The Astrophysical Journal, 701:1426–1435, 2009 August 20 doi:10.1088/0004-637X/701/2/1426C© 2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

ULTRAVIOLET RADIATION INSIDE INTERSTELLAR GRAIN AGGREGATES. III. FLUFFY GRAINS

Adriano Cacciola1, Cesare Cecchi-Pestellini

2, Rosalba Saija

1, Maria Antonia Iatı

3,

Ferdinando Borghese1, and Paolo Denti

11 Dipartimento di Fisica della Materia e Ingegneria Elettronica, Universita di Messina, Salita Sperone 31, 98166 Messina, Italy

2 INAF-Osservatorio Astronomico di Cagliari, Strada n.54, Loc. Poggio dei Pini, 09012 Capoterra (CA), Italy3 Istituto per i Processi Chimico-Fisici del CNR, Salita Sperone, 98158 Faro Superiore, Messina, Italy

Received 2008 October 10; accepted 2009 June 24; published 2009 July 31

ABSTRACT

We study the problem of light depolarization within fluffy interstellar dust grains, in which coagulation generatesinterstitial cavities partly filled with icy condensed gas. A significant amount of elliptical polarized ultravioletradiation may be generated in situ by geometry-induced depolarization within the cavities. The behavior of thedepolarization is studied both by changing the orientation of the aggregates and by changing chemical compositionand size of the subunits forming the aggregate. We found that a considerable amount of depolarization occurs withinthe ice mantles of the subunits, provided their thickness is not too large. We discuss the implications of these resultsfor chiral selection in space.

Key words: dust, extinction – polarization – radiative transfer

1. INTRODUCTION

Dust grains are important constituents of many astrophysicalregions, as they are the dominant source of opacity in the inter-stellar medium, dimming and reddening starlight. The knowl-edge of the dust properties is constrained by continuum absorp-tion and emission features in the infrared, by observational dataon elemental depletions with respect to solar or other standardsalong various lines of sight, and by the results of models ofgrain optics describing interstellar extinction, polarization, andscattering. It appears that the larger grains, at least, must benonspherical in order to be capable of being partially alignedby, e.g., the interstellar magnetic field by inclusions of magneticmaterial (Martin et al. 1999), and porous in order to fulfill abun-dance constraints (Voshchinnikov et al. 2006). Indeed, a numberof methods have been developed to calculate the optical prop-erties of irregularly shaped particles such as the discrete dipoleapproximation (Purcell & Pennypacker 1973) and the T-matrixmethod (Waterman 1971). Internal field effects have been, ingeneral, ignored. However, there are instances, such as, e.g.,the computation of the internal heating (Dusel et al. 1979), inwhich it is necessary to know the electromagnetic field withinthe particle.

The knowledge of internal fields and their polarization pat-terns may also have implications in space chemistry. Interstellardust grain surfaces are known to be the site of H2 formationand to be implicated in a much wider range of reactions in-volving atoms, molecules, and radicals (e.g., Herbst & Cuppen2006), giving rise to a profuse organic chemistry. During thesequence that brings matter from a molecular cloud to a fullydeveloped planetary system, complex organic molecules areformed, transformed and preserved until they become incor-porated into comets and meteorites. Some of these moleculesare frozen onto grain mantles while the protostar slowly losesits envelope and material collects along the equatorial plane ofthe protoplanetary disk. Grains are then incorporated into plan-etesimals and, ultimately, into planets and small bodies. Thepeculiar isotopic fractionation (e.g., deuteration), imprinted indust molecular mantles, retains a memory of the prior eras. Dustgrain aggregates formed during the early history of the solar sys-

tem were likely porous and fluffy with much of their internalvolume being partly filled with molecular material as a resultof ice percolation and coagulation of smaller-size grains. In thisphase, the interior of dust grain aggregates offers the possibilityof a peculiar in space, terrestrial type, chemistry induced bycosmic rays, and/or high energy radiation (Duley 2000). As aconsequence, a range of molecular species much wider than theordinary gas phase and surface chemistry in molecular clouds,e.g., very complex organic molecules, is expected.

Indeed, complex chiral organics, such as amino acids andsugars, have been found in some carbonaceous meteorites (e.g.,Pizzarello et al. 2003), providing a unique natural record ofprebiotic chemical evolution. Some amino acids also showL-enantiomeric excess, and suggest the intriguing possibilitythey may have contributed to terrestrial homochirality by directinput of meteoritic materials to the early Earth. Several contro-versial theories have been developed to explain an abiogenicorigin of the chiral homogeneity in terms of the physicochemi-cal processes involved (Bonner 1991). Among the many scenar-ios proposed, one involves the asymmetric photolysis of aminoacids in space, triggered by Ultraviolet Elliptical Polarized Light(UV EPL; Bailey 2001, and references therein). It has been ex-perimentally proved that an enantiomeric excess is producedwhen amino acids in racemic mixtures are exposed to UV EPL(Flores et al. 1977; Bonner & Bean 2000). The differential pho-tolysis by UV EPL has its basis on the unequal absorption of thetwo enantiomers of a chiral molecule. Observations of youngstellar objects reveal the existence of elliptical polarized radi-ation in the infrared, with a substantial degree of polarization(about 20%) detected only in massive star formation regions(Bailey et al. 1998). Although model calculations seem to indi-cate that similar circular polarization levels should be obtainedin the IR and UV domains (Lucas et al. 2005), the presence ofUV EPL appears uncertain at the present state of the art. Re-views of possible sources of UV EPL in space are presentedby Bailey (2001), Jorissen & Cerf (2002), and Meierhenrich &Thiemann (2004).

