Stars as Laboratories for Fundamental Physics - MPP Theory Group

The Energy-Loss Argument 19
ing to the phase-space element d 3 p is g ω (e ω/T − 1) −1 d 3 p/(2π) 3 where
g is the number of polarization degrees of freedom and (e ω/T − 1) −1
is the thermal boson occupation number. With an angular integration
d 3 p becomes 4πp 2 dp where p = |p|. Using p = (ω 2 − m 2 ) 1/2 one finds
p dp = ω dω so that
B ω =
g ω 2 (ω 2 − m 2 ) 1/2
. (1.17)
2π 2 e ω/T − 1
The total energy flux is found by integrating over all frequencies,
∫ ∞
F = − 1 ∇T dω β
3 ω l ω ∂ T B ω , (1.18)
m
where ∇B ω = ∂ T B ω ∇T was used with ∂ T = ∂/∂T .
For photons, one usually writes F = −(3κ γ ρ) −1 ∇aT 4 where aT 4
is the total energy density in photons (a = π 2 /15). Together with
Eq. (1.18) this defines the photon Rosseland mean opacity κ γ . For
other bosons one defines a corresponding quantity,
1
κ x ρ ≡ 1 ∫ ∞
4aT 3 m
dω l ω β ω ∂ T B ω . (1.19)
The “exotic” opacity thus defined appears in the stellar structure equation
in the way indicated by Eq. (1.7).
The production and absorption of bosons involves a Bose stimulation
factor. This effect is taken account of by including a factor
(1 − e −ω/T ) under the integral in Eq. (1.19). The “absorptive” opacity
thus derived is usually referred to as the reduced opacity κ ∗ which is
the quantity relevant for energy transfer. In practice, it is not very
different from κ as ω is typically 3T for massless bosons.
In the large-mass limit (m ≫ T ) the reduction factor may be ignored
entirely and one finds to lowest order,
1
κ x ρ = g 15 ( ) m 3
e
−m/T
4π 4 T
∫ ∞
0
dy l ω (y) y e −y , (1.20)
where y ≡ β 2 ωm/2T was used so that the energy of a nonrelativistic
boson is given by ω = m + yT .