Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
y
Φ
C
x
Θ
dΦ
E
dΘ
Figure D.1: Geometric picture of the neutrino bulb model.
Consequently, the number density of neutrinos can be written in two ways:
Using Eq.(5.9) and the identities
dnνα(q) = jνα(q) cosϑ0R 2 ν
ϑ
d(cos Θ)dΦ
(l − l0) 2
(D.4a)
= jνα(q) d(cosϑ)dφ, (D.4b)
dΦ = dφ, (D.5)
cosϑ0Rν dΘ = (l − l0) dϑ. (D.6)
we can easily check the equality between Eq.(D.4a) and Eq.(D.4b).
We now have to express the fundamental quantity jνα(q). Using Eq.(D.4a) and
integrating it over all the neutrino sphere and over all the possible emission
angles (i.e over the range of ϑ), we will obtain the total number of να with energy
q emitted per unit time at the neutrino sphere:
Nνα(q) =
1
d(cos ϑ0)
2π
dφ
1 2π
0
1
0
2π
0
1
0
dNνα,E
2π
= d(cos ϑ0)
0
0
dφ d(cos Θ)
0
0
dΦjνα(q) cosϑ0R 2 ν
= 8π 2 R 2 νjνα(q) 1
cosϑ0 d(cos ϑ0)
0
= 4π 2 R 2 νjνα(q) (D.7)
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