Kronoseismology. IV. Six Previously Unidentified Waves in Saturn's Middle C Ring

This analysis focuses on stellar occultation data obtained by the Visual and Infrared Mapping Spectrometer (VIMS) on board the Cassini Spacecraft (Brown et al. 2004). While in its standard operating mode VIMS obtains spatially resolved spectra of various objects in the Saturn system, this instrument can also operate in a mode where it repeatedly measures the spectrum of a star as the planet or its rings pass between the star and the spacecraft. In this occultation mode, a precise time-stamp is appended to each spectrum to facilitate reconstruction of the observation geometry.

As with previous occultation studies, here we will only consider data obtained at wavelengths between 2.87 and 3.00 microns, where the rings are especially dark and so provide a minimal background to the stellar signal. Using the appropriate SPICE kernels (Acton 1996), we use the timing information encoded with the occultation data to compute both the radius and inertial longitude in the rings that the starlight passed through. Note that the information encoded in these kernels has been determined to be accurate to within one kilometer, and fine-scale adjustments based on the positions of circular ring features enable these estimates to be refined to an accuracy of order 150 m. For this analysis, we use the latest estimates of these offsets from French et al. (2017); we have verified that these new offsets do not change the results described in Hedman & Nicholson (2014).

The VIMS instrument has a highly linear response function (Brown et al. 2004), so the raw data numbers returned by the spacecraft are directly proportional to the apparent brightness of the star. We can therefore estimate the transmission through the rings T as simply the ratio of the observed signal at a given radius to the average signal in regions outside the rings. From this transmission, we can compute the ring's optical depth τ using the standard formula $\tau =-\mathrm{ln}(T)$. Both T and τ depend on the observation geometry, but for relatively low-optical depth regions like the middle C ring we can define the normal optical depth ${\tau }_{n}=\tau /| \sin (B)|$ (B being the ring opening angle to the star), which should have nearly the same value for all the occultations considered here.

We will consider three different groups of occultations for this study:

• A.  
  For our initial investigation of the W81.02 patterns (Section 3), we use the same 23 occultation cuts obtained between 2007 and 2014 listed in Table 2 of Hedman & Nicholson (2014).
• B.  
  For our initial search for waves with m = −6 through −9 (Section 4), we consider the subset of occultations that use the star γ Crucis and were obtained between Revs 71 and 102 (i.e., 2008–2009) listed in Table 1.
• C.  
  Finally, for our more in-depth analysis of the wave candidates (Section 5), we utilize a larger set of 56 occultations summarized in Table 1. This set includes all occultations obtained prior to Rev (i.e., Cassini orbit) 270 that cover the relevant wave features and satisfy the following requirements:

  • (a)  
    Do not have any data gaps larger than 1 km in the region of interest for the wave (see Table 2).
  • (b)  
    Have mean normal optical depth values within the region of interest (see Table 2) that are within 0.1 of the median value for all occultations. This eliminates occultations which have unstable signal levels and other instrumental issues that could impact the relevant signals.
  • (c)  
    Have rms normal optical depth variations smaller than 0.015 on radius scales of 0.1 km in nearby featureless ring regions (80700–80800 km, 81200–81300 km, 82300–82350 km, 83,700–83800 km or 84100–84200 km). This removes occultations with low signal-to-noise.

Table 2.  Summary of Wave Properties

Wave Name	Other	Previous	Figures	Radii Considered	Wavelengths	m	Resonant	Pattern Speed	Peak Wave
	Designationsa	Analysisb			Considered		Radiusd		Amplitudee
W80.98	Baillié 13, Rosen e	HN13	10, 12	80978–80994 km	1–3 km	−4	80986.15 km	$1660\buildrel{\circ}\over{.} 36\pm 0\buildrel{\circ}\over{.} 02$/day	0.386
W81.02a	Baillié 14	HN14c	2, 13, 15, 22	81016–81025 km	1–3 km	−5	81023.15 km	159363 ± 002/day	0.118
W81.02b	Baillié 14	HN14c	3, 14, 15, 22	81019–81028 km	1–3 km	−11	81024.17 km	145050 ± 001/day	0.140
W81.43			15, 18, 20, 22	81422–81432 km	1–3 km	−6	81429.55 km	153824 ± 004/day	0.056
W81.96			15, 17, 22	81955–81965 km	1–3 km	−7	81962.45 km	149246 ± 002/day	0.082
W82.00	Baillié 15	HN13, HN14	7, 12	81995–82010 km	1–3 km	−3	82007.75 km	173665±002/day	0.281
W82.06	Baillié 16, Rosen f	HN13, HN14	8, 12	82043–82058 km	1–3 km	−3	82059.40 km	173500±002/day	0.458
W82.21	Baillié 17, Rosen g	HN13, HN14	9, 12	82195–82210 km	1–3 km	−3	82207.50 km	173029±002/day	0.555
W82.53			15, 19, 21, 22	82522–82532 km	1–3 km	−8	82528.75 km	145422±004/day	0.050
W83.09			15, 16 , 22	83038–83093 km	1–3 km	−9	83090.65 km	142184±001/day	0.099
W83.63	Baillié 18, Rosen h	HN14	11, 12	83623–83628 km	1–3 km	−10	83632.02 km	139406±001/day	0.440
W84.64	Baillié 19, Rosen i	HN13, HN14	6, 12	84630–84640 km	1–3 km	−2	84643.20 km	186075 ± 003/day	0.471

