Degenerate matter

Degenerate matter is called degenerate matter when a significant fraction of the pressure comes from the Pauli exclusion principle, which establishes that two fermions cannot have the same quantum numbers.

Depending on the conditions, the degeneracy of different particles can contribute to the pressure of a compact object, such that a white dwarf is sustained by electron degeneracy, while a neutron star does not collapse due to the combined effect of the degenerate neutron pressure and the pressure due to the repulsive part of the strong interaction between baryons.

These restrictions in the quantum states cause the particles to acquire very high moments since they do not have other positions in the phase space where they can be located, it can be said that the gas, unable to occupy more positions, is forced to spread out in space of moments with speed limitation c. Thus, since matter is so compressed, low-energy states are quickly occupied, so many particles have no choice but to place themselves in very energetic states, which entails additional pressure of quantum origin. If matter is sufficiently degenerate, such pressure will dominate by far all other contributions. This pressure is also independent of temperature and only depends on density.

Great densities are needed to reach the degenerate states of matter. For the degeneration of electrons, a density of around 10⁶ g/cm³ will be required, (1000 kg/cm³) for that of neutrons, much more will be required, approximately 10¹⁴ g/cm³ (100,000,000 Tons/cm³).

Mathematical treatment of degeneration

To calculate the number of particles in the fermionic world as a function of their momentum, the Fermi-Dirac distribution will be used (see Fermi-Dirac statistics) as follows:

${displaystyle n(p)dp=2cdot {frac {4pi p^{2d}{h^{3}}}}}{cdot {frac {1}{1+exp left({frac {E_{p}}}{KT}}}}-psi right)}}}}}}}}}}}}}}$

Where n(p) is the number of particles with momentum p. The initial coefficient 2 is the double spin degeneracy of the fermions. The first fraction is the volume of the phase space at a moment differential divided by the volume of a cell in said space. The h³ is Planck's constant cubed which, as has been said, means the volume of those cells in which up to two particles with positive or opposite spins can fit. The last fractional term is the so-called filling factor. K is the Boltzmann constant, T the temperature, E_p the kinetic energy of a particle with momentum < i>p and ψ the degeneracy parameter which is dependent on density and temperature.

• The filling factor indicates the likelihood that a state is full. Its value is between 0 (all empty) and 1 (all full).

• The parameter degeneration indicates the degree of degeneration of particles. If you take large and negative values the matter will be in an ideal gas regime. If it's close to 0 degeneration starts to notice. The material is said to be partially degenerated. If the value is large and positive, the material is highly degenerated. This happens when densities are high or also when temperatures are low.

From this equation we can deduce the integrals of the number of particles, the pressure they exert and the energy they have. These integrals can only be solved analytically when the degeneracy is complete.

${displaystyle n=int _{0}^{infty }n(p)dpqquad P={frac {1}{3}}int _{0}{infty }n(p)v_{p}{pdpqquad U=int _{0}{infty }{p}n(p)d)$

The value of the energy of the particles will depend on the velocity of the particles, that is to say, if there is a relativistic gas or not. In the first case, Einstein's equations will be used and in the second, the classical approximation will be valid. As can be seen, the energy-pressure relationships vary significantly, the pressures obtained with the complete non-relativistic degeneracy being higher. It is logical since relativistic matter is hotter.

• Non-relativist degenerate (NR): ${displaystyle v variable!}{cqquad p=m_{e}vqquad E_{p}={frac {p^{2}}{2m_{e}}}}}}}{qquad U={frac {3}{2}}}$
• Extremely relativistic degenerate (ER): ${displaystyle vsimeq cqquad psimeq m_{e}cqquad E_{p}simeq pcqquad U=3P}$

Typical stars with degeneracy are white dwarfs, brown dwarfs supported by electrons, and neutron stars supported by degenerate neutrons. It is considered that their temperature tends to 0 since they do not have any heat source. We will assume said bodies with a degeneracy parameter tending to +infinity.

Degenerate gases

Degenerate gases are gases composed of fermions such as electrons, protons, and neutrons instead of molecules of ordinary matter. The electron gas in ordinary metals and in the interiors of white dwarfs are two examples. Following the Pauli exclusion principle, there can only be one fermion occupying each quantum state. In a degenerate gas, all quantum states are filled up to the Fermi energy. Most stars are supported against their own gravitation by the normal thermal pressure of the gas, while in white dwarf stars the lift comes from the degeneracy pressure of the electron gas within. In neutron stars, the degenerate particles are neutrons.

A fermion gas in which all quantum states below a certain energy level are full is called a totally degenerate fermion gas. The difference between this energy level and the lowest energy level is known as the Fermi energy.

Electron degeneracy

In an ordinary fermionic gas in which thermal effects dominate, most of the available electron energy levels are unfilled, and electrons are free to move into these states. As particle density increases, electrons progressively fill lower energy states and additional electrons are forced to occupy higher energy states even at low temperatures. Degenerate gases strongly resist further compression because electrons cannot move to already filled lower energy levels due to the Pauli exclusion principle. Since electrons cannot give up energy when going to lower energy states, thermal energy cannot be extracted. However, the momentum of the fermions in the fermion gas creates pressure, called the "degeneracy pressure".

