Scattering by aggregates and layers of closely packed particles

Sanaz Vahidinia

Jeffrey Cuzzi, Bruce Draine, Frank Bridges

Overview• Introduction

– Light scattering in nature

• Saturn’s F ring aggregates– Scattering from diffuse particle layer– Independent scattering– Cassini VIMS data, modeling, and results

• Regolith scattering– Scattering from compact particle layers– Interference effects– motivation, current models, our new model– results

• Future work

1. diffuse layer

2. diffuse aggregate layer

3. compact layer

Part 1: Studies of Saturn’s F ring particles As aggregates

ISS

RSS, Radar

CIRSUVIS

Ultraviolet imaging spectrograph (occultations, mapping)

MIMI

Composite infrared Spectrometer (thermal emission)

(WA, NA cameras)

(neutrons ! )

(vis, near-IR mappingspectrometer)

Cassini’s Telescopes

RPWS(AM, FM)

ISS approach color composite

A

B

C

C D

F

D

or ???

what kind of particles could the F ring particles be? solid, porous, or aggregates

solid homogeneous particle Mie theory

porous homogeneous particleMie theory + EMT

aggregateDDA

solid carbon particles vs. solid ice particles

solid ice particlesvs. data

or ???

what kind of particles could the F ring particles be? solid, porous, or aggregates

solid homogeneous particle Mie theory

porous homogeneous particleMie theory + EMT

aggregateDDA

Discrete Dipole Approximation

• Model each dielectric grain with an array of N polarizable dipoles with polarizability .

• Assign polarizability such that the dipole lattice exhibits the macroscopic dielectric constant of the material.

• Each dipole is driven by an incident field and the field of all other oscillators in the lattice.

• The steady state polarization or field in the material is found by an iterative method.

Pj = Ej = ·(Ei + k2 (Pj * G) )

- 1 = 4N / (1- 4N/3)

2m(d / ) < 1/2 DDA Constraint

extinction efficiency (Qext) = Qa + Qs = ext / r2

albedo (w) = Qs / Qext

I / F = Qext * w * p * n * r2phase (p) = phase function

The “toy model”

Part 1 conclusions:

• F ring particles are not well modeled by uniform density spheres, even if arbitrary density is allowed

• Aggregates fit better and allow fit to overall shape of spectral Forward scattering I/F AND depth of single observed absorption feature

• Aggregates are more effective scatterers per unit mass;toy model explains why (interfaces)

• Single abs feature has been explained as Christiansen frequency and allows us to discriminate between crystallineand amorphous ice

• F ring aggregate particles have a moderately narrow size distribution

Part 2: Regolith radiative transfer:A new approach with applications

To many granular surfaces

1. diffuse layer

2. diffuse aggregate layer

3. compact layer

wavelength, microns

wavelength (microns)

R

IRTF

HST

model

Saturn’s entire B ring

Groundbased reflectance spectrum: water ice bands and reddish material

Certain regions get more “polluted”

F. Poulet et al. (2003)

98% water ice, 2% carbon, 1% tholins

• Roll off is the transition region between good absorbers and perfect reflectors.

• This region is difficult to model due to closely packed grains and grain sizes comparable to .

Wavelength (m)10 100 1000

0.5

1.0

Spilker et al.(2005) Ring emissivities of the A,B, and C rings as a function of wavelength.

Grains are good absorbers of short

Grains start to scatter radiation in this region

Ring particles become good scatterers in the microwave region

BAC

Data tabulated in Esposito et al 984

Thermal IR and microwave: change from emitting to scattering behavior

Most ring particles are wavelength size (cm-m)and

Composed almost entirely (>90%) of water ice;Transition region (100 - few mm) poorly characterized

Review of current RRT modelsand sample calculation of Quartz emissivity using current model…

motivation fordevelopment of our model

0

0

0

I0

It

Ir

P = particle phase function, which gives the angular distribution and polarization of scattered light for any polarization of incident radiation.

g = cos P11 d/4, anisotropy parameter characterizing the shape of the phase function.

w = single scattering albedo. Defined as the fraction scattered, of the total energy removed from the incident beam.

Methods of solving the Transfer Equation for plane parallel layers

Equation of Transfer

€

dI (τ ,μ ,μ 0 ,φ)

dτ= −I (τ ,μ ,μ 0 ,φ) + w P(∫∫ μ ,μ ' ,φ) I (τ ,μ ' ,μ 0 ,φ) dμ 'dφ +

w

4exp(−

τ

μ 0

)P(μ ,μ 0 ,φ)I 0

Current RRT models

Some of the current transfer solutions can be summarized in two parts

Calculate grain Albedo (w) using: Mie theory … Spherical particles, well separated (Conel, Hansen,..) Ray optics … Particles large compared to wavelength, allows for irregular particles (Hapke) Multiple scattering calculation for layer:

Isotropic scattering in a homogenous semi infinite layer can be solved by Chandrasekhar's semi-analytic H-function. The H function is a nonlinear integral equation used to numerically solve the reflection from an infinite layer.

