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Astron. Astrophys. 364, 835-844 (2000)

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3. Plasma diagnostics

Without a knowledege of the density, temperature, and elemental abundances in astrophysical plasmas, almost nothing can be said regarding the physical processes taking place in them. Thus, a considerable effort has been made in the past years in developing diagnostic techniques to infer plasma temperature, density, and elemental abundances for solar and other astrophysical plasmas, especially by means of optically thin emission-line spectra (e.g., Dwivedi 1994; Mason & Monsignori-Fossi 1994, and references therein). A fundamental property of hot solar plasmas is their inhomogeneity. The emergent intensities of spectral lines from optically thin plasmas are determined by integrals along the line of sight through the plasma. Spectroscopic diagnostics of the temperature and density structure of hot optically thin plasmas using emission-line intensities is usually described in two ways. The line-ratio method uses an observed line intensity ratio to determine density or temperature from theoretical density or temperature-sensitive line-ratio curves, calculated taking account of physical processes for the formation of lines, in the assumption that both lines are emitted by the same plasma volume. The line-ratio method is stable, leading to well-defined values of T or Ne, but in realistic cases of inhomogeneous plasmas the results are hard to interpret. The more general "differential emission measure" (DEM) method (e.g., Mason & Monsignori-Fossi 1994) recognizes that observed plasmas are better described by distributions of temperature or density along the line of sight, and poses this problem in inverse form. Derivation of DEMfunctions, while more generally acceptable, is unstable to noise and errors in spectral and atomic data. The exact relationship between the two approaches has never been explored in depth, although particular situations were discussed by Brown et al. (1991). The mathematical relationship between these two approaches has recently been reported by McIntosh et al. (1998).

However, since off-limb plasma outside active regions has been found to be nearly isothermal (Feldman et al. 1998), the DEM approach loses its meaning, and a line ratio approach is sufficient for our purposes. However, the presence of many plasma structures in the field of view complicates the analysis for the cooler lines in our dataset. Since SUMER lacks the capability of spatially resolving the individual structures, we have to consider the results obtained with the Mg/Ne ratios as averages over several different structures.

The power P per unit volume (in energy units) emitted in an optically thin spectral line emanating from an upper level (j) to a lower level (i) is given by


where [FORMULA] is the transition probability, [FORMULA] is the population of the upper level j of the ion X[FORMULA]. [FORMULA] can be further parametrized as


where [FORMULA] is the relative upper level population,
[FORMULA] is the ion fraction of the ion [FORMULA], [FORMULA] is the abundance of the element relative to hydrogen, which varies in different astrophysical plasmas and also in different solar features, [FORMULA] is usually assumed to be around 0.8 for a fully ionized plasma.

Combining Eqs. (1) and (2), we can write


where [FORMULA] is evaluated at the plasma electron density and temperature, and [FORMULA] is evaluated at the plasma temperature.

In low-density plasmas, the collisional excitation processes are generally faster than ionization and recombination timescales, and therefore the collisional excitation is dominant over ionization and recombination in populating excited states. The low-lying level populations can then be treated separately from the ionization and recombination processes. The number density population of level j is calculated by solving the statistical equilibrium equations for a number of low-lying levels, taking account of all the important collisional and radiative excitation and deexcitation mechanisms.

In order to obtain relative abundances of two elements X and Y, ratios of spectral line intensities from the two elements are used. The line intensity ratio R of two lines is given by


If the intensities of the two lines have a similar temperature dependence, then the lines are presumably formed in the same plasma volume and at the same density. For determining the relative element abundance between X and Y, we can then compute a theoretical line intensity ratio assuming equal abundances for X and Y, and subsequently deduce [FORMULA] from the observed line intensity ratio [FORMULA].

Theoretical line ratios have been computed using the CHIANTI database (Dere et al. 1997; Landi et al. 1999) and the density and temperature values measured from the observations, assuming unity elemental abundances. Unless otherwise specified, ion fractions come from Mazzotta et al. (1998).

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001