Stars as Laboratories for Fundamental Physics - MPP Theory Group

130 Chapter 4
4.3.2 Bremsstrahlung Emission of Neutrino Pairs
In order to calculate the neutrino pair emission rate explicitly turn
first to the nondegenerate limit for which one uses Γ σ and s(x) given
in Eq. (4.7) and the integrals of Tab. 4.1. The average energy of a
neutrino pair is (s 4 /s 3 ) T ≈ 4.36 T and the total energy-loss rate is
Q ND
νν = 2048
385 π 7/2 C2 AG 2 Fα 2 π
n 2 BT 11/2
,
m 5/2
N
ϵ ND
νν = 2.4×10 17 erg g −1 s −1 ρ 15 T 5.5
MeV, (4.23)
with ρ 15 = ρ/10 15 g cm −3 , T MeV = T/MeV, and ϵ ND
νν
= Q ND
νν /ρ is the
energy-loss rate per unit mass.
For the degenerate rate one uses Eqs. (4.9). The average energy of
a neutrino pair is (s 4 /s 3 ) T ≈ 5.78 T and the total energy-loss rate is 21
Q D νν = 41π
4725 C2 A G 2 F α 2 π p F T 8 ,
ϵ D νν = 4.4×10 13 erg g −1 s −1 ρ −2/3
15 T 8 MeV. (4.24)
One can correct for a nonvanishing value of the pion mass by virtue of
Eq. (4.12).
For a mixture of protons and neutrons the same remarks as in
Sect. 4.2.6 apply. Apart from a small correction the neutrino coupling
is isovector (C p A ≈ −CA) n so that α 0 ≈ 0 while the other α’s are
approximately equal (Appendix B). For degenerate conditions, with a
small modification the dependence on the proton concentration is the
same as that shown in Fig. 4.4. Again, the absolutely dominating contribution
is from np collisions unless protons are so rare that they are
nondegenerate. 22
21 Friman and Maxwell’s (1979) total energy-loss rate is 2/3 of the one found here.
Apparently they did not include the crossterm in the squared matrix element, i.e.
the third term in Eq. (4.2).
22 This conclusion, based on the work of Brinkmann and Turner (1988), is in
conflict with the results of Friman and Maxwell (1979). They found that Q νν was
proportional to the proton Fermi momentum which is relatively small in neutronstar
matter. On the other hand, for small proton concentrations Brinkmann and
Turner’s p F in Eq. (4.24) approaches the neutron Fermi momentum. I am in no
position to decide between these conflicting results.