1. UVIS spectrometry of Saturn’s rings
Todd Bradley
1/7/2008

2. Investigation summary
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

3. Review
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

4. Present physical picture of the micro-structure of the rings
Ring particle composed of many grains (multiple scattering between grains)
Regolith ice grain (model as single scattering)
Incident photon
Emission of photon from ring particle

5. 4 models have been tried
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

6. Single scattering model
Use Hapke formulation of scattering efficiency, Qs, that includes the mean path length
Assume Qs = single scattering albedo
Free parameter is the mean path length

7. Single scattering model
n,k = complex indices of refraction.
D = mean path length
Assume the scattering efficiency = single scattering albedo

8. Hapke-Van de Hulst model
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

9. Hapke-Van de Hulst model
n,k = complex indices of refraction.
D = mean path length

10. Ring particle albedo (Hapke-Van de Hulst)
Assume Qs = single scattering albedo (ῶl) and let g = the asymmetry parameter
Then from Van de Hulst:

11. Functional form of I/F using Hapke-Van de Hulst ring particle albedo

12. Shkuratov model 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

13. Shkuratov model Slab model of regolith grain Poulet et al., 2002
Re = average external reflectance coefficient which = average backwards reflectance coefficient (Rb) + average forward reflectance coefficient (Rf)
Ri = average internal reflectance coefficient
Te = average transmission from outside to inside
Ti = average transmission from inside to outside
Wm = Probability for beam to emerge after mth scattering
= 4pkS/l
k = imaginary index of refraction
Slab model of regolith grain
Poulet et al., 2002

14. Shkuratov model
Use real part of indices of refraction (n) to determine Re, Rb, and Ri. Empirical approximations from Shkuratov (1999) give:
Re ~ (n-1)2 / (n + 1)
Rb ~ (0.28 n – 0.20)Re
Ri ~ 1.04 – 1/n2
Shkuratov assumes W2 = 0 and Wm = 1/2 for m > 2. Then adding all the terms shown in the last figure becomes a geometric series and gives:
rb = Rb + 1/2TeTiRi exp(-2t)/(1 – Ri exp(-t))
rf = Rf + Te Ti exp(-t) + 1/2 Te Ti Ri exp(-2t)/(1 – Ri exp(-t))
where rb + rr is assumed to be the single scattering albedo of a regolith particle (Poulet et al., 2002)

15. Ring particle albedo (Shkuratov)
Denote “q” as the volume fraction filled by particles. Then:
rb = q * rb
rf = q*rf + 1 – q

16. Functional form of I/F using Shkuratov ring particle albedo

17. Hapke-H function model
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

18. Hapke-H functions n,k = complex indices of refraction.
D = mean path length

19. Ring particle albedo (Hapke-H function)
Assume Qs = single scattering albedo (ῶl)
Make the argument that the only the H functions and the phase function affect the spectral shape of the curve.

20. Functional form of I/F using Hapke-H function model
Presently using power law phase function:

21. Single scattering delta function

22. Single scattering and Hapke-Van de Hulst

23. Single scattering, Hapke-Van de Hulst, and Shkuratov

24. Single scattering, Hapke-Van de Hulst, Shkuratov, and Hapke-H functions

25. Retrieved mean path length for 4 models from a single observation

26. Normalized mean path lengths for 4 models from a single observation

27. Path length results from Shkuratov model

28. Path length results from Hapke-Van de Hulst model

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

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

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

32. Path length results from Hapke-H function model, n = 5

33. Scatter plot of I/F average (1800 Å – 1900 Å) vs. mean path length

34. Contaminant abundance
Use the estimate of the mean path length to estimate the contaminant fraction times the contaminant reflectance.
where “fraction” is the fraction of water ice and Rc is the reflectance of the contaminant

35. (1 – fraction) * Rc from Hapke-H function model

36. Contaminant-phase angle scatter plot

37. 1850/1570 Å color ratio

38. Color ratio for phase angle ~ 20°

39. Estrada and Cuzzi, 1996
G = 563 nm
V = 413 nm
UV = 348 nm

40. Estrada and Cuzzi, 1996
G = 563 nm
V = 413 nm
UV = 348 nm

41. Results 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