Notes.

^aDesignations from Baillié et al. (2011) and Rosen et al. (1991). ^bHN13 = Hedman & Nicholson (2013), HN14 = Hedman & Nicholson (2014). ^cDiscussed, but not identified. ^dDerived from pattern speed. ^eNo error is provided on these quantities because their uncertainties are dominated by systematic errors in the reconstruction that are difficult to rigorously quantify.

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For the purposes of this study, the two most important parameters associated with these waves are the number of spiral arms $| m|$ and the pattern speed Ω_p at which these density variations rotate around the planet in inertial space (Shu 1984). For waves generated by planetary normal-mode oscillations, the number of spiral arms equals the mode's azimuthal wavenumber, while the pattern speed equals the mode's propagation rate around the planet in inertial space. In addition, a wave with a given number of spiral arms and pattern speed can only be generated at resonant locations r_L where the ring-particles' orbital mean motion n_L and radial epicyclic frequency κ_L satisfy the following relationship:

$$\begin{eqnarray}&&m({n}_{L}-{{\rm{\Omega }}}_{p})={\kappa }_{L}.\end{eqnarray} \tag{ 1 }$$

Note that in this expression we allow m to be a signed quantity, such that m > 0 corresponds to cases where the pattern speed is slower than the mean motion (i.e., Inner Lindblad Resonances), and m < 0 corresponds to cases where the pattern speed is faster than the mean motion (i.e., Outer Lindblad Resonances). If a wave is observed at a given radius, the corresponding orbital frequencies n_L and κ_L can be determined from the planet's gravitational field; hence, there is a discrete set of pattern speeds the wave could have, one for each possible value of m.

Fortunately, both m and Ω_p can be estimated by comparing wave profiles observed at different times and longitudes. A generic density wave with $| m|$ arms and pattern speed Ω_p causes the surface mass density of the ring σ to vary with radius r, longitude λ and time t as follows:

$$\begin{eqnarray}&&\sigma ={\sigma }_{0}{\mathfrak{R}}[1+A(r){e}^{i[{\phi }_{r}(r)+| m| (\lambda -{{\rm{\Omega }}}_{p}t)+{\phi }_{0}]}],\end{eqnarray} \tag{ 2 }$$

where σ₀ and ϕ₀ are constants, A(r) is a slowly varying function of radius, and ${\phi }_{r}(r)$ is the radius-dependent part of the wave's phase, which has the following form at sufficiently large distances from the resonant radius r_L (so long as $m\ne 1$, Shu 1984):

$$\begin{eqnarray}&&{\phi }_{r}(r)=\displaystyle \frac{3| m-1| {M}_{P}{(r-{r}_{L})}^{2}}{4\pi {\sigma }_{0}{r}_{L}^{4}},\end{eqnarray} \tag{ 3 }$$

where M_P is the planet's mass. Note that waves with m > 0 can only propagate exterior to the resonant radius, while waves with m < 0 can only propagate interior to the resonant radius.

For the entire region of interest here, the ring's optical depth appears to be directly proportional to its surface mass density (Hedman et al. 2011), so in any given occultation, the ring's optical depth will vary quasi-sinusoidally, but the positions of the peaks and troughs will vary with longitude and time in a manner that is sensitive to m and ${{\rm{\Omega }}}_{p}$. Indeed, wavelet-based methods allow us to quantify the phase differences between optical depth profiles, and thereby determine the values of m and Ω_p for a large number of waves that appear to be generated by structures inside the planet (Hedman & Nicholson 2013, 2014; French et al. 2016, 2019).

