Icarus 225 (2013) 726–739
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Icarus
journal homepage: www.elsevier.com/locate/icarus
Scattering properties of Saturn’s rings in the far ultraviolet from Cassini
UVIS spectra
E. Todd Bradley a,⇑, Joshua E. Colwell a, Larry W. Esposito b
a
b
Department of Physics, University of Central Florida, 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
Article history:
Received 13 August 2012
Revised 22 February 2013
Accepted 7 April 2013
Available online 27 April 2013
Keywords:
Saturn, Rings
Ices
Ultraviolet observations
Ices, UV spectroscopy
a b s t r a c t
We use Cassini UVIS data to determine the scattering properties of Saturn’s ring particles in the FUV. We
have replaced the scattering function from the classical Chandrasekhar single scattering radiative transfer
equation for reflectance with a ring wake model for the A and B rings derived from stellar occultations.
The free parameters in this model are the ring particle Bond albedo, AB, and the ring particle asymmetry
parameter, g, which equals the cosine of the most probable scattering angle of a photon from a ring particle. The spectrum of Saturn’s rings from 140 to 190 nm shows an absorption feature due to water ice
shortward of 165 nm. We compare our model values for I/F to lit-side data at 155 nm and at 180 nm
for regions in both the A and B rings. We used the unmodified Chandrasekhar model for the C ring
and Cassini Division, and in all cases we determined AB and g in the FUV for the first time. Values of AB
vary between 0.04 and 0.091 at 180 nm and between 0.012 and 0.019 at 155 nm. The variations across
the ring of AB at 180 nm is consistent with a greater abundance of non-ice contaminant in the C ring
and Cassini Division and a minimum in contaminant abundance in the outer B ring. There is little variation in AB at 155 nm across the rings, which suggests that the reflectance of the water ice and non-water
ice material shortward of the 165 nm absorption edge are about the same. Values of g vary between
0.68 and 0.78 at 180 nm and between 0.63 and 0.77 at 155 nm showing that the ring particles
are highly backscattering in the FUV. We find that the wavelength of the absorption feature varies with
ring region and viewing geometry indicating a different photon mean path length, L, through the outer
layer of the ring particle (Bradley, E.T., Colwell, J.E., Esposito, L.W., Cuzzi, J.N., Tollerud, H., Chambers, L.
[2010]. Icarus 206 (2), 458–466). We compared I/F from 152 to 185 nm to a radiative transfer spectral
model developed by Shkuratov et al. (Shkuratov, Y., Starukhina, L., Hoffmann, H., Arnold, G. [1999]. Icarus
137, 235–246) and modified by Poulet et al. (Poulet, F., Cuzzi, J.N., Cruikshank, D.P., Roush, T., Dalle Ore,
C.M. [2002]. Icarus 160, 313–324). We find that L is positively correlated with phase angle, which we
attribute to multiple scattering within the particle on length scales comparable to L. We extrapolate L
to zero phase angle and find values of L at zero phase ranging from 2 to 3 lm. This provides a direct
measure of the distance from the surface of a ring particle to the first scattering center. L at zero phase
is roughly constant across the rings suggesting the outermost 1.25 lm of the ring particles have the same
structural properties in all ring regions. We azimuthally binned and interpolated observations of the unlit
side of the A ring taken during Saturn orbit insertion to a 100 km resolution radial profile. We see halos
(enhanced brightness) surrounding the Janus 4:3 and Janus 5:4 density waves. We also computed I/F
across the A ring using the SOI observational geometry along with AB and the power-law index, n, derived
from the retrieval approach from lit side observations. I/F determined by this technique agrees with
results from the lit side analysis for the A2 ring but diverge for the inner and outer A ring, which we attribute to multiple scattering effects.
Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction
The Cassini Ultraviolet Imaging Spectrograph (UVIS) has obtained spectra of Saturn’s rings at a range of viewing and illumina⇑ Corresponding author.
E-mail address: tbradley@physics.ucf.edu (E.T. Bradley).
0019-1035/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.icarus.2013.04.008
tion angles over the course of the Cassini nominal and equinox
missions. These data provide the first spatially resolved information on the spectral and photometric properties of Saturn’s rings
in the far ultraviolet (FUV). FUV reflectance spectra of Saturn’s
main rings contain an absorption feature shortward of 165 nm that
is characteristic of water ice, first observed in Saturn’s rings in 12
IUE observations from 1982 to 1985 (Wagener and Caldwell,
E.T. Bradley et al. / Icarus 225 (2013) 726–739
1988). Analysis of the brightness of the rings longward or shortward of the absorption edge provides information on the ring particle Bond albedo, AB, and the ring particle phase function. The
angular distribution of photons scattered from the ring particles
is characterized by the asymmetry parameter, g, which is the cosine of the average scattering angle of photons from a particle
(Henyey and Greenstein, 1941). In addition, the spectral location
of the absorption feature provides information on the mean distance that a photon travels within a ring particle before exiting,
which we quantitatively describe by the photon mean path length,
L. AB is the fraction of incident photons scattered from a body back
into space. Each of the constituents that comprises a ring particle
has its own reflectance properties making AB dependent on the relative abundance and composition of the constituents. Epstein et al.
(1984) estimated the fractional abundance of water ice in the rings
to be greater than 90% from microwave observations with the
remaining few percent comprised of a contaminant that is primordial and/or due to meteoroid bombardment (Cuzzi and Estrada,
1998). Constraining AB is crucial to identifying the composition of
the non-water ice material comprising the particle. The abundance
of non-water ice material throughout the rings helps to constrain
evolution models. Results from Elliott and Esposito (2011) suggest
that darker rings are either less massive or exposed longer to meteoritic bombardment, and probably both for the C ring.
