Classnotes 14

Final States

Stars evolve from the main sequence to their deaths. For single stars, there are three possible final states:

Initial Mass	Final State
0.8M < M < 8M	White Dwarf (M WD < 1.4 M but M - 0.6M is common)
8 < M < 40(?)M	Neutron Star (M NS 3 M)
M 40(?)M	Black Hole MBH 3M

Objects with M 0.08M are not stars: they do not burn H to He. They approach the main sequence, fail to ignite H, and cool and fade slowly. They do not continue to collapse because they are supported by the pressure of degenerate electrons.

Objects with masses between 0.08 and 0.8 M burn H to He and must eventually exhaust their supply of H. But even a 0.8M star remains on the main sequence for an age greater than the age of the Universe.

White dwarfs and neutron stars are Final States in the sense that gravity's desire to squash a star is permanently and finally thwarted by a pressure that does not depend on temperature. This is the pressure of degenerate electrons (white dwarfs) or degenerate neutrons (neutron stars). WD and NS cool, and fade but do not collapse further. A WD has a radius about that of the Earth. A NS has a radius of about 10 km (the size of Austin).

Degenerate electrons provide a limited pressure. The mass of a WD cannot exceed 1.4 M), a limit known as Chandrasekhar's mass. The mass of a NS cannot exceed about 2 to 3 M (the actual value is uncertain) owing to the maximum pressure provided by degenerate neutrons.

Above 2 to 3 M), the stellar remnant continues to collapse. This is a Black Hole.

Degenerate electrons also play a part in the evolution of stars prior to their attaining their final states. Seeds offers a brief introduction (Box 10- 1) to degenerate matter. Here is a more complete account.

Degenerate Electrons and Neutrons

Two fundamental and secure principles of physics govern the behavior of a gas at very high densities.

Consider a gas composed of electrons. We may say that an electron requires a minimum volume to call its own. This volume is defined by the Heisenberg Uncertainty Principle, see Appendix. As I shall consider a one-dimensional gas, I state this principle in that form: where is the smallest length to which an electron can be confined and is the smallest increment in momentum (energy, for our purposes) that can be given an electron. is a constant (a small number).

The second principle is the Pauli Exclusion Principle. This states that no more than 2 electrons can have the same energy and position. "Same" means within and of each other. The restriction to 2 is related to the spin of the electron. The electron may spin 'up' or 'down' and the 2 electrons in the same 'box' must have opposite spin.

Consider now an 1D example. We can locate each electron on a momentum (energy) - position diagram. The squares are by in size. No square may contain more than 2 electrons.

Normal (non-degenerate) Gas

--We note many empty boxes among the lowest energy boxes. --All filled boxes are at the lowest energys. --Electrons can move pretty freely to left and right without coming up agains the 2 per box limit. --Electrons can collide and exchange energy as empty boxes are available. --We can compress the gas; i.e., take the 11 boxes in the first diagram and squeeze it down to seven boxes; some boxes will still not have their full quota of 2. --This gas can lose energy out through the sides -- higher energy electrons can fine a home in a lower energy box after transferring some energy to the outside.

A Degenerate Gas

At high densities, the boxes fill up. We then say the electrons are degenerate. In the same 1-dimensional picture, this would be like

-- All low energy boxes (states) are filled.

-- Further compression requires that higher energy states be filled.

This Requires Hard Work!! --Low energy electrons cannot exchange energy easily because the adjacent boxes are filled. --The mobile electrons are those at the highest energies and the highest velocity and these gain energy easily but not lose very much.

This property explains why metals are good conductors of heat (energy) and electricity (electrons carry charge.)

--The pressure that a degenerate electron gas can exert is a permanent pressure. This high energy electrons cannot lose their energy because all the lower energy boxes are filled.

Contrast this with a normal gas:

So, a degenerate electron gas can permanently withstand the self-gravitation -- its energy cannot be radiated away.   Examples include, -white dwarfs -- they cool off but never collapse to zero, -black dwarfs -- either low mass stars (brown dwarfs) that never ignited H or exhausted white dwarfs, -cores of low mass red giants.

Can gravitation ever win when the electrons try the degenerate 'trick'?

