Modeling Porous Dust Grains with Ballistic Aggregates. II. Light Scattering Properties

1.4 MB·English

by  Yue Shen

DRAFTVERSIONJANUARY21,2009
PreprinttypesetusingLATEXstyleemulateapjv.05/04/06
MODELINGPOROUSDUSTGRAINSWITHBALLISTICAGGREGATES.II.
LIGHTSCATTERINGPROPERTIES
YUESHEN,B.T.DRAINE,ANDERICT.JOHNSON
PrincetonUniversityObservatory,Princeton,NJ08544
DraftversionJanuary21,2009
ABSTRACT
9
We studythelightscatteringpropertiesofrandomballistic aggregatesconstructedin Shenetal. (PaperI).
0
Usingthediscrete-dipole-approximation,wecomputethescatteringphasefunctionandlinearpolarizationfor
0
randomaggregateswithvarioussizesandporosities,andwithtwodifferentcompositions: 100%silicateand
2
50%silicate-50%graphite.Weinvestigatethedependenceoflightscatteringpropertiesonwavelength,cluster

n size and porosity using these aggregatemodels. We find that while the shape of the phase functiondepends
a mainly on the size parameter of the aggregates, the linear polarization depends on both the size parameter
J
andtheporosityoftheaggregates,withincreasingdegreeofpolarizationastheporosityincreases.Contraryto

5 previousstudies,wearguethatmonomersizehasnegligibleeffectsonthelightscatteringpropertiesofballistic
1
aggregates, as long as the constituentmonomer is smaller than the incidentwavelength up to 2πa /λ∼1.6
0

] where a0 is the monomer radius. Previous claims for such monomer size effects are in fact the combined
P effects of size parameter and porosity. Finally, we present aggregate models that can reproduce the phase
E function and polarization of scattered light from the AU Mic debris disk and from cometary dust, including
the negative polarization observed for comets at scattering angles 160◦ .θ <180◦. These aggregateshave
.
h moderateporosities,P ≈0.6,andareofsub-µm-sizeforthedebrisdiskcase,orµm-sizeforthecometcase.
p
Subjectheadings:dust,extinction–polarization–scattering–circumstellarmatter–comets–interplanetary
-
o medium–stars: individual(AUMic,GJ803)
r
t
s
a 1. INTRODUCTION ature (e.g., West 1991; Kozasaetal. 1992, 1993; Ossenkopf
[ 1993; Kimuraetal. 2006; Bertinietal. 2007; Lasueetal.
Interplanetary dust particles (IDPs) collected in the
2009),thenewly-introducedBAM1andBAM2clustershave
1 Earth’s stratosphere by high-flying aircraft (Brownlee 1985;
geometriesthatarerandombutsubstantiallyless“fluffy”than
v Warrenetal. 1994) usually have irregular shapes and
theBA clusters. TheeffectiveporosityP (eq.12inPaperI)
7 fluffy structures. Similar structures have been pro-
7 duced in laboratory and microgravity experiments of dust increases from BAM2→BAM1→BA and covers a wide dy-
1 particle interactions (Wurm&Blum 1998; Blum&Wurm namicalrange,allowingustoinvestigatetheeffectsofporos-
2 2000; Krause&Blum 2004). It has also been sug- ityontheopticalpropertiesoftheaggregatesinasystematic
. way. Using these aggregation rules, we can construct grain
1 gested that interstellar dust grains may consist primarily
modelswith varioussizes and compositions. In Paper I, we
0 of such aggregate structures (e.g., Mathis&Whiffen 1989;
computedtotalscatteringandabsorptioncrosssectionsforthe
9 Dorschner&Henning1995),withamixtureofvariouschem-
threetypesofaggregates(BA, BAM1andBAM2), forthree
0 icalcompositionsandvacuum.
