Stars as Laboratories for Fundamental Physics - MPP Theory Group

214 Chapter 6
Fig. 6.4. Electromagnetic dispersion relations in the classical limit according
to Eq. (6.45) for v ∗ = (5T/m e ) 1/2 = 0.2. The shaded area indicates the
“width” of ω(k) in the longitudinal case due to Landau damping.
with ω = k directly from Eq. (6.37)
m 2 T = 4α π
∫ ∞
0
p 2
dp f p
E . (6.46)
Limiting cases are
⎧
1 Classical,
[
m 2 ⎪⎨ 3
T
= 1 − 1 − ( ) ] v2 F 1 + vF
log
Degenerate,
ωP
2 2vF
2 2v F 1 − v F
⎪⎩ 3
Relativistic.
2
(6.47)
In Fig. 6.5 (ω 2 − k 2 ) is shown for several values of v ∗ as a function
of k. It is quite apparent how the transverse mass is asymptotically
approached.
The dispersion relation for longitudinal modes is more interesting in
several regards. First, according to Eq. (6.44) the oscillation frequency
is only a function of v ∗ k and so the natural scale for k is ω P /v ∗ . In
Fig. 6.6 I show ω 2 − v 2 ∗k 2 as a function of v ∗ k.