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Density waves in Cassini UVIS stellar occultations
Icarus 200 (2009) 574–580
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Density waves in Cassini UVIS stellar occultations 1. The Cassini Division J.E. Colwell a,∗ , J.H. Cooney a , L.W. Esposito b , M. Sremˇcevic´ b a b
Department of Physics, University of Central Florida, 4000 Central Florida Blvd., Orlando, FL 32816-2385, USA LASP, University of Colorado, 392 UCB, Boulder, CO 80309-0392, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 June 2008 Revised 8 December 2008 Accepted 10 December 2008 Available online 7 January 2009
We analyze density waves in the Cassini Division of Saturn’s rings revealed by multiple stellar occultations by Saturn’s rings observed with the Cassini Ultraviolet Imaging Spectrograph. The dispersion and damping of density waves provide information on the local ring surface mass density and viscosity. Several waves in the Cassini Division are on gradients in the background optical depth, and we find that the dispersion of the wave reflects a change in the underlying surface mass density. We find that over most of the Cassini Division the ring opacity (the ratio of optical depth to surface mass density) is nearly constant and is ∼5 times higher than the opacity in the A ring where most density waves are found. However, the Cassini Division ramp, a 1100-km-wide, nearly featureless region of low optical depth that connects the Cassini Division to the inner edge of the A ring, has an opacity like that of the A ring and significantly less than that in the rest of the Cassini Division. This is consistent with particles in the ramp originating in the A ring and being transported into the Cassini Division through ballistic transport processes. Damping of the waves in the Cassini Division suggests a vertical thickness of 3–6 m. Using a mean opacity of 0.1 cm2 /g we find the mass of the Cassini Division, excluding the ramp, is 3.1 × 1016 kg while the mass of the Cassini Division ramp, with an opacity of 0.015 cm2 /g, is 1.1 × 1017 kg. Assuming a power-law size distribution for the ring particles, the larger opacity of the main Cassini Division is consistent with the largest ring particles there being ∼5 times smaller than the largest particles in the ramp and A ring. © 2009 Elsevier Inc. All rights reserved.
Keywords: Saturn, rings Planetary, rings
1. Introduction The many small moons near and within Saturn’s rings excite spiral density waves in the rings at the locations of Lindblad resonances where the radial frequency of a ring particle, κ , equals an integer multiple of the pattern speed of the perturbing potential in the frame of the ring particle (Goldreich and Tremaine, 1982) m(n − Ω p ) = ±κ ,
(1)
where m is an integer, n is the mean motion of a ring particle, and Ω p is the pattern speed in the inertial frame. The pattern speed is given by (Murray and Dermott, 2000) mΩ p = mnmoon + kκmoon + p νmoon ,
(2)
where k and p are integers and nmoon , κmoon , and νmoon are the azimuthal, epicyclic, and vertical frequencies of the Moon. Because the gravity of the ring particles acts as a restoring force on the wave, the dispersion of the wave provides a means of measuring the local surface mass density of the rings. In the linear density
*
Corresponding author. E-mail address: [email protected] (J.E. Colwell).
0019-1035/$ – see front matter doi:10.1016/j.icarus.2008.12.031
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2009 Elsevier Inc. All rights reserved.
wave theory of Shu (1984), the wavelength, λ, decreases with increasing separation from resonance, r L , such that their product is proportional to the background surface mass density, σ :
(r − r L )λ =
4π 2 G σ r L
DL
(3)
,
where
D L = 3(m − 1)n2 + J 2
RS rL
2
21 2
9 − (m − 1) n2 , 2
(4)
R S = 60330 km, and J 2 is the spherical harmonic coefficient of Saturn’s gravitational field. Esposito et al. (1983) and Spilker et al. (2004) analyzed density waves in the A ring in Voyager PPS stellar occultation data to arrive at surface mass densities. Nicholson et al. (1990) and Rosen et al. (1991) also analyzed Voyager PPS and radio occultation data, matching observed wave profiles to the linear theory of Shu. Tiscareno et al. (2007) analyzed high resolution Cassini images of primarily weaker waves that were not detected in the Voyager PPS data, including two in the Cassini Division. Esposito et al. (1983) and Tiscareno et al. (2007) also analyzed the damping distance of the waves. Collisions between the ring particles ultimately damp the wave, so the distance of propagation of
Density waves in the Cassini Division
the wave provides a means of determining the viscosity of the ring and hence typical collision velocities, c. These can in turn be used to constrain the ring thickness, H , by assuming H = c /n. Density waves have therefore been a particularly valuable probe of local ring properties. Here we report on an analysis of 5 density waves in the Cassini Division and one at the inner edge of the A ring using more than 30 stellar occultations observed by the Cassini Ultraviolet Imaging Spectrograph (UVIS). Four of these waves (3 in the Cassini Division) have not been previously analyzed, providing a more complete picture of the surface mass density variations in the Cassini Division. These waves generally have low amplitudes and are in regions where self-gravity wakes are not prominent. This makes a linear analysis of the dispersion of the waves and a calculation of the mass extinction coefficient (or opacity, K = τ /σ ) more straightforward than for most waves in the A ring where the presence of the self-gravity wakes affects the measurement of τ . One of the waves we analyze here, the Pandora 9:7 density wave, is located on a narrow ringlet, and the wave propagates to the edge of the ring. The assumption of constant background surface mass density is therefore not valid for this wave, and it is likely not valid for several of the other Cassini Division waves as well. We present an analysis of how the dispersion of these waves varies across the gradient in background optical depth. Two of the Cassini Division waves occur at regions where the optical depth is comparable to that in the central A ring allowing a direct comparison of the mass extinction coefficient, or opacity, K = τ /σ , for the Cassini Division and A ring. We find that the opacity in the Cassini Division is 3 to 5 times higher than that in the A ring except for the Cassini Division ramp which has an opacity like that of the A ring. This suggests that the particle size distribution in the ramp is more like that of the A ring and that particles in the ramp may have drifted there from the A ring, consistent with models of ballistic transport evolution of the inner edge of the A ring (Durisen et al., 1989, 1992). We also find vertical thicknesses of the Cassini Division at the locations of three waves of between 3 and 6 m. In the next section we describe the stellar occultation observations that were analyzed and present some wave profiles, and in Section 3 we describe the analysis techniques. Results and discussion on the local surface mass densities, opacities, total Cassini Division mass, and viscosities are in Section 4. 2. Observations The Cassini UVIS has a high speed photometer (HSP) used for observing stellar occultations (Esposito et al., 1998, 2004). The HSP has recorded more than 60 partial and complete occultations of stars by the rings, usually with spatial resolution in the ring plane of ∼20 m, though the resolution varies between occultations and also within occultations as the viewing geometry changes. Detailed descriptions of the first set of ring occultations observed by UVIS are in Colwell et al. (2007). The occultations used in the analysis presented here are listed in Table 1. Not all occultations obtained by UVIS spanned the Cassini Division waves discussed here, and others did not show the waves due to low signal. Others were not included in order to limit the total time span of the data analyzed and thus limit the effect of changes in resonance locations due to orbital changes of the moons. Because we are analyzing the dispersion and damping of a wave signal in the rings, our density wave results do not depend on the background signal or the HSP instrumental response to bright stars that affects determinations of absolute optical depths (Colwell et al., 2007). All occultations show the same optical depths in the Cassini Division, regardless of viewing geometry, suggesting a near total absence of self-gravity wakes. We are thus able to obtain
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Table 1 List of occultations. Occultation star (Rev)
Date (year-day)
|B|
λ Cet (28)
2006-256 2006-269 2006-269 2006-285 2006-286 2006-313 2006-313 2006-325 2006-337 2006-337 2006-337 2006-337 2006-350 2006-352 2006-363 2006-363 2006-364 2007-003 2007-015 2007-020 2007-020 2007-022 2007-049 2007-056 2007-063 2007-073 2007-080 2007-080 2007-082 2007-092 2007-092 2007-179
15.3 32.2 41.7 17.2 47.4 54.4 47.4 54.4 17.2 17.2 44.5 44.5 48.5 54.4 20.3 20.3 54.0 51.0 54.0 51.0 51.0 61.0 54.0 47.5 35.1 39.3 35.1 35.1 54.0 48.5 48.5 2.7
α Sco B (29) λ Sco (29)
α Vir (30) γ Lup (30) α Ara (32) γ Lup (32) α Ara (33) α Vir (34I) α Vir (34E) η Lup (34I) η Lup (34E) κ Cen (35) α Ara (35) γ Peg (36I) γ Peg (36E) δ Per (36)
ε Lup (36) δ Per (37)
ε Lup (37I) ε Lup (37E) γ Ara (37) δ Per (39)
χ Cen (39) γ Gru (40) 3 Cen (40)
γ Gru (41I) γ Gru (41E) δ Per (41)
κ Cen (42I) κ Cen (42E) ζ Ori (47)
R (m)
I0 (counts)
b (counts)
19 20 6.5 42 8.7 10 15 10 52 52 7.0 7.0 10 6.5 15 15 18 9.3 15 5.3 5.3 6.8 7.0 6.2 13 8.3 11 11 20 1.8 1.8 30
4.8 7.0 570 1070 153 77.7 144 72.4 1020 988 96.2 89.3 90.5 75.8 148 132 27.0 62.9 27.4 63.6 59.8 51.9 25.0 26.2 14.5 9.5 16.2 15.1 24.3 84.6 81.7 336
0.2 0.1 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.3 0.1 0.2 0.6 0.6 0.8 0.4 0.1 0.3 0.3 0.3 0.2 0.3 0.3 0.2 0.4 0.3 0.8 0.3 0.3 2.4
Notes: R is given at 120,000 km and is the radial separation of two consecutive points in the HSP data. I 0 is the unocculted star signal in counts per integration period, and b is the background signal. Both I 0 and b vary over the course of an occultation in general (Colwell et al., 2007). Values reported here are based on measurements of the Huygens Gap for the sum I 0 + b and opaque regions in the B ring for b.
