Chromospheric Height and Density Measurements in a Solar Flare Observed with RHESSI – I. Theory

We obtain a theoretical description of the height (z) distribution of flare hard X-rays in the collisional thick-target model as a function of photon energy ε. This depends on the target atmosphere density structure n(z) and on the beam spectral index δ. We show that by representing the data in terms of the 1-D function z(ε) defining where the emission peaks as a function of ε it is possible to derive n(z) from data on z(ε). This is done first on the basis of a simple stopping depth argument then refined to allow for the dependence on spectral index δ. The latter is worked out in detail for the case of a parameterization n(z)=n0 (z/z0)−b which yields numerical results for z(ε) well fit by z(ε)∼ε−α, with α dependent on δ, which is also found to fit well to actual observations. This enables derivation of flare loop n(z) in terms of n0,b from RHESSI data in an entirely novel way, independent of other density diagnostic methods, and also of how n(z) varies with time in flares such as by evaporation, as detailed in companion Paper II.