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Degenerate Objects

Degenerate Dwarf Stars

The final evolutionary state for an intermediate-mass star is as a degenerate dwarf star, also called a white dwarf. Degenerate dwarfs are corpses, no longer able to generate energy despite being composed of light elements such as carbon and oxygen. They are stars cooling to invisibility. Small in radius—they are comparable to Earth in size despite having masses similar to the Sun's—these stars are dim, even when they are still young and hot. Degenerate dwarfs are the second most numerous stars in the universe, outnumbered only by the low-mass main-sequence stars.

A star becomes degenerate when the pressure at its core is dominated by quantum effects acting through the electrons. A fundamental feature of quantum mechanics is that no more than two electrons can reside in a given energy state; this principle is called the Pauli exclusion principle. Because of it, one cannot have an arbitrary number of free electrons at rest in a star. Instead, in a cold star, one has two electrons at rest, two electrons with a small amount of energy, two more electrons with slightly more energy, on up until all of the available energy levels are filled. Electrons that carry momentum exert a pressure on their surroundings. Because the electrons cannot lose their energy when lower energy levels are filled, the electrons carry momentum, and therefore exert pressure, even when the temperature is zero. This pressure is called degeneracy pressure.

The pressure exerted by a degenerate, cold, non-relativistic gas of electrons is determined solely by the density of the electrons; the pressure rises as the density to the 5/3 power. It is this property that determines the characteristic radius of a degenerate dwarf star. The characteristic radius, which is of order 109 cm for a 1 solar mass star, falling as the mass rises, with the radius proportional to the mass to the −1/3 power.

Degeneracy of the electrons within a solar mass star does not occur at the densities found at the core of the Sun; solar mass stars do not enter a degenerate-electron state until they contract with the depletion of their core hydrogen, and they do not settle into a permanent degenerate-electron state until they consume their helium. But these stars do stop contracting before carbon fusion can commence. These stars are the carbon-oxygen degenerate dwarfs. Single main-sequence stars with masses between about half a solar mass and 8 solar masses end their lives in this state. Stars more massive than 8 solar masses and less massive than about 10 solar masses can contract until carbon-fusion commences. These stars become permanently degenerate once the carbon is depleted, but before oxygen fusion commences, producing the oxygen-neon-magnesium degenerate dwarfs. Anything larger than about 10 exhausts all of its thermonuclear fuel and then collapses to something much more compact than a degenerate dwarf—either a neutron star or a black hole.

The age of our Galaxy has another implication for degenerate dwarfs: only single stars of more than about half a solar mass have had enough time to complete their thermonuclear fusion of hydrogen and become a degenerate dwarf star. All stars less massive than this are still burning hydrogen; they are still on the main sequence. This limited time for stellar evolution means that isolated degenerate dwarfs composed of helium do not yet exist, because only a low mass star will support itself through electron degeneracy before shrinking to a density that supports helium fusion. There do appear to be, however, helium degenerate dwarfs in orbit with other stars, particularly in orbit with millisecond pulsars. These degenerate dwarfs are made rather than grown. Starting life as a large star, the progenitor of one of these degenerate dwarf stars rapidly burns through its hydrogen supply. Once it completes its main-sequence life and transitions to the red-giant phase, the progenitor is stripped of its atmosphere by its companion star. These stars are low mass, less than 0.5 solar masses.

Not too surprising given the evolution outlined above, the mean mass of detected degenerate dwarfs is around 0.6 solar masses. The masses are narrowly dispersed, with a standard deviation from the mean of only 0.14 solar masses. These values reflects the prevalence of small stars over large stars and the inability of very small stars to complete hydrogen fusion in time since stars first formed.

While a degenerate dwarf will eventually cool to the point that it no longer produces visible light, but instead only radiates infrared radiation, this process takes many tens of billions of years. This means that all degenerate dwarfs created in our galaxy are still emitting visible light. Find the least luminous of these stars, and we have a timer for the age of our Galactic disk. Of course, this calculation depends on how well we understand the structure of degenerate dwarfs, so such a study has some uncertainty. But when astronomers search for the dimmest degenerate dwarfs, they do see a low cutoff in the power generated. From this cutoff, an age of between 8 and 14 billion years has been estimated for the age of the Galactic disk.

All of the degenerate dwarfs within 13 parsecs of the Sun are believed to have been found, and they number 46. Astronomers have identified 109 degenerate dwarfs within a 20 parsecs of the Sun, but there remain some degenerate dwarfs within this volume to be found.

The best known white dwarf is Sirius B, the companion star to Sirius, the brightest star in our night sky; despite being bluer, and therefore hotter, than Sirius, Sirius B emits only about 10−4 of the power of its companion main-sequence star. Sirius B has an apparent magnitude of only 8.3, far below the limiting magnitude of the human eye, versus brilliant Sirius, with an apparent magnitude of −1.5. This difference in power output is simply a consequence a difference in surface area between the two stars. Sirius B is the nearest degenerate dwarf to Earth, with a distance of only 2.64 parsecs. It is the hottest and one of the most massive of the nearby degenerate dwarfs, with a mass of slightly more than 1 solar mass.

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