Stars as Laboratories for Fundamental Physics - MPP Theory Group

106 Chapter 3
The factors F + and F − are of order unity. Dicus et al. (1976) derived
analytic expressions in terms of a screening scale and the Fermi
velocity of the electrons. In fact, because F − is always much smaller
than F + and C 2 V − C 2 A is much smaller than C 2 V + C 2 A, the “minus”
term may be neglected entirely. Therefore, the Dicus et al. (1976) result
is identical with that of Festa and Ruderman (1969). Either one
is correct only within a factor of order unity because Eq. (6.61) was
used as a screening prescription with the Thomas-Fermi wave number
as a screening scale. However, in a degenerate medium electrons never
dominate screening. The most important effect is from the ion correlations
which, in a weakly coupled plasma (Γ ∼ < 1), can be included by
Eq. (6.72) with the Debye scale k i of the ions as a screening scale. While
it is easy to replace k TF with k i in these results, the modification of the
Coulomb propagator according to Eq. (6.72) cannot be implemented
without redoing the entire calculation.
A systematic approach to include ion correlations (i.e. screening
effects) was pioneered by Flowers (1973, 1974) who showed clearly how
to separate the ion correlation effects in the form of a dynamic structure
factor from the matrix element of the electrons and neutrinos. This
approach also allows one to include lattice vibrations when the ions
form a crystal in a strongly coupled plasma. In a series of papers Itoh
and Kohyama (1983), Itoh et al. (1984a,b), and Munakata, Kohyama,
and Itoh (1987) followed this approach and calculated the emission rate
for all conditions and chemical compositions.
As an estimate, good to within a factor of order unity, one may use
F + = 1 and F − = 0. Moreover, inspired by the axion results one can
guess a simple expression which can be tested against the numerical
rates of Itoh and Kohyama (1983). I find that F − = 0 and
F + ≈ ln
( 2 + κ
2
κ 2 )
+ κ2
2 + κ 2 (3.42)
is a reasonable fit even for strongly coupled conditions (Appendix C).
3.5.4 Neutron-Star Crust
The degenerate bremsstrahlung emission of νν pairs is relevant in such
diverse environments as the cores of low-mass red giants, white dwarfs,
and in neutron-star crusts. Pethick and Thorsson (1994) noted that
in the latter case the medium is so dense that band-structure effects
of the electrons become important. The band separations can be up