Asymmetric photolysis in space has been questioned on thebasis that every proposed mechanism induced by an externalsource of UV EPL has to meet with the Kuhn–Condon rule

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No. 2, 2009 UV RADIATION INSIDE INTERSTELLAR GRAIN AGGREGATES. III. 1427

Table 1Coordinates (in nm) of the Centers of the Spheres that Compose the Planar

Aggregate Depicted in Figure 1

α x y α x y

0 0.000 0.000 9 0.000 100.0001 50.000 0.000 10 −43.301 75.0002 0.000 50.000 11 −93.301 25.0003 −25.000 −43.301 12 −85.355 −35.3554 −50.000 0.000 13 −110.360 −78.6575 18.301 −68.301 14 110.360 −78.6576 85.355 −35.355 15 118.300 68.3017 93.301 25.000 16 −118.300 68.6578 43.301 75.000

0

1

2

3

4

5

6

7

8

910

11

12

1314

1516

y

x

Figure 1. Geometry of the planar aggregate of spheres of silicates. The radiusof all spheres is � = 25 nm, and sphere 0 contains a centered empty cavity ofradius �c = 6.25 nm = 0.25 �.

(Cerf & Jorissen 2000). UV EPL sources operate only at a sin-gle photochemically active circular dichroic band, which is notshared by all biogenic amino acids. However, asymmetric catal-ysis and chirality transfer (Soai et al. 1995; Bandyopadhyayet al. 2007) are changing the astronomical perspective, defo-cusing the quest for biogenic amino acid selection to the es-tablishment of an enantiomeric excess in space. The very highenantiomeric excess of L-isovaline (�15%) measured in theMurchison and Murray meteorites (Pizzarello et al. 2003) sug-gests that asymmetric UV photolysis was not the sole causationof the enantiomeric excess (cf. Bailey 2001, Table 1).

In previous papers (Cecchi-Pestellini et al. 2005; Saija et al.2005; Borghese et al. 2005), we proposed a scenario in whichchiral molecules formed in the cavities of interstellar aggregatesare exposed to asymmetric photolysis induced by an effectiveUV EPL generated in situ by geometry-induced depolarizationof incident linearly polarized waves. A noticeable amount ofdepolarization occurs in the UV frequency range, whereas nodepolarization occurs in the visual and in the infrared range.As a consequence, cavities inside dust grains might provide asuitable environment in which interstellar molecules can evolveand be selected into complex prebiotic chiral species of a givensymmetry.

In the present paper, we present a natural extension ofour study to the case in which dust grains are modeled as a

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16−0.05

0

0.05

0.1

0.15

VS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16−0.05

0

0.05

0.1

0.15

NS

λ = 0.10 μm

λ = 0.22 μm

Figure 2. Vs within the cavity contained in sphere 0 of the aggregate of Figure 1,as a function of the number of the added spheres, for two different wavelengthsof the incident wave.

fluffily substructured collection of stratified spheres composedby a mineral core covered by an icy mantle. As such amodel includes the presence of interstitial voids generatedby coagulation of particles, we extend the analysis of Saijaet al. (2005) to the state of polarization of the UV field thatpropagates through such interstitial cavities. In our previousanalysis of field depolarization, we found that both the extentof depolarization and the sense of rotation of the fields dependon the position of cavities with respect to the incident wave. Inother words, for such a mechanism to work, grain aggregatesshould be at least partly aligned. Therefore, in the present workwe perform a study of the stability of polarization patternsas functions of the orientation of the aggregate with respectto the incident field. The study will be confined to likelyrotations of the aggregate when embedded in a protoplanetarydisk. By studying the behavior of grain particles in a gasflow, Nakamura & Hidaka (1998) were able to show that,even in a subsonic regime, aggregates preferentially assumean aligned configuration. Such alignment is also favored bythe exchange of angular momentum between the incidentradiation and the particles (e.g., Draine & Weingartner 1996;Borghese et al. 2007b; Hoang & Lazarian 2009). The fieldconsidered in the present study is the near field whose stateof polarization, for reasons fully explained elsewhere (Saijaet al. 2005; Borghese et al. 2005, 2007a), cannot be describedby the customary set of Stokes parameters. Therefore, we haveto resort to a more general description of the polarization. Thefield scattered in the near zone, and in particular within theinterstitial cavities, by an aggregate of spherical scatterers ofknown geometry, is computed through a technique described inBorghese et al. (1984) and validated at first by comparison withexperimental data (Borghese et al. 1989) and further validated ina number of applications (e.g., Sindoni et al. 2006). An accuratedescription of the far-field optical properties of such particles,e.g., absorption and scattering cross sections, can be found inIatı et al. (2004). Interested readers are referred to Borgheseet al. (2007a).

