3 Spectroscopic techniques for the determination of densities, temperatures and elemental abundances

The flux $I(\lambda_{ij})$ (ergs cm^-2 s^-1), of an optically thin spectral line of wavelength $\lambda_{ij}$ is

$$% I(\lambda_{ij})={h \nu_{ij} \over{4\pi d^2} }\;{\int_V \; N_j(X^{+r}) \;A_{ji}\;{\rm d}V}$$ (1)

where: i, j are the lower and upper levels; A_ji is the spontaneous transition probability; N_j(X^+r) is the number density of the upper level j of the emitting ion X^+r; d is the star's distance; dV is the volume element, and V is the entire source volume. The flux can be written:

$$% I(\lambda_{ij})= {1 \over d^2} ~ \int_V \; G(N_{\rm e}, T, \lambda_{ij}) \; N_{\rm e} N{\rm _H} \;{\rm d}V$$ (2)

having defined $N_{\rm e}$ and $N_{\rm H}$ (cm^-3) as the electron and hydrogen number densities, and the contribution function $G(T,\lambda_{ij},N_{\rm e})$ (ergs cm³ s^-1) of each line:

$$% G(N_{\rm e},T,\lambda_{ij}) = A_{ji} ~{h \nu_{ij} \over{4\p... ...N_j(X^{+r})\over N_{\rm e} N(X^{+r})} { N(X^{+r}) \over N(X)}$$ (3)

where: $A_{\rm b}(X)=N(X) / N_{\rm H}$ is the element abundance relative to hydrogen; N_j(X^+r) / N(X^+r)is the population of level j relative to the total N(X^+r)number density of the ion X^+r and is a function of the electron temperature and density; N(X^+r) / N(X) is the ionisation ratio, and is predominantly a peaked function of the temperature, confining the emission to a limited temperature range. It is common to assume that the abundance of the element $A_{\rm b}(X)$ is constant over the source volume, and to define a differential emission measure $DEM (T) = ~N_{\rm e} N_{\rm H} {{\rm d}h \over {\rm d}T}$ (cm^-5 K^-1) in order to write:

$$% {I(\lambda_{ij})}= {A_{\rm b}(X)}{\int ~{C(T,\lambda_{ij},N_{\rm e})} ~DEM (T) {\rm d}T}$$ (4)

where we have defined $G(T, N_{\rm e}) = {A_{\rm b}(X)} \times {C(T, N_{\rm e})}$, and assumed filling factors of unity and spherical symmetry, i.e. d $V=4\pi R^2_* {\rm d}h$(h is the coordinate along the line of sight and R_* is the star's radius). Given a set of observed fluxes, the problem is to invert a set of integral equations like Eq. (4). In this paper, the DEM analysis was performed using a modified version of the Arcetri inversion code (Monsignori Fossi & Landini 1991). The CHIANTI atomic database (Dere et al. 1997) has been used to calculate the contribution functions of the observed lines, assuming collisional ionisation equilibrium. In particular, Version 3 of the database (Dere et al. 2001) was used, together with an update for O V that is described below.

Note that the determination of the DEM(T) distribution is an ill-posed problem (see, e.g. Craig & Brown 1986; McIntosh 2000, and references therein) where solutions are not unique. However, improvements can be made with a careful selection of lines, and the larger uncertainties become those associated with either the observational data or the atomic physics calculations. Also note that the DEM(T) does not necessarily have a direct physical significance (in terms of e.g. a temperature gradient), unless other factors are taken into account. For example, filling factors (cf. Judge 2000) are known to be small in the transition region, and probably different from those of the low corona.

Once the DEM is derived from a set of observed fluxes, it is possible to calculate an emission measure value $EM_{\Delta T}(T_i)$ for each temperature T_i, once a temperature interval $\Delta T$ is defined:

$$% EM_{\Delta T} (T_i) \equiv \int_{T_i- {\Delta T \over 2}}^{T_i+{\Delta T \over 2}} DEM(T)\; {\rm d}T .$$ (5)

Note that many different definitions of DEM and EM are found in the literature, that, together with some confusion in the terminology adopted, can create some difficulties when comparing results from different papers. For the sake of clarity, we therefore now briefly summarise other common definitions. In order to avoid the use of spherical symmetry and of the star's radius (a parameter not always well known), a volume differential emission measure $DEM (T) = ~N_{\rm e} N_{\rm H} {{\rm d}V \over {\rm d}T}$ (cm^-3 K^-1) and a corresponding emission measure EM are often defined. The ionisation state is often assumed known by assigning a value to the ratios of the electron and hydrogen number densities (e.g. ${N_{\rm H} / N_{\rm e}} \simeq 0.8$ by assuming that H and He are fully ionised and $A_{\rm b}{\rm (He)}=0.1$).

