Physical State of the

Due to the high temperature the gas in the stars is almost completely ionized. The interactions between individual particles are small, so that, to a good approximation, the gas obeys the perfect gas equation of state, k

where k is Boltzmann's constant, ¡x the mean molecular weight in units of mH, and mH the mass of the hydrogen atom.

The mean molecular weight can be approximately calculated assuming complete ionization. An atom with nuclear charge Z then produces Z +1 free particles (the nucleus and Z electrons). Hydrogen gives rise to two particles per atomic mass unit; helium gives rise to three particles per four atomic mass units. For all elements heavier than hydrogen and helium it is usually sufficient to take Z +1 to be half the atomic weight. (Exact values could easily be calculated, but the abundance of heavy elements is so small that this is usually not necessary.) In astrophysics the relative mass fraction of hydrogen is conventionally denoted by X, that of helium by Y and that of all heavier elements by Z, so that

(The Z occuring in this equation should not be confused with the nuclear charge, which is unfortunately denoted by the same letter.) Thus the mean molecular weight will be

At high temperatures the radiation pressure has to be added to the gas pressure described by the perfect gas equation. The pressure exerted by radiation is (see p. 239)

where a is the radiation constant. Thus the total pressure is

The perfect gas law does not apply at very high densities.

The Pauli exclusion principle states that an atom with several electrons cannot have more than one electron with all four quantum numbers equal. This can also be generalized to a gas consisting of electrons (or other fermions). A phase space can be used to describe the electrons. The phase space is a 6-dimensional space, three coordinates of which give the position of the particle and the other three coordinates the momenta in x, y and z directions. A volume element of the phase space is

AY = AxAyAzA px A pyA pz

From the uncertainty principle it follows that the smallest meaningful volume element is of the order of h3. According to the exclusion principle there can be only two electrons with opposite spins in such a volume element. When density becomes high enough, all volume elements of the phase space will be filled up to a certain limiting momentum. Such matter is called degenerate.

Electron gas begins to degenerate when the density is of the order 107kg/m3. In ordinary stars the gas is usually nondegenerate, but in white dwarfs and in neutron stars, degeneracy is of central importance. The pressure of a degenerate electron gas is (see p. 239)

where me is the electron mass and N/V the number of electrons per unit volume. This equation may be written in terms of the density p = NßemH/V , where is the mean molecular weight per free electron in units of mH. An expression for /xe may be derived in analogy with (10.10):

For the solar hydrogen abundance this yields

-e = 2/(0.71 +1) = 1.17 . The final expression for the pressure is

This is the equation of state of a degenerate electron gas. In contrast to the perfect gas law the pressure no longer depends on the temperature, only on the density and on the particle masses.

In normal stars the degenerate gas pressure is negligible, but in the central parts of giant stars and in white dwarfs, where the density is of the order of 108 kg/m3, the degenerate gas pressure is dominant, in spite of the high temperature.

At even higher densities the electron momenta become so large that their velocities approach the speed of light. In this case the formulas of the special theory of relativity have to be used. The pressure of a relativistic degenerate gas is h ( N f = hc (-L. f

In the relativistic case the pressure is proportional to the density to the power 4/3, rather than 5/3 as for the nonrelativistic case. The transition to the relativistic situation takes place roughly at the density 109 kg/m3.

In general the pressure inside a star depends on the temperature (except for a completely degenerate gas), density and chemical composition. In actual stars the gas will never be totally ionized or completely degenerate. The pressure will then be given by more complicated expressions. Still it can be calculated for each case of interest. One may then write

giving the pressure as a known function of the temperature, density and chemical composition.

The opacity of the gas describes how difficult it is for radiation to propagate through it. The change dI of the intensity in a distance dr can be expressed as d I = —Ia dr , where a is the opacity (Sect. 4.5). The opacity depends on the chemical composition, temperature and density of the gas. It is usually written as a = Kp, where p is the density of the gas and k the mass absorption coefficient ([k] = m2/kg).

The inverse of the opacity represents the mean free path of radiation in the medium, i. e. the distance it can propagate without being scattered or absorbed. The different types of absorption processes (bound-bound, bound-free, free-free) have been described in Sect. 5.1. The opacity of the stellar material due to each process can be calculated for relevant values of temperature and density.

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