optical depths at the locations of the analyzed waves from any occultation, including those presented in Colwell et al. (2006, 2007). For the low optical depths in most of the Cassini Division, occultations of stars with a low incidence angle, B, provide the best sensitivity. The star with the lowest value of B observed by UVIS to date is the bright star ζ Orionis (Rev 047, Table 1). However, ζ Orionis is a double star, and the HSP data include a signal from both components of the binary system. The projected radial separation of the two stars in the ring plane in the Cassini Division for this occultation is 10 km which is longer than the wavetrain of some of the waves analyzed here. That occultation is thus useful for analysis of the wave dispersion and damping, but optical depth profiles are complicated by light from the secondary star passing through a different region of the rings. The other binary stars in Table 1 are γ Lupi, κ Centauri, λ Scorpii, and ε Lupi. α Scorpii is also a binary star, but the HSP only detects α Scorpii B because α Scorpii A is a red giant and therefore too cool to be detected in the FUV bandpass of the HSP. The HSP signal levels of the two components of the γ Lupi system are nearly identical, and the radial separation of the two stars in the ring plane is 2.9 km on Rev 030 and 1.7 km on Rev 032. The radial ring plane separations for κ Centauri (Revs 035 and 042), η Lupi (Rev 034), ε Lupi (Revs 036 and 037) and λ Scorpii (Rev 029) are less than 100 m which is less than the shortest wavelength of the density wavetrains analyzed here. While the γ Lupi separation is comparable to the density wave wavelengths studied, the signal from two stars puts a constant wavelength signal on the data where the density waves have a characteristic dispersion. In addition, our filtering process removes most of the
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Fig. 1. Optical depth profile of the Cassini Division from the α Leo Rev 009I occultation (Table 1). The density waves with measured surface mass densities in the Cassini Division are labeled by their resonances: Pa = Pan, Pd = Pandora, Pr = Prometheus, At = Atlas, and Ja = Janus/Epimetheus. The Cassini Division ramp is the region between 120,900 km and 122,040 km where there is a linear increase in optical depth with increasing ring plane radius. The Atlas 6:5 wave is on the inner edge of the A ring which begins at the outer edge of the ramp.
Fig. 2. Optical depth profile of the Prometheus 9:7 density wave in the α Leo Rev 009I occultation. The profile has been boxcar smoothed by 7 points from the full resolution data to better show the wave which extends from resonance at R ≈ 118,066 km to R ≈ 118,078 km.
signal from the beating due to the binary stars. Analysis of individual wave profiles in the γ Lupi occultation data, where possible, gave results consistent with those from the other occultations (Section 3). We also checked our derived quantities without the γ Lupi occultation data and found no significant difference. The star at the lowest incidence angle after ζ Orionis is α Leo with B = 9.5◦ . Fig. 1 shows the optical depth profile of the Cassini Division from the α Leo Rev 009 ingress (Rev 009I) occultation. Optical depths were computed using the calibration for this occultation described in Colwell et al. (2007). Figs. 2–7 show profiles for the individual waves at higher resolution from the α Leo (Rev 009I) occultation for the waves at low optical depth and from the α Virginis (Rev 034E) occultation for two of the three waves at higher optical depths. For optical depths from α Virginis (Rev 034E) we assume a background count rate of 2000 counts/s and I 0 = 5.2 × 105 counts/s based on the measured signal in the outer gaps of the Cassini Division and in optically thick regions in the outer B ring. Although the Atlas 6:5 wave shows up clearly in frequency space (Section 3), no individual raw occultation profile shows the wave clearly.
Fig. 3. Optical depth profile of the Pan 6:5 density wave from the α Leo Rev 009I occultation. The profile has been boxcar smoothed by 9 points from the full resolution data to better show the wave which extends from resonance at R ≈ 118,453 km to R ≈ 118,469 km. This wave is on a strong gradient in the background optical depth.
Fig. 4. Optical depth profile of the Atlas 5:4 density wave from the α Leo Rev 009I occultation. The profile has been boxcar smoothed by 9 points from the full resolution data to better show the wave which extends from resonance at R ≈ 118,830 km to R ≈ 118,848 km.
Fig. 5. Optical depth profile (solid line) and surface mass density (dashed line) of the Pandora 9:7 density wave from the α Virginis Rev 034E occultation. The resonance location is near the inner edge of the ringlet, and the wave is observed in this and other occultations to propagate down the gradient in optical depth at the outer edge of the ringlet. The derived surface mass density tracks the declining optical depth of the ringlet where the wave is located, with a radial offset.
Density waves in the Cassini Division
Fig. 6. Optical depth profile of the Pan 7:6 density wave from the α Virginis Rev 034E occultation. The profile has been boxcar smoothed by 5 points from the full resolution data to better show the wave which extends from resonance at R ≈ 120,668 km to R ≈ 120,695 km. Although this wave is in the middle of the three-peaked broad ringlet in the outer Cassini Division, the wave itself is in a relatively uniform section of the ringlet.
Fig. 8. Morlet wavelet scalogram (solid line) and surface mass density (dashed line) of the Atlas 5:4 density wave in the α Leo Rev 009I occultation. The wavelet transform is zeroed on either side of lines drawn by hand on the scalogram on either side of the density wave signal.
wavelet power contour plot (Fig. 8). We then select the range in ring plane radii over which we fit the linear dispersion relation to the peak power in the scalogram. We next find the peak power within the selected radial range. This gives an effective λ(r ). The surface mass density at each point along the wave is then calculated from Eq. (3):
σ (r ) =
Fig. 7. Optical depth profile of the Janus/Epimetheus density wave from the α Leo Rev 009I occultation. The profile has been boxcar smoothed by 13 points from the full resolution data to better show the wave which extends from resonance at R ≈ 121,260 km to R ≈ 121,320 km. The effect of the overlapping waves from the two satellites can be seen in the modulation of the amplitude and the wavelength increasing again at R > 121,300 km.
