Far Ultraviolet Spectral Properties of Saturn's Rings
from Cassini UVIS
E. Todd Bradleya, Joshua E. Colwella, Larry W. Espositob, Jeffrey
N. Cuzzic, Heather Tollerudd, Lindsey Chamberse
a
Department of Physics, University of Central Florida, Orlando, FL, 32816-2385, USA
b
LASP, University of Colorado, 392 UCB, Boulder, CO 80309-0392, USA
c
Space Science Division, NASA Ames Research Center, Mail Stop 24503, Moffett Field,
CA, 94035, USA
d
Department of Geosciences, Pennsylvania State University, University Park, PA 16802,
USA
e
Department of Earth and Planetary Sciences, University of California - Santa Cruz,
1156 High St, Santa Cruz, CA, 95064, USA
This paper is dedicated to the memory of Michel Festou whose many contributions to the
UVIS investigation included conceiving of the path length analysis presented here.
Submitted to Icarus
June 9, 2009
Suggested Running Head: Ultraviolet Spectrum of Saturn’s Rings
Address Editorial Correspondence to:
Todd Bradley, Department of Physics, University of Central Florida, Orlando, FL,
32816-2385, USA
tbradley@physics.ucf.edu
2
Abstract
Spectra taken by the Cassini Ultraviolet Imaging Spectrograph (UVIS) of Saturn’s C
ring, B ring, Cassini Division, and A ring have been analyzed in order to characterize ring
particle surface properties, and water ice abundance in the rings. UVIS spectra sense the
outer few microns of the ring particles. Spectra of the normalized reflectance (I/F) in all
four regions show a characteristic water ice absorption feature near 165 nm. Our analysis
shows that the fractional abundance of surface water ice is largest in the outer B ring and
decreases by over a factor of 2 across the inner C ring. We calculate the mean path length
of UV photons through icy ring particle regolith and the scattering asymmetry parameter
using a Hapke reflectance model and a Shkuratov reflectance model to match the location
of the water ice absorption edge in the data. Both models give similar retrieved values of
the photon mean, however the retrieved asymmetry (g values) are different. The photon
mean path lengths are nearly uniform across the B and A rings. Shortward of 165 nm the
rings exhibit a slope that turns up towards shorter wavelengths, while the UV slope of
180/150 nm (reflectance outside the water absorption ratioed to that inside the absorption
band) tracks I/F with maxima in the outer B ring and in the central A ring. Retrieved
values of the scattering asymmetry parameter show the regolith grains to be highly
backscattering in the FUV spectral regime.
3
1. Introduction
Saturn’s rings are known to be composed mostly of water ice (Cuzzi et al. 1984,
Esposito et al. 1984). However, a decrease in reflectivity shortward of 600 nm, which
gives the rings their characteristic color, implies the presence of a non-icy constituent
(Lebofsky et al. 1970, Estrada and Cuzzi 1996, Cuzzi and Estrada 1998). Microwave
observations of the rings have constrained estimates of the fractional abundance of water
ice in the rings to be greater than 90% (Epstein et al. 1984). The remaining fraction is a
contaminant that is primordial and/or due to meteoroid bombardment (Cuzzi and Estrada
1998). Water ice contains a distinctive absorption feature near 165 nm that was first
observed in Saturn’s rings in 12 IUE observations from 1982 to 1985 (Wagener and
Caldwell 1988). Near-infrared spectra taken with the Cassini Visual and Infrared
Mapping Spectrometer (VIMS) have also shown the main rings to be composed mostly
of water ice, with a decrease in reflectance at wavelengths below 600 nm suggesting an
organic contaminant (Nicholson et al. 2008). While typical ring particle sizes have been
shown to range in size from centimeters to several meters (e.g. Zebker et al. 1985),
ultraviolet reflectance spectra contain information concerning the micro-structural surface
properties of the particles themselves. The analysis in this paper assumes the rings to be
composed of large (> 1 cm) particles, each of which is covered with a regolith of ice
grains. Following the assumptions by Cuzzi and Estrada (1998), this analysis assumes
that each regolith grain has its own single scattering albedo and that multiple scattering
occurs from one regolith grain to another.
The wavelength of the water ice absorption feature is governed by the distance
photons travel through water ice and the absorption properties of the regolith ice grain.
4
The more ice that photons traverse, the more absorption there is in the water ice, and this
causes greater absorption at longer wavelengths, increasing the wavelength of the
absorption edge (Hendrix and Hansen 2008). Physical properties of ring particles that
could affect the mean photon path length include the distance between cracks or
scattering centers in the ring particles, mean spacing between contaminants in a water-ice
matrix, and the size of individual ice grains on the surface of larger ring particles.
