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

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

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

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

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

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

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

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

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

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

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

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

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

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)2 + 0.05 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)

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

Functional form of I/F using Shkuratov ring particle albedo

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

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

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.

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

Single scattering delta function

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

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

(1 – fraction) * Rc 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

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