3. Analysis technique We obtain the surface mass density at the location of each density wave by analyzing the dispersion of the wave. We analyzed the waves in two ways. When an individual occultation profiles provided a clear signal of a density wave, we analyzed the dispersion as well as the damping of the wave following a procedure similar to that of Tiscareno et al. (2007) using the discrete wavelet analysis tool of Torrence and Compo (1998). For all waves we also computed a weighted wavelet Z (WWZ) transform (Foster, 1996) of each occultation and then summed the transforms. This strengthens the signal from the wave relative to that from noise in the data. 3.1. Individual wavetrain analysis Following Tiscareno et al. (2007), we use the complex Morlet wavelet transform on our occultation data. We filter the data by zeroing the wavelet transform on either side of lines drawn on the
577
λD L (r − r L ) 4π 2 Gr L
.
(5)
We calculated the mean value of σ from σ (r ) for each wave. Because the selection of the points to filter and the selection of the endpoints of the data to be fit require a subjective selection, two of us (JEC and JHC) independently repeated this analysis for 31 profiles of the Atlas 5:4 density wave. The mean values of σ from the two sets of analyses were identical to three significant figures. This technique was used for all the waves except the Pandora 9:7 density wave, though only 18 and 8 occultations were analyzed for the Prometheus 9:7 and Atlas 6:5 waves, respectively, because of the weak signature of those waves in some occultations. The Pandora 9:7 density wave exhibits the strongest gradient in surface mass density of the waves analyzed, so the selection of the radial range strongly affects the average value derived. Consequently for this wave we calculate σ (r ) from the averaged WWZ transform (Section 3.2). The Prometheus 9:7 wave and the Atlas 6:5 wave are too weak in most occultation profiles to provide a strong signal, so for those waves as well we use the averaged WWZ transform. The Janus 7:5 density wave in the Cassini Division ramp is complicated by interference from the weaker Epimetheus 7:5 wave and by the 4-year synodic period of these coorbital moons. This results in a change in the resonance locations and limits the distance of propagation of the wave. Tiscareno et al. (2006) modeled the time-dependent superposition of the waves from the coorbitals and found σ = 11.5 g cm−2 and ξd = 8, where ξd is the damping length of the wave in units of the dimensionless distance from resonance,
ξ=
DL r L r − r L 2π G σ
rL
.
(6)
We follow the same procedure as Tiscareno et al. (2007) to obtain ξd for the three waves we analyzed individually using the Morlet wavelet transform. In addition, we also fit the cumulative phase of these three waves, again following Tiscareno et al. (2007), to have yet another measure of the surface mass density. The vertical thickness of the rings, H , is estimated from the wave damping length by assuming that particle random veloci-
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Fig. 9. Wavelet power (contours) and surface mass density (dashed line) for the Pandora 9:7 density wave obtained from summing the WWZ transforms of 30 occultation profiles. The upward curvature of the peak power ridge reflects the declining surface mass density.
Fig. 10. σ (r ) vs τ (r ) for the 5 waves analyzed here that are in the main part of the Cassini Division. The Atlas 6:5 wave, also analyzed here, has a surface mass density outside the range of this plot. The dashed line shows the relationship between σ and τ assuming a constant opacity, κ , equal to the average value for the 5 waves.
ties, c, are not correlated and using (Goldreich and Tremaine, 1978; Tiscareno et al., 2007)
Table 2 Summary of density wave results.
c=
2ηn 1 + τ 2 = Hn,
τ where η is the ring viscosity given by (e.g. Shu, 1984)
η≈
9 7nξd3
(2π G σ )3 r L . DL
Wave
(7)
(8)
These approximations for viscosity and ring thickness are justified in the Cassini Division where optical depths are low and particles have not been observed to clump strongly into self-gravity wakes. 3.2. Averaged wavetrain analysis We take the weighted wavelet Z-transform (WWZ) of each occultation profile in Table 1. The WWZ transform is a modification of the Morlet transform implemented by Torrence and Compo (1998) used in the previous section and by Tiscareno et al. (2007) that can handle data with uneven sampling in time (Foster, 1996). By weighting points by time interval rather than the temporal density of points, the WWZ transform technique of Foster (1996) allows the transforms from multiple data sets to be co-added, providing equal weight to each data set. Co-adding the weighted Z-transform is equivalent to summing probabilities, which is the best way to treat data sets of different qualities (as opposed to adding raw power). Although our data are regularly sampled in time, the radial ring plane sampling is different between occultations, and we are analyzing periodic structure in the ring radial dimension rather than the temporal dimension. In fact, the ring plane sampling is not constant within an occultation, but the change in radial resolution over the length of the density waves described here is small enough to be negligible and does not affect the Morlet wavelet analysis described above. Before summing the transforms we adjust the radius scale of each occultation profile to minimize offsets between the profiles due to trajectory uncertainties. We used the presumedcircular fiduciary ring feature #11 from Nicholson et al. (1990) at 120,245 km to coalign the occultations before taking the WWZ transforms. The summed WWZ transform for the Pandora 9:7 from 30 occultations and the resulting average σ (r ) are shown in Fig. 9. We computed summed WWZ transforms for 5 Cassini Division waves (excluding the Janus/Epimetheus 7:5 wave) and the Atlas 6:5 wave on the inner edge of the A ring. We then computed σ (r ) from these average WWZ transforms using the same filtering and
Prometheus 9:7 Pan 6:5 Atlas 5:4 Pandora 9:7 Pan 7:6 Atlas 6:5 [2]
σ (Morlet) (g cm−2 )
σ (phase) (g cm−2 )
1.08 1.11 1.30
1.13 1.15 1.35
3.45 14.7
3.55 15.4
σ (WWZ) (g cm−2 ) 1.10 0.98 1.31 5.76 3.51 15.4
κ
ξd
0.111 0.083 0.068 0.118 [1] 0.086 0.033
6.6 ± 1.3 7.0 ± 1.4 7 .1 ± 0.9
4 3 6
7.8 ± 1.1 6 .2 ± 0.8
5 20
(cm g−1 )
H (m)
Values of κ use the radially averaged σ from the WWZ analysis and the radially averaged τ . [1] Pandora 9:7 values are for the range 120,055 to 120,073 km. [2] Values of σ from the Morlet analysis of this wave are from 8 occultations, and the damping length is the average of 6 occultations.
analysis used for the Morlet transform described in the previous section. 4. Results and conclusions The waves we analyze are not in regions of constant background optical depth. We took the measured optical depth of the Cassini Division from the UVIS stellar occultations, and smoothed it to remove the fluctuations due to the individual density waves. This gives us τ (r ) corresponding to σ (r ) obtained from the WWZ analysis. In Fig. 10 we show σ (τ ) for the 5 density waves within the Cassini Division analyzed here. Within each wave there is a correlation between σ and τ , with the greatest variation in τ occurring in the Pandora 9:7 wave. For that wave, the measured change in σ is about 34% (Fig. 9 where the background optical depth changes by more than a factor of two (Fig. 5). However, most of the variation in τ for this wave occurs over a radial span of only ∼5 km and a few wavelengths of the wave. The correspondence between σ and τ is not perfect within each wave, but does demonstrate for the first time that variations in wave dispersion within a single density wave are affected by variations in the ring property over radial scales greater than the wavelength. The Pandora spiral waves in the A ring were found to be modulated by the 3:2 near-corotation resonance between Pandora and Mimas. This likely affects the Pandora 9:7 wave analyzed here as well. We compute the radially-averaged opacity, κ = τ (r )/σ (r ), for each density wave analyzed here. Our results for σ , κ , ξd , and thickness, H , are summarized in Table 2. Fig. 11 shows our values of κ together with those for the A ring and the Cassini Division ramp. The opacities in the A ring are calculated from the α Ara (Rev 033) A ring optical depth and surface mass densities from Tiscareno et al. (2007). We use τ = 0.17 for the Cassini Di-
Density waves in the Cassini Division
Fig. 11. Opacities derived for the 6 density waves analyzed here (diamonds), the waves in the A ring analyzed by Tiscareno et al. (2007), and the Janus 7:5 density wave in the Cassini Division ramp. The opacity shows a sharp and consistent increase between the ramp and the rest of the Cassini Division.