: Porous,compositeaggregatesareoftenmodeledasaclus- differentcompositions(50%silicate and50%graphite;50%
v silicate and50%amorphouscarbonAC1,Rouleau&Martin
terofsmallspheres(“spherules”or“monomers”),assembled
Xi under various aggregation rules. The optical properties of 1991,and100%silicate),andforwavelengthsfrom0.1µmto
4µm. Thepurposeofthispaperistoinvestigatethedetailed
r these aggregatescan be calculated using numericalschemes
a such as the generalized multisphere Mie (GMM) solution lightscatteringpropertiesofthese aggregates,i.e., the phase
functionandthelinearpolarization.
(Mackowski 1991; Xu 1997) or the discrete dipole approx-
The paper is organized as follows: in §2 we recapitulate
imation (DDA) method (e.g., Purcell&Pennypacker 1973;
our aggregatemodels; the scattering phase functionand lin-
Draine&Flatau 1994). These methods have been used to
earpolarizationforvariousballisticaggregatesarepresented
study the optical properties of different kinds of aggregates
in §3, where we explore the dependence of light scattering
during the past decade (e.g., West 1991; Lumme&Rahola
properties on aggregate properties; we present examples of
1994; Petrovaetal. 2000; Kimuraetal. 2006; Bertinietal.
aggregates that can be applied to circumstellar debris disks
2007;Lasueetal.2009); mostofthosestudiesarededicated
and cometarydust in §4, and we show that moderate poros-
to interpretthe phase functionandpolarizationof lightscat-
ityaggregatescanreproducetheobservedscatteringandpo-
teredbycometarydust.
larizationpropertiesofdustin bothsolarsystem cometsand
Inacompanionpaper(Shenetal.2008,hereafterPaperI),
extrasolardebrisdisks. Wesummarizeourresultsin§5.
we constructed aggregates using three specific aggregation
rules: ballistic agglomeration (BA), ballistic agglomeration 2. AGGREGATEMODELS
withonemigration(BAM1)andballisticagglomerationwith
A detailed description of the target generation algorithms
twomigrations(BAM2).Wedevelopedasetofparametersto
and resulting geometric properties of the BA, BAM1 and
characterizetheirregularstructureoftheseaggregates.While
BAM2clusterscanbefoundinPaperI.Herewereviewsome
the BA clusters are essentially the Ballistic Particle-Cluster
of the basic conceptsthat will be used in the following sec-
Agglomeration(BPCA) clusters frequentlyused in the liter-
tions.
2 SHEN,DRAINE&JOHNSON
In the present paper we study the scattering properties of
TABLE1 aggregateswithtwocompositions:100%silicate,or50%sil-
CLUSTERGEOMETRIES icate+50%graphite(volumefractions). InPaperIwefound
thataggregatesconsistingof50%silicate+50%AC1amor-
cluster P R/aeff Ndip ndip Figs. phouscarbonhadscatteringpropertiesintermediatebetween
BA.256.1 0.8598 1.9249 115656 451.8 6 the100%silicateand50%silicate+50%graphiteaggregates.
BA.256.2 0.8525 1.8925 105855 413.5 6
Calculations are performed using DDSCAT version 7.0
BA.256.3 0.8553 1.9050 102725 401.3 6
BAM1.32.4 0.6188 1.3791 13903 434.5 4ab (Draine&Flatau 2008). DDSCAT is a code based on the
BAM1.256.1 0.7060 1.5038 103921 405.9 6 discretedipoleapproximation(Purcell&Pennypacker1973;
BAM1.256.2 0.7412 1.5693 107160 418.6 6 Draine&Flatau 1994), designed to compute scattering and
BAM1.256.3 0.6980 1.4904 107047 418.2 6
absorptionofelectromagneticwavesbytargetswitharbitrary
BAM2.256.1 0.5632 1.3179 113696 444.1 1,3,4cd,5,6,7
BAM2.256.2 0.5781 1.3333 103509 404.3 1,3,4cd,5,6,7 geometry and composition, for targets that are not too large
BAM2.256.3 0.5818 1.3372 107332 419.3 1,3,4cd,5,6,7 comparedtothewavelengthλ. Foreachclustertype(defined
BAM2.512.14 0.6127 1.3719 211211 412.5 4ab by N, aggregation rule, and composition) we generally av-
BAM2.1024.1 0.6386 1.4039 414977 405.3 4cd,8
erage over three random realizations and 54 orientationsfor
BAM2.1024.2 0.6476 1.4158 412077 402.4 4cd,8
BAM2.1024.3 0.6387 1.4041 430184 420.1 4cd,8 eachrealization.2 DDSCAT7.0allowsustotreatthegraphite
NOTE. —Namingconvention: “BA.256.1”meansrealization 1of monomersasrandomly-orientedsphereswiththeanisotropic
theN=256BAclusters. dielectrictensorofgraphite.
