Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
Appendix D
Some details concerning the
neutrino-neutrino calculations
In this appendix, we develop the different calculation steps that are needed to
describe and interpret the neutrino-neutrino interaction and its consequences on
the neutrino evolution in the supernova environment.
D.1 The differential number density of neutrinos
We consider the differential number density of neutrinos dnνα(q) at radius r which
has the contribution from all να with energy q which propagate in directions
within the range between ˆq and ˆq + dˆq.
Following [52] if we define jνα(q) as the number flux of να with energy q
emitted in any direction at the neutrino sphere. The number of neutrinos going
outwards radially through the differential area R2 ν d(cos Θ)dΦ on the neutrino
sphere surface per unit time is
dNνα,E = jνα(q)R 2 ν d(cos Θ)dΦ (D.1)
Since we follow neutrinos going through this differential area but going towards
the point P on Fig.(5.2), we need to multiply this expression by a geometric
factor cosϑ0 and consequently we obtain:
dNνα,P = jνα(q) cosϑ0R 2 ν
d(cos Θ)dΦ (D.2)
Another way to express this quantity is to look at the number of neutrinos arriving
on point P with angle ϑ with respect to the z–axis within the range of the
differential area (l − l0) 2 d(cosϑ)dφ, and we obtain:
dNνα,P = jνα(q)(l − l0) 2 d(cosϑ)dφ = (l − l0) 2 dnνα(q) (D.3)
175