Stars as Laboratories for Fundamental Physics - MPP Theory Group

98 Chapter 3
Beginning with the case of scalars, the low-energy cross section is
constant at σ ∗ = 4 3 παα′ /m 2 e, leading to
ϵ scalar = 8αα′
π
Y e T 4
m u m 2 e
= α ′ 5.7×10 29 erg g −1 s −1 Y e T 4 8 , (3.22)
where T 8 = T/10 8 K. Vector bosons carry an extra factor of 2 for their
polarization states.
Turning to pseudoscalars, the low-energy cross section was found to
be σ ∗ = 4 3 (παα′ /m 2 e) (ω/m e ) 2 , i.e. p = 2, so that
ϵ pseudo =
160 αα′
π
Y e T 6
m u m 4 e
= α ′ 3.3×10 27 erg g −1 s −1 Y e T 6 8 . (3.23)
The average energy is ⟨ω⟩ ≈ 5T or ⟨ω⟩/m e ≈ 0.08 T 8 .
Finally, turn to neutrino pair production for which the NR cross
section was given in Eq. (3.13). The energy-loss rate is
ϵ νν = (C 2 V + 5C 2 A) 96 α
π 4
G 2 Fm 6 e
m u
Y e
( T
m e
) 8
= (C 2 V + 5C 2 A) 0.166 erg g −1 s −1 Y e T 8 8 . (3.24)
Because p = 4 for this process, ⟨ω⟩ ≈ 7T . Relativistic corrections become
important at rather low temperatures. For example, at T = 10 8 K
the true emission rate is about 25% smaller than given by Eq. (3.24).
3.2.6 Applying the Energy-Loss Argument
After the derivation of the energy-loss rates it is now a simple matter
to apply the energy-loss argument. In Chapter 2 it was shown that the
most restrictive limits obtain from the properties of globular-cluster
stars; two simple criteria were derived in Sect. 2.5 which amount to the
requirement that a novel energy-loss rate must not exceed 10 erg g −1 s −1
for the typical conditions encountered in the core of a horizontal-branch
star, and in the core of a red giant just before helium ignition which
both have T ≈ 10 8 K.
The Compton process is suppressed by degeneracy effects in a dense
plasma as discussed above—see Eq. (3.18). Therefore, at a fixed temperature
the emissivity per unit mass decreases with increasing density.
Because a red-giant core is nearly two orders of magnitude denser than
the core of an HB star it is enough to apply the argument to the latter
case.