In Section 2, we present a brief outline of the method.Section 3 contains a collection of results. Our conclusions arepresented in Section 4.

1428 CACCIOLA ET AL. Vol. 701

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0

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30

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0

10

20

30

x

y

z

34

21

Figure 3. Geometry of the aggregate of spheres of equal radius � = 12 nm. The position of the centers of the spheres has been chosen in order to generate themaximum number of interstitial cavities. The darker spheres have a core of carbon, whereas the lighter ones have a core of silicates. All the spheres are covered by amantle of water ice with thickness 25% �. Vertical lines indicate the position of resulting interstitial cavities.

0.1 0.15 0.2 0.25 0.3−0.02

0

0.02

0.1 0.15 0.2 0.25 0.3−0.02

−0.01

0

0.1 0.15 0.2 0.25 0.30

0.2

0.4

λ(μm)

Vs

y−direction

x−direction

z−direction

Figure 4. Vs within cavity 1 contained in the aggregate of Figure 3. Labels referto the direction of propagation of the incident wave. The polarization is alongthe y-axis for incidence in the x-direction and along the x-axis for incidenceboth along the y- and the z-direction.

2. THEORY

The model aggregates consist of a number of clusteredstratified spheres, numbered by the index α, whose centers lieat Rα , and whose radius and complex refractive index are �α

and nα , respectively. The field incident on the aggregate is thelinearly polarized plane wave of circular frequency ω,

EI = E0 eI exp (ikI · r) , (1)

where eI is the polarization vector, kI = nkv kI, in which n isthe real refractive index of the surrounding medium, kv = ω/c,and kI is the direction of propagation. The field scattered bythe whole aggregate is computed, without invoking any far zone

−0.15 −0.1 −0.05 0 0.05 0.1 0.15−0.1

0

0.1

0.2

0.3

−0.15 −0.1 −0.05 0 0.05 0.1 0.15−0.2

0

0.2

0.4

0.6

δz

VS

S

I

I IC

I C

CI

ICS

λ = 0.22 μm

λ = 0.1 μm

Figure 5. Vs within cavity 1 contained in the aggregate of Figure 3, as a functionof the penetration of the radiation into the aggregate, for incidence along thez-axis, and polarization along the x-axis. δz is the z coordinate scaled by the sizeof the aggregate. S: silicate sphere; I: ice layer; IC: interstitial cavity; C: carbonsphere.

approximation, as the superposition of the fields scattered byeach sphere,

ES(r) =∑

α

ES α . (2)

Then, we expand all the fields in a series of vector multipolefields: for the incident plane wave we have

EI =∑plm

J (p)lm (r, k) W

(p)lm (eI, kI) (3)

whereas, for the field scattered by the αth sphere,

ES α =∑plm

H (p)lm (rα, k)A(p)

αlm . (4)

No. 2, 2009 UV RADIATION INSIDE INTERSTELLAR GRAIN AGGREGATES. III. 1429

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020

4060

80

−40

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0

20

40

−40

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0

10

20

30

40

x

y

z

132

Figure 6. Geometry of the aggregate of spheres of different radii. The darker spheres have a core of carbon, whereas the lighter ones have a core of silicates. All thespheres are covered by a mantle of water ice whose thickness is 25% of the corresponding radius. Vertical lines indicate the position of resulting interstitial cavities.

The amplitudes A(p)αlm depend both on the polarization and the

direction of incidence of EI, and rα = r − Rα . In Equations (3)and (4), we define the multipole fields (Rose 1956)

J (1)lm(r) = jl(kr)X lm(r), J (2)

lm(r) = 1

k∇ × J (1)

lm (5)

that are regular at the origin, and the multipole fields H (p)lm (r)

satisfying the radiation condition at infinity. The latter areidentical to the J multipole fields except for the substitutionof the spherical Hankel functions of the first kind hl(kr) inplace of the spherical Bessel functions jl(kr). The X lm arevector spherical harmonics (Jackson 1975) and the superscriptp is a parity index that distinguish the magnetic multipolefields (p = 1) from the electric ones (p = 2). The multipoleamplitudes of the incident plane wave W

(p)lm (eI, kI) are defined

asW

(p)lm (eI, kI) = 4πip+l−1 eI · Z(p)∗

lm (kI) , (6)

where

Z(1)lm(r) = X lm(r), Z(2)

lm(r) = X lm(r) × r (7)

are transverse harmonics (Fucile et al. 1997). The field withinthe αth sphere is expanded as

ET α = E0

∑plm

J (p)lm (rα, kα)C(p)

α lm (8)

with kα = nαkv, if the spheres are homogeneous, while minormodifications are required in the case of stratified or of radiallynonhomogeneous spheres (Iatı et al. 2008). The amplitudesA(p)

αlm

and C(p)αlm are determined by imposing to the field the boundary

conditions across the surface of each of the spheres in theaggregate.

