Seasonal variation of Saturn's Lyman-$\alpha$ brightness

We examine Saturn's non-auroral (dayglow) emissions at Lyman-$\alpha$ observed by the {Cassini/UVIS} instrument from 2004 until 2016, to constrain meridional and seasonal trends in the upper atmosphere. We separate viewing geometry effects from trends driven by atmospheric properties, by applying a multi-variate regression to the observed emissions. The Lyman-$\alpha$ dayglow brightnesses depend on the incident solar flux, solar incidence angle, emission angle, and observed latitude. The emissions across latitudes and seasons show a strong dependence with solar incidence angle, typical of resonantly scattered solar flux and consistent with no significant internal source. We observe a bulge in Ly-$\alpha$ brightness that shifts with the summer season from the southern to the northern hemisphere. We estimate atomic hydrogen optical depths above the methane homopause level for dayside disk observations (2004-2016) by comparing observed Lyman-$\alpha$ emissions to a radiative transfer model. We model emissions from resonantly scattered solar flux and a smaller but significant contribution by scattered photons from the interplanetary hydrogen (IPH) background. During northern summer, inferred hydrogen optical depths steeply decrease with latitude towards the winter hemisphere from a northern hemisphere bulge, as predicted by a 2D seasonal photochemical model. The southern hemisphere mirrors this trend during its summer. However, inferred optical depths show substantially more temporal variation between 2004 and 2016 than predicted by the photochemical model.