The hot core is initially supported by the degenerate electrons and the hot nuclei contained in the core. The star follows a well-defined track in the Hertzprung-Russell diagram.

• Initially, the core slowly cools and loses pressure support (because the nuclei are not completely negligible). This causes the core to contract and to get hotter and the bare core moves to the left in the HR diagram.

• After the nuclei cool to the point where they do not contribute much pressure support, the degenerate electron pressure essentially halts the contraction and the core stops heating.

• The hot core now cools without losing pressure support. (This is not strictly true, however, as the nuclei stil contribute a little tiny amount of pressure support which means that pressure still decreases a touch and so the star contracts a touch but does not lead to significant change in the radius of the star; the white dwarf contracts more or less at constant radius The cooling process is exceedingly slow and takes billions or years

• When the white dwarf becomes cool enough, it can crystallize.

• At an arbitrary point when the white dwarf becomes very cold, we declare it to be a black dwarf

• An interesting possibility for white dwarf evolution concerns white dwarfs which are in short orbital period (P ~ hours) binary star systems. Such systems are so small that the white dwarf is actually able to steal material from its companion star. Such binary systems are known as cataclysmic variables.

Properties of White Dwarfs

White dwarfs are the endpoints of the evolution of low mass stars. They are interesting objects in that they are supported by degenerate electron pressure and thus do not need internal nuclear energy sources. White dwarfs radiate because they are born hot and because they slowly contract releasing gravitational energy as they cool.

White dwarfs cannot be more massive than 1.4 M(sun) (Chandrasekhar Limit, see below) and they have radii on the order of the radius of the Earth, R(wd) ~ 10,000 kilometers. Comment -- this means that white dwarfs are extremely dense; densities on the order of 200,000 grams per cc to 100,000,000 grams per cc. Recall that the density of lead is ~ 11 grams per cc. A sugar cube of white dwarf material would weigh anywhere from 400 pounds to 200 tons at the surface of the Earth!

• Mass-Radius Relationship
  • There is a well-defined relationship between the mass of a white dwarf and its radius. The relationship is not intuitive in that
    

    that is, the larger the mass of the white dwarf, the smaller is its radius!!

  • This can be understood by noting the size of the degenerate pressure depends on the density of the gas in the sense that the pressure is greater, the greater the density. That is, the higher the density, the harder it is further compress the white dwarf.

  • If one thinks a little about the mass-radius relationship, a plausible scenario arises. Since the radius of a white dwarf must be small for a massive white dwarf, for a very massive white dwarf, the radius must be tiny. Is there a limit on how long one can keep making the radius of a white dwarf smaller to compensate for an increase in the mass of a white dwarf. Yes, there is a limit. Performing a detailed analysis, once can show that for a white dwarf of mass ~ 1.4 M(sun), the the radius of the white dwarf must be 0 in order for the the degenerate electron pressure to counter-act the force of gravity. Huh. Say what??

    Effectively this means that there is an upper limit to the mass of a white dwarf. The limit is ~ 1.4 M(sun) and is referred to as the Chandrasekar Limit