We apply two photometric models to interpret spectra from the C ring, B ring,
Cassini Division, and A ring and establish the mean photon path length, L, and the
scattering asymmetry parameter, g, for a selected set of observations. The technique is
based solely on the spectral shape of the absorption feature and not on the absolute
magnitude of the normalized reflectance, I/F, where πF is the incident solar flux at
Saturn. The two models we use are the Hapke model (Hapke, 1981, Hapke, 2002) and the
Shkuratov model (Shkuratov et al, 1999, Poulet et al, 2002). We made the assumption
that any contaminant present is gray and is spectrally flat in the FUV. This assumption is
based on results from Khare et al (1984) where the imaginary component of the indices
of refraction of Titan tholins vary by less than an order of magnitude over the spectral
range of the FUV while the imaginary component of the index of refraction of water ice
varies by over 8 orders of magnitude for the same spectral range (Warren and Brandt,
2008). The real part of the index of refraction for both water ice and carbon are relatively
flat over the FUV spectral range, thus any spectral variations in the reflectance of a
contaminant is assumed to be small and not treated in this investigation.
We assess the relative water ice abundance across the rings by taking the UV
slope, i.e., the ratio of the average I/F from 180-190 nm to the average I/F from 140-160
5
nm. Calculations made by (Hendrix and Hansen, 2008) show typical bidirectional
reflectances of several contaminants to be on the order of 0.008 at a phase angle of 42°,
which are more in the range of I/F averaged from 140-160 nm and considerably lower
than I/F between 180-190 nm. Thus any contaminant present in an otherwise pure water
ice regolith will serve to decrease the value of I/F for the long wavelengths. The presence
of contaminant may increase or decrease I/F at the short wavelengths, depending on the
relative magnitudes of the two. The result is that the UV slope will decrease for
increasing contaminant levels providing a relative measure of the fractional abundance of
the contaminant across the rings.
In Section 2 we describe the observations and data reduction techniques used to
determine I/F. In Section 3 we describe spectral modeling of the data using spectrally
dependent terms from Hapke’s radiance factor model and from Shkuratov’s albedo
model. Results are presented in Section 4 with estimates for the uncertainty in the photon
path lengths and asymmetry parameters, and we discuss implications of our results for
ring particle properties and composition in Section 5.
2. Observations and data reduction
The Cassini UVIS is a spectrograph that contains two channels, an FUV channel
(111.5 -191.2 nm) and an EUV channel (56.3-118.2 nm, see Esposito et al. 2004 for a
complete description of UVIS). Here we analyze the water absorption edge in the FUV
and thus only consider FUV observations. We have analyzed hundreds of observations in
the FUV taken by the Cassini UVIS from 2005 to 2008. From these, we selected
observations on the lit side of the rings with phase angles between 5° and 30°, with broad
6
spatial coverage of the rings and high signal-to-noise spectra with a projected smeared
pixel size of no more than 4000 km in the radial direction (Table 1).
The FUV channel has 3 selectable entrance slits of different widths (highresolution, low-resolution, and occultation). These result in different projected fields of
view in the spectral direction as well as different spectral resolutions. For the FUV
channel the high-resolution configuration results in a spectral resolution of 0.275 nm with
a projected field of view of 0.75 mrad in the spectral direction. The low-resolution
configuration results in a spectral resolution of 0.48 nm with a projected field of view of
1.5 mrad in the spectral direction. None of our selected observations used the occultation
slit. The projected field of view in the spatial direction is 64 mrad for all 3 slit
configurations. Since there are 64 pixels in the spatial direction, this results in an angular
resolution of 1 mrad in the spatial direction. After light passes through the entrance slit
and is dispersed by the grating, it is imaged onto a 1024 spectral × 64 spatial element
CODACON detector, thus providing independent spectra of 64 different spatial locations.
The ends of the detector in the spatial dimension are partially blocked by the detector
mount resulting in 60 usable spectra for each measurement. Time series observations
were made that built up a data cube with 2 spatial dimensions and 1 spectral dimension.
One of the spatial dimensions results from moving the field of view from one location to
another. Depending on the rate of the scan and the integration period, the spatial
resolution in the direction perpendicular to the slit is generally different from the spatial
resolution along the spatial axis of the slit.
7
1
2
3
4
Observation Name
Maximum
phase angle
(degrees)
Minimum
phase angle
(degrees)
UVIS_008RI_SUBML20LP002_CIRS
UVIS_014RI_SUBML15LP001_CIRS
UVIS_036RI_SUBML17LP001_CIRS
UVIS_054RI_SUBML08LP001_CIRS
20.0
16.8
18.9
36.6
0.950
4.38
4.60
19.4
Average
projected field
of view size
(km)
250 X 2460
100 X 2660
365 X 2740
700 X 3375
Table 1. Summary of geometrical characteristics of observations used in this
investigation. For the analysis of observations, only pixels with phase angles between 5°
and 30° were used.
We produced an image of the raw spectra for each ring observation to select data
for this initial analysis. Spatial coverage and count rates are used as selection criteria to
categorize the quality of the observations. We selected observations based on good
spatial coverage and counting rates as well as consistent viewing geometry between
observations (lit face, low phase angle).
Data reduction consists of first subtracting a background from the raw counts. The
background arises primarily from detector dark counts introduced by Cassini’s three
radioisotope thermoelectric generators (RTGs). We then multiply the data by a
calibration factor that converts the counts to a radiance in units of kilo-Rayleighs/Å,
where 1 kilo-Rayleigh = 109 photons/sec/cm2 emitted over 4π steradians. We use a solar
continuum flux (πF) measured by the Solar EUV Experiment (SEE) on board the TIMED
spacecraft (Woods et al 2005). The solar spectra are available on the TIMED-SEE
website (http://lasp.colorado.edu/see/l3_data_page.html). SEE has been making daily
measurements of EUV and FUV solar irradiance since 2002. Solar continua that have
been corrected for the rotation of the Sun in order to correspond to the face of the Sun
8
towards Saturn at the time of the observation were smoothed to the bandpass of the
instrument and corrected for the Sun-Saturn distance. The solar flux was divided by π to
get the radiance of a perfect Lambert surface and the observed radiance divided by that to
produce I/F.
