The relation between line ratio and emission measure analyses

Spectroscopic diagnosis of the temperature and density structure of hot optically thin plasmas from emission line intensities is usually described in two ways. The simplest approach, the 'line ratio' method, uses an observed ratio of emission line intensities to determine a 'spectroscopic mean' value of electron temperature 〈T_e〉 or electron density 〈n_e〉. The mean value is chosen to be the theoretical value of T_e or n_e which matches the observed value. The line ratio method is stable, leading to well defined values of 〈T_e〉 or 〈n_e〉 for each line pair but, in the realistic case of inhomogeneous plasmas, these are hard to interpret since each line pair yields different mean parameter values. The more general 'differential emission measure' (DEM) method recognizes that observed plasmas are better described by distributions of temperature or density along the line of sight, and poses the problem in inverse form. It is well known that the DEM function is the solution to the inverse problem, which is a function of T_e, n_e, or both. Derivation of DEM functions, while more generally applicable, is unstable to noise and errors in spectral and atomic data. The mathematical relation between these two approaches has never been precisely defined. In this paper we demonstrate the formal equivalence of the approaches, and discuss some potentially important applications of methods based upon combining the line ratio and DEM approaches.