WD progenitors
| 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 | ||||||||||||||||||||
| Schönberg-Chandrasekhar limit Core contracts rapidly, H shell burns hotter | |||||||||||||||||||||
| Envelope expands - red giant phase | |||||||||||||||||||||
| He ignition | Helium flash if original M<8Mo | ||||||||||||||||||||
| 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
| Cooling white dwarf, mostly Carbon and Oxygen.
| | |||||||||
Some models: M=Mo, R=0.016Ro= 11,300km=1.745Re
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!
| 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 C²) |
| 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 |
For a completely degenerate nonrelativistic electron gas,
In terms of the Sun's mass Mo and with μe=2 we have
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
where γ=4/3 corresponds to a polytrope of index n=3. This is a funny polytrope since the mass-radius relation for polytropes
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
The Schwarzschild limit RS=2GM/c² 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
