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10.1. Available codes

There are several spectral codes available that can be used to model hot plasmas. Without being complete, we mention here briefly the following codes. For a more extensive review, see Raymond (2005).

First, there are codes which calculate predominantly spectra in Collisional Ionisation Equilibrium:

  1. CHIANTI 1, (named after the wine region near Firenze, Landi et al. 2006) evolved from early work by Landini & Monsignori Fossi (1970); originally, it focussed on the calculation of Solar spectra and hence EUV lines are well represented in this code.

  2. APEC 2 (Astrophysical Plasma Emission Code, (Smith et al., 2001) evolved from the original Raymond & Smith (1987) code; originally, the main goal was to calculate the emissivity and cooling curve properly. This model is available in the popular fitting package XSPEC 3, (Arnaud, 1996).

  3. Mekal (after MEwe, KAastra & Liedahl; see Mewe et al., 1995) evolved from early work by Mewe and collaborators starting from 1970 onwards. The Mekal model is also incorporated in XSPEC.

The Mekal model is now part of the SPEX package 4, (Kaastra et al., 1996b). SPEX is a spectral fitting package developed at SRON Utrecht based on the original CIE models of Mewe and collaborators. It has grown significantly and also contains NEI and a few simple PIE models; more models are under development. The package offers the opportunity to plot or tabulate also various physical parameters of interest of the model, apart from the spectrum, and the atomic physics used in the Mekal model has been significantly updated in the SPEX version as compared to the older version available in XSPEC. A majority of all plots shown in this paper were produced using SPEX.

There are also models focussing upon photo-ionised plasmas. We mention here two codes (apart from SPEX):

  1. XSTAR 5 evolved from the original work of Kallman (1982), focussing upon the X-ray band. It is relatively straightforward to use and there are options to use it with XSPEC.

  2. Cloudy 6 (Ferland et al., 1998) evolved from work by Ferland and collaborators that originally was meant to model optical emission line regions. It gradually evolved to the X-ray band. It is somewhat more complicated to use, but has more options than XSTAR.

10.2. Important parameters

In order to calculate X-ray spectra for collisional plasmas, there is a set of parameters that determine the spectral shape and flux. The two most important parameters are the electron temperature Te and emission measure Y of the gas. The emission measure is sometimes not well or even incorrectly defined. We follow here the definition used in SPEX, namely Y = integ ne nH dV where ne and nH are the electron and hydrogen density (whether ionised or not). Sometimes people use "n2" in this expression, but it is not always obvious whether ion density, baryon density, electron density, total particle density or something else is meant. The next important parameters are the abundances of the elements; these do not only determine the line emission spectrum, but also affect the continuum emission (Sect. 5.1). Note that the abundances also affect the precise values of ne / nH and ni/nH. For a fully ionised plasma with proto-solar abundances (Lodders, 2003) these ratios are 1.201 and 1.097, respectively (for the older Anders & Grevesse (1989) abundances these ratios are 1.209 and 1.099).

For ionising plasmas (NEI) also the parameter U = integ ne dt (see Eqn. 28) is important, as this describes the evolution of a shocked or recombining gas towards equilibrium.

When high spectral resolution observations are available, other parameters are important, like the ion temperature and turbulent velocity. These parameters determine the width of the spectral lines (see Sect. 5.2.2 and Sect. 6.3). Note that in non-equilibrium plasmas, the electron and ion temperatures are not necessarily the same. In an extreme case, the supernova remnant SN 1006, Vink et al. (2003) measured an oxygen ion temperature that is 350 times larger than the electron temperature of 1.5 keV. SPEX has also options to approximate the spectra when there are non-Maxwellian tails to the electron distribution (Sect. 9). These tails may be produced by non-thermal acceleration mechanisms (Petrosian et al., 2008 - Chapter 9, this volume).

In general, for the low densities encountered in the diffuse ISM of galaxies, in clusters and in the WHIM, the absolute electron density is not important (except for the overall spectral normalisation through the emission measure Y). Only in the higher density regions of for example stars X-ray line ratios are affected by density effects, but this is outside the scope of the present paper.