vision ramp at the location of the Janus/Epimetheus 7:5 density wave (R = 121,260–121,290 km) from the α Ara (Rev 033) occultation. With the surface mass density of 11.5 g cm−2 found for this wave by Tiscareno et al. (2006), this gives κ = 0.015 cm2 g−1 . The opacities of the five Cassini Division density waves analyzed here are significantly higher than those found in the A ring and in the Cassini Division ramp. If the particle size distribution and particle densities are constant across the ramp, the opacity there corresponds to a total mass of the Cassini Division ramp of 1.1 × 1017 kg. In contrast to the typical A ring and ramp opacity of κ < 0.02 cm2 g−1 , the mean opacity of the 5 waves in the rest of the Cassini Division is κ = 0.093 ± 0.02 cm2 g−1 . Using this average opacity and the measured optical depth profile of the Cassini Division, this corresponds to a mass of 3.1 × 1016 kg for the Cassini Division excluding the ramp, equivalent to an icy moon with a density of 1 g cm−3 and a radius of 19 km. The increase in the opacity of roughly a factor of five from the A ring and Cassini Division ramp to the rest of the Cassini Division suggests that the particles in the ramp are more like those in the A ring than those in the rest of the Cassini Division. The optical depths at the Pandora 9:7 and Pan 7:6 waves are comparable to the optical depths in the central A ring, strengthening the conclusion that this is a change in particle properties and not an effect that is proportional the particle packing or optical depth. Compositional measurements of this region by the Cassini VIMS instrument show a smooth transition in water ice band depths from the inner A ring across the ramp and into the triple-peaked ringlet where the Pan 7:6 wave resides (Nicholson et al., 2008). The red color of the ramp in the near-IR (0.85–1.4 μm) is more like that of the rest of the Cassini Division than it is like the A ring. This spectral difference between the ramp and the A ring indicates different compositional evolution between the two regions or a mixture of redder Cassini Division particles with icy A ring particles in the ramp. The larger opacity of the Cassini Division compared to the A ring indicates smaller particles on average. The Voyager radio science occultation at 3.6 cm and 13 cm shows no significant change in the differential optical depth between the Cassini Division and the ramp, consistent with the change in the size distribution being at the upper size cutoff. French and Nicholson (2000) found comparable opacities for the Cassini Division and A ring based on stellar occultation measurements, but their model did not include the effect of self-gravity wakes on observed optical depth, and the Cassini Division was the least well-constrained region of the rings
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due to its relatively narrow width. If the particle size distribution can be described by a power-law with differential size index = −3, then the difference in opacity between the A ring and the Cassini Division implies the largest particles in the Cassini Division are ∼5 times smaller than the largest particles in the A ring and the Cassini Division ramp. The lower mass of the Cassini Division than the A ring makes it both easier to create from satellite disruption and easier to erode through micrometeoroid bombardment. Colwell (1994) calculated satellite disruption timescales and found a disruption time ∼109 years for a 19 km-radius moon at the outer edge of Saturn’s rings, consistent with calculations by Colwell et al. (2000) for the lifetime of similar-sized moons in the ring systems of Uranus and Neptune. The micrometeoroid flux delivers mass to the Cassini Division particles at the rate of ∼103 g s−1 , where we have increased the flux of Cuzzi and Estrada (1998) by a factor of 1.5 after accounting for gravitational focusing and the low optical depth of the Cassini Division, and the total area of the Cassini Division (excluding the ramp) is ∼1.6 × 1019 cm2 . The time to add 10% exogenic material, by mass, to the Cassini Division is then also ∼109 years. On the other hand, the time to physically erode the particles through micrometeoroid bombardment is considerably shorter because ejecta mass yields are ∼3 × 104 times the mass of the impactors. This gives a time to erode pristine ring particles to a depth of ∼10 cm of a few million years Esposito and Elliott (2007). Disruption of a moon massive enough to produce the Cassini Division would reset the brightness of a 1000-km-wide swath of the A ring (Esposito et al., 2005). Our results support a common origin for most of the particles in the ramp and those in the A ring, consistent with models of ballistic transport producing the ramp as angular momentum is removed from the inner edge of the A ring by asymmetric absorption of meteoroid ejecta (Durisen et al., 1989, 1992). They also suggest that the remainder of the Cassini Division has a different origin than the A ring (and ramp), or has had a different evolution of the size distribution of the particles within it, or some combination of the two. Acknowledgments This work was supported by NASA through the Cassini project. We thank Matt Tiscareno and Nicole Rappaport for their helpful reviews of this paper. References Colwell, J.E., 1994. The disruption of planetary satellites and the creation of planetary rings. Planet. Space Sci. 42, 1139–1149. Colwell, J.E., Esposito, L.W., Bundy, D., 2000. Fragmentation rates of small satellites in the outer Solar System. J. Geophys. Res. 105, 17589–17599. ´ M., 2006. Gravitational wakes in Saturn’s Colwell, J.E., Esposito, L.W., Sremˇcevic, A ring measured by stellar occultations from Cassini. Geophys. Res. Lett. 33, doi:10.1029/2005GL025163. L07201. ´ M., Stewart, G.R., McClintock, W.M., 2007. Colwell, J.E., Esposito, L.W., Sremˇcevic, Self-gravity wakes and radial structure of Saturn’s B ring. Icarus 190, 127–244. Cuzzi, J.N., Estrada, P.R., 1998. Compositional evolution of Saturn’s rings due to meteoroid bombardment. Icarus 132, 1–35. Durisen, R.H., Cramer, N.L., Murphy, B.W., Cuzzi, J.N., Mullikin, T.L., Cederbloom, S.E., 1989. Ballistic transport in planetary ring systems due to particle erosion mechanisms. I. Theory, numerical methods, and illustrative examples. Icarus 80, 136–166. Durisen, R.H., Bode, P.W., Cuzzi, J.N., Cederbloom, S.E., Murphy, B.W., 1992. Ballistic transport in planetary ring systems due to particle erosion mechanisms. II. Theoretical models for Saturn’s A- and B-ring inner edges. Icarus 100, 364–393. Esposito, L.W., Elliott, J.P., 2007. Regolith growth and pollution on Saturn’s moons and ring particles. Bull. Am. Astron. Soc. 39, 421. Abstract 7.09. Esposito, L.W., O’Callaghan, M., West, R.A., 1983. The structure of Saturn’s rings: Implications from the Voyager stellar occultation. Icarus 56, 439–452.
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Esposito, L.W., Colwell, J.E., McClintock, W.M., 1998. Cassini UVIS observations of Saturn’s rings. Planet. Space Sci. 46, 1221–1235. Esposito, L.W., Barth, C.A., Colwell, J.E., Lawrence, G.M., McClintock, W.E., Stewart, A.I.F., Keller, H.U., Korth, A., Lauche, H., Festou, M.C., Lane, A.L., Hansen, C.J., Maki, J.N., West, R.A., Jahn, H., Reulke, R., Warlich, K., Shemansky, D.E., Yung, Y.L., 2004. The Cassini ultraviolet imaging spectrograph investigation. Space Sci. Rev. 115, 299–361. Esposito, L.W., Colwell, J.E., Larsen, K., McClintock, W.E., Stewart, A.I.F., Hallett, J.T., Shemansky, D.E., Ajello, J.M., Hansen, C.J., Hendrix, A.R., West, R.A., Keller, H.U., Korth, A., Pryor, W.R., Reulke, R., Yunk, Y.L., 2005. Ultraviolet imaging spectroscopy shows an active saturnian system. Science 307, 1251–1255. Foster, G., 1996. Wavelets for period analysis of unevenly sampled time series. Astron. J. 112, 1709–1729. French, R.G., Nicholson, P.D., 2000. Saturn’s rings. II. Particle sizes inferred from stellar occultation data. Icarus 145, 502–523. Goldreich, P., Tremaine, S., 1978. The velocity dispersion in Saturn’s rings. Icarus 34, 227–239. Goldreich, P., Tremaine, S., 1982. The dynamics of planetary rings. Annu. Rev. Astron. Astrophys. 20, 249–283. Murray, C.D., Dermott, S.F., 2000. Solar System Dynamics. Cambridge University Press, Cambridge.
