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Astron. Astrophys. 322, 266-279 (1997)

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2. Method

In order to examine in detail the formation of the Na I D doublet in cool dwarfs, rather than making an extended study across the HR diagram, we focussed our attention on a specific case. Starting from a photosphere representative of an early M dwarf, we then added a grid of model chromospheres (Sect.  2.2). For each model atmosphere we computed the emerging Na I spectrum using a non-LTE radiative transfer code, MULTI (Carlsson 1986), version 2.0. The code has been modified in several respects, mostly to deal with the overlapping (severe in late-type stars) of the fine structure components in most of the Na I lines, including the resonance doublet at 5890/5896 Å. The treatment of the boundary conditions (Sect.  2.3) and of the background atmospheric opacity (Sect.  2.4) has also been improved. For the line scattering process, the code employs the approximation of complete frequency redistribution of photons within the line profile (CRD).

For each model chromosphere, i.e. for each temperature structure, the electron density, proton density etc., were set to a first guess. A self-consistent solution was then found by solving the non-LTE problem for hydrogen and iterating to include the equation of hydrostatic equilibrium, taking into account the departures from LTE of sodium. As a by-product of this procedure, along with the sodium lines, we simultaneously obtain the hydrogen spectrum. This provides the opportunity to compare the Na I lines with other chromospheric diagnostics, such as H [FORMULA]. The results are discussed in Sect.  3; in the following sections we will describe the procedures employed in the calculations.

2.1. Model atoms

The H model atom comprises 9 levels plus continuum. The transfer of radiation in all the radiative transitions (36 lines and 9 continua) was explicitly computed in non-LTE. The collisional bound-bound and bound-free cross-sections used in the rate equations are from Johnson (1972).

The Na model atom used in our calculations has been derived essentially from CGS and Bruls et al. (1992); its term structure includes 10 bound levels (up to n=7; some of the highest lying terms were grouped together), plus the ground states of [FORMULA] and [FORMULA].

It is usually assumed that electrons give the main contribution to collisional transition rates in stellar atmospheres. However, in cool atmospheres the electron density can be so low that other perturbers may become important, hydrogen in particular. We have followed CGS in their adaptation of the cross sections of Kaulakys (1985, 1986) for the excitation of the first 6 levels of Na by collisions with hydrogen atoms. The Kaulakys formula, valid for Rydberg atoms, is expected to give fair estimates for alkali atoms. An alternative approach following Drawin (1968, 1969), generalised by Steenbock & Holweger (1984) and, for forbidden transitions, by CGS, leads to cross sections that are usually much larger, even by a factor [FORMULA]. However, there are indications that the latter cross-sections may be strongly overestimated (CGS, Pavlenko & Magazzù 1996, and references therein). We therefore adopt the cross-sections from the Kaulakys approach, but in Sect.  3.3 we will also compare the results obtained with the Drawin formulae for a sample model. For a more detailed discussion on the Kaulakys and Drawin rates we refer to CGS. We feel we should only mention here that their Eq. 1 presents several typographical errors, that however do not reflect errors in their actual calculations.

One of the main differences with the previous Na model atoms, is the fact that we accounted for the fine structure of almost all the transitions in the model atom. The oscillator strengths for the fine structure components in the line have been computed from the total oscillator strength of the transition assuming LS coupling. Moreover, in the calculation of the line profiles we assumed that the relative populations of the sublevels are proportional to their degeneracies. This approximation also considerably simplifies the rate equations.

Another difference is the inclusion of a further stage of ionisation ([FORMULA]). Given the combined effect of the low ionisation potential of [FORMULA] (5.139 eV) and of the high ionisation potential of [FORMULA] (47.286 eV), sodium is mostly in the form of [FORMULA] in the photosphere and chromosphere of solar-type stars. However, in active stars, the wavelength range below the photoionisation threshold for [FORMULA] (262 Å) is rich with coronal emission lines. Since it is not evident a priori that such a coronal illumination has no influence on the ionisation balance of sodium in the chromosphere, we have studied this effect in some of the models. Hence the necessity of including at least the ground state of [FORMULA] in the model atom. For the ground state of [FORMULA], the photoionisation cross-section was obtained from the Opacity Project database (e.g. Seaton 1987; calculations by Scott 1996), while the collisional ionisation rate is from Landini & Monsignori Fossi (1990);

As for the treatment of the transfer of line radiation in the Na I D lines, the approximation of CRD is adequate, at least in main-sequence stars (Kelch & Milkey 1976). From a computational point of view, this is a considerable simplification with respect to the more general case of partial redistribution (PRD). This fact may be regarded as an advantage for the D lines over other more widely used chromospheric diagnostics (e.g Ca II or Mg II resonance lines), for which CRD is not as good an approximation.