However, if the density becomes extremely low, such as in the lowest density parts of the WHIM, the gas density matters, as in such situations photoionisation by the diffuse radiation field of galaxies or the cosmic background becomes important. But even in such cases, it is not the spectral shape but the ionisation (im)balance that is affected most. In those cases essentially the ratio of incoming radiation flux to gas density determines the photoionisation equilibrium.

10.3. Multi-Temperature emission and absorption

Several astrophysical sources cannot be characterised by a single temperature. There may be various reasons for this. It may be that the source has regions with different temperatures that cannot be spatially resolved due to the large distance to the source, the low angular resolution of the X-ray instrument, or superposition of spectra along the line of sight. Examples of these three conditions are stellar coronae (almost point sources as seen from Earth), integrated cluster of galaxies spectra, or spectra along the line of sight through the cooling core of a cluster, respectively. But also in other less obvious cases multi-temperature structure can be present. For example Kaastra (2004) showed using deprojected spectra that at each position in several cool core clusters there is an intrinsic temperature distribution. This may be due to unresolved smaller regions in pressure equilibrium with different densities and hence different temperatures.

The most simple solution is to add as many temperature components as are needed to describe the observed spectrum, but there are potential dangers to this. The main reason is temperature sensitivity. Most spectral temperature indicators (line fluxes, Bremsstrahlung exponential cut-off) change significantly when the temperature is changed by a factor of two. However, for smaller temperature changes, the corresponding spectral changes are also minor. For example, a source with two components of equal emission measure Y and kT = 0.99 and 1.01 keV, respectively, is almost indistinguishable from a source with the emission measure 2Y and temperature 1.00 keV. Obviously, the most general case is a continuous differential emission measure distribution (DEM) D(T) ident dY(T) / dT. Typically, in each logarithmic temperature range with a width of a factor of two, one can redistribute the differential emission measure D(T) in such a way that the total emission measure as integrated over that interval as well as the emission-measure weighted temperature remain constant. The resulting spectra are almost indistinguishable. This point is illustrated in Fig. 16.

Figure 16a Figure 16b

Figure 16. Reconstructed DEM distribution for a simulated spectrum containing two equally strong isothermal components of 0.6 and 2.4 keV (left panel) or a uniform distribution between 0.6 and 2.4 keV (right panel). Thick solid line: the input model; thin solid line: regularisation method; dot-dashed line: polynomial method; dashed line: genetic algorithm; dotted line: two broadened Gaussian components. The corresponding best-fit spectra differ in chi2 by less than 2 for each case. Obviously, small scale detail cannot be recovered but bulk properties such as the difference between a bimodal DEM distribution (left panel) or a flat distribution (right panel) can be recovered. From (Kaastra et al., 1996a). See that paper for more details.

There are several methods to reconstruct D(T). We refer here to Kaastra et al. (1996a) for more details. All these methods have been implemented in SPEX. They have been applied to stellar coronae or Solar spectra, but recently also to clusters of galaxies. See for example Kaastra (2004). In the last case, an analytical power-law approximation D(T) ~ T 1 / alpha has been used.

Also for photo-ionised plasmas a similar technique can be applied. Instead of the temperature T, one uses the ionisation parameter xi (XSTAR users) or U (Cloudy users). In some cases (like Seyfert 2 galaxies with emission spectra) one recovers D(xi) from the observed emission spectra; in other cases (like Seyfert 1 galaxies with absorption spectra), a continuum model is applied to the observed spectrum, and D(xi) is recovered from the absorption features in the spectrum. Most of what has been said before on D(T) also applies to this case, i.e. the resolution in xi and the various methods to reconstruct the continuous distribution D(xi).

The authors thank ISSI (Bern) for support of the team "Non-virialized X-ray components in clusters of galaxies". SRON is supported financially by NWO, the Netherlands Organization for Scientific Research.

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