Astron. Astrophys. 333, 219-230 (1998)

4. Results

The statistical equilibrium of Mg I in the solar photosphere is similar to that found for Al I (Baumüller & Gehren 1996). This is not unexpected since both atomic systems are dominated by their low-level photoionization in the UV. In fact, taking into account that Mg I is mainly photoionized from the triplet ground state which is connected to the singlet ground state by an intercombination line whereas Al I is photoionized directly from its ground state, the similarity in the population pattern found for the levels of different excitation is striking. The completeness of the atomic model has been investigated with a variety of different model atoms, and the result is the atomic model presented in Fig. 1. Additional tests were devoted to the formulation of the hydrogen collision rate scaling factor  . The results refer to the following models,

• (A) LTE populations
• (B) statistical equilibrium with hydrogen collisions according to the original formula of Drawin (1969), and
• (C) statistical equilibrium with hydrogen collisions from Drawin (1969) but scaled exponentially with upper level excitation energy as described in Sect. 2.2.2.
• (D) statistical equilibrium with no hydrogen collisions

Model (C) is the non-LTE reference model which is our standard unless stated otherwise. Its implications will be discussed in more detail below (see Sect.  5).

4.1. Departure coefficients

Fig. 4 shows the typical run of some of the more important level populations with optical depth. Here, the departure coefficients are defined as usual as the ratio between statistical equilibrium and thermal (Saha-Boltzmann) number densities. Note that in spite of relatively strong collisional interaction - including neutral hydrogen atoms - all level populations already deviate from their LTE values at optical depths near unity. This underpopulation is the result of the large photoionization rate which is known to dominate the near-UV spectra of cool stars such as the Sun. It is important to recognize that the deviations among the departure coefficients of different levels are reduced by increasing the contributions of the hydrogen collisions. Such effects as well as corresponding changes of the electron collisions have been discussed by Baumüller & Gehren (1996).

Fig. 4. Departure coefficients for some levels of Mg I in the solar atmospheric model GRS88.

4.2. Line profiles

The synthetic line profiles shown in Figs. 5 and 6 are all calculated on the basis of the standard line data set used also for the determination of the statistical equilibrium (cf. Table 1). The lines are compared with solar flux observations taken from the Kitt Peak Atlas (Kurucz et al. 1984) and with solar intensity profiles at three disc positions from the infrared atlases of Farmer & Norton (1989) and Brault and Noyes (1983).

Fig. 5. Synthetic flux profiles of selected Mg I lines compared with the observed spectrum of the Kurucz et al. (1984) solar flux atlas (continuous line). The line profiles refer to LTE (Model A, dot-dashed), and to the final statistical equilibrium model (C) with hydrogen collisions scaled exponentially with excitation energy (dashed). (top left): intercombination line 3s  S - 3p    at 4571 Å . (top right): excited line 3p   - 4s  S at 11828 Å . (middle left): 3d  D  - 6f  transition at 8213 Å . (middle right): excited line (4s  S - 5p  ) at 8923 Å . (bottom left): 3p    - 5d  D transition at 4703 Å . (bottom right): 3p    - 4d  D transition at 5528 Å .

Fig. 6. Synthetic flux profiles of selected Mg I lines compared with the observed spectrum of the Kurucz et al. (1984) solar flux atlas (continuous line). (top): 3p    - 3d  D  transition at 8806 Å . (left): line profiles for LTE and non-LTE reference model (C). (right): line profiles for non-LTE model with hydrogen collisions according to Drawin's (1969) original formula (model B, dot-dashed), and with no hydrogen collisions (model D, dashes). All other line profiles refer to LTE (Model A, dot-dashed), and to the final statistical equilibrium model (C) with hydrogen collisions scaled exponentially with excitation energy (dashed). (middle left): Mg I b line (3p   - 4s  S) at 5167 Å  including strong Fe blend. (middle right): Mg I b line at 5172 Å . (bottom left): Mg I b line at 5183 Å . (bottom right): 4s  S  - 6p  triplet near 6319 Å