Nicholson, P.D., Cooke, M.L., Pelton, E., 1990. An absolute radius scale for Saturn’s rings. Astron. J. 100, 1339–1362. Nicholson, P.D., Hedman, M.M., Clark, R.N., Showalter, M.R., Cuikshank, D.P., Cuzzi, J.N., Filacchione, G., Capaccioni, F., Cerroni, P., Hansen, G.B., Sicardy, B., Drossart, P., Brown, R.H., Buratti, B.J., Baines, K.H., Coradini, A., 2008. A close look at Saturn’s rings with Cassini VIMS. Icarus 193, 182–212. Rosen, P.A., Tyler, G.L., Marouf, E.A., 1991. Resonance structures in Saturn’s rings probed by radio occultation. 1. Methods and examples. Icarus 93, 3–24. Shu, F.H., 1984. Waves in planetary rings. In: Greenberg, R., Brahic, A. (Eds.), Planetary Rings. Univ. of Arizona Press, Tucson, pp. 513–561. Spilker, L.J., Pilorz, S., Lane, A.L., Nelson, R.M., Pollard, B., Russell, C.T., 2004. Saturn A ring surface mass densities from spiral density wave dispersion behavior. Icarus 171, 372–390. Tiscareno, M.S., Nicholson, P.D., Burns, J.A., Hedman, M.M., Porco, C.C., 2006. Unravelling temporal variability in Saturn’s spiral density waves: Results and predictions. Icarus 651, L65–L68. Tiscareno, M.S., Burns, J.A., Nicholson, P.D., Hedman, M.M., Porco, C.C., 2007. Cassini imaging of Saturn’s rings II. A wavelet technique for analysis of density waves and other radial structure in the rings. Icarus 189, 14–34. Torrence, C., Compo, G.P., 1998. A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79, 61–78.
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Density waves in Cassini UVIS stellar occultations 1. The Cassini Division J.E. Colwell a,∗ , J.H. Cooney a , L.W. Esposito b , M. Sremˇcevic´ b a b
Department of Physics, University of Central Florida, 4000 Central Florida Blvd., Orlando, FL 32816-2385, USA LASP, University of Colorado, 392 UCB, Boulder, CO 80309-0392, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 June 2008 Revised 8 December 2008 Accepted 10 December 2008 Available online 7 January 2009
We analyze density waves in the Cassini Division of Saturn’s rings revealed by multiple stellar occultations by Saturn’s rings observed with the Cassini Ultraviolet Imaging Spectrograph. The dispersion and damping of density waves provide information on the local ring surface mass density and viscosity. Several waves in the Cassini Division are on gradients in the background optical depth, and we find that the dispersion of the wave reflects a change in the underlying surface mass density. We find that over most of the Cassini Division the ring opacity (the ratio of optical depth to surface mass density) is nearly constant and is ∼5 times higher than the opacity in the A ring where most density waves are found. However, the Cassini Division ramp, a 1100-km-wide, nearly featureless region of low optical depth that connects the Cassini Division to the inner edge of the A ring, has an opacity like that of the A ring and significantly less than that in the rest of the Cassini Division. This is consistent with particles in the ramp originating in the A ring and being transported into the Cassini Division through ballistic transport processes. Damping of the waves in the Cassini Division suggests a vertical thickness of 3–6 m. Using a mean opacity of 0.1 cm2 /g we find the mass of the Cassini Division, excluding the ramp, is 3.1 × 1016 kg while the mass of the Cassini Division ramp, with an opacity of 0.015 cm2 /g, is 1.1 × 1017 kg. Assuming a power-law size distribution for the ring particles, the larger opacity of the main Cassini Division is consistent with the largest ring particles there being ∼5 times smaller than the largest particles in the ramp and A ring. © 2009 Elsevier Inc. All rights reserved.
Keywords: Saturn, rings Planetary, rings
1. Introduction The many small moons near and within Saturn’s rings excite spiral density waves in the rings at the locations of Lindblad resonances where the radial frequency of a ring particle, κ , equals an integer multiple of the pattern speed of the perturbing potential in the frame of the ring particle (Goldreich and Tremaine, 1982) m(n − Ω p ) = ±κ ,
(1)
where m is an integer, n is the mean motion of a ring particle, and Ω p is the pattern speed in the inertial frame. The pattern speed is given by (Murray and Dermott, 2000) mΩ p = mnmoon + kκmoon + p νmoon ,
(2)
where k and p are integers and nmoon , κmoon , and νmoon are the azimuthal, epicyclic, and vertical frequencies of the Moon. Because the gravity of the ring particles acts as a restoring force on the wave, the dispersion of the wave provides a means of measuring the local surface mass density of the rings. In the linear density
*
Corresponding author. E-mail address: [email protected] (J.E. Colwell).
0019-1035/$ – see front matter doi:10.1016/j.icarus.2008.12.031
©
2009 Elsevier Inc. All rights reserved.
wave theory of Shu (1984), the wavelength, λ, decreases with increasing separation from resonance, r L , such that their product is proportional to the background surface mass density, σ :
(r − r L )λ =
4π 2 G σ r L
DL
(3)
,
where
D L = 3(m − 1)n2 + J 2
RS rL
2
21 2
9 − (m − 1) n2 , 2
(4)
R S = 60330 km, and J 2 is the spherical harmonic coefficient of Saturn’s gravitational field. Esposito et al. (1983) and Spilker et al. (2004) analyzed density waves in the A ring in Voyager PPS stellar occultation data to arrive at surface mass densities. Nicholson et al. (1990) and Rosen et al. (1991) also analyzed Voyager PPS and radio occultation data, matching observed wave profiles to the linear theory of Shu. Tiscareno et al. (2007) analyzed high resolution Cassini images of primarily weaker waves that were not detected in the Voyager PPS data, including two in the Cassini Division. Esposito et al. (1983) and Tiscareno et al. (2007) also analyzed the damping distance of the waves. Collisions between the ring particles ultimately damp the wave, so the distance of propagation of
Density waves in the Cassini Division
the wave provides a means of determining the viscosity of the ring and hence typical collision velocities, c. These can in turn be used to constrain the ring thickness, H , by assuming H = c /n. Density waves have therefore been a particularly valuable probe of local ring properties. Here we report on an analysis of 5 density waves in the Cassini Division and one at the inner edge of the A ring using more than 30 stellar occultations observed by the Cassini Ultraviolet Imaging Spectrograph (UVIS). Four of these waves (3 in the Cassini Division) have not been previously analyzed, providing a more complete picture of the surface mass density variations in the Cassini Division. These waves generally have low amplitudes and are in regions where self-gravity wakes are not prominent. This makes a linear analysis of the dispersion of the waves and a calculation of the mass extinction coefficient (or opacity, K = τ /σ ) more straightforward than for most waves in the A ring where the presence of the self-gravity wakes affects the measurement of τ . One of the waves we analyze here, the Pandora 9:7 density wave, is located on a narrow ringlet, and the wave propagates to the edge of the ring. The assumption of constant background surface mass density is therefore not valid for this wave, and it is likely not valid for several of the other Cassini Division waves as well. We present an analysis of how the dispersion of these waves varies across the gradient in background optical depth. Two of the Cassini Division waves occur at regions where the optical depth is comparable to that in the central A ring allowing a direct comparison of the mass extinction coefficient, or opacity, K = τ /σ , for the Cassini Division and A ring. We find that the opacity in the Cassini Division is 3 to 5 times higher than that in the A ring except for the Cassini Division ramp which has an opacity like that of the A ring. This suggests that the particle size distribution in the ramp is more like that of the A ring and that particles in the ramp may have drifted there from the A ring, consistent with models of ballistic transport evolution of the inner edge of the A ring (Durisen et al., 1989, 1992). We also find vertical thicknesses of the Cassini Division at the locations of three waves of between 3 and 6 m. In the next section we describe the stellar occultation observations that were analyzed and present some wave profiles, and in Section 3 we describe the analysis techniques. Results and discussion on the local surface mass densities, opacities, total Cassini Division mass, and viscosities are in Section 4. 