Outer Solar System icy bodies have been shown to have a negative value of g, implying backscattering behavior (Verbiscer et al.,
1990). This is in contrast to terrestrial water ice, which typically
exhibits forward scattering behavior. According to Verbiscer et al.
this is possibly due to more complex internal textures of outer Solar System ices due to low-temperatures and low-gravity conditions. Values of L in the FUV spectral regime have already been
shown to be less than 10 lm (Bradley et al., 2010). Longer and
shorter wavelengths may probe to deeper and more shallow
depths, respectively, simply because shorter wavelengths in order
not to be absorbed must be reflected from a shorter distance within the ice grains. These turn out to be the photons that are observed
by the instrument.
AB and g have been determined for visible and infrared spectral
regimes by other researchers. Porco et al. (2005) showed AB for a
broad range of radial distances covering the main rings at discrete
wavelength intervals from 338 to 862 nm. For all ring regions AB
decreased with decreasing wavelength with minimum values of
0.2–0.4 at 338 nm. Porco et al. (2005) used a Monte Carlo ray tracing model incorporating self-gravity wakes to model images obtained by the Cassini Imaging Science System (ISS). For the C ring
and outer A ring they found AB = 0.32 and AB = 0.63, respectively.
Using a power-law phase function, Porco et al. obtained powerlaw indices equal to 2.624 and 3.092 for the outer A and C rings,
respectively. The power-law phase function formulation presented
in Dones et al. (1993) describes the relation between the powerlaw index, n, and the asymmetry parameter, g and has the following form:
P ¼ C n ðp aÞn
Z
1 p
g¼
PðaÞ cos a sin ada
2 0
727
scattered light. The degree of scattering depends on the viewing
geometry and is not an intrinsic property of individual ring particles. The geometry of a particular observation, i.e., the solar elevation angle (Bo), observer elevation angle (B), and phase angle (a),
affects the observed brightness and must be accounted for before
the absolute value of AB can be retrieved. The effect that the observational geometry and scattering behavior have on the brightness
of an atmosphere or body is characteristic of any atmosphere or
surface where an instrument measures solar radiation scattered
by the body. Radiative transfer equations that account for these effects have been derived for both classical atmospheres (Chandrasekhar, 1960) and surface reflectance models for bodies such as the
Moon or icy satellites where the atmosphere, if any, is negligible
(Hapke, 1981, 1993; Shkuratov et al., 1999). Saturn’s rings do not
typically behave like a classical atmosphere or a solid surface, making it difficult to interpret the meaning of absolute values of the
reflectance. This is caused in part by the self-gravity wake structure of the A and B rings (Colwell et al., 2006, 2007; Hedman
et al., 2007) as well as by volume filling fractions greater than zero
(a classical atmosphere) but less than one (a solid surface that is
extremely tightly packed). Even the use of a radiative transfer code
that is suitable for an optically thick atmosphere e.g., doubling
adding, where multiple scattering is carefully modeled, is inappropriate for application to the rings since the volume filling factor is
not negligible.
To retrieve AB and g we have employed a single scattering analytical technique for dealing with the scattering properties of the
rings that is based on a self-gravity wake model derived from stellar occultation data (Colwell et al., 2006, 2007). Because the selfgravity wakes are non-axisymmetric, the optical depth of the rings
to incoming sunlight is not necessarily the same as the optical
depth for the emerging rays observed by Cassini, even after taking
into account the incidence and elevation angles. Therefore we use a
modified Chandrasekhar model with the two optical depths treated independently. We assume, as did Cuzzi and Estrada (1998),
that a ring particle is a centimeter to meter-sized object that is covered by a regolith of icy grains. Esposito (1979) showed that mutual shadowing between ring particles has only a small effect on
multiply scattered photons. Furthermore, Cuzzi et al. (1984) argued that in the backscattering, low ring tilt angle geometry occurring during Voyager observations, I/F is well approximated by
neglecting multiple scattering. Since this analysis deals with backscattered light with relatively low ring tilt angle, with the exception of Saturn orbit insertion data presented in Section 4, we
neglect multiple scattering between ring particles; however multiple scattering between regolith ice grains covering a ring particle
must not be neglected. Below we refer to multiple scattered light
as light that is scattered within the regolith of a ring particle and
not to scattering between ring particles. Present within the regolith
are non-ice contaminants that are intermixed with the ice grain
although we do not make an attempt to retrieve the composition
or abundance of the non-ice component. The classical Chandrasekhar model for singly-scattered light reflected from the rings is given by:
ð1Þ
where a is the phase angle, Cn is a normalization constant, n is the
power-law index, and g is the asymmetry parameter. Specifying n
uniquely defines the asymmetry parameter, g. Using this approach
the values of n obtained by Porco et al. translate to g values of
0.5 and 0.55 for the A and C rings, respectively.
Retrieval of AB, g, and L requires that the scattering properties of
the rings be understood and modeled correctly. For instance, since
L is a measure of the amount of ice a photon travels through, then
multiply-scattered light will have a larger value of L than singly-
I
lo
¼ AB P ½1 expðsn =lÞ expðsn =lo Þ
F
4ðl þ lo Þ
ð2Þ
where AB is the ring particle Bond albedo, P is the ring particle phase
function, l and lo are the cosines of the emission and incidence angles, respectively, and sn is the normal optical depth. AB, is the number of photons that exit the ring particle divided by the number of
incident photons that interact with the ring particle. This implies
that both singly and multiply-scattered photons may contribute
to the total number of photons that exit the particle. The problem
with using this model for the rings is that the rings contain wake
728
E.T. Bradley et al. / Icarus 225 (2013) 726–739
structure (for the A and B rings) and non-zero volume filling, rendering the problem different than for a classical atmosphere. Specifically, the optical depth terms, which describe the scattering
behavior of the medium, relate the photon scattering to the normal
optical depth. While scattering from the isolated particles within
the gaps between wakes could be modeled in this manner, the
wakes are opaque structures with a preferential orientation. Thus
light scattering from a region of the rings that encompasses both
wakes and gaps presents a complex problem of light scattering.
