Saturn's Seismic Rotation Revisited

Normal mode seismology is a promising means of measuring rotation in gas giant interiors, and ring seismology presents a singular opportunity to do so at Saturn. We calculate Saturn's normal modes of oscillation and zonal gravity field, using nonperturbative methods for normal modes in the rigidly rotating approximation, and perturbative methods for the shifts that Saturn's deep winds induce in the mode frequencies and zonal gravity harmonics. The latter are calculated by solving the thermogravitational wind equation in an oblate geometry. Comparing many such models to gravity data and the frequencies of ring patterns excited by Saturn's normal modes, we use statistical methods to estimate that Saturn's cloud-level winds extend inward along cylinders before decaying at a depth 0.125-0.138 times Saturn's equatorial radius, or 7530-8320 km, consistent with analyses of Cassini's gravity and magnetic field data. The seismology is especially useful for pinning down Saturn's poorly constrained deep rotation period, which we estimate at 2 pi/omega(S) = 634.7 minutes (median) with a 5/95% quantile range of 633.8-635.5 minutes. Outstanding residuals in mode frequencies at low angular degree suggest a more complicated deep interior than has been considered to date. Smaller but still significant residuals at high angular degrees also show that our picture for the thermal, composition, and/or rotation profile in Saturn's envelope is not yet complete.