Stars as Laboratories for Fundamental Physics - MPP Theory Group

512 Chapter 13
rate Γ σ which, in the nondegenerate limit, was given in Eq. (4.7) on the
basis of a perturbative one-pion exchange (OPE) calculation. For the
protoneutron star model displayed in Fig. 13.2 the profile of this Γ σ /T
is shown in Fig. 13.5. In Fig. 4.8 the axion emission rate was shown as a
function of Γ σ , revealing that for the conditions of interest one is in the
neighborhood of the maximum of the solid curve. In a realistic nuclear
medium, the true spin fluctuation rate may be smaller than the OPE
calculated value, taking one perhaps somewhat to the left of the maximum.
Therefore, the true axion emission rate corresponds to the naive
one (dashed line in Fig. 4.8) at Γ σ /T ≈ 3−5 which at temperatures
around 30 MeV corresponds to around 20% nuclear density.
Fig. 13.5. Profile for the nondegenerate spin-fluctuation rate Γ σ of Eq. (4.7)
in the protoneutron star model S2BH 0 of Keil, Janka, and Raffelt (1995)
shown in Fig. 13.2.
Given the overall uncertainties involved in this discussion it is best
to derive a plausible limit on g a by the analytic criterion Eq. (13.8). The
relevant average temperature is about 30 MeV for which the maximum
emission rate corresponds to the naive one at about 5×10 13 g cm −3 .
From Eq. (4.8) one finds an approximate axion energy-loss rate of
g 2 a 1×10 38 erg g −1 s −1 . Then Eq. (13.8) indicates that one needs to require
g a ∼ < 3×10 −10 , about a factor of 3 less restrictive than the nominal
bound from the numerical calculations above. Altogether one may
adopt
3×10 −10 ∼ < g a ∼ < 3×10 −7 (13.11)
as a range excluded by the SN 1987A cooling-time argument.