Stars as Laboratories for Fundamental Physics - MPP Theory Group

126 Chapter 4
scale for the proton and neutron Fermi momenta which are p n,p
F =
(3π 2 n B ) 1/3 Yn,p 1/3 . According to a result of Brinkmann and Turner (1988)
the effective coupling is then
α a → α n Y 1/3
n + α p Y 1/3
p + ( 28 3 α 0 + 20 3 α 1) (Y 2/3
n + Y 2/3
p ) 1/2
× 1
2 √ 2
(2 −
|Y
2/3
n − Y 2/3
p |
Yn 2/3 + Yp
2/3
)
. (4.14)
The third term is from np collisions; it has the remarkable feature that it
does not vanish as Y p → 0. For equal couplings (C n = C p ) the variation
of the emission rate is shown in Fig. 4.4 as a function of Y p = 1 − Y n .
For all proton concentrations the np contribution dominates.
Fig. 4.4. Variation of the degenerate nucleon-nucleon bremsstrahlung rate
with Y p (proton number fraction) according to Eq. (4.14) with equal couplings
so that α n = α p = α 0 and α 1 = 0.
4.3 Neutrino Pair Emission
4.3.1 Structure Function
Neutrino pair emission in nucleon-nucleon collisions (Fig. 4.5) is analogous
to that of axions. Therefore, instead of embarking on a new
calculation it is worth understanding their common features—the neutrino
rates can be obtained for free on the basis of the axion ones.
This approach amounts to defining the dynamical structure functions