At high densities, matter becomes a degenerate gas when all the electrons are knocked out of its atoms. The core of a star, once the burning of hydrogen in nuclear fusion reactions is stopped, becomes a collection of positively charged ions, mostly helium and carbon nuclei, floating in a sea of electrons, which have been stripped of the cores. The degenerate gas is an almost perfect conductor of heat and does not obey the ordinary gas laws. White dwarfs are luminous not because they generate energy, but because they have trapped a large amount of heat that is gradually radiated away. Normal gas exerts more pressure when it is heated and expanded, but the pressure in a degenerate gas does not depend on temperature. When gas is supercompressed, the particles pack against each other to produce a degenerate gas that behaves more like a solid. In degenerate gases the kinetic energies of the electrons is quite high and the collision rate between electrons and other particles is quite low, so degenerate electrons can travel great distances at speeds approaching the speed of light. Instead of temperature, the pressure in a degenerate gas depends only on the velocity of the degenerate particles; however, adding heat does not increase the speed of most electrons, because they are trapped in fully occupied quantum states. The pressure is increased only by the mass of the particles, which increases the gravitational force that pulls the particles closer to each other. Therefore, the phenomenon is the opposite of what normally occurs in matter, where if the mass of matter is increased, the object becomes larger. In degenerate gas, as the mass increases, the particles are spaced further apart due to gravity (and pressure increases), so the object gets smaller. The degenerate gas can be compressed to very high densities, with typical values on the order of 10,000 kilograms per cubic centimeter.

There is an upper limit to the mass of an electron degenerate object, the Chandrasekhar limit, beyond which electron degeneracy pressure cannot hold the object against collapse. The limit is about 1.44 solar masses for objects with typical compositions expected for white dwarf stars (carbon and oxygen with two baryons per electron). This mass limit is appropriate only for a star supported by an ideal electron degeneracy pressure under Newtonian gravity; in general relativity and with realistic Coulomb corrections, the corresponding mass limit is around 1.38 solar masses. The limit can also change with the chemical composition of the object, since it affects the relationship between mass and number of electrons present. The rotation of the object, which counteracts the gravitational force, also changes the limit for a particular object. Celestial objects below this limit are white dwarf stars, formed by the gradual contraction of the cores of stars that run out of fuel. During this contraction, an electron-degenerate gas forms in the nucleus, providing sufficient degeneracy pressure as it is compressed to resist further collapse. Above this mass limit, a neutron star (supported mainly by neutron degeneracy pressure) or a black hole may instead form.

Neutron degeneracy

Neutron degeneracy is analogous to electron degeneracy and is demonstrated in neutron stars, which are partially supported by the pressure of a degenerate neutron gas. Collapse occurs when the core of a white dwarf exceeds about 1.44 solar masses, which is the Chandrasekhar limit, above which the collapse is not stopped by the pressure of degenerate electrons. As the star collapses, the Fermi energy of the electrons increases to the point where it is energetically favorable for them to combine with protons to produce neutrons (via inverse beta decay, also called electron capture). The result is an extremely compact star composed of nuclear matter, which is predominantly degenerate neutron gas, sometimes called neutronium, with a small mixture of degenerate proton and electron gases.

Neutrons in a degenerate neutron gas are much more widely spaced than electrons in a degenerate electron gas because the most massive neutron has a much shorter matter wavelength at a given energy. In the case of neutron stars and white dwarfs, this phenomenon is compounded by the fact that the pressures inside neutron stars are much higher than those inside white dwarfs. The increase in pressure occurs because the compactness of a neutron star makes the gravitational forces much greater than in a less compact body with a similar mass. The result is a star with a diameter on the order of one thousandth that of a white dwarf.

There is an upper limit for the mass of a neutron-degenerate object, the Tolman-Oppenheimer-Volkoff limit, which is analogous to the Chandrasekhar limit for electron-degenerate objects. The theoretical limit for non-relativistic objects supported by an ideal neutron degeneracy pressure is only 0.75 solar masses; however, with more realistic models that include baryonic interaction, the precise limit is unknown, as it depends on the equation of state of nuclear matter, for which a very precise model is not yet available. Above this limit, a neutron star can collapse into a black hole or other possible dense forms of degenerate matter.

Proton Degeneracy

Sufficiently dense matter containing protons experiences proton degeneracy pressure, similar to the electron degeneracy pressure in electron-degenerate matter: protons confined to a small enough volume have a large uncertainty in their momentum due to the Heisenberg's uncertainty principle. However, since protons are much more massive than electrons, the same momentum represents a much lower velocity for protons than for electrons. As a result, in matter with approximately the same number of protons and electrons, the proton degeneracy pressure is much less than the electron degeneracy pressure, and proton degeneracy is usually modeled as a correction to the equations of state of the electron degenerate matter.

Quark degeneracy

At densities greater than those supported by neutron degeneracy, quark matter is expected to be produced. Several variations of this hypothesis representing degenerate quark states have been proposed. Strange matter is a quark degenerate gas that is often assumed to contain strange quarks in addition to the usual up and down quarks. Colored superconducting materials are quark degenerate gases in which the quarks pair up in a similar way to the Cooper pairing in electrical superconductors. The equations of state for the various proposed degenerate matter forms of quarks vary widely, and are often poorly defined, due to the difficulty of modeling strong force interactions.

Quark-degenerate matter may be present in the cores of neutron stars, depending on the neutron-degenerate matter equations of state. It can also appear in hypothetical quark stars, formed by the collapse of objects above the Tolman-Oppenheimer-Volkoff mass limit for neutron-degenerate objects. The formation of quark degenerate matter in these situations depends on the equations of state of both neutron degenerate matter and quark degenerate matter, both of which are poorly understood. Quark stars are considered an intermediate category between neutron stars and black holes.