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Astron. Astrophys. 334, 953-968 (1998)

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3. Micro-physical inputs

Since comparisons with former results by different authors will turn out somehow elusive due to our previously untested evolution/convection/mixing combination, we felt obliged to at least summarize and shortly describe the micro-physical inputs in ATON 2.0. In the Appendix, also the main numerical features of the code will be addressed.

3.1. Thermodynamics

Thermodynamics tables are given as a function of temperature and pressure. In ATON 2.0 they are built, for any given metal abundance Z, in three steps.

First, the tables by Magni & Mazzitelli (MM 1979, recently updated in full ionization regime) are stored for the five H -abundances: [FORMULA] ; [FORMULA] ; [FORMULA] ; [FORMULA] ; [FORMULA]. Since MM tables were computed for [FORMULA], an "average" metal is interpolated from the pure-carbon table by Graboske et al. (1973). This latter procedure satisfactorily mimicks an EOS with metals at least as long as [FORMULA]. A former release of MM EOS has been however shown by Saumon (1994) somewhat biased in high-T, low- [FORMULA] regions, while the [FORMULA] relation performs fairly well in the low-T, high- [FORMULA] non-ideal gas regime.

A severe warning is here necessary: contrarily to some more recent EOS' for non-ideal gas (e.g. Saumon et al. 1995), MM thermodynamics has been computed not only for pure H or He chemical compositions, but also for [FORMULA] mixtures. In spite of its age, it is then still more physically updated than other EOS' in pressure dissociation and ionization domain, where interpolation (with the additive volumes law or any other such scheme) between pure elements is bound to give unreliable results, due to the physical influence of H -ions on the He -bound states.

As a second step, the tables (covering the [FORMULA] -T plane up to densities where the lattice ion quantum effects begin to show up) are partially overwritten by the Mihalas et al. (MHD, 1988) equation of state, for the chosen Z and in the range Log [FORMULA] ; Log [FORMULA]. MHD EOS is in fact biased (Saumon 1994) at larger densities, where the MM EOS is instead still quite good. With this second step, correct allowance for Z is obtained in the low-T, nearly ideal gas region, but for the unique [FORMULA] ratio for which the MHD EOS are provided.

Third and final step: the above tables are partially overwritten by the OPAL EOS tables (Rogers et al. 1996) where available ([FORMULA]), for the same five [FORMULA] ratios as in MM. In this way, residual anomalies (if any) at high-T in the MM EOS are forgotten, and variable [FORMULA] ratios are restored in all the cases of interest.

Lastly, one pure-carbon and one pure-oxygen table, computed in full ionization regime with an improved MM-like scheme, are addedd. Bicubic logaritmic interpolation on both P and T are performed on the tables at fixed H-abundances, and linear interpolation on the chemical composition follows, to evaluate the various thermodynamical quantities.

3.2. Opacity

A procedure similar to the above one is performed also for the opacities. For each given Z, OPAL opacities (Rogers & Iglesias 1993), linearly extrapolated ([FORMULA] vs. [FORMULA]) in the high- [FORMULA] region, and harmonically added to conductive opacities (Itoh & Kohyama, 1993), form the ground level.

At lower temperatures ([FORMULA] K), Alexander & Ferguson's (1994) molecular opacities (plus electron conduction when in full ionization) complete the tables, for the same 10 [FORMULA] ratios as in OPAL's case. At variance, Kurucz (1993) low temperatures ([FORMULA] K) opacities can be used, but for one [FORMULA] ratio only.

For [FORMULA] mixtures, interpolation on fifteen out of the 60 OPAL tables (plus conductive opacities) is performed. Comparisons with the full set of 60 tables show that, with the present reduced set, interpolation on the chemical composition always gives agreement better than [FORMULA] and, in the vast majority of cases, better than [FORMULA]. Bicubic logaritmic interpolation on both [FORMULA] and T are performed on the tables at a fixed [FORMULA] ratio, and quadratic interpolation on H follows to get the final opacity value.

3.3. Nuclear network

It includes the 14 elements: [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA]. We explicitly consider the following 22 reactions:

[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]
[FORMULA]

The reactions in parentheses are so fast that they are not explicitated (in terms of mixing, the half-lifes of the elements are so short that they are always in local equilibrium). The cross sections for the other reactions are from Caughlan & Fowler (1988); low, intermediate and strong screening coefficients are from Graboske et al. (1973). The [FORMULA] reaction has been given a fictitious end in [FORMULA], since ATON 2.0 code can follow [FORMULA] -ignition, but has not been set to deal with more advanced evolutionary phases.

Logaritmic nuclear cross sections are in tables with a very thin spacing in Log T, and are cubically interpolated.

3.4. Neutrinos

Pair, photo, bremsstrahlung and plasma neutrino have been taken from Itoh et al. (1992). Recombination neutrinos have not been included, since they are of interest only for more advanced (pre-supernova) evolutionary phases, which presently lie out of the domain of ATON 2.0.

Due to the computing time required to evaluate the various neutrino fluxes with the fitting formulae in the literature, we built up logaritmic tables of neutrino rates for various elements. Bicubic logaritmic interpolation on both [FORMULA] and T are performed on the tables at a fixed chemistry, and linear interpolation on the chemistry follows to compute the neutrino emission.

3.5. [FORMULA] sedimentation

Also gravitational settling and chemical and thermal diffusion of [FORMULA] is optionally included in ATON 2.0. To avoid confusion with the term diffusion, for which we prefer to maintain the meaning of mixing due to turbulent convection, in the future, we will refer to all the processes leading to [FORMULA] settling as sedimentation tout court.

The following approximations are made, consistently with Muchmore (1984) and Paquette et al. (1986):

  • radiative force can be neglected
  • all particles, including the electrons, have approximately Maxwellian distribution
  • mean thermal velocities are much larger than the diffusion velocities
  • magnetic fields are absent.

The sedimentation velocities then satisfy (Proffitt & Michaud 1991; Paquette et al. 1986):

[EQUATION]

where [FORMULA] are the partial pressure, mass density, number density, mean charge, and sedimentation velocity for species a (in our case, respectively: [FORMULA], [FORMULA] and electrons), and E is the electric field induced by the gradients in ion densities (Aller & Chapman 1960). The resistance coefficients [FORMULA] are taken from a fit (Iben & Mc Donald 1985) of the numerical results by Fontaine & Michaud (1979), while for the total thermal diffusion we adopt [FORMULA] (Alcock & Illarianov 1980).

In terms of temperature gradient and number density, Eq. (5) may be rewritten as:

[EQUATION]

[EQUATION]

where [FORMULA] is the molecular weight for the [FORMULA] species, while [FORMULA] is the hydrogen mass.

To complete the set of equations we apply the condition for no net mass flow relative to the center of mass and no electrical current. To proceed, we separate out the velocity therm [FORMULA] as:

[EQUATION]

where [FORMULA] is relative to the gravity components and temperature gradient, while the summatory is relative to the components due to gradients in number densities, that is:

[EQUATION]

where all the symbols have the same meaning as in Iben & McDonald (1985) but for H (due to thermal diffusion, not present there):

[EQUATION]

We then use these expressions in the continuity equation:

[EQUATION]

which can be solved by a conservative, semi-implicit finite difference method of first-order in time and second-order in the spatial variable r, always following the procedure by Iben & Mc Donald (1985).

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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