We replace the optical depth terms in the classical Chandrasekhar equation with terms derived from the granola-bar self-gravity
wake model of stellar occultation data by Colwell et al. (2006,
2007). The reasoning behind this is as follows: the optical depth
terms in the classical equation account for attenuation of photons
due to atmospheric species, which may be considered to have a
zero volume filling factor and are uniformly distributed. The normal optical depth, sn, is the absorption coefficient multiplied by
unit length. Due to the presence of self-gravity wakes in the A
and B rings, a simple relation between optical depth and an absorption coefficient is not possible. However stellar occultation experiments directly measure the optical depth of the rings at
different radial distances. These optical depths depend on the
width, W, separation, S, and height, H, of the self-gravity wakes,
the optical depth of the inter-wake gaps, sgap, as well as the azimuthal angle relative to the wakes (/ /wake) and line-of-sight angle to the ring plane (B). The simple ‘‘granola bar’’ model of Colwell
et al. (2006, 2007) successfully explains the variation of optical
depth with B and / so that optical depths at arbitrary viewing
geometries can be calculated. The transparency of the rings in
terms of the wake parameters is given by (Colwell et al. (2007)):
T
¼ expðsn =lÞ
½S=W H=Wj sinð/ /wake Þj cot B
expðsgap =lÞ
S=W þ 1
¼ expðsn =lo Þ
¼
To
¼
ð3Þ
½S=W H=Wj sinð/o /wake Þj cot Bo expðsgap =lo Þ
S=W þ 1
In the C ring and Cassini Division the observed normal optical depth
is insensitive to viewing geometry, and we use the unmodified
Chandrasekhar model in those regions. The exponentials in Eq. (2)
represent the transmission of the incoming solar flux and the outgoing scattered flux. For a homogeneous atmosphere these are each
described simply by the normal optical depth of the atmosphere
and a single angle specifying the slant path through the atmosphere. For rings with self-gravity wakes, however, there are additional geometric factors that determine the transmission. We use
Colwell et al. (2007) to get Eq. (3). Then putting this in Eq. (2)
we have the following modified Chandrasekhar granola bar model
for I/F:
I
lo
½S=W H=Wj sinð/ /wake Þj cot B
expðsgap =lÞ
¼ AB P 1
F
S=W þ 1
4ðl þ lo Þ
½S=W H=Wj sinð/o /wake Þj cot Bo ð4Þ
expðsgap =lo Þ
S=W þ 1
with P given by Eq. (1).
Table 1
Ring regions.
Region
Lower ring plane radius (km)
Upper ring plane radius (km)
C4
B1
B2
B3
B4
CD2
A1
A2
A3
A4
88,000
92,000
99,000
104,500
110,000
120,000
122,000
125,400
131,000
133,800
92,000
99,000
104,500
110,000
117,550
122,000
125,200
130,500
133,300
136,700
Table 2
Observation list.
Observation
Ring
coverage
a
Bo
B
UVIS 000RI RINGSCANA001 SP
UVIS_007RI_LATPHASE001_VIMS
UVIS_007RB_VERTLMP002_CIRS
UVIS_008RI_SUBML20LP002_CIRS
UVIS_008RI_FP34INTEG001_CIRS
UVIS_013RB_BMOVIE1001_VIMS
UVIS_013RB_BMOVIE1002_VIMS
UVIS_013RI_LATPHASE002_VIMS
UVIS_014RI_LATPHASE003_VIMS
UVIS_014RI_SUBML15LP001_CIRS
UVIS_029RI_AZSCAN001_ISS
UVIS_035RI_SUBML25MP001_CIRS
UVIS_036RB_SPOKE001_VIMS
UVIS_036RI_SUBML17LP001_CIRS
UVIS_037RI_IMPACT013_CIRS
UVIS_037RI_IMPACT014_CIRS
UVIS_037RI_RPXMOVIE001_VIMS
UVIS_039RI_SUBML59MP001_CIRS
UVIS_040RI_TEMPL40LP001_CIRS
UVIS_041RB_SPOKE001_VIMS
UVIS_041RC_VERTLLP001_CIRS
UVIS_060RI_SUBML10LP003_CIRS
UVIS_062RI_VERTLLP001_CIRS
UVIS_064RB_BMOVIE001_VIMS
UVIS_065RI_SUBML39LP001_CIRS
UVIS_066RI_LATPHASE001_VIMS
UVIS_071RI_SPOKEMOV001_ISS
UVIS_074RI_SHADLMP001_CIRS
UVIS_074RI_SUBML25LP002_CIRS
UVIS_075RB_BMOVIE4001_VIMS
UVIS_075RI_LATPHASE001_VIMS
UVIS_077RI_SHADLMP001_CIRS
UVIS_077RI_SUBML24MP001_CIRS
UVIS_082RI_TEMPS20LP001_CIRS
UVIS_085RI_VCASLSLP001_CIRS
UVIS_089RI_SUBMS45LP001_CIRS
UVIS_089RI_APOMOSL107_VIMS
UVIS_090RI_SUBMS10LP001_CIRS
UVIS_096RI_SUBMS10LP001_CIRS
UVIS_102RI_LATPHASE001_VIMS
UVIS_102RI_SPKFMLFLP001_ISS
A
A, B
B
A, B, C, CD
B
A, B, CD
A, B, CD
A, B, CD, C
A, B
A, B, C, CD
A
B
B
A, B, C, CD
B
A, B
A, CD
A, B
A, B, CD
B
B, C
B, C
A, B
A, B, CD
B
B, C
A, B, CD
B
A, B, C, CD
B, CD
B
A,
A, B, C, CD
B, C, CD
A
A, B
A, B, C
A, B
B
B
A, B, CD
59.2
32.8
38.2
9.29
74.8
34.4
25.8
21.67
36.7
6.05
54.9
77.5
31.6
11.3
29.1
23.2
33.2
57.4
30.5
61.3
21.2
21.5
34.3
18.1
51.0
12.6
18.8
37.8
33.8
27.9
14.0
35.5
36.4
26.6
53.2
55.8
23.3
20.9
23.8
58.7
37.9
24.5
21.9
21.8
21.6
21.6
20.7
20.7
20.7
20.5
20.5
15.7
14.6
14.4
14.4
14.2
14.2
14.1
13.7
13.4
13.2
13.2
8.09
7.77
7.48
7.34
7.18
6.58
6.27
6.26
6.15
6.14
5.94
5.93
5.37
5.04
4.58
4.57
4.45
3.72
2.89
2.87
27.1
20.1
19.8
22.2
10.1
20.6
21.8
20.9
15.1
16.2
62.1
56.8
45.9
13.2
43.2
36.4
7.43
58.8
42.1
49.2
32.1
5.15
18.1
17.3
35.4
6.26
18.1
40.3
36.4
29.0
7.39
38.7
22.9
23.6
40.1
56.2
15.1
11.4
11.5
49.5
25.8
Note: The date and UCT time of the observation is within the name of each observation file. The solar elevation angles and the observer elevation angles are measured with respect to the ring plane where positive and negative values are on the
northern and southern hemisphere sides, respectively. All angles are in degrees.