Anisotropic scattering can be solved using van de Hulst’s Similarity relations. This method approximates complex phase functions in terms of solutions for simpler phase functions (specifically isotropic phase functions). (Cuzzi & Estrada, Shkuratov, Conel…)

The Doubling/Adding method is a numerical approach where the overall transfer properties of one layer is calculated by equating the emergent radiation from an arbitrary layer to the incident radiation upon the layer in question. (Hansen, Eddington…)

Emissivity of a layer of particles

Black body surface at Tc

• Black body radiation gets absorbed, scattered, and re-emitted by the layer.• The radiation contributions from the layer can be broken up into the following: 1) Direct transmission of black body radiation 2) Diffusely scattered radiation transmitted

through the layer . through the layer. 3) Diffusely scattered radiation reflected by the layer. 4) Emitted radiation from the layer.

Bc

Bc

Bc=Black body radiation intensity

~

€

Bν = Iν + Bν exp(−τ /μ ) +Bν

4πμ(S +T )dμ 0dφ0∫∫

ε (μ ) =Iν

Bν

= 1− exp(−τ /μ ) −1

4πμ(S +T )dμ 0dφ0∫∫

€

1−1

4πμS dμ 0dφ0∫∫

Conservation of Energy

Outline of Mie/similarity calculation for quartz emissivity data

A = S(, ) d do = ring particle spherical albedo

using van de Hulst’s similarity parameter

S = [(1 - w) / (1 - w•g)]1/2

A = (1- S)(1-0.139 S) / (1+1.175 S)

Emissivity = 1 - A

• Mie scattering for grain albedo(w) and g• Similarity transformation for particle (or integrated) albedoIntegrated emissivity = 1 - Integrated Albedo

Data courtesy J. Michalski

Current models make various simplifying assumptions and do not account for close packing, interference

effects, and irregular particles.

We need to develop an RRT model that takes all theseeffects into account

Current models don’t match the emissivity data.

Discrete Dipole Approximation

• Model dielectric grain with an array of N polarizable dipoles with polarizability .

• Assign polarizability such that the dipole lattice exhibits the macroscopic dielectric constant of the material.

• Each dipole is driven by an incident field and the field of all other oscillators in the lattice.

• The steady state polarization or field in the material is found by an iterative method.

Pj = Ej = ·(Ei + k2 (Pj * G) )

- 1 = 4N / (1- 4N/3)

2m(d / ) < 1/2 DDA Constraint

Discrete Dipole Approximation applied to grainy surface

Layer radiative transfer

s

T

{s + T} ()

Scattered field in the shadow zone

Scattered field on a plane in the near field(either reflection or transmission)

E(x,y) = k2 (Pj * G)FFT

E(, )

x

y

Note periodic boundary conditions

Angular Spectrum

€

G =exp(ikrpq )

k 2rpq

× k 2 (ˆ r pq ˆ r pq − Ι) +exp(ikrpq ) −1

rpq2

(3ˆ r pq ˆ r pq − Ι) ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥p≠q

€

vE (x, y) = k 2 Pq ∗G∑

€

vE (x, y) = Ei (x, y)ˆ i + E j (x, y) ˆ j + Ek (x, y) ˆ k

€

vE (x, y; z) = ∑ f (kx ,ky )∑ exp(−ikx x − iky y − ikzz)dkxdky F (kx ,ky ) = f (kx ,ky )∗exp(−ikzz)

F (kx ,ky ) = ∑ Ei (x, y)ˆ i + E j (x, y) ˆ j + Ek (x, y) ˆ k [ ] exp(−ikx x − iky y)dxdy∑⇒ Fi (kx ,ky )ˆ i + F j (kx ,ky ) ˆ j + Fk (kx ,ky ) ˆ k

2-D vector field

Fourier transform vector field to k-space

€

F (kx ,ky )2

= Fi (kx ,ky )2

+ F j (kx ,ky )2

+ Fk (kx ,ky )2k-space power

spectrum

€

k 2 = kx2 + ky

2 + kz2

G = Green’s function (J. D. Jackson)

x

y

z

Angular distribution of scattered intensity

€

vk

€

k 2 = kx2 + ky

2 + kz2

€

kz2 = k 2 − (kx

2 + ky2)

evanescent waves occur when:

€

kx2 + ky

2 > k 2

€

sinθ cos φ =kx

k, sinθ sinφ =

ky

k

Resolution Specifications

€

mkd =1

2⇒ DDA, sampling interval = D ; If D = d Then

Spatial frequency bandwidth =1

d,

Sampling int erval (Δk) =bandwidth

sampled po int s=

1

Nd,

Satisfy : Δkd < kd

€

kx

€

ky

€

kx2 + ky

2 > k 2

• Dielectric slab test

Simulate a homogeneous dielectric layer with DDA-PBC code. Calculate reflection and transmission from the layer with the near field method. Compare R and T with Fresnel coefficients.

