Degenerate Objects

The Neutrino Cooling of Degenerate Dwarfs

Among the giant gaseous planets, the lightest are the coolest.  Saturn has had time to cool to full electron degeneracy over its 4.5 billion year life, but Jupiter has not.  The brown dwarfs, having many times the mass of Jupiter, are far from full electron degeneracy.  One would think that this trend of  the more massive electron degenerate objects being farther from full electron degeneracy would continue to the degenerate (white) dwarfs, but this is not the case.  The degenerate dwarfs, including very young white dwarfs like Sirius B, are generally close to full degeneracy.  The reason is that they have a source of cooling that is not available to the lighter electron degenerate objects; the degenerate dwarfs can radiate way energy as neutrinos.

Neutrinos are ghost-like particles that pass almost unseen through the universe.  They are created in the thermonuclear reactions within the stars, and they escape freely from the stars into space.  The neutrinos created by the proton-proton nuclear reactions within the Sun escape freely into space, carrying away about 3% of the nuclear power generated at the Sun's core.  The Earth is bathed by these neutrinos, most of which pass untouched through the Earth.

Most commonly, neutrinos arise in interactions involving a force called the weak force and a type of particle called the lepton.  There are three lepton particles: the electron, the muon, and the tau.  Each of these leptons has its own associated neutrino: the electron neutrino, the mu neutrino, and the tau neutrino.  Within a star, electron neutrinos are created in thermonuclear reactions and when unstable atomic nuclei undergo beta decay.  For instance, a neutrino is released when two hydrogen atoms combine to form deuterium.  A neutrino is also released when the isotope boron-8 undergoes beta decay to beryllium-8.  The weakness of the interaction involving neutrinos and leptons causes these fusion and beta decay reactions to be slow, and it causes the interaction between neutrinos and matter to be weak, enabling the neutrinos to freely escape most astronomical bodies, including degenerate dwarfs.

Neutrinos freely escape the interior of all electron degenerate objects, so whatever energy a neutrino is born with is the energy that is carry away into space.  The question of whether neutron emission can cool the interior of an electron degenerate object is then a question of whether the object can generate a large numbers of neutrinos.  Stars, of course, generate some neutrinos through thermonuclear processes and beta decay, and these neutrinos carry away only a small fraction of the nuclear power generated by a star; most of the power released by a star is radiated away at the photosphere.  This is also the case with giant gaseous planets and brown dwarfs.  With no thermonuclear fusion taking place in the giant gaseous planets, and only a minimal amount of thermonuclear fusion occurring in brown dwarfs, neutrinos are not being created through fusion processes, and nearly all of the energy at the cores of these objects is transported to their photospheres and released as light into space.

This picture changes dramatically, however, for the degenerate dwarfs.  While thermonuclear fusion and beta decay are the principal processes in fusion-powered stars creating neutrinos, many other processes are also creating them.^[1]  Within white dwarfs, neutrinos are predominately created from photons through a process called the plasmon neutrino process.^[2]  In this process, a single photon that is in a plasma decays into a neutrino and an antineutrino?the antiparticle of the neutrino:

In this reaction, * represents, in the jargon of theorists working on white dwarfs, a ?plasmon,? which is a photon moving through a plasma, is an electron neutrino, and ^? (in non-standard notation) is an electron antineutrino.

This process is exotic, but it relies on a very common physical fact: when light moves through a material, it interacts with that material in a way that slows its motion, as though the photon has acquired a small mass.  This slowed motion is the reason that a light ray is bent by a lens; the light moves more slowly in glass than in air, so a light wave that passes through the thick center of the lens lags behind the light wave that passes through the thin outer edges of the lens, causing the light wave to have a concave shape that converges to a point.  The reason the light acts as though it has a small mass is that the photon gives some of its energy to the material it is moving through; for a photon moving through a plasma, this energy is carried by the free electrons of the plasma.

The interesting point for degenerate dwarfs is that a photon in a plasma?a plasmon?is unstable in the same way that a radioactive particle is unstable; a plasmon can decay into a particle and its antiparticle.  This is only possible because it has an effective rest mass.  A photon in a vacuum has no mass, and it cannot decay into two particles, because the decay would violate the conservation of energy and momentum.  By having a mass, the plasma photon can decay into two particles without violating the conservation laws.  While the decay can be to a wide variety of particles, the important decay in a degenerate dwarf is to a neutrino and an antineutrino.  These particles escape the interior of the degenerate dwarf, carrying away the energy that was in the plasmon.

Two factors affect the rate at which neutrinos are radiated by a degenerate dwarf: the density and the temperature of the core.  The density sets the effective mass of the plasmon.  The mass is proportional to the square root of the density (technically, the plasmon mass is the electron plasma frequency times the Planck constant), so as the density increases, plasmons become more massive, and they decay into more energetic neutrinos and antineutrinos.  The temperature, on the other hand, sets the density of photons in the plasma, with the plasmon density rising rapidly as the temperature rises above the plasmon mass.  Because the density and Fermi energy of a degenerate dwarf is much higher than those of a brown dwarf or a giant gaseous planet, a white dwarf that is not fully degenerate generates a strong neutrino flux, while a giant gaseous planet or a brown dwarf in the same state generates a negligible neutrino flux.  White dwarfs are therefore able to cool to full degeneracy in a much shorter time than can their lighter cousins.

This cooling from the inside through neutrino emission gives the degenerate dwarfs an unusual thermal structure; the core loses energy through neutrino emission faster than the outer layers can cool by emitting light, so the core becomes cooler than the outer layers.  Unlike a main-sequence star, the energy in a white dwarf flows from the outer layers to the cooler core, where neutrinos are created and carry the energy into space.  For this reason, a degenerate dwarf cools much faster than one would expect from its surface radiation.  The real power of a degenerate dwarf is hidden from sight.