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 micro-structure 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 (Hapke-Van 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
~ (n-1) 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
(Hapke-H 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 Hapke-Van de Hulst

Single scattering, Hapke-Van de
Hulst, and Shkuratov

Single scattering, Hapke-Van de Hulst,
Shkuratov, and Hapke-H 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 Hapke-Van de Hulst model

Path length results from Hapke-H function model, 2 < n < 6

Path length results from Hapke-H function model, n = 3

Path length results from Hapke-H function model, n = 4

Path length results from Hapke-H 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 Hapke-H function model

Contaminant-phase 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

• Hapke-H function model gives best fit to data
• A multiple valued exponent for the phase function may be more appropriate for the Hapke-H function model
• Hapke-Van de Hulst and Shkuratov models give similar fits to the data
• Hapke-Van de Hulst mean path length ~ 2X
Shkuratov value, but very similar radial variation
• Hapke-H function mean path length ~ 6X
Shkuratov value
• Single scattering model neglects multiple scattering and thus only models an ice grain