White Dwarf (wd) Stars

History:
F. Bessell (1844) proper motions of Sirius and Procyon exhibit a wobble indicating they are astrometric binaries. But the secondaries are unseen, perhaps "dark stars".
Alvin Clark (1861) spots the companion to sirius, Sirius B, at Dearborn Observatory. Procyon B was spotted in 1895 at Lick Observatory.
The spectrum of another wd, 40 Eri B, was observed indicating spectral class A so T>9000K indicated R<0.01Ro. Then Adams (1915) found Sirius B is also an A star with M=Mo, R=Re - "white dwarf"
Eddington realized the Gas must be fully ionized so nuceli could acheive such density but since the star can't expand against gravity it is --"too hot to cool down"-- the atoms cannot recombine as the star cools.
Fowler (1926) recognized the role of "degeneracy pressure".
Chandrasekhar (1935) discovers upper limit to mass supported by electron degeneracy, 1.4Mo, due to limit of velocity of light.
Zwicky (1935) investigates neutron degeneracy, neutron stars.
Hewish and Bell (1967) discover the crab pulsar.

WD progenitors

WD are made by stars in the mass range 0.08-8Mo
(Mass unit is the solar mass, Mo, 2×1033kg.) The formation of a WD is the natural end of a chain of events starting with the collapse of an interstellar cloud that results in a compact cloud of gas or protostar with a mass in excess of about 0.08Mo but less than 5-9Mo. Objects with mass less than 0.08 never reach a central temperature and density sufficient to start hydrogen fusion, the signature of stardom. Objects above the 5-9Mo limit will produce cores that exceed the Chandrasekhar limit and produce a supernova with a neutron star or black hole remnant.
Presumed history of single star with 0.5≤M≤8
Lifetime 12 billion years over mass to the 2.6 power
Hayashi track to Zams Protostar, Gravitational contraction, convective
ZAMS H ignites in core. Star is homogeneous.
Zams to core H depletion   Main sequence, 4H→He (pp, CNO)
H Shell burning Red giant. He core contracts.
He core mass increases Thick H burning shell, slowly contracting He core
Sch�nberg-Chandrasekhar limit Core contracts rapidly, H shell burns hotter
Envelope expands - red giant phase
He ignition Helium flash if original M<3Mo
(He core expands,H shell slows, 3α may shut off)
If M>2.3Mo He burning resumes when nondegenerate core is 0.1M
If M<2.3Mo degenerate core must grow to 0.4 or 0.5M before He core flash,
the core expands, removes degeneracy, and continues to burn:
He core+H shell He burning main sequence/ Horizontal branch
He core depletion Carbon core, C(α,γ)O produced
He shell + H shell Asymptotic giant branch
Thermal pulses, rapid mass loss
S-process? dredge-ups? most C, O? in galaxy produced here
Central star planetary nebula
Cooling white dwarf, mostly Carbon and Oxygen.
Final mass is about 0.34+0.14M(initial mass)
wd may have a thin layer of helium (M<0.01Mo, about 250km) and a thinner layer of hydrogen (M<0.0001Mo, about 30km)

Some models: M=Mo, R=0.016Ro= 11,300km=1.745Re

If Te= 5777K then Mb=+4.74+8.98=+13.72 (the +4.74 is Mo, +8.98 is R/Ro)
If Te=11444K then Mb=13.72-3.01=+10.71 (the -3.01 is from doubling Te)

Minimum mass for helium 3α ignition is about 0.45 Mo (Mazzitelli 1989). Universe not old enough to produce single He wd, but Find He wd in binaries where stripping can occur eg, 0.2Mo He WDs in compact binaries in globular clusters.
In the 8Mo to 11Mo range some additional nuclear reactions may result in O-Mg-Ne white dwarfs.
The 10Mo to 40Mo stars give rise to SNe with a neutron star remnant.
Above 40Mo the star goes supernove with a black hole remnant.
Watch for changes in the above ranges as theory improves!

WD Classification
White dwarf (wd or D) classification scheme (Allen's A.Q. 4thed.)
Scheme is "Dabn" where the D is for degenerate, a=ABOCQ=spectrum,
b=PHV=secondary features, n=50400/Te=temperature index.  
DA   Balmer H lines dominate (pure H atmosphere) 80%
DB Neutral He lines dominate (pure He atmosphere)20%
DO Ionized He lines dominate, may have weak He I, hint of H.
DC no lines deeper than 5% of continuum*
DQ carbon features (atomic if hot, molecular if cool -was C�)
DZ metallic lines dominate, esp Ca II no H, He (was DF, DG)
DX unidentified features ?due to magnetic field?
 secondary symbols
P polarized light
H magnetic field from Zeeman splitting
V variable
PEC spectral peculiarities
 examples
DA1 wd with Balmer lines and 37,500<Te<100,000K
DAO1 as DA1 but weak He II present
DOZ1 strong He II, weak He I, H and N V features, Te=70,000K
DBAQ4  He I > Balmer > carbon features, Te=12,000K
DXP5 magnetic, polarized, unidentified features, Te=10,000K
DZA7 metallic lines with weak Balmer, Te=8500K
DC9 featureless spectrum, Te=5500K

WD interiors

For a completely degenerate nonrelativistic electron gas,

P = Kρ5/3 = Kργ
where
K=(3h�/πmH)2/3 / 20μe5/3memH=9906439/μe5/3.