2. Observations The Cassini UVIS has a high speed photometer (HSP) used for observing stellar occultations (Esposito et al., 1998, 2004). The HSP has recorded more than 60 partial and complete occultations of stars by the rings, usually with spatial resolution in the ring plane of ∼20 m, though the resolution varies between occultations and also within occultations as the viewing geometry changes. Detailed descriptions of the first set of ring occultations observed by UVIS are in Colwell et al. (2007). The occultations used in the analysis presented here are listed in Table 1. Not all occultations obtained by UVIS spanned the Cassini Division waves discussed here, and others did not show the waves due to low signal. Others were not included in order to limit the total time span of the data analyzed and thus limit the effect of changes in resonance locations due to orbital changes of the moons. Because we are analyzing the dispersion and damping of a wave signal in the rings, our density wave results do not depend on the background signal or the HSP instrumental response to bright stars that affects determinations of absolute optical depths (Colwell et al., 2007). All occultations show the same optical depths in the Cassini Division, regardless of viewing geometry, suggesting a near total absence of self-gravity wakes. We are thus able to obtain
575
Table 1 List of occultations. Occultation star (Rev)
Date (year-day)
|B|
λ Cet (28)
2006-256 2006-269 2006-269 2006-285 2006-286 2006-313 2006-313 2006-325 2006-337 2006-337 2006-337 2006-337 2006-350 2006-352 2006-363 2006-363 2006-364 2007-003 2007-015 2007-020 2007-020 2007-022 2007-049 2007-056 2007-063 2007-073 2007-080 2007-080 2007-082 2007-092 2007-092 2007-179
15.3 32.2 41.7 17.2 47.4 54.4 47.4 54.4 17.2 17.2 44.5 44.5 48.5 54.4 20.3 20.3 54.0 51.0 54.0 51.0 51.0 61.0 54.0 47.5 35.1 39.3 35.1 35.1 54.0 48.5 48.5 2.7
α Sco B (29) λ Sco (29)
α Vir (30) γ Lup (30) α Ara (32) γ Lup (32) α Ara (33) α Vir (34I) α Vir (34E) η Lup (34I) η Lup (34E) κ Cen (35) α Ara (35) γ Peg (36I) γ Peg (36E) δ Per (36)
ε Lup (36) δ Per (37)
ε Lup (37I) ε Lup (37E) γ Ara (37) δ Per (39)
χ Cen (39) γ Gru (40) 3 Cen (40)
γ Gru (41I) γ Gru (41E) δ Per (41)
κ Cen (42I) κ Cen (42E) ζ Ori (47)
R (m)
I0 (counts)
b (counts)
19 20 6.5 42 8.7 10 15 10 52 52 7.0 7.0 10 6.5 15 15 18 9.3 15 5.3 5.3 6.8 7.0 6.2 13 8.3 11 11 20 1.8 1.8 30
4.8 7.0 570 1070 153 77.7 144 72.4 1020 988 96.2 89.3 90.5 75.8 148 132 27.0 62.9 27.4 63.6 59.8 51.9 25.0 26.2 14.5 9.5 16.2 15.1 24.3 84.6 81.7 336
0.2 0.1 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.3 0.1 0.2 0.6 0.6 0.8 0.4 0.1 0.3 0.3 0.3 0.2 0.3 0.3 0.2 0.4 0.3 0.8 0.3 0.3 2.4
Notes: R is given at 120,000 km and is the radial separation of two consecutive points in the HSP data. I 0 is the unocculted star signal in counts per integration period, and b is the background signal. Both I 0 and b vary over the course of an occultation in general (Colwell et al., 2007). Values reported here are based on measurements of the Huygens Gap for the sum I 0 + b and opaque regions in the B ring for b.
optical depths at the locations of the analyzed waves from any occultation, including those presented in Colwell et al. (2006, 2007). For the low optical depths in most of the Cassini Division, occultations of stars with a low incidence angle, B, provide the best sensitivity. The star with the lowest value of B observed by UVIS to date is the bright star ζ Orionis (Rev 047, Table 1). However, ζ Orionis is a double star, and the HSP data include a signal from both components of the binary system. The projected radial separation of the two stars in the ring plane in the Cassini Division for this occultation is 10 km which is longer than the wavetrain of some of the waves analyzed here. That occultation is thus useful for analysis of the wave dispersion and damping, but optical depth profiles are complicated by light from the secondary star passing through a different region of the rings. The other binary stars in Table 1 are γ Lupi, κ Centauri, λ Scorpii, and ε Lupi. α Scorpii is also a binary star, but the HSP only detects α Scorpii B because α Scorpii A is a red giant and therefore too cool to be detected in the FUV bandpass of the HSP. The HSP signal levels of the two components of the γ Lupi system are nearly identical, and the radial separation of the two stars in the ring plane is 2.9 km on Rev 030 and 1.7 km on Rev 032. The radial ring plane separations for κ Centauri (Revs 035 and 042), η Lupi (Rev 034), ε Lupi (Revs 036 and 037) and λ Scorpii (Rev 029) are less than 100 m which is less than the shortest wavelength of the density wavetrains analyzed here. While the γ Lupi separation is comparable to the density wave wavelengths studied, the signal from two stars puts a constant wavelength signal on the data where the density waves have a characteristic dispersion. In addition, our filtering process removes most of the
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Fig. 1. Optical depth profile of the Cassini Division from the α Leo Rev 009I occultation (Table 1). The density waves with measured surface mass densities in the Cassini Division are labeled by their resonances: Pa = Pan, Pd = Pandora, Pr = Prometheus, At = Atlas, and Ja = Janus/Epimetheus. The Cassini Division ramp is the region between 120,900 km and 122,040 km where there is a linear increase in optical depth with increasing ring plane radius. The Atlas 6:5 wave is on the inner edge of the A ring which begins at the outer edge of the ramp.
Fig. 2. Optical depth profile of the Prometheus 9:7 density wave in the α Leo Rev 009I occultation. The profile has been boxcar smoothed by 7 points from the full resolution data to better show the wave which extends from resonance at R ≈ 118,066 km to R ≈ 118,078 km.
signal from the beating due to the binary stars. Analysis of individual wave profiles in the γ Lupi occultation data, where possible, gave results consistent with those from the other occultations (Section 3). We also checked our derived quantities without the γ Lupi occultation data and found no significant difference. The star at the lowest incidence angle after ζ Orionis is α Leo with B = 9.5◦ . Fig. 1 shows the optical depth profile of the Cassini Division from the α Leo Rev 009 ingress (Rev 009I) occultation. Optical depths were computed using the calibration for this occultation described in Colwell et al. (2007). Figs. 2–7 show profiles for the individual waves at higher resolution from the α Leo (Rev 009I) occultation for the waves at low optical depth and from the α Virginis (Rev 034E) occultation for two of the three waves at higher optical depths. For optical depths from α Virginis (Rev 034E) we assume a background count rate of 2000 counts/s and I 0 = 5.2 × 105 counts/s based on the measured signal in the outer gaps of the Cassini Division and in optically thick regions in the outer B ring. Although the Atlas 6:5 wave shows up clearly in frequency space (Section 3), no individual raw occultation profile shows the wave clearly.
Fig. 3. Optical depth profile of the Pan 6:5 density wave from the α Leo Rev 009I occultation. The profile has been boxcar smoothed by 9 points from the full resolution data to better show the wave which extends from resonance at R ≈ 118,453 km to R ≈ 118,469 km. This wave is on a strong gradient in the background optical depth.
Fig. 4. Optical depth profile of the Atlas 5:4 density wave from the α Leo Rev 009I occultation. The profile has been boxcar smoothed by 9 points from the full resolution data to better show the wave which extends from resonance at R ≈ 118,830 km to R ≈ 118,848 km.
Fig. 5. Optical depth profile (solid line) and surface mass density (dashed line) of the Pandora 9:7 density wave from the α Virginis Rev 034E occultation. The resonance location is near the inner edge of the ringlet, and the wave is observed in this and other occultations to propagate down the gradient in optical depth at the outer edge of the ringlet. The derived surface mass density tracks the declining optical depth of the ringlet where the wave is located, with a radial offset.
