Stars as Laboratories for Fundamental Physics - MPP Theory Group

What Have We Learned from SN 1987A 507
mostly with a time later than 0.5−1 s where the outer core has settled
and the shock has begun to escape. The density of the protoneutron
star falls within a thin shell from supranuclear levels to nearly
zero, causing the “photosphere” radius r x of the new particles to be
essentially the radius R ≈ 10 km of the settled compact star. With a
“photosphere” temperature T x of the new objects their luminosity is
4πr 2 σTx
4 with the Stefan-Boltzmann constant σ which is gπ 2 /120 in
natural units with g the effective number of degrees of freedom (2 for
photons). Therefore, one must demand that
T x ∼ < 8 MeV g −1/4 , (13.9)
in order to stay below the total neutrino luminosity of 3×10 52 erg s −1 .
It is nontrivial, however, to determine the temperature T x which
corresponds to about unit optical depth. Following the approach of
Turner (1988) who carried this analysis through for axions one may
assume a simple model for the run of temperature and density above
the settled inner core. A simple power-law ansatz is ρ(r) = ρ R (R/r) n
with the density ρ R = 10 14 g cm −3 at a radius R ≈ 10 km. A plausible
ansatz for the temperature profile is T (r) = T R [ρ(r)/ρ R ] 1/3 with T R
(temperature at radius R) of around 10 MeV. From the opacity κ as a
function of density and temperature one may then calculate the optical
depth as τ(r x ) = ∫ ∞
r x
κρ dr. From the condition τ(r x ) ≈ 2 one can
3
determine the “photosphere” radius r x and thus its temperature T x .
The opacity of axions for a medium of nondegenerate nucleons was
given in Eq. (4.28). One may define τ R ≡ κ R ρ R R so that κρR =
τ R (ρ/ρ R ) 2 (T R /T ) 1/2 where Eq. (4.28) yields τ R = ga 2 3.4×10 16 . Then
one finds for Turner’s model an optical depth τ a at the axion-sphere
temperature T a
τ a = τ R ( 11 6 n − 1) (T a/T R ) 11/2−3/n . (13.10)
Because n is a relatively large number such as 3−7 the criterion τ a ∼ < 2
3
yields τ R ∼ > n (T R /T a ) 6 . With the requirement T a ∼ < 8 MeV and with
T R ≈ 20 MeV as taken by Turner one finds g a ∼ > 2×10 −7 , not in bad
agreement with what one would conclude from the numerical results
shown in Fig. 13.1.
Still, this argument is rather sensitive to the detailed model assumptions
concerning the protoneutron star structure. Also, as axions
contribute to the transfer of energy within the star, a self-consistent
model must take this effect into account. Moreover, for novel fermions
such as r.h. neutrinos one must distinguish carefully between their neutrino
sphere (from where they can escape almost freely) and the deeper