2. Observations
A complete description of the UVIS instrument is given by
Esposito et al. (2004). The observations processed in Section 3 were
selected for their high signal-to-noise and spatial resolution (less
than 4000 km per pixel) and are all taken from the sunlit side of
the rings between 2005 and 2009 with a range of Bo from 3° to
22°. In this investigation we have selected regions from each of
the A, B, and C rings as well as the Cassini Division. The selection
was based on regions with relatively little variation in normal optical depth throughout the radial extent of the region. Table 1 shows
the lower and upper ring plane radii of the regions, and Table 2
lists the observations used in this investigation along with coverage and relevant geometrical information. Regions investigated in
this analysis begin with the C4 region. The inner three regions of
the C ring (C1–C3) are not discussed here due to large amounts
of off-axis light and Saturn-shine in the signal. Similarly, data from
E.T. Bradley et al. / Icarus 225 (2013) 726–739
729
Fig. 1. I/F (averaged from 175–185 nm) with error bars. I/F depends strongly on a, Bo, and /. The dependence on a is responsible for the overall decline of I/F with increasing a
while the dependence on Bo and / is responsible for variations in I/F at a particular a.
the inner Cassini Division are contaminated by light from the outer
B ring.
Data reduction consists of the same technique described in
detail in Bradley et al. (2010) and summarized here. We subtract
the background counts from the raw count rate and calculate the
radiance (I) in kilo-Rayleighs using an instrument calibration factor. Last we compute I/F by using the solar continuum flux (pF)
from the SOLSTICE instrument on board the SORCE spacecraft
(Sparn et al., 2005) that has been corrected for the rotation of
the Sun to correspond to the face of the Sun towards Saturn at
the time of the observation. We resample the UVIS data to the
bandpass of SORCE SOLSTICE II (see Esposito et al., 2004;
McClintock et al., 2005 for discussion on the spectral bandpass
of each instrument), divide by p to get the radiance of a Lambertian surface and then divide this into the radiance, producing I/F.
Uneven sampling in both the radial and azimuthal directions requires data to be azimuthally binned and then resampled to an
evenly distributed one-dimensional radial array. We bin the data
azimuthally over 30° increments to insure that the observer azimuth angle does not vary greatly for a set of data points. Resampling the data to an evenly distributed array also insures that
the average data over a radial bin is not weighted towards one
particular sub-region within the bin due to larger numbers of
pixels occurring within that sub-region. After this the evenly distributed grid of data points were averaged across each of the regions for this investigation.
Fig. 1 shows I/F versus phase angle for the C4, B3, CD2, and A3
ring regions and is color-coded according to Bo, which varies from
1° to 22°. Variations in I/F are dominated by changes in phase angle where I/F is inversely correlated to phase angle. A less dominant effect is the positive correlation of I/F with B’. Dependency
on B was checked and found to be negligible. Variations in I/F with
/ and /wake may be significant due to non-axisymmetric self-gravity wakes as discussed in Section 1. We attach error bars to the data
points, which we calculate by taking the inverse of the square root
of the total raw counts at each data point. Assuming Poisson statistics this gives the fractional uncertainty, r. We then multiply r
determined at each data point by the value of I/F at each data point
resulting in the fractional uncertainty of I/F.
3. Results
Results are broken into two broad categories. The first is the
retrieval of AB and g by comparison of the Chandrasekhar-granola bar model to discretely averaged values of I/F. We averaged
I/F from 152 to 158 nm and from 175 to 185 nm and compared
the averages to the model. The second is the retrieval of L by
comparison of the Shkuratov et al. (1999) spectral model to I/F
from 152 to 185 nm. This step must come second since the retrieved values of g from the comparison of the I/F to the Chandrasekhar-granola bar model are used as input to the
Shkuratov model. We scaled the magnitude of the Shkuratov
model to the value of I/F above and below the absorption edge.
We chose the value of L that resulted in the best fit of the model
to the data as the retrieved value of L.
730
E.T. Bradley et al. / Icarus 225 (2013) 726–739
Fig. 2. I/F data points (black stars) for the four B ring regions and the model (red diamonds) using the retrieved parameters AB and g. (For interpretation of the references to
color in this figure legend, the reader is referred to the web version of this article.)
3.1. Retrieval of AB and g
The free parameters within the Chandrasekhar-granola bar
model given by Eq. (4) are AB, and the power-law index, n, which
is converted to the ring particle asymmetry parameter, g, through
Eq. (1). We computed I/F for each data point within a ring region
for a specified value of AB and n. We varied AB between 0.005
and 0.4 in 0.005 increments and n between 0.25 and 20 in increments of 0.25, which corresponds to g values from 0.07 to
0.95. Values outside of the above limits consistently result in
worse fits of the model to the data. Best-fit values of AB and g were
found by minimizing D where
D¼
N
1X
ðmi di Þ2 ;
N i¼1
ð5Þ
N is the number of data points within a ring region, di is the measured I/F of measurement i, and mi is the model value of I/F.
