Analysis of Density Waves
in UVIS Ring Stellar Occultations
Josh Colwell
January 2008 UVIS Team Meeting
Introduction
• Problem: Thousands of density wave
profiles in occultation data contain
information on ring mass density and
viscosity. Data are of variable quality,
and wave signals are on top of other
features in the rings.
• Goal: Find a procedure with minimal
decision-making to extract ring
properties from thousands of wave
profiles.
January 8, 2008
UVIS Team Meeting. Colwell.
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Derivation of Surface Mass Density
Following Tiscareno et al. (2007):
• Density wave phase depends quadratically on
distance from resonance far from resonance.
r  rL
x
rL
 DL rL 

x
 2 G 
(1)
2
 RS   21 9

2
DL  3(m  1) L  J 2     (m  1)  2L
 rL   2 2

 DW  0  4  12  2  A  Bx 2

DL rL
4 GB
(2)
(3)
(4)
L subscript refers to location of Lindblad resonance. RS is radius of
Saturn,  is the mean motion,  is the surface mass density, and m
is the azimuthal parameter of the resonance.
January 8, 2008
UVIS Team Meeting. Colwell.
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Synthetic Wave
• Technique is based on the linear density
wave theory of Shu (1984).
• Generate a synthetic wave data set with
the parameters of the Prometheus 10:9
density wave:
– RL=13004 km,
– =40 g cm-2
– D=6
January 8, 2008
UVIS Team Meeting. Colwell.
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Synthetic Wave Profile
January 8, 2008
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Wavelet Transform of Synthetic Data
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Isolate the Wave by Filtering
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Fit Cumulative Phase
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Chi-Squared Values
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Amplitude of Filtered Wave
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Determine Damping Length
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Compare Signal and Model
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Recovered Parameters
• Surface mass density: 39.96 g cm-2 (40
input)
• Damping Length: 5.87 (6 input)
• Resonance Location: fixed at 130004
km.
• Initial phase determined from satellite
location. Derived phase required to fit
initial phase.
• Get better fits with wrong resonance
location.
January 8, 2008
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With Search for RL…
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Wrong RL and  Found
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Cumulative Phase Fits, but not Phase
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Allow Initial Phase to be Unconstrained
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This Locates Correct Resonance Location
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Then Run with RL and 0 Fixed
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Still Get a Good Fit
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Mass Density from Wave Dispersion
 = 45 g cm-2 (40 input)
(Data generated from model)
January 8, 2008
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Mass Density from Wave Dispersion
 = 42 g cm-2 (40 input)
(Data generated from model)
January 8, 2008
UVIS Team Meeting. Colwell.
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Synthetic Wave Dispersion Analysis
Gives  = 46 g cm-2
Calculate sigma at each point from
wavelength at that point.
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Alpha Leo (9) Ingress: Pr10:9
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Alpha Leo (9) Ingress: Pr10:9
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Alpha Leo (9) Ingress: Pr10:9
Fit appears to be offset.
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Alpha Leo (9) Ingress: Pr10:9
So readjust RL to get
January 8, 2008
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Alpha Leo 9I Wave Dispersion Analysis
=35 g cm-2
Significantly lower than model fitting
or cumulative phase suggest.
January 8, 2008
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Alpha Leo 9I Wave Dispersion Analysis
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Alpha Leo 9I Phase Analysis
Unconstrained initial phase
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Alpha Leo 9I Phase Analysis
Constrained initial phase (offset by 2PI)
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Model Fit with Mean Dispersion Value
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Model Fit with Phase Value
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Conclusion (1)
Forget about removing human inspection
of the wave and the fit when extracting
ring properties from density wave.
January 8, 2008
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Analysis of Actual Density Waves
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Pandora 9:7
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Pandora 9:7
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Pandora 9:7
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Variation in Background 
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Variation in Background 
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Variation in Background 
Pandora 5:4
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Variation in Background 
Pandora 5:4
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Variation in Background 
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Variation in Background 
Janus 2:1
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Variation in Background 
Janus 2:1
Hiccups probably due to J/E swap
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Status
• See variations in wave dispersion
across wavetrain.
• Offset in dispersion from underlying
optical depth.
• Change in surface mass density not
proportional to change in optical depth
in Pandora 5:4. Anti-correlated with
optical depth in Janus 2:1.
January 8, 2008
UVIS Team Meeting. Colwell.
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Next Steps
• Finish a systematic extraction of surface
mass density and viscosity from density
waves using a combination of
techniques, validated by visual
inspection of fit of model to data.
• Determine mass of Cassini Division.
• Look for waves with variable surface
mass density and links between mass
and optical depth.
January 8, 2008
UVIS Team Meeting. Colwell.
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