1.3.1 Electron temperature and density
It is well known, and mentioned in all textbooks, that some line ratios (e.g. the ratios of the lines labelled A1 and N2 in Table 1.11 in the appendix) are strongly dependent on the temperature, since they have different excitation energies. If the critical densities for collisional de-excitations are larger than the density in the medium under study, these line ratios depend only on the temperature and are ideal temperature indicators. The most frequently used is the [O Ill] A4363/5007 ratio.
On the other hand, in collisionally excited lines that arise from levels of similar excitation energies, their ratios depend only on the density. The commonest density indicator in the optical is the [SII] A6716/6731 ratio. Other ones can easily be found by browsing in Table 1.11. Rubin (1989) gives a convenient list of optical and infrared line-density indicators showing the density range where each of them is useful.
Similar plasma diagnostics are now available in the X-ray region (Porquet & Dubau 2000, Delahaye et al. 2006, see also Porter & Ferland 2006).
There are basically four methods to derive the chemical composition of ionized nebulae. The first one, generally thought to be the "royal way", is through tailored photoionization modelling. The second is by comparison of given objects with a grid of models. These two methods will be discussed in the next section. In this section, which deals with purely empirical methods, we will discuss the other two: direct methods, which obtain an abundance using information directly from the spectra, and statistical methods, which use relations obtained from families of objects.
18.104.22.168 Direct methods
In these methods, one first derives ionic abundance ratios directly from observed line ratios of the relevant ions:
Ii'j't' J n(Xj, )neei>fV(Te,ne)dV, where the Is are the intensities and the es are given by eijl = eijln(X3i )ne Therefore
where Tl and nl are, respectively, the electron temperature and density representative of the emission of the line l.
Assuming that the chemical composition is uniform in the nebula, one obtains the element abundance ratios:
n(Xi' ) eiji(Ti,ni)/ei'j'i' (Ti' ,nv ) where ICF is the ionization correction factor,
In a case where several ions of the same element are observed, one can use a "global" ICF adapted to the ions that are observed (e.g. ICF(O+ + O++) for planetary nebulae in which oxygen may be found in higher ionization states). Note that in HII regions (except those ionized by hot Wolf-Rayet stars) ICF(O+ + O++) = 1.
The application of direct methods requires a correct evaluation of the Tls and nls as well as a good estimate of the ionization correction factor.
For some ions, the Tls can be obtained from emission-line ratios such as [O iii] A4363/5007 and [NII] A5755/6584. For the remaining ions, the Tls are derived using empirical relations with T(4363/5007) or T(5755/6584) obtained from grids of photoion-ization models. The most-popular empirical relations are those listed by Garnett (1992). A newer set of relations, based on a grid of models that reproduces the properties of H II galaxies, is given by Izotov et al. (2006). It must be noted, however, that observations show larger dispersion about those relations than predicted by photoionization models. It is not clear whether this is due to underestimated observational error bars, or to additional processes not taken into account by photoionization models. At high metallicities^ the relevance of any empirical relation among the various Tls is even more questionable, due to the existence of large temperature gradients in the nebulae, which are strongly dependent on the physical conditions.
t Throughout, the word "metallicity" is used with the meaning of "oxygen abundance". This is common practice in nebular studies. Although oxygen is not a metal according to the definition given by chemistry, in nebular astronomy the word metal is often used to refer to any element with relative atomic mass >12. The use of the O/H abundance ratio to represent the "metallicity" - as was first done by Peimbert (1978) - can be justified by the facts that oxygen represents about half of the total mass of the "metals" and that it is the major actor - after hydrogen and helium - in the emission spectra of nebulae. Note that, for stellar astronomers, the word "metallicity" is related to the iron abundance, rather than to the oxygen abundance, so the two uses of the word "metallicity" are not strictly compatible, since the O/Fe ratio changes during the course of chemical evolution.
The ionization correction factors that are used are based either on ionization potential considerations or on formulae obtained from grids of photoionization models. For HII galaxies, a set of ICFs is given by Izotov et al. (2006). For planetary nebulae, a popular set of ionization correction factors is that from Kingsburgh & Barlow (1994), which is based on a handful of unpublished photoionization models. Stasinska (2007, in preparation) gives a set of ICFs for planetary nebulae based on a full grid of photoion-ization models. It must be noted, however, that theoretical ICFs depend on the model stellar atmospheres that are used in the photoionization models. Despite the tremendous progress in the modelling of stellar atmospheres in recent years, it is not yet clear whether predicted spectral energy distributions (SEDs) in the Lyman continuum are correct.
Finally, note that the line-of-sight ionization structure, in the case of observations that sample only a small fraction of the entire nebula, is different from the integrated ionization structure. This is especially important to keep in mind when dealing with trace ionization stages.
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