The eight panels of Fig. 2 show I/F at 155 nm and 180 nm versus
phase angle for the four B ring regions. Phase angles for observations used in the analysis of the B ring varied between 5° and
75°. The variation with Bo is clearly present in all regions of the B
ring for both the short and long wavelength spectral regimes.
The eight panels of Fig. 3 show I/F and model results for the four
A ring regions. For the A ring the phase angles for available observations used in this analysis varied between 5° and 60°. Brightness dependency on Bo is not as prevalent in the A ring as for the
B ring. The four panels of Fig. 4 show I/F and model results for
the C4 and for the CD2 regions. The observations available for
the Cassini Division spanned the least phase angle coverage in this
analysis from 5° to 40° while those from the C ring varied between 5° and 70°. The brightness dependency on Bo that is seen
in the B and A rings is minimal in both the C ring and Cassini
Division.
Fig. 5 shows contours of D corresponding to four regions in the
rings. For consistency, values of the inner contour for each panel
E.T. Bradley et al. / Icarus 225 (2013) 726–739
731
Fig. 3. I/F data points (black stars) for the four A ring regions and the model (red diamonds). Best-fit values of AB and g for each data set are given on the corresponding panel.
(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
were chosen to be twice the minimum value of D for that region,
which offered a reasonable visual estimate of the uncertainties in
the determination of AB and g across all regions of the rings and
for the two wavelengths investigated. The broader minima of the
D contours show that the model fit is less constrained for the short
wavelength reflectance than it is for the long wavelength reflectance. We attribute this to lower signal levels at the short wavelengths leading to increased uncertainty in the measured I/F. For
all rings regions the contours show a closed contour around a unique solution of AB and g with smaller uncertainty at 180 nm than
for 155 nm.
Figs. 6 and 7 show the retrieved values of AB and g, respectively,
versus ring plane radius. At 180 nm AB peaks at 0.091 in the B3 region, remaining nearly the same for the B2 and B4 regions. The A
ring regions as well as the B1 region have AB = 0.06–0.08 while
the C4 and Cassini Division regions have the darkest particles with
AB between 0.04 and 0.05. At 155 nm AB has less variation across
the rings with values between 0.012 and 0.019. This suggests that
at 180 nm AB is sensitive to variations in the non-water ice fractional abundance. However, the relatively smaller variation of AB
ratios at 155 nm suggests that at wavelengths below the absorption feature the reflectance of water ice and non-water ice components of a ring particle are comparable. Based on the quality of fit
shown by the chi-square analysis displayed in Fig. 5, we can estimate the typical errors of determination of the parameters AB of
±20% and g of ±30%, from the extent of the first contour around
the best fit. These values are consistently larger than the errors
associated purely with counting statistics, and give an empirical
estimate of the quality of our retrievals.
Fig. 7 shows the g values the rings to be highly backscattering
across the rings in the FUV spectral regime with values that vary
between 0.68 to 0.78 for 180 nm and from 0.63 to 0.77 at
155 nm. For the 180 nm values the C ring and Cassini Division values are the lowest, which could possibly be related to non-water
ice material in those regions; however there is not enough of a correlation between g and AB in other regions of the rings to argue for
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E.T. Bradley et al. / Icarus 225 (2013) 726–739
Fig. 4. The top two panels are the I/F and model results for the C4 region while the two bottom panels are for the CD2 region.
a relation between g values and non-water ice abundance. The
180 nm A ring g values increase with radius, which is consistent
with the analysis that Dones et al. (1993) found from analyzing
Voyager visible wavelength images.
3.2. Comparison of AB to other data sets
We compared AB from the A, B, C rings and Cassini Division to values from longer wavelength investigations. Fig. 8 is a composite that
includes four AB values retrieved in this investigation along with AB
near the same ring plane radii presented in Porco et al., 2005. The
data points from Porco et al. are at wavelengths of 338 nm,
451 nm, 568 nm, 650 nm, and 862 nm. AB for this investigation extend the spectral coverage into the FUV to wavelengths of 155 nm
and 180 nm. The Porco et al. data from different rings regions show
considerable spread at the longest wavelengths and converge towards similar values as the wavelength decreases. The FUV data
points appear to support this trend with AB from all of the ring’s regions converging to values from 0.01 to 0.02 at 155 nm
3.3. Retrieval of L
Fig. 9 shows I/F spectra taken from the B3 region at four different phase angles. The water ice absorption feature near 165 nm
shows a shift in the absorption edge to longer wavelengths for larger phase angle. Typically a spectral shift in the absorption edge is
coincident with a change in L, where an increase in L results in the
absorption edge shifting towards longer wavelengths. This is because photons traveling further in ice experience more absorption,
thus decreasing the value of I/F at longer wavelengths.
We fit a spectral albedo model (Shkuratov et al., 1999) to I/F
from 152 to 185 nm in order to retrieve L as a function of phase angle. L is the only free parameter in the Shkuratov model; however
Poulet et al. (2002) modified the Shkuratov model to also allow for
the ring particle asymmetry parameter, g, to be a free parameter.
This modified version of the Shkuratov model was used with retrieved g values from the previous section to infer the value of L.
We used water ice optical constants from Warren (1984) and Warren and Brandt (2008). The technique involved scaling the Shkuratov model in magnitude to the average of I/F at wavelengths above
and below the absorption edge from each observation and fitting
the model to the data over a range of L values from 0.1 to 10 lm
in 0.1 lm increments. Initially a much larger range of L values were
used, however it was found that values of L outside of 0.1–10 lm
resulted in progressively worse fits. The selected value of L was
found from taking the value of L corresponding to the minimum
value of D given by Eq. (4) where for this case mi and di are the
spectral components of the scaled Shkuratov model and I/F, respectively, from 152 to 185 nm. An example of the best fit for the Shkuratov model to I/F from the B3 region is shown in Fig. 10.