Our previous analyses of the planet-generated waves employed wavelet-based algorithms which estimated the average phase differences between pairs of wave profiles and then compared those differences with those expected given different assumptions about m and ${{\rm{\Omega }}}_{p}$ (Hedman & Nicholson 2013, 2014; French et al. 2016, 2019). While powerful, these algorithms have two important limitations: they assume there is a single wave in the analyzed region of interest, and they assume the wave signal dominates the optical depth variations in individual profiles. Here we will relax these assumptions by adapting a different set of algorithms to isolate waves with particular pattern speeds and number of spiral arms within the occultation data. The details of this approach are described in Hedman & Nicholson (2016), but for the sake of completeness, we will summarize the important aspects of the procedures here.

We begin by taking each occultation profile and interpolating the transmission values onto a regular grid of radii with a spacing of 100 m, converting the profile to normal optical depth,⁴ and then transforming the profile into a wavelet using the IDL wavelet routine (Torrence & Compo 1998) with a Morlet mother wavelet and parameter ω₀ = 6. This yields a complex wavelet for each profile ${{ \mathcal W }}_{i}={{ \mathcal A }}_{i}{e}^{i{{\rm{\Phi }}}_{i}}$ where ${{ \mathcal W }}_{i},{{ \mathcal A }}_{i}$ and Φ_i are all functions of both radius r and wavenumber k. For the signal from a density wave the wavelet phase Φ_i is equivalent to the wave phase in Equation (2). Hence, given the observed longitude λ_i and observation time t_i for each occultation, we can compute the phase parameter ${\phi }_{i}=| m| [{\lambda }_{i}-{{\rm{\Omega }}}_{p}{t}_{i}]$ for any chosen values of $| m|$ and Ω_p. For a wave with the selected parameters, the phase difference ${{\rm{\Phi }}}_{i}-{\phi }_{i}={\phi }_{r}(r)+{\phi }_{0}$ for every occultation, so we can define a phase-corrected wavelet:

$$\begin{eqnarray}&&{{ \mathcal W }}_{\phi ,i}={{ \mathcal W }}_{i}{e}^{-i{\phi }_{i}}={{ \mathcal A }}_{i}{e}^{i({{\rm{\Phi }}}_{i}-{\phi }_{i})}.\end{eqnarray} \tag{ 4 }$$

For signals with the selected m and Ω_p, the phase will be the same for all the occultation profiles, so the average phase-corrected wavelet

$$\begin{eqnarray}&&\langle {{ \mathcal W }}_{\phi }\rangle =\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}{{ \mathcal W }}_{\phi ,i}\end{eqnarray} \tag{ 5 }$$

will be nonzero, while any signal without these properties will average to zero. Thus, only the desired signal should remain in the power of the average phase-corrected wavelet

$$\begin{eqnarray}&&{{ \mathcal P }}_{\phi }(r,k)=| \langle {W}_{\phi }\rangle {| }^{2}={\left|\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}{{ \mathcal W }}_{\phi ,i}\right|}^{2},\end{eqnarray} \tag{ 6 }$$

while all other signals are only seen in the average wavelet power:

$$\begin{eqnarray}&&\bar{{ \mathcal P }}(r,k)=\langle | {W}_{\phi }{| }^{2}\rangle =\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}{| {{ \mathcal W }}_{\phi ,i}| }^{2}.\end{eqnarray} \tag{ 7 }$$

Hence, the ratio of these two powers ${ \mathcal R }(r,k)={{ \mathcal P }}_{\phi }/\bar{{ \mathcal P }}$ (which ranges between 0 and 1, see Hedman & Nicholson 2016) provides a measure of how much of the signal is consistent with the assumed m and Ω_p.

The average phase-corrected wavelet can also be used to produce a reconstructed profile of the part of the signal with the selected m-number and pattern speed. This is accomplished by taking the average phase-corrected wavelet and applying the inverse wavelet transformation. In practice, we only consider a finite range of wavenumbers when performing this inversion to remove residual high-frequency noise and slow background trends. The resulting profile is complex, but the real and imaginary parts of the profile just correspond to absolute wave phases of 0 and π/2, respectively. Since the wavelets were computed from the normal optical depth profiles, this reconstructed profile gives the normal optical depth variations associated with the wave. We divide these variations by the average optical depth profile to create a profile of the fractional optical depth variations. These fractional optical depth variations are easier to compare among different waves. We will therefore plot the real part of these optical depth variations, and report the peak amplitude of these variations as the maximum value of the square root of the sum of the squares of the real and imaginary parts of the profile.

