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Degenerate Objects

Degeneracy Pressure

The size of an astronomical object has a floor.  It is defined by planets, brown dwarfs, and degenerate dwarfs for objects with less than 1.4 solar masses.  For objects between about 1.4 and 2.5 solar masses, a much lower floor is defined by neutron stars.  Above about 2.5 solar masses, the floor is set by black holes.  The scales of fundamental physics define these three floors.  The first is set by the quantum mechanics of the electron, the second is set by the quantum mechanics of the proton and neutron, and the third is set by gravity and the speed of light.

Quantum mechanics sets the scale of a large planet or a degenerate dwarf through its fracturing of an electron's kinetic energy into discrete states and its imposition of an exclusionary rule in populating these states with electrons.  These two features of quantum mechanics are why atoms have distinctive chemical properties.

The basic features of quantum mechanics are most clearly displayed by the hydrogen atom.  An electron bound to a hydrogen atom has discrete values of kinetic energy of −13.6 eV/n2, where the integer n ranges from 1 to infinity;  the negative sign expresses the fact that the energy states are bound states.  This quantization of kinetic energy reflects a quantization of the characteristic distance of the electron from the atomic nucleus.  For low values of n, the electron is close to the nucleus, and the energy states available to the electron are far apart.  For high values of n, the electron is far from the nucleus, and the energy states are close together.  At very large values of n, the states are so close together that their separation in energy are unmeasurable; the motion of the electron around an atom in these states is accurately described by Newton physics.  The lowest energy state is the ground state (n = 1), while the remaining states are excited states, since they are at higher energies than the ground state, and an electron in one of them often jumps from it to the ground state, emitting a photon in the process that carries away the excess energy of the excited state over the ground state.

The radial states of the hydrogen atom are further divided into angular momentum states.  The ground state for hydrogen has 1 angular momentum state; the first excited state has 4 angular momentum states.  The number of angular momentum states increases as n2.  In the hydrogen atom, all angular momentum state for a given value of n have very nearly the same energy—there are slight shifts in energy from the equation given above among the different angular momentum states because of special relativity, the spin of the electron, and other weak effects.  In other atoms, however, the inner electrons can counteract the charge of the atomic nucleus seen by the outer electrons, giving different angular momentum states of a given radial state n much different energies.

The Pauli exclusion principle states that only two electrons can simultaneously occupy a given quantum state.  This means that the ground state of an atom can contain no more than 2 electrons, and the first excited level can contain no more than 8 electrons.  In a cool environment, the electrons fill all of the lowest energy states, and this creates an ordered sequence of filling specific radial and angular momentum states as one moves from elements with a small number of electrons to those with a large number of electrons.  For instance, helium, with its 2 electrons, fills its ground state, while neon, which has 10 electrons,  fills its ground state with 2 electrons and its first excited radial state with 8 electrons.

The presence of quantum states that hold only 2 electrons carries over to electrons moving freely through space.  The energy states of the unbound electrons are quantized, with the number of quantum states per unit volume below a given value of the kinetic energy rising as that energy to the 3/2 power.  Because of the Pauli exclusion principle, no more than 2 electrons can occupy any one of these energy states.  Usually the energy states below a given energy far outnumber the electrons.  For example, within the core of the Sun, which have a temperature of 15 million degrees Kelvin, there are about 1026 energy states per cubic centimeter below 15 million degrees Kelvin available for occupation by free electrons, but there are less than 1024 electrons per cubic centimeter.  In such a state, the quantization of energy has no effect on the pressure exerted by the electrons; the pressure is simply proportional to the temperature times the density.  At a lower temperature or a higher density, however, all of the states become filled.  For instance, a degenerate dwarf like Sirius B, with its density of nearly 1030 electrons per cubic centimeter, needs an internal temperature below about 6 billion degrees Kelvin for all of the lowest energy states to be filled, which is the case.

Three types of astronomical object are supported by electron degeneracy pressure: giant gaseous planets, brown dwarfs, and degenerate dwarfs, listed in order from low mass to high mass.  Electrons, however, are not the only source of degeneracy pressure; the Pauli exclusion principle applies to a broad class of particles called fermions.  Among the fermions are electrons, neutrons, and protons.  Normally, neutrons and many protons are locked within the nuclei of atoms heavier than hydrogen, but at very high density, they are freed, just as electrons are freed from atoms at very high densities.  In this state, the neutrons and protons exert a degeneracy pressure.  The only objects supported by nucleon degeneracy pressure are the neutron stars.

The energy at which the density of lower-energy states equals the number density of fermions is called the Fermi energy, and its value is set by the density of fermions.  When the fermions have a temperature that is below the Fermi energy, they are said to be degenerate, and the majority of the states below the Fermi energy are  filled with fermions.  When fermions have zero temperature, they are fully degenerate, and they fill all of the quantum states below the Fermi energy and none of the states above it.  To compress a fully-degenerate gas, which raises the Fermi energy, some fermions must be moved to higher, unfilled energy states, which requires one to put energy into the gas.  This need for energy under compression is why a degenerate electron gas exerts a pressure despite having zero temperature: exerting a force against this pressure to compress a gas causes work to be done on the gas, which places the energy into the gas required to lift fermions to higher energy levels.  The pressure exerted by a fully-degenerate gas is called degeneracy pressure.

The dependence of degeneracy pressure on fermion density is generally simple, although it changes as the Fermi energy approaches the fermion rest-mass energy.  At low densities, the highest energy fermions have kinetic energies far smaller than the fermion rest-mass energy, so all the fermions are described by non-relativistic quantum mechanics.  In this state, the pressure is proportional to the density to the five-thirds power (ρ5/3).  As the density increases, however, and fermions are shoved into higher energy states, and the most energetic fermions are relativistic, having kinetic energies that are larger than the fermion rest-mass energy.  At extremely high densities, when most of the fermions are relativistic, the pressure becomes proportional to the density to the four-thirds power (ρ4/3).

This transition in the dependence of pressure on density at a high Fermi energy may seem trivial, since a change of 1/3 is rather small, but it has a profound effect on the astronomical objects in the universe.  The reason is that this small change makes a degenerate object unstable to gravitational collapse.  This instability occurs when a degenerate object becomes too massive, and the gravitational force drives the degenerate material into the relativistic regime.  The onset of instability is why electron degeneracy can only support bodies of less than 1.4 solar masses, why degenerate dwarfs that pull gas from companion stars can become unstable, collapse, and explode, and why neutron stars larger than 2.5 solar masses collapse to black holes.

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