Modelling H$_{3}^{+}$ in planetary atmospheres: effects of vertical gradients on observed quantities

Since its discovery in the aurorae of Jupiter ~30 years ago, the H$_{3}^{+}$ ion has served as an invaluable probe of giant planet upper atmospheres. However, the vast majority of monitoring of planetary H$_{3}^{+}$ radiation has followed from observations that rely on deriving parameters from column-integrated paths through the emitting layer. Here, we investigate the effects of density and temperature gradients along such paths on the measured H$_{3}^{+}$ spectrum and its resulting interpretation. In a non-isothermal atmosphere, H$_{3}^{+}$ column densities retrieved from such observations are found to represent a lower limit, reduced by 20% or more from the true atmospheric value. Global simulations of Uranus' ionosphere reveal that measured H$_{3}^{+}$ temperature variations are often attributable to well-understood solar zenith angle effects rather than indications of real atmospheric variability. Finally, based on these insights, a preliminary method of deriving vertical temperature structure is demonstrated at Jupiter using model reproductions of electron density and H$_{3}^{+}$ measurements. The sheer diversity and uncertainty of conditions in planetary atmospheres prohibits this work from providing blanket quantitative correction factors; nonetheless, we illustrate a few simple ways in which the already formidable utility of H$_{3}^{+}$ observations in understanding planetary atmospheres can be enhanced.