4.1.5. Cie: collisional ionisation equilibrium model

This model calculates the spectrum of a plasma in collisional ionisation equilibrium (CIE). It consists essentially of two steps, first a calculation of the ionisation balance and then the calculation of the X-ray spectrum. The basis for this model is formed by the mekal model, but several updates have been included (see Plasma model in SPEX 3.0).

4.1.5.1. Temperatures

Some remarks should be made about the temperatures. SPEX knows three temperatures that are used for this model.

First there is the electron temperature T_{\mathrm e}. This is usually referred to as “the” temperature of the plasma. It determines the continuum shape and line emissivities and the resulting spectrum is most sensitive to this.

Secondly, there is the ion temperature T_{\mathrm i}. This is only important for determining the line broadening, since this depends upon the thermal velocity of the ions (which is determined both by T_{\mathrm i} and the atomic mass of the ion). Only in high resolution spectra the effects of the ion temperature can be seen.

Finally, we have introduced here the ionization balance temperature T_{\mathrm b} that is used in the determination of the ionization equilibrium. It is the temperature that goes into the calculation of ionization and recombination coefficients. In equilibrium plasmas, the ratio R_{\mathrm b} \equiv T_{\mathrm b} / T_{\mathrm e} is 1 by definition. It is unphysical to have R_{\mathrm b} not equal to 1. Nevertheless we allow for the possibility of different values of R_{\mathrm b}, in order to mimick out of equilibrium plasmas. For R_{\mathrm b}<1, we have an ionizing plasma, for R_{\mathrm b}>1 a recombining plasma. Note that for ionizing plasmas SPEX has also the nei model (Neij: non-equilibrium ionisation jump model), which takes into account explicitly the effects of transient (time dependent) ionization.

It is also possible to mimic the effects of non-isothermality in a simple way. SPEX allows for a Gaussian emission measure distribution:

Y(x) = \frac{Y_0}{\sqrt{2\pi} \sigma_T} e^{\displaystyle{-(x-x0)^2/2\sigma_T^2}}

where Y_0 is the total, integrated emission measure. By default x\equiv \log T and x_0\equiv \log T_0 with T_0 the average temperature of the plasma (this is entered at the “T” value in SPEX). However, this can be changed to x\equiv T and x_0\equiv T_0 by setting logt to 0. If the parameter sup is set > 10^{-5}, then the Gaussian emission measure distribution model becomes asymmetric. The sig parameter determines the slope of the low-temperature part of the Gaussian and sup determines the high-temperature side. Usually (default) \sigma_T = 0 and in that case the normal isothermal spectrum is chosen. Note that for larger values of \sigma_T the cpu time may become larger due to the fact that the code has to evaluate many isothermal spectra.

The actual emission measure distribution that SPEX uses can be shown using Ascdump: ascii output of plasma and spectral properties. The subcommand for this is dem:

ascdump ter 1 1 dem

(if your cie model is in sector 1 and component 1). SPEX creates a temperature grid (either logarithmic or linear) and for each grid point, it calculates an emission measure following the Gaussian distribution. The sum of all the emission measures is equal to the total emission measure Y_0 (norm).

4.1.5.2. Line broadening

Apart from line broadening due to the thermal velocity of the ions (caused by T_{\mathrm i} > 0) it is also possible to get line broadening due to (micro)turbulence. Since SPEX version 3.06, we use the RMS velocity broadening in km/s in our models. For more information about this change see: Definition of the micro-turbulent velocity in SPEX.

4.1.5.3. Density effects

It is also possible to vary the hydrogen density n_{\mathrm H} of the plasma. This does not affect the overall shape of the spectrum (i.e., by changing n_{\mathrm H} only SPEX still uses the previous value of the emission measure Y \equiv n_{\mathrm H} n_{\mathrm e} V), but spectral lines that are sensitive to the hydrogen density will get different intensities. Usually this occurs for higher densities, for example under typical ISM conditions (n_{\mathrm H} = 1 cm^{-3}) this is not an important effect.

4.1.5.4. Non-thermal electron distributions

The effects of non-thermal electron distribution on the spectrum can be included. See Non-thermal electron distributions for more details.

4.1.5.5. Abundances

The abundances are given in Solar units. Which set of solar units is being used can be set using the abun command (Abundance: standard abundances). For spectral fitting purposes it is important to distinguish two situations.

In the first case there is a strong thermal continuum. Since in most circumstances the continuum is determined mainly by hydrogen and helium, and the X-ray lines are due to the other elements, the line to continuum ratio is a direct measurement of the metal abundances compared to H/He. In this situation, it is most natural to have the hydrogen abundance fixed and allow only for fitting of the other abundances (as well as the emission measure).

In the other case the thermal continuum is weak, but there are strong spectral lines. Measuring for example the Fe abundance will give large formal error bars, not because the iron lines are weak but because the continuum is weak. Therefore, all abundances will get rather large error bars, and despite the fact of strong O and Fe lines, the formal error bars on the O/Fe ratio becomes large. This can be avoided by choosing another reference element, preferentially the one with the strongest lines (for example Fe). Then the O/Fe ratio will be well constrained, and it is now only the H abundance relative to Fe that is poorly constrained. In this case it is important to keep the nominal abundance of the reference element to unity. Also keep in mind that in general we write for the normalisation n_{\mathrm e}n_{\mathrm X}V in this case; when the reference element is H, you mays substitute X=H; however when it is another element, like O, the normalisation is still the product of n_{\mathrm e}n_{\mathrm X}V where X stands for the fictitious hydrogen density derived from the solar O/H ratio.