Density waves in the Cassini Division
Fig. 6. Optical depth profile of the Pan 7:6 density wave from the α Virginis Rev 034E occultation. The profile has been boxcar smoothed by 5 points from the full resolution data to better show the wave which extends from resonance at R ≈ 120,668 km to R ≈ 120,695 km. Although this wave is in the middle of the three-peaked broad ringlet in the outer Cassini Division, the wave itself is in a relatively uniform section of the ringlet.
Fig. 8. Morlet wavelet scalogram (solid line) and surface mass density (dashed line) of the Atlas 5:4 density wave in the α Leo Rev 009I occultation. The wavelet transform is zeroed on either side of lines drawn by hand on the scalogram on either side of the density wave signal.
wavelet power contour plot (Fig. 8). We then select the range in ring plane radii over which we fit the linear dispersion relation to the peak power in the scalogram. We next find the peak power within the selected radial range. This gives an effective λ(r ). The surface mass density at each point along the wave is then calculated from Eq. (3):
σ (r ) =
Fig. 7. Optical depth profile of the Janus/Epimetheus density wave from the α Leo Rev 009I occultation. The profile has been boxcar smoothed by 13 points from the full resolution data to better show the wave which extends from resonance at R ≈ 121,260 km to R ≈ 121,320 km. The effect of the overlapping waves from the two satellites can be seen in the modulation of the amplitude and the wavelength increasing again at R > 121,300 km.
3. Analysis technique We obtain the surface mass density at the location of each density wave by analyzing the dispersion of the wave. We analyzed the waves in two ways. When an individual occultation profiles provided a clear signal of a density wave, we analyzed the dispersion as well as the damping of the wave following a procedure similar to that of Tiscareno et al. (2007) using the discrete wavelet analysis tool of Torrence and Compo (1998). For all waves we also computed a weighted wavelet Z (WWZ) transform (Foster, 1996) of each occultation and then summed the transforms. This strengthens the signal from the wave relative to that from noise in the data. 3.1. Individual wavetrain analysis Following Tiscareno et al. (2007), we use the complex Morlet wavelet transform on our occultation data. We filter the data by zeroing the wavelet transform on either side of lines drawn on the
577
λD L (r − r L ) 4π 2 Gr L
.
(5)
We calculated the mean value of σ from σ (r ) for each wave. Because the selection of the points to filter and the selection of the endpoints of the data to be fit require a subjective selection, two of us (JEC and JHC) independently repeated this analysis for 31 profiles of the Atlas 5:4 density wave. The mean values of σ from the two sets of analyses were identical to three significant figures. This technique was used for all the waves except the Pandora 9:7 density wave, though only 18 and 8 occultations were analyzed for the Prometheus 9:7 and Atlas 6:5 waves, respectively, because of the weak signature of those waves in some occultations. The Pandora 9:7 density wave exhibits the strongest gradient in surface mass density of the waves analyzed, so the selection of the radial range strongly affects the average value derived. Consequently for this wave we calculate σ (r ) from the averaged WWZ transform (Section 3.2). The Prometheus 9:7 wave and the Atlas 6:5 wave are too weak in most occultation profiles to provide a strong signal, so for those waves as well we use the averaged WWZ transform. The Janus 7:5 density wave in the Cassini Division ramp is complicated by interference from the weaker Epimetheus 7:5 wave and by the 4-year synodic period of these coorbital moons. This results in a change in the resonance locations and limits the distance of propagation of the wave. Tiscareno et al. (2006) modeled the time-dependent superposition of the waves from the coorbitals and found σ = 11.5 g cm−2 and ξd = 8, where ξd is the damping length of the wave in units of the dimensionless distance from resonance,
ξ=
DL r L r − r L 2π G σ
rL
.
(6)
We follow the same procedure as Tiscareno et al. (2007) to obtain ξd for the three waves we analyzed individually using the Morlet wavelet transform. In addition, we also fit the cumulative phase of these three waves, again following Tiscareno et al. (2007), to have yet another measure of the surface mass density. The vertical thickness of the rings, H , is estimated from the wave damping length by assuming that particle random veloci-
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Fig. 9. Wavelet power (contours) and surface mass density (dashed line) for the Pandora 9:7 density wave obtained from summing the WWZ transforms of 30 occultation profiles. The upward curvature of the peak power ridge reflects the declining surface mass density.
Fig. 10. σ (r ) vs τ (r ) for the 5 waves analyzed here that are in the main part of the Cassini Division. The Atlas 6:5 wave, also analyzed here, has a surface mass density outside the range of this plot. The dashed line shows the relationship between σ and τ assuming a constant opacity, κ , equal to the average value for the 5 waves.
ties, c, are not correlated and using (Goldreich and Tremaine, 1978; Tiscareno et al., 2007)
Table 2 Summary of density wave results.
c=
2ηn 1 + τ 2 = Hn,
τ where η is the ring viscosity given by (e.g. Shu, 1984)
η≈
9 7nξd3
(2π G σ )3 r L . DL
Wave
(7)
(8)
These approximations for viscosity and ring thickness are justified in the Cassini Division where optical depths are low and particles have not been observed to clump strongly into self-gravity wakes. 3.2. Averaged wavetrain analysis We take the weighted wavelet Z-transform (WWZ) of each occultation profile in Table 1. The WWZ transform is a modification of the Morlet transform implemented by Torrence and Compo (1998) used in the previous section and by Tiscareno et al. (2007) that can handle data with uneven sampling in time (Foster, 1996). By weighting points by time interval rather than the temporal density of points, the WWZ transform technique of Foster (1996) allows the transforms from multiple data sets to be co-added, providing equal weight to each data set. Co-adding the weighted Z-transform is equivalent to summing probabilities, which is the best way to treat data sets of different qualities (as opposed to adding raw power). Although our data are regularly sampled in time, the radial ring plane sampling is different between occultations, and we are analyzing periodic structure in the ring radial dimension rather than the temporal dimension. In fact, the ring plane sampling is not constant within an occultation, but the change in radial resolution over the length of the density waves described here is small enough to be negligible and does not affect the Morlet wavelet analysis described above. Before summing the transforms we adjust the radius scale of each occultation profile to minimize offsets between the profiles due to trajectory uncertainties. We used the presumedcircular fiduciary ring feature #11 from Nicholson et al. (1990) at 120,245 km to coalign the occultations before taking the WWZ transforms. The summed WWZ transform for the Pandora 9:7 from 30 occultations and the resulting average σ (r ) are shown in Fig. 9. We computed summed WWZ transforms for 5 Cassini Division waves (excluding the Janus/Epimetheus 7:5 wave) and the Atlas 6:5 wave on the inner edge of the A ring. We then computed σ (r ) from these average WWZ transforms using the same filtering and
Prometheus 9:7 Pan 6:5 Atlas 5:4 Pandora 9:7 Pan 7:6 Atlas 6:5 [2]
σ (Morlet) (g cm−2 )
σ (phase) (g cm−2 )
1.08 1.11 1.30
1.13 1.15 1.35
3.45 14.7
3.55 15.4
σ (WWZ) (g cm−2 ) 1.10 0.98 1.31 5.76 3.51 15.4
κ
ξd
0.111 0.083 0.068 0.118 [1] 0.086 0.033
6.6 ± 1.3 7.0 ± 1.4 7 .1 ± 0.9
4 3 6
7.8 ± 1.1 6 .2 ± 0.8
5 20
(cm g−1 )
H (m)
Values of κ use the radially averaged σ from the WWZ analysis and the radially averaged τ . [1] Pandora 9:7 values are for the range 120,055 to 120,073 km. [2] Values of σ from the Morlet analysis of this wave are from 8 occultations, and the damping length is the average of 6 occultations.