Yes! Einstein - special relativity - says velocity cannot exceed the speed of light (c), i.e., there is an energy ceiling and hence, an upper limit to the pressure provided by degenerate electrons. If gravity exceeds this pressure, the star will collapse in spite of the degenerate electrons. Chandrasekhar showed that this ceiling was reached with a degenerate star with a mass of 1.44 M - now known as Chandrasekhar's Limit (exact numerical value depends on the type of nuclei mixed in within the electrons). This is the limit for helium (why do we choose He and not hydrogen?).

The nuclei generally escape the box filling problem because their boxes are much smaller. (Also, one He nucleus is matched by 2 electrons so nuclei are less numerous).

Neutron Stars

For stellar cores above 1.44 M, collapse is not halted by the degenerate electrons. Also for lower masses, it may be possible to achieve more squashing; for example, the core may be part of a more massive star.

There can come a time when electrons and nuclei (positively charged, of course) are forced so close that they can cancel out their charges.

For example:
(At these high densities, the He nuclei produced in an earlier stage of nuclear burning of H are broken down into protons and neutrons). Then the core becomes a neutron star.

At normal densities, neutrons are unstable and decay n --> p + e^-+ v in a little over 11 minutes. At these high densities, neutron formation dominates over decay.

But neutrons obey a form of the Pauli Neutrino principle too, so we can (and do) have a degenerate neutron gas and a mass limit
equivalent to Chandrasekhar's. Actually, the limit is set by the fact that neutrons can be broken down into even more elementary particles: this is likely to occur as the high energy neutrons near the top of the neutron sea collide and fragment. Here, the physicist is on a frontier of his/her knowledge and opinions vary, but M_{Neutron Star} < 3 M is a best guess.

The figure shows what a neutron star is thought to look like. Note three things:  1. The radius of the star is small (10km).  2. The crust contains nuclei. Neutrons exist below the crust.  3. The ? in the center. These mean no one really knows what goes on there. Quarks are the elementary particles that make up neutron and protons.

Nuclear Reactions in Degenerate Electron Gas

In a star, the energy input comes from nuclear reactions. Consider now the ignition of He in a low mass red giant. He burning starts and the core is heated. The core does not expand because it is (electron) degenerate. Nuclei are not degenerate - they continue to heat up and reat faster and not degenerate. For example, in the interior of main sequence stars, injection of heat causes the gas (electrons and nuclei) to
expand. Expansion lowers the density and temperature and the release of nuclear energy from 4H --> He is slowed.

1. Degenerate electron gas (and associated nuclei) cannot expand and cool when energy is put in.

 

2. Inside star, electrons are degenerate, but the atomic nuclei are not. Energy input comes from nuclear reactions (collisions between nuclei).

3. Energy input from reactions has no effect on electrons, but nuclei respond by moving faster.

4. When nuclei move faster, there are more nuclear reactions.

5. Therefore, more energy is put into the gas and the nuclei move faster still.

 

6. More nuclear reactions --> more energy into gas --> nuclei move faster --> more nuclear reactions.

7. Eventually, sufficient energy is given to the electrons to "lift" their degeneracy.

This is an example of a runaway process. This example leads to an explosion. If the explosion occurs in the core of a star, the outer cooler layers may contain the explosion and prevent the star from blowing itself apart.

Appendix
Discussions of the uncertainty principle are often put in the form of presenting a duffer (and long live all such duffers) who attempts to do an experiment which will deny the predictions of the principle; he retires, of course, bruised from the ring. Heisenberg, whose principle this is, presented such a duffer in order to show that all such gedanken experiments (thought experiments) must fail. His jester used a microscope to measure the position of the particle and, in order to do so with increasing precision, selected one operating with ever shortening wavelength of light. But the shorter the wavelength the more momentum each photon carries and since at least one photon must be scattered into the microscope aperture in order for the position to be determined, it is clear that the very act of observation imparts a momentum to the particle. An analysis of the experiment, taking into account aperture-diffraction effects and momentum transfers on light scattering, concludes that the uncertainty product is indeed not less than .-In a classical world the jester would laugh last, because would be zero and there would be no intrinsic limitation on the precision.