# 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)¶

Here is the emission measure in units of , where and are the electron and Hydrogen densities and the volume of the source.

For , we obtain the isothermal model, for large a steep temperature decline is recovered while for 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 instead of for the normalisation, and instead of its inverse , so that we can test isothermality by taking . The emission measure distribution for the model is binned to bins with logarithmic steps of 0.10 in , 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 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 directly as the lower temperature cut-off, we use a scaled cut-off such that .

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

Warning

Take care that . For , the model becomes isothermal, regardless the value of . The model also becomes isothermal for =0, regardless of the value of .

Warning

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

The parameters of the model are:

`norm`

: Integrated emission measure between and
`t0`

: Maximum temperature , in keV. Default: 1 keV.`p`

: Slope . Default: 0.25 ().`cut`

: Lower temperature cut-off , in units of
. Default value: 0.1.The following parameters are the same as for the cie-model:

`hden`

: Electron density in `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).