White Dwarfs
White dwarfs are a class of stars that are approximately the same size as the Earth but have roughly the same mass as our Sun. They are about a thousand times dimmer than our Sun, although their surface temperatures are about twice the Sun's. Though they are called "white," the light emitted by the stars is actually a bluish color. A white dwarf is a stage of evolution in a star's life cycle commonly known as the star's "death," because the star is cooling and will eventually stop radiating light energy. Scientists estimate that about 10% of the stars in our galaxy are white dwarfs.
One of the earliest discovered and most famous white dwarfs is Sirius B, which is the binary companion to the bright star Sirius. By observing the orbit of Sirius and using Kepler's Laws, the mass of Sirius B was determined to be between 0.75 and 0.95 solar masses, and its brightness was determined to be about 1/360 of that of the Sun. Its temperature and radius were also estimated in 1914 by looking at its radiation spectrum. However, it was not until 1926, after the invention of quantum mechanics and Fermi-Dirac statistics, that scientists began to understand why Sirius B and other white dwarfs did not collapse.
In a "normal" star like our Sun, gravitational forces compress gases (mostly hydrogen) until nuclear fusion reactions begin to take place. Visible light and heat are emitted in the fusion process, and these exert an outward pressure. Eventually, the star settles down into an equilibrium state where the pressure due to gravity is equal to the radiation pressure, and it stays relatively stable until its nuclear fuel is exhausted.
Once the nuclear fuel is used up, there is no more radiation pressure and the gravitational forces compress the star further. The core of the star starts to collapse and the outer shell is blown off into space, forming a planetary nebula. Then a quantum-mechanical effect comes into play, called the Pauli exclusion principle, which says that no two spin-1/2 particles (such as electrons) may have the same quantum numbers (such as energy, momentum, and spin direction). The exclusion principle causes an outward pressure which counteracts the pressure due to gravity, and halting the collapse of the core, thus stabilizing the white dwarf.
As the white dwarf cools, the atoms radiate their remaining thermal energy as blackbody radiation until their energy becomes close to the Fermi energy. At this point they stop radiating and simply contribute to the stability of the white dwarf. The calculated time scale for a white dwarf to stop radiating completely is 109 years, so the process happens very slowly, which explains why we can still observe white dwarfs in the galaxy today.
One of the interesting features of white dwarfs is that they have a maximum mass, first discovered in 1930 by Chandrasekhar and later called the Chandrasekhar limit. In the case where the Fermi energy of the electrons is much greater than their mass-energy, we must treat the problem using special relativity. The Fermi energy only depends on the number of electrons and the size of the star in this case, and as a result, there is a maximum number of electrons that allows the white dwarf to be stable. Multiplying this maximum number by the mass of a nucleon and the number of nucleons per electron (since the nucleons are so much heavier than the electrons) yields a rough limit of 1.5 solar masses. For the case of a Fermi energy much lower than the electron mass-energy, the mass of the star will always be less than a solar mass. Therefore, all white dwarf stars must have a mass less than the Chandrasekhar limit of about 1.5 solar masses.
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