Another definition that is commonly used (see, e.g. Griffiths & Jordan 1998, and references therein) involves differentiation over the logarithm of the temperature: $DEM_{\rm log} (T) = N_{\rm e} N_{\rm H} {{\rm d}V \over {\rm d}\; {\rm log} \,T}$. This definition is more convenient when an approximation for the volume emission measure is sought. In fact, if one defines an average temperature T₀ such as log $T_0 = \int G(T) \log\, T {\rm d} \log\, T\,/\,\int G(T) {\rm d} \log\, T$, and assumes that $DEM_{\rm log} (T)$ varies linearly with $\log\, T$, the total emission measure over an interval $\Delta {\rm log}\, T=0.3$ centred at T₀ can be approximated with what is generally termed the EM(0.3) value (cf. Griffiths & Jordan 1998):

$$% EM_{\Delta {\rm log}\, T=0.3} (T_0) \simeq EM(0.3) \equiv ... ...d^2 \; I_{\rm ob} \over \int G(T) \; {\rm d}\, {\rm log}\, T }$$ (6)

where $I_{\rm ob}$ is the observed flux at Earth. The EM(0.3) points are often used instead of the $EM_{\Delta T}$ values. Note that in the definition of EM(0.3) in Eq. (8) of Griffiths & Jordan (1998) F_* should read $4\pi d^2 F_{\rm obs}$ according to their previous definitions. Also note that the EM(0.3) values are generally plotted at the temperature $T\mbox{$\rm _{max}$ }$ where C(T) has a maximum, and not at T₀, as they should.

Another different and common approach (see Fig. 8) is to plot the ratio $I\mbox{$\rm _{ob}$ } / G(T)$ for each line as a function of temperature and consider the loci of these curves to constrain the shape of the emission measure distribution. In fact, for each line and temperature T_i the value $I\mbox{$\rm _{ob}$ } / G(T_i)$ represents an upper limit to the value of the emission measure at that temperature (assuming that all the observed emission $I\mbox{$\rm _{ob}$ }$ is produced by plasma at temperature T_i).

   
3.1 Anomalous behaviour of Li- and Na-like ions

It should be noted that the ionisation equilibrium plays a major role not only in the derivation of the DEM, but also in that one of the elemental abundances.

The anomalous behaviour of the ions of the Li and Na isoelectronic sequences has been known for more than 30 years. For anomalous behaviour we mean that once a DEM analysis is performed using lines from different isoelectronic sequences, the theoretical intensities of the lines of the anomalous ions are consistently under- or over-estimated by large factors. Burton et al. (1971) reported this anomalous behaviour based on a solar spectrum from a rocket flight. Dupree (1972) confirmed this anomaly, using OSO-IV solar spectra. A possible explanation of this anomaly resides in the density dependence of the ionisation fractions. Burgess & Summers (1969) showed that the density dependence of the dielectronic recombination, together with the collisional ionisations from metastable levels produce significant changes in the ionisation fractions. Vernazza & Raymond (1978) took into account these effects and showed that significant increases in the line intensities of the Li-like ions occur at high densities. The same applies to the Na-like ions, and, to a lesser extent, to other ions as well. Raymond & Doyle (1981a, 1981b) applied the ionisation equilibrium calculations of Vernazza & Raymond (1978) to Skylab data and found a good agreement between lines of different isoelectronic sequences. The problem appeared to be resolved.

However, the same authors (Doyle & Raymond 1984), among others, presented cases where these density effects were not enough to explain the anomalous behaviour. At that time, uncertainties in the atomic data and instrument's calibration were a major concern. Judge et al. (1995) presented a DEM analysis of a rocket spectrum of the entire Sun, obtained with a good photometric accuracy of 15%. They used more recent atomic data, and calculated the ionisation equilibrium taking into account the density effects. They found "very significant and systematic differences'' between the line intensities of the Li and Na isoelectronic sequences and those of the other ions. C IV, N V, and O VI lines were underestimated by factors of 2 to 5. Regarding the Na sequence, only Si IV was observed, with an underestimation by a factor of 2.

Another difficulty in investigating these effects was due to the fact that the majority of the strong lines that could be used for DEM analyses were from the anomalous class. For example, Raymond & Doyle (1981b) used Li-like lines (O VI, Ne VIII, Mg X, Si XII, S XIV), with the exception of Ar VII and Ca X, to constrain the million-degree peak of the DEM. The same argument applies to the Judge et al. (1995) results.

