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Astron. Astrophys. 326, 287-299 (1997)

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2. Computational method

2.1. Atmospheric models

Because the calculation of moderate resolution line opacity over a range of [FORMULA] A is computationally intensive, we consider a restricted sample of six chromospheric models taken from the grid presented by ADB. These models have as their photospheric base a radiative equilibrium model calculated with PHOENIX (Allard & Hauschildt 1995 ). This photospheric model corresponds to a star of [FORMULA] K, log [FORMULA], [FORMULA], and [FORMULA] km s-1. These parameters correspond to a star of spectral type dM0 or dM1 (Lang 1992 ; Mihalas & Binney 1981 ). The PHOENIX model atmosphere code includes many important diatomic and triatomic molecules such as TiO and H2 O in the equation of state and opacity data and has been shown to provide realistic models of early M stars (Allard & Hauschildt 1995 ).

From each of the two model series of ADB labeled [FORMULA] and [FORMULA], we take models with the smallest, largest, and an intermediate value of [FORMULA], which is the mass loading at the onset of the transition region. These two model series differ in the value of [FORMULA] in the chromosphere, with the series [FORMULA] models having a steeper chromospheric temperature rise. Therefore, comparison of spectral diagnostics computed with these two model series allows an assessment of the sensitivity to the location of [FORMULA] and the steepness of the chromospheric gradient (or, equivalently, the thickness of the chromosphere). The constancy of the chromospheric [FORMULA] in these models is in keeping with the results of previous semi-empirical modeling of the outer atmospheres of a large variety of late-type stars (cf. Eriksson et al. 1983 ; Basri et al. 1981 ; Kelch et al. 1979 and other papers in those series). The value of [FORMULA] in the transition region, [FORMULA], is also constant in these models, and we have chosen to use the sub-set of the grid that has a value of -6.5. The temperature at the top of the chromosphere where the transition region begins, [FORMULA], is fixed at 8500 K following Houdebine & Doyle (1994 ). We only consider models that are in radiative equilibrium below [FORMULA]. Therefore, by fixing the value of [FORMULA], we also fix the value of [FORMULA], which is the mass loading at [FORMULA]. Table 1 shows the chromospheric parameters of the grid models, and Fig. 1 shows their temperature structure.


Table 1. Parameters of grid models

[FIGURE] Fig. 1. Temperature structure of models in grid. [FORMULA] series: thin dashed line, [FORMULA] series: thick dashed line.

The increase in temperature throughout the chromosphere has an associated increase in the micro-turbulent velocity, [FORMULA]. In these models, [FORMULA] rises exponentially with decreasing [FORMULA] in the chromosphere to a value of [FORMULA] km s-1 at [FORMULA], then rises rapidly in the transition region to a value of [FORMULA] km s-1. A value of 10 km s-1 at the top of the chromosphere is typical of values found for other late-type stars (see, for example, Eriksson et al. (1983 ) and papers in that series).

2.2. Line opacity

We have used PHOENIX to compute for our grid of models the mean of the total mass absorption due to lines, [FORMULA], in [FORMULA] intervals from 500 to [FORMULA], and in [FORMULA] intervals from 25000 to [FORMULA]. The line lists used for the calculation incorporate those of Kurucz (1990 ), which contain about 58 million lines due to atoms and diatomic molecules, as well as the most recent and comprehensive line lists for molecules of particular importance in M stars such as TiO and H2 O. The total line list contains [FORMULA] million lines. The PHOENIX code was originally developed to compute atmospheric models of nova and supernova, therefore, it been proven over a wider range of temperature and density than most codes that are optimized for the atmospheric modelling of M stars. Furthermore, PHOENIX calculates the line opacity by direct Opacity Sampling rather than by pre-tabulated Opacity Distribution Functions. Therefore, we were able to compute [FORMULA] at all depths from the base of the photosphere to a point in the lower transition region around a temperature of [FORMULA] K.

The temperatures and densities in the upper chromosphere and lower transition region in our models correspond to the partial ionization of H I, which is, therefore, the main [FORMULA] donor at those heights. The calculation of the ionization equilibrium of H I in the chromosphere is complicated by severe non-LTE effects. Therefore, the [FORMULA] structure used in the calculation of [FORMULA] is calculated from a multi-level non-LTE solution of the coupled radiative transfer and statistical equilibrium equations for the first five levels of H I using an operator splitting/accelerated lambda iteration procedure that is incorporated into PHOENIX. The [FORMULA] structure and the value of [FORMULA] in the chromosphere may also be affected by the non-LTE ionization of various metals. Therefore, we have also treated in non-LTE some of those species for which PHOENIX incorporates detailed atomic models: the first five levels of Mg II and Ca II, the first ten levels of He I and He II, and the first three levels of Na I.

2.3. Non-LTE hydrogen

We have used the code MULTI (Carlsson 1986 ) to solve the combined radiative transfer and statistical equilibrium equations for an atomic model that incorporates the lowest nine levels of H I and the H II state. Because the chromospheric [FORMULA] density structure is determined by the H I /H II ionization balance, we iterate the non-LTE solution and the equation of hydrostatic equilibrium to convergence. The radiative transfer problem is solved in detail for all [FORMULA] transitions connecting the nine H I states and for the [FORMULA] transitions of these states. The calculation of the background radiation field includes the addition of the PHOENIX line opacities, [FORMULA], to the continuous opacity normally computed by MULTI. These were incorporated using a modified version of MULTI that was presented in ADB. For the [FORMULA] transitions, [FORMULA] was included in the Ly [FORMULA], H [FORMULA], and Pa [FORMULA] lines as a straight arithmetic mean in [FORMULA] intervals. We included blanketing in the first two of these because of their important role in the statistical equilibrium of H I, and in the latter because we wish to develop the detailed line shape of Pa [FORMULA] as a chromospheric diagnostic (Pa [FORMULA] is generally too contaminated by telluric absorption to be a useful). For the [FORMULA] transitions, [FORMULA] was included as a harmonic mean in [FORMULA] intervals. An interval width of [FORMULA] is necessary because normally a sparse frequency sampling of the [FORMULA] continua is used in MULTI calculations in order to control the computation time. A harmonic mean was used because occasional strong lines have a disproportionately large effect on the straight mean in a large wavelength interval.

ADB also included background line opacities computed with PHOENIX in their non-LTE H I/II calculation with the same models. However, they were able to compute [FORMULA] for a photospheric model only. For their chromospheric models they gradually ramped the value of [FORMULA] down to zero, ad hoc, over a decade in column mass density above [FORMULA]. In order to compare our results with previous studies, we have used our procedure to calculate [FORMULA] for a radiative equilibrium model and ramped it down to zero just above [FORMULA] using the same procedure as ADB. We then re-calculated the non-LTE H I/II and [FORMULA] solution using these photospheric [FORMULA] tables. For clarity, we henceforth designate as [FORMULA] the line opacity that reflects the chromospheric and transition region temperature structure, and as [FORMULA] the line opacity that is purely photospheric.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998