STUDY OF SCATTERING PROPERTIES OF AN ENSEMBLE OF RECTANGULAR PRISMS OF DIFFERENT COMPOSITION,
SIZE DISTRIBUTION, AND ASPECT RATIO:
A POSSIBLE APPLICATION TO STUDY THE COMETARY DUST GRAINS
1
2
23
R. Vilaplana , F. Moreno , A. Molina
1 Dpto. de Física Aplicada, Universidad Politécnica de Valencia, E.P.S.A., 03801 Alcoy, Spain.
2 Instituto de Astrofísica de Andalucía, CSIC, 18008 Granada, Spain.
3 Dpto. de Física Aplicada, Universidad de Granada, 18010 Granada, Spain.
THE OBSERVED POLARIZATION PROPERTIES OF COMETARY DUST
MODEL OF COMET NUCLEI
ƒ
Most of current models of comet nuclei assume that they are essentially aggregates of interstellar dust particles.
ƒ
Our knowledge about cometary dust grains is mainly inferred from their interaction with light.
ƒ
The dust grain is assumed to be composed of several particles of size of about tenth of a micrometer, each of which contains a
silicate core, a layer of complex organic material, and an outer layer of ices in which different molecules are embedded [1].
ƒ
The scattered light is partially polarized.
ƒ
All of those components have been observed in comet comae in one way or another.
ƒ
Different grain morphologies have been proposed for cometary grain but there is no a
Measurements from Earth show curves of the variation of the degree of linear polarization
ƒ
and phase function versus phase angle for different comets [3-4].
They show variations depending on the observed comet but they have certain common
ƒ
definite conclusion about the interstellar grain morphologies [2].
characteristics, as the persistence of a negative branch at small phase angles.
STUDY OF THE COMETARY GRAINS FROM THE SCATTERED LIGHT
EARTH
ƒ The measurements of linear polarization and phase function have information
ƒ
about the system (cometary grains).
ƒ These measurements are related to the scattering matrix elements, F11 and F12.
ƒ The fundamental problem when studying the scattered light by cometary grains is to obtain the
scattering matrix of the system for each scattering angle.
Iinc
SUN
θ
θ
φ
θ scattering angle
φ = 180º-θ phase angle
Isca
STOKES VECTOR
COMET
Isca
Qsca
Usca
Vsca
The sun light which is non polarized (incident light) interacts with cometary grains (medium).
These particles are of the different sizes and have irregular shapes.
These grains have random orientations.
The grains are separated enough among them, so that single scattering is dominant.
The sunlight is deviated in all directions and is partially linearly polarized (scattered light).
ƒ
ƒ
ƒ
ƒ
ƒ
I ~ Intensity
I
F11(θ)
F12(θ)
0
0
∝
Iinc
Qinc
Uinc
Vinc
F12(θ) 0
0
F22(θ) 0
0
0
F33(θ) F34(θ)
0 -F34(θ) F44(θ)
Q
Î
Q~
Linear polarization
U
U~
V
V ~ Circular polarization
F11(θ) is the phase function
P = -isFthe
F11(θ)
is the degree of the linear polarization of the scattered light
12(θ)/
F11(θ)
phase
function
P = - F12(θ)/ F11(θ) is the degree of the linear polarization of the scattered light
SCATTERING MATRIX
INITIAL VALUES OF THE MODEL PARAMETERS
A computational model is used to obtain the scattering properties of the cometary grains. The
model parameters are chosen according to cometary sources.
DISTRIBUTIONS
PARTICLES
SHAPES Æ constant distribution of rectangular prisms with different aspect ratios between
the main axis: 5:5:1, 5:1:1, 1:1:1, 5:3:1, 5:2:1, 2:1:2, 4:3:4, 5:4:2, 5:4:1
COMPOSITION (Heterogeneous)
MODEL PARAMETERS
SIZES Æ power law of negative index: n( r ) ≈ r - α assumed to be the same for each shape
with α1 = 1.8 for req [0.1 – 1.0] µm and α2 = 3.8 for req [10 – 100] µm
SHAPES (Compact rectangular prisms)
i)
The shape, size and chemical composition of the particles
ii)
The wavelength of the incident light.
