Electron Degeneracy Pressure

Author:
robdjeff
This graph shows the pressure due to fully degenerate electrons in a white dwarf as a function of gas density (both expressed in logarithmic units - i.e. 11 is equivalent 10^11). The composition of the white dwarf can be varied using the slider which changes μ_e, the "mean mass per electron" in atomic mass units (=1 for H; =2 for He, C,O; = 2.15 for Fe). If you move the red point along the curve you can see the value of the instantaneous slope, which is equivalent to the "polytropic index" in the relationship P=Kρ^α. You can also see the value of the relativity parameter (the Fermi momentum/mc). At high densities, the Fermion (electron) gas becomes increasingly relativistic, because the Fermi momentum, p_F, depends on density^(1/3). As it does so, the polytropic index tends towards 4/3. At low densities we have non-relativistic degeneracy, p_F/mc <<1, and the polytropic index is 5/3. Typical densities in a white dwarf are 10^8-10^14 kg/m^3, so you can see that we traverse regions where neither the non-relativistic or ultra-relativistic approximations will apply! Note also that changing μ_e makes only a small difference.
This plot assumes that: the electrons are completely degenerate; that the pressure due to the positive ions is negligible; and that electrons are non-interacting. The first two of these are usually valid, but the third requires further consideration (inverse β decay and electrostatic corrections being the most important additions). Note that the relationship between P and the density, temperature and composition of a gas is called The Equation of State. The equation of state for a degenerate gas does not depend on temperature!