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(T _e), between temperatures characteristic of the low transition region and the low corona. The true emission measure distribution, Emd _t(T _e), is determined by the energy balance and geometry adopted and, with a numerical model, can be used to predict Emd _app(T _e), 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(T _e) 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 T _o and a maximum at the mean coronal temperature T _c (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 T _o and T _c depend on only three parameters: the stellar surface gravity and the values of T _o and T _c. The results are tested against full numerical solutions for ε Eri (K2 V) and are applied to Procyon (α 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.