A Useful Approximation to the Cooling Coefficient of Trace Elements

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Abstract
Radiative cooling is an important ingredient in hydrodynamical models involving evolution of high temperature plasmas. Unfortunately, calculating an accurate cooling coefficient generally requires the solution of over a hundred differential equations to follow the ionization. We discuss here a simple 2-parameter approximation for the cooling coefficient due to elements heavier than H and He, for the temperature range $T= 10^{4}-10^{8}$K. Tests of the method show that it successfully tracks the ionization level in severe dynamical environments, and accurately approximates the non-equilibrium cooling coefficient of the trace elements, usually to within 10% in all cases for which cooling is actually important. The error is large only when the temperature is dropping so rapidly due to expansion that radiative cooling is negligible, but even in this situation, the ionization level is followed sufficiently accurately. The current approximation is fully implemented in publicly available FORTRAN code. A second paper will discuss general approaches to approximation methods of this type, other realizations which could be even more accurate, and the potential for extension to calculations of non-equilibrium spectra.
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