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Astron. Astrophys. 327, 1230-1241 (1997)

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2. The theoretical method

2.1. The plasma emissivity

The number of photons emitted in a bound-bound transition from upper level j to lower level i per unit volume and time is

[EQUATION]

where [FORMULA] is the population of the level j of the ion of charge +m of element X and [FORMULA] the spontaneous radiative transition probability of the [FORMULA] transition.

It is usual to write

[EQUATION]

where [FORMULA] describes the population of level j relative to the total ion, [FORMULA] is the relative abundance of the ion [FORMULA] assuming ionization equilibrium, [FORMULA] is the abundance of the X element relative to Hydrogen and [FORMULA] is about 0.8 in a completely ionized plasma of cosmic composition.

The number of photons that we receive at distance d from each pixel of the image covering an area S, is the contribution along the line of sight h

[EQUATION]

Because [FORMULA] is the solid angle covered by the pixel the intensity is

[EQUATION]

We can define the [FORMULA] as follows:

[EQUATION]

For resonance lines in very low density limit

[EQUATION]

the [FORMULA] reduces to the usual form

[EQUATION]

and the line intensity is

[EQUATION]

To evaluate the [FORMULA] use have been done of some recent ionization equilibrium computations (Arnaud and Rothenflug 1985, Arnaud and Raymond 1992). They do not include density dependent processes, which are expected to affect only dielectronic recombination for rather high density (Jordan 1969).

When forbidden or intercombination lines are considered, the population of all the relevant levels must be computed assuming statistical balance among the most important processes, mainly electron collisional excitation and decay and radiative decay. In this case [FORMULA] may depend remarkably on density and very smoothly on temperature.

2.2. The diagnostic technique

The suggested diagnostic is performed in a very simple way: the ratio of the measured intensity of each line of a selected ion, over the emissivity [FORMULA] at a proper temperature [FORMULA] is plotted versus [FORMULA]. All the ratios meet within a restricted region of the diagram depending on the errors as shown for instance in Fig.1.

[FIGURE] Fig. 1. The ratio of the observed intensity to the [FORMULA] is plotted versus the logarithmic density for all the lines of Si X. For each line the observed intensity plus and minus the error is considered.

This example is obtained using five lines of SiX measured by SERTS89 and given in [FORMULA] and evaluating the [FORMULA], in [FORMULA], for each line with the CHIANTI database. The coordinates of the meeting point are the density at [FORMULA] and the weighted emission measure for temperature [FORMULA], (see eq 18).

A better example is given by the simulation of Fig. 2 where the synthetic flux is evaluated for seven lines of FeXIII using a sharp dem and electron density [FORMULA] and used as "observed intensity" in the plot.

[FIGURE] Fig. 2. The synthetic flux of several Fe XIII lines evaluated for density [FORMULA] is used to show that all the lines cross at the correct density

In this case, when "observed data" are simulated and no error is added, all the curves cross the same point at the density used for the simulation.

In the following of this section, we will show:

  • the reason why all the curves meet at the same density, when no strong inhomogeneity occurs along the line of sight
  • how the ordinate of of the crossing point is related to the d.e.m.
  • how to evaluate the proper temperature [FORMULA]

Basically this method rests on the observation that the dependence of the [FORMULA] on Ne and T is such that we can express [FORMULA] as the product of two distinct functions depending respectively on electron density and electron temperature, and on temperature alone:

[EQUATION]

The variation of the [FORMULA] functions is mainly due to the ionization equilibrium and is the same for all the lines of the same ion while [FORMULA] is determined mainly by the population of the upper level. The dependence of [FORMULA] on electron density is mainly due to the collisional population and depopulation processes determining the statistical equilibrium term [FORMULA] of the upper atomic level j. The dependence of [FORMULA] on temperature comes both from the ionization equilibrium term [FORMULA] for the ion [FORMULA] and from the population of the upper atomic
level j.

