3 Degeneracy

Before considering the physics of white dwarfs, let’s pause to look at the concept of degeneracy. In this context, degeneracy means that there is more than one quantum state with the same energy.

In quantum mechanics, all particles display wave-like behaviour. In particular, any particle has a characteristic wavelength, known as the de Broglie wavelength, defined by

where h is Planck’s constant ( or ) and p is the particle’s momentum.

Now, inside a star, a particle’s kinetic energy can be written as , where T is the temperature and k_B is Boltzmann’s constant (). Another expression for the kinetic energy of a particle (with mass m and speed v) is simply . So equating these two we obtain or . (Strictly, particles exist with a range of speeds, but is close to the average.) The momentum of a particle is given by , so . Hence the de Broglie wavelength of a particle may be written as

Degeneracy becomes relevant when particles are packed so closely together that their separation is similar to their de Broglie wavelength. It turns out to be important at various stages in the lives of stars. Firstly, it determines the lower mass limit for stars: below a certain mass (about 0.075 times that of the Sun) matter in the star’s core becomes degenerate as the star collapses and hydrogen fusion cannot begin. Such a star will instead become a brown dwarf. Secondly, degeneracy also determines how helium fusion begins: for stars with a mass below about 2.25 times that of the Sun, after hydrogen fusion finishes in the core, matter in the star’s core becomes degenerate and helium fusion begins explosively in an event called a helium flash. Both these phenomena are therefore due to degeneracy. This concept will now be explored in detail as we consider the compact remnants left behind at the end of a star’s life.