EachaggregateiscomposedofNsphericalmonomerswith
radius a . We define the “effective radius” of a cluster, a , 3. SCATTERINGPHASEFUNCTIONANDPOLARIZATION
0 eff
to be the radius of an equal-volume solid sphere; thus our The phase function and linear polarization of the scat-
aggregateshave tered light as functions of scattering angle θ can be re-
aeff=N1/3a0. (1) trieved from the elements of the 4×4 Muller matrix Sij
(e.g., Bohren&Huffman 1983). For unpolarized incident
The structure of the cluster is characterized by a porosity
light, the scattered light phase function is proportional to
parameterP (seeeq.12ofPaperI)andacharacteristicradius S =(4π2/λ2)dC /dΩ (where dC /dΩ is the differential
R≡aeff/(1- P)1/3(seeeq.11ofPaperI),whichdependson sc1a1ttering cross sseccation for unpolarsiczaed incident light) and
P andis typically1- 2timesaeff. Tables1 and2of PaperI the linear polarizationparameteris p=- S21/S11. By defini-
givetabulatedmeanvaluesofP andR/aeffforthethreetypes tion,thepolarizationisperpendicular/paralleltothescattering
ofaggregateswith23≤N≤216. ForagivenvalueofN,the planewhen pispositive/negative.
BAclustershavethehighestP,whiletheBAM2clustershave We have obtained S (θ) and p(θ) for our realization-
11
thelowestP. Informationforthespecificclustergeometries and orientation-averaged aggregates for wavelengths 0.1 ≤
employedinthispapercanbefoundinTable1,includingthe λ/µm≤4. For illustrative purposes, in most cases we will
porosityP, thenumberNdip ofdipolesrepresentingthereal- presentthe results for the BAM2 aggregates– the aggregate
ization,andthenumberofdipolespersphere,ndip=Ndip/N. geometrywiththelowestporosity.Orientation-averagedscat-
Theactualgeometry(includingimages)oftheseandotherre- teringpropertiesfortheclustersstudiedinthispaper(includ-
alizationsofBA,BAM1,andBAM2clusterscanbeobtained ing wavelengths not shown in the figures) are available on-
online.1 line.3
InPaperIweconsideredthreedifferentcompositions:50%
silicate+50%graphite,50%silicate+50%AC1,and100%
3.1. WavelengthDependence
silicate. Silicatematerialaccountsforperhaps2/3ofthetotal
mass of interstellar dust, and it is naturalto assume that sil- We first show the wavelength dependence of S11 and lin-
icateswillalso providethe bulkoftherefractorymaterialin ear polarization for N =256-monomer BAM2 clusters with
comets or debris disks. Interstellar silicates are amorphous; monomerradiusa0=0.02µminFigure1,forselectedwave-
andamorphoussilicatesarebelievedtodominatethesilicate lengths. For the N =256 BAM2 case, aeff =0.127 µm, the
masseveninthecaseofcometsorcircumstellardiskswhere porosityP ≈0.58(seeTable1),andthecharacteristicradius
crystallinesilicateshavebeendetected.Weusethe“astrosili- R≈1.334aeff=0.17µm . The phase function shows a rela-
cate”dielectricfunction(Draine&Lee1984;Draine2003). tively smooth dependenceon wavelength λ: for λ.0.5µm
Carbonaceousmaterialprovidesasignificantfractionofthe (x≡2πR/λ>2),itshowsastrongpeakintheforwardscat-
total mass of interstellar grains, and this may also be true teringandamildbackscatteringenhancement,withtheover-
of dust in comets and debris disks. The smallest carbona- all forward-backward asymmetry decreasing monotonically
ceous particles in the ISM consist primarily of polycyclic as the incident wavelength increases. For linear polariza-
