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Astron. Astrophys. 364, 879-886 (2000)

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4. Thermal equilibrium quantities

For simplicity, we only employ the partition functions in the ground state for the dominant elements [FORMULA] and [FORMULA]. Considering the correction of nonideality and electron degeneracy to thermodynamic quantities, the modified Saha equation can be written as (Christensen-Dalsgaard 1977; Pols et al. 1995):

[EQUATION]

[EQUATION]

and

[EQUATION]

where the total correction to the electron chemical potential is

[EQUATION]

[FORMULA] is the configuration energies in unit of electron volts, i.e.,

[EQUATION]

For the number [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] of neutral and ionized hydrogen and helium particles, we introduce a set of dependent variables, i.e., [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA], being the occupation numbers of particles divided by the number of nuclei of the respective [FORMULA] or [FORMULA] species. With these definitions, we have the occupation number for hydrogen species

[EQUATION]

where the partition functions are given according to

[EQUATION]

with the normalization condition

[EQUATION]

For helium species

[EQUATION]

where the partition functions are

[EQUATION]

with the normalization condition

[EQUATION]

Under conservation constraints, the number of free electrons per unit mass is given by

[EQUATION]

Since the present model considers six species [FORMULA], Eqs. (56) and (58)-(62) are six simultaneous equations for the six unknown [FORMULA] with a unique unknown variable, i.e., the electron degeneracy [FORMULA]. Here [FORMULA] defined in Eq. (54) is also an explicit function of the electron degeneracy [FORMULA] by means of the free electron number [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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