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Astron. Astrophys. 347, 617-629 (1999)

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5. Results

5.1. Investigated range of parameters

Density and temperature: The hydrogen particle density [FORMULA] and the kinetic temperature of the gas T are varied within the intervals [FORMULA] and [FORMULA]K, respectively. The chosen ranges should cover most of the conditions present in the outer atmospheres, the chromospheres, and the inner parts of the circumstellar envelopes of both non-pulsating as well as pulsating stars, where shock waves can heat up the gas considerably.

Continuous background radiation field: The gas is assumed to be optically thin in the continuum and to be exposed to diluted star light, which is regarded as the typical situation for the gas in stellar winds. For the general purpose of this paper, we will consider a diluted Planckian background radiation field

[EQUATION]

which roughly describes the spectral radiative energy distribution close to a stellar surface. [FORMULA] is the Planck function, [FORMULA] a radiation temperature and [FORMULA] a dilution factor. The latter may be identified with the fraction of the solid angle occupied by the stellar disk with radius [FORMULA] at radial distance r

[EQUATION]

In this case, [FORMULA] where [FORMULA] is the stellar temperature. If continuum optical depths effects are important in the circumstellar envelope, [FORMULA] is usually smaller than [FORMULA] and [FORMULA] is larger than predicted by Eq. (22), but Eq. (21) still provides a reasonable fit. We consider radiation temperatures between 0 and 5000 K in this paper.

For reasons of comprehensiveness, we will focus our interest in the present paper on a particular radial zone around [FORMULA] ([FORMULA]). This region has proven to be important for the generation and the acceleration of cool star winds in many publications (e. g. Bowen 1988, Fleischer et al. 1992, Linsky et al. 1998). The depicted results are, however, similar to those obtained with other choices for r.

Velocity gradient: In spherical symmetry, the mean velocity gradient [FORMULA], which enters into the determination of the escape probabilities, can be expressed in terms of [FORMULA] and [FORMULA] (Woitke et al. 1996). Characteristic values in stationary stellar winds are [FORMULA] where x is of the order of unity at the sonic point and decreases to [FORMULA] 0.0067 at [FORMULA]. In the shocked atmospheres and envelopes of pulsating stars, the velocity gradients are much larger, roughly [FORMULA], where [FORMULA] is the velocity of a nearby shock wave and [FORMULA] is a typical distance between the shock waves. We will mostly consider a value of [FORMULA], which is a typical value found in models for long-period variables (e. g. Fleischer et al. 1992). The influence of this parameter is discussed in Sect. 5.5.

5.2.Results as function of n(H), T, and Trad

Figs. 4, 5 and 6 show the resulting concentrations and heating/cooling rates of Fe I and Fe II in the (temperature/density)-plane for three different choices of the background radiation field.

1) For negligible continuous radiation fields (Fig. 4), iron is found to be predominately neutral up to temperatures of about 9000 K, triggered by the balance of collisional ionisation and photo-recombination. Since the rates of both processes scale linearly with [FORMULA], fractional ionisation [FORMULA] is mainly a function of temperature which results in essentially horizontal contour lines in the upper row of plots in the figure. The maximum possible concentration of Fe I or Fe II is [FORMULA]. The degree of ionisation of the gas [FORMULA] (middle plot, upper row) shows a similar behaviour: The gas is found to be almost completely neutral below about 3000 K and about fully ionised beyond 10000 K. In comparison to LTE, the fractional ionisation of the gas is always lower, because the rates of photo-ionisation are zero, but the rates of photo-recombination are non-zero. At high densities and very low temperatures, iron is mainly present as Fe(OH)2 according to the model.

According to these results, the radiative cooling by iron in the case without background radiation field is dominated by Fe I up to temperatures of about 9000 K (see lower row of plots in Fig. 4). The magnitude of the cooling rate is remarkable: For example, at [FORMULA] and [FORMULA]K we find a cooling rate of [FORMULA], which translates to a rate of temperature decrease of [FORMULA]K/day due to Fe I-cooling alone.