During an observation the field of view of a pixel projected in the ring plane is
affected by the spacecraft motion and the angle and orientation at which the line of sight
intersects the ring plane. This results in pixels that are non-uniform both in size and
distribution in the ring plane. Azimuthally binning pixels within relatively large radial
bins may bias the result towards regions within bins where there is a higher concentration
of pixels. In order to deal with this issue, we resample the data to generate an evenly
distributed grid. In anticipation of data being azimuthally binned over some radial
increment, we only consider the radial direction when resampling the data. We divide the
rings into a 100 km radial grid and for each grid element, we select all pixels that
intersect that element. The fraction, f, of each pixel within the element is multiplied by
the value of I/F for that pixel and summed according to
fi =
100
Ri
n
∑ (I / F)
I / F(resampled) =
i
* fi
Eq 1
i= 1
n
∑f
i
i= 1
where Ri is the length of the ith smeared pixel in the radial direction and n is the total
€ for which some fraction falls within the grid element. All of the pixels
number of pixels
were larger than 100 km so that none of the pixels fit completely within an element. The
result is an evenly weighted radial distribution of I/F values in 100 km increments.
9
After the data were evenly gridded, we then binned the results in larger radial
increments in order to improve the signal-to-noise ratio. The sizes of the radial
increments were determined by partitioning the rings into broad regions based on an
optical depth profile from a UVIS stellar occultation profile. Figure 1 shows the optical
depth profile along with the boundaries chosen for radial binning.
The upper panel of Figure 2 shows I/F measured for four regions in the rings from
118-190 nm for observation 2. The spectra clearly show the water ice absorption feature
present in all four regions. Lyman-α at 121.5 nm scatters out in the wings at longer
wavelengths. However, the raw counts from Lyman-α scattered to pixels corresponding
to wavelengths greater than 140 nm is almost a factor of 3 lower in magnitude than the
Lyman-α counts at line center. For Lyman-α present in the ring observations this
corresponds to count rates lower than the RTG background implying that scattered
Lyman-α photons are negligible for wavelengths greater than 140 nm. Thus the spectral
slope of the rings shortward of 160 nm is a real spectral characteristic of the rings and not
an artifact of Lyman-α background. The lower panel of Figure 2 shows the averages of
I/F from 140-160 nm and 175-190 nm as a function of ring plane radius. Error bars are
shown that represent the 1-σ uncertainty due to photon counting and rtg background. The
reason the uncertainty is so small is due to the large number of pixels that are used to
generate each data point. At the low phase angles studied here, the long wavelength
average of I/F shows that the outer B ring is significantly brighter than the rest of the
rings. I/F changes with azimuth angle due to self-gravity wakes (Colwell et al, 2007),
where the azimuth angle is defined as the angle between a vector from Saturn center to
the intersection of the field of view with the ring plane and the line of sight vector
10
projected onto the ring plane. The azimuthal dependence is detectable in the data for
observations with a large range of azimuth angles. For observation 3, the value of I/F in
the A ring changed from 0.055 to 0.065 for a change in azimuth angle from 105° to 45°.
3. Spectral modeling
3.1 Radiative transfer for the rings
We begin by expressing I/F for the rings according to Cuzzi et al (2002):
I / F = APparticleO
Eq 2
where A is the ring particle albedo, Pparticle is ring particle phase function, and O
€
incorporates geometry and optical depth as a scattering function. In the case where
multiple scattering between ring particles may be ignored, O is separable from A and
Pparticle and is thus a multiplicative constant for a given observational geometry. We used
a numerical ray-tracing Monte-Carlo model (Chambers, 2008) based on Salo and
Karjalainen (2003) to investigate the relative contributions of single and multiple
scattering from ring particles for solar photons incident on the rings. The code models
photon interactions with ring particles, includes Saturnshine, and a variable volume
filling factor (D). The result of the code is a relation between albedo and I/F due to
contributions from multiple orders of scattering; thus for measured I/F the contributions
from both single and multiple scattering are determined. The code was benchmarked
against results from Salo and Karjalainen (2003) as well as against Cassini data (see
Chambers (2008) for details). We used the Callisto power-law phase function for the ring
particles, and a modified Barkstrom’s law phase function was used for Saturn, which is
based on that used by Dones et al. (1993). The Barkstrom phase function coefficients
11
were empirically changed from the coefficients used from Voyager data to match
seasonal changes seen at Saturn at the time of Cassini. We performed tests at optical
depths of 0.5, 1.5, 0.1 and 0.1 for the A-Ring, B-Ring, C-Ring and Cassini Division,
respectively. Two suites of simulations were run with different values for the volume
filling factor: D = 0.001 (classical thick ring) and D = 0.1 (thinner ring). For the FUV
spectral regime and low phase angle observing geometry multiple scattering from ring
particles was found to contribute from 2-15% to the reflected photon flux for the Cassini
Division and B ring, respectively. Therefore in the following analysis we neglect multiple
scattering between ring particles, which is further complicated by the presence of selfgravity wakes (Porco et al. 2008). Furthermore, if Pparticle is assumed to be spectrally
independent over the wavelength range of the UVIS data, then P becomes a
multiplicative constant for a given observational geometry.