iii)
The shape and size distributions of the particles
SIZES
req [0.1 , 100] µm
COMPUTATIONAL TECHNIQUES
WAVELENGHT RADIATION
req [0.1, 1] µm Æ Discrete Dipole Approximation (DDA)
req [10 , 100] µm Æ Ray Tracing (RT)
λ = 0.6µm
RESULTS
Composition 1:
5
33% n1 (λ)+67% n2 (λ)
0.35
10
α1=0.5 α2=3.8
α1=1.3 α2=3.8
α1=1.8 α2=3.8
α1=2.3 α2=3.8
0.5
(1:1:1)
(5:5:1)
(5:1:1)
(5:2:1)
(5:4:2)
(4:3:4)
(4:2:4)
(5:3:1)
(5:4:1)
1.2
10
15
20
25
50
75
100
Linear polarization
1.2
0.3
0.15
0.39
0.76
0.92
1.70
2.00
10.0
1
0.6
0.4
0.8
0
Linear polarization
0.2
Fig. 1
−0.2
0
20
40
60
80
100
120
Scattering angle(º)
140
160
180
1.2
0.10
0.31
0.37
0.42
0.52
0.63
0.74
0.92
1.00
1
Linear polarization
0.8
0.6
Linear polarization
1
0.8
10
−0.2
0
80
100
120
Scattering angle(º)
140
160
0
20
40
60
80
100
Scattering angle(º)
120
140
160
180
20
40
60
80
100
Scattering angle(º)
120
140
160
180
0
20
40
60
80
100
Scattering angle(º)
120
140
160
180
−0.1
0
20
40
60
80
100
Scattering angle(º)
120
140
160
180
Fig. 5 and Fig. 6 Averaged size and aspect ratio distribution of phase functions and linear polarization curves for a power-law size distribution function having power indices of α =3.8 for sizes between 10 and 100
2
µm and α varying from 0.5 to 2.5 for sizes between 0.1 and 1.0 µm. The shape distribution is assumed to be constant. Also shown are several measurements of the linear polarization in comets: Hyakutake, Hale1
Bopp, Halley, Bradfield+Liller+Levy, Austin.
Fig. 4 The size-averaged linear polarization resulting for each aspect ratio, for a power-law size distribution function having
a power index of α =1.8 for sizes between 0.1 and 1.0 µm.
180
Fig. 6
Fig. 5
1
−0.2
0
Fig. 3
60
0.1
−0.05
−0.1
0
40
0.15
0.05
10
Fig. 2
x Hyakutake
+ Hale Bopp
* Halley
v Bradfield+Liller+Levy
o Austin
2
10
0
0.4
0.2
20
3
10
0.1
0.2
0
0.25
0.2
0.2
0.6
0.4
−0.2
0
0.3
α1=0.5 α2=3.8
α1=1.3 α2=3.8
α1=1.8 α2=3.8
α1=2.3 α2=3.8
4
Linear polarization
0.4
F11
n1(λ)= 1.65 + 0.05 i ( silicates with additions of carbon )
n2(λ)= 1.31 + 0.01 i ( dirty ice )
1
Fig. 1, Fig. 2 and Fig. 3: Linear polarization curves for the heterogeneous rectangular prism with aspect ratio 5:5:1 having equivalent
radii varying from 0.1 µm to 100 µm.
A similar behaviour for all rectangular prisms occurs, as it is
shown in Fig.1, Fig.2 and Fig.3:
req << λ Æ Rayleigh limit
req ~ λ Æ Oscillating curves (negative branches)
req >> λ Æ Fresnel limit
n1(λ)= 1.88 + 0.71 i ( carbon )
n2(λ)= 1.58+ 0.003 i ( silicate )
2
10
0
50
100
150
0.4
0.2
0
−0.2
200
0
Scattering angle(º)
100
150
0.25
200
1
F33/F11
F22/F11
50
0.5
x Hyakutake
+ Hale Boop
* Halley
v Brafield+Lille+Levy
o Austin
0.3
Scattering angle(º)
1
ƒ An increment of the index α1 implies an increase of the maximum of positive polarization Pmax , the negative branch
remain without change. The values of α1 and Pmax are related [5].
