3 Comparing FCS spectra with the MEKAL model

3.1 Fitting procedure

We analyzed the two flare spectra by splitting the total range covered in each case into several smaller intervals. This facilitated the comparison with model spectra through the fact that the large instrumental background varies with wavelength in an irregular way over the whole range but over smaller ranges the variation can be characterized by a simple dependence on wavelength. Table 1 shows the wavelength ranges covered by each small interval and the notation we will use in the following.

In order to test the theoretical MEKAL model we generated a synthetic spectrum for each wavelength interval and convolved this with the FCS instrument response using version 2.0 of the SPEX package. We used an optically thin collision-ionization-equilibrium (CIE) model with two temperature components. Each component is defined by temperature (T₁ and T₂) and emission measure (EM₁ and EM₂). We added a third component to represent the instrumental background which we found could be adequately represented over each spectral interval by a power law, i.e. background flux $=N_{\rm pl} E^{-\gamma}$ where E is the photon energy in keV. Values of appropriate parameters are given in Tables 2 and 3. Note that temperatures are expressed in keV (1 keV = 11.6 MK), emission measures in 10⁴⁸ cm^-3, and the parameter $N_{\rm pl}$ is in units of 10³⁴ photons s^-1 keV^-1 at 1 keV (for a distance of 1 AU). Sometimes the first component has an unphysically low temperature ($T_1 \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}0.1$ keV with high emission measure; in such cases, this component does not contribute significantly to the spectrum but merely serves to obtain a better statistical fit. Solar photospheric abundances [1, (Anders & Grevesse 1989)] were assumed, except in one case (scan s12) for O for which the relative abundance was taken to be $0.559 \times$ Anders & Grevesse's value to fit the O VIII Ly-$\alpha$ and Fe XVII lines. We assumed that the lines have profiles which are a convolution of a thermal Doppler broadening (Gaussian) profile, in which the ion temperature is equal to the corresponding electron temperature T₁ or T₂, and the instrumental profile determined by the crystal rocking curve which was assumed to be Lorentzian. We neglected any additional broadening mechanisms, e.g. due to plasma turbulence. Some rebinning was applied to improve the statistics of the weaker lines. A broad Gaussian component centred at 22.4 Å was included to fit the instrumental background emission which greatly increases at $\lambda \gt 20$ Å.

Using the existing atomic data in MEKAL without any corrections, a value for $\chi^2$ was derived from the comparison of the synthesized spectrum with the observed data, with account taken of the photon count statistics. We repeated the process with different fit parameters until a minimum value for $\chi^2$ was derived. Table 2 and Table 3 give the details of the fits to the spectra of the two flares we analyzed, in which the fit parameters for this first trial are labelled by the letter "u". This did not result in a good fit, e.g., for the s11 scan of the 1980 August 25 flare $\chi^2$ = 70188 for 643 spectral bins (Table 2).

3.2 Line wavelength adjustments

In the range covered by the two flare spectra analyzed here (5-22 Å), the MEKAL code has about 3100 lines with associated wavelengths and excitation data from various sources. For the lines from this set that occur in the SMM flare spectra with sufficient intensity we made small corrections to the wavelengths in about 450 cases to match the SMM wavelengths. As will be mentioned, we also had recourse to laboratory (EBIT and PLT) data for confirmation or in some cases slight correction of the SMM solar data, enabling us to make a better judgement of the wavelength shift needed.

The original line wavelengths from the MEKAL code [24, (Mewe et al. 1995)] and those after applying small shifts are given in Appendix I, along with the SMM and laboratory wavelengths, intensity estimates, and transition details. The spectral interval that each line occurs in is also listed, following the notation of Table 1. The data are given in eleven tables, each covering a particular wavelength range of the 1980 August 25 and 1985 July 2 flares. The energy level notation used in specifying the transitions in the last two columns of Tables 1 to 8 of Appendix I is that of [22, Mewe et al. (1985)] except for the ions Fe XVII to Fe XXIV where the notation is based on a convention used by one of us (D.A.L.) in calculations made with the HULLAC code. The energy level scheme is given in the seven tables making up Appendix II. The notation used by [30, Wargelin et al. (1998)] for the Fe ion lines observed from the PLT tokamak is added to the last column of Tables 9 to 11 of Appendix I.

Some second-order (N=2) lines are sometimes significant. The strongest are lines of Mg XI at 9.170 Å (appearing at 18.323 Å) and 9.315 Å (at 18.612 Å). (The lines have apparent wavelengths not exactly double the true values because of crystal refractive index and flatness corrections.) Details are given in Appendix I.