2.2. Model atmospheres

The model photosphere, from Allard & Hauschildt (1995b), corresponds to a star with effective temperature [FORMULA]  K, [FORMULA] (g, gravity acceleration, in cm s-2) and solar metallic abundances. The model belongs to version 5 (in preparation) of the grid obtained with Allard & Hauschildt's code PHOENIX. Such a grid of models differs from the previous one (Allard & Hauschildt 1995a) in several aspects, notably in improved molecular opacities.

We have superimposed to that base photosphere a grid of model chromospheres, each one described by an ad hoc thermal structure [FORMULA] (m is the column mass, in g cm-2). At a given position of the onset of the transition region, specified by the pair [FORMULA], we have considered four types of chromospheric thermal stratification, as in Fig. 1.

[FIGURE] Fig. 1. Four chromospheric structures we have grafted to the photospheric structure of Allard & Hauschildt (1995b). For these models, the transition region starts at [FORMULA]. Each chromospheric structure is labeled in the close-up plot; see text for an explanation of labeling. The base model photosphere is designated by a thick solid line.

Models [FORMULA] and [FORMULA] are constructed using segments where the temperature varies linearly with [FORMULA]. Models B ([FORMULA] and [FORMULA]) are identical to the corresponding A models in the upper chromosphere, while the temperature gradient in the lower chromosphere has been steepened, causing the temperature minimum to be pushed further out. The main reason for this choice of models is that we want to probe the sensitivity of Na I lines to the structure of the lower chromosphere.

In the construction of the models, another relevant parameter is the temperature gradient in the transition region. Again, we have chosen to assume a linear dependence of T with [FORMULA], or a constant gradient [FORMULA]. For a given chromospheric structure, we have considered the values [FORMULA] and 7, thus bringing to eight the number of model atmospheres at a given position of the onset of the transition region.

As for [FORMULA], we fixed its value to 8500 K, following Houdebine & Doyle (1994). More important in determining the "activity" of a model chromosphere is its pressure. Following the procedure outlined by Andretta & G iampapa (1995), we generate more "active" states from a given "quiet" chromosphere by translating the temperature structure toward higher mass columns. Thus, the variation of the pressure in the chromosphere and transition region can be conveniently parametrised by the mass loading at the top of the chromosphere, [FORMULA]. More specifically, we considered the models with [FORMULA] as being representative of a quiet chromosphere. We then generated a series of more active models by increasing the mass loading up to [FORMULA]. In particular, nine values of [FORMULA] spanning this interval were considered. This procedure has been followed for each one of the eight "basic" models we have already described, thus obtaining eight series of model atmospheres, for a total of 72 chromospheric structures.

As an example, Fig. 2 shows three sample models belonging to the series with chromosphere structure of type [FORMULA] and [FORMULA].

[FIGURE] Fig. 2. Run of temperature for three models belonging to the series of model chromospheres generated (see text) from a chromosphere structure of type [FORMULA] (see Fig. 1) and [FORMULA]. The three models shown here correspond to [FORMULA] (solid line), -4.8 (dotted line) and -3.8 (dashed line).

Note that, in hydrostatic equilibrium, the gradient [FORMULA] is related to the gradient [FORMULA] (h is the height above the photosphere) via the equation: [FORMULA], where H is the pressure scale height, which depends only on temperature and on the molecular mean weight. Therefore, the adopted scaling with [FORMULA] preserves [FORMULA] in the transition region and, less accurately, in the chromosphere (the mean molecular weight in the transition region is practically constant, whereas in the upper chromosphere it depends, among the other things, on the non-LTE ionisation equilibrium of hydrogen).

