``Dwarfs are very upsetting.""- Stephen Sondheim

Key Concepts

Stars with initial masses M sun willend as white dwarfs. White dwarfs are compact objects supported by degenerate-electronpressure. Newly formed white dwarfs are surrounded by emission nebulaecalled ``planetary nebulae"".(1) Stars with initial masses M sun willend their lives as white dwarfs.The life stages of a low-mass star (one with M sun): Fusion of H into He in the star"s core:main sequence Fusion of H into He in a shell outside the core:red giant Fusion of He to C in the core, H to He in a shell:horizontal branch Fusion of He to C in a shell, H to He in a larger shell:asymptotic giant branch No more fusion: WHITE DWARFA white dwarf is the end state of stars which startwith a mass less than four times that of the Sun.White dwarfs have no internal power source; theysimply radiate away their heat energy into outer space,slowly cooling off like an unplugged iron.When the radii and masses of white dwarfs such as Sirius Bwere first computed, astronomers were flabbergasted. Starswith masses comparable to that of the Sun were scrunched downinto a volume comparable to that of the Earth. What isthe source of the pressure which keeps white dwarfs fromcollapsing under their own strong gravitational force?A white dwarf is supported by a differenttype of pressure (not dependent onthe temperature of the white dwarf):degenerate-electron pressure. (2) White dwarfs are compact objects supported by degenerate-electronpressure.White dwarfs are very small (R = 0.01 Rsun =1 Rearth) comparedto a main sequence star, even though they have masses whichare comparable to that of a main sequence star. Thus, whitedwarfs must be very dense compared to an everyday main sequencestar. The density of a white dwarf is approximatelya ton per cubic centimeter. A teaspoonful of white dwarfstuff would be as massive as an elephant.Under the extreme conditions which prevail within a white dwarf,the laws of quantum mechanics become important.Quantum mechanics is the study of how subatomicparticles (such as electrons, protons, and neutrons) behave.Subatomic particles do not always obey the same laws as largeobjects. Hence, the laws of quantum mechanics sometimes seemcontrary to common sense.One rule of quantum mechanics (known as the Pauli exclusion principle)is this:Two identical electrons, located in the same region of space,cannot have the same velocity.In a dense white dwarf, where the electrons are packedclose to each other, some of the electrons are forcedto have high velocities, and hence provide a high pressure.In a degenerate object such as a white dwarf, the fast-movinghigh-energy electrons provide a pressure which is independent of temperature. Even as the temperatureof a white dwarf falls toward absolute zero, the Pauliexclusion principle demands that the high-velocity electronskeep moving at the same speed. Hence, the pressure exertedby the electrons remainsconstant as the temperature falls. (3) Newly formed white dwarfs are surrounded by planetarynebulae.An asymptotic giant branch star (a red giant star whichis about to run out of fuel) is not very stable. It undergoesthermal pulses during which the outer layers of the star areejected into space. Finally, when the star totally exhastsit fuel supply, its core collapses and heats up. Thecore becomes a very hot white dwarf, with a surfacetemperature of 100,000 Kelvin, or more.The ejected outer layers, heated by thehot new white dwarf, form an emission nebula.An emission nebula ofthis sort - ejected gas which is being excited bya hot white dwarf - is called a planetarynebula. (This confusing name goes back tothe 18th century; viewed through a small telescope,the fuzzy disk of a planetary nebula looks a bit like thefuzzy disk of a planet like Uranus. Viewed with theSpace Telescope, however, planetary nebulae like the``Spirograph Nebula"" and the``Eskimo Nebula"" show a wealth of fine detail.)
The above picture is of the Ring Nebula, aplanetary nebula in the constellation Lyra. (Click on theimage for a higher-resolution version.) The blue lightin the center of the nebula is emitted by ionized helium.In the cooler outer regions of the nebula, the dominantsources of emission are hydrogen and oxygen.The central hot white dwarf is visible as a point oflight in the center of the nebula.(Image credit: N. Lame and R. Pogge )Measuring the Doppler shifts of planetary nebulaereveals that they are expanding. A typical middle-agedplanetary nebula will be about a light year across.A planetary nebula will last for about 50,000 yearsbefore fading into invisibility.After the planetary nebula fades, the white dwarfwill still be visible. White dwarfs shine becausethey are hot; although a white dwarf has no internalpower source, it takes billions of years fora white dwarf to cool down. Thermal energy in theinterior of a white dwarf is carried to the surfaceby conduction, then radiated away.As the temperature T of the white dwarf"s surfacedecreases, the radius R remains constant. (Rememberthe degenerate-electron pressure which supportsa white dwarf is not dependent on T; thus, hydrostaticequilibrium is maintained even as the white dwarf cools.)Since T decreases and R is constant, the luminosityL decreases. The oldest, coldest white dwarfs haveL = 0.0001 Lsun and T = 5000 Kelvin.In the future, the eventual fate of a white dwarfwill be to become a black dwarf (not to beconfused with a black hole). A black dwarf is an extremely coldcompact object supported by degenerate-electron pressure.There is an UPPER LIMIT to the permitted mass of a white dwarf.White dwarfs with larger masses have smaller radii. The pressure within a white dwarf depends onlyon density, not on temperature; to maintain the tremendous pressuresrequired to support a massive white dwarf, the white dwarf musthave a very great density. At a mass of M = 1.4 Msun (amass known as the Chandrasekhar limit, after the man who discovered it),the radius of the white dwarf is squeezed down to nothing, and thedensity shoots up to infinity. In practical terms, this means thata white dwarf more massive than 1.4 solar masses doesn"t havea degenerate-electron pressure large enough to maintain hydrostaticequilibrium.You can"t have a white dwarf more massive than1.4 Msun.Asymptotic giant branch starslose matter into space at a rapid rate.It is possible for fairly massive starsto slim down to below the Chandrasekhar limit by the timethey collapse into white dwarfs. A star with a main sequence massof 4 Msun, for instance, will lose about 2.6 Msuninto outer space, and will end as a 1.4 Msun whitedwarf. Stars which are more massive than about 4 Msun duringtheir main sequence lives will NOT be able to lose enough massto become white dwarfs.

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