4.1.5.6. External magnetic field

External magnetic field resolves the degeneracy of the magnetic sublevels of a given level and allows mixing with neighbour levels of the same magnetic quantum number and parity, which could not happen in the field-free case. The appearance and intensities of these otherwise forbidden transitions depend on the magnetic field.

The magnetic field strength is given in unit of Gauss. For very strong fields ($B\geq 10^{9}$ Gauss), the calculation likely becomes less reliable as the Lorentzian field versus Coulomb field strength ratio reaches or goes beyond unity.

4.1.5.6.1. Example

Suppose the Solar O abundance is 1E-3 (i.e. there is 0.001 oxygen atom per hydrogen atom in the Sun). Take the reference atom oxygen (Z=8). Fix the oxygen abundance to 1. Fit your spectrum with a free hydrogen abundance. Suppose the outcome of this fit is a hydrogen abundance of 0.2 and an emission measure 3 (in SPEX units). This means n_{\mathrm e}n_{\mathrm X}V = 3\times 10^{64} \mathrm{m}^{-3}. Then the “true” emission measure n_{\mathrm e}n_{\mathrm H}V = 0.2 \times 3\times 10^{64} \mathrm{m}^{-3}. The nominal oxygen emission measure is n_{\mathrm e}n_{\mathrm O}V = 0.001 \times 3\times 10^{64} \mathrm{m}^{-3}, and nominally oxygen would have 5 times overabundance as compared to hydrogen, in terms of solar ratios.

4.1.5.7. Parameter description

Warning

When you make the temperature too low such that the plasma becomes completely neutral, the model will crash. This is because in that case the electron density becomes zero, and the emission measure is undefined. The nominal temperature limits that are implemented in SPEX usually avoid that condition, but the results may depend somewhat upon the metallicity because of the charge exchange processes in the ionisation balance.

Warning

In high resolution spectra, do not forget to couple the ion temperature to the electron temperature, as otherwise the ion temperature might keep its default value of 1 keV during spectral fitting and the line widths may be wrong.

Warning

Some people use instead of the emission measure Y \equiv n_{\mathrm H} n_{\mathrm e} V, the quantity Y^\prime = n_{\mathrm e}^2 V as normalisation. This use should be avoided as the emission is proportional to the product of electron and ion densities, and therefore use of Y^\prime makes the spectrum to depend nonlinear on the elemental abundances (since an increase in abundances also affects the n_{\mathrm e} / n_{\mathrm H} ratio).

Warning

The default line broadening is just Doppler broadening. This is fine and self-consistent for the ‘old’ line calculation. To incorporate the natural line broadening for the ‘new’ calculations, the user must use the var dopp 4 option to get Voigt profiles. This is physically better but takes more computation time.

The parameters of the model are:

norm : the normalisation, which is the emission measure Y \equiv n_{\mathrm H} n_{\mathrm e} V in units of 10^{64} \mathrm{m}^{-3}, where n_{\mathrm e} and n_{\mathrm H} are the electron and Hydrogen densities and V the volume of the source. Default value: 1.
t : the electron temperature T_{\mathrm e} in keV. Default value: 1.
sig : the width \sigma_T of the gaussian emission measure profile. Default value: 0. (no temperature distribution i.e. isothermal)
sup : the width \sigma_T of the high-temperature part of the gaussian emission measure profile. If larger than 10^{-5} keV, the sig parameter becomes the sigma value for the low-temperature end. Default value: 0
logt : Switch between linear and logarithmic temperature scale for the gaussian emission measure profile. Default value: 1 (logarithmic)
hden : the Hydrogen density n_{\mathrm e} in units of 10^{20} \mathrm{m}^{-3} (or 10^{14} \mathrm{cm}^{-3}). Default value: 10^{-14}, i.e. typical ISM conditions, or the low density limit.
it : the ion temperature T_{\mathrm i} in keV. Default value: 1
rt : the ratio of ionization balance to electron temperature, R_{\mathrm b} = T_{\mathrm b} / T_{\mathrm e} in keV. Default value: 1.
vrms : RMS Velocity broadening in km/s (see Definition of the micro-turbulent velocity in SPEX)
ref : reference element. Default value 1 (hydrogen). See above for more details. The value corresponds to the atomic number of the reference element.
01 : Abundance of hydrogen (H, Z=1) in Solar units. Default 1.
02 : Abundance of helium (He, Z=2) in Solar units. Default 1.
\ldots 30 : Abundance of zinc (Zn, Z=30) in Solar units. Default 1.
file : Filename for the non-thermal electron distribution. If not present, non-thermal effects are not taken into account (default).
b : External magnetic field strength in Gauss. Default value: 0.

Recommended citation: Kaastra et al. (1996).