As mentioned previously, the region of overlap between FCS channels 1 and 2 in the range 13.10-14.94 Å for the 1980 August 25 flare gives us a means of checking some line wavelengths. For some lines, we considered the channel 1 wavelengths (from scan s11) to be more precise since the statistical quality of the observed line feature was better than for channel 2, but for other lines the better spectral resolution of channel 2 in this region (scan s23) meant that the wavelengths from this channel were more precise. As indicated already, the channel 1 scan was taken at an earlier stage in the flare decay than the channel 2 scan, a fact which can be illustrated to some extent by the two-temperature coronal models for the s11 and s23 scans that cover this overlap region; although the two temperatures T₁ and T₂ are similar, the emission measure ratio EM₁/EM₂ is much lower for s11, showing the presence of hotter material in the case of the s11 scan. This was taken into account when identifying lines in each range.

In the last column of Table 2 and Table 3 ("Remarks''), fits with the MEKAL code with wavelengths from [24, Mewe et al. (1995)] are labelled "u'' (for uncorrected) and the fits with corrected wavelengths are labelled "c''. As can be seen, the fit as measured by the value of $\chi^2$ is significantly better. Thus, for the s11 scan $\chi^2$ is reduced from 70188 to 35421 (Table 2).

Apart from blends of three or more lines, the magnitudes of the shifts applied to the MEKAL wavelengths are less than or equal to 20 mÅ for the 1980 August 25 scans s41, s42, s31, and s21 (i.e. wavelengths less than about 13 Å), but up to 35 mÅ for a few lines in the s11 scan and 40 mÅ for a very weak line in the s12 scan. The maximum shift applied in the case of lines in the 1985 July 2 flare scans is 11 mÅ. Table 4 gives values of root-mean-square (rms) wavelength differences between either the [26, Phillips et al. (1982)] or [, Fawcett et al. (1985)] wavelengths, adjusted to the wavelength scales defined by the SSW software, and the MEKAL [24, (Mewe et al. 1995)] wavelengths. For observed line features with multiple identifications, the theoretical line with wavelength nearest to the observed wavelength is the one used to calculate the rms shift.

  

Table 4: Root-mean-square wavelength differences

$\sqrt{\mid \Delta \lambda \mid ^2/N}$ is the root-mean-square difference (in mÅ) of the Ph82 or Fa87 wavelengths and those in the original MEKAL code (Me95) for observed line features which are composed of not more than two identified lines. N is the number of lines in each scan.

  

Figure 2: Illustration of how an observed solar (SMM FCS) spectrum (part of s11 in Table 1) was fitted with the MEKAL code (dotted line) before (left panel) and after (right panel) with wavelength corrections described in the text, and with thirteen lines which were observed but apparently not included or too weakly predicted in the MEKAL code fitted with Gaussian profiles

As well as the need for wavelength shifts in the MEKAL code, there are a number of observed line features that are either much more intense than corresponding lines in MEKAL at or near the appropriate wavelengths or are without identifications altogether. These mismatches are likely to be due to incorrectness in our emission measure model since it can be expected that the theoretical formation rates of some lines are not accurate enough to match the observed ones and since most other lines are reasonably well fitted. We have therefore attempted to improve the $\chi^2$ values by fitting these observed line features with Gaussian profiles; we will refer to these as "missing" lines. In SPEX we can combine a maximum of 16 different models. Thus, in addition to the two thermal components and the power-law background component, we may add a number $n_{\rm G}$, maximum 13, of Gaussian lines. These fits are labelled "cc'' in the last column of Table 2 and Table 3. For example, if we fit the s11 spectrum with the model extended by $n_{\rm G}=13$ Gaussian lines centred at the wavelength positions of the strongest missing lines, we reduce $\chi^2$ to 21492 (cf. Table 2). There is hence a significant further improvement in the fit.

An illustration of the improvement is shown in Figs. 2a and 2b. The section s11 of the 1980 August 25 flare is shown (histogram) in each of these figures with the uncorrected MEKAL fit ("u'') superimposed (dotted curve) in Fig. 2a, then with the corrected MEKAL fit together with Gaussian lines added in Fig. 2b ("cc'').

In Fig. 3 and Fig. 4 we present the spectra measured in the eight spectral intervals of the 1980 August 25 flare ([26, Phillips et al. 1982]) and in Fig. 5 and Fig. 6 the spectra in the five spectral intervals of the 1985 July 2 flare ([12, Fawcett et al. 1987]). In addition, part of the July 2 flare scans smm3, smm4 and smm5 are shown on an expanded wavelength scale. The figures show the SMM FCS spectra (histograms) together with the best-fit model spectra (dotted lines) using the MEKAL model with corrected line wavelengths and "missing lines'' added. Identifications of the strongest lines are indicated on these figures by the emitting ion, although these are sometimes omitted where there are several contributions or the feature includes a "missing" line (e.g. at $\lambda$ 10.770 Å).

