Rick's Cosmology Tutorial: Appendix A0 Abstract

Derivation of the Black Body Radiation Formulae for Photons and Relativistic Fermions

In this opening Appendix we derive from first principles formulae for the black body spectrum, energy density, particle density, entropy, radiation pressure and radiation flux. Initially this is done for photons (i.e. for electromagnetic radiation). This derivation is expressed in such a way that it applies to any boson field in the limit that virtually all particles have extreme relativisitic energies (kT >> mc^2), so the particles may be approximated as massless.

The derivation is then repeated for fermions. Again the formulae are developed for the extreme relativistic (or massless) limit. It is shown that the energy density and entropy of a black body fermion field is a factor 7/8 less than that for a boson field (per particle species and per spin state). The particle density for relativistic fermions is a factor 3/4 smaller than that for relativistic bosons (per particle species and per spin state). The derivations are carried out so that the generalisation to non-relativistic energies, when the formulae become dependent upon particle mass, is clear. Closed form expressions are not derived for this general case only because the integrals involved are not conducive to this.

Read Appendix A0 (pdf): Derivation of the Black Body Formulae

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