Towards a Constraint on Vertical
Thermal Transport across
Saturn’s Rings
S. Brooks
UVIS Team Meeting
16 – 18 June 2014
Berlin, Germany
Thermal Throughput Across the Rings
• At its most opaque, B ring
optical depths are 5 and
higher!
• At such optical depths the
unlit side of the rings
should be effectively cut
off from the Sun.
• But CIRS data show that
the unlit side receives solar
forcing … somehow (see
Pilorz et al.,submitted;
Ferrari et al., 2013).
Fig. 1 Lit temperatures ()
compared to unlit temperatures (*)
as Saturn advances towards
equinox. One might expect little to
no variation in unlit temperatures.
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incident sunlight
transmitted sunlight
Fig. 2 To zeroth order, the difference in the energy balance of the lit
and the unlit sides of the rings is due to the difference in the incident
solar flux.
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IR Observations & Rings Geometry
• But what about intermediate t?
• Retrieved CIRS ring temperatures are dependent
on viewing geometry (Spilker et al., 2005).
• In order to compare lit temperatures to unlit
temperatures we can:
– model temperature variations with geometry (difficult)
or
– take lit and unlit observations at otherwise similar
geometries (also difficult, but for different reasons).
• TDIFs are an attempt to do the latter.
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Fig. 2 TDIF opportunities were found by determining when the phase angles and
(absolute values of) the sub-spacecraft latitudes on the lit and unlit sides of the rings
were similar. The portion of the orbit below the ring plane (- -) is compared to the rest
of the orbit (—). Regions where they cross (—) were identified as TDIF opportunities.
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CIRS_186RI_TDIFNORTH001_PIE
We had another pair of
TDIFs: NORTH in rev
191, SOUTH in rev194.
CIRS_186RI_TDIFSOUTH001_PIE
We have another pair of
TDIF PIEs scheduled in
IN-2.
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reviewed and determined not to contain export controlled technical data.
Fig. 4 The emission angle and the phase angle for the lit and unlit CIRS
radial scans are shown for comparison. The TDIF pair was designed to
limit the difference in these parameters, which have been shown to be
correlated with differences in retrieved ring temperature.
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reviewed and determined not to contain export controlled technical data.
Fig. 5 Observed fluxes from the TDIFS plotted for comparison.
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Making a Comparison
• Of course, comparing radial scans taken at
different times is not straightforward.
• The location and spacing of consecutive Q5
points is not consistent.
• The range to the rings is necessarily different, the
projected size of the FP1 footprint, and hence
the point spread function, necessarily differs.
• We need a way to perform a consistent applesto-apples comparison between scans.
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Concept and Method
• Postulate: there exists a matrix P which maps the
“true” temperature profile, x, into the profile
measured by CIRS, y:
y=P×x
• Then postulate a matrix P+, mapping the measured
profile back to the “true” temperature profile:
x = P+ × y
• By determining P+, the pseudoinverse of P, we can
derive the rings’ true temperature profile.
• P is related to the instrument point spread function.
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Fig. 6 The PSF over an FP1
footprint (blue) is
mapped onto n radial
bins. The m rows of P
contain the total
contribution from the FP1
PSF in each radial bin.
This is how P is computed.
Note that the calculation
is not performed over the
footprint mapped as
shown at the right! The Q
points are used to fit an
ellipse that describes the
FP1 field of view. It is that
ellipse that is used to
calculate the PSF
contribution to each
radial bin.
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Singular Value Decomposition
• P+ can be obtained through the singular value
decomposition of P into matrices U, V and S.
P = U × S × VT
and
P+ = V × S × UT
• S is a diagonal matrix of singular values. We zero
out the singular values above a threshold that
produces the best results.
• But direct application of this method is flawed.
P
P as computed for
CIRS_186RI_TDIFSOUTH001_PIE
Non-zero off-diagonal
entries indicate that the
SVD is not returning the
input matrix.
PSVD = U S VT
Modifying the Deconvolution
• This “ringing” arises because the number of rows
in P does not match the number of columns; the
problems is inherently underconstrained.
• To get around this, we use a Lagrange multiplier
y = (P + l B ) × x
where B encodes for some additional constraint.
• Typically, this additional constraint is that the
solution to the above be smooth within the
confines of the error bars on our data.
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Fig. 8 The fit to CIRS_186RI_TDIFNORTH001_PIE utilizing the Lagrange multiplier
method is shown. The radial profile has been deconvolved to a resolution of 50 km.
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Fig. 9 The fit to CIRS_186RI_TDIFSOUTH001_PIE utilizing the Lagrange multiplier
method is shown. Minor problems with the implementation must be addressed.
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Fig. 10 The differential flux between the lit side of the rings and the unlit side of the
rings is plotted as a function of distance from Saturn.
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Fig. 11 The differential flux between the lit side of the rings and the unlit side of the
rings is plotted as a function of optical depth.
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Fig. 12 The discrepancy between the observed and the expected flux differentials
suggests significant contact between the lit and unlit sides nearly everywhere.
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Summary
• The method of Lagrange multipliers holds out
reasonable hope of allowing us to deconvolve
CIRS FP1 scans.
• More work needs to be done with:
– deconvolution errors
– determining limits on the deconvolution process
– exploring deconvolution parameter space to refine
the process (l B, etc. …).
• We plan to use this method to study fine radial
features (e.g. halos), map out thermal inertia …
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reviewed and determined not to contain export controlled technical data.