Without knowledge of the exact power law index for Pparticle or the dependence of
the scattering function, O, on optical depth, the absolute magnitude of I/F is not modeled
in the following analysis. However, based on the spectral location and shape of the water
ice absorption feature, we are able to investigate the photon mean path length and the
asymmetry parameter of the ice grains in the ring particle regolith. We use the spectrally
dependent portions of two models, the Hapke model (1993, 2002) and the Shkuratov
model (Shkuratov et al, 1999, Poulet et al, 2002) and scale them to the measured values
of I/F in order to constrain the photon mean path length and the asymmetry parameter for
the rings.
Since both models are based on the indices of refraction for the constituents of the
ring particle regolith and we are only using the indices of refraction for water ice in this
12
investigation, then scaling the model to the data requires consideration of the functional
form of A that includes a term for non-ice contaminants that is not modeled. For an areal
mixture of ice and one grey contaminant, A may be written as:
A = f w Aw + (1 − f w ) * Ac
Eq 3
where fw is the fractional abundance of water ice, Aw is the albedo of water ice, and Ac is
€
the albedo of the contaminant. Substituting Equation 3 into Equation 2 and rearranging
terms gives
I / F = f w Aw PparticleO + (1 − f w ) Ac PparticleO
Eq 4
Using the average I/F at two wavelength intervals (180-190 nm and 153-160 nm) along
with the average€Aw from either of the two models over the same two wavelength
intervals results in two simultaneous equations that may be solved for factors that scales
the model to the data on either side of the absorption feature as shown below.
I / F( λ 2 ) = f w Aw( λ2 ) PparticleO + (1 − f w ) Ac PparticleO
I / F( λ1 ) = f w Aw( λ1 ) PparticleO + (1 − f w ) Ac PparticleO
Eq 5
€ 1 refer to the averages of the two wavelength intervals. Solving these two
where 2 and
equations simultaneously gives
f w PparticleO =
( I / F( λ 2 ) − I / F( λ 2 ) )
(A
w( λ 2 )
− Aw( λ1 ) )
Eq 6
(1 − f w ) Ac PparticleO = I / F( λ 2 ) − f w Aw( λ2 ) PparticleO
€
Substituting Equation
6 back into Equation 4 gives
13
Mscaled =
( I / F( λ 2 ) − I / F( λ1 ) ) * A
Aw( λ 2 ) − Aw( λ1 )
w
Eq 7
( I / F( λ 2 ) − I / F( λ1 ) ) * A
+ I / F( λ 2 ) −
w( λ )
Aw( λ 2 ) − Aw( λ1 )
2
€
where Mscaled implies that the model is scaled in magnitude to I/F at wavelengths above
and below the absorption edge. This technique incorporates the functional form of a two
component areal mixed ring particle regolith. Therefore, although the contaminant albedo
or fraction of water ice is not known, the water ice model scales to the same magnitude as
it would if those quantities (along with Pphase and O) were known and put directly into
Equation 4.
3.2 Hapke model
The next step is to model the water ice albedo. I/F, which is referred to as the
radiance factor by Hapke (1993), is given as follows:
I / F = RADF =
ϖ o µo
{[ 1 + B(g) ] p( φ ,g)
4 µo + µ
Eq 8
+ H( ϖ o , µ o )H( ϖ o , µ ) − 1}R(Θ )
€ I/F), is the ratio of the bidirectional reflectance of a surface to that of a
where RADF (i.e.,
perfectly diffuse surface illuminated at an incidence angle = 0, ϖο is the regolith grain
single scattering albedo, µο is the cosine of the solar incidence angle, µ is the cosine of
the emission angle, B(g) is the opposition effect term, p(φ,g) is the regolith grain phase
function for phase angle (φ) and asymmetry parameter (g), H(ϖο,µ) and H(ϖο,µο) are
functions that arise from the contributions to the reflected light due to isotropic multiple
14
scattering between regolith grains (Chandrasekhar 1960), and
is the large-scale
surface roughness term. The opposition effect only becomes significant for phase angles
smaller than the observations analyzed here so B(g) is neglected. The roughness term is
used to deal with the roughness of the particulate surface on scales larger than the size of
the regolith grains. This paper assumes that the ring particle surface is relatively smooth
compared with the regolith grain size, and thus the roughness term is neglected.
