Astron. Astrophys. 347, 401-408 (1999)

3. Results

An example of the total emissivity curve as resulting from the present version of the Arcetri Code calculation is displayed in Fig. 1. It has been evaluated assuming cm^-3 and adopting the Arnaud & Raymond 1992 (Fe ions) and Arnaud & Rothenflug 1985 (other ions) ion fractions and the Feldman 1992 element abundances. The calculation has been performed for temperatures in the  K range, but it is important to note that for temperatures smaller than K opacity effects play an important role in line and continuum radiation formation. As these effects are not accounted for in the present calculation, the total emissivity curve and the radiative losses curves should be taken with caution below this temperature limit.

Fig. 1. Total radiated power per unit as a function of electron temperature, calculated at  cm-3.

Fig. 2 shows the contribution of the most abundant elements to the total emissivity curve. Hydrogen is responsible for nearly all the radiative losses at chromospheric temperatures, while iron provides most of the output energy at high temperature. Continuum radiation may be neglected for all temperatures lower than a few million degrees, but at very high temperature free-free continuum radiation dominates the total emissivity.

Fig. 2. Contribution of the most abundant elements to the total emissivity curve.

Figs. 3 and 4 display the total emissivity for some of the most abundant elements in astrophysical plasmas together with the strongest lines of each of the element's ions. For some temperatures the total emissivity of some elements is dominated by the emission of a very small number of very strong lines; these are some of the strongest spectral features observed in solar and stellar spectra.

Fig. 3. Contribution Function of the strongest line of each ion to the total emissivity of the most abundant elements.

Fig. 4. Contribution Function of the strongest line of each ion to the total emissivity of the most abundant elements.

3.1. Effect of the electron density

Both continuum and line radiation may be electron density dependent, and this may cause the radiative losses and the total emissivity curve to be density dependent as well. As in literature the total emissivity curve is usually given as a function of temperature only, it is important to check the density dependence of this curve.

Continuum radiation electron density dependence stems from the two-photons continuum process: the populations of the H-like level and of the He-like level, decaying to the ground level through a two-photon process, may be altered by collisional de-excitation when electron density reaches a critical value. However the two-photons continuum represents a minor contribution to the continuum radiation at coronal densities and temperatures. Line radiation electron density dependence is given by the role played by collisional excitation and de-excitation into level population; this dependence may provide precious diagnostic tools for determining the electron density of the emitting plasma. Another source of density dependence is given by density effects on ionization and recombination coefficients. Summers 1972 and 1974 calculated density-dependent ionization equilibrium finding that ion fractions change as a function of density, mostly because of the density dependence of the dielectronic recombination coefficient; Vernazza & Raymond 1979 also find that under coronal condition ion fractions are density dependent, mostly due to collisional ionization and dielectronic recombination. Plasma microfields also may have a significant effect on dielectronic recombination, giving a further density dependence to ion fractions. Badnell et al. 1993 carried out quantal calculations for dielectronic recombination of [ C iv] in an electric field, finding that the dielectronic recombination rate could change by 40%.

However, in the literature ion fractions are usually reported as a function of electron temperature only, so in the present work it is not possible to check the effects of their density dependence on the total emissivity curve due to ionization balance.

In order to assess the density dependence due to level population we have performed the theoretical calculation of this curve assuming four different values of the electron density: cm^-3. Outside this density range line radiation is density insensitive: for higher densities ion level populations for the most important lines have reached Boltzmann equilibrium, while for lower densities collisional de-excitation becomes negligible compared to radiative decay and the Coronal Model Approximation (yielding density insensitive line Contribution Functions ) may be adopted.

Fig. 5 displays the percentual difference

(with , and cm^-3) between total emissivity curves calculated at different densities as a function of electron temperature. As expected, the greatest differences are found with the curves at cm^-3, which are very similar, because density-dependence affects line emissivity mostly between and cm^-3. Differences are always smaller than 25% and show a marked temperature dependence, being highest at transition region and coronal temperatures and decreasing down to zero at the edges of the selected temperature range.

Fig. 5. Percentual differences between total emissivity curves calculated assuming different values of the electron density. Full line : 108 vs. 1010 cm-3; Dashed line : 108 vs. 1012 cm-3; Dash-dotted line : 108 vs. 1014 cm-3.

The maximum at coronal temperatures is given by the presence of a host of strong density dependent lines formed in quiet corona, mainly from Fe, Mg and Si ions. The high temperature tail is dominated by strong, density insensitive lines and free-free continuum; the low temperaure tail is dominated by density insensitive transition region and chromospheric lines and for this reason there are small differences between computations carried out assuming different density values.

3.2. Effect of different datasets and approximations in level population computation

Level populations are strongly sensitive to any change or problem in the atomic parameters, collision strengths and transition probabilities as well as in the approximation adopted for their calculations, and this affects line radiation. It is therefore important to check the effects of different transition probabilities datasets on the resulting total emissivity curve.