The problems of the paucity of the observed lines and uncertainty in the instrument's calibration have been overcomed with the spectroscopic instruments on board SOHO. Del Zanna (1999) has presented many DEM analyses of different solar regions, using SOHO/Coronal Diagnostic Spectrometer (CDS) spectra. The CDS instrument covers almost entirely the 150-800 Å spectral region, and is rich in emission lines from a large number of highly ionised ions of many different isoelectronic sequences. The CDS instrument has a good in-flight calibration (see Del Zanna et al. 2001a and references therein). The lines from the following ions have presented significant deviations (factors of 2 to 10): Ne VIII, Na IX, Mg X, Al XI, Si XII (Li-like); Ca X (Na-like). Del Zanna (1999) also found similar deviations with re-analyses of solar spectra from various instruments (including SERTS, Skylab). These discrepancies cannot be ascribed to elemental abundance anomalies, nor to instrumental effects.

Furthermore, progress in the theoretical models for ionisation and recombination processes has led to significant changes in the ionisation and recombination rates over the years. The compilation of Mazzotta et al. (1998) is the best available to date, computed for the low density limit. The Mazzotta et al. results have been used here. They are close to those of Arnaud & Raymond (1992) for the Fe ions. For the other elements the differences from the older calculations of Arnaud & Rothenflug (1985) are quite substantial, as discussed by Del Zanna (1999). Del Zanna et al. (2001b) have reanalysed Skylab data using the Mazzotta et al. (1998) ionisation fractions and up-to-date atomic data to show the extent of the anomalous behavior. The results are that lines of the Li-like N V and C IV are underestimated by factors of 3 and 10, while those of Ne VIII and Mg X are overestimated by factors of 5 and 10, respectively. The S VI 933.3 Å (Na-like) is also underestimated by a factor of 3.

We conclude that all the spectral lines of the Li and Na isoelectronic sequences observed in EUV solar spectra present significant deviations from other isoelectronic sequences. The analyses presented in Del Zanna (1999) also indicate a new characteristic. All lines in the lower transition region (e.g. C IV) are consistently underestimated, while those in the upper transition region (e.g. Ne VIII) are consistently overestimated. The deviations vary, depending on the source region observed. This suggests to us that one of the main inaccuracy in the ion balance calculations is the temperature corresponding to the peak emission. Until the reasons for these discrepancies are fully understood, and all the relevant physical effects taken into account, the lines of the Li and Na isoelectronic sequences should not be used for DEM, density and elemental abundance analyses.

   
3.2 Elemental abundances

Stellar coronal abundances are still poorly known, but are important to measure, because they affect our understanding of the stellar atmospheres. Comparisons with the solar case are difficult, since a large variety of solar coronal abundances have been reported, with variations from the photospheric values up to an order of magnitude (cf. the reviews of Feldman 1992; Raymond et al. 2001). These differences appear to be related to the first ionisation potential (FIP) of the various elements. The abundances of elements with low FIP (<10 eV, e.g. Fe) appear enhanced compared to those of the high FIP (>10 eV, e.g. Ne) elements, using the solar photospheric values as a standard reference. Sulfur is an important element since it lies in the middle of these two classes. This "FIP effect'' can be used as an important diagnostic of the physical processes that occur in the chromosphere and TR of active stars.

In principle, it is possible to derive a relative element abundance $A_{\rm b}(X_1)/A_{\rm b}(X_2)$ of two elements X₁ and X₂ from any observed intensity ratio $I\mbox{$\rm _{1}$ } / I\mbox{$\rm _{2}$ }$:

$$% {A_{\rm b}(X_1) \over A_{\rm b}(X_2)} = { {I_1 \times ~\int... ..._2 \times ~\int C_1(T, N_{\rm e}) ~DEM (T) \,{\rm d}T} }\cdot$$ (7)

In practice, the ionisation balance, the selection of lines, and the spectroscopic method used can each account for a variation of a factor of two or more in the derived element abundances. Various examples are given in Del Zanna (1999) and Del Zanna et al. (2001b).

In the investigation of elemental abundances and the FIP effect it has been common to use ratios of low- vs. high-FIP lines that are emitted over the same temperature range and have similar contribution functions. In the solar case lines of Mg VI and Ne VI have been widely used. In the FUSE spectra the following ions present some temperature overlap: S III, S IV vs. N III; C III vs. N III; Si III vs. S III and C III; Ne VI vs. O VI; Ne V vs. S VI. The FUSE observations are complementary to those of Chandra and XMM, which observe coronal ions.