Furthermore, since we are not concerned with the absolute magnitude of I/F, scalar
multipliers may be neglected as well. Comparison of the data with the spectrally
dependent terms from the Hapke model, the water ice albedo, Aw, from Equations 4 and 7
is modeled according to:
Aw = ϖ o [ p( φ ,g) + H( ϖ o , µ )H( ϖ o , µ o ) − 1]
Eq 9
€
where Hapke (2002)
gives
−1


 1 − 2ro x  1 + x 
H( ϖ o ,x) = 1 − ϖ 0 x ro + 
 ln

 2 
x 


Eq 10
€
where x in Equation
10 refers to either µο or µ and
ro =
1− 1− ϖ0
1+ 1− ϖ0
Eq 11
€
15
We use a single-lobed Henyey-Greenstein phase function (Henyey and
Greenstein, 1941):
p( φ ,g) =
1 − g2
(1 + g 2 + 2g cos φ ) 3 / 2
Eq 12
for the grain phase function, where φ is the phase angle and g is the single scattering
€ The asymmetry parameter, g, ranges from -1 (complete backward
asymmetry parameter.
scattering) to +1 (complete forward scattering). The Hapke model assumes that the
regolith grain size is much larger than the wavelength in the FUV. Under this assumption
the scattering efficiency, Qs = the regolith grain single scattering albedo. Qs is given by
Hapke (1993) as
Qs = Se + ( 1 − Se )
( 1 − Si ) Θ
1 − Si Θ
Eq 13
where the external scattering coefficient (Se) ad the internal scattering coefficient (Si) are
given by
€
2
(n − 1) + k 2 + 0.05
Se =
2
(n + 1) + k 2
4
Si = 1 −
2
n( n + 1)
Eq 14
where and n and k are the real and imaginary components of the complex index of
€
refraction. The internal-transmission
factor Θ is the total fraction of light entering the
particle that reaches another surface after one transit and is given by
Θ=
(
1 + r exp( −
)
α (α + s) L )
ri + exp − α (α + s) L
i
Eq 15
where
€
16
ri =
1 − α /(α + s)
1 + α /(α + s)
Eq 16
€
and
Eq 17
α = 4πk / λ
€
L is the photon mean path length and s is the internal scattering coefficient inside the
grain. We use a value of s = 1e-17 after Roush (1994). Figure 3 shows the complex index
of refraction for water ice in the FUV. Equations 9 through 17 provide a model for the
shape of the ring reflectivity spectrum that can be fit to the measured values of I/F with
two free parameters: the mean photon path length L, and the grain phase function
asymmetry parameter g. Equations 9-17 along with averages of I/F on either side of the
absorption feature are used in Equation 4 to determine Mscaled for the Hapke case.
3.3 Shkuratov model
Modeling Aw by the Shkuratov method uses the formulation from Shkuratov et al
(1999) and Poulet et al (2002). For the albedo of a particulate surface the Shkuratov et al
(1999) gives:
Aw =
1 + ρ b2 − ρ 2f
2ρ b
2
 1 + ρ b2 − ρ 2f 
− 
 −1
2ρ b


Eq 18
where
ρ b = q ⋅ rb
ρ f = q ⋅ rf + 1 − q
€
Eq 19
€
17
with q being the volume fraction filled by regolith grains and where rb and rf are the
fractions of light scattered by a grain into the backwards and forwards hemispheres,
respectively. rb and rf are given by
1
TeTi Ri exp(−2τ ) /(1 − Ri exp(−τ ))
2
1
r f = R f + TeTi exp(−τ ) + TeTi Ri exp(−2τ ) /(1 − Ri exp(−τ ))
2
rb = Rb +
Eq 20
where the€transmission and reflection coefficients are given by
Te = 1 − Re
Ti = 1 − Ri
R f = Re − Rb
Rb ≈ (0.28 ⋅ n − 0.20)Re
Re ≈ ro + 0.05
Eq 21
Ri ≈ 1.04 − 1/ n 2
€ indices of refraction are denoted by n and k, respectively, and τ =
The real and imaginary
4πkL/λ. Following the example of Poulet et al (2002), we define the asymmetry
parameter, g =
r f − rb
and the single scattering albedo, ω = rb + r f . Then for initial
r f + rb
values of rf and rb, we fix g and solve for new values of rf and rb and then proceed to
€
€ Ashk. In this manner g is a free variable and allows us to investigate both the
calculate
asymmetry and the photon mean path length as we did using the Hapke formulation.
Equations 18-21 along with averages of I/F on either side of the absorption feature are
used in Equation 4 to determine Mscaled for the Shkuratov case.
18
3.4 Determination of L and g
Figure 2 (upper panel) clearly shows that the absorption edge near 165 nm. The
wavelength and slope of the absorption edge depends on the values of L and g. We
generated a family of model curves from Equation 9 (Hapke model) and Equation 18
(Shkuratov model) over a range of L from 0.3-100 microns in 0.1 micron increments and
over a range of g from -1.0 to 1.0 in increments of 0.05 and scaled them in magnitude by
the procedure described in Section 3.1. After scaling the model to the data, we quantify
the agreement between model and data by the quantity
Eq 22
where N is the number of data points used. After calculating D for each model curve over
the range of L and g, the value of L and g corresponding to the minimum value of D gives
the mean photon path length and single scattering asymmetry parameter. The difference
between the normalized reflectance and the model fit from 150-190 nm is less than 10%
at a spectral resolution of 1.5 nm.
