp’!À
fÀ C" , (66)
4
E³,min ³rs"t 4Àc2
where ³rs is the Lorentz factor of the reverse shock. For the afterglow, the relevant time
-1/2
scale is 10-100 seconds and the luminosity is L³ " E³ [176]. E³,min, the minimum photon
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energy to produce pions via the "-resonance, is given by:
2
³rs(m2 - m2)
" p
E³,min = . (67)
4Ep
Therefore 1021 eV protons can kinematically produce À s on photons with energy as low as
10 eV. Combining Eq. 66 and Eq. 67 we find
10 eV Ep 1/2 300 5 20sec
fÀ C" .003 × . (68)
E³,min 1021eV ³rs "t
Note that above keV energy, the photon luminosity follows the broken spectrum with L³ "
-1 1/2
E³ and, therefore, fÀ " Ep rather than fÀ " Ep .
Associating the accelerated beam with the observed ultra high-energy cosmic ray flux,
-2
dNp/dEp
the resulting neutrino flux is given by
dN½ dNp
(E½)
dE½ dEp
dN½ 10eV 20E½ 1/2 300 5 20sec
(E½)
dE½ E³,min 1021eV ³rs "t
dN½ 2 10eV 300 5 20sec
1/2
E½(E½)
dE½ E³,min ³rs "t
It is important to note that if the burst occurs in a region of higher density gas, as can
be the case for a collapsing star, reverse shocks are produced earlier and, therefore, with
smaller Lorentz factors. This results in fÀ C" 1. Then,
dN½ dNp
(E½) C" (Ep = 20E½) × 1 C" A × (20E½)-2, (72)
dE½ dEp
dN½ 2
E½(E½)
dE½
In either case, the result is only valid above the threshold energy required to generates pions
via the "-resonance,
2
2
³rs(m2 - m2)
³rs 1keV
" p
min min
E½ C" .05Ep C" .05
max
4E³ 300 Emax
³
Below this threshold, ultra high-energy protons may still interact with non-thermal MeV
photons, however.
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The event rate in a neutrino telescope is calculated following Eq. 11. In the high-energy
appoximation,
0.5
P½’!µ C" 1.2 × 10-2E½,obs(EeV). (75)
This yields
5×1010GeV
-2 0.5
Nevents
7×108GeV
This is a very small rate indeed. The neutrino energy is, however, above the threshold for
EeV telescopes using acoustic, radio or horizontal air shower detection techniques. This
mechanism may represent an opportunity for detectors with very high threshold, but also
large effective area to do GRB physics.
11. The Decoupling of Neutrons: GeV Neutrinos
The conversion of radiation into kinetic energy in the fireball will accelerate neutrons along
with protons, especially if the progenitor involves neutron stars. Protons and neutrons are
initially coupled by nuclear elastic scattering. If the expansion of the fireball is sufficiently
rapid the neutrons and protons will no longer interact. Neutrons decouple from the fireball
while protons are still accelerated. Protons and neutrinos may then achieve relative velocities
sufficient to generate pions which decay into GeV neutrinos [206, 207]. We define the ratio
of neutrons to protons as,
nn
¾ a" , (77)
np
which initially remains constant during expansion. The fraction of neutrons which generate
pions is calculated in the same way as in Eq. 47,
"R2
fÀ C" . (78)
»pn
We can relate the density of nucleons to the density of photons by the dimensionless entropy,
³E³
np+n C" n³ . (79)
·mpc2
Following the arguments used in our discussion of PeV neutrinos, we arrive at
L³ Ãnp
fÀ C" , (80)
mp·³3"t 4Àc2(1 + ¾)
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where Ãnp C" 3 × 1026cm2 is the neutron-proton cross section for pion production. As the
neutrons and protons decouple, fÀ approaches unity. Using the fact that ³ asymptotically
approaches · at the end of expansion, we see that decoupling occurs for
1/4 1/4
LÃnp 1/4 L 100 km 1/4 2
>
· ·np a" C" 400 . (81)
4ÀR0mpc3(¾ + 1) 1052erg/s R0 ¾ + 1
In fact, the requirement for exceeding the threshold for À production is · e" 1.2 ·np [206].
The scattering time is therefore longer than the expansion time by a factor 1.24 C" 2.1 and
1 - e-2.1
=
more than 99% of the neutrons would scatter. It is therefore a reasonable approximation
to assume that all neutrons produce À s as long as · is above threshold. The number of
neutrons in the fireball is large with
E ¾ 1 E 2¾ 500
Nn C"
mpc2 1 + ¾ · 1053ergs 1 + ¾ ·
Above the pion threshold, every neutron interacts with a proton producing one of the fol-
lowing:
" p + n ’! p + p + À- ’! ½µ + µ- ’! e- + ½e + ½µ + ½µ
¯ ¯ ¯
" p + n ’! n + n + À+ ’! ½µ + µ+ ’! e+ + ½e + ½µ + ½µ
¯
" p + n ’! p + n + À0 ’! ³ + ³
Thus on average, each interaction produces two 30-50 MeV muon neutrinos and two 30-50
MeV electron neutrinos or two 70 MeV photons. The observed neutrino energy is
³ ³ 2
E½,obs C" 30 - 50 MeV × C" 6 - 10 GeV × . (83)
1 + z 400 1 + z
This energy is below the threshold of neutrino telescopes with the possible exception of
Baikal and ANTARES provided it is built with a sufficiently dense arrangement of the
photomultipliers.
P½’!µ(E½
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