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Astron. Astrophys. 358, 575-582 (2000)

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2. Construction of models and details of the calculations

2.1. Basic atmosphere

There exist two schools of thoughts for constructing `semi-empirical' models of stellar atmospheres. One school follows a strict semi-empirical approach (VAL, Giampapa et al. 1982, Mauas & Falchi 1996) which means that they guess a starting solution for the atmosphere and then compare the output from the radiative transfer calculations to the observed spectral lines/fluxes. With the advent of faster computers there has emerged a new school which basically calculates grids of models (Houdebine & Doyle 1994, Short et al. 1997) by varying some of the important parameters in the atmosphere. After calculating the output of the models they compare the calculated results with observation and try to find which model gives the best agreement. We can call the latter approach `schematic'.

In order to investigate the different levels of the non-thermal velocities through the model atmosphere we kept the temperature structure versus column mass identical for all models. The temperature stratification is achieved using the `standard approach', choosing the position of the temperature minimum and transition region. Up to the temperature minimum we keep the photospheric model as calculated in radiative equilibrium, from the temperature minimum to the top of the chromosphere we consider a linear temperature rise with the logarithm of the column mass and in the transition region we keep [FORMULA]T versus [FORMULA]m constant.

The photospheric model used is the `Next Generation' model of Hauschildt et al. (1999) with the effective temperature of [FORMULA] and [FORMULA] and solar metallicity. It is interesting that some authors use a lower effective gravity that we find to be of `historical' origin - in the eighties the best photospheric models for late type stars were models of Mould (1976) which were between [FORMULA] and 5.75. We choose a photospheric model which corresponds to a dwarf star of spectral type M0-M1.

From the temperature minimum, situated at [FORMULA], to the transition region, which onsets at temperatures around 8,200 K (Houdebine & Doyle 1994, Short & Doyle 1998) and in our test case at the column mass [FORMULA], we keep [FORMULA] constant. From the onset of the transition region to the top of the atmosphere at 300,000 K we keep [FORMULA]. The temperature dependence on the column mass is shown in Fig. 1. Each atmospheric structure has one hundred depth points. This model is similar to the model used by Short et al. (1998) describing a so called zero-activity M dwarf.

[FIGURE] Fig. 1. The temperature dependence of the column mass for our test model.

2.2. Models for the non-thermal velocities

To test the influence of the non-thermal velocity on the electron density stratification and the line formation we use twelve different model distributions. These can be divided into three groups:

  • A constant non-thermal velocity throughout the atmosphere for three different levels of [FORMULA] and 5 km s-1.

  • Models where the non-thermal velocity depends on the sound velocity defined by

    [EQUATION]

    where [FORMULA] is the generalized adiabatic exponent:

    [EQUATION]

    (see e.g. Mihalas and Mihalas 1984, p.52). [FORMULA] is the ionization energy from the first level of the Hydrogen atom and x is the degree of ionization.

    We consider five levels of the non-thermal velocity in the atmosphere namely [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] and we limit the maximum non-thermal velocity to 30 km s-1.

  • Models where we keep the non-thermal velocity constant (1 km s-1) up to the temperature minimum, then linearly increasing to (2, 3, 5 and 10 km s-1) at the top of the chromosphere and then to (5, 10, 20, 30 km s-1) at the top of the atmosphere.

fcfc For the electron density calculation it is important to take into account turbulent pressure, therefore the hydrostatic equilibrium equation is written as:

[EQUATION]

where g is the gravity acceleration, m the column mass, [FORMULA], [FORMULA] and [FORMULA] are the gas, electron and radiative pressure and [FORMULA] is the turbulent pressure.

2.3. Details of calculation

We use the radiative transfer code MULTI (Carlsson 1986, 1992) to solve simultaneously the equations of radiative transfer, and the statistical and hydrostatic equilibrium equations for Hydrogen. Our Hydrogen model atom consists of 15 bound levels plus continuum. The oscillator strengths were from Green et al. (1957), the transition probabilities from Reader et al. (1980) and the values for the Stark broadening from Sutton (1978). Our Sodium atom has 10 bound levels plus two continuum levels with the atomic data taken from Bashkin & Stoner (1982). We use extensively the radiative-collisional switching technique of Hummer & Voels (1988) to achieve convergence of the population levels. The population of the levels in the Hydrogen and Sodium atom are iterated up to the level when changes are less than [FORMULA].

In this calculation we have used opacities from the Uppsala opacity package. Andretta et al. (1997) and Short et al. (1997) have shown that it is necessary to include a better treatment of background opacities. At the moment we are working on the inclusion of a depth dependant micro-turbulent velocity in PHOENIX. Thus to have a self-consistent treatment, we presently use only Uppsala opacities.

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© European Southern Observatory (ESO) 2000

Online publication: June 8, 2000
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