The properties of very dense matter are so peculiar that you may not be familiar with the underlying physical principles. In order to appreciate the implications for astrophysics of these principles we give here a brief outline of the behaviour of matter at high densities. Naturally the presentation involves some elementary algebra. All of the elements can solidify at extremely low temperatures ; that is to say, if their atoms are made to move slowly enough, they lock into a solid structure (they freeze). There must, therefore, be some attractive force between the atoms which can bind them together in the form of a crystal if the atoms are not too energetic. Just as important, there must also be a second, repulsive, force which prevents the crystal, once formed, from collapsing altogether under the influence of that attraction. What are these forces ?

Gravitation is far too weak and nuclear forces have too short a range. Although each atom is, as a whole, electrically neutral, the attractive force is in fact electrostatic. The motions of the outer electrons of each atom are related in such a way that the nucleus of each atom sees another nucleus as attractive. Counteracting this is the dominant repulsive force which, especially in dense matter, is caused by the degeneracy pressure, of the electrons.

Heisenberg’s UNCERTAINTY PRINCIPLE states that the uncertainty in a measurement of the, momentum, of (for example) an electron and the uncertainty in the measurement of its position, are related thus:
(momentum uncertainty) X (position uncertainty)
is greater than 6.6 X 10-34 Js

where 6.6x 10~34 Js is PLANCK’S CONSTANT and will be denoted by h. The meaning of this famous uncertainty principle is that if we devise an experiment to measure at some instant the speed of an electron to a high degree of accuracy (that is to say the momentum error is small), then we shall riot be able to predict the position of the electron at that instant very accurately (because the positional error is then large). Conversely, if we fix an electron in space with sufficient precision, its momentum cannot be specified. This principle is fundamental to microphysics, and the uncertainties mentioned occur even in an ideal experiment. Since Planck’s constant is so small, we are unaware of the uncertainty principle in everyday life. For example, the average inaccuracy due to the uncertainty principle incurred in dropping a pebble onto a given spot from a height of? 10 metres is 10-16m – about a tenth of the radius of a nucleus! But, as we shall now see, there are circumstances in physics when the uncertainty principle is critically important.

The position of an electron that is under the influence of an atomic nucleus is well specified, and consequently its momentum is ill-defined. The probability of our observing the electron at rest under such circumstances is vanishingly small. This ceaseless motion of the electron exerts a pressure, termed DEGENERACY PRESSURE, on its surroundings, just as ordinary gas pressure is caused by the continual motion of the gas molecules. (As a simple but valid analogy we may consider the trapping of an unwilling creature in a box with movable walls. The more it is confined, the more wildly it rushes around, the more frequently it collides with the walls, and the more pressure it exerts on the walls of the box.) We should stress that although the bound electron is in motion and has energy, this energy cannot be removed, since doing so would decrease its momentum, and thus degenerate matter may be thought of as cold.

The pressure, P, of a collection of particles may be, written approximately as P= nmv2 where n is the number of particles per unit volume, m is the mass of a particle and v the average velocity. Consider the degeneracy pressure of the electrons in a lump of solid hydrogen of volume V which contains N atoms. Clearly the number, n, of particles per unit volume is N/V. Because of PAULI s EXCLUSION PRINCIPLE, which states that no two electrons (or protons or neutrons) can occupy the same state, the effective volume available to any one electron is not the total volume, but is rather V/N, the volume occupied by an atom. Thus each electron is effectively confined to a volume V/N, or to a small distance (V/N)3. The momentum is mv which, by the uncertainty principle, is equal to h/(distance uncertainty). If we now gather these relations

together, we have v = h/m(n/v)1/3

Our pressure equation for a degenerate electron gas is therefore given by:

P=h2/m2(n/v)5/3

where me is the mass of an electron. We can make a further modification by noting that the density of the matter, which we shall denote by p is given by (number of particles) x (particle mass)/(volume), that is p = Nm e /V, and consequently:

P=h2me -8/3 ? 5/3

This is known as the EQUATION OF STATE of such a gas. We note that, unlike ordinary pressure, degeneracy pressure is independent of temperature, and depends just on the density and the mass of the degenerate particle.
But how important is degeneracy pressure? In the Earth’s atmosphere the air pressure is almost all due to ordinary heat pressure, with degeneracy pressure contributing less than one part in 105. In the central core of a red-giant star, however, the density exceeds 108kgnrr3 and degeneracy pressure can dominate despite the high temperature (108K). In an ordinary gas at a fixed temperature, the, pressure; is proportional to the density. Since degeneracy pressure depends on the five-thirds power of the density, it, will become; more, important than the gas pressure at sufficiently high densities, whatever the temperature. We have somewhat laboured this point because the extraordinary proper-tics of dense matter are due to unfamiliar laws of physics. We have one final complication to discuss before returning to the astronomical aspects of dense matter. If the density is sufficiently high the momentum of an electron becomes so large that its velocity approaches the speed of light. The motion is governed by Einstein’s theory of special relativity, and the electrons are said to become relativistic. Then the pressure of a relativistic degenerate electron gas is given by :

P=hc(n/v)4/3 =hcme -4/3 ? 4/3

The important point here is that once again the pressure is in dependent of temperature, but in this case it is proportional to the four thirds power of the density.

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