Stars as Laboratories for Fundamental Physics - MPP Theory Group

Characteristics of Stellar Plasmas 601
As a neutron star cools it becomes transparent to neutrinos. In this
case their chemical potential vanishes which yields
µ νe = 0 Free neutrino escape (D.20)
as the other extreme additional condition.
D.2.3
Cold Nuclear Matter
The limit T → 0 relevant for old neutron stars is particularly simple
because it allows one to express the Fermi-Dirac distributions as stepfunctions.
One may express all Fermi momenta in units of the effective
nucleon mass, i.e., x j ≡ p j F/m ∗ N. Then baryon conservation is
x 3 B = x 3 p + x 3 n,
(D.21)
where n j = (p j F) 3 /3π 3 was used, and
x B ≡ (3π 2 n B ) 1/3 /m ∗ N = 0.255 ρ 1/3
14 (m N /m ∗ N), (D.22)
with ρ 14 the baryonic mass density in units of 10 14 g cm −3 .
Because the star is transparent to neutrinos one may use µ νe = 0.
Then the equation of β-equilibrium becomes µ e + µ p − µ n = 0 or
x p + (1 + x 2 p) 1/2 − (1 + x 2 n) 1/2 = 0,
(D.23)
where x e = x p was used from the condition of charge neutrality. This
is easily solved to yield (Shapiro and Teukolsky 1983)
x 2 p =
x 4 n
4 (1 + x 2 n) . (D.24)
This result may be expressed in terms of the usual composition parameters
Y p and Y n which give the number of protons and neutrons per
baryon; Y p + Y n = 1. Then Y n,p = (x n,p /x B ) 3 so that
( xB
Y p =
2
) 3 (1 − Yp ) 2
. (D.25)
[1 + x 2 B(1 − Y p ) 2/3 ]
3/2
When x B ≪ 1 this is Y p = (x B /2) 3 = 2.1×10 −3 ρ 14 (m N /m ∗ N) 3 so that
the proton fraction is small—hence the term “neutron star”—although
the exact Y p for a given density depends sensitively on the nucleon
dispersion relation. For infinite density (x B → ∞) a maximum of
Y p = 1 is reached.
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