3.3 Sources of error

Despite the improvements in the fits, our spectral synthesis code would not appear to fit the high-resolution SMM spectra in a statistically acceptable way for most of the spectral scans analyzed in the sense that the "reduced" $\chi^2$ (i.e. $\chi^2$ divided by the number of degrees of freedom which is approximately the number of spectral bins) greatly exceeds unity. This is due to inaccuracies in atomic parameters other than wavelengths, e.g. ionization fractions and excitation rate coefficients as a function of temperature, the approximation made for the emission measure distribution, and finally the fit to the instrumental background.

Ionization fractions for Fe ions as given by [3, Arnaud & Rothenflug (1985)] are considerably different from those in the later work of [2, Arnaud & Raymond (1992)] as a result of improved ionization and recombination rates. It is most likely that the ionization fractions of other elements are in need of revision from the currently used [3, Arnaud & Rothenflug (1985)] data (cf. recent calculations of the ionization balance by [20, Mazzotta et al. 1998]). At present, in the current version of SPEX we use as a default [2, Arnaud & Raymond (1992)] (ARa) for the ion fractions of Fe and [3, Arnaud & Rothenflug (1985)] (ARo) for the other elements, but the code has the option to use ARo for Fe or to use the recent data of [20, Mazzotta et al. (1998)].

Collision strengths and rate coefficients of the Fe L lines in the 7-19 Å range have been calculated by [18, Liedahl et al. (1995)] with HULLAC, including radiative and dielectronic recombination contributions. These results were recently compared with EBIT measurements of Fe XXIII and Fe XXIV spectra [14, (Gu et al. 1999)], and found to be in very good agreement. The calculations of electron impact excitation cross-sections did not include resonant excitation, which however was shown in the experiment to contribute $\leq 5$% to the total line power. In the present version of MEKAL, only the strongest Fe L lines as calculated by HULLAC are included, however, and in rather a crude way, with the emissivity of a given line approximated by taking it to be that of maximum emissivity scaled to other temperatures by assuming the same temperature dependence as was used in the MEKA code for the strongest line of the same ion [24, (Mewe et al. 1995)]. Therefore, the correct temperature dependence of the line emissivity away from the maximum abundance of a given ionization stage may differ somewhat from that given in the present version of MEKAL. Hence, the current excitation rates of the Fe L lines (and also of other lines) in the present MEKAL code need revision.

For most of the spectral ranges (see Tables 2 and 3), the values of T₁ and T₂ are fairly representative of the most intense lines in those ranges, so we expect the emission measure models for these ranges to be reasonably accurate. However, for some ranges (e.g. s31 and s42 in the 1980 August 25 flare), the value of T₁ is unphysically small (< 0.1 keV) and does not represent any of the lines present, but is used only to adjust to the background. The fits for these ranges are essentially single-temperature, which is reasonable in view of the small temperature range of the strongest lines present (e.g. s42 is dominated by Si XIII lines).

Possibly the most important factor in the reduced $\chi^2$ still being much more than unity is the imperfect fit to the instrumental background, assumed to be a power law in photon energy. Sometimes the fitted curve appreciably departs from the observed background even though the fits to the line emission can be considered to be very satisfactory. This will have the effect of increasing $\chi^2$, considerably so if the background is at a high level and varying in an irregular manner.

Note that this benchmark study is a very stringent one: the SMM FCS spectral resolution is very high, so that even comparatively small (e.g. >5 mÅ) errors in the MEKAL line wavelength data are very important for the purposes of this work. However, the typical wavelength resolution of the instruments on board Chandra and XMM is an order of magnitude larger, so such errors would not be of significance for comparison with data from these instruments.

