Friday, January 25, 2013

Degenerate Matter: White Dwarfs

This post deals with white dwarfs and their composition. For an introduction to Degenerate Matter, see here.

Stellar remnants, which no longer produce the huge amounts of energy necessary to counteract the compressive force of gravity, succumb to it. A star becomes a stellar remnant when nuclear fusion, and therefore energy output, ceases. At what point this occurs depends on the size of the star.

For the smallest stars, fusion ends when all of the hydrogen is consumed and converted into helium.

The composition of a low-mass star.

The core has become rich in helium after the completion of hydrogen fusion. When the outer layers are shed, a so-called white dwarf of pure helium remains. It is estimated that, for stars with less than .5 solar masses, this type of white dwarf is the eventual fate. Such stars, during their lifetime, never achieve the temperature necessary (100 million Kelvin) to form elements heavier than helium. However, such small stars also burn their hydrogen fuel very slowly, so such a star would have a lifetime greater than 13.7 billion years, the current age of the Universe. As such, none of this type of dwarf should be found in the Universe. Despite this, some have been observed, and it is proposed that such objects form when another body sucks the outer hydrogen layers from a low-mass star, making it too small to continue fusion and abruptly ending its lifetime.

However, a slightly heavier star, such as our Sun, is able to fuse helium near the end of its life, forming heavier elements such as carbon and oxygen. The white dwarf that will be the end stage of our Sun's lifetime will be composed mainly of these elements.

The size of a white dwarf also depends on its mass, but not in the expected fashion. More massive white dwarfs actually have smaller radii. To be more precise, the cube of the radius varies inversely with the mass, or, for some proportionality constant k, R3 = k/M.

A mass-radius diagram for a white dwarf (click to enlarge).

A Fermi gas is a gas composed of particles that exert degeneracy pressure. The name distinguishes it from an ideal gas; models assuming gases to be ideal ignore particle size and atomic forces. In the case of a white dwarf, the electrons exert the degeneracy pressure, and they are the particles of the Fermi gas. The curve labeled "Non-relativistic Fermi gas" models the inverse cube law that relates the mass and radius. A typical white dwarf (from the mass-radius chart), therefore has a density hundreds of thousands of times greater than that of Earth.

At first, it is nearly coincident with the actual variation, that modeled by relativity (the green curve). The gradual, and then rapid, disassociation of these curves stems from the velocity of the electrons. By Heisenburg's uncertainty principle, the velocity of the electrons that cause the degeneracy pressure increases as the radius of the white dwarf decreases, due to the increased certainty of position. However, as the speed of the electrons approaches c, the speed of light, the theory of relativity predicts that their mass will actually increase. This, from a relativistic frame of reference, augments the mass of the white dwarf and causes it to contract further before stabilizing due to increased degeneracy pressure.

However, this can only continue up to a certain point before the relativistic model would spiral out of control: the added degeneracy pressure caused by the contraction of the white dwarf would not be enough to combat the increased velocity, and therefore mass, of the electrons. The contraction would then be self-reinforcing, and the radius would shrink to zero, as illustrated on the graph.

Therefore, there is an upper limit on white dwarf mass, known as the Chandrasekhar Mass Limit (marked CM on the diagram), equal to about 1.44 solar masses. Some white dwarfs have been observed as being more massive, but this may be due to another factor: spin. If a white dwarf is rotating, the centrifugal force is added to the degeneracy pressure to combat the force of gravity. Theoretically, a rapid spin could allow a white dwarf to even exceed 2 solar masses.

Such an event is somewhat rare though, and in the case of a parent star over 8 solar masses, the remnant is usually too massive to become a white dwarf. In this event, however, the stellar remnant does not simply contract to nothing; it becomes another type of degenerate matter, which will be discussed in the next post.

Sources: White Dwarf at Wikipedia,,

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