4.4A. Blackbody radiation
In almost all stellar environments, we may assume a strict thermodynamic equilibrium applies. STE implies the radiation follows the Planck distribution.
Elementary statistical mechanics leads to the following expression for the number of particles of species i in phase-space volume d 3r d 3p :
ni(p) d 3r d 3p = (d 3r d 3p / h3) ∑ j gj {e [−μi + εj + ε(p)] / kT ± 1}−1 .
For isotropic photons g = 2 and the number density in the range dp becomes
nγ(p) dp = (4πp 2dp / h3) 2 {e pc / kT − 1}−1 /cm3 .
The radiation pressure and internal energy become
Prad = (1/3)
∫ nγ(p) pc 4πp2 dp =
aT 4 / 3
Erad =
∫ nγ(p) pc 4πp2 dp =
aT 4 = 3Prad .
Exercise: perform the above integral.
Radiation follows a γ-law equation of state with γ = 4/3, with potentially disasterous results...
We usually work with the frequency ν = cp/h (or wavelength) distribution uν, not the momentum distribution: dν = cdp/h.
uν dν = (8πhν3dν / c3) {e hν / kT − 1}−1 erg cm−3 Hz−1 Hz .