0.5
0
x
+
*
v
o
0.3
α1=0.5 α2=3.8
α1=1.3 α2=3.8
α1=1.8 α2=3.8
α1=2.3 α2=3.8
0.25
0.2
0.15
0.1
0.05
Hyakutake
Hale Bopp
Halley
Bradfield+Liller+Levy
Austin
50
100
150
−1
ƒ From Fig.8 we can see that the compositional model of carbon+ silicates
produces a h parameter giving a better fits than that given by the model having
ice+silicates. However, the negative branch is better reproduce in the later case.
0.2
0.15
ƒ Fig.9 shows that a non-constant shape distribution produces the best results. This
shape distribution contains 50% of rectangular prisms with aspect ratio of 5:1:1
and the same proportions of all the others aspect ratio prisms.
0.1
0.05
200
0
0
Scattering angle(º)
0.5
F44/F11
1
0.1
0
−0.2
0
50
100
50
100
150
150
Fig. 8
−0.05
0
Fig. 9
20
40
60
80
100
120
140
160
180
Scattering angle(º)
0
−0.05
0
20
40
60
80
100
120
140
0
Scattering angle(º)
50
100
150
SUMMARY AND CONCLUSIONS
9 The main contribution to the negative polarization branch comes from absorbing compact
rectangular prisms with req ~ λ.
200
Scattering angle(º)
1
(1:1:1) Î (5:5:4) Î (5:5:3) Î (5:5:2) Î (5:5:1)
0
0
50
100
150
200
50
100
150
0
−1
200
50
100
150
−0.2
200
0
Scattering angle(º)
50
100
150
0
50
Scattering angle(º)
100
150
−1
200
0.5
0.1
0.5
−0.1
−0.2
−0.5
0
50
100
150
Scattering angle(º)
200
−1
Fig. 8
F44/F11
0.1
F34/F11
1
F44/F11
0.2
0
−0.1
0
50
100
150
200
−0.2
50
100
150
Scattering angle(º)
Scattering angle(º)
200
0
50
100
150
Fig. 9
50
100
150
200
50
100
150
Scattering angle(º)
0
200
50
100
150
0.2
0
−0.2
0
0.2
0.1
0
0
50
100
150
Scattering angle(º)
100
1
200
−0.5
0
50
100
150
200
Scattering angle(º)
References:
[1] Greenberg J.M. “Making a comet nucleus”, Astronomy and Astrophysics, 330, 375-380 (1998).
0.5
[2] Li A., Greenberg J.M. “Mid-infrared spectropolarimetric constraints on the core-mantle interstellar dust model”, The Astrophysical Journal, 577,789-794 (2002).
0
[3] Kiselev N.N. “Are there two populations of comets based on polarimetric properties of dust particles?”, Evolution and source regions of asteroids and comets proceedings of the 173rd
colloquium of the International Astronomical Union, 24-28 (1998).
−0.5
−1
0
50
100
150
Scattering angle(º)
Fig.10, Fig.11 and Fig.12 Averaged size scattering properties for a power-law size distribution function having power indices of α =3.8 for sizes between 10 and 100 µm and α = 1.8 for sizes between 0.1 and 1.0 µm for different rectangular prisms ranging in shape from (5:1:1) to (1:1:1) , (1:1:1) to (5:5:1) and (5:5:1) to (5:1:1).
2
150
Scattering angle(º)
9 Larger values of Pmax and abundance of small grain are related. This suggest that “dusty”
comets shows high values of Pmax while do not show any change in the negative branch.
9 The width and depth of the negative branch depend on the size and shape distributions.
Shape distributions having an overabundance of elongated particles give a better fits to the
cometary measurements.
9 The parameter h of the linear polarization curve decrease if the albedo of the grain
increase. This suggest that the composition 2 of the cometary grains is more probable than
the composition1 or either the “dirty” ice exists in another proportion in composition 1.
