Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particles Interacting with Electrons and Baryons 97
initial- and final-state electrons have the same momentum, reducing the
calculation to an average of the Pauli blocking factor over all electrons
F deg = 1 n e
∫
2 d 3 (
)
p 1
1
1 −
, (3.16)
(2π) 3 e (E−µ)/T + 1 e (E−µ)/T + 1
where µ is the electron chemical potential and E 2 = m 2 e + p 2 . Then,
F deg = 1
n e π 2 ∫ ∞
m e
p E dE
e x
(e x + 1) 2 , (3.17)
where x ≡ (E − µ)/T . For degenerate conditions the integrand is
strongly peaked near x = 0 so that one may replace p and E with the
values p F and E F at the Fermi surface (x = 0), and one may extend the
lower limit of integration to −∞. The integral then yields T so that
F deg = 3E F T/p 2 F, (3.18)
where n e = p 3 F/3π 2 was used.
Returning to the nonrelativistic, nondegenerate limit note that the
cross sections are of the form σ = σ ∗ (ω/m e ) p so that
Q = σ ∗n e T p+4
π 2 m p e
∫ ∞
0
dx xp+3
e x − 1 = (p + 3)! ζ p+4 σ ∗ n e T p+4
. (3.19)
π 2
Here, ζ n = ζ(n) is the Riemann zeta function which shall be set equal
to unity. 16 In a medium of mass density ρ the electron density is n e =
Y e ρ/m u where Y e is the electron number fraction per baryon and m u
the atomic mass unit. Therefore, the energy-loss rate per unit mass is
ϵ =
(p + 3)! Y e σ ∗ T p+4
π 2 m u m p . (3.20)
e
The average energy of the photons which are converted into weakly
interacting particles is
⟨ω⟩ = (p + 3) T. (3.21)
For p = 0 one recovers ⟨ω⟩ = 3T for the average energy of blackbody
photons. 17
16 ζ n → 1 rather quickly with increasing n; for example ζ 4 = π 4 /90 ≈ 1.082.
Therefore, the error is small if one takes ζ n = 1 which corresponds to using a
Maxwell-Boltzmann rather than a Bose-Einstein distribution. They differ at small
ω where an effective photon mass in the plasma should be taken into account anyway.
Therefore, at the crude level of accuracy where one uses massless photons nothing
is gained by using the Bose-Einstein distribution.
17 With a Bose-Einstein distribution it is ⟨ω⟩/T = 3 ζ 4 /ζ 3 = π 4 /30ζ 3 ≈ 2.70.
m p e