The expansion of the scattered field includes multipoles fieldswith different origins. However, using the appropriate addition

theorem for vector multipole fields (Borghese et al. 1980), thefield scattered by the whole aggregate at the surface of the αthsphere turns out to be

ES = E0

∑α

∑plm

[H (p)

lm (rα, k)A(p)αlm

+∑α′

∑p′l′m′

J (p′)l′m′(rα, k)H(p,p′)

αlmα′l′m′ A(p′)α′l′m′

], (9)

where the quantities H(p,p′)αlmα′l′m′ , whose explicit expression is

given in Borghese et al. (2007a), effect the translation of originof the H multipole fields from Rα′ to Rα . Also, the multipolefields that enter the expansion of the incident plane wave areexpressed in terms of multipole fields centered at rα with theresult

EI = E0

∑plm

∑p′l′m′

J (p)lm (rα, k)J (pp′)

αlm0l′m′ W(p′)l′m′ , (10)

where the quantities J (pp′)αlm0l′m′ are the analogous of the quantities

H(p,p′)αlmα′l′m′ for the J multipole fields.We are now able to impose the customary boundary condi-

tions across the surface of each one of the spheres and get foreach α four equations among which the elimination of the am-plitudes of the internal field is trivial. The procedure leads to thesystem of linear nonhomogeneous equations for the amplitudesof the scattered field∑

α′

∑p′l′m′

M(pp′)αlmα′l′m′ A(p′)

α′l′m′ = −W (p)αlm , (11)

where we define

W (p)αlm =

∑p′l′m′

J (pp′)αlm0l′m′ W

(p′)l′m′ (12)

1430 CACCIOLA ET AL. Vol. 701

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0

0.1

0.2

0.1 0.15 0.2 0.25 0.3−0.1

0

0.1

0.2

λ (μm)

VS

x−direction

y−direction

z−direction

0.1 0.15 0.2 0.25 0.30

0.05

0.1 0.15 0.2 0.25 0.3−0.04

−0.02

0

0.1 0.15 0.2 0.25 0.30

0.5

1

λ (μm)

VS

x−direction

y−direction

z−direction

0.1 0.15 0.2 0.25 0.3

−0.04

−0.02

0

0.1 0.15 0.2 0.25 0.3

−0.6

−0.2

0

λ (μm)

VS

0.1 0.15 0.2 0.25 0.3

0 0.005

0.015 x−direction

y−direction

z−direction

Figure 7. Vs as a function of wavelength within the three cavities of the aggregateshown in Figure 6. Cavity 1: upper panels; cavity 2: middle panel; cavity 3:lower panel. Labels refer to the direction of propagation of the incident wave.The polarization was chosen to be along the y-axis for propagation along thex-direction, and along the x-axis for propagation both along the y- and thez-direction.

and

M(pp′)αlmα′l′m′ = (

R(p)αl

)−1δαα′δpp′δll′δmm′ + H(p,p′)

αlmα′l′m′ . (13)

Note that the quantities R(p)αl are the Mie coefficients for the

scattering by the αth sphere.

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06−0.1

0

0.1

0.2

0.3

δz

VS

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06−0.1

−0.05

0

0.05

0.1

I

λ= 0.22 μm

C IC I C

I I C

λ= 0.1 μm

C carbon − I ice − IC interstitial cavity

C IC

−0.1 −0.05 0 0.050

0.01

0.02

0.03

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δz

VS

C I IC I S

λ = 0.1 μm

ICC I I S

λ = 0.22 μm

−0.05 0 0.05 0.1−0.2

−0.15

−0.1

−0.05

0

VS

−0.05 0 0.05 0.1−0.8

−0.6

−0.4

−0.2

0

δz

λ=0.22 μm

λ=0.1 μm

S IIC

I C

I IC IS C

Figure 8. Vs within cavities 1, 2, and 3 (top to bottom) contained in the aggregateof Figure 6, as a function of the penetration of the radiation into the aggregate,for incidence along the z-axis and polarization along the x-axis. δz is the z

coordinate scaled by the size of the aggregate. S: silicate sphere; I: ice layer;IC: interstitial cavity; C: carbon sphere.

Once system (11) has been solved, Equations (2) and (4)yield the scattered field at any point in space. Moreover, theboundary conditions yield, in terms of the A(p)

αlm, the amplitudesC(p)

αlm needed in order to compute the field within the spheres.

No. 2, 2009 UV RADIATION INSIDE INTERSTELLAR GRAIN AGGREGATES. III. 1431

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3−0.08

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−0.04

−0.02

0

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

0.05

0.1

0.15

Tice

(nm)

Vs

λ=0.1μm

λ=0.22μm

Figure 9. Vs as a function of the thickness of the ice mantle within cavity 1for incidence along the z-axis and polarization along the x-axis. The solid anddashed lines report the results at the nearest and the farthest point of the cavitymet by the z-axis, respectively.

