White Dwarf

When the triple-alpha process in a red giant star is complete, those evolving from stars less than 4 solar masses do not have enough energy to ignite the carbon fusion process. They collapse, moving down and to the left of the main sequence until their collapse is halted by the pressure arising from electron degeneracy. An interesting example of a white dwarf is Sirius-B, shown in comparison with the Earth's size below. The sun is expected to follow the indicated pattern to the white dwarf stage.

1 teaspoon of a white dwarf would weigh 5 tons. A white dwarf with solar mass would be about the size of the Earth.

At left may be a future white dwarf in Helix Nebula. At right is hot white dwarf NGC2440. Both are surrounded by "cocoons" of the gas they ejected in their collapse toward the white dwarf stage.

Sirius-B

The white dwarf Sirius-B was not seen until 1862, but was predicted in 1844 from the motion of Sirius-A. The blackbody spectrum of Sirius-B peaks at 110 nm, corresponding to a temperature of 26,000 K. From the known absolute magnitude, the radius is calculated to be just 4200 km. Smaller than the Earth, it is almost as massive as the Sun.

Electron Degeneracy

Electron degeneracy is a stellar application of the Pauli Exclusion Principle, as is neutron degeneracy. No two electrons can occupy identical states, even under the pressure of a collapsing star of several solar masses. For stellar masses less than about 1.44 solar masses, the energy from the gravitational collapse is not sufficient to produce the neutrons of a neutron star, so the collapse is halted by electron degeneracy to form white dwarfs. This maximum mass for a white dwarf is called the Chandrasekhar limit. As the star contracts, all the lowest electron energy levels are filled and the electrons are forced into higher and higher energy levels, filling the lowest unoccupied energy levels. This creates an effective pressure which prevents further gravitational collapse.

Sirius-B gives an example of the size of a white dwarf. Electron degeneracy halts the collapse of this star at the white dwarf stage. Though comparable in mass to the Sun, its white dwarf stage is smaller than the Earth.

Sirius-A

The star Sirius, referred to as Sirius-A, is perhaps most notable for the study of the "companion of Sirius" or Sirius-B which was the first example of a white dwarf star to be studied. Sirius itself is one of the brightest stars in the sky, being only 8.6 light-years away from us.

It is also notable for being the subject of one of the first serious studies of the carbon cycle of nuclear fusion. It is much hotter than our Sun and it was clear that some process other than proton-proton fusion was taking place to produce all that energy.

The Chandrasekhar Limit for White Dwarfs

The calculation of the maximum mass of 1.44 solar masses for a white dwarf was done by Subrahmanyan Chandrasekhar on a ship on the way from India to England to begin graduate study in physics at Cambridge University! This initial calculation was done when he was only 20 and carefully refined by the time he was 22! The naming of the limit for its discoverer seems particularly appropriate in light of the intense personal story which surrounds it. Chandrasekhar was interested in the final states of collapsed stars as determined by electron degeneracy and had used the work of Arthur S. Eddington and Ralph H. Fowler to begin his calculations. He realized that they hadn't included relativity in their calculations. When he revised their equations to include relativity, he found that above a certain limit there was no solution. This implied that for masses above 1.44 solar masses there could be no balance between electron degeneracy and the crushing gravitational force and that the star would continue to collapse.

The poignancy of the situation for this young, essentially self-taught, physicist was that Eddington strongly resisted his ideas for years! Eddington's public and vocal opposition made Chandrasekhar's life so difficult that at age 29 he wrote a definitive book on the subject of stellar structure, determined to close that subject and pursue other interests. In the process, he produced a work which defined the subject for years afterward and is regarded as a classic.

To Eddington's credit, he later acknowledged the value and correctness of Chandrasekhar's work as he wrote about the remarkable white dwarf Sirius-B: "The message of the Companion of Sirius when it was decoded ran:'I am composed of material 3,000 times denser than anything you have come across; a ton of my material would be a little nugget that you could put in a matchbox.' What reply can one make to such a message? The reply that most of us made in 1914 was - 'Shut up. Don't talk nonsense.'"

Chandrasekhar himself had no idea what would happen when the limit of 1.44 solar masses was exceeded, except that the star would continue to collapse. Our present understanding is that the collapse will continue until it is stopped by neutron degeneracy with the formation of a neutron star. But even that is not the ultimate limit, since neutron degeneracy can also be overcome by masses greater than 3 solar masses and the ultimate collapse is toward a black hole.