The depicted dependency of [FORMULA] [erg/g/s] on temperature and density is complex. At large densities ([FORMULA]) and temperatures 1000 K ... 4000 K, we find approximately horizontal contour lines for [FORMULA], which means that the iron cooling rates per mass are roughly density-independent in this regime: [FORMULA], which is a typical LTE-feature (see Sect. 5.3). At small densities, however, the contour lines are roughly vertical: [FORMULA]. The reason for this behaviour is the insufficient collisional energy supply for the emitting states, which linearly reduces the cooling rates per mass with decreasing density (see Sect 6). The iron cooling is found to have a maximum efficiency around [FORMULA] at high temperatures, but around [FORMULA] at low temperatures, which indicates that different types of lines are responsible for the cooling at different temperatures. Consequently, neither [FORMULA], nor [FORMULA], nor some other simple formula provides a reasonable approximation in the whole (temperature/density)-plane.

2) Fig. 5 shows the results for a diluted 3000 K-background radiation field. The concentrations are quite different from the case discussed before. Only at large densities [FORMULA] and low temperatures [FORMULA], iron is still predominantly neutral. In the remaining part of the (temperature/density)-plane, iron is present mainly in form of Fe II. Accordingly, the electron concentration is much higher as compared to the case without continuous background radiation. A minimum electron concentration of about [FORMULA] to [FORMULA] is preserved even at very low temperatures due to photo-ionisation of metal atoms like Ca, Mg, Al, Si and Fe, depending on the density.

The cooling rates at high temperatures ([FORMULA]K) are similar to those depicted in the previous figure, indicating that the background radiation only slightly disturbs the main energy fluxes between the various iron levels at high temperatures. According to its higher concentration, Fe II is the main coolant in most parts of the (temperature/density)-plane. Its maximum cooling efficiency is located around [FORMULA] at high temperatures, and around [FORMULA] at low temperatures. Due to the background radiation, now both Fe I and Fe II cause radiative heating at low temperatures (see dashed contour lines). The complicated slope of the ([FORMULA])-lines (thick full lines in the lower plots) are difficult to understand. Apparently, some especially efficient heating by forbidden Fe I-lines 6 takes place at [FORMULA], whereas for Fe II, some especially strong fine-structure cooling around [FORMULA] is active 7. In any case, the heating/cooling by Fe I and Fe II will drive the gas towards the respective ([FORMULA])-line.

3) Fig. 6 shows the results for a diluted 5000 K-blackbody radiation field. The Fe II and electron concentrations are again higher when compared to the previous cases. Fe I can only exist (in considerable amounts) under high density conditions. This result is rather independent from the gas temperature, which indicates that collisional ionisation and three-body recombination are unimportant, unless the temperature exceeds 15000 K where the results are in fact similar to the cases discussed before.

[FIGURE] Fig. 4. Contour plots of the results without background radiation field as function of total hydrogen density [FORMULA] and temperature T. The upper row of three contour plots shows the logarithmic concentrations of Fe I, electrons, and Fe II, respectively, with respect to hydrogen, e. g. [FORMULA]. The lower two contour plots depict the cooling rates per unit mass of the gas caused by Fe I and Fe II, respectively, e. g. [FORMULA]. The small numbers indicate the values of the contour lines. Additional parameter: [FORMULA].

[FIGURE] Fig. 5. Same as Fig. 4, but including a background radiation field with radiation temperature [FORMULA] [FORMULA] [FORMULA]K and radial distance [FORMULA] [FORMULA] 2. The thick full line in the lower two diagram indicates Q [FORMULA] 0. Underneath this line, the dashed contour lines indicate radiative heating rates , e. g. [FORMULA]

[FIGURE] Fig. 6. Same as Fig. 4, but including a background radiation field with [FORMULA]K and radial distance [FORMULA].

The Fe II heating/cooling rate, which mostly dominates over Fe I, shows an interesting shift with respect to density between maximum cooling (at [FORMULA], solid lines) and maximum heating (at [FORMULA], dashed lines), once more suggesting that different types of lines are responsible for the heating (at low T) and the cooling (at high T), respectively. Furthermore, regarding the depicted contour lines of [FORMULA] in Fig. 6, the cooling decreases faster than the heating with decreasing density. Concerning Fe II, the heating rate only varies by 2-3 orders of magnitude, whereas the cooling rate varies by [FORMULA] 6 orders of magnitude. Consequently, the temperature where heating and cooling balance each other [FORMULA] is density-dependent and has its lowest value at high densities for both Fe I and Fe II (disregarding densities [FORMULA]), where the cooling is strongest. This general behaviour is also found for the other atomic heating/cooling rates (see Table 1).

