Helioseismically resolvable weak nonideality of the solar plasma
Astrophysikalisches Institut Potsdam (Germany)
Recently, even nonideal pressure modifications below 1% are being considered in helioseismology, especially due to the use of different partition functions. Here only the electron partial pressure is studied, and thus no partition function is taken into account. Instead, radial pressure profiles are calculated beyond the usually applied Debye-Hückel theory, and quantum-mechanical electron-exchange is taken into account very carefully. It is shown that the total electron pressure at distances from the solar centre larger than 0.83Rsun, if it may be described by equilibrium thermodynamics, is smaller than the ideal-gas value because of the dominating Debye-Hückel screening. On the contrary, at distances smaller than 0.83Rsun electron exchange increases the total pressure up to values above the ideal pressure. It was found that the higher-order interactions beyond the Debye-Hückel screening cause pressure modifications of even up to one per cent. It should be mentioned that already in 1994 Perez obtained pressure modifications of different signs for three points in the sun.
The observations of solar oscillations over many lifetimes of the modes (lasting up to months and about a year), especially the study of the low frequency waves, gives e.g. the possibility
That nonideal effects have to be considered in the equation of states (EOS) of the solar convection zone, this was already shown by Gough (1984), Christensen-Dalsgaard and Däppen (1992), Vorontsov et al. (1994), Antia and Basu (1994), Kosovichev (1995). Under nonideal effects one understands both, the screening of the electrostatic interaction between freely moving charges, and the pressure ionization because of the variation of bound states by the sourrounding charged particles. Now it is standard to take into account non-ideal effects in Debye-Hückel approximation (Stix and Skaley 1990, Christensen-Dalsgaard and Däppen 1992). This approximation corresponds to virial series up to the density order 3/2. Here it is shown, that it is necessary to consider virial series of higher orders in the electron density.
In this work, the weekly coupled, weakly degenerated solar plasma was assumed to be a point-like electron-proton one. The interaction parts of the free energy and of the partial pressure pint of the electrons were considered in density order 5/2 (Kraeft et al. 1986, Alastuey et. al 1995). Besides quantum-statistical effects were taken into account. Heisenberg quantum effects as well as electron exchange were calculated with an accuracy of almost 10-2 %. For the plasma parameters of the solar interior, values of a standard solar model tabulated by Stix (1991) were used.
Fig. 1: Parameters of the solar plasma as function of the ideal part of the gas pressure. Red -- mean particle distance, blue -- Born interaction parameter, black -- de Broglie wavelength of the electrons.
Fig. 2: The quantum-statistical virial function Ko introduced by Ebeling (1976) in dependence on the Born parameter. Red - exact solution, black - electron-exchange neglected.
Fig. 3.:Ratio of the nonideal part of the electron pressure to the ideal part vs. ideal electron pressure pid. Black -- Debye-Hückel approximation, red -- solution in density order 5/2. Calculations with exact virial function Ko.
Fig. 4: Ratio of the nonideal part of the electron pressure to the ideal part vs. ideal electron pressure pid. Black -- Debye-Hückel approximation, red -- solution in density order 5/2. Results found neglecting the electron exchange in Ko.
Fig. 5: Born parameter vs. ideal electron pressure pid.
Fig. 6: Plasma nonideality parameter vs. ideal electron pressure pid.
Fig. 7: Ratio of the nonideal part of the electron pressure to the ideal part vs. plasma nonideality parameter. Black -- Debye-Hückel approximation, red -- solution in density order 5/2 \blue neglecting electron exchange in Ko.
Fig. 8: Ratio of the nonideal part of the electron pressure to the ideal part vs. plasma nonideality parameter. Calculations done with exact virial function Ko. The red branch shows the region where the Fermi energy of the electrons EF is larger than the electrostatic potential energy Ep.
Fig. 9: Fermi energy of the electrons EF and electrostatic potential energy Ep vs. ideal electron pressure pid. The red branch shows the region where the Fermi energy of the electrons EF is larger than the electrostatic potential energy Ep.
The authors would like to thank the Minesterium für Wissenschaft, Forschung und Kulture des Landes Brandenburg for financial support by the HSP III-Project 24-4/055; 1999 "Space Plasma Physics". Moreover, they gratefully acknowledge support of the present work by the "Deutsche Forschungsgemeinschaft" under DFG grant STA 351/5-1.