Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 143
is uncorrelated one may ignore the cross terms in the correlator. With
the single-nucleon spin operator σ one finds then
S σ (ω) = 1 3
∫ +∞
−∞
dt e iωt ⟨ σ(t) · σ(0) ⟩ . (4.59)
With Eq. (4.21) one obtains
dI ˙ ( ) 2
a
dω = CN ω 4 ∫ +∞
2f a 12π 2 −∞
dt e −iωt ⟨ σ(t) · σ(0) ⟩ (4.60)
for the differential axion energy-loss rate (radiation power) per nucleon.
Because of collisions with other nucleons, and because of a spin dependent
interaction potential caused by pion exchange, the nucleon spins
evolve nontrivially so that the correlator has nonvanishing power at
ω ≠ 0, allowing for axion emission.
4.6.5 Axion Emission in the Classical Limit
The correlator representation Eq. (4.60) of the axion emission rate is
extremely useful to develop a general understanding of its main properties
without embarking on a quantum-mechanical calculation. To this
end the nucleon spin σ is approximated by a classical variable, basically
a little magnet which jiggles around under the impact of collisions with
other nucleons. With an ergodic hypothesis about the spin trajectory
on the unit sphere one may replace the ensemble average in Eq. (4.60)
by a time average. The radiation intensity (time-integrated radiation
power) emitted during a long (infinite) time interval is then
( ) 2
dI a
dω = CN ω 2 ∣∫ ∣∣∣ +∞
2f a 12π 2 −∞
dt e iωt ˙σ(t)
∣
2
, (4.61)
where two powers of ω were absorbed by a partial integration with
suitable boundary conditions at t = ±∞.
In this form the energy-loss rate is closely related to a well-known
expression for the electromagnetic radiation power from a charged particle
which moves on a trajectory r(t). The nonrelativistic limit of a
standard result (Jackson 1975) is
dI γ
dω = 2α
3π
∫ +∞
∣ dt e iωt 2
a(t)
∣ , (4.62)
−∞
where a(t) = ¨r(t) is the particle’s acceleration on its trajectory.