Fig. 3 elucidates the above considerations; as an example, the emissivity of FeXIII 318.127  Å  for different densities is divided by the G(T) function of FeXIII 251.95  Å , in order to remove the common temperature dependence g(T); this line is very weakly density sensitive and its G(T) has been evaluated at [FORMULA]. The electron density [FORMULA]  labels each curve. Clearly the [FORMULA] of line 318.127 is strongly density dependent between [FORMULA] and [FORMULA] and very weakly dependent on temperature for any density. A straight line fits each curve within a few percent. This property holds quite usually and in any case may be easily verified before the method is applied. When comparison is made with observations, such as in Sect. 3, this procedure has been applied to all the lines of the same ion, assuming as g(T) for each ion, the G(T) of the less density sensitive line.

[FIGURE] Fig. 3. Emissivity functions for line 318.127 Å  of Fe XIII referred to line 251.950 Å  of the same ion for electron density [FORMULA] ; each curve is labeled with log of electron density; a straight line fits each curve to within a few percent

In order to take into account the non isothermal nature of the plasma along the line of sight, as usual, the [FORMULA] is introduced

[EQUATION]

and the intensity of a line can be rewritten

[EQUATION]

[EQUATION]

Since [FORMULA] is nearly a linear function of LogT for all the temperatures of interest in a stellar corona, we can expand the [FORMULA] function as a power series of Log T around the point Log [FORMULA] and consider only the first term of this expansion.

[EQUATION]

[EQUATION]

The [FORMULA] function is not known but if a "trial" d.e.m. [FORMULA] is known, (see next section) the "true" [FORMULA] may be written

[EQUATION]

If [FORMULA] is a reasonable approximation to [FORMULA], [FORMULA] is likely to be a slow function of T very near to 1 and may be put as

[EQUATION]

[EQUATION]

If the temperature region where the line is formed is not inhomogeneous , one may assume that only one mean electron density [FORMULA] occurs and neglecting contributions of second order, the previous equations allow the observed intensity to be expressed as a sum of two distinct contributions:

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] represents the electron density of the emitting plasma. If we choose the temperature [FORMULA] as

[EQUATION]

the contribution from the linear term [FORMULA] vanishes and the expression reduces to

[EQUATION]

It is very important to note that in this final equation the dependence of [FORMULA] on the electron density [FORMULA] is separated from the dependence on the electron temperature T.

This property allows to define the function [FORMULA] as the ratio between the observed line intensity and the [FORMULA] [FORMULA] calculated as a function of Ne at the temperature [FORMULA]:

[EQUATION]

[EQUATION]

We note that the integral gives the same result for all the spectral line belonging to the same ion [FORMULA] since [FORMULA] is the same for any line of the same ion. Thus, for all the spectral lines of the same ion the [FORMULA] function can be reduced to the form

[EQUATION]

where

[EQUATION]

When [FORMULA] the ratio [FORMULA] becomes equal to 1 and the [FORMULA] function has the value

[EQUATION]

the same for all the lines of the same ion.

The measured [FORMULA] allows [FORMULA] to be evaluated , to obtain a better approximation of [FORMULA] and start an iterative procedure, if the values of [FORMULA] are significantly different from 1.

In this way when the [FORMULA] functions of an ion are plotted versus density in the same diagram, all the curves meet in the same point [FORMULA] if a mean electron density exists over the temperature region where the ion is formed. The value of the abscissa at the crossing point [FORMULA] is the value of the electron density that fits all the observations and can be taken as the value of the electron density at [FORMULA].

This density diagnostic method is still suitable in the case that the observed intensity [FORMULA] is given by the sum of the contributions of [FORMULA] unresolved spectral lines belonging to the same ion. In this case we can define a new [FORMULA] function as follows:

[EQUATION]

Since all the lines belong to the same ion, the temperature sensitive part of the [FORMULA] [FORMULA] is the same for all spectral lines. Using the same approximations for the function [FORMULA] we obtain

[EQUATION]

[EQUATION]

Again, a plot of the new [FORMULA] versus the electron density will show that for [FORMULA] the curve will meet the same point [FORMULA] as any other fully resolved transition of the spectrum of the ion [FORMULA].