Table 1. Atomic data for Mg I line synthesis

The synthetic flux profiles have been convolved with a rotation velocity of km/s and a Gaussian macroturbulence distribution of km/s. This profile is derived from a series of fits to solar lines of different strength. Line broadening is partly treated in a semi-empirical mode where the constants are evaluated from solar line fits. Damping due to the quadratic Stark effect has only been included for the 12 µm IR line profiles. When computing broadening constants according to the Lindholm theory (cf.  Hunger 1960), the quadratic Stark effect becomes the dominant source of line broadening for most Mg I transitions involving , except for S or P terms. This is at variance with the semi-classical impact approximation of Cowley (1971) or Freudenstein and Cooper (1978) which at solar atmospheric temperatures leads to damping parameters roughly an order of magnitude smaller (a factor of 15 for the 12 µm line; see also Chang & Schoenfeld 1991 and Dimitrijevic & Sahal-Bréchot 1996). Note also that the often used approximation of a mean interacting nearby level such as the introduced by Freudenstein and Cooper (their Eq. (22)) may lead to substantially smaller values for for nearly all levels. For the 7i I term the corresponding factor is 600. Whereas in most other transitions van der Waals damping rules, the 12 µm lines are dominated by the quadratic Stark effect for which we have used the values given in Table 1. The same broadening constants were used for all models (A) to (D).

In the following we discuss the properties of selected groups of lines.

4.2.1. Intercombination line

In the solar photosphere the intercombination line of Mg I at 4571 Å  has been inferred from NLTE model synthesis to be formed completely under thermal excitation conditions (e.g.  Altrock & Cannon 1972; Altrock & Canfield 1974; Mauas et al. 1988). Carlsson et al. (1992) point out that the 4571 Å  line is the only line in the optical spectrum of the Sun producing a line centre emission reversal at the solar limb (where the line-of-sight crosses the temperature minimum). We confirm that the difference between the LTE and non-LTE line profiles in model (C) with the exponentially scaled hydrogen collisions is negligibly small, corresponding to dex. As a result of the strong photoionization in the statistical equilibrium the opacity of the intercombination line is reduced by %, and the line is therefore formed slightly deeper in the atmosphere under non-LTE conditions compared to that calculated assuming LTE. This explains the small difference in the line core and the resulting abundance change evident from Fig. 5.

4.2.2. Excited lines

Inspection of the synthetic line profiles displayed in Figs. 5 and 6 reveals that - mainly as a consequence of decoupling of the level populations due to the strong photoionization - the excited Mg I lines show quite different non-LTE effects, predominantly in the line cores. Whereas most of the lines such as 11828, 4703, 5528, and 8806 Å  have line cores that are too weak, those of 8923 and 6319 Å  are too strong as compared with the solar flux spectrum. The latter group of lines seems to arise from the more excited levels above 5 eV; therefore they provide an important test for the Mg I atomic model.

11828 Å : The Mg I 11828 Å  line displayed in Fig. 5 marks the transition from level 3p    to 4s  S. Together with 8806 Å  and 8923 Å  it strongly emphasizes the need to include hydrogen collisions in the Mg I model. Model calculations with constantly increased electron collision rates do not produce the necessary run of the departure coefficients since electron collisions are more important in the inner part of the photosphere whereas the profile diverges from LTE in the line core which is formed at optical depths around   = 0.001 . The explanation for the dominance of hydrogen collisions in these outer regions is buried in the fact that the density ratio increases by an order of magnitude between   = 1 and   = 0.01. The non-LTE calculation with the exponentially scaled hydrogen collisions (C) thus provides the best fit to the profile (see also the 8806 Å  line in Fig. 6), and any replacement introducing enhanced electron collisions instead would require an individual adjustment of many lines which increases the number of free parameters even more. It should also be noted here that increasing the electron collision rates will lead away from the impact approximation as discussed in Sect. 2. Which of the two alternatives represents the real plasma can in principle be decided by analyses of cool metal-poor stars that have a significantly greater density ratio in their atmospheres, if the lines under consideration could be observed there. Such an investigation has been followed in the case of Al I (Baumüller & Gehren 1997), and a corresponding analysis is planned for Mg I. The difference in the abundance determination between the LTE and non-LTE calculations amounts only to dex, because the deficiency of the LTE profile in the line core is mostly compensated by slightly stronger LTE line wings.

