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Analytical approximations to numerical solutions of theoretical emission measure distributions

Emission line fluxes from cool stars are widely used to establish an apparent emission measure distribution, Emd-app(Te), between temperatures characteristic of the low transition region and the low corona. The true emission measure distribution, Emd-t(Te), is determined by the energy balance and geometry adopted and, with a numerical model, can be used to predict Emd-app(Te), to guide further modelling. The scaling laws that exist between coronal parameters arise from the dimensions of the terms in the energy balance equation. Here, analytical approximations to numerical solutions for Emd-t(Te) are presented, which show how the constants in the coronal scaling laws are determined. The apparent emission measure distributions show a minimum value at some temperature (T0) and a maximum at the mean coronal temperature Tc (although in some stars, emission from active regions can contribute). It is shown that, for the energy balance and geometry adopted, the analytical values of the emission measure and electron pressure at T0 and Tc, depend on only three parameters: the stellar surface gravity and the values of T0 and Tc. The results are tested against full numerical solutions for Epsilon Eri (K2 V) and are applied to Procyon (alpha CMi; F5 IV/V). The analytical approximations can be used to restrict the required range of full numerical solutions, to check the assumed geometry and to show where the adopted energy balance may not be appropriate.