Stars as Laboratories for Fundamental Physics - MPP Theory Group

596 Appendix D
A Fermi gas becomes degenerate when a typical thermal (kinetic)
energy is on the order of ˆµ. Therefore, the degeneracy parameter
η ≡ ˆµ/T = (µ − m)/T
(D.10)
is frequently used to characterize the fermions. They are degenerate
for η larger than a few, and nondegenerate for η < 0. For electrons,
contours of η in the ρ-T -plane are shown in Fig. D.4. A comparison
with the loci of the stellar models of Fig. D.1 reveals that electrons in
stellar plasmas are often at least partially degenerate.
D.1.4
Plasma Frequency
Other characteristic properties of a plasma refer to the behavior of
electromagnetic waves. The photon dispersion relation (Sect. 7.4) is
characterized by the plasma frequency which at T = 0 is
ω 2 0 = 4πα n e /E F = (4α/3π) p 3 F/E F .
(D.11)
Nonrelativistically, it takes on the familiar form ω 2 0 = 4πα n e /m e . Numerically
it is
ω 0 = 28.7 eV
(Y e ρ) 1/2
, (D.12)
[1 + (1.019×10 −6 Y e ρ) 2/3 ]
1/4
where the mass density ρ is in units of g cm −3 .
D.1.5
Screening Scale
An electric test charge will be screened by the polarization of the
plasma. If the plasma is weakly coupled (see below) the screened
Coulomb potential takes the form of a Yukawa potential r −1 e −k S r where
k S is the screening scale; its inverse is the screening radius. The plasma
is polarized by the test charge because the positive constituents of the
plasma are repelled while the negative ones are attracted, or the reverse.
Therefore, both electrons and ions contribute to screening. If
both are nondegenerate the total contribution is kS 2 = kD 2 + ki
2 where
the electron contribution is known as the Debye scale,
k 2 D = 4παn e /T = (4α/3π) p 3 F/T.
(D.13)
The ions (charge Z j e, atomic weight A j ) contribute
k 2 i
= 4πα
T
∑
j
Z 2 j n j = 4πα
T
ρ
m u
∑
j
X j Z 2 j
A j
. (D.14)