# DDA 2015 – The Titan -1:0 bending wave in Saturn’s C ring

Philip D. Nicholson (Cornell)

#### Abstract

In 1988 Rosen & Lissauer identified an unusual wavelike feature in Saturn’s inner C ring as a bending wave driven by a nodal resonance with Titan (Science 241, 690) This is sometimes referred to as the -1:0 resonance since it occurs where the local nodal regression rate is approximately equal to $-n_T$, where $n_T = 22.577$ deg/day is Titan’s orbital mean motion. We have used a series of 44 stellar occultation profiles of this wave observed by the Cassini VIMS instrument to test their hypothesis. We find that, as predicted, this wave is an outward-propagating m=1 spiral with a leading orientation and a retrograde paRern speed equal to $-n_T$. Applying the standard linear dispersion relation (Shu 1984), we find a mean background surface mass density of $0.7\ g/cm^2$, similar to previous estimates for the inner C ring.

But the most intriguing feature of the wave is a narrow, incomplete gap which lies ~7 km outside the resonance. This gap varies noticeably in width and is seen in roughly 3/4 of the occultation profiles, appearing to rotate with the wave in a retrograde direction. We have developed a simple, kinematical model which accounts for the observations and consists of a continuous but very narrow gap (radial width = 0.5 km), the edges of which are vertically distorted by the propagating bending wave as it crosses the region. Differences in viewing geometry then largely account for the apparent width variations. We find a vertical amplitude of 3.8 km for the inner edge and 1.2 km for the outer edge, with nodes misaligned by ~110 deg. Moreover, both edges of the gap are slightly eccentric, with pericenters aligned with Titan, suggesting that the eccentricities are forced by the nearby Titan apsidal resonance. We hypothesize that the gap forms because the local slope of the ring becomes so great that nonlinear effects result in the physical disruption of the ring within the first wavelength of the bending wave. However, the vertical relief on the gap edges is ~10 times the predicted amplitude of the bending wave, so this story may be incomplete.

#### Notes

• Stellar occultation with VIMS
• Small region of interest:
• resolution ~2 km
• bending wave
• nodal precession = rate of Titan’s motion: -1:0 MMR
• Colombo ringlet (in Colombo Gap)
• Titan 1:0 MMR
• pericenter of ring locked to position of Titan
• That -1:0 bending wave:
• wave amplitude varies occultation to occultation
• (angle of view)
• resonance location just inside of wave
• episodic appearance of a ~1-5 km gap!
• about half the time, there’s a density peak instead of a gap!
• variation appears to be due to viewing geometry
• $\rightarrow$ leading spiral density wave
• Adjust for viewing geometry, and regular pattern emerges
• gap features associated with bending wave
• $W(\lambda,t) = W_0\, – \Delta z(\theta) \cos (\lambda\, – \lambda_{star})/\tan (B_{star})$
• (B = star-ring plane angle)
• pretty decent fit to peaklets & gaplets
• Allow each gap edge to be eccentric:
• 10 parameters to fit
• eccentric at ~1 km amplitude
• vertical displacements: about $110^{\circ}$ out of phase
• So what’s going on?
• bending wave propagating outward
• gap forms when local slope of wave first exceeds unity
• beyond gap, wave re-establishes itself with a smaller amplitude
• Don’t know why
• gap is probably a nonlinear response of ring to the steep local slope, leading to vertical ‘tearing’ of the ring surface