Stars as Laboratories for Fundamental Physics - MPP Theory Group

212 Chapter 6
Fig. 6.2. Function G(x) according to Eq. (6.41).
The most astonishing observation of Braaten and Segel (1993) is
that Eq. (6.38) is a good approximation for all conditions, not only for
the limiting cases for which it was derived. As the approximation is
much better than 1%, which is the approximate accuracy of an O(α)
result, these representations can be taken to be exact to this order.
6.3.5 Dispersion Relations
In order to determine the photon dispersion relation for specific conditions
one must determine ω P and v ∗ corresponding to the temperature
T and chemical potential µ of the electrons. In Fig. 6.3 contours for
v ∗ and γ ≡ ω P /T are shown in the T -ρ-plane of a plasma. Analytic
limiting cases are (Braaten and Segel 1993)
⎧
⎪⎨ (5T/m e ) 1/2 Classical,
v ∗ = v F
Degenerate,
(6.42)
⎪⎩
1 Relativistic,
⎧
4πα n e
(1 − 5 )
T
Classical,
m e 2 m e
⎪⎨
ωP 2 4πα n
=
e
= 4α
E F 3π p2 Fv F Degenerate,
(6.43)
4α (
⎪⎩ µ 2 + 1 3
3π
π2 T 2) Relativistic,
where v F = p F /E F is the velocity at the Fermi surface, “classical” refers
to the nondegenerate and nonrelativistic limit, and “relativistic” is for
any degree of degeneracy.