As big improvements have been done in the present version of the Code versus the older version described in Landini & Monsignori Fossi 1990, we have performed a comparison between the present results and those obtained using the 1990 version of the Arcetri Code. The adopted element abundances are from Allen 1973. There are three main differences between the two versions of the Arcetri Code: (a) the old 1990 Code calculated all line intensities using the Coronal Model Approximation , (b) the collision rates were calculated using Gaunt factors and (c) radiative data came from different literature sources than in the present version of the Code.

Thus, the present comparison allows to check also the effects of different assumptions in level population calculations on the resulting total plasma emissivity.

Fig. 6 displays the percentual difference

between the two versions of the Code as a function of electron temperature. It is possible to see that rather high differences (up to 60%) are found at transition region temperatures, and smaller discrepancies occur at coronal temperatures. In the positive section of Fig. 6 the older version of the Arcetri Code has higher total emissivity than the more recent version at transition region temperatures. This is due to the presence of few very bright transitions from [ O iv], [ O v], [ C iv] whose emissivities have very different values in the two versions of the Code; their difference is due both to the use of different datasets and to the different approximations used in level population calculation leading to an overestimation of line emissivity for these transitions in the old version of the Code. The negative section of the diagram is due to the much larger number of lines included in the new version.

Fig. 6. Percentual difference between the total emissivity curve obtained with the old and new version of the Arcetri Spectral Code. The adopted electron density is 1010 cm-3.

3.3. Effect of ionization equilibrium

Ion fractions are necessary to both line and continuum calculation and any difference in their values are usually reflected into the total emissivity curve. We have checked the changes between curves calculated adopting different ion fractions datasets. All these calculations have been carried out assuming ionization equilibrium, and ion fractions come from Shull & Steenberg 1982 (SS, but with H and He ion fractions coming from Arnaud & Rothenflug 1985), Arnaud & Rothenflug 1985 (RO), Arnaud & Rothenflug 1985 plus Arnaud & Raymond 1992 for the Fe ions (RA), Mazzotta et al. 1998 (MA).

Fig. 7 displays the percentual differences

between the results obtained adopting RA ion fractions and those obtained with the other three datasets. The overall differences are smaller than 40%, and the greater differences are found with RO ion fractions. These are due to Fe ion fractions, dominating the high temperature tail of the total emissivity curve (the other elements' ion fractions being the same). Differences with the SS and MA results are much smaller.

Fig. 7. Percentual difference between total emissivity obtained with RA ion fractions and: Full line : MA ion fractions; Dashed line : RO ion fractions; Dash-dotted line : SS ion fractions. The adopted electron density is 1010 cm-3, element abundances are from Feldman 1992.

On the overall, the effect of the use of different ion fractions onto the total emissivity curve may rise up to a maximum of 40%, and are smaller than 20% at transition region and chromospheric temperatures below  K.

3.4. Effect of element abundances

Variation of the chemical composition of the emitting plasma may change the total emissivity curve by very large amounts.

It has been long acknowledged that element abundances change in solar plasmas, and their values seem to be associated to magnetic structures in the solar atmosphere (see the reviews of Feldman et al. 1992, Feldman 1992, Mason 1995). These variations seem to be related to the First Ionization Potential (FIP) of the emitting elements (e.g. Haisch et al. 1996). Cook et al. 1989 determined the radiative loss function using photospheric, chromospheric and coronal abundaces and found huge differences in the  K temperature range; they also found that these changes have serious effects on loop models.

Also Bhringer & Hensler 1989 and Sutherland & Dopita 1993 have studied the effects of metallicity variations on total emissivity curve, finding huge differences as metallicity decreases from the solar value. This is due to the importance of line radiation from elements with at temperatures between and  K.

The importance of abundance changes pointed out by these authors has led us to check the effect of different element abundance values on the resulting total emissivity curve. As this curve is dominated by the emission of some elements, these effects are expected to be very large. In order to check these effects total emissivities have been calculated assuming several different sets of element abundances: Allen 1973 (AL), Feldman 1992 (FE), Grevesse & Anders 1991 (GA), Meyer 1985 (ME) and Waljeski et al. 1994 (WA).

Fig. 8 displays the percentual differences

found between the emissivity curve calculated using FE abundances and those calculated adopting the other datasets. There are huge variations (up to a factor 2.5) between FE and WA total emissivities, while differences up to 70% are found between FE and the other three sets of abundance values.

Fig. 8. Percentual difference between total emissivity obtained adopting Feldman 1992 abundances and those obtained with Allen 1973, Grevesse & Anders 1991, Meyer 1985 and and Waljeski et al. 1994 datasets.

Considering the results shown in Fig. 8, and that differences up to factor 9 have been observed in the solar atmosphere between distinct structures very close to each other (e.g. Young & Mason 1997), abundance variations are a key factor for the evaluation of total plasma emissivity and need to be carefully chosen in order to be able to properly determine the plasma radiative losses.

© European Southern Observatory (ESO) 1999

Online publication: June 18, 1999
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