0 100 200 300−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

θ (degree)

Vs

0 100 200 300−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

θ (degree)

λ=0.1μm

λ=0.22μm

λ=0.1μm

λ=0.22μm

Figure 10. Vs as a function of the angle of rotation of the aggregate aroundthe x-axis, θ , for two different wavelengths of an incident wave propagatingalong the z-direction. Vs is computed at the location x = −56.303 nm,y = z = 0 nm, i.e., on the rotation axis of the aggregate. Left panel: wavepolarization along the x-axis; right panel: wave polarization along the y-axis.

The multipole expansions are truncated at an order l to ensurethe convergency at the required degree of approximation.

According to Equation (2) the scattered field, including itstime dependence, has the general form

E(r, t) = ES(r) exp(−iωt) . (14)

Since E is monochromatic, the direction of propagation of theassociated electromagnetic energy is given by the time-averagedPoynting vector S = Re[S], where

S = c

8π

[ES × H∗

S

](15)

is the complex Poynting vector (Jackson 1975). Both the electricand magnetic fields are orthogonal to S, so that, in this sense,the electromagnetic waves are transverse (Born & Wolf 2002).Nevertheless, the transversality of the field does not imply that,at any point, the plane swept, e.g., by the electric field be

0.1 0.15 0.2 0.25 0.3−0.2

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0.2

0.1 0.15 0.2 0.25 0.3−0.4

−0.2

0

λ (μm)

VS

Cavity 3

Cavity 2

Cavity 1

z−direction

Figure 11. Vs as a function of wavelength within the three cavities of theaggregate shown in Figure 6, in which all subunits are made of carbonaceousmaterials covered by an ice mantle. Cavity 1: upper panels; cavity 2: middlepanel; cavity 3: lower panel. The incidence is along the z-direction and thepolarization is along the x-axis.

0.1 0.15 0.2 0.25 0.3−0.2

−0.1

0

0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.1 0.15 0.2 0.25 0.3−0.5

0

0.5

λ (μm)

VS

Cavity 2

Cavity 1

Cavity 3

z−direction

Figure 12. Same as in Figure 11, for a silicate aggregate.

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Figure 13. Geometry of the reduced aggregate that bounds the interstitialcavity 1.

1432 CACCIOLA ET AL. Vol. 701

0.1 0.15 0.2 0.25 0.3−0.02

0

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0.04

VS

0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.1 0.15 0.2 0.25 0.3−0.2

0

0.2

λ (μm)

z−direction

y−direction

x−direction

Figure 14. Vs as a function of wavelength within the interstitial cavity of thereduced aggregate. Labels refer to the direction of propagation of the incidentwave. The choice of the polarization is the same as in Figure 7.

orthogonal to the direction of S. The orthogonality is granted forhomogeneous plane waves, but not for monochromatic wavesof the general form. However, it is not difficult to show that amonochromatic wave of the general form is elliptically polarized(Borghese et al. 2005). Since the plane of the polarization ellipseshould contain both ES and its complex conjugate E∗

S, the realvector (Carozzi et al. 2000)

V = i ES × E∗S (16)

individuates the normal to that plane and, according to thepreceding definition, ES rotates counterclockwise with respectto V . Moreover,

|V | = 2ab = 2

πA , (17)

where a and b are the semiaxes of the polarization ellipse of areaA. Thus, |V | vanishes for linear polarization, but when |V | �= 0,

V × S �= 0 , (18)

so that the plane of the polarization ellipse is not orthogonal toS. Therefore, we assume that the quantity

Vs = V · S/|S| , (19)

i.e., the projection of V along the direction of propagation ofthe electromagnetic energy, gives the sense of rotation of thefield with respect to S. Note that, since V and S can neverbecome mutually orthogonal, Vs = 0 only for linear polarization(|V | = 0).

The definition of V can be derived in a natural way from thetheory of the spectral density tensor (Mishchenko et al. 2004)that yields the most general description of the polarization ofthe electromagnetic field.

3. RESULTS

In previous papers (Saija et al. 2005; Borghese et al. 2005) wefound that the occurrence of depolarization within cavities em-bedded into a homogeneous sphere is a geometrical asymmetryeffect. In fact, given a linearly polarized incident field, V = 0when the spherical cavity is centered within a homogeneous

sphere, while V �= 0 when the cavity becomes eccentric. Thisresult holds true for whatever choice of the material of the hostsphere.

