Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 121
factor to compensate for double counting of identical fermions in the
initial and final state. The occupation numbers f 1,2 and the Pauli
blocking factors (1 − f 3,4 ) are for the nucleons while the axions are
assumed to escape freely so that a Bose stimulation factor as well as
backreactions (axion absorption) can be neglected.
In the nonrelativistic limit the nucleon mass m N is much larger than
all other energy scales such as the temperature or Fermi energies. The
nucleon momenta are then much larger than the momentum carried by
the radiation. A typical nonrelativistic nucleon kinetic energy is E kin =
p 2 /2m N so that a typical nucleon momentum is p = (2m N E kin ) 1/2 .
In a bremsstrahlung process, the radiation typically takes the energy
E kin with it, less in a degenerate medium, so that a typical radiation
momentum is k a = E kin ≪ p. Therefore, one may ignore the radiation
in the law of momentum conservation so that Eq. (4.4) is simplified
according to
δ 4 (P 1 + P 2 − P 3 − P 4 − K a ) →
→ δ(E 1 + E 2 − E 3 − E 4 − ω a ) δ 3 (p 1 + p 2 − p 3 − p 4 ). (4.5)
In this case, the second integral expression in Eq. (4.4), which “knows”
about axions only by virtue of the energy-momentum transfer K a in
the δ function, is only a function of the axion energy ω a .
Therefore, in terms of a dimensionless function s(x) of the dimensionless
axion energy x = ω a /T one may write the energy-loss rate in
the form (baryon density n B )
Q a =
(
CN
2f a
) 2
n B Γ σ
∫
= α an B Γ σ T 3
4π m 2 N
∫ ∞
0
d 3 k a
2ω a (2π) 3 ω a s(ω a /T ) e −ωa/T
dx x 2 s(x) e −x . (4.6)
Γ σ will turn out to represent the approximate rate of change of a nucleon
spin under the influence of collisions with other nucleons.
4.2.3 Nondegenerate Limit
To find Γ σ and s(x) turn first to an evaluation of Eq. (4.4) in the
nondegenerate limit. The initial-state nucleon occupation numbers f 1,2
are given by the nonrelativistic Maxwell-Boltzmann distribution f p =
(n B /2) (2π/m N T ) 3/2 e −p2 /2m N T so that ∫ 2f p d 3 p/(2π) 3 = n B gives the
nucleon (baryon) density where the factor 2 is for two spin states. Pauli
blocking factors are omitted: (1 − f 3,4 ) → 1.