Stars as Laboratories for Fundamental Physics - MPP Theory Group

124 Chapter 4
The total energy-loss rate is
Q D a
= α a α 2 π
31π 2
945
p F T 6
,
m 2 N
ϵ D a = α a 1.74×10 31 erg g −1 s −1 ρ −2/3
15 T 6 MeV, (4.10)
(Iwamoto 1984; Brinkmann and Turner 1988).
The degenerate and nondegenerate rates are best compared in terms
of a parameter ξ ≡ p 2 F/(2π m N T ) which approaches η/π in the degenerate
limit (degeneracy parameter η),
Q D a
Q ND
a
= 31π4
1536 √ 2 ξ−5/2 ≈ 1.39 ξ −5/2 . (4.11)
Therefore, they are equal for ξ ≈ 1, or a degeneracy parameter of
η ≈ 3.5. This defines the dividing line between the regimes where these
approximations can be reasonably used.
In the degenerate limit the nucleon phase-space integrals can be
done analytically with the inclusion of a nonzero m π . The m π = 0 rates
must be supplemented with a factor 20 (Ishizuka and Yoshimura 1990)
G(u) = 1 − 5u ( ) 2
6 arctan u 2
+
u 3(u 2 + 4) +
+
u 2
( √ )
2
6 √ 2u 2 + 4 arctan 2u2 + 4
, (4.12)
u 2
where u = m π /p F . For only one species of nucleons (as approximately
in a neutron star) p F = 515 MeV ρ 1/3
15 so that u = 0.26 ρ −1/3
15 with ρ 15
the mass density in 10 15 g/cm 3 . For this case G is shown as a function
of ρ in Fig. 4.3 (solid line).
4.2.5 Bremsstrahlung Emission of Scalars
The previous results equally apply to pseudoscalars with a coupling
ig aN ψ N γ 5 ψ N ϕ with α a ≡ g 2 aN/4π if a derivative pion-nucleon interaction
is used (Sect. 14.2.3). However, for scalars which couple according
20 Eq. (4.12) differs from the corresponding result of Friman and Maxwell (1979)
which is identical with that of Iwamoto (1984) who apparently did not take the
third term of the matrix element Eq. (4.2) properly into account.