white dwarf - Formation, Characteristics, Mass and radius relationship

A small, dim star in the final stages of its evolution. The masses of known white dwarfs do not exceed 1·4 solar masses. They are defunct stars, collapsed to about the diameter of the Earth, at which stage they stabilize, with their electrons forming a degenerate gas, the pressure of which is sufficient to balance gravitational force.

A white dwarf is an astronomical object which is produced when a low or medium mass star dies. The white dwarf is supported only by electron degeneracy pressure. The maximum mass of a white dwarf, beyond which degeneracy pressure can no longer support it, is about 1.4 solar masses. A white dwarf which approaches this limit (known as the Chandrasekhar limit), typically by mass transfer from a companion star, may explode as a Type Ia supernova via a process known as carbon detonation.

Eventually, over hundreds of billions of years, white dwarfs will cool to temperatures at which they are no longer visible. However, over the universe's lifetime to the present (about 13.7 billion years) even the oldest white dwarfs still radiate at temperatures of a few thousand kelvins.

As a class, white dwarfs are fairly common; they comprise roughly 6% of all stars in the solar neighborhood.(RECONS estimate)

Formation

Almost all small and medium-size stars will end up as white dwarfs, after nearly all the hydrogen they contain has been fused into helium.

A typical white dwarf has half the mass of the Sun yet is only slightly bigger than Earth; this makes white dwarfs one of the densest forms of matter (10), surpassed only by neutron stars, black holes and hypothetical quark stars. The higher the mass of the white dwarf, the smaller the size. There is an upper limit to the mass of a white dwarf, the Chandrasekhar limit (about 1.4 times the mass of the Sun). Carbon-oxygen white dwarfs avoid this fate by undergoing a runaway nuclear fusion reaction (leading to a Type Ia supernova explosion) prior to reaching the limiting mass.

Despite this limit, most stars end their lives as white dwarfs since they tend to eject most of their mass into space before the final collapse (often with spectacular results—see planetary nebula). It is thought that even stars eight times as massive as the Sun will in the end die as white dwarfs, cooling gradually to become black dwarfs.

Characteristics

Many white dwarfs are approximately the size of the Earth, typically 100 times smaller in diameter than the Sun; their average mass is about 0.5-0.6 solar masses, though there is quite a bit of variation.(see link for discussion) Their compactness implies that the same amount of matter is packed in a volume that is typically 1003 = 1,000,000 times smaller than the Sun and so the average density of matter in white dwarfs is 1,000,000 times greater than the average density of the Sun. for instance, white dwarfs grow smaller—and thus their densities increase—with higher mass (see "further reading"). In the 1930s this was explained as a quantum mechanical effect: the weight of the white dwarf is supported by the pressure of electrons (electron degeneracy), which only depends on density and not on temperature. Few stars are in the low-brightness-hot-color region (the white dwarfs), but most stars follow a strip, called the main sequence. They look red and are called red dwarfs or (even cooler) brown dwarfs.

Most white dwarf stars are extremely hot; This heat is a remnant of that generated from the star's collapse, and is not being replenished (unless the white dwarf accretes matter from other nearby stars). However, since white dwarfs have an extremely small surface area from which to radiate this heat, they remain hot for a long period of time.

Eventually, a white dwarf will cool into a black dwarf. no black dwarfs are thought to exist, and the coolest white dwarfs found have surface temperatures around 3900 K.(see below) and the cooling is slower as it progresses. A white dwarf may cool from 20,000K to 5,000K in the same amount of time it takes to cool from 5,000K to 4,000K. In all, a 0.5 solar mass white dwarf starting at 20,000K would require approximately 25 billion years to cool to ambient.

Many nearby, young white dwarfs have been detected as sources of soft X-rays (i.e.

White dwarfs cannot independently exceed 1.4 solar masses (the Chandrasekhar limit). Most white dwarfs form with a mass close to 0.6 solar masses, but there is a working method to get them close to this limit. White dwarfs in binary systems can steadily accrete material from a companion star. If the accreted material were to push the mass of the white dwarf beyond the 1.4 solar mass limit, degeneracy pressure would no longer support the star, and collapse would ensue. The fusion reaction is unregulated because the white dwarf is supported against gravity by quantum degeneracy pressure, not by thermal pressure. Initiation of fusion thus increases the temperature of the star's interior without increasing the pressure, so the white dwarf does not expand and cool in response.

When accretion does not push the white dwarf close to the Chandrasekhar limit, hydrogen-rich accretion material on the surface may still light up in a thermonuclear explosion. In general, binary systems with a white dwarf accreting matter from a companion are called cataclysmic variables.

Mass and radius relationship

To find a relationship between the mass of a white dwarf and its radius, one can start from the hydrostatic equilibrium condition:

where is the rate of change in pressure as a function of radius G is the gravitational constant M is the mass inside a specific radius, r ρ is the density as a function of radius

This derivation will show that higher-mass white dwarfs will have a smaller radius. First, one makes the very rough estimation of an average constant density, given by the mass of the white dwarf divided by its volume:

Putting that into the hydrostatic equilibrium equation and then integrating, one obtains an equation for pressure inside the center of the star to be:

Now, for a degenerate gas (which is what makes up a white dwarf), pressure is also proportional to density by:

So setting these two equations of pressure proportional:

Now this is a relationship between mass of a white dwarf to its radius. So, drop all the constants to see it more clearly:

After this rough derivation, what has been shown is that as mass of a white dwarf increases, its radius decreases. As more white dwarfs were found, astronomers began to discover that white dwarfs are common in our galaxy. In 1917 Adriaan Van Maanen discovered Van Maanen's Star, the second known white dwarf.

After the discovery of quantum mechanics in the 1920's, an explanation for the density of white dwarfs was found in 1926. 81-82) in an article called "The maximum mass of ideal white dwarfs" that no white dwarf can be more massive than about 1.4 solar masses.

NASA's Spitzer Space Telescope has recently spotted what may be comet dust sprinkled around the white dwarf star G29-38, which died approximately 500 million years ago.