The aim of the present study is to assess the presence of trendsand regularities in the wave depolarization pattern, exploiting amore representative model for grain aggregates. First, we wantto determine whether the depolarization within a centered cav-ity can be induced by the asymmetry of the environment. Westart our study by considering a sphere of astronomical sili-cates (Draine 1985), hereafter sphere 0, with radius � = 25 nmcontaining a centered spherical void of radius �c = 6.25 nm(�c = 0.25 �). We then progressively add to this sphere 16other homogeneous spheres of silicates whose radius is still� = 25 nm and whose centers lie in the xy plane of a frameof reference with origin at the center of sphere 0. The finalaggregate, after all the spheres have been added, is shown inFigure 1, whereas the coordinates of the centers of the spheresare reported in Table 1. We consider a linearly polarized planewave and directions of incidence both in and perpendicular tothe plane of the aggregate (i.e., along the z-axis), and we calcu-late Vs in several points within the cavity contained in sphere 0.We found that, as expected for symmetry reasons, no depo-larization occurs for incidence in the plane of the aggregate,both for polarizations along the z-axis and in the plane of theaggregate, as the xy plane is a plane of reflection. Quite dif-ferent results are obtained when the incidence occurs along thez-axis. We report in Figure 2 the value of Vs as a function ofthe number of the added spheres for wavelengths λ = 0.1 μmand λ = 0.22 μm. Depolarization values are reported at a pointof coordinates x = 0, y = 0, z = −4.1 nm, for polarizationalong the x-axis. For both the wavelengths we consider, whensphere 1 is added, the value of Vs becomes slightly differentfrom 0. In fact, the breakdown of the spherical symmetry iscomparatively small, as the addition of sphere 1 still leaves acylindrical symmetry around the x-axis, which makes virtuallyequivalent any direction of incidence in the plane yz. The addi-tion of sphere 2 yields a noticeable depolarization of the fieldas the aggregate becomes much less symmetric. Adding morespheres the polarization of the field remains nonvanishing untilthe aggregate, while growing, approaches an overall cylindricalsymmetry around the z-axis. Ultimately, our results show thatdepolarization within a centered cavity can occur provided theenvironment yields enough asymmetry.

We now address the problem of field depolarization within in-terstitial cavities arising from coagulation of spherical subunitsto form an aggregate. Since planar aggregation cannot yield in-terstitial cavities, we aggregate 25 spheres of radius 12 nm, cov-ered with a mantle of water ice (Warren 1984) of thickness 25%of the total radius, the position of the centers of the spheres be-ing chosen, in three dimensions, so as to generate the maximumnumber of interstitial cavities. The core of 12 randomly selectedspheres is of silicates (Draine 1985), and the remaining 13 coresare of carbon (Rouleau & Martin 1991). The resulting aggregateis shown in Figure 3, whereas the coordinates of the centers ofthe spheres are reported in Table 2. Note that the coordinate axesare principal axes of inertia for the aggregate, so that rotationsaround these axes are dynamically stable. Aggregation leads tofour interstitial cavities that are indicated by vertical lines inFigure 3. We calculated Vs at several points within the cavitiesfor incidence along all coordinate axes (which is equivalent toa rotation of the target grain), in the wavelength range from 0.1to 0.3 μm. As the results are quite similar in all the cavities,we report in Figure 4 the values of Vs in a point close to the

No. 2, 2009 UV RADIATION INSIDE INTERSTELLAR GRAIN AGGREGATES. III. 1433

Table 2Coordinates (in nm) of the Centers of the Spheres that form the Aggregate of

Figure 3

α x y z

1* 28.435 12.236 0.2282* 44.276 −11.303 18.8563* 43.974 −9.872 −20.3094* 10.529 −25.258 −1.0045* −26.970 8.606 −19.0876* −34.664 19.303 0.9727* 22.617 −0.977 19.4008* 48.218 25.821 0.5729* 74.019 3.792 −0.43110* 6.776 22.562 0.77211* −73.836 23.299 −0.28212* −100.170 5.558 0.97413 −13.006 8.977 0.42814 8.652 −1.348 −0.11515 30.311 −11.674 −0.65916 50.094 1.910 −0.31617 −76.248 7.439 0.85918 −5.311 −1.719 −19.63119 −11.130 −14.932 −0.45920 22.315 0.453 −19.76521 −5.010 −3.149 19.53322 −26.668 7.175 20.07723 −56.714 −6.488 0.19924 51.969 −21.999 −1.20425 −32.788 −4.606 0.084

Notes. All the spheres have radius � = 12 nm and are composed either ofsilicates or carbon. The asterisk denotes the spheres of silicates.

Table 3Coordinates (in nm) of the Centers and Radii of the Spheres Composing the

Aggregate of Figure 6

α x y z �

1* −63.071 −12.575 −1.582 12.0002* −62.864 11.422 −1.923 12.0003* −1.8398 −16.077 0.318 15.0004* −10.682 −0.85711 10.285 5.2275* −33.611 −34.815 −0.370 14.0006* 29.897 −1.3386 1.063 20.0007* −81.398 −0.42558 −2.311 10.0008* 49.660 −27.748 2.043 13.0009* 7.261 −1.3015 −10.075 5.22710* 72.971 −27.868 7.751 11.00011* −34.284 −0.35692 31.128 12.00012* −32.502 −1.2108 −27.838 7.00013 −33.318 −0.82002 −0.853 20.00014 −56.303 −0.50909 7.054 4.30915 −55.783 −0.75808 −10.140 4.30916 −1.580 13.919 −0.108 15.00017 −10.051 −1.1595 −10.600 5.22718 −32.998 36.175 −1.379 17.00019 50.145 28.426 1.244 16.00020 6.630 −0.999 10.810 5.22721 59.882 −1.584 1.973 10.00022 65.481 28.610 23.466 11.00023 66.718 28.017 −17.469 9.00024 30.923 −1.830 −32.917 14.00025 −40.503 −1.287 −38.085 6.000

