Fundamental Limitations of Emission-Line Spectra as Diagnostics of Plasma Temperature and Density Structure
We discuss the problem of determining plasma structure from optically thin emission lines whose emission coefficients and frequency-integrated intensities are dependent on temperature T and electron density n. We cast the problem into the inverse form discussed by Hubeny & Judge (1995).Three properties of the kernels in the integral equations lead to fundamental limitations in trying to determine the source term μ(T, n), the ``emission measure differential in temperature and density,'' from a set of emission-line intensities. First, the kernels are rather weakly dependent on n. Second, they have asymptotically identical dependencies on n. The inverse problem is therefore very poorly conditioned in the density dimension. Third, the kernels cannot (and may never) be calculated with an accuracy better than +/-10%. These properties set limits on the accuracy of all solutions, independent of the accuracy of observed line intensities. This concurs with earlier but less general work by Brown et al. (1991). We try to determine solutions for μ(T, n), using specific target sources and numerical algorithms. Using realistic uncertainties, calculations indicate that meaningful inverse solutions for μ(T, n) cannot be obtained owing to the severe propagation of kernel errors, irrespective of the quality of observational data. Solutions for the ``emission measure differential in temperature'' ξ(T) = \smallint μ(T, n)dn are more robust against instabilities driven by poor conditioning. Since traditional ``emission-line diagnostic ratios'' can only be defined through μ(T, n) (or some generalization thereof), our analysis casts doubt on the meaning of plasma properties derived from such line ratios, and illustrates the severe nonuniqueness of any equivalent ``inverse'' solution. This work may be important for studying a wide variety of atomic and ionic emission-line spectra, including work with instruments on SOHO and the Hubble Space Telescope.