Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
C
A E
Θ
Rν
B
Neutrinosphere
Θmax
I
ϑ0
P
ϑ
ϑmax ν
Figure 5.2: 2 dimension geometric picture of the neutrino bulb model. An arbitrary
neutrino beam (solid line) is shown emanating from a point on the neutrino sphere
with polar angle Θ. This beam intersects the z–axis at point P with angle ϑ. Because
neutrinos are emitted from the neutrino sphere of radius Rν, point P sees only neutrinos
traveling within the cone delimited by the dotted lines. One of the most important
geometric characteristics of a neutrino beam is its emission angle ϑ0, defined with
respect to the normal direction at the point of emission on the neutrino sphere ϑ0 =
Θ+ϑ). All other geometric properties of a neutrino beam may be calculated using the
radius r = CP and ϑ0.
here. Secondly, a cylindrical symmetry is also present for the neutrino flux at
any given point on the z–axis. Consequently different neutrino beams possessing
the same polar angle with respect to the z–axis and with the same initial physical
properties (flavor, energy, etc...) will have identical flavor evolution histories.
One may choose this polar angle to be ϑ, the angle between the direction of the
beam and the z–axis. This angle varies between 0 (a beam of neutrinos emitted
along the z–axis) and ϑmax ≡ arcsin
Rν the maximum angle with respect to
r
the z–axis where neutrinos can come from (since we postulated that any given
point of the neutrino sphere emits isotropically2 ). A posteriori, below the neutrino
sphere, a beam could be specified by the polar angle Θ giving the emission
position of the beam on the neutrino sphere (see Fig. 5.2). Θ is the ”symmetric”
of ϑ, while the former describes an emission like it was starting from below the
neutrino sphere, the latter describes a beam emitted at the surface of the neutrino
sphere. Therefore, Θ varies between 0 and Θmax ≡ arcos
Rν . A third option
r
would be to define the emission angle ϑ0 with respect to the normal direction
at the point of emission on the neutrino sphere (see Fig. 5.2). That definition
may be useful since it is an intrinsic geometric property of the beam, and does
not depend on the distance from the center. Indeed, this angle varies from 0
2 Actually, we can consider that the emission is semi-isotropic in the sense where the emission
is directed towards the outside of the neutrinos sphere. Since neutrinos emitted towards the
inside of the neutrino sphere will come out from another point of the neutrino sphere and will
be counted for this particular point.
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z