Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
Since we want in the new referential the same equations than previously without
matter, we have the following relation:
and consequently
Pu = cos α(t) Px − sin α(t) Py (D.20)
Pv = sin α(t) Px + cosα(t) Py
Pu
˙ = ( ˙ Px − ˙αPy) cosα(t) − ( ˙αPx + ˙ Py) sin α(t) (D.21)
Pv
˙ = ( ˙ Px − ˙αPy) sin α(t) + ( ˙αPx + ˙ Py) cosα(t)
Expressing ˙ Puu and ˙ Pvv in the previous referential and using Eq.(D.16), we
obtain
Pu
˙ cos α(t) + ˙ Pv sin α(t) = ˙ Px + λ ˙ Py
(D.22)
which yields:
˙α(t) = −λ and therefore α(t) = −λ t+cst. Since the two referentials coincide initially,
i.e α(t = 0) = 0, we have: α = −λ t. Consequently, in the new referential,
the equations of motion would be exactly the same than without matter:
∂tPω = +ωB(t) + µ Pω − ¯
Pω × Pω ,
∂t ¯ Pω = −ωB(t) + µ Pω − ¯
Pω × Pω
¯ . (D.23)
except that the magnetic field B(t) is now time dependent as:
⎛
⎞
sin(2θ0) cos(−λt)
B(t) = ⎝ sin(2θ0) sin(−λt) ⎠. (D.24)
− cos(2θ0)
We name this new referential, the rotating frame.
D.3 The corotating frame and the adiabaticity
The corotating frame
To express the new E.O.M. in this frame, we perform a similar derivation than
the one made previously to make disappear the matter effect. We define the new
frame such that D is a constant vector in this frame.
⎛
∂tD − ωcB × D = ⎝
( ˙ Dx + ωcDy)x
( ˙ Dy − ωcDx)y
˙Dzz
⎞
⎛
⎠ = ⎝
0u
0v
0w
⎞
⎛
⎠ = ⎝
˙Duu
˙Dvv
˙Dww
⎞
⎠ . (D.25)
Using the same relation to move from one basis to another than before (see
Eq.(D.18)) with α(t) = −ωct one obtains:
⎛ ⎞ ⎛
Pxx ˙ (
⎝ Pyy ˙ ⎠ = ⎝
Pzz ˙
˙ Pu + ˙αPv)u
( ˙
⎞ ⎛ ⎞
˙αPvu
Pv − ˙αPu)v ⎠ + ⎝ − ˙αPuv ⎠ . (D.26)
Pww ˙
0w
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