Stars as Laboratories for Fundamental Physics - MPP Theory Group

336 Chapter 9
Fig. 9.2. Dependence of the flavor relaxation rate on η = µ νx /µ νe for the
case of a “large” ∆m 2 Eq. (9.66) and a “small” ∆m 2 Eq. (9.69). F e is
the contribution of electron targets (νe → eν) while F p is from protons
(ep ↔ nν e ).
and numerically
3G 2 F
(
) 5/3 ( ) 2
5π n pµ 2 ν e
= 1.0×10 9 s −1 Y p ρ µνe
, (9.67)
10 14 g cm −3 µ e
which sets the time scale for flavor conversion.
For the case of a “small” ∆m 2 one finds from Eq. (9.60) and from
the production and scattering rates of the previous section
τ −1 = θ 2 0
where F p = 1 and
(∆m 2 ) 2
8πn p
[
(C 2 V + 3C 2 A) F p + µ ν e
µ e
F e
]
, (9.68)
F e =
9
5 η−1 − 137
40 + 3 4 (1 + η4 ) log η + 5 3 η + η3 − 101
120 η4 − 1 5 η5
8
3 (1 − η3 )
(Fig. 9.2, right panel). Numerically,
(9.69)
(∆m 2 ) 2
8πn p
( 10
= 1.4×10 2 s −1 14 g cm −3 ) ( (∆m 2 ) 1/2 ) 4
. (9.70)
Y p ρ 1 keV
This time scale is independent of Fermi’s constant because a factor G 2 F
from the scattering rate cancels against G −2
F from θ 2 E.