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# The Masses of Degenerate Objects

An astronomical object must satisfy two conditions to be a stable, electron-degenerate body.  First, the object must be dense enough for the electrons to be free rather than bound to atoms.  Second, the object must be tenuous enough that the most energetic electrons have kinetic energies that are much less than the electron rest-mass energy.  These conditions set a lower limit and an upper limit, respectively, on the mass of an electron-degenerate object.  These limits define a mass range that spans the giant gaseous planets, the brown dwarfs, and the degenerate dwarfs.

Within the smaller planets, such as Mars, Earth, and Uranus, pressure is provided by atoms, but within the giant gaseous planetsSaturn and Jupiter—pressure is provided by degenerate free electrons.  When the pressure within an object is able to counter the gravitational field generated by that object, the temperature of the particles providing the pressure becomes equal to the gravitational potential energy of the particles providing the mass.  As the mass of the body increases, this internal temperature rises, and eventually the temperature of the atoms providing the pressure exceeds the binding energy of electrons to the atoms.  When this happens, the electrons in the atoms become free, and the pressure  within the body is provided by the degeneracy pressure of these electrons.

When an object is in an electron-degenerate state, the Fermi energy equals the gravitational potential energy of the atomic nuclei in the body.  By setting the binding energy of an electron to a hydrogen atom equal to the fermi energy of the free electrons, one finds the minimum mass of an electron-degenerate body in terms of several basic physical constants:

Md = 0.17 mp−2 (e2/G)3/2 = 4×1029 g = 0.0002 Ms

In this equation, mp is the mass of the proton, e is the charge of the electron, and G is the gravitational constant.  The last value is the mass in units of solar masses (Ms).  This simple equation, which relates the properties of the electron and proton to the gravitational constant, gives a minimum mass for an electron-degenerate object that is close to Saturn's mass of 5.68×1029 g.

While the binding energy of the electron in a hydrogen atom defines the low-mass end of the mass range for electron-degenerate objects, the transition from a non-relativistic to a relativistic value for the Fermi energy defines the high-mass end of the range.  This transition defines the mass at which an electron-degenerate object becomes unstable to gravitational collapse.  Like the transition from the pressure exerted by atoms to the degeneracy pressure, the transition from stability to instability is set by fundamental constants of physics.  For an object composed of an element with 1 electron for every 2 nucleons (helium, carbon, oxygen, etc.), the transition mass is of order the following:

Mi = 0.05 mp−2 (hc/G)3/2 = 3×1033 g = 1.5 Ms

In this equation, h is the Planck constant and c is the speed of light.  There are several interesting things about this relationship.  First, the mass and charge of the particle providing the degeneracy pressure is absent.  The only mass is the mass of the heaviest particle in the body, the particle that is generating the gravitational field.  This means that this  relationship roughly applies to objects supported by neutron and proton degeneracy pressure as well as electron degeneracy pressure.   For this reason, degenerate dwarfs and neutron stars become unstable to gravitational collapse at of order the same mass, despite being supported by particles—electrons in the former, neutrons and protons in the latter—that differ in mass by a factor of 2000.

This equation assumes that there are two nucleons (protons and neutrons) providing mass for every electron proving pressure.  Degenerate dwarfs, which are composed of carbon and oxygen, satisfy this criterion, and therefore should be unstable for masses above 1.5 solar masses.  A more careful calculation gives the better limit of 1.4 solar masses.  This limit is called the Chandrasekhar limit.

Application of the upper limit estimate to neutron stars is more challenging, because the internal composition is believed to be more complex than simply free neutrons and protons.  First, each nucleon providing mass is also providing a degeneracy pressure, so that there is one particle providing pressure for every particle providing mass; this condition should raise the mass limit for neutron star itstability above the Chandrasekhar limit by a factor of 23/2.  Second, the material within a neutron star may contain other particles besides nucleons.  For instance, one theory hypothesizes that a fundamental particle called a pion contributes to the mass and the pressure of a neutron star.  Other theories introduce other fundamental particles to provide mass and pressure.  Even without these particles, the interactions among protons and neutrons can alter the pressure provided by the degenerate protons and neutrons.  Finally, because neutron stars have radii close to the black hole of the corresponding mass, the effects of general relativity alter the stability of the star.  For these reasons, calculations of the upper mass at which a neutron star becomes unstable range from 1.4 to 2.5 solar masses.  Despite the additional complications, the upper limit on the mass of a neutron star is not much different from the Chandrasekhar limit.

An interesting feature of objects that are supported by electron degeneracy is that the ratio of the most massive objects to the least massive objects is set purely by a single well-known fundamental constant, the fine structure constant, which is given by α = 2πe2/hc = 1/137.04.  The ratio of the upper limt for a body composed of carbon to the lower limit for a body composed of hydrogen is of order the following:

Mi/Md = 2.8 α−3/2 = 4,400.

This value is not too far from the ratio of the Chandrasekhar limit to Saturn's mass, which is 4,900.

The fine structure constant come from the detailed description of how electrons move around the nucleus of an atom.  It is roughly the ratio of the energy of the electron's electric field to the electron's mass.  This constant sets the first-order spacing of the hydrogen's energy states through the formula

En = −me c2 α2/2 n2 = −13.6 eV/n2,

where me is the mass of the electron and n is an integer ranging from 1 to infinity.  The fine structure constant also describes the splitting of these energy states into groups of closely-spaced states, which appear in the light emitted by an atom as multiple closely-spaced spectral lines (the fine structure of the lines in an atom's radiation spectrum) separated in frequency by of order α4.

If α were much smaller than 1/137, then in the microscopic world an electron falling to the hydrogen ground state would emit visible light rather than an ultraviolet light, and in the astronomical world the span in mass of the electron-degenerate bodies would exceed 100,000.   On the other hand, if α were approximately 1, an electron falling to the hydrogen ground state would emit a gamma-ray, and electron-degenerate bodies would cease to exist.   In this way, the observed properties of the hydrogen atom are directly tied to the observed properties of electron-degenerate bodies, so the fine structure constant describes both the detailed character of atomic structure and the gross character of astronomical objects.