Study Astronomy Online at Swinburne University
All material is © Swinburne University of Technology except where indicated.
Degenerate matter is matter whose properties are determined by quantum mechanics. This matter can behave very differently to the matter that we are familiar with; you can call degenerate matter the counterpart of matter that respects the ideal gas law. Matter becomes degenerate under the enormous pressures of very dense stellar remnants.
If the core of a star is compressed, the atoms and the electrons in the core get closer and closer to each other. Normally such a compression happens if the temperature in the core decreases and as a consequence the thermal pressure that counteracts the force of gravity decreases too. But the electrons cannot get closer to each other than the laws of quantum mechanics allow. According to these laws it is not possible that more than two electrons in the same atom can have the same quantum state. This means that if matter is compressed to a certain degree, the Pauli Exclusion Principle generates a counter pressure called degeneracy pressure which prevents further contraction of the matter. Due to this degeneracy pressure the matter can support a much higher pressure than it would be able to support without degeneracy. One important consequence is that the pressure of degenerate matter does not depend on temperature any more as is the case for normal matter when increasing its temperature causes a gas to also increase its pressure (see ideal gas law). Amongst others this fact plays a crucial role with supernova Type Ia. Now, by further increasing the pressure on degenerate matter there is still no further compression due to the Pauli Exclusion Principle, but instead the speed with which the electrons move increases. This way the pressure on degenerate matter can still be increased without collapse of the matter. Anyhow, if pressure is increased to a certain point (the Chandrasekhar limit) the electrons approach the speed of light. There is nothing in this universe that can reach or exceed the speed of light. At this point degeneracy pressure is no longer able to support the pressure of the matter and the matter suddenly collapses. In astronomy this happens with core collapse supernovae (Type II, Type Ib and Type Ic supernova).
The next "stop" of the matter is a neutron star. Most of the electrons and protons of the progenitor star have been literally pressed into each other by the enormous pressure, forming neutrons. You can imagine the neutron star as a gigantic nucleus consisting of almost nothing but neutrons. It is extremely compact. To understand this better, imagine the size of a normal atom to be as big as a football stadium. In such an atom the nucleus would be the size of a grain of rice at its centre. All the other space would be empty with just the even smaller electrons flying around in it. A neutron star is so compact because the former atom (or football stadium) is now squeezed into the size of just the nucleus (or rice grain) itself. In effect, the neutron star still consists of single atomic nuclei, but packed together almost seamlessly.
Now, the Pauli Exclusion Principle is not only valid for electrons but for any particles called fermions (which are all half integer spin-particles like protons, neutrons, muons and others). So, instead of electron degeneracy the neutron star is held up against collapse from neutron degeneracy with the main difference that the neutron degeneracy pressure is much higher. The same Pauli Exclusion Principle applies; a neutron must occupy its own quantum state (or space) and cannot be compressed further. If we apply even higher pressure again by adding more matter to the neutron star, this leads to higher speeds of the vibrating neutrons. If the mass of a neutron star reaches 3 solar masses the speed of the neutrons again reaches the speed of light and the neutron star cannot support its own weight anymore. This limit is called the Tolman-Oppenheimer-Volkhof limit; it's analogous to the Chandrasekhar limit of electron degenerate matter. The neutron star collapses to a theoretical object called a quark star. These quark stars have not been discovered and are only a hypothesis at this time.
You can read more about the formation of neutron stars and black holes in our supernova article; more information about a neutron star can be found in the corresponding neutron star article on Sun.org.
All text and articles published by Sun.org are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
The test to see whether degeneracy pressure is going to be significant is to compare $kT$ with the Fermi energy $E_F$
The Fermi energy is the energy level up to which all energy states would be occupied in a completely degenerate fermion gas. It is given by (for non-relativistic conditions) $$ E_F = \frac{h^2}{2m}\left(\frac{3}{8\pi}\right)^{2/3} n^{2/3},$$ where $m$ and $n$ are the mass and number density of the fermions (in this case, electrons).