Notes. The numbers marked with an asterisk denote the spheres of silicates, theother spheres are of carbon. All the spheres are covered by a layer of water icewhose thickness is 25% of the total radius.

center of cavity 1 as a representative example of the behaviorof the field depolarization. Figure 4 shows that there is a sig-nificant degree of depolarization only for z-incident waves, andthat Vs �= 0 even for wavelengths larger than 0.2 μm, contraryto our expectation based on the results of Saija et al. (2005). Thepolarization of the incoming wave is along the x-axis. Similarresults, but with opposite sign, are found for polarization alongthe y-axis. The rotation of the polarization direction is equiva-lent to a physical rotation of the aggregate around the z-axis. Wehave also investigated the behavior of depolarization along a linecrossing the interstitial cavity. We consider three lines parallelto the axes and crossing the cavity approximately at its center,and two representative wavelengths λ = 0.1 and 0.22 μm. Onlythe line along the z-axis shows non-negligible values of Vs notonly within the interstitial cavity but also within the ice layer(see Figure 5 for the wave polarization along the x-axis). Inthe ice layer, which is the only relevant region for chemistry,significant depolarization occurs only at λ = 0.22 μm. Whenthe polarization becomes parallel to the y-axis equivalent to arotation of 90◦ about the z-axis of the aggregate, the profile ofVs changes maintaining roughly the same shape with invertedsign.

Since the depolarization is geometrical in nature, we alsoexploited a more complex aggregate in which the radii of the25 subunits are randomly chosen. The resulting geometry isshown in Figure 6, whereas both coordinates of the centersand radii of the composing spheres are reported in Table 3.Even in this case, the coordinate axes are the principal axesof inertia of the aggregate. The aggregate shown in Figure 6forms three interstitial cavities, in which we computed Vsfor incidences along the coordinate axes in the wavelengthrange 0.1 � λ � 0.3 μm. The results that are reported inFigure 7 show that a much lower depolarization occurs forincidence along the x-axis, as well as along the y-axis. Actually,the polarization is indicated in the panels of Figure 7. Weconclude our analysis by reporting in Figure 8 the results ofour computations of Vs as a function of the penetration of thefield within the aggregate. Again we choose three lines parallelto the coordinate axes, crossing the cavities approximately attheir centers, and we show the results for the two representativewavelengths λ = 0.1 and 0.22 μm. Again, as seen in Figure 5,in the ice layer significant depolarization occurs at only onewavelength. Different choices for the polarization of the incidentwave produce similar results. We also changed the thicknessof the water ice mantle of the subunits, and we studied thevariations induced in the depolarization. In Figure 9, we presentVs as a function of the thickness of the ice layer at two oppositelocations intercepted by the direction of propagation of the waveon the inner surface of cavity 1. Results in Figure 9 clearly showthat Vs decreases with increasing mantle width.

All these results show that the sign of Vs may change forrotations of the aggregate with respect to a fixed direction ofincidence and of polarization. In order to derive the behaviorof Vs for rotations of the aggregates, we perform calculationsby rotating them around the x-axis in steps of 30◦. The wavepropagates along the z-direction. We consider a point withincavity 1 located on the rotation axis of the aggregate. Results areshown in Figure 10, for two different choices of the polarization.It is evident that Vs increases with increasing asymmetry.

We now explore the dependence of the above results on thechemical composition. We thus left intact the geometry but

1434 CACCIOLA ET AL. Vol. 701

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06−0.1

−0.05

0

0.05

0.1

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06−0.2

0

0.2

0.4

0.6

δz

Vs

All Silicate C−C Si−Si Si−C All Carbon

I IC I

I IC I

Figure 15. Vs as a function of δz for two wavelengths, λ = 0.1 and λ = 0.22 μm. The incident field is x polarized and the results refer to the reduced aggregate shownin Figure 13. Homogeneous composition: full circles, 1, 2, 5, 7, 11, 13, 14, 15, 18 (silicates); empty circles, 1, 2, 5, 7, 11, 13, 14, 15, 18 (carbon). Mixedcomposition: squares, 1, 2, 5, 7, 11, 13, 14, 15, 18; x, 1, 2, 5, 7, 11, 13, 14, 15, 18; crosses, 1, 2, 5, 7, 11, 13, 14, 15, 18.

considered chemically homogeneous aggregates (either carbonand silicates). Also, in this case each sphere is covered by anice mantle of thickness 25% of the total radius. In Figures 11and 12 results for carbon and silicate aggregates, respectively,are shown. The depolarization as a function of wavelength, forincidence along the z-axis and polarization along the x-axis, hasbeen computed as in the case of the heterogeneous aggregate.Both figures, when compared with the corresponding ones inFigure 7, show trends in the field depolarization similar up toλ � 0.15 μm. At longer wavelengths, the field depolarizationtends to increase with increasing wavelength in the case ofcarbon aggregate, while vanishes in the case of aggregatecomposed by silicates. This effect was not present in the resultsof Saija et al. (2005), where different optical constants forcarbonaceous material were used (i.e., Suh 2000). Althoughdepolarization is expected when the internal field crosses anyinterface, details depend on the involved materials.