• Granular runs Run a granular layer with various

porosities. Examine transmission and reflection as a function of particle

packing.

Io R

T

Specular reflection: reflected beam is localized in frequency space at three emission angles corresponding to three incident zenith angles 20o, 40o, and 60o.

Specular beam in k-space

Granular layers for various porosities

Optical depth studies of granular layers

• Calculate Q from the optical depth and compare with current mie models• Use optical depth for the adding code

z€

Q = Extinction Efficiency =Extinction cross section

Geometric cross section

€

= nπr 2Qz

€

I T = I 0 exp(−τ

sinθ)

Next step in future work………

Adding/Doubling of Grain layer Scattering

By using the adding/doubling technique we can propagate the transmitted radiation throughout the layers and obtain the final reflectivity of the regolith.

Reflectivity

s

T

Part 2 Conclusion:

I have developed an end-to-end approach for regolith radiative transfer for monomers

Of arbitrary size, shape, and packing density

Approach has been tested in several regimes

Intriguing results are already apparentin three different layer porosities

(extinction does not scale simply as “tau” of independentparticles, spacing does matter..)

Next steps for future work:

Detailed spectral modeling of quartz data showing the effect of porosity, monomer shape, asperities, edges, etc on spectral bands

Adding doubling method for building thicker layers

The End.

Regolith = A layer of loose, heterogeneous material covering a surface.

Regolith Saturn’s ring particles?

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Scattered field in the shadow zone

E(x,y) = k2 (Pj * G)

y

x

FFT

E(, )

Densely packed ring due to inelastic collisions?

30m thick

Optically thick, Physically thin

Salo et al, French et al

Gentle inelastic collisions and weak gravity between particlesgive the rings the quality of a viscous fluid

Goldreich & Tremaine 1978, Bridges et al 1984 et seq., Wisdom & Tremaine 1988, Salo 1992

Cassini RSS 3 wavelength occultationBlue: small particles; red; larger particles, white: high tau

Marouf et al

CIRS measurement of ring Particle temperatures

70K 115K

Cassini VIMS results

Detailed, spatially resolved spectra show spatially variable near-IR slope interpreted as “Fe2+” (silicate) content

More in regions where particles are darker

Courtesy P. Nicholson & M. Hedman

Regional composition variations

OverviewIntro to scattering, why is scattering of light important in nature: fun pictures..blue sky, snow bank, saturns rings,

…etccontinuum medium scatteringwhat is a regolith? (regolith = A layer of loose, heterogeneous material covering a surface.)made up of grains, grains can form aggregates..interesting optical regimes: volume scattering, surface scattering, Christiansen frequency, restrahlen bands, …

etc Lead to FringFring intro:

Cassini spacecraft Ring system overview Ring particle size and compositionhigh phase geometry and dataice indicesmodelingmie emtddatoy modelconclusions

Layer Radiative Transfer review

Some unanswered questions:Quartz powder emissivity data and modelingCassini-CIRS (far infrared) data

Regolith Radiative Transfer (RRT) Model1. Discrete Dipole ApproximationIntroductionPBCNear field - FFTSlab testgranular layer

Conclusions and future goals We will use the new RRT model to determine the composition and extent of impurities imbedded inside

Saturn’s grainy ring particles.or any other regolith (space weathering, icy moons, etc)2. Adding/Doubling code• Emissivity4. can check a lot of currently only guessed-at properties of regolith scattering.

Just a few bullets here, of course..

Adding/Doubling Calculations using the matrix operator theory

Calculate transmission and reflection for a combined layer from the known transmission and reflection operators from two separate layers.

2

0

€

S2 = S1 +T1 Ι − S1S1( )−1

T1 , T2 = S1 +T1 Ι − S1S1( )−1

S1T1

€

S1 , T1

€

S1 , T1

€

S2 , T2

Emissivity calculation and comparison with CIRS

I = B (Tp) (1 - e-/ )

~

€

1−1

4πμS dμ 0dφ0∫∫

CIRS measures intensity I

Conclusion• Current RRT models do not account for close packing, coherent scattering, and irregular

particles.

• CIRS data can probe deeper into the particle and with our model we can get an idea on how impurities are buried within the regolith.

• This model has many applications in remote sensing of granular surfaces of solar system objects (icy satellites, comets, planetary surfaces,..).

Current and future work

• Obtain S,T functions for regolith layer with DDA using the near field calculation.

• Given the S & T functions from DDA, use the adding/doubling code to build an optically thick regolith.

• Calculate emissivity and vary refractive indices to find best fit with CIRS data.

• Use the refractive indices to infer composition.

• Use laboratory data to test new model against prior models (Hapke, Mie, etc..).