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

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1. Introduction

The study by spectroscopic means of the chromospheres of late type stars relies heavily on strong lines of hydrogen or of abundant metals. The fact that chromospheres in solar-type stars consist of a rather tenuous plasma normally causes the source function of these lines to decouple from the local thermal pool in large parts of the line-forming regions. Hence significant departures from local thermodynamic equilibrium (LTE) affect the transfer of radiation in practically all chromospheric diagnostics.

From the classical theory of non-LTE transfer of line radiation, assuming the equality of the line absorption and emission profiles, the source function can be written (Thomas 1957 , 1965) as


In this expression, the term [FORMULA], the specific intensity averaged over all directions and over the line profile, accounts for the non-coherent photon scattering, while the terms [FORMULA] and [FORMULA] represent, respectively, the photon "source" and "sink" terms. In particular, the term [FORMULA] accounts for the creation of line photons by direct collisional excitation: it thus expresses the coupling of the source function with the local thermal pool ([FORMULA] is the Planck function relative to the local kinetic temperature T). Conversely, the term [FORMULA] in the denominator of Eq.  1 is due to the destruction of photons by direct collisional de-excitation. The terms [FORMULA] and [FORMULA] include all other indirect processes. Usually, these latter terms are dominated by bound-free processes, and [FORMULA] can be expressed as the Planck function at the radiation temperature of the relevant continuum. Thus, if [FORMULA] and [FORMULA] the line is said to be "photoionisation dominated"; vice versa, when the opposite inequalities hold, the line is "collision dominated". All other cases lead to "mixed domination".

It is clear from Eq.  1 that whenever direct collisional processes dominate over scattering and indirect excitation/de-excitation processes, the line source function approaches LTE. On the other hand, the decoupling from the chromospheric thermal pool, i.e. the departure of the line source function from [FORMULA], can be rather extreme. For example, the strong resonance lines classified as photoionisation dominated are rather insensitive to the chromospheric temperature rise, even if their cores are formed in the chromosphere: on the contrary, they are controlled by a photoionising radiation field that forms in other atmospheric layers, normally the photosphere. In the solar atmosphere, the Na I D resonance doublet at 5890/5896 Å is usually regarded as an example of such a "bad" chromospheric diagnostic, as opposed, for instance, to the collision dominated resonance lines of Ca [FORMULA] or Mg [FORMULA]. In this case, the radiation field that dominates the formation of the line is the photospheric ultraviolet radiation shortward 2413 Å, the photoionisation threshold of [FORMULA]. This is because the main source term in Eq.  1, [FORMULA], is due to the feeding of level [FORMULA] (the upper level of the D doublet) through the path [FORMULA]. It is perhaps this kind of consideration that justifies the relative paucity of work done on the Na I D lines in the study of the chromospheres in late type stars (notable exceptions are Sakhibullin 1987 and Thatcher 1994).

Nevertheless, it should be noted that, as the physical properties of the stellar atmosphere change, a line can switch category with respect to its solar classification. In particular, the UV radiation field in very cool stars is much less intense than in the Sun, and that may make possible a collisional control of the D lines. Moreover, the relatively high density of the chromosphere of cool dwarfs may tend to enhance this effect. It is true that, even in some M stars, low resolution spectra of the D lines do not show core emission (Pettersen & Hawley 1989), the signature of the chromospheric temperature rise in collisionally controlled lines. But high spectral resolution observations of active M dwarfs do in fact reveal the presence of an emission in the core of the Na I doublet (see e.g. Pettersen 1989, or Panagi et al. 1991). This chromospheric feature is not as spectacular as the emission of, say, H [FORMULA] or Ca II H & K; on the other hand, the Na I D doublet presents some distinct advantages as a diagnostic that we will investigate in this paper (see also last paragraph of Sect  2.1).

While a more detailed comparison with chromospheric diagnostics other than hydrogen lines is outside the scope of this study, from an observational standpoint an obvious advantage of the Na I doublet is that it lies in a region where M dwarfs are considerably brighter (even an order of magnitude or more) than at the wavelengths of the Ca II and Mg II resonance lines. Also, most current ground-based instrumentation is more sensitive in the V band. Finally, it is worth mentioning that the interesting chromospheric/transition region He I [FORMULA] line ([FORMULA]) happens to be very close in wavelength to the Na I doublet. Indeed, we feel that observations of the D lines together (Na I [FORMULA] and He I [FORMULA]) provide a relatively easy-to-get and potentially very useful data-set.

We will, however, compare more in detail the formation of the Na I D doublet with the hydrogen line spectrum. As an introductory remark, here it is opportune to note only that there is a consensus on the fact that H [FORMULA] in cool dwarfs is completely uncoupled from the thermal structure of the region at the temperature minimum (Cram & Mullan 1979, henceforth: CM). In fact, in M dwarfs that line is a good diagnostic of the upper chromosphere (Houdebine & Doyle 1994). The Na I D doublet in M dwarfs on the other hand displays well developed photospheric wings. It is therefore reasonable to expect that its profile can map the entire atmosphere from the photosphere up to the chromosphere, including the region of temperature minimum not covered by H [FORMULA]. It has in fact been shown (Caccin et al. 1993, henceforth: CGS) that in sunspot umbrae (i.e. in conditions that approach those of late-type stars), the Na I D doublet is a good diagnostic of the thermal structure close to the temperature minimum.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998