Todd Bradley
1/7/2008
• Analyzed multiple observations in FUV
• Observations were all of lit side
• Phase angles ranged from 6° to 25°
• Fit I/F with 4 different models
• Found photon mean path length in water ice grains to be model dependent
• FUV observations of
Saturn’s rings typically show a water ice absorption feature
• Spectral location of absorption feature is dependent on mean path length of photon in ice
• Goal so far has been to find mean path length
• Attempted 4 different models to retrieve mean path length
Present physical picture of the microstructure of the rings
Incident photon
Emission of photon from ring particle
Regolith ice grain
(model as single scattering)
Ring particle composed of many grains (multiple scattering between grains)
• Single scattering model with different distributions of mean path length
• Hapke model for single scattering regolith grain and Van de Hulst approximation for ring particle albedo (Cuzzi and Estrada, 1998, Van de Hulst,
1980)
• Shkuratov model (Shkuratov et al., 1999, Poulet, et al., 2002)
• Hapke model for single scattering regolith grain and H functions for ring particle albedo
• For all 4 models, use minimum least squares analysis over the free parameters to determine the mean path length
• Use Hapke formulation of scattering efficiency, Q s
, that includes the mean path length
• Assume Q s
= single scattering albedo
• Free parameter is the mean path length
S e
n n
1
1
2
2
k 2 k 2
0 .
05
S i
Q s
1
S e n
n
4
1
2
1
S e
1
1
S i
S i
e
4
kD /
n,k = complex indices of refraction.
D = mean path length
Assume the scattering efficiency = single scattering albedo
• Determine scattering efficiency and assume this is equal to single scattering albedo of a single grain
• Use single scattering albedo in a Van de
Hulst (1980) approximation to determine ring particle albedo
• Free parameters are the mean path length and asymmetry parameter
S e
n n
1
1
2
2
k 2 k 2
0 .
05
S i
Q s
1
S e n
n
4
1
2
1
S e
1
1
S i
S i
e
4
kD /
n,k = complex indices of refraction.
D = mean path length
Ring particle albedo (HapkeVan de
Hulst)
Assume Q s
= single scattering albedo (ῶ and let g = the asymmetry parameter
)
Then from Van de Hulst: s
A
1
1
1
s
g
1
1
0 .
139
1 .
17 s
s
Functional form of I/F using Hapke
Van de Hulst ring particle albedo
I
F
A
P
,
,
o
,
A
P
O is
,
the ring is
particle albedo
,
the ring
o
,
is particle phase function all of the geometrica l and optical depth term s
• Geometrical optics model
• First determine albedo of a single grain
• Use albedo of a single grain along with porosity to determine the ring particle albedo
• Free parameters are the mean path length and porosity
• Phase function asymmetry is not a free parameter
Shkuratov model
Slab model of regolith grain
Poulet et al., 2002
R e
= average external reflectance coefficient which = average backwards reflectance coefficient (R b
) + average forward reflectance coefficient (R f
)
R i
= average internal reflectance coefficient
T e
= average transmission from outside to inside
T i
= average transmission from inside to outside
W m
= Probability for beam to emerge after m th scattering
= 4
kS/
k = imaginary index of refraction
Use real part of indices of refraction (n) to determine Re, Rb, and Ri.
Empirical approximations from Shkuratov (1999) give:
R e
~ (n1) 2 / (n + 1) 2 + 0.05
R b
~ (0.28 n – 0.20)R e
R i
~ 1.04 – 1/n 2
Shkuratov assumes W
2
= 0 and W m
= 1/2 for m > 2. Then adding all the terms shown in the last figure becomes a geometric series and gives: r b
= R b
+ 1/2T e
T i
R i exp(2
)/(1 – R i exp(
)) r f
= R f
+ T e
T i exp(
) + 1/2 T e
T i
R i exp(2
)/(1 – R i exp(
)) where r b
+ r r is assumed to be the single scattering albedo of a regolith particle (Poulet et al., 2002)
Denote “q” as the volume fraction filled by particles. Then: r b
= q * r b r f
= q*r f
+ 1 – q
A
1
r b
2
2 r b
r
2 f
1
r b
2
2 r b
r f
2
1
Functional form of I/F using
Shkuratov ring particle albedo
I
F
A
P
,
,
o
,
A
P
O is
,
the ring is
particle albedo
,
the ring
o
,
is particle all of phase function
the geometrica l and optical depth term s
• Determine scattering efficiency and assume this is equal to single scattering albedo of a single grain
• Multiply single scattering albedo by H functions plus phase function to determine a scaled ring particle albedo that spectrally fits the data
• Free parameters are the mean path length and phase function
S e
n n
1
1
2
2
k 2 k 2
0 .
05
S i
Q s
1
S e n
n
4
1
2
1
S e
1
1
S i
S i
e
4
kD /
n,k = complex indices of refraction.
D = mean path length
Ring particle albedo
(HapkeH function)
Assume Q s = single scattering albedo (ῶ
)
Make the argument that the only the H functions and the phase function affect the spectral shape of the curve.
A
( scaled )
( P (
)
H
H
o
)
H x
cos
emission angle
,
o
1
1
2 x
2
x
, x
,
o cos
incidence angle
,
1
Functional form of I/F using Hapke
H function model
I / F
P ( s )
H
H
o
1
O
,
,
o
,
Presently using power law phase function: s
P ( s )
C n
S n
scattering angle

C n
normalizat ion constant n
a positive constant, generally between 2 and 6 for the rings, Dones, et al., 1993
Single scattering and HapkeVan de Hulst
Single scattering, HapkeVan de
Hulst, and Shkuratov
Single scattering, HapkeVan de Hulst,
Shkuratov, and HapkeH functions
Retrieved mean path length for 4 models from a single observation
Normalized mean path lengths for
4 models from a single observation
Path length results from Shkuratov model
Path length results from HapkeVan de Hulst model
Path length results from HapkeH function model, 2 < n < 6
Path length results from HapkeH function model, n = 3
Path length results from HapkeH function model, n = 4
Path length results from HapkeH function model, n = 5
Scatter plot of I/F average (1800 Å
– 1900 Å) vs. mean path length
Use the estimate of the mean path length to estimate the contaminant fraction times the contaminant reflectance.
I
F
I
F water
* fraction
( 1
fraction ) R c where “fraction” is the fraction of water ice and R c reflectance of the contaminant is the
(1 – fraction) * R c from HapkeH function model
Contaminantphase angle scatter plot
1850/1570 Å color ratio
Color ratio for phase angle ~ 20 °
Estrada and
Cuzzi, 1996
G = 563 nm
V = 413 nm
UV = 348 nm
Estrada and
Cuzzi, 1996
G = 563 nm
V = 413 nm
UV = 348 nm
• HapkeH function model gives best fit to data
• A multiple valued exponent for the phase function may be more appropriate for the HapkeH function model
• HapkeVan de Hulst and Shkuratov models give similar fits to the data
• HapkeVan de Hulst mean path length ~ 2X
Shkuratov value, but very similar radial variation
• HapkeH function mean path length ~ 6X
Shkuratov value
• Single scattering model neglects multiple scattering and thus only models an ice grain