8213 Å : Mg I 8213 Å  couples 3d  D  with the 6f  F level. 8213 Å  is similar to 8923 Å  and to the 6319 Å  triplet in that it displays an LTE line profile that is notably stronger than its non-LTE counterpart. This is a reversal of the behaviour of the 11828 and 8806 Å  lines, and Fig. 4 outlines that it is due to the different coupling of the and terms to the terms at lower energies and to the and terms at higher energies. As explained above the and terms are depopulated by strong photoionization, however, with collisonal coupling becoming less important (a) for optical depths decreasing, and (b) for energy increasing. In particular the coupling to highly excited energy levels is considerably less effective than that between the and (or ) terms, and there will be a net cascade of electrons that populate the and terms without fully thermalizing the term. However, the 8213 Å  line is not only affected at the line core but over a significant fraction of its wings (see Fig. 5). Consequently, the abundances determined from non-LTE and LTE differ by dex.

8923 Å : The Mg I 8923 Å  line arises from the transition of 4s  S to 5p  P. Its line core displays a dominant non- LTE influence. Again, 8923 Å  has a deeper LTE line core than that of the non-LTE calculation (see Fig. 5). The abundance difference between non-LTE and LTE results is also large with dex.

4703 Å : Mg I 4703 Å  couples the first excited singlet level 3p    with 5d  D. Though it is not as strong, this line has a similar behaviour to the Mg I b triplet. The 4703 Å  profile fit is shown in Fig. 5. The difference between the abundance determinations using either non-LTE or LTE calculations is small, with dex.

5528 Å : The Mg I 5528 Å  line marks the transition from 3p    to 4d  D. This line displays a clear increase in the deviation from LTE at the line core as compared with the 4703 Å  line. The 5528 Å  profile fit is shown in Fig. 5. The difference between the abundances determined in the non-LTE and LTE calculations is dex.

8806 Å : Mg I 8806 Å  line is the transition from level 3p    to 3d  D, i.e.  the leading line of the singlet series. This line therefore shows the strongest deviation from LTE in the line core (see Fig. 6). 8806 Å  has an exceptionally strong isotopic shift, and the three components corresponding to ²⁴ Mg , ²⁵ Mg and ²⁶ Mg have intensities in the ratio of 10:2:2 with wavelengths at 8806.7678, 8806.7358 and 8806.7032 Å , respectively (Meißner 1938). The line plays an important rôle in the determination of the collision rates with neutral hydrogen. We see clearly in Fig. 6 that the Mg I atomic model either without hydrogen collisions (D) or with hydrogen collisions using Drawin's (1969) standard formula (model B) does not fit the observed flux profile. The corresponding scaling factors are   = 0 or 1 for models (D) and (B), respectively, but 1.6 for model (C). Since the line core is formed in the uppermost photospheric layers, it is particularly sensitive to collisions, and a factor of 2 in collision rates both thermalizes the line source function near log   = -2.5 and decreases the line center optical depth thus avoiding strong contributions from upper photospheric layers where the departure coefficient falls below that of . However, the difference in the abundance determinations from the equivalent width using either non-LTE or LTE calculations with model (C) is very small, dex.

Mg b triplet: The Mg I b triplet at 5167, 5172 and 5183 Å , arises from transitions between the first excited level 3p    to 4s  S . These lines are formed from near LTE level populations except for a slight deviation from LTE in the line core; the core, however, cannot be synthesized without introducing a model chromosphere. The line profile fits of the Mg I b lines are reproduced in Fig. 6. The synthetic flux profiles with dot-dashed lines refer to the LTE calculation (model A), whereas the dashed lines are non-LTE calculations with exponentially scaled hydrogen collisions of model (C). The very cores differ from the observed solar flux profiles by approximately 2%. Similar to the Na D lines the chromospheric contributions to the line cores therefore may be affected by small-scale velocity fields. The difference in the abundance determinations between non-LTE and LTE calculations amounts to dex.