The values of L and g corresponding to the minimum value of D for observation 2
at r = 123,600 km were used to calculate Mscaled and plotted over I/F for that observation
(Figure 4). For the Hapke model L = 3.8 microns and g = -0.5 and for the Shkuratov
model L = 3.9 microns and g = -0.7. The fits from both models are similar to each other
for other ring plane radii. The uncertainty in of L and g are difficult to show by means of
error bars and may be irrelevant given that the uncertainty of how accurately the models
describe the physical properties of the rings is unknown. The uncertainty in of L and g is
best demonstrated by plotting additional model curves for values of L and g different
from the optimum fit values. Figure 5 (upper left panel) shows the variation in the
19
Shkuratov model curves for 3 g values plotted over I/F from observation 2 at r = 123,600
km. L is kept fixed at the optimum value of 3.9 microns while g is given the values -0.5, 0.7, and -0.9, with g = -0.7 being the retrieved g value for this observation. The upper
right panel shows the ratio of I/F to each of the model curves and demonstrates that in the
spectral region from 160-170 nm that the curve with g = -0.7 is closest to one, while g
values that are +/- 0.2 from the retrieved value are in error from 5-10%. The lower panels
of Figure 5 shows a similar analysis for variation in L vales with errors in the fit
compared to the model from 5-25% for L values that are +/- 2 microns from the retrieved
value of 3.9 microns.
4. Results and analysis
4.1 UV slope
The UV slope has been computed by taking the ratio of the average of I/F from
180-190 nm to the average of I/F from 150-160 nm and is shown in Figure 6. In order to
make a comparison between multiple observations, the higher spatial resolution data had
to be degraded to the resolution of the lowest spatial resolution data. For three of the
observations, C ring data are not shown because there is off- axis light contamination
from the dayside of the planet. The UV slope is an indicator of the fraction of
contaminant in the rings as discussed in Section 1. As the fraction of contaminant
increases, I/F for wavelengths greater than the absorption feature decreases in magnitude
while I/F for wavelengths shorter than the absorption feature either increase or decrease,
resulting in a lower value for the UV slope. Figure 6 shows that the outer B ring has the
highest fractional abundance of water ice, while the C ring is the most contaminated.
20
4.2 Retrieved mean path length and asymmetry parameter
Figure 7 shows the retrieved values of L and g for observation 2 for both the
Hapke and Shkuratov models. The values of L (upper panel) vary by over a factor of 2
from the C ring to the inner B ring. The curve is relatively flat from the mid B ring
through the outer A ring. For a radial bin, the retrieved L values from the Shkuratov and
Hapke models agree to within a few percent, with the Shkuratov model producing the
larger L values. The single scattering asymmetry parameter (lower panel) shows that the
rings are strongly backscattering, with the C ring being the most backscattering. In order
to test the g retrieval technique on other data sets, we performed a fit to I/F in the FUV
taken by the UVIS of Enceladus at a low phase angle using the Hapke model approach
and obtained a g value from -0.6 to -0.7 depending on assumed values of the
backscattering parameters. The agreement between the two models is not as good for g as
it was for L. The g values retuned from the Hapke model are consistently higher than for
the Shkuratov model, with both displaying similar trends from the C ring through the
Cassini division. However, the g values from the Hapke model show a slight increase
from the inner to the outer A ring whereas the g values from the Shkuratov model show a
decrease across the same region. Furthermore, there does not appear to be a correlation
between L and g across the rings. Figure 8 shows a scatter plot of L and g to the UV
slope. The linear Pearson correlation coefficient was computed to determine if there is a
correlation between retrieved values of L and g with the UV slope. The correlation
coefficients for L with the UV slope are 0.178 and 0.087 for the Hapke and Shkuratov
models, respectively. The correlation coefficients for g and the UV slope are 0.276 and
0.437 for the Hapke and Shkuratov models, respectively, where the first four data points
21
were excluded since they all had the same value. This implies that the photon mean path
length is not dependent on the UV slope; however, the asymmetry is strongly correlated
with UV slope.
5. Disscusion
Our analysis of low-phase FUV reflectance spectra of Saturn’s rings demonstrates
that the reflectance of the rings in the FUV from 175-190 nm (longward of the water ice
absorption feature near 165 nm) increases from the inner C ring through the outer edge of
the B ring before dropping in the Cassini Division and then peaking again in the central A
ring. Shortward of the absorption edge from 140-160 nm there is a similar trend of I/F
with radius, but with much smaller amplitude variations. Although we cannot resolve
much of the structure in the rings, I/F in the FUV does not generally track with ring
optical depth except in the optically thin regions. Considering I/F averaged from 175-190
nm, we find, for example, a much larger value of I/F in the outer B ring (B4) than the
central B ring (B2-B3) where the ring is nearly opaque, and we find the largest values of
I/F in the central A ring where the optical depth for that ring is smallest. I/F across the B
ring varies by more than a factor of two from the inner B ring to the outer B ring, whereas
in the visible regime (Estrada and Cuzzi, 1996, Estrada et al. 2003, Cuzzi et al, 2002) the
variation is less than 1.5. Furthermore the FUV I/F in the outer B ring is slightly less than
a factor of two larger than I/F from the Cassini Division whereas in the visible regime the
ratio of the outer B ring to Cassini Division I/F is greater than 3. Visible and infrared data
presented in Cuzzi et at., 2002 show the same overall trend of I/F with radius but with
much less variation. For instance, at a wavelength of 555 nm, Cuzzi et al. show I/F to go
22
from about 0.5 in the inner B ring to 0.65 in the outer B ring, whereas for the FUV I/F
averaged from 175-190 goes from 0.06-0.12 over the same radial range. Thus the rings
become less red from the inner to the outer B ring. The opposite is true from the inner A
ring through the outer A ring where the rings become more red with increasing radius.
