.. SPDX-FileCopyrightText: 1992-2026 NWO-I/SRON Space Research Organisation Netherlands .. .. SPDX-License-Identifier: CC-BY-4.0 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: .. math:: \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. :label: demalpha Here :math:`Y` is the emission measure :math:`Y \equiv n_{\mathrm H} n_{\mathrm e} V` in units of :math:`10^{64}` :math:`\mathrm{m}^{-3}`, where :math:`n_{\mathrm e}` and :math:`n_{\mathrm H}` are the electron and Hydrogen densities and :math:`V` the volume of the source. For :math:`\alpha\rightarrow\infty`, we obtain the isothermal model, for large :math:`\alpha` a steep temperature decline is recovered while for :math:`\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 :eq:`demalpha` by using the integrated emission measure :math:`Y_{\mathrm{tot}}` instead of :math:`c` for the normalisation, and instead of :math:`\alpha` its inverse :math:`p=1/\alpha`, so that we can test isothermality by taking :math:`p=0`. The emission measure distribution for the model is binned to bins with logarithmic steps of 0.10 in :math:`\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 :math:`\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 :math:`T_{\min}` directly as the lower temperature cut-off, we use a scaled cut-off :math:`\beta` such that :math:`T_{\min} = \beta T_{\max}`. From the parameters of the wdem model, the emission weighted mean temperature :math:`kT_{\mathrm{mean}}` can be calculated `de Plaa et al. (2006) `_: .. math:: T_{\mathrm{mean}} = \frac{(1+\alpha)}{(2+\alpha)} \frac{(1 - \beta^{2+\alpha})}{(1 - \beta^{1+\alpha})} ~T_{\mathrm{max}} .. warning:: Take care that :math:`\beta<1`. For :math:`\beta=1`, the model becomes isothermal, regardless the value of :math:`\alpha`. The model also becomes isothermal for :math:`p`\ =0, regardless of the value of :math:`\beta`. .. warning:: For low resolution spectra, the :math:`\alpha` and :math:`\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 :math:`T_{\min}` and :math:`T_{\max}` | ``t0`` : Maximum temperature :math:`T_{\max}`, in keV. Default: 1 keV. | ``p`` : Slope :math:`p=1/\alpha`. Default: 0.25 (:math:`\alpha = 4`). | ``cut`` : Lower temperature cut-off :math:`\beta` , in units of :math:`T_{\max}`. Default value: 0.1. The following parameters are the same as for the cie-model: | ``hden`` : Electron density in :math:`10^{20}` :math:`\mathrm{m}^{-3}` | ``it`` : Ion temperature in keV | ``vrms`` : RMS Velocity broadening in km/s (see :ref:`sect:turbulence`) | ``ref`` : Reference element | ``01...30`` : Abundances of H to Zn | ``file`` : Filename for the nonthermal electron distribution *Recommended citation:* `Kaastra et al. (2004) `_.