If $E_F \gg kT$, then the gas can be considered completely degenerate and the electrons exert a non-relativistic degeneracy pressure. If $E_F \sim 10 kT$ the electrons are partially degenerate. In both these circumstances, the pressure exerted by the electrons is much higher than they would have in a perfect gas, because a large fraction of electrons occupy high energy (and momentum) states.
Degeneracy can be achieved either by having a low temperature or a high number density of fermions. In a white dwarf, electrons are completely degenerate because the number density of electrons is extremely high. In a gas giant, the electron density is nowhere near as high, but then the temperature is a lot lower and so they reach a state of partial degeneracy.
The pressure of a partially degenerate gas is higher than that of a perfect gas at the same density and temperature. More importantly, the pressure is only very weakly dependent on temperature. As a gas giant radiates away its potential energy, it is able to do so whilst only contracting very slightly and the core does not become hot enough to initiate nuclear fusion.
Note that degeneracy pressure and thermal pressure are not two different things. Both arise merely as a consequence of particles having a distribution of momenta and obeying (in this case) Fermi-Dirac statistics to determine how they occupy the possible energy states. The expression $P = nkT$ is simply a convenient approximation for fermions that holds only when $E_F \ll kT$ and the quantum nature of the particles is not apparent.
In a degenerate gas of fermions, the fermions fully occupy momentum states from zero up to a momentum corresponding to the Fermi energy. It is the momentum of the fermions that leads to degeneracy pressure.
As long as the kinetic energy of particles at the Fermi energy is much less than $kT$, then the fermions can be considered completely degenerate, so that the situation above applies and there are no fermions that occupy energy states higher than the Fermi energy. The Fermi energy only depends on the density of fermions.
The pressure is given by the following integral $$ P = \frac{1}{3}\int g(p) F(p) v\ \mathrm{d}p,$$ where $g(p) = 8\pi p^2/h^3$ is the density of available momentum states, $F(p)$ is the occupation number of those states, and $v$ is the particle velocity. For a degenerate gas, the integral is easy because $F(p)=1$ up to the Fermi momentum and zero thereafter. What this means is that temperature does not feature in the integrand or in its limits. Therefore the pressure is independent of temperature.
If the gas is heated (for instance, nuclear fusion reactions are present), then initially the temperature can rise with no increase in the pressure. It is not until $kT$ approaches the Fermi energy that a significant number of energy states above the Fermi energy become occupied and the pressure becomes temperature dependent again.
The work done to compress a gas is $PdV$, whether it is degenerate or not. For a given density of particles, the pressure of a degenerate gas is lower than that of a perfect gas. So from that point of view it is easier to compress. On the other hand if heat can escape from the gas and the compression can be done isothermally, then the pressure of a perfect gas increases with density, but the pressure of a (non-relativistic) gas increases as density to the power 5/3, so is harder to compress.
In compact stars, the significance is that a gas can collapse and then cool and settle into a degenerate state at high density and thereafter maintain a constant pressure. This means that white dwarfs and neutron stars can cool without shrinking. In the cores of low-mass stars, or in white dwarfs, these properties mean that fusion reactions can ignite in an explosive, runaway fashion, since the nuclear reaction rates are highly temperature dependent, but the degeneracy pressure does not respond to an increasing pressure.
EDIT: I feel I need to refine this answer in the light of comments and an answer contributed by Ken G in Why is the release of energy during the He-flash in stars almost explosive?
The answer to your headline question actually ought to be that in isolation, degenerate gases do expand if you add enough heat to them. However, the point is that by the time you have added enough heat to make them expand significantly then they can no longer be considered degenerate gases. This is because the heat capacity of a degenerate gas is very small, so for a given amount of added heat, the temperature can increase enormously, hence lifting the degeneracy as explained above.
In white dwarfs and the cores of low-mass stars, this is prevented from happening initially because the electrons that provide most of the pressure are not the only species present. Most of the heat energy from thermonuclear reactions is actually deposited in the non-degenerate ions. These however make only a small contribution to the total pressure and hence the electrons remain degenerate and the gas does not expand significantly·