Microturbolence, [FORMULA], is another ingredient in the construction of the model atmospheres, if less important than the temperature structure. In the photosphere we adopted a constant value [FORMULA]  km/s. In the chromosphere we gradually increased the magnitude of the microturbolence field as [FORMULA], where [FORMULA] is the location of the temperature minimum, and [FORMULA] is the corresponding value of temperature. This function roughly matches the distribution in the standard solar model chromosphere from Vernazza et al. 1981, but it is otherwise arbitrary. For most part of the model chromospheres, the resulting microturbolence does not exceed a few kilometers per second, up to about 5 km/s at [FORMULA] in models 1B, 2A and 2B. For the non-quiescent models the same scaling with [FORMULA] as for the temperature distribution has been adopted.

2.3. Coronal back-radiation

As anticipated in Sect.  2.1, the presence of often vigorously active coronae in chromospherically active stars can potentially affect the ionisation equilibrium of sodium. In fact, the structure of the transition region and upper chromosphere in M dwarfs can be altered by coronal back-radiation (Cram  1982). However, we will not deal explicitly with the problem of determining the structure of an X-ray illuminated atmosphere. Rather, we will limit ourselves to the simpler problem of determining the ionisation equilibrium of sodium in the presence of an XUV flux capable of ionising [FORMULA].

In order to study this problem, parallel to the model calculations with the temperature structures determined as described in the previous section with no coronal illumination, we have also considered the case of a non-zero coronal flux as a boundary condition atop the model atmosphere.

The spectrum of the coronal flux in the XUV below 262 Å, the photoionisation threshold for [FORMULA], has been obtained from the observed solar irradiance at activity minimum, as modelled by Tobiska (1991). These intensities, representative of an "average" solar quiet corona, have been multiplied by a factor [FORMULA] ; the result is shown in Fig. 3.

[FIGURE] Fig. 3. Adopted coronal XUV distribution; the photoionisation threshold for [FORMULA] is also marked.

The corresponding surface total flux in the soft X-band 0.15-4.0 keV is [FORMULA] ([FORMULA] in ergs cm-2 s-1); assuming a radius [FORMULA], and a corona homogeneously distributed over the stellar surface, the resulting total luminosity in the same band is [FORMULA], while its ratio to the bolometric luminosity (for [FORMULA]  K) is [FORMULA]. These values correspond to a corona of a considerably active M star (Doyle & Butler 1985). Therefore they are intended to be used for upper-limit estimates of the effect of coronal radiation on the sodium spectrum.

2.4. Background opacity

We have already emphasised that the photospheric radiation field in the ultraviolet can be crucial in determining the formation of the Na I D line cores and, more generally, of the Na I spectrum in the presence of a chromosphere. Moreover, the resonance doublet exhibits remarkably broad wings in M dwarfs, that can span tens of Å ngströms, an interval over which noticeable changes in the photospheric emission can occur. It is thus important to correctly evaluate the background photospheric emission.

The non-LTE code used here, MULTI, evaluates the photospheric radiation field from a given model atmosphere using a variety of continuum opacity sources. In M dwarfs, however, a very large number of atomic and molecular lines gives an important, even dominant contribution, to the plasma opacity. Therefore, we have added to the "standard" MULTI calculations, opacities from atomic and molecular lines in the form of an opacity table, provided again by Allard & Hauschildt (1995b) along with their atmospheric model (see Sect.  2.2).

In the table, the opacity dependence upon wavelength is given at each depth point of the model atmosphere with a step of 2 Å. Such a wavelength sampling is adequate for the calculations of photoionisation and photoexcitation rates, and for a comparison of the resulting synthetic spectra with low resolution observations.

One difficulty is that the opacity data used here are model-specific. However, in the model atmospheres constructed as described in Sect.  2.2, the photosphere is left unaltered. Therefore, in that region it is still correct to use the tabulated additional opacities. For the higher chromospheric temperatures, the standard MULTI background opacities were used. Thus, in our calculations we have included the additional background opacities only in the photosphere, that is, in the region below the temperature minimum. Above that region, the additional opacities have been ignored, except for a smooth "transition" region, spanning a decade in column mass. Such a smooth zeroing of the additional line opacities has been introduced in order to mimic the dissociation of molecular species in the chromosphere (for this specific model, molecules are the main contributor to the total opacity in the temperature minimum region and for most parts of the photospheric spectrum).

Another problem that requires a proper treatment of plasma opacity, is the transfer in the chromosphere of XUV photons below the [FORMULA] photoionisation threshold. In addition to the H and He opacities already included in the code, we have taken into account inner-shell photoionisation by the most abundant metals, following Avrett & Loeser (1988).

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998