This γ=5/3 corresponds to a polytrope of index n=1.5 for which the mass-radius relationship is
(1.5)=132.3843)

M1/3R = 2.5 Kω(1.5)1/3 / (4π)4/3G = 3.49974E17/μe5/3 [MKSA]

In terms of the Sun's mass Mo and with μe=2 we have

R = 8765 / (M/Mo)1/3 kilometres.

Chandrasekhar realized that since, as radius decreases with increasing mass, a regime will be reached where the electron speeds (momenta) approach the speed of light, the gas becomes relativistically degenerate, and the equation of state changes to the form

P = Kρ4/3 = Kργ
where
K3=3h3c3/512πmH4μe4

where γ=4/3 corresponds to a polytrope of index n=3. This is a funny polytrope since the mass-radius relation for polytropes

M(n-1)/nR(3-n)/n=(n+1)K1ω(n)(n-1)/n
gives for n=3
M2/3=4K1ω(3)2/3
where ω(3)=2.01284 so
M=μe-2(4hc/8GmH)3/2(3/πmH)1/2(4π)-1/2×2.01284
=5.72μe-2 solar masses

which, for μe=2 gives 1.43Mo. It seems that any radius including zero is possible for this mass! Chandrasekhar integrated the equation of hdrostatic equilibrium using the equation of state for a completely relativistic electron gas
P=(m4c5 / π2h3)F(x)   x~1
to show that in the transiton to complete relativistic degeneracy the radius decreases more and more rapidly and approaches zero when the limiting mass is approached, the Chandrasekhar limit.

Neutron "stars"
So what happens to the core of a star whith M>10Mo that can burn He in a shell without thermal pulsing ejecting a lot of mass and so the C/O core can burn to heavier elements and finally to iron/nickel which is the endpoint of the fusion energy source. As the fully catalyzed core mass increases beyond the Chandrasekhar limit the density increases and higher and higher electron momentum states are occupied. Eventually the electrons at the top of the Fermi sea reach energies of 782 keV which is the energy difference (mass defect) between a neutron and a hydrogen atom. These electrons can "tunnel" through the barrier (set by the uncertainty principle) into protons forming a neutron, losing a bit of energy via a neutrino, and removing a bit of the degeneracy pressure contributed by the electron. Once the process begins a catastropic collapse occurs. Interestingly, all the fusion energy produced by the star is "paid back" by gravitational energy release during this process ("deflagration" and "neutronification"), with considerable interest in the form of energetic neutrinos. With the loss of the electron degeneracy core support, the outer layers free fall, releasing more gravitational energy, and the process of a supernova is underway.

Meanwhile, the core becomes -what? The neutron energies get higher and higher as the collapse continues until nuclear densities are reached. People are still working on what happens next but if we assume we can use the same expression as for electron degeneracy with the mass of the electron replaced by the mass of the neutron and μe replaced by μn=1 which results in a radius a factor of μemn/me~500 times smaller than the same mass white dwarf and a corresponding limiting mass of 5.72Mo.

The above limit is an upper limit, ignoring the fact that neutrons are attracted by the strong nuclear force while electrons are repelled by Coulomb forces and at high densities the neutrons may no longer be neutrons, rather a quark "soup". But the Quarks are Fermions so will put up a degeneracy pressure. Work in progress. Also we have ignored general relativity and at neutron star mass and density GR must be included. Recall the Schwarzschild radius for a one solar mass object is about 3km.

In fact, the measured neutron star masses fall in a remarkably small range, 1.35±0.05Mo! These masses are from binary pulsars and are some of the best determined masses in astronomy. Too bad we don't really know about their radii which are expected to lie in the range 10 to 13.5km.

Hubble discovered an isolated neutron star RX J1856.5-3754 which radiates like a 700,000K black body with a radius of 11km, confirmed by CHANDRA.

The crab pulsar was discovered accidentally in 1967 by Jocelyn Bell while looking in that direction with the (Nuffield) Radio Astronomy Observatory at Cambridge. Now over a thousand radio pulsars are known suggesting there are 200,000 in the galaxy. The lifetime of a pulsar is about 20,000,000 years suggesting a new pulsar is born about every 100 years, about the same as the supernova production rate.

See also
The CSIRO ATNF Pulsar education page,
The Jodrell Bank Pulsar group,
The Princeton Pulsar Group,
The

Black holes
The Schwarzschild limit RS=2GM/c� sets a limit on the mass of a neutron star, if we put RS in the M3/2R=... relation for neutron star polytrope n=1.5 (nonrelativistic) we find M=3.42Mo to be the upper mass limit of a neutron star. We are spared the embarrassment of trying to work out an exact "special relativity Chandrasekhar limit" by the "general relativity Einstein limit". When relativistic considerations are explicitly included in the analysis ("real" neutron star models not classical polytropes) the limit comes down to 3.2Mo. With fancy particle physics some workers (Glendenning) have brought the limit down to 1.5Mo.

Link to White Dwarf Research Corporation