Figures 6–11 show the results of the wavelet analysis for the previously identified waves W84.64, W82.00, W82.06, W82.21, W80.98, and W83.63. For the sake of conciseness, these figures just show the power ratio ${ \mathcal R }$, the parameter that illustrates the wave signatures most clearly. The upper panels in these figures show ${ \mathcal R }$ as a function of radius and wavenumber for the selected best-fit pattern speed, while the lower left-hand panels show the peak power ratio between wavelengths of 1 and 3 km as a function of radius and pattern speed (expressed as offsets in the assumed resonant radius and pattern speed). The lower right-hand panels show a profile of the maximum power ratio between wavelengths of 1 and 3 km and between the two radii indicated by the vertical dashed lines. These profiles show clear peaks at the selected pattern speed for each wave, thus demonstrating that those pattern speeds best organize the signals associated with each wave. Many of these plots also show secondary maxima offset from the main peak by around 005/day. These peaks arise because the occultations are not evenly distributed in time, but instead come from three distinct time periods (2007–2010, 2012–2014, and 2016–2017) separated by 2–4 years. The secondary maxima correspond to one extra cycle of the pattern between these times. Fortunately, the three time periods are long enough and the gaps between them are short enough that there is no ambiguity in the best-fit solution for any of these waves.

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The width of the peaks also provides an estimate of the uncertainty in each wave's pattern speed. To be conservative, we will here report uncertainties that correspond to the full width of the peaks in the profiles. Hence, W84.64 has a pattern speed of 186075 ± 003/day, W82.00 has a pattern speed of 173665 ± 002/day, W82.06 has a pattern speed of 173500 ± 002/day, W82.21 has a pattern speed of 173029 ± 002/day, W80.98 has a pattern speed of 166036 ± 002/day, and W83.63 has a pattern speed of 139406 ± 001/day. It is important to note that the actual peak positions can be determined to significantly higher precision than this. However, providing statistically rigorous uncertainties on these parameters is not practical at this time because we are considering maximum values of the power ratio over ranges of radii and wavenumbers, and propagating the appropriate uncertainties is beyond the scope of this report.

Figure 12 shows the reconstructed wave profiles derived from the average phase-corrected wavelets, specifically wavelengths between 0.5 and 5 km. Note that these profiles have been normalized so that they represent the fractional optical depth variations associated with each wave. The peak amplitudes of all these waves are all greater than 0.25, so the optical depth variations associated with these waves are a large fraction of the mean optical depth. Standard linear density wave theory is therefore not strictly appropriate for these waves. The nonlinear aspects of these waves do not affect their symmetry properties and pattern speeds, but they do affect the detailed shape of the density variations (for example, they give rise to the asymmetries in the shapes of individual peaks and troughs). More importantly, the standard expressions relating the wave amplitude to the strength of the applied perturbation do not hold, complicating any effort to translate the relative amplitudes of these waves into information about the relative amplitudes of the oscillations inside the planet.

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The results of the wavelet analyses of the two components of the W81.02 wave are shown in Figures 13 and 14. Consistent with the analysis presented in Section 3, there are clear signals in the power ratios for both m = −5 and m = −11. The m = −5 component has a peak signal at a pattern speed of 159663 ± 002/day, while the m = −11 component has a peak signal at a pattern speed of 145050 ± 001/day. (Again, the uncertainties are conservative estimates based on the full widths of the peaks in the power ratio profiles). These two pattern speeds correspond to resonant radii separated by only about 1 km. Also, the signals seen at the appropriate pattern speeds have sensible trends in wavenumber-radius space, with the strongest signals being seen just interior to the nominal resonance location, and the signal occurring at higher wavenumbers further inwards from the resonance.

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The reconstructed wave profiles for both these waves derived from the Group C occultations are shown in the top and bottom panels of Figure 15. These waves clearly have much lower signal-to-noise than any of the waves described in the previous subsection, and only a few wave cycles are visible for each of these waves. Still, as will be demonstrated below, these signals are consistently found among subsets of the data with peak amplitudes between 0.1 and 0.15. Hence, W81.02a and W81.02b are both valid waves but are also weaker than any of the previously identified waves.

