3.5 The pressure of an ultra-relativistic degenerate gas

We can also develop a similar expression for the equation of state for ultra-relativistic particles (i.e. when v ~ c). In this case, the total energy of the gas may be expressed as

Because, in this case, the rest-mass energy is negligible, the kinetic energy per particle is . Furthermore, the kinetic energy per unit volume is (once again) the kinetic energy per particle multiplied by the number of particles per unit volume. So the kinetic energy per unit volume is .

Now, it turns out that the pressure provided by ultra-relativistic particles is 1/3 of the kinetic energy per unit volume, so

Then using Equation (8) we have

The equation of state for ultra-relativistic degenerate electrons may therefore be written as follows:

Notice that the equations of state for degenerate electrons, whether non-relativistic or ultra-relativistic (or, indeed, somewhere in between), show that the pressure is independent of temperature and depends in each case only on the number density of electrons. In both the non-relativistic and ultra-relativistic cases, it is the independence of the pressure from the temperature that helps give a degenerate gas its interesting properties.

Whether the gas of electrons is considered to be non-relativistic or ultra-relativistic, it is clear that white dwarfs are supported against further collapse by electron degeneracy pressure. Notice that the equations of state for a degenerate gas (whether non-relativistic or ultra-relativistic) show that the electron degeneracy pressure does not depend on temperature. This is quite different from a so-called ideal gas (which is a good approximation for most normal gases) where the equation of state is , where n is the particle number density, k_B is Boltzmann’s constant and T is the temperature. This decoupling of pressure and temperature in a degenerate gas is what is responsible for many of the unusual properties of white dwarfs.