The values of L were found as a function of phase angle for the
same data points as those shown in Figs. 2–4. Results, shown in
Fig. 11, indicate that there is a small upward slope in L with
increasing phase angle that is attributed to multiple scattering effects. A linear fit was made to the values of L versus phase angle to
quantify the y intercept, Lo, and the slope. In all cases the slope was
positive to within one sigma uncertainty in the fit consistent with
the assumption that L is positively correlated with phase angle due
to increased multiple scattering effects. It follows that Lo, which
predicts L at opposition, is a measure of twice the single scattering
photon mean path length to the nearest scattering centers to the
surface. Lo across the rings show very little variation (Fig. 12) with
values from 2 to 3 lm.
4. Analysis of SOI data
On July 1, 2004 Cassini entered orbit around Saturn. Here we
present a new analysis of the SOI (Spacecraft Orbit Insertion) data
(Esposito et al., 2005) consistent with the new calibration, flatfielding and parameter-retrieval techniques used for lit-face data
in this paper and in Bradley et al. (2010). SOI data were taken over
a 30 min segment of the orbit insertion phase where the spacecraft
flew within 18,000 km of the unlit side of the rings and at a phase
angle of 59.3°. The instantaneous size of a spatial pixel projected
onto the ring plane was 45 km per side; however the motion of
the spacecraft throughout the integration period resulted in a
skewed spatial pixel 220 km in the radial direction. We apply
the filtering and interpolation technique described in Bradley
et al. (2010) to the data in order to azimuthally bin and create a
one dimensional evenly distributed radial profile of I/F for the A
ring. Fig. 13 shows the results for a 100 km interpolated radial
E.T. Bradley et al. / Icarus 225 (2013) 726–739
733
Fig. 5. Contours of D for the four ring regions with the star in each panel denoting the retrieved values of AB and g. The inner contour of each of the 175–185 nm contours are
0.28 105, 3.2 105, 1.0 105, 1.1 105, for the C4, B3, CD2, and A3 contours, respectively. The inner contour of each of the 152–158 nm contours are 4.5 106,
1.9 106, 3.0 106, 2.3 106 for the C4, B3, CD2, and A3 contours, respectively. Each contour is one-half the value of next largest contour.
profile of I/F. The data (solid line) show slight increases in brightness in the surrounding regions of the Janus 4:3, Janus 5:4, and Mimas 5:3 density waves with decreases in I/F right at the density
wave due to the increased optical depth and decreased transmission through the rings at those locations. The dotted line is an optical depth profile from a UVIS stellar occultation of a Arae that
shows the increase in optical depth at these density waves. The regions of increased brightness extend a few hundred kilometers
from the density wave, which is consistent with halos of greater
brightness at 1.08 lm reported by Nicholson et al. (2008) in the
simultaneous and co-aligned infrared SOI observation by the Visual
and Infrared Mapping Spectrometer (VIMS).
We used AB and n retrieved for the four regions in the A ring
from Section 3 in the model for single-scattered transmission to
determine I/F for the geometry corresponding to the SOI observations. SOI observations were at a finer spatial resolution that the
four regions determined for the lit side analysis so within each region we used the high resolution geometry from the SOI observations. The equation for single-scattered transmission is given by:
I
lo
¼ AB P ½expðsn =lÞ expðsn =lo Þ
F
4ðl lo Þ
ð6Þ
where we used the self-gravity wake expressions for ring transmission (Eq. (3)) in place of the exponential terms. Fig. 14 shows the
model I/F, which is I/F from Eq. (6) for the SOI geometry using the
self-gravity wake parameters determined from stellar occultations,
compared to I/F determined from the actual data from SOI observations. I/F determined from data and the model agrees well for the A2
region; however, the model I/F determined for regions A1, A3, and
A4 are consistently lower than the data, especially for the A4 region.
We expect an increase in I/F for the outer A ring based on Voyager
observations of the lit side analyzed by Dones et al. (1993). We tried
replacing the power-law phase function with a double lobed Henyey–Greenstein phase function to investigate the possibility of forward scattering ring particles. However since this was used in a
single scattering model and the phase angle is 60° this did not result in better agreement between the model and data.
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E.T. Bradley et al. / Icarus 225 (2013) 726–739
Fig. 6. Retrieved values of AB as a function of ring plane radius. Lower AB at 180 nm for the C ring and Cassini Division are consistent with more contaminated ring particles
whereas the larger values of AB in the mid to outer B ring are consistent with more pure water ice. The flat AB values at 155 nm suggest that the reflectance is not very sensitive
to the abundance of contaminants in this spectral regime. The inner contour of the A3 region of Fig. 5 gives an uncertainty in AB of ±0.003 for AB = 0.062 at 180 nm and ±0.002
for AB = 0.017 at 155 nm.
Fig. 7. Retrieved values of g as a function of ring plane radius showing that the ring particles are highly backscattering in the FUV. The inner contour of the A3 region of Fig. 5
gives an uncertainty in AB of ±0.05 for g = 0.71 at 180 nm and ±0.16 for g = 0.63 at 155 nm.
There are several possible explanations for this discrepancy
including multiple scattering effects, scattering from the edges of
self-gravity wakes, and three dimensional effects of self-gravity
wakes that is not captured in the model. We discuss this in more
detail in Section 5.
5. Discussion and conclusions
Following the arguments of Cuzzi et al. (1984) we assumed a
single scattering model between ring particles to retrieve AB and
g. It is possible that there could be a small contribution from multiple scattering between ring particles that we did not include. This
implies that our retrieved values of AB and g are unique only to the
extent that single scattering is valid. Retrieved values of AB longward of the water ice absorption edge show a peak in the central
and outer B ring and a minimum in the C ring and Cassini Division.
AB throughout the A ring are similar to the B1 region. We interpret
this as a measure of the relative abundance of the non-water ice
material in the ring particles where the fraction is inversely correlated with AB. The comparison of AB retrieved from this investigation to the values reported by Porco et al. (2005) shows that AB
converges to similar values across the rings below the FUV absorption edge. We conclude that the reflectance of water ice and nonwater ice materials are comparable below the absorption edge
(at least to our low absolute precision at these low reflectances),
thus the fractional abundance of the water ice and non-water ice
materials are indistinguishable in the short wavelength regime.