It is worth underlining some properties of the L function:

  • Density insensitive lines must have identical L functions within the experimental uncertainties.
  • Lines that depend on the electron density in the same way must have identical L functions within the experimental uncertainties.
  • Lines originated from the same upper level must have identical L functions.

2.3. The differential emission measure evaluation

The only problem of the density diagnostic method lies in the calculation of the temperature [FORMULA], since its definition involves a "trial" [FORMULA] of the emitting source. In literature several different methods for calculating the [FORMULA] have been developed, using different algorithms and approximations; a comprehensive description of the most important ones and a critical assessment and comparison of their reliability can be found in Harrison and Thompson (1992).

In this paper we present a new method to evaluate the d.e.m. which adopts the same iterative procedure described in the previous section and uses density independent lines.

We assume that a trial d.e.m. [FORMULA] is known; using a Correction Function [FORMULA], the true [FORMULA] is

[EQUATION]

as before

[EQUATION]

[EQUATION]

The intensity of the emitted line then can be expressed as

[EQUATION]

[EQUATION]

If we define the temperature [FORMULA] as

[EQUATION]

then the contribution of the linear term to the total intensity of a density independent line vanishes and we can express the total observed intensity [FORMULA] for the transition as

[EQUATION]

from which the correction [FORMULA] for each line may be computed and the first approximation [FORMULA] evaluated. A spline function is drawn through the [FORMULA] and the procedure is repeated until the [FORMULA] are all equal to 1 within the errors, or the best [FORMULA] is reached.

The procedure for d.e.m. evaluation proved to be very quick since only few steps are required in order to reach the condition of constant Correction Function. The convergence of the results to the true d.e.m. has been checked extensively using simulated observed intensities.

As usual for diagnostic procedures, a lot of information may be obtained by the analysis of [FORMULA] for differnt lines; disagreement among [FORMULA] values for lines of the same ion, may suggest problems of atomic physics, blending with other lines, inadequate calibrations. Systematic disagreement of [FORMULA] values for different ions of the same element may be used to check the ionization balance; systematic disagreement of [FORMULA] values belonging to ions of different elements may allow the chemical composition to be verified.

2.4. Discussion

This method is particularly suitable when high resolution spectra, such as those provided by SERTS or CDS and SUMER spectrographs on SOHO, are available and several lines of the same ions may be observed simultaneously. Nevertheless very accurate radiative and collisional transition probabilities are necessary since these data are essential for the calculation of the [FORMULA] for each transition. The quality of the theoretical data adopted for the calculation of the theoretical [FORMULA] functions is very critical because it affects the analysis of the observed spectrum and the measurements of the electron density. In the present work we have made use of the Arcetri spectral code updated with the CHIANTI database (Dere et al. 1996). This databank is very extended and includes very carefully assessed atomic data both of radiative transition probabilities and electron-ion collision strengths.

The main advantage for the use of this method lies in the fact that it makes possible to study directly the behavior of each line of the spectrum comparing it with the behavior of all the other lines of the same ion. In a single plot all the informations concerning density sensitive and insensitive lines of the same ion may be shown. All the curves meet in a narrow range of values around the point [FORMULA] and a measure of the mean electron density of the emitting plasma is provided.This method is more convenient than the traditional line pairs ratio and the problem to find different solutions of the measured N [FORMULA] using different line pairs is then overcome. Lines not sharing the common meeting point are immediately identified.

As for the d.e.m. analysis ,but in a more complete way, they may suggest either misidentifications, or blendings, or inaccurate atomic physics. Systematic effects are easily recognized and must be accurately investigated; disagreements common to the same range of wavelength may indicate calibration problems; disparities common to ions of the same element point to chemical composition; discrepancies occurring for consecutive ions show that the ionization balance may be questionable.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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