The state of the gaseous material in a star can be described by various physical quantities such as its pressure, P, temperature, i, and density, p. These quantities do not vary independently. If we change the pressure at some point in a star we also alter the temperature and/or density. Clearly, if we are to give an account of how a star works, we must write an equation that tells us how pressure, temperature, and density interrelate. The relationship between these quantities is given by an EQUATION OF STATE An example of one of these is the perfect gas law which can be written

where R is the gas constant and µ is the mean molecular weight of stellar material. This relation is none other than the familiar P = RT/V or PV = RT, in which the volume, V, has been replaced by H/p; this substitution simply takes account of the fact that stellar material is composed of several different elements in different states of ionization. The perfect gas law holds reasonably well under the extreme densities and pressures in stars mainly because, at the high temperatures, ionization has stripped the atoms of their electrons so they have a much smaller size than usual. On Earth, for example, at a density of 103kgnr3 (water) a material is a liquid at room temperature and the perfect gas law does not hold. In a star, however, at such density the temperature is about 106K and the stellar material is highly ionized.

However, in stellar material the relatively simple equation of the perfect gas law may not always hold. Of special importance are the contributions of RADIATION PRESSURE and DEGENERACY. Radiation pressure is the pressure exerted by photons. It has a particularly simple form :
Prad = (1/3)a T4

where a is the radiation density constant (a = 7. 56 X 10-16Jm 3 K-4). It is clear that radiation pressure is very sensitive to temperature, since it depends on the fourth power, and it only becomes important at relatively high temperatures. For example, at the density of water the temperature must be over 10 million degrees Kelvin for radiation pressure to be greater than gas pressure. How-ever in the interiors of many stars high enough temperatures are encountered for radiation pressure to be very important.

Degeneracy (see chapter 6) is a quantum mechanical effect whereby only two electrons, of opposite spins are permitted to be in the same very small volume of phase-space. The size of the small volume is set by the Uncertainty Principle. At a temperature of 106K and at the density of ordinary matter (typical stellar quantities) the electrons can easily fit into the available space and degeneracy is of no consequence ; at very high densities there are many more electrons per unit volume and degeneracy becomes important. The set of electrons is forced, by the constraints of the Uncertainty Principle, to take on a different distribution of velocities than it would have had at smaller densities. The different form of the distribution of velocity gives rise to a different pressure and hence a different (degenerate) equation of state. Typically degeneracy is important in stellar interiors at densities of order 107kgm-3. Such d are not usually encountered in ordinary stars but they are most important in very dense objects such as white dwarfs and Neutron stars

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