Stars as Laboratories for Fundamental Physics - MPP Theory Group

220 Chapter 6
(∂ t J = 0) are not screened. The magnetic field associated with a stationary
current is the same at a distance whether or not the plasma
is present.
Not so for the electric field associated with a charge. In the static
limit one finds
π L (0, k) = 4α π
∫ ∞
0
dp f p p (v + v −1 ). (6.55)
Because this expression does not depend on k it can be identified with
the square of a fixed wave number k S , leading to Poisson’s equation in
the form
(
k 2 + k 2 S
)
Φ(k) = ρ(k). (6.56)
Because for a point-like source this gives a Yukawa potential
Φ(r) ∝ r −1 → r −1 e −k Sr
(6.57)
electric charges are screened for distances exceeding about k −1
S .
Evaluating Eq. (6.55) explicitly in the classical limit reproduces the
well-known Debye screening scale (Debye and Hückel 1923; for a textbook
discussion see Landau and Lifshitz 1958)
k 2 S = k 2 D = 4πα n e
T
= m e
T ω2 P. (6.58)
At this point one recognizes that kD
2 is independent of the electron
mass, in contrast with the plasma frequency ωP. 2 Therefore, it is no
longer justified to ignore the ions or nuclei; they contribute little to
dispersion because of their reduced Thomson scattering amplitude, but
they contribute equally to screening. Therefore, one finds kS 2 = kD 2 + ki
2
with
k 2 i
= 4πα
T
∑
n j Zj 2 , (6.59)
j
where the sum is over all species j with charge Z j e.
Comparing Eq. (6.55) with the corresponding expression for the
plasma frequency Eq. (6.40) reveals that in the relativistic limit (v = 1)
k 2 D = 3ω 2 P. It is clear that apart from a numerical factor they must
be the same because a relativistic plasma has only one natural scale,
namely a typical electron energy.