5.3. Comparison with limiting cases

For hydrodynamical calculations it is important to know where certain limiting cases can be applied in order to minimise the numerical efforts for the proper calculation of the energy exchange rates between matter and the radiation field. It is very convenient, for example, to use the LTE approximation

[EQUATION]

where [FORMULA] is the local mean intensity and [FORMULA] the Planck function. [FORMULA], in principle, is the full frequency-dependent true absorption coefficient which includes all lines, free-free and bound-free continua of the gas, but no scattering. It is difficult, though, to use Eq. (23) directly, because often (i) only a grey radiative transfer can be performed and detailed knowledge of [FORMULA] is not available, or (ii) only a static radiative transfer can be performed, which ignores the Doppler-shifts of the lines thereby introducing errors in the calculation of [FORMULA] just around those wavelength-positions where [FORMULA] is strongest (at the wavelengths positions of optically thick lines).

When only a grey radiative transfer is available, the usage of mean opacities [FORMULA] (Rosseland or Planck means) is a common procedure

[EQUATION]

For instance, Fokin e. g. (1992) applied Rosseland means in the energy equation concerning various non-linear pulsation models for RR Lyrae, [FORMULA] and [FORMULA] Cepheids, and RV Tauri stars. Gauger et al. (1993) used Rosseland means in the energy equation for models of long-period variables. Höfner et al. (1998) used Planck means for dynamic model atmospheres of AGB stars.

Other authors prefer to use pre-calculated cooling laws, as for example Bowen (1988) and Cuntz (1990), where the assumption [FORMULA] is often made. However, this assumption is only valid in the low-density limiting case (see Sect. 6), where the cooling rates are limited by the amount of energy transfered to the emitters via collisions. If extrapolated to larger densities, this approach ignores optical depths effects in the lines.

In order to study the applicability of these two limiting cases, we have recalculated the Fe I and Fe II heating/cooling rates with the following modifications:

  • i) The LTE-heating/cooling rate [FORMULA] is obtained by assuming a Boltzmann distribution within the bound levels. The escape probabilities are included.

  • ii) The heating/cooling rate in the optically thin limit [FORMULA] is calculated by putting all [FORMULA], obtaining a new solution of the statistical Eqs. (1) and calculating the radiative heating/cooling rates for this case.

In both cases, the electron density and the Fe I/Fe II ratio have been left unchanged.

Fig. 7 shows the results as function of density for a particular choice of temperature, radiation field and velocity gradient. In this case, the LTE-assumption yields approximately the same heating/cooling rate like the full non-LTE calculations at large densities [FORMULA] (corresponding to [FORMULA]) concerning both Fe I and Fe II. Good agreement with the results in the optically thin limit are achieved for low densities [FORMULA] (Fe I) and [FORMULA] (Fe II), respectively. It is difficult, however, to generalise these findings to other conditions or other species, although the LTE-assumption seems to work reasonably fine for electron densities [FORMULA]. If the velocity gradient is increased by a factor of a 100, this "critical" electron density increases by a factor of about 10.

[FIGURE] Fig. 7. Comparison to the results obtained in the LTE limiting case [FORMULA] (including optical depth effects) and in the non-LTE optically thin case [FORMULA]. Left hand side : Fe I. Right hand side : Fe II. The upper diagrams shows the concentrations with respect to hydrogen. The thick full grey lines in the lower diagrams indicate the heating/cooling rates Q from the full non-LTE calculations including optical depths effects. Contributions by permitted, forbidden, fine-structure and bound-free transitions to this rate are also shown. Note the two-fold logarithmic scaling of the y-axis for heating and cooling as indicated. Fixed parameters are [FORMULA]K, [FORMULA]K, [FORMULA] and [FORMULA].

It is absolutely necessary to account for the optical depths effects in the lines. Ignoring these effects in the non-LTE case leads to an overestimation of the cooling rates by up to 5 orders of magnitude (see Figs. 7 and 9). When applying the LTE approximation and ignoring the optical depths effects (i. e. if both LTE and [FORMULA] is assumed) 8, the deviations are even larger as a roughly constant value of [FORMULA] is found in this case. These results cannot even be depicted in Fig. 7.