Also, since others have shown the rings to be mostly water ice
(Cuzzi et al., 2009, 2010) and the long wavelength AB of Porco
et al. (2005) for the B3 region is 0.85, then we conclude that
the non-water ice material must be very dark (comparable to
water ice) at shorter wavelengths.
Lit side observational analysis by Dones et al. (1993) found that
the single scattering albedo of ring particles remains relatively
constant across the A ring with the exception of the trans-Keeler
regime where the particle single scattering albedo increases by
E.T. Bradley et al. / Icarus 225 (2013) 726–739
735
Fig. 8. AB, using values from Porco et al. (2005) at wavelengths longer than 300 nm and from this work for the FUV wavelengths. AB from different rings regions appears to
converge to similar values at very short wavelengths.
Fig. 9. I/F from the B3 region at four different phase angles. The data are normalized to the I/F at a phase angle of 34°. The shift in the spectral location of the absorption edge is shows an
obvious trend with phase angle. Since L is related to the spectral location of the absorption feature this merited an investigation of L with phase angle using the Shkuratov model.
20%. We do not see an increase in AB from the lit side analysis of
Section 3.1 for the outer A ring; however the A4 region extends
from just outside the Encke gap to the ring edge so we do not resolve the trans-Keeler region for the analysis of Section 3.1.
Fig. 8 shows that AB drops significantly below 300 nm and is
attributed to non-water ice material. Spectral modeling of AB from
300–4050 nm by Poulet and Cuzzi (2002) suggests that the ring
particle surface is consistent with 93% water ice and 7% carbonaceous material. The carbonaceous material is thought to be primordial and/or due to meteoroid bombardment (Cuzzi and Estrada,
1998). Alternatively, Hendrix et al. (2010) have shown that NH3
and a small amount of tholin fit the far ultraviolet to visible wavelength spectra (up to 1000 nm) from Enceladus; although this analysis does not cover the absorption features above 1000 nm.
However, ultraviolet to mid-infrared laboratory measurements
(350–15,500 nm) of water ice samples contaminated with organic
compounds (Clark et al., 2009) as well as modeling using optical
constants of organics and iron based constituents (Clark et al.,
2012) have not been able to discriminate between the relative
abundance of organics, iron, tholins, and other material responsible
Fig. 10. I/F spectra from the B3 region at a phase angle of 6° clearly showing the
water ice absorption edge. The thick line is the best fit Shkuratov model to the data.
For an increase in the photon mean path length the absorption edge shifts towards
longer wavelengths. As the photons travel further in ice they experience more
absorption leading to a decrease in the value of I/F at longer wavelengths.
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E.T. Bradley et al. / Icarus 225 (2013) 726–739
Fig. 11. Retrieved L values in microns from comparing the Shkuratov–Poulet model to I/F. The value of g was fixed to the average of the short and long wavelength retrieved
values of g resulting from comparing the granola bar model to I/F. A linear fit to the values of L (solid line) shows a positive slope with the value and one sigma uncertainty
shown in the upper left corner. The slope still remains positive within the lower limit of the one sigma uncertainty.
for the non-water ice spectral features present in spectra of Saturn’s rings and icy moons. One hypothesis for the steep decrease
in reflectance below 550 nm measured by VIMS (Filacchione
et al., 2012) is that small amounts (parts per thousand) of nanophase iron (less than 0.2 lm grain size) may be responsible for
Rayleigh absorption in this spectral regime (Clark et al., 2012),
although laboratory spectra have not been obtained to validate
this. Modeling efforts by Clark et al. (2012) using Cassini VIMS data
from observations of dark material on Iapetus suggests nano-phase
iron and hematite to explain some unusual spectral features. This
does not rule out the possibility of larger abundances of tholins
and carbonaceous material, which is cosmochemically more likely,
comprising the non-icy material as long as iron and hematite remain spectrally dominant. New laboratory spectra are required
to determine whether or not icy mixtures containing nano-phase
iron and hematite are spectrally dominant in the FUV.
There is little variation of g values across the rings indicating no
radial dependence on the scattering properties of ice grains. The
lowest values of g occur in the C ring and Cassini Division, which
could possibly be related to the higher fractional abundance of
E.T. Bradley et al. / Icarus 225 (2013) 726–739
737
Fig. 12. Lo versus ring plane radius from the Shkuratov retrieval technique for each of the regions investigated. L at zero phase value provides a direct measure of twice the
distance from the surface of a ring particle to the first scattering center.
Fig. 13. I/F of the unlit side of the A ring from data taken during orbit insertion in 2004 (solid line). Pixels were azimuthally binned and interpolated to 100 km evenly
distributed pixels. The dotted line is an optical depth profile from a UVIS stellar occultation of a Arae.
non-water ice material. However there seems to be no relation between g and AB for the other regions so we conclude that the scattering asymmetry is insensitive to variations in ring composition.
This may argue in favor of the morphological features of the ring
particles and ice grain regolith dominating the scattering asymmetry. As pointed out by Verbiscer et al. (1990) the very cold temperatures at which outer Solar System ices exist could lead to the
formation of complex structures during an event where the ice is
melted and then freezes rapidly. Baragiola (2003) points out that
cracks in ice induced by stresses are the main reason for low-temperature ices to be strongly backscattering.
The g values across the A ring increase slightly from the A1 to
A4 region, implying that the outer A ring is less backscattering. This
is consistent with results from Dones et al. (1993), who attribute
this to either collisional packing of the ring particles or that the
rings are physically thicker in the outer A ring than the rest of
the A ring.
The ring I/F spectra shown in Fig. 9 shows that the absorption
edge shifts to longer wavelengths as the phase angle increases.
Comparison of I/F with the Shkuratov model showed L to be positively correlated with phase angle for all of the regions investigated. Under the assumption that the ring particle regolith is
backward scattering, low phase angles result in mostly single scattered photons received by the instrument. At larger phase angles,
there is an increase in the multiple scattered contribution to the
signal. Consequently the multiple scattered photons travel a greater total distance within ice leading to an increase in the value of L.
