Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 129
vector available from which a spatial tensor can be constructed. Thus,
the structure function has the most general form
S ij (ω) = S σ (ω) δ ij . (4.20)
For nonrelativistic nucleons all axion or axial-vector neutrino processes
involve only one scalar function S σ (ω) of the energy transfer.
The contraction of δ ij with K a µ Ka ν for axion emission yields k 2 a =
ωa.
2 (The spatial Kronecker δ should be viewed as a Lorentz tensor
with a zero in the 00 position.) Therefore, S µν K a µ Ka ν → ωaS 2 σ (−ω a )
in Eq. (4.19). For neutrino emission, K 1 · K 2 = ω 1 ω 2 − k 1 · k 2 and
the contraction of δ ij with g µν is −3. Therefore, the contraction of
δ ij with N µν yields 8ω 1 ω 2 (3 − cos θ) where θ is the angle between the
ν and ν momenta. The neutrino phase-space integration will always
average cos θ to zero so that it may be dropped. Therefore, S µν N µν →
24 ω 1 ω 2 S σ (−ω 1 −ω 2 ) in Eq. (4.19). One may then immediately perform
the integration over one of the neutrino energies and is left with an
integration over the energy transfer.
Thus, in the nonrelativistic limit the energy-loss rates Eq. (4.19) are
of the form
( ) C
N 2 ∫
Q νν = A G
√ F n ∞ B
dω ω 6 S 2 20π 4 σ (−ω),
0
Q a =
( ) 2 ∫ CN n ∞ B
2f a 4π 2 0
dω ω 4 S σ (−ω). (4.21)
The medium properties are embodied in a common function which may
be expressed in the form
S σ (ω) = Γ σ
ω 2 s(ω/T ) × { 1 for ω > 0,
e ω/T for ω < 0.
(4.22)
Because of the “detailed-balance relationship” between positive and
negative energy transfers to be discussed more fully below, s(x) must
be an even function. In the nondegenerate and degenerate limits Γ σ
and s(x) have been determined above. They allow one to calculate the
νν emission rate without any effort.