It is important to assess the influence of the environmenton the occurrence of depolarization. Thus, we disassemblethe aggregate of Figure 6 retaining only the spheres thatsurround cavity 1. The resulting reduced cluster of subunitsis depicted in Figure 13 and turns out to be composed of thespheres numbered 1*, 2*, 5*, 7*, 11*, 13, 14, 15, and 18 inTable 3. Vs is reported at the same point at which it has beencalculated in cavity 1 for directions of incidence along thecoordinate axes. The results are shown in Figure 14 for thesame choice of the polarization of the top panel of Figure 10.When we compare Figure 14 with the top panel of Figure 7, thesimilarity of the results appears quite evident. Thus, the behaviorof the depolarization appears to be essentially determinedby the geometry of the nearest environment. Keeping thisreduced geometrical configuration, we studied the behaviorof depolarization in cavity 1 as the material of the subunitschanged. Five different chemical compositions are compared inFigure 15, showing that the materials of the assembling subunitsare an important factor that may determine both the strength andthe sign of depolarization.

4. CONCLUSIONS

In this work, we address the problem of field depolarizationwithin interstellar dust aggregates, in which coagulation gener-ates interstitial cavities partly filled with icy condensed gas. Inprevious papers (Saija et al. 2005; Borghese et al. 2005), we dealtwith a similar case, i.e., the occurrence of depolarization withincavities embedded into a homogeneous sphere, and found that,to large extent, the depolarization effect is of purely geometricalnature, depending on the existence of an asymmetric interfacebetween regions of different refractive indexes. Here, exploit-ing a much more complex description for the grain aggregate(see Figures 3 and 6) we confirm the central role of geometricalasymmetry in generating depolarization, although details, suchas the variation with wavelength, may depend on the choice ofthe interface materials.

The main results of this study are summarized as follows.

1. In all examined cases a net depolarization is present.2. Depolarization in a given cavity depends exclusively on the

environment, i.e., morphology and chemical compositionof the aggregate close to the cavity (compare the top panelof Figure 7, and Figure 14).

3. Any time there is significant depolarization in an interstitialcavity, a significant amount occurs in its icy boundaries,the most interesting region for photochemistry; moreover,in all examined cases, the depolarization depends on thewavelength, with Vs varying both in sign and/or in value,ranging from λ = 0.1 to 0.22 μm. This effect is particularlyevident in the case of homogeneous sphere + eccentriccavity (Saija et al. 2005); this behavior fulfills, incidentally,the requirements posed by the Kuhn–Condon sum rule(compare Cerf & Jorissen 2000).

4. Significant depolarization occurs when the projected sec-tion of the target aggregate on a plane perpendicular to thedirection of incidence of the wave bears enough asymme-try; in fact, the depolarization of a wave approaching theaggregate along the x-axis, the axis of maximum elongation

No. 2, 2009 UV RADIATION INSIDE INTERSTELLAR GRAIN AGGREGATES. III. 1435

of the aggregate, is always negligible (lower than few per-cent); in general, along only one direction of incidencesignificant depolarization is achieved.

5. The sign of Vs changes erratically from cavity to cavity.6. Rotations can affect the sign of Vs, but, on average, a net

depolarization of a given sign is always present, i.e., theintegral of Vs over the rotation angle is never zero.

Our calculations show that a net depolarization effect isalways present in all cavities. Since the sign of Vs does notnecessarily stay the same in different cavities of the sameaggregate, it is not clear whether or not the depolarization, i.e.,the presence of an effective EP field within the ice layers, mightbe relevant to the selection of chiral molecules. This prevents theestablishment of a net enantiomeric excess of a given symmetry.

A possible solution to the problem raised by the inconstancyof the sign of Vs may seem to consist of considering a muchlarger region of the parameter space, i.e., in the orientation ofthe aggregate with respect to the field, the degree of chemical ho-mogeneity, and the morphology. Nevertheless, such an extendedexploration would require heavy calculations, the interpretationof whose results could not be univocal. Thus, the strategy ofthe exploration should be carefully planned. At present, wecan state that the results of preliminary numerical simulationsperformed exploiting aggregate shapes simpler than those usedhere, as in Saija et al. (2005), suggest that there are configu-rations in which the required constraint on the sign of Vs isfulfilled.

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