In the literature it is common to assume that any intensity ratio is directly proportional to the relative abundances of the two elements involved. A common approach, proposed by Widing & Feldman (1989), is to approximate Eq. (4) by defining for each spectral line a single differential emission measure $DEM_{\rm L}$ value for the temperature range over which the line forms:

$$% DEM_{\rm L} \equiv \left\langle N_{\rm e} N_{\rm H} {{\rm d... ...m _{ob}$ } \over A_{\rm b}(X) ~ {\int ~{C(T) {\rm d}T }}}\cdot$$ (8)

The relative elemental abundances are then adjusted in order to make the $A_{\rm b}(X) ~ DEM_{\rm L} = I\mbox{$\rm _{ob}$ } / {\int ~{C(T) {\rm d}T }}$ points lie along a common smooth curve. These points are displayed at the temperatures $T\mbox{$\rm _{max}$ }$, defined as the temperature where C(T) has a maximum. This method can be misleading. First, the use of $T\mbox{$\rm _{max}$ }$ is problematic, since this temperature can in some cases be quite different from the temperature where the bulk of the plasma is emitted. An effective temperature $T_{\rm eff}$, defined as $\log\, T_{\rm eff} = \int C{\left({T}\right)}~ DEM{\left({T}\right)} \log\, T~{\rm d}T / ({\int C{\left({T}\right)}~DEM{\left({T}\right)}~{\rm d}T}$) is more appropriate. Second, this method neglects the shape of the DEM distribution (which is not known a priori) and can therefore lead to a large under- or over-estimation of the relative abundance. This can occur if for example the DEM(T) has a steep gradient in the temperature range where C₁(T)and C₂(T) have some differences (see Del Zanna et al. 2001b for details).

The approach adopted here is to use the largest possible number of lines, calculate the line emissivities at the measured TR densities, and then determine the relative element abundances and the DEM(T) at the same time. The scaling between the high and low-FIP elements is done using lines that overlap in temperature. More details of the technique can be found in Del Zanna & Bromage (1999). In this way, most of the uncertainties will be reduced, and any systematic effects, such as those occurring with the Na- and Li-like lines will be highlighted. The use of line ratio methods for abundance diagnostics of solar and stellar coronae, in particular when lines of the Na and Li isoelectronic sequences are used, as recently suggested by e.g. Laming & Feldman (1999), is inadvisable.

   
3.3 Electron density

The most accurate density diagnostics are those obtained by considering the ratios of lines emitted by the same ion which are sensitive to density. However, owing to the paucity of the observed density-sensitive line pairs from the same ion, other methods have often been adopted in solar and stellar physics. One method is to compare the emission measures of the forbidden lines with those of the allowed ones. One of the first applications of this method was due to Feldman et al. (1977). A variation of this method is to compare the emission measure loci of the two groups of lines, and is the most commonly used (see, e.g., Brown et al. 1984b; Byrne et al. 1987; Jordan et al. 1987; Linsky et al. 1989; Maran et al. 1994; Linsky et al. 1995; Griffiths & Jordan 1998; Pagano et al. 2000). In most cases, large inconsistencies between the densities obtained using this method with those derived from line ratios have been reported. These inconsistencies are caused by the use of lines with anomalous behaviour, as shown below in Sect. 5.3.

Diagnostics based on line ratios have been extensively used in solar and stellar physics. In many cases, discrepancies between densities derived using different line ratios or different ions have been reported, thus leading to suggestions that the derived values might not have any physical significance (Judge & McIntosh 1999). It is well known that significant effects on the line ratios can be caused by inhomogeneities in the atmosphere (see e.g., Doschek 1984) or by plasma flows (cf. Raymond & Dupree 1978). However, we would like to stress that some of these discrepancies could also be due to un-accounted blending and/or inaccurate atomic calculations, as shown below in Sect. 5.2 with some examples. In addition, that misleading results can easily be obtained by using line ratios that have a very small sensitivity at the observed densities or when temperature effects are not taken into account. In some cases, it is possible to obtain consistent measurements, although it should be born in mind that the derived densities are in any case averaged values over the entire stellar disk and along the line of sight.

Finally, it is common in the literature to use only a few well-behaved line ratios, without considering all the observed lines. A different approach, preferable when more than two lines from an ion are observed, is to plot the values of

$$% f_{ji}= { I(\lambda_{ij}) N_{\rm e} \over {h \nu_{ij}}\; N_j(N_{\rm e}) \;A_{ji}}$$ (9)

as a function of electron density, calculated at a fixed temperature. All the curves should cross at one point, if the plasma is isodensity (a description of this method, termed L-function, can be found in Landi & Landini 1998). The f_ji curves should be calculated at the effective temperature, i.e. at the temperature where the bulk of the ion emission is. This can be quite substantially different from the temperature of the maximum ionisation fraction, in particular when the DEM has a steep gradient at the temperatures where the ion is emitted. This method has the following advantages over the line ratio technique: a) it gives an overall view of all lines; b) it shows any discrepancies even when considering lines that are not density-sensitive; c) it clearly shows which lines (and not ratios) are more suitable in a particular density regime; d) it clearly shows when lines are at the limit of their density sensitive regime. We use this method in Sect. 5.2 to discuss individual cases.