4.1.40. Wdem: power law differential emission measure model

This model calculates the spectrum of a power law distribution of the differential emission measure distribution. It appears to be a good empirical approximation for the spectrum in cooling cores of clusters of galaxies. It was first introduced by Kaastra et al. (2004) and is defined as follows:

(1)\frac{ {\mathrm d}Y }{ {\mathrm d}T } = \left\{
\begin{array}{ll}
0            & \qquad \mathrm{if} \quad T \leq \beta T_{\max} ; \\
cT^{\alpha}  & \qquad \mathrm{if} \quad \beta T_{\max} < T < T_{\max} ;\\
0            & \qquad \mathrm{if} \quad T \geq T_{\max} .
\end{array} \right.

Here Y 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.

For \alpha\rightarrow\infty, we obtain the isothermal model, for large \alpha a steep temperature decline is recovered while for \alpha=0 the emission measure distribution is flat. Note that Peterson et al. (2003) use a similar parameterisation but then for the differential luminosity distribution). In practice, we have implemented the model (1) by using the integrated emission measure Y_{\mathrm{tot}} instead of c for the normalisation, and instead of \alpha its inverse p=1/\alpha, so that we can test isothermality by taking p=0. The emission measure distribution for the model is binned to bins with logarithmic steps of 0.10 in \log T, and for each bin the spectrum is evaluated at the emission measure averaged temperature and with the integrated emission measure for the relevant bin (this is needed since for large \alpha the emission measure weighted temperature is very close to the upper temperature limit of the bin, and not to the bin centroid). Instead of using T_{\min} directly as the lower temperature cut-off, we use a scaled cut-off \beta such that T_{\min} = \beta T_{\max}.

From the parameters of the wdem model, the emission weighted mean temperature kT_{\mathrm{mean}} can be calculated de Plaa et al. (2006):

T_{\mathrm{mean}} = \frac{(1+\alpha)}{(2+\alpha)}
                   \frac{(1 - \beta^{2+\alpha})}{(1 - \beta^{1+\alpha})} ~T_{\mathrm{max}}

Warning

Take care that \beta<1. For \beta=1, the model becomes isothermal, regardless the value of \alpha. The model also becomes isothermal for p=0, regardless of the value of \beta.

Warning

For low resolution spectra, the \alpha and \beta parameters are not entirely independent, which could lead to degeneracies in the fit.

The parameters of the model are:

norm : Integrated emission measure between T_{\min} and T_{\max}
t0 : Maximum temperature T_{\max}, in keV. Default: 1 keV.
p : Slope p=1/\alpha. Default: 0.25 (\alpha = 4).
cut : Lower temperature cut-off \beta , in units of T_{\max}. Default value: 0.1.

The following parameters are the same as for the cie-model:

hden : Electron density in 10^{20} \mathrm{m}^{-3}
it : Ion temperature in keV
vrms : RMS Velocity broadening in km/s (see Definition of the micro-turbulent velocity in SPEX)
ref : Reference element
01...30 : Abundances of H to Zn
file : Filename for the nonthermal electron distribution

Recommended citation: Kaastra et al. (2004).