Since L is larger for multiple scattered photons, the result is that
the absorption edge shifts towards longer wavelengths.
The extrapolated value of L at zero phase, Lo, may be thought of
as the single scattering photon mean path length. Physically this
means that Lo is twice the average distance the photon travels into
an ice grain before being backscattered. The small variation of Lo
with ring plane radius despite large variations in (long wavelength)
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E.T. Bradley et al. / Icarus 225 (2013) 726–739
Fig. 14. I/F calculated from SOI observations of the unlit side of the A ring (stars) using AB and n retrieved in each of the four rings regions from the lit side analysis of
Section 3.1. I/F was calculated using the Chandrasekhar-granola bar model for single scattering for transmission. Also shown is I/F from the lit side retrieval analysis of
Section 3.1.
AB across the rings suggests that the FUV photons are sampling
scattering centers that are all about the same distance from the
surface. Scattering may occur because of contaminants located
within the ice matrix, a crack within the grain, or because the photons travel completely across a grain and scatter off the other side.
L may not necessarily reflect the size of the ice grains since scattering may occur at some point inside the grain. Typical Lo values of
3 lm translate to 1.5 lm from the surface to the scattering center and the wavelengths we are dealing with are 0.165 lm. Nicholson et al. (2008, Fig. 4) report a best fit to the A and B ring using a
grain size of about 10 lm from analysis of data from the Visual
Infrared Mapping Spectrograph (VIMS) on Cassini. Alternatively
Filacchione et al. (2012) retrieve grain sizes that are somewhat larger than 10 lm, which may be the result of different modeling approaches. Nevertheless grain sizes determined from VIMS spectra
results in a mean path length (or grain size) 10 times that retrieved from the FUV data, while their wavelength regime is about
10 times longer than UVIS. Thus it may be that both instruments
are merely sampling to a depth governed by the wavelengths of
the observed photons. This implies that the ice grains do not contain significantly large scattering centers in the outer few microns
since the long wavelength investigations of VIMS show a larger
photon path length; therefore photons in the FUV are probing
the extreme surface layers of the grain.
Halos have been detected by others (Nicholson et al., 2008;
Dones et al., 1993). According to Nicholson et al. halos appear as
regions of increased brightness in reflected light for phase angles
below 30° and as minima at phase angles from 50 to 70°. Nicholson
et al. who were reporting on SOI observations (unlit side transmitted light at 59° phase) made by VIMS corresponding to the same
SOI data presented here, saw increased regions of brightness surrounding strong resonances, consistent with our results. Esposito
et al. (2005) suggested density waves released brighter material
from the particles with the bits diffusing away from the sources
in the density wave region. This is caused by streamline crowding
(Lewis and Stewart, 2005), which damps the interparticle velocity
and allows temporary clumps to grow. These clumps lead to an increase in the velocity of smaller particles surrounding these
clumps, which then impact the clumps and excavate purer ice
(see the predator–prey model of Esposito et al., 2012).
The single scattering transmission model that made use of AB
and n from the lit side analysis of Section 3 compared well with
the SOI I/F from the A2 region but over-predicted I/F for the
other regions of the A Ring. Since the single scattering reflectance model fit the data quite well in the lit side analysis of Section 3 we conclude that multiple scattering is negligible for the
lit side analysis therefore the values of AB and g derived in Section 3 should be reasonably accurate. The cause of the poor
model fit to the unlit side SOI data for the A1, A3, and A4 regions may be a combination of different effects. One is that multiple scattering contributions may be more significant in
transmission for the A1, A3, and A4 regions, which our single
scattering model shown in Eq. (5) does not account for. Salo
and Karjalainen (2003) performed Monte-Carlo radiative transfer
simulations and showed that multiple scattering is typically a
greater fraction of the total transmitted light than for reflected
light. The average of gap optical depths for the A1–A4 regions
determined by UVIS occultation measurements are 0.128,
0.095, 0.131, and 0.181, respectively. Nicholson et al. (2008) suggested that all the light seen on the unlit side observations must
be scattered by particles in the gaps between the self-gravity
wakes because the wakes themselves are opaque. The gap optical depth in A2 is smaller than elsewhere in the A ring, so the
contribution from multiple scattering should be smallest in this
region. However as the gap optical depth increases by a factor
of 2 in the A4 region the single scattering model of Eq. (5)
underestimates I/F as the contribution from multiple scattering
grows. The discrepancies between the single-scattering ‘‘classical-granola bar’’ model and the SOI data in Fig. 14 are qualitatively consistent with this explanation of a correlation between
the multiple-scattered contribution and the optical depth of
the gaps between the self-gravity wakes.
A second explanation deals with the morphology of self-gravity
wakes. The wakes should be three-dimensional structures resembling long flattened, ellipsoids (Meinke et al., 2012). At the moderate phase angles of SOI, the light will diffuse between these
structures, and the scattered light is more like diffuse reflection
from the edge regions. The self-gravity wake margins will appear
to UVIS FUV like the ‘‘silver lining’’ of clouds on Earth. Those parts
of the rings viewed from the dark side will light up if they have the
E.T. Bradley et al. / Icarus 225 (2013) 726–739
right geometry and optical thickness, which may be the case for
the geometry of the SOI observations.
A third explanation deals with the optical depth of the single
scattering model we are using. The Chandrasekhar-granola bar
model has only two optical depth values and miss the threedimensional nature of the rings, which may be more prominent
in the outer A ring, or in the regions of overstability. These geometrical effects and the fact that the Toomre critical length may being
smaller in the outer A ring due to the lower surface mass density
might have a larger effect on the results than multiple scattering.
Acknowledgments
This material is based upon work supported by the National
Aeronautics and Space Administration under Grant No.
NNX10AF20G issued through the Cassini Data Analysis Program
and the Cassini Project and UVIS team.
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