Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particles Interacting with Electrons and Baryons 105
β F → 1. Hence in the relativistic limit
F = 2 + κ2
2
ln
( 2 + κ
2
κ 2 )
− 1, (3.39)
somewhat different from what Iwamoto (1984) found who used the
Thomas-Fermi wave number as a screening scale.
For a strongly coupled, degenerate plasma typical for white dwarfs
the factor F was calculated numerically by Nakagawa, Kohyama, and
Itoh (1987) and Nakagawa et al. (1988) who also gave analytic approximation
formulae for the axion emission rate, applicable to nonrelativistic
and relativistic conditions. For a 12 C plasma with densities in the
range 10 4 −10 6 g cm −3 and temperatures of 10 6 −10 7 K it is found that
F = 1.0 within a few tens of percent. Therefore, for simple estimates
this value is a satisfactory approximation.
Altherr, Petitgirard, and del Río Gaztelurrutia (1994) have calculated
the bremsstrahlung process with the methods of finite temperature
and density (FTD) field theory. The main point is that one computes
directly the interaction of the electrons with the electromagnetic
field fluctuations which are induced by the ambient charged particles.
The result for the emission rate is similar to the one derived above.
3.5.3 High Degeneracy: Neutrino Pairs
Neutrino pair bremsstrahlung (Fig. 3.5) was the first nonnuclear neutrino
emission process ever proposed (Pontecorvo 1959; Gandel’man
and Pinaev 1959). A detailed calculation of the energy-loss rate in
a degenerate medium, relativistic and nonrelativistic, was performed
by Festa and Ruderman (1969) while conditions of partial degeneracy
were studied by Cazzola, de Zotti, and Saggion (1971). The Festa and
Ruderman calculation was extended by Dicus et al. (1976) to include
neutral-current interactions. After a calculation very similar to the one
presented above for pseudoscalars the emission rate for neutrino pairs is
Q = 2πα2
189 G2 FT 6 ( ∑
j
n j Z 2 j
) [ 1
2 (C2 V + C 2 A)F + + 1 2 (C2 V − C 2 A)F −
]
.
(3.40)
With a single species of nuclei (charge Z, atomic mass A) the energyloss
rate per unit mass is
ϵ = 0.144 erg g −1 s −1 (Z 2 /A) T 6 8 [ . . . ], (3.41)
where T 8 = T/10 8 K and the square bracket is from Eq. (3.40). The
temperature dependence is steeper than for axions by two powers.