In collisional ionization equilibrium (CIE), the X-ray spectrum from a plasma depends on the distribution of emission measure over temperature (DEM). Due to the well-known ill conditioning problem, no precisely resolved DEM can be inverted directly from the spectrum, so often only a gross parametrization of the DEM is used to approximate the data, in hopes that the parametrization can provide useful model-independent constraints on the heating process. However, ill conditioning also introduces ambiguity into the various different parametrizations that could approximate the data, which may spoil the perceived advantages of model independence. Thus, this paper instead suggests a single parametrization for both the heating mechanism and the X-ray sources, based on a model of impulsive heating followed by complete cooling. This approach is similar to a ``cooling flow approach, but allows injection at multiple initial temperatures, and applies even when the steady state is distribution of different shock strengths, as for a standing shock with a range of obliquities, or for embedded stochastic shocks that are only steady in a statistical sense. This produces an alternative parametrization for X-ray spectra that is especially streamlined for higher density plasmas with efficient radiative cooling, and provides internal consistency checks on the assumption of impulsive heating followed by complete cooling. The result is no longer model independent, but the results are more directly interpretable in terms of useful physical constraints on the impulsive heating distribution.
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