6319 Å  triplet: This triplet, arising from the 4s  S  (i.e.  the upper level of the Mg b lines) to 6p  P transition, is located at 6318.75, 6319.20 and 6319.43 Å . The 6319 Å  triplet shows abnormal behaviour compared with the other triplet lines since the synthetic LTE flux profile is deeper than that of the non-LTE calculation in the central part of the profiles (see Fig. 6). The difference between the abundances determined with LTE or non-LTE calculations amounts to dex.

4.2.3. Emission lines near m

The existence of two emission lines in the solar spectrum near 12 µm was announced by Murcray et al. (1981). Chang & Noyes (1983) identified these lines as transitions between highly excited levels of Mg I, m or 818.058 cm^-1 (3s7h - ) and 12.3 µm or 811.578 cm^-1 - H , respectively. Brault & Noyes (1983) were the first to study these Mg I lines and demonstrate their large diagnostic potential, with line profile observations from different areas on the solar disk. Carlsson et al. (1992) carried out non-LTE calculations for the Mg I emission lines and they were able to reproduce the emission feature. As was also pointed out by Lemke & Holweger (1992) these lines are formed in the photosphere ; therefore modelling the chromospheric temperature rise is not required. Even small deviations from LTE of the corresponding level populations create a relative population inversion that produces an outward increase of the line source function sufficient to form an emission profile. The reason lies in the increasing importance of stimulated emission in this wavelength region. A more detailed description of other investigations of these famous Mg I emission lines can be found in Carlsson et al. (1992) and further references therein. Similar lines are found for Al I, and they have been analyzed by Baumüller & Gehren (1996).

The calculated line intensity profiles of the 7h - 6g transition at 12.2 µm and the 7i - 6h transition at 12.3 µm are compared with the spectrum observed at the solar disc centre (Brault & Noyes, 1983) in Fig. 7, from which it can be seen very clearly that LTE synthesis from both the GRS88 model or the HM74 model of course cannot reproduce the Mg I emission feature. The non-LTE calculations with our Mg I model can reproduce the observed m emission peak. While the line fit using neutral hydrogen collisions calculated according to the standard Drawin formula (model B) is not very convincing, we obtain excellent agreement with the observed profile when treating the same collisions with the empirical correction factor to Drawin's formula given in Sect. 2.2.2. A similarly good fit is naturally obtained in this particular case for model (D) since   = 0 is not significantly different from   = , the value obtained from our hydrogen scaling formula.

Fig. 7. Line intensity profiles of two Mg I Rydberg transitions near m calculated for different atmospheric and atomic models, and compared with the spectrum observed at the centre of the solar disk (Brault & Noyes 1983) (solid line). LTE from GRS88 model atmosphere: long dashes. LTE from HM74 model: dash-dots. Non-LTE model with full hydrogen collisions: dash-three dots. Non-LTE model (C) with exponentially scaled hydrogen collisions: dashes. (left): 6g - 7h transition at 12.22 µm. (right): 6h - 7i transition at 12.32 µm.

The centre-to-limb variation of both line profiles is plotted in Fig. 8. Our fit is exceptionally precise; in fact it is better than that of some of the pure absorption lines. The two lines thus offer a unique opportunity to study the influence of hydrogen collisions in statistical equilibrium systems. The exponential scaling formula proposed in Sect. 2.2.2 is fixed at its upper end (at high excitation energies) by comparison with these Rydberg transitions, and in the middle or at its lower end by transitions such as the 8806 Å  line. The exponential scaling of the Drawin (1969) formula produces results similar in quality to the step-like scaling used by Baumüller & Gehren (1996), but it requires the additional differentiation of excitation energies to produce acceptable line fits.

Fig. 8. Centre-to-limb variation of Mg I Rydberg transitions near m comparing the solar observations of Brault & Noyes (1983, solid line) with the profiles arising from the non-LTE model with exponentially scaled hydrogen collisions (dashes). (left column): 6g - 7h transition, (right column): 6h - 7i transition. From top to bottom the spectra refer to , and 0.2, respectively

© European Southern Observatory (ESO) 1998

Online publication: April 15, 1998
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