We have ignored the effects of self-gravity wakes for this analysis, though the azimuthal
brightness asymmetry is visible in the A ring and outer B ring in our data. However, we
have azimuthally averaged our data, minimizing the effects of this azimuthal asymmetry
since it has a smaller amplitude than the radial variations shown here.
Results from the UV slope show that the Cassini division is more similar to the
inner B ring and outer A ring than the outer B ring and Inner A ring. This differs from the
slopes obtained by Estrada et al (2003) where the slopes in the inner B ring and outer A
ring were larger than that in the outer B ring and inner A ring, respectively. However,
Estrada et al studied a different spectral regime. Additionally the spatial resolution of the
FUV data covered the entire Cassini Division, whereas the data presented by Estrada et al
(2003) were able to resolve the Cassini Division and showed that the outer Cassini
Division was comparable to the A ring and the inner Cassini Division was comparable to
the C ring. The UV slope follows the same radial trend as I/F, which seems consistent
with the idea that contaminants decrease I/F at wavelengths longer than the absorption
edge and have only a small effect on I/F for wavelengths shorter than the absorption
edge. The results presented here show the outer B ring to be the least contaminated and
becoming more contaminated towards the inner B ring and outer A ring. The Cassini
division shows the same level of contamination as the inner B ring and outer A ring. The
C ring shows the highest degree of contamination; however the off axis light associated
23
with the C ring may actually be making the level of contaminant seem less than it really
is under the assumption that off axis light scattered from Saturn may be more intense at
the longer wavelengths than at the shorter wavelengths.
The fits from the Hapke and Shkuratov models to I/F were almost identical. The
upper left panel of Figure 5 shows that as g increases (that is, as the grains go from being
highly backscattering at g = -0.9 to g = -0.5) the steepness of the water ice absorption
edge decreases with only minimal affect on the spectral location of the absorption edge.
The lower left panel of Figure 5 shows that adjusting the mean photon path length, L,
does not affect the steepness of the absorption edge but and increase in L increases the
wavelength of the absorption edge. The steepness of the absorption is an independent
constraint on the particle scattering properties while the spectral location of the
absorption edge constrains the photon mean path length. The Hapke and Shkuratov
models gave similar results for retrieved L values, however the g values were somewhat
different. One possible explanation for this is that for the Hapke model the g value
manifests itself only in the single scattering term in the Hapke formulation, whereas it
applies to all orders of scattering in the Shkuratov model.
We have retrieved information concerning the directionality of scattering through
the single scattering asymmetry parameter, g. Across all of the rings g was negative,
implying that backscattering from regolith grains dominates at FUV wavelengths. The g
values retrieved from the Hapke model showed a gradual increase towards less negative
values (-0.85 to -0.45) from the C ring to the outer A ring. The g values retrieved from
the Shkuratov model showed a similar overall trend from the C ring to the inner A ring
(-1 to -0.75) but then dramatically decreased to ~ -0.9 for the outer A ring. These values
24
are more negative than results obtained for icy moons. For instance Verbiscer and
Veverka (1994) obtained a g value = -0.399 +/- 0.005 for Enceladus using Hapke’s
(1986) photometric equation. This result was based on data at much longer wavelengths
than ours. The discrepancy between the g values of the rings and g values obtained by
others for icy moons may be attributable to both wavelength dependencies and
differences in ice-contaminant grain structure. If particles are “irregular” to the nm
wavelength scale, as might be true for airless bodies at temperatures much less than the
melting point of water (i.e., not terrestrial snows and frosts which are constantly
annealed) then the more negative g’s might be physically understood. Longer
wavelengths may not “sense” the fine structure of irregularly shaped un-annealed grains
that are retrieved by the FUV analysis presented here.
We have retrieved a measure of the amount of water ice that UV photons traverse
before leaving the rings from the location of the absorption edge in the data. This mean
photon path length, L, represents the total path through water ice on the surface of a ring
particle. It is not a measure of the size of a ring particle or regolith grain, but is related to
the distance between scattering centers within those regolith grains. These could be the
far side of a small grain, cracks or other flaws in the ice, or non-ice contaminants within
the grain. The retrieved value of L is relatively uniform from the mid B ring to outer A
ring, while the minimum occurs in the inner C ring. One possible explanation is that the
mean distance photons travel could be highly dependent on the level of contaminant, with
more contaminant implying a shorter path length. A gray contaminant added to the water
ice model would serve to reduce the albedo at wavelengths longer than the water ice
absorption feature and slightly increase or decrease the albedo at wavelengths shorter
25
than the absorption feature; however, this would not change the spectral location of the
absorption edge. A non-gray contaminant would cause a shift in the absorption edge and
will be considered in future work.
