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Probably all stars with initial masses up to about eight solar masses finally  end up as white dwarfs. Stars with more than about 8 solar masses explode as type II supernovae after a lifetime of only a few million years and become neutron stars or black holes. 90% of all stars finally become white dwarfs when  their nuclear energy generation has ceased. With typically 0.6 solar masses and  radii of about 109 cm (0.01 solar radii) the mean densities of white dwarfs are  of the order of 105-106 g/cm3 so that these stars can be considered as  laboratories for matter at extreme densities and pressures. White dwarfs are  stabilized against a gravitational collapse by the pressure of the degenerate electron gas in the interior. If the central density rho rises because of an  increasing mass of the star, the electrons become relativistic, since the Pauli  exclusion principle forces the majority of the electrons to reach relativistic  velocities, even at low temperatures; by this effect the equation of state becomes "soft" (P~ρ4/3). The consequence is that no white dwarfs above the Chandrasekhar limiting mass of about 1.4 solar masses can be stabilized against  a gravitational collapse.
Investigation of white dwarfs provide important constraints on the theory of stellar evolution. E.g. the existence of white dwarfs in young stellar clusters  (with turn-off masses of about 5 solar masses ) show that massive stars must loose the bulk of their matter during their lifetime until they become a white dwarf with about 1.15 solar masses. The majority of white dwarfs have masses of about 0.6 solar masses and stems from progenitors with about 2 solar masses. The  mass-loss processes during the stellar evolution is not well understood; the  largest quantity is lost on the asymptotic giant branch (AGB) when the energy in  a star is produced by alternating ignitions of a hydrogen and helium burning shell ("thermal pulses") in the interior. At the end of this phase about 0.1-0.3  solar mases are lost when the outer layers are blown away; this matter  subsequently becomes visible as a planetary nebula. The nebula emits light since it is ionized by the hot (typically 50-200 kK) remnant. This central star of a  planetary nebula is the direct progenitor of a white dwarf. After about 30000  years the nebula becomes invisible and the white dwarf cools down before they eventually become invisible. The coolest white dwarfs known have an age of about  9-10 billion years; the existence of these stars is a measure for the age of the  disc of our galaxy.
With the exception of very cool objects the equation of state in the atmospheres of white dwarfs is the one for an ideal gas. For a white dwarf with an effective temperature of 25000K the particle densities and the temperatures range from 1015-1017 cm10-3 and 15-50 kK, respectively.
The chemical composition of white dwarfs is rather peculiar compared to main sequence stars: Normally only one element shows up in the optical spectra. The  reason for this almost mono-elemental composition has been known for a long  time. In 1949 Schatzman has shown that the huge gravitational acceleration  (typically 108 cm sec-2) together with the electric field leads to a downward diffusion of the heavier elements on time scales which are rather short compared to the evolutionary time scale.
The stars with spectral type DA are defined by the fact that their optical spectra exhibit the Balmer lines of hydrogen only. About 80% of all known white  dwarfs belong to this class. Nearly all atmospheres of the non-DAs are dominated by helium. Depending on the effective temperature He II (spectral type DO, 45-120000K) or HeI (type DB, 11-28000K) lines are the strongest. On the cool end of the cooling sequence (below 11000K) the high energy needed to excite spectral  lines leads to a continuous spectrum in the optical (type DC). In some stars with helium atmospheres, lines of carbon (type DQ) or other metals (Ca, Mg, Fe:  spectral type DZ) show up in the spectra, probably due to accretion from the interstellar medium.
While many details and physical processes in the atmospheres of white dwarfs are well understood, the most obvious observational fact, namely that the white dwarfs are basically divided into two major groups --- the DAs and the non-DAs  ---, is still unexplained. One possible explanation is that the spectral type is  determined by the pre-white dwarf evolution. It has been suggested that the  exact phase, when the star leaves the AGB, or a late thermal pulse in a post-AGB  star determines whether significant amounts of hydrogen are left over in the  atmosphere or not.
It is, however, very certain that transitions between the DA and the non-DA  sequence must occur: between 28000 K and 45000 K not a single non-DA star exists. Since there is no physical process known by which the effective temperature of a DO with 45000 K should jump down to 28000 K the DO stars must become hydrogen rich  at about 45kK for some, yet unknown reason.