Astronomy and Astrophysics – Astrophysics
Scientific paper
Nov 1980
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1980a%26a....91..212d&link_type=abstract
Astronomy and Astrophysics, vol. 91, no. 1-2, Nov. 1980, p. 212-220.
Astronomy and Astrophysics
Astrophysics
19
Equations Of State, Free Energy, Helium Plasma, Hydrogen Plasma, Space Plasmas, Stellar Evolution, Equilibrium Equations, Numerical Integration, Stellar Models, Thermodynamic Properties
Scientific paper
In the free energy method statistical mechanical models are used to construct a free energy function of the plasma. The equilibrium composition for given temperature and density is found where the free energy is a minimum. Until now the free energy could not be expressed analytically, because the contributions from the partially degenerate electrons and from the inner degrees of freedom of the bound particles had to be evaluated numerically. In the present paper further simplifications are made to obtain an analytic expression for the free energy. Thus the minimum is rapidly found using a second order algorithm, whereas until now numerical first order derivatives and a steepest- descent method had to be used. Consequently time-consuming computations are avoided and the analytical version of the free energy method has successfully been incorporated into the stellar evolution programmes at Geneva Observatory. No use of thermodynamical tables is made, either. Although some accuracy is lost by the simplified analytical expression, the main advantages of the free energy method over simple ideal-gas and Sacha-equation subprogrammes (as used in the stellar programmes mentioned) are still kept. The relative errors of the simplifications made here are estimated and they are shown not to exceed 10% altogether. Densities up to those encountered in low-mass main-sequence stars can be treated within the region of validity of the method. Higher densities imply less accurate results. Nonetheless they are consistent so that they cannot disturb the numerical integration of the equilibrium equation in the stellar evolution model. The input quantities of the free energy method presented here are either temperature and density or temperature and pressure, the latter require a rapid numerical Legendre transformation which has been developed here.
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