Four radial trends are characterized in this investigation: I/F, UV slope, g values
and L values. The lack of correlation of L with the UV slope seems to imply that L
depends on regolith grain characteristics other than contamination. One possibility is that
photons scattered in the direction of the instrument did not encounter contaminants while
photons that do encounter contaminants are absorbed. The same conclusion may be stated
concerning the retrieved g values, which are only moderately correlated with the UV
slope. If the photons reaching the instrument are indeed those photons that did not
encounter contamination, then the retrieved g values are directly related to the structure
of the ice grains on the ring particles. While pure water ice should be transparent to
photons of wavelengths longer than the FUV absorption edge, the presence of internal
facets or edges of grains in a complex structure of ice regolith would result in scattering
incident photons. Other outer solar system ices have been shown to be backscattering
(Verbiscer, 1994), although the values in this investigation show a larger degree of
backscattering. This may be due to both wavelength differences and structure of the ring
particle regolith ice grains.
26
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28
Figure 1. Optical depth profile of the C through A rings measured with the Cassini UVIS
High Speed Photometer (HSP) observing the β Centauri Rev 077 ingress occultation.
Optical depths were calculated following the procedure in Colwell et al. (2007). The
vertical arrows show the boundaries of the bins between which FUV spectra were
averaged after the spectra had been re-sampled to an evenly distributed grid. Bin
boundaries were chosen to match regional boundaries in the rings where possible
(Colwell et al. 2009). The four main regions of the B ring are labeled.
29
Figure 2. I/F from observation 2 for 4 regions in the rings (upper panel). The center radial
positions are 86000, 107250, 119965, and 127950 km for the C ring, B ring, Cassini
Division, and A ring, respectively. The lower panel shows the average I/F from 1400160 nm and 175-190 nm plotted against ring plane radius. The small error bars represent
the
1-σ uncertainty in the photon counting statistics and the rtg background that have
been propagated through to I/F.
30
Figure 3. Complex index of refraction for hexagonal water ice taken from Warren and
Brandt (2008). The real part (top panel) is shown on a linear plot while the imaginary part
(lower panel) is shown as the log of the value. The absorption of water ice is determined
by the imaginary part of the index of refraction. The strong decrease of k longward of
150 nm indicates that water ice becomes less absorbing for longer wavelengths and is
thus the basis of the technique to determine the characteristics of water ice from the FUV
reflectance.
31
Figure 4. Example of fit of Hapke model (blue curve) and Shkuratov model (red curve) to
I/F from observation 2 for r = 123,600 km. For the Hapke model L = 3.8 microns and g =
-0.5 and for the Shkuratov model L = 3.9 microns and g = -0.7, which were retrieved
from the minimum value of D in Equation 22. The difference between the model and I/F
was less than 10% at 1.5 nm resolution. Data below 153 nm were not modeled due to the
additional structure in the spectrum that is not included in our Hapke model.
32
Figure 5. Shkuratov model curves compared to I/F for observation 2 at a ring plane radius
= 123,600 km. The red curve in upper left and lower left panels is the optimum fit from
Figure 4 with L = 3.9 microns and g = -0.7. For the upper left panel, additional model
curves are plotted where L is kept fixed at 3.9 microns and g is given values of -0.9 and 0.5. Varying g values alter the slope of the model in the fall off region. The upper right
panel is the ratio of I/f to each of the model curves from the upper left panel. The region
from 160-170 nm shows a ~5-10% error for g that is 0.2 larger or smaller than the
optimum value. For the lower left panel, additional model curves are plotted where g is
fixed at -0.7 and L is given values of 1.9 and 5.9 microns and demonstrates the spectral
variation of the absorption edge with L. The lower right panel shows a ~ 5-25% error for
L that is 2 microns different from the optimum value.
33
Figure 6. UV slope for four different observations. The ratio is obtained by dividing the
average I/F from 175-190 nm by the average I/F from 140-160 nm. Off-axis light
contaminated the signal from the C ring, with observation 2 containing the least amount
of contamination, thus being the only observation shown for the C ring. The field of view
of each pixel projected in the ring plane for observations 3 and 4 was from 2-3 times
larger than for observations 1 and 2. For the purpose of comparison between observations
the spatial quality of observations 1,2, and 3 was degraded.
34
Figure 7. Retrieved value of L (upper panel) and g (lower panel) from observation 2. L is
relatively flat from the mid B ring through the outer A ring within the uncertainties
established in the analysis shown in Figure 5. The regolith grains in the ring particles are
strongly backscattering in the FUV, but less so in the inner and central A ring and outer B
ring where FUV reflectance and UV slope are largest.
35
Figure 8. Scatter plot of g (upper panel) and L (lower panel) to the UV slope from
observation 2. The retrieved value of L is independent of the radiance. The linear Pearson
correlation coefficient was computed to determine if there is a correlation between
retrieved values of L and g with the UV slope. The correlation coefficients for g with the
UV slope are 0.276 and 0.437 for the Hapke and Shkuratov models, respectively, where
the first four data points are excluded since they are all the same value. The correlation
coefficients for L with the UV slope are 0.178 and 0.087 for the Hapke and Shkuratov
models, respectively.
36