analysis used for the Morlet transform described in the previous section. 4. Results and conclusions The waves we analyze are not in regions of constant background optical depth. We took the measured optical depth of the Cassini Division from the UVIS stellar occultations, and smoothed it to remove the fluctuations due to the individual density waves. This gives us τ (r ) corresponding to σ (r ) obtained from the WWZ analysis. In Fig. 10 we show σ (τ ) for the 5 density waves within the Cassini Division analyzed here. Within each wave there is a correlation between σ and τ , with the greatest variation in τ occurring in the Pandora 9:7 wave. For that wave, the measured change in σ is about 34% (Fig. 9 where the background optical depth changes by more than a factor of two (Fig. 5). However, most of the variation in τ for this wave occurs over a radial span of only ∼5 km and a few wavelengths of the wave. The correspondence between σ and τ is not perfect within each wave, but does demonstrate for the first time that variations in wave dispersion within a single density wave are affected by variations in the ring property over radial scales greater than the wavelength. The Pandora spiral waves in the A ring were found to be modulated by the 3:2 near-corotation resonance between Pandora and Mimas. This likely affects the Pandora 9:7 wave analyzed here as well. We compute the radially-averaged opacity, κ = τ (r )/σ (r ), for each density wave analyzed here. Our results for σ , κ , ξd , and thickness, H , are summarized in Table 2. Fig. 11 shows our values of κ together with those for the A ring and the Cassini Division ramp. The opacities in the A ring are calculated from the α Ara (Rev 033) A ring optical depth and surface mass densities from Tiscareno et al. (2007). We use τ = 0.17 for the Cassini Di-
Density waves in the Cassini Division
Fig. 11. Opacities derived for the 6 density waves analyzed here (diamonds), the waves in the A ring analyzed by Tiscareno et al. (2007), and the Janus 7:5 density wave in the Cassini Division ramp. The opacity shows a sharp and consistent increase between the ramp and the rest of the Cassini Division.
vision ramp at the location of the Janus/Epimetheus 7:5 density wave (R = 121,260–121,290 km) from the α Ara (Rev 033) occultation. With the surface mass density of 11.5 g cm−2 found for this wave by Tiscareno et al. (2006), this gives κ = 0.015 cm2 g−1 . The opacities of the five Cassini Division density waves analyzed here are significantly higher than those found in the A ring and in the Cassini Division ramp. If the particle size distribution and particle densities are constant across the ramp, the opacity there corresponds to a total mass of the Cassini Division ramp of 1.1 × 1017 kg. In contrast to the typical A ring and ramp opacity of κ < 0.02 cm2 g−1 , the mean opacity of the 5 waves in the rest of the Cassini Division is κ = 0.093 ± 0.02 cm2 g−1 . Using this average opacity and the measured optical depth profile of the Cassini Division, this corresponds to a mass of 3.1 × 1016 kg for the Cassini Division excluding the ramp, equivalent to an icy moon with a density of 1 g cm−3 and a radius of 19 km. The increase in the opacity of roughly a factor of five from the A ring and Cassini Division ramp to the rest of the Cassini Division suggests that the particles in the ramp are more like those in the A ring than those in the rest of the Cassini Division. The optical depths at the Pandora 9:7 and Pan 7:6 waves are comparable to the optical depths in the central A ring, strengthening the conclusion that this is a change in particle properties and not an effect that is proportional the particle packing or optical depth. Compositional measurements of this region by the Cassini VIMS instrument show a smooth transition in water ice band depths from the inner A ring across the ramp and into the triple-peaked ringlet where the Pan 7:6 wave resides (Nicholson et al., 2008). The red color of the ramp in the near-IR (0.85–1.4 μm) is more like that of the rest of the Cassini Division than it is like the A ring. This spectral difference between the ramp and the A ring indicates different compositional evolution between the two regions or a mixture of redder Cassini Division particles with icy A ring particles in the ramp. The larger opacity of the Cassini Division compared to the A ring indicates smaller particles on average. The Voyager radio science occultation at 3.6 cm and 13 cm shows no significant change in the differential optical depth between the Cassini Division and the ramp, consistent with the change in the size distribution being at the upper size cutoff. French and Nicholson (2000) found comparable opacities for the Cassini Division and A ring based on stellar occultation measurements, but their model did not include the effect of self-gravity wakes on observed optical depth, and the Cassini Division was the least well-constrained region of the rings
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due to its relatively narrow width. If the particle size distribution can be described by a power-law with differential size index = −3, then the difference in opacity between the A ring and the Cassini Division implies the largest particles in the Cassini Division are ∼5 times smaller than the largest particles in the A ring and the Cassini Division ramp. The lower mass of the Cassini Division than the A ring makes it both easier to create from satellite disruption and easier to erode through micrometeoroid bombardment. Colwell (1994) calculated satellite disruption timescales and found a disruption time ∼109 years for a 19 km-radius moon at the outer edge of Saturn’s rings, consistent with calculations by Colwell et al. (2000) for the lifetime of similar-sized moons in the ring systems of Uranus and Neptune. The micrometeoroid flux delivers mass to the Cassini Division particles at the rate of ∼103 g s−1 , where we have increased the flux of Cuzzi and Estrada (1998) by a factor of 1.5 after accounting for gravitational focusing and the low optical depth of the Cassini Division, and the total area of the Cassini Division (excluding the ramp) is ∼1.6 × 1019 cm2 . The time to add 10% exogenic material, by mass, to the Cassini Division is then also ∼109 years. On the other hand, the time to physically erode the particles through micrometeoroid bombardment is considerably shorter because ejecta mass yields are ∼3 × 104 times the mass of the impactors. This gives a time to erode pristine ring particles to a depth of ∼10 cm of a few million years Esposito and Elliott (2007). Disruption of a moon massive enough to produce the Cassini Division would reset the brightness of a 1000-km-wide swath of the A ring (Esposito et al., 2005). Our results support a common origin for most of the particles in the ramp and those in the A ring, consistent with models of ballistic transport producing the ramp as angular momentum is removed from the inner edge of the A ring by asymmetric absorption of meteoroid ejecta (Durisen et al., 1989, 1992). They also suggest that the remainder of the Cassini Division has a different origin than the A ring (and ramp), or has had a different evolution of the size distribution of the particles within it, or some combination of the two. Acknowledgments This work was supported by NASA through the Cassini project. We thank Matt Tiscareno and Nicole Rappaport for their helpful reviews of this paper. References Colwell, J.E., 1994. The disruption of planetary satellites and the creation of planetary rings. Planet. Space Sci. 42, 1139–1149. Colwell, J.E., Esposito, L.W., Bundy, D., 2000. Fragmentation rates of small satellites in the outer Solar System. J. Geophys. Res. 105, 17589–17599. ´ M., 2006. Gravitational wakes in Saturn’s Colwell, J.E., Esposito, L.W., Sremˇcevic, A ring measured by stellar occultations from Cassini. Geophys. Res. Lett. 33, doi:10.1029/2005GL025163. L07201. ´ M., Stewart, G.R., McClintock, W.M., 2007. Colwell, J.E., Esposito, L.W., Sremˇcevic, Self-gravity wakes and radial structure of Saturn’s B ring. Icarus 190, 127–244. Cuzzi, J.N., Estrada, P.R., 1998. Compositional evolution of Saturn’s rings due to meteoroid bombardment. Icarus 132, 1–35. Durisen, R.H., Cramer, N.L., Murphy, B.W., Cuzzi, J.N., Mullikin, T.L., Cederbloom, S.E., 1989. Ballistic transport in planetary ring systems due to particle erosion mechanisms. I. Theory, numerical methods, and illustrative examples. Icarus 80, 136–166. Durisen, R.H., Bode, P.W., Cuzzi, J.N., Cederbloom, S.E., Murphy, B.W., 1992. Ballistic transport in planetary ring systems due to particle erosion mechanisms. II. Theoretical models for Saturn’s A- and B-ring inner edges. Icarus 100, 364–393. Esposito, L.W., Elliott, J.P., 2007. Regolith growth and pollution on Saturn’s moons and ring particles. Bull. Am. Astron. Soc. 39, 421. Abstract 7.09. Esposito, L.W., O’Callaghan, M., West, R.A., 1983. The structure of Saturn’s rings: Implications from the Voyager stellar occultation. Icarus 56, 439–452.
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