  

Figure 3: Observed SMM FCS spectra (histograms) and fitted spectra using the corrected version of the MEKAL model (dotted curve) of the 1980 August 25 flare. The intensity scale ("counts") is arbitrary and should only be considered as an indication of relative intensity. Principal lines are identified with the emitting ion name. The spectral scans are (from top to bottom) s41, s42, s31, s32

  

Figure 4: Observed and fitted spectra of the 1980 August 25 flare. The spectral scans are (from top to bottom) s21, s11, s23, s12. For other details, see caption to Fig. 3

  

Figure 5: Observed and fitted spectra of the 1985 July 2 flare. The spectral scans are (from top to bottom) smm5, smm5 (detail), smm1, smm2. For other details, see caption to Fig. 3

  

Figure 6: Observed and fitted spectra of the 1985 July 2 flare. The spectral scans are smm3, smm3 (detail), smm4, smm4 (detail). For other details, see caption to Fig. 3

3.4 Laboratory measurements

As mentioned, we compared the SMM FCS data to sets of laboratory data from the Lawrence Livermore EBIT device and the PLT tokamak. Using flat-crystal spectrometers [6, (Beiersdorfer & Wargelin 1994)] at the EBIT facility, extensive measurements of the iron L-shell emission have been performed. EBIT has the advantage of a tunable electron beam energy making it possible to isolate one ionization stage from neighbouring ionization stages. This helps to identify the ion unambiguously producing a particular set of lines. Measurements of nearly thirty Fe XVII lines were recently reported by [7, Brown et al. (1998)]. The wavelength accuracy of these measurements was about 3 mÅ or better. EBIT measurements of the higher ionization stages of iron, Fe XVIII through Fe XXIV, are in progress. For our comparison, we used preliminary results [5, (Beiersdorfer et al. 1996)], which have a wavelength accuracy of about 7 mÅ. Results with higher accuracy are expected to become available in the future [8, (Brown et al. 1999)]. The data sets from EBIT cover the line emission in the range from about 10 to 18 Å. Measurements from the PLT tokamak [4, (Beiersdorfer 1988)] provide line identifications in the range from 7 to 9 Å, e.g., for the high-n ($\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}6$) iron L-shell transitions [30, (Wargelin et al. 1998)]. The wavelength accuracy of the PLT data is very high, reaching 0.1 mÅ in some cases.

The EBIT and PLT laboratory measurements help in line identification of the SMM data. They help especially in identifying the contributions from different ions to line blends. Generally there is extremely good agreement between the wavelengths of the laboratory measurements and the SMM values, validating the line identification and wavelength values obtained from the solar measurements. For the Fe XVII lines in the FCS channel 1 spectrum (which has the lowest spectral resolution of the seven FCS channels) there is some tendency, as has been noted by [7, Brown et al. (1998)], for the SMM wavelengths in the 1980 August 25 flare as given by [26, Phillips et al. (1982)] to be slightly less than EBIT wavelengths for this ion. However, the SMM wavelengths on the wavelength scale as defined by the more recent SSW software package are up to 5 mÅ longer, so there is now better agreement (see Appendix I). For the PLT measurements of Fe ion lines with n=4-2 or larger transitions reported by [30, Wargelin et al. (1998)], it is particularly noteworthy that the agreement between the SMM wavelengths for the 1985 July 2 flare [12, (Fawcett et al. 1987)] and those from PLT is mostly to within 2 or 3 mÅ. More significant differences arise where there are blends, i.e. where the composition of the solar line features cannot be clearly ascertained. We can also state the converse, namely, a significant disagreement between SMM and laboratory wavelengths indicates that the solar line feature is not the same line identified in the laboratory measurements.

The analysis of the 1985 July 2 flare by [12, Fawcett et al. (1987)] established the identifications of several n=4-2 and n=5-2 transitions in Fe ions not previously published, this being done by runs of the Cowan Hartree-Fock code. Nearly all these identifications are now confirmed by the analysis of the PLT spectra [30, (Wargelin et al. 1998)]. In addition, there are a number of lines which were not included by Fawcett et al. Some are present in a section of the July 2 flare spectrum from 7.34 to 7.62 Å that was not analyzed in that work. A number of Fe ion lines between 7.83 Å and 10.02 Å which we now regard as real were ignored by Fawcett et al. because of their weakness. Finally, the intense He-like Mg (Mg XI) lines and their associated dielectronic satellites were not considered. The present analysis (given in the tables of Appendix I) includes the lines in the 7.34-7.62 Å section, the previously ignored weak lines, and the Mg lines, and assigns identifications from the PLT spectra of [30, Wargelin et al. (1998)] and other sources.

3.5 Identification of missing lines

As indicated earlier, the fits to the SMM FCS spectra were improved not only by adjusting line wavelengths in the MEKAL code but also by adding lines either not previously in the MEKAL code or, if present, augmenting their fluxes. This was done by fitting the line features or the required additional flux with Gaussian profiles. Most of these missing lines appear in fact to be lines already in MEKAL but whose intensities are predicted to be much smaller than observed, possibly because of inaccurate ionization fractions or excitation rate coefficients as discussed earlier.