Stars as Laboratories for Fundamental Physics - MPP Theory Group

216 Chapter 6
are
k1
2
ωP
2
⎧
1 + 3T/m e Classical,
⎪⎨ [ ( ) ]
3 1 1 + vF
=
log − 1 Degenerate,
vF
⎪⎩
2 2v F 1 − v F
∞
Relativistic.
(6.49)
Then, for k > k 1 the four-momentum is space-like, ω 2 − k 2 < 0.
As discussed in Sect. 6.2.2 there is nothing wrong with a space-like
four-momentum of an excitation. In media with electron resonances
such as water or air even (transverse) photons exhibit this behavior
which allows kinematically for their Cherenkov emission e → eγ or
absorption γe → e. In a plasma, transverse excitations are always
time-like and thus cannot be Cherenkov absorbed. Their lowest-order
damping mechanism is Thomson scattering γe → eγ which is not included
because it is an O(α 2 ) effect. Longitudinal excitations with
k > k 1 , in contrast, can and will be Cherenkov absorbed by the ambient
electrons, leading to an O(α) damping rate. It corresponds to an
imaginary part of the dispersion relation (an imaginary part of π L ).
In the expression Eq. (6.36) this damping effect corresponds to a
vanishing denominator, essentially to P ·K = 0, which occurs when the
intermediate electron in Compton scattering “goes on-shell.” Evidently,
P · K = Eω − p · k can never vanish for k < ω while for k > ω there are
always some electrons, even in a nonrelativistic plasma, which satisfy
this condition. When the phase velocity ω/k becomes of the order of
a typical thermal velocity the number of electrons which match the
Cherenkov condition becomes large, and then the damping of plasmons
becomes strong. Because v ∗ measures a typical electron velocity this
occurs for k > ∼ ω/v ∗ (Fig. 6.4). Therefore, while nothing dramatic
happens where the dispersion relation crosses the light cone, it fizzles
out near the “electron cone.” For k > ∼ ω/v ∗ there are no organized
oscillations of the electrons—longitudinal modes no longer exist.
This damping mechanism of plasma waves was first discussed by
Landau (1946) and is named after him. A calculation in terms of Cherenkov
absorption was performed by Tsytovich (1961). In the classical
limit the Landau damping rate (the imaginary part of the frequency) is
Γ L
ω P
=
√ π
8
(
kD
k
) 3
e − k2 D
2k 2
= √ π
( 5
2
) 3/2 ( ) ωP 3
− 5
e 2
v ∗ k
ω 2 P
v 2 ∗k 2 , (6.50)
where k D = 4πα n e /T is the Debye screening scale. (Note that ω 2 P/k 2 D =
T/m e = v 2 ∗/5.) For a given wave number a plasmon must be viewed