Stars as Laboratories for Fundamental Physics - MPP Theory Group

148 Chapter 4
For axion bremsstrahlung, the energy-loss rate of the medium is
given by ϵ a = Q a /ρ ∝ ∫ ∞
0 dω ω 4 S σ (ω) so that with Eq. (4.72) one needs
to evaluate
ϵ a ∝ Γ σ
∫ ∞
0
dω ω4 e −ω/T
ω 2 + Γ 2 /4
(4.73)
with Γ determined from the normalization condition. In the dilute
limit (Γ ≈ Γ σ ≪ T ) one may ignore Γ in the denominator, so that
ϵ a ∝ Γ σ . This is indeed what one expects from a bremsstrahlung process
for which the volume energy-loss rate is proportional to the density
squared, and thus ϵ a proportional to the density which appears in the
spin-fluctuation rate Γ σ .
In the high-density limit (Γ ≈ Γ σ /2 ≫ T ) the denominator in
Eq. (4.73) is dominated by Γ because the exponential factor suppresses
the integrand for ω ≫ T so that one expects ϵ a to be a decreasing
function of Γ σ . In Fig. 4.8 the variation of ϵ a with Γ σ /T is shown
(solid line), taking Eq. (4.72) for the spin-density structure function.
The dashed line shows the “naive rate,” based on Γ σ /ω 2 which ignores
multiple-scattering effects, and which violates the normalization condition.
Fig. 4.8 illustrates that even very basic and global properties
of S σ (ω) reveal an important modification of the axion emission rate
at high density: they saturate with an increasing spin-fluctuation rate,
Fig. 4.8. Schematic variation of the axion emission rate per nucleon with
Γ σ /T , taking Eq. (4.72) for the spin-density structure function. The dashed
line is the naive rate without the inclusion of multiple-scattering effects, i.e.
it is based on S σ (|ω|) = Γ σ /ω 2 .