Fig. 7 also shows the individual contributions of bound-free transitions and permitted, forbidden and fine-structure lines. Note that for large densities, the bound-free transitions of Fe I actually dominate, whereas for medium densities, forbidden and permitted lines are more important concerning both Fe I and Fe II. At low densities, fine-structure (and forbidden) lines cool the gas at the same time as permitted lines heat it, which indicates that the upper levels of the cooling transitions are radiatively pumped by the permitted lines.

5.4. Which lines are important?

Fig. 8 indicates where in the ([FORMULA])-plane the different types of radiative transitions contribute most to the heating and cooling by Fe I and Fe II, depending on the background radiation field. In the case [FORMULA], heating cannot occur and, consequently, the upper panels for heating are empty in Fig. 8. Considering trajectories of roughly decreasing temperature and density through the plane, we find in this case a certain sequence of the most important cooling process: bound-free - permitted - forbidden - fine-structure. In cases where a background radiation field is included, the results are more complex and the permitted lines seem to gain ground especially as heating agents. However, at small densities, the energy absorbed by the atoms in permitted and even in forbidden lines is partly re-emitted in other lines, usually at longer wavelength. Therefore, heating and cooling is important at the same time. Analysis of the rates in such cases reveal rich fluorescent pumping effects, e. g. the upper levels of fine-structure transitions are pumped by absorption in a permitted line followed by re-emission in another line of the same multiplet.

[FIGURE] Fig. 8. Sites in the ([FORMULA])-plane where the indicated type of radiative transition dominates the cooling (r.h.s.) and the heating (l.h.s.) by iron. Left column: Fe I - right column: Fe II. Blank regions indicate that none of the radiative processes is important for cooling or heating, respectively. The parameter [FORMULA] is varied from zero (upper row) to 5000 K (lower row). Fixed parameters are [FORMULA] and [FORMULA].

5.5. Influence of the velocity gradient

The influence of the velocity gradient [FORMULA] on the results for the total cooling rate is demonstrated for Fe I in Fig. 9. At large densities, the cooling rates caused by line emission are strongly affected by self-absorption in the lines, which efficiently blocks the photons. At large Sobolev optical depth [FORMULA], this blocking reduces the cooling rates linearly with increasing density (see Eq. 6), since the escape probability scales as [FORMULA] and [FORMULA]. The figure indicates that this blocking is already relevant at [FORMULA], although the full effect develops at larger densities. This suggests that the cooling at medium densities is a superposition of optically thin and optically thick lines. At large densities, all the relevant lines become optically thick, and the blocking acts at full strength.

[FIGURE] Fig. 9. Influence of the velocity gradient on the total cooling rate of Fe I. The labels indicate [FORMULA] in units of [FORMULA]. The ([FORMULA])-curves are calculated by assuming [FORMULA] in each case (see Sect. 5.3). Other parameters are [FORMULA] K, [FORMULA]K and [FORMULA].

5.6. Results as a function of the size of the model atoms

Fig. 10 shows the cooling rates of Fe II as function of the size N of the model atom. If only very few levels are included ([FORMULA]), the model atom actually comprises only (parts of) the ground state quintet [FORMULA] and, consequently, only accounts for fine-structure cooling. As the number of levels is increased, first the forbidden lines come into play, which are most effective for the chosen parameters in this plot. At [FORMULA], the first permitted lines enter into competition. However, these are not as important as the forbidden lines. A further increase of N beyond a value of about 60 does not lead to any significant changes in the cooling rates, although the total number of lines still increases about quadratically with increasing N. We note that the energy of the 60[FORMULA] level [FORMULA], which may serve as a rough guide on how much levels are to be included for a proper calculation of Q. However, the convergence of the results as a function of N depends on the relative importance on the different contributions of fine-structure, forbidden and permitted lines, and therefore depends on the physical conditions (see Fig. 8). For instance, if the permitted lines are most important (e. g. at high temperatures), more levels have to be included to reach convergence.

[FIGURE] Fig. 10. Contributions of permitted, forbidden and fine-structure lines to the total Fe II cooling rate Q as function of the number of included levels N of the Fe II model atom. Parameters for this plot are [FORMULA] K, [FORMULA], [FORMULA]K, [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: June 30, 1999
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