Stars as Laboratories for Fundamental Physics - MPP Theory Group

102 Chapter 3
however, it is important even in environments which are approximately
nondegenerate and so I begin with this simple case.
The calculation amounts to a straightforward evaluation of the matrix
element corresponding to the amplitude of Fig. 3.5 and an integration
over the Maxwell-Boltzmann distributions of the electrons. In
order to account for screening effects the Coulomb propagator is modified
according to Eq. (6.72), |q| −4 → [q 2 (q 2 + kS)] 2 −1 where k S is a
screening wave number (Sect. 6.4). For the emission of pseudoscalars
one then finds for the energy-loss rate per unit volume to lowest order
in kS 2 (Krauss, Moody, and Wilczek 1984; Raffelt 1986a)
Q = 128
45 √ π
× ∑ j
α 2 α ′
m e
( T
m e
) 5/2
n e
(
√
n j
[Zj
2 2 1 − 5 8
kS
2 ) (
+ Z j 1 − 5 m e T
4
kS
2 )]
. (3.28)
m e T
Here, the sum is extended over all nuclear species with charges Z j e and
number densities n j ; note that n e = ∑ j Z j n j . The term quadratic in Z j
corresponds to electron-nucleus collisions while the linear term is from
electron-electron scattering which yields a nonnegligible contribution
under nondegenerate conditions. 19
Raffelt (1986a) incorrectly used Eq. (6.61) as a modification of the
Coulomb propagator, a procedure which enhances the terms proportional
to kS 2 by a factor of 2. Either way, screening is never an important
effect. The screening scale is kS 2 = kD 2 + ki
2 because both electrons
and nuclei (ions) contribute. Then
kS
2
m e T = 4πα (n
m e T 2 e + ∑ j
n j Z 2 j
)
, (3.29)
which is about 0.12 at the center of the Sun and 0.17 in the cores of
horizontal-branch (HB) stars.
Ignoring screening effects, the energy-loss rate per unit mass is
ϵ = α ′ 5.9×10 22 erg g −1 s −1 T 2.5
8 Y e ρ ∑ j
Y j
(
Z
2
j + Z j / √ 2 ) , (3.30)
where T 8 = T/10 8 K, ρ is in g cm −3 , and Y j = X j /A j is the number
fraction of nuclear species j relative to baryons while X j is the mass
fraction, A j the mass number.
19 Note that e − e − → e − e − γ vanishes to lowest order because two particles of
equal mass moving under the influence of their Coulomb interaction do not produce
a time-varying electric dipole moment because their center of mass and “center of
charge” coincide. However, the emission of pseudoscalars corresponds to M1 rather
than E1 transitions and so it is not suppressed.