Outline

4. Equation of State and Related Descriptions

From Hansen and Kawaler.

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) pcp2 dp = aT 4 / 3
Erad = ∫ nγ(p) pcp2 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 = (8π3 / c3) {e / kT − 1}−1 erg cm−3 Hz−1 Hz .