The electronic pressure is negative, because the electrostatic energy is negative (7.106). The pressure PES is to be added to the other sources of pressure, such as gas and radiation, Pot = Pg + Pad + Pes + Numerically, one has

with q in g cm-3 and T6 = T/106 K. In the Sun, the ratio of the electrostatic to gas pressure is PES/Pg ~ -0.015. The ratio q/T3 does not vary much inside a star (cf. 3.101), thus the effect of the electrostatic corrections changes little with depth. For a standard composition, the formal equality

This is shown in Fig. 7.8 with a dashed-dotted line. The above developments apply only for (PES) /Pg C 1, thus for T > 1.37 x 105q1/3. The electrostatic pressure becomes relatively more important for lower mass stars. For very low-mass stars and brown dwarfs, a more detailed treatment of the equation of state is necessary.

7.6.3 Ionization by Pressure

The ion and its electronic cloud form a bound system. The energy of an electron in the cloud is Eess = -e2/rD (7.105). Thus, the energy of a "free" electron in the cloud (in the continuum) is below zero. Thus, the ionization potential is reduced with respect to that of an isolated atom.

There is another effect: the energy levels of the bound states are also modified because the potential does not behave strictly as 1 /r. An electron bound to a nucleus of charge Ze is moving in a potential

r rD

The second term is the potential resulting from the charge Z - 1 of the ion and electron due to the interaction of this resulting charge with the other ones. The energy of the fundamental level is thus

r rD

The energy of the bound state is shifted upward by an amount (Z - 1) e2/rD, while the energy of the continuum is lowered by -e2/rD. As a result, the effective ioniza-tion potential becomes

where I0 is the theoretical value for an isolated atom. Ionization is favored when the gas becomes denser and electrostatic effects increase.

The excited atomic levels are also shifted toward higher energies, so that the highest levels lie in the continuum. This simplifies the calculation of the partition functions (Sect. 7.1.2). The summation no longer concerns an infinite series of levels, it can often be made with a limited number of terms. The changes of ionization potentials and energy levels also influence the calculation of radiative opacities.

For dense media, the Debye radius rD is reduced and so does the effective ion-ization potential (7.115). Thus at a given temperature, there is a density value above which the medium is essentially ionized by pressure effect. Such a limit is shown for hydrogen in Fig. 7.8 by the dotted line which separates H+ from H (close to the limit between the degenerate and non-degenerate domains). In practice, one often considers [285] that ionization is complete due to pressure effects, when Saha equation gives degrees of ionization which start decreasing toward the interior. The equation of state and the thermodynamic functions of fully ionized electron-ion plasmas have been established [107] for a wide range of physical conditions.

7.6.4 Crystallization

For high densities (e.g., q > 102 g cm-3 at T = 106 K; see Fig. 7.8), the electrostatic forces between ions start dominating over the thermal energy. If the density still further increases, at some stage the ions are no longer subject to thermal motions, but are stuck to the nodes of a lattice determined by the repulsive forces between ions. The Coulomb plasma thus crystallizes.

Let us consider a medium of temperature T, electron and ion concentrations ne and ni, of charge Z and atomic mass number A. The Wigner-Seitz radii ae and ai are the mean inter-electron and inter-ion distances

The condition of electric neutrality implies that ne = ni Z, thus, ai = aeZ3 • (7.117)

The Coulomb plasma is characterized by the ratios of the Coulomb energy to the thermal energy for the electrons and ions. These ratios are, respectively, e2 (Ze)2 5

ae kT ai kT

Numerically, the value of re is re = 2^693 x 10-3 T • (7.119)

with ne given in cm-3 and T in K. The electron concentration is related to the density by (7.42), which may be expressed as ne = (q/mu)(< Z > /< A >), where < Z > and < A > are the average charge and mass numbers. From (7.117) and (7.119), one has the following relation between the temperature and density for a given value of r

2^27 x 105 5 1 ( < Z >\ 1 m Z5 Q3 —— , (7.120)

with q in g cm-3 and T in K. For Fl = 1, one has a gas where the electrostatic and thermal energies are of comparable importance (Fig. 7.8). For higher values, the medium progressively becomes a Coulomb liquid and for a value of r > 175, the liquid turns into a stable Coulomb crystal [475]. Thus, putting this value in (7.120) provides an approximate limit for crystallization for a medium of nuclei characterized by average Z and A values. An illustration of the location of this limit is shown in Fig. 7.8.

7.7 Degenerate Gases

The fermions, which are particles with spin 1/2, 3/2, ..., obey the Fermi-Dirac statistics (Appendix C.5). The exclusion principle of Pauli tells us that the extension A3qi A3pi in the phase space of position qi and momentum pi with r = 1,2,3 obeys the inequality

This already shows that if the density of a medium increases, i.e., if the space volume interval A3qi decreases, the particles get a higher momentum. These high momenta are a source of pressure of quantum origin called the pressure of degeneracy and the gas (of electrons or neutrons) is said degenerate.

For an ionized medium with electrons of mass me and nuclei of mass mN, one has energy equipartition. In the non-relativistic case, this implies

0 0

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