9 A more advanced version of the model will be performed by including irregular particles and
fluffy particles.
0
1
200 Fig. 10
50
0.5
−1
200
Scattering angle(º)
0.3
−0.1
0
0.4
1
0.5
0
200
Scattering angle(º)
0
−1
0
(5:1:1)
(5:2:1)
(5:3:1)
(5:4:1)
(5:5:1)
0.6
Scattering angle(º)
−0.5
0
10
200
1
Scattering angle(º)
1
0
150
0
0.2
0
100
−0.5
0
200
50
Scattering angle(º)
1
0.5
0.5
2
10
0
0
Scattering angle(º)
−0.5
0
0
1
F22/F11
F33/F11
F22/F11
0.5
0
10
Scattering angle(º)
1
0.5
Lineal polarization
−0.2
200
4
10
0
F33/F11
150
0.2
F44/F11
100
0.4
0
Scattering angle(º)
(1:1:1)
(5:5:4)
(5:5:3)
(5.5:2)
(5.5:1)
0.6
2
10
F22/F11
50
0.2
F34/F11
0
1
10
(5:1:1)
(5:2:2)
(5:3:3)
(5:4:4)
(1:1:1)
0.4
Lineal polarization
0
10
4
0.6
F33/F11
Lineal polarization
F11
2
F11
4
(5:5:1) Î (5:4:1) Î (5:3:1) Î (5:2:1) Î (5:1:1)
F11
(5:1:1) Î (5:2:2) Î (5:3:3) Î (5:4:4) Î (1:1:1)
10
180
1
Fig. 7 The size-averaged scattering properties resulting for each aspect ratio, for a power-law size distribution function having a power
index of α =1.8 for sizes between 0.1 and 1.0 µm.
10
160
Scattering angle(º)
Fig. 8 and Fig. 9 Averaged size and aspect ratio distribution of linear polarization curves for a power-law size distribution function having power indices of α =3.8 for sizes between 10 and 100 µm
2
and α varying from 0.5 to 2.5 for sizes between 0.1 and 1.0 µm. The shape distribution is assumed to be constant (right panel) and non constant (left panel).
−0.5
−1
200
0
200
Scattering angle(º)
0.2
−0.1
F34/F11
ƒ We observe from Fig.7, Fig.10, Fig.11 and Fig.12 that the precise shape of the
negative branch depends on the aspect ratio of the rectangular prisms. The
rectangular prisms that produces the closest match to the cometary observations
are those having a 5:1:1 aspect ratio.
α1=0.5 α2=3.8
α1=1.3 α2=3.8
α1=1.8 α2=3.8
α1=2.3 α2=3.8
−0.5
0
F34/F11
ƒ The magnitude of the maximum of the positive branch changes with the power index. The negative branch is found to be
nearly independent of the index of the power law distribution.
0.35
0.35
(1:1:1)
(5:5:1)
(5:1:1)
(5:2:1)
(5:4:2)
(4:3:4)
(4:2:4)
(5:3:1)
(5:4:1)
0.6
Linear polarization
Lineal polarization
F11
10
0
ƒ The obtained linear polarization (Fig.6) shows a negative branch at scattering angles larger than 160º.
ƒ A maximum of polarization of less than about 40% at
intermediate scattering angles in all cases.
13% n1 (λ)+87% n2 (λ)
4
10
ƒ We observe in Fig.5 an increment in the backscattering direction for all index values in the phase function as it is
observed in comets.
ƒ A negative polarization branch appears in most cases at
scattering angles larger than about 160º.
Lineal polarization
Composition 2:
Figure 4 shows:
200
[4] Levasseur-Regourd A.C., Hadamcik E., Renard J.B.”Evidence for two classes of comets from their polarimetric properties at large phase angles”, Astronomy & Astrophysics, 313, 327-333
(1996).
[5] Vilaplana R., Moreno F. , Molina A. “Computations of the single scattering properties of an ensemble of compact and inhomogeneous rectangular prisms: implications for cometary dust “ ,
Journal of Quantitative Spectroscopy and Radiative Transfer, 88, 219-231 (2004).