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Among the four new wave signatures revealed by our search, the $m=-7$ and m = −9 signals designated W81.96 and W83.09 are stronger and more robust, and so we will consider them first. Figure 16 shows the wavelet analysis of the m = −9 signal W83.09, which shows a clear peak at 142184 ± 001/day, corresponding to a resonant radius of 83090.65 km. The signal is also perfectly consistent with an inward-propagating density waves, falling just inside the expected resonant radius and showing a clear increase in wavenumber with distance from the resonance. Furthermore, the reconstructed wave profile for this signal, shown in Figure 15, looks like a sensible inward-propagating density wave with a peak amplitude of around 0.10, which is not much smaller than the components of W81.02. We can therefore be fairly confident that this signal comes from a real m = −9 density wave.

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The wavelet analysis of W81.96 shows a comparably clear $m=-7$ signal with a pattern speed of 149246 ± 002/day, corresponding to a resonant radius of 81962.45 km (see Figure 17). This signal also appears to be quite consistent with an inwardly propagating density wave, with another clear trend where the wavenumber increases inwards. The reconstructed wave profile for W81.96, while having a slightly lower amplitude than W83.09, still preserves multiple wave cycles and looks like a reasonable inward-propagating density wave. Thus, we can conclude that W81.96 is indeed an m = −7 density wave.

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Finally, we must consider the candidate m = −6 and m = −8 waves W81.43 and W82.53. Figures 18 and 19 show wavelet analyses of these signals based on the full suite of occultations. For W81.43 there does appear to be a peak is the power ratio when m = −6 at a pattern speed of 153824 ± 002/day, corresponding to a resonant radius of 81429.55 km, which falls just outside the region with the strongest signal. For W82.53, there are multiple peaks in the power ratio profile, but the strongest falls at 145422 ± 001/day, which corresponds to a resonant radius of 82528.75 km, a sensible location just outside the observed signal. The signal associated with this pattern speed also exhibits sensible trends in wavenumber-radius space, with higher wavenumbers being found at increasing distances from the resonance.

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Turning to the reconstructed wave profiles for these regions (shown in Figure 15), we find that both these waves have extremely small amplitudes and are just barely above the background noise fluctuations. W82.53 appears to be a scaled down version of the other waves and so is perhaps more convincing. By contrast, W81.43 only preserves a cycle or two and so does not look particularly wavelike. In both cases, one can reasonably ask whether these are real wave signatures or just a chance alignment of random noise in the various profiles.

To address these concerns, we sought to establish whether these two signals could be seen throughout the Cassini mission. We therefore divided the occultations into two groups based on whether they were observed before or after 2010. This produced two separate data sets with roughly comparable signal-to-noise. Figures 20 and 21 show the results of the wavelet analyses for these two time periods for each of the wave candidates W81.43 and W82.53, while Figure 22 shows the reconstructed profiles for all the weak waves derived from these two time periods.

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For both W81.43 and W82.53, the data obtained before 2010 shows a relatively clear peak in the power ratio at the expected pattern speed that strongly resembles the signal seen in the full data set. By contrast, the data obtained after 2010 do not show such a unique signal. While the expected pattern speed does correspond to a peak in the power ratios, there are multiple peaks of comparable strength at other pattern speeds and locations. The strongest signals therefore appear to be restricted to the early part of the Cassini mission. However, both the early and late data for each wave candidate do show signals at the same radii and wavenumbers for the selected pattern speeds. This at least hints that the signal seen prior to 2010 did persist to later times. Also, if one examines the reconstructed profiles for these two time periods (shown in Figure 22), one finds that the individual peaks and troughs associated with these wave candidates do line up, unlike the other features in these profiles. Furthermore, while the amplitudes of the structures seen after 2010 are lower than those seen before 2010, this appears to be a general trend common to all these wave signals.

It is still unclear why the wave signals seem to be stronger in the earlier data. As shown in Table 1, many of the occultations observed before 2010 used the star γ Crucis. While the high elevation angle of γ Crucis above the rings makes these occultations especially useful for probing high-optical-depth regions like the B ring (Hedman & Nicholson 2016), it is not so obvious what would make γ Crucis occultations especially sensitive to waves in low-optical depth regions like the C ring. It could be that the more heterogeneous occultations used later in the mission somehow reduced the efficacy of the phase-correction techniques, but we have thus far been unable to identify any evidence for this. Hence, the question of whether these differences reflect a subtle artifact of our data processing algorithms or a real temporal variation in the waves, must be left as an open question for future work.

The evidence for W81.43 and W82.53 being real density waves is certainly weaker than the other signals considered in this report. However, the relatively unambiguous signals in the pre-2010 data, along with the consistent patterns seen before and after 2010 are sufficient for us to regard these features as reasonable candidate m = −6 and m = −8 density waves.