Stars as Laboratories for Fundamental Physics - MPP Theory Group

144 Chapter 4
The same result can be found with the above methods applied to the
spatial part of the vector current. Easier still, it can be obtained directly
from Eq. (4.61). To this end note that photon emission by an electron
involves nonrelativistically (e/m e ) p = ev so that we must substitute
˙σ → ˙v = a. The role of k (axions) is played by the polarization
vector ϵ (photons) so that k 2 = ω 2 must be replaced by ϵ 2 = 1. With
(C A /2f a ) → e, using α = e 2 /4π, and inserting a factor of 2 for two
photon polarization states completes the translation.
In order to understand the radiation spectrum consider a single
“infinitely hard” collision with ˙σ(t) = ∆σ δ(t). The radiation power is
dI a
dω = (
CN
2f a
) 2
ω 2
12π 2 |∆σ|2 . (4.63)
For photons one obtains the familiar flat bremsstrahlung spectrum
dI γ /dω = (2α/3π)|∆v| 2 which is hardened, for axions, by the additional
factor ω 2 from their derivative coupling.
In the form Eq. (4.63) the total amount of energy radiated in a single
collision is infinite. In practice, collisions are not arbitrarily hard,
and the backreaction of the radiation process on the emitter must be
included. This is not rigorously possible in a classical calculation, it requires
a quantum-mechanical treatment. Therefore, a classical analysis
is useful only for the soft part of the spectrum where backreactions can
be ignored, i.e. for radiation frequencies far below the kinetic energy
of the emitter. In a thermal environment, a classical treatment then
appears reasonable for ω < ∼ T .
Next, consider a large random sequence of n hard collisions with a
spin trajectory
n∑
˙σ(t) = ∆σ i δ(t − t i ). (4.64)
i=1
This yields the average radiation intensity per collision of
d ˙ I a
dω = (
CN
2f a
) 2
ω 2
12π 2 Γ coll ⟨(∆σ) 2 ⟩ F (ω), (4.65)
where Γ coll is the average collision rate and ⟨(∆σ) 2 ⟩ is the average
squared change of the spin in a collision. Further,
1
n∑ ∆σ i · ∆σ j cos[ω(t i − t j )]
F (ω) = 1 + lim
, (4.66)
n→∞ n
⟨(∆σ) 2 ⟩
i,j=1
i≠j
where the first term (the “diagonal” part of the double sum) gives the
total radiation power as an incoherent sum of individual collisions. The