Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
3.2 Our supernova model
3.2.1 The density profile
Here we present the same description that is used in [25]. One can assume that at
sufficiently large radius above the heating regime there is hydrostatic equilibrium
[29]:
dP
= −GMNSρ
dr r2 , (3.14)
where P is the hydrostatic pressure, G is Newton‘s constant, MNS is the mass of
the hot proto-neutron star, and ρ is the matter density. Using the thermodynamic
relation for the entropy at constant chemical potential, µ,
δP
Stotal = , (3.15)
δT µ
and integrating Eq. (3.14) we can write entropy per baryon, S, as
TS = GMNSmB
, (3.16)
r
where mB is the average mass of one baryon, which we take to be the nucleon
mass. The entropy per baryon can be written in the relativistic limit since the
material in the region above the neutron star is radiation dominated:
S
k
= 2π2
45
gs
ρB
kT
c
where the statistical weight factor is given by
gs =
gb + 7
8
bosons
fermions
3
, (3.17)
gf. (3.18)
Assuming a constant entropy per baryon, Eqs. (3.16) and (3.18) give the baryon
density, ρB, in units of 10 3 g cm −3 as
ρB ∼ 38
gs
11/2
1
S4 100r3 , (3.19)
7
where S100 is the entropy per baryon in units of 100 times Boltzmann‘s constant,
r7 is the distance from the center in units of 10 7 cm, and we assumed that
MNS = 1.4 M⊙. This density going as 1/r 3 , is the profile used in our numerical
calculations for the first two works (see chapters 4 and 6). Note that the third
work includes a temporally evolving density profile that includes shock-waves. In
Figure 3.2, we present the matter density profile. Several values of S100 can be
used to describe stages in the supernova evolution. Smaller entropies per baryon,
S100 0.5, provide a better description of shock re-heating epoch, while larger
values, S100 1, describe late times in supernova evolution.
58