Stars as Laboratories for Fundamental Physics - MPP Theory Group

104 Chapter 3
by n e = p 3 F/3π 2 . This also implies that |q| 2 ≈ |p 1 − p 2 | 2 ≈ 2p 2 F(1 − c 12 )
where c 12 is the cosine of the angle between p 1 and p 2 . With these
approximations and the velocity at the Fermi surface β F ≡ p F /E F =
p F /(m 2 e + p 2 F) 1/2 one finds
where
∫
F =
Q = π2 α 2 α ′
15
dΩ2
4π
∫
dΩa
4π
T 4 ( ∑
m 2 e j
n j Z 2 j
)
F, (3.33)
(1 − β 2 F) [2 (1 − c 12 ) − (c 1a − c 2a ) 2 ]
(1 − c 1a β F ) (1 − c 2a β F ) (1 − c 12 )(1 − c 12 + κ 2 )
(3.34)
with κ 2 ≡ k 2 S/2p 2 F.
For a single species of nuclei with charge Ze and atomic weight A
the energy-loss rate per unit mass is
ϵ = π2 α 2 α ′
15
Z 2
A
T 4
F
m u m 2 e
= α ′ 1.08×10 27 erg g −1 −1 Z2
s
A T 8 4 F, (3.35)
where again T 8 = T/10 8 K. Because F is of order unity for all conditions,
the bremsstrahlung rate mostly depends on the temperature and
chemical composition, and ϵ is not suppressed at high density.
Expanding Eq. (3.34) in powers of β F for nonrelativistic or partially
relativistic electrons one finds
F = 2 ( ) [ ( ) 2 + κ
2 2 + 5κ
2 2 + κ
2
3 ln + ln − 2 ]
β
κ 2 15 κ 2 F 2 + O(β 4
3
F).
(3.36)
Therefore, in contrast to the nondegenerate calculation this expression
would diverge in the absence of screening.
Another approximation can be made if one observes that Coulomb
scattering is mostly forward, i.e. the main contribution to the integral
is from c 12 ≈ 1 which implies c 1a ≈ c 2a . With c 2a = c 1a only in the
denominator one obtains
F = 2 ( ) [ ( ) ]
2 + κ
2 2 + 3κ
2 2 + κ
2
3 ln + ln − 1 f(β
κ 2 6 κ 2 F ) (3.37)
with
− 3 (1 − β2 F)
2β 3 F
( ) 1 + βF
. (3.38)
1 − β F
f(β F ) = 3 − 2β2 F
βF
2 ln
The function f(β F ) is 0 at β F = 0 and rises monotonically to 1 for