aromatic hydrocarbon material, but the form of the carbon tion,thesituationismorecomplicated. Thepolarizationnear
in the larger grains (where most of the carbon resides) re- θ≈90◦ firstdecreasesasλincreases,reachesaminimumat
mains uncertain: Pendleton&Allamandola(2002) conclude λ≈R, and then rises again with increasing λ, approaching
that the hydrocarbonmaterial is ∼85% aromatic (ring-like) p(90◦)=100%intheRayleighlimit. Thewavelengthλmin.pol
and15%aliphatic(chain-like),butDartoisetal.(2004)claim where p(90◦) is minimum is well-defined for the pure sili-
thataliphaticmaterialpredominates,withatmost15%ofthe cate, with λmin.pol ≈0.17µm. For the graphite-silicatecom-
carbon in aromatic form. To explore the effect of material positionitis less well-defined, with minimanear ∼0.17µm
thatisstronglyabsorptiveinthevisible,weusethedielectric
tensor of graphite for the carbon in our mixed-composition 2 9valuesoftheangleΘbetweentheclusterprincipalaxisˆa1andˆx(the
direction oftheincident light), and6values oftherotation angleβ ofthe
aggregates. clusteraroundˆa1.WeuseasinglevalueoftherotationangleΦofˆa1around
ˆxbecauseweaverageover4scatteringplanes.
1 http://www.astro.princeton.edu/∼draine/agglom.html 3 http://www.astro.princeton.edu/∼draine/SDJ09.html
AGGREGATESASPOROUSDUSTGRAINS.II. 3
and ∼ 0.45µm. The increase of polarization with increas- I showed that EMT-Mie calculations can be used to obtain
ingwavelengthin the opticalbandis knownasthe polariza- moderatelyaccuratetotalextinctionandscatteringcrosssec-
tioncoloreffectincometaryscatteredlightobservations(e.g., tions, Figure 3 shows that the scattering phase function and
Chernovaetal.1996;Levasseur-Regourd&Hadancik2001). polarizationestimatedusingEMT-Miecalculationsdonotac-
Thereversebehaviorofincreasingpolarizationwithdecreas- curatelyreproducethescatteringpropertiesofirregularclus-
ingwavelengthintheUVband,however,ismorecomplicated ters.
tointerpret. Itcouldbecausedbythechangeinthesize pa-
rameterx≡2πR/λ,orchangesinthedielectricfunctionasλ 3.2. DoesMonomerSizeMatter?
varies,orboth. Wewillreturntothispointin§3.3. There is another parameter that might affect S and po-
11
ItwasshowninPaperIthattheEMT-Miemodelprovides larization: the monomer size. There have been claims that
a good approximation for the total extinction cross section large monomer size is crucial in decreasing the polarization
as a function of λ, providedthat the vacuum fraction f is andinproducingthenegativepolarizationbranchobservedin
vac
set to f ≈P. We now test to see if the EMT-Mie model cometary dust (e.g., Petrovaetal. 2000; Bertinietal. 2007).
vac
reproducesthescatteringphasefunctionS (θ)andpolariza- However, in these previous studies, variations of monomer
11
tion p(θ). For the EMT-Mie calculationswe use an optimal size were always coupled with changes in porosity P and
valueofvacuumfraction f =0.55andthesameamountof clustersizeR,henceeffectsattributedtovaryingthemonomer
vac
solid material as in the N =256 BAM2 clusters. For the ef- size mayin factbe dueto variationsin P orR. We haveal-
fectivedielectricpermittivityǫ weusetheBruggemanrule readyseeninPaperIthattheapparenteffectsofmonomersize
eff
(seeBohren&Huffman1983) ontotalcrosssectionsareessentiallytheeffectsofvaryingP
ǫ - ǫ orR.
f i eff =0, (2)
X iǫ +2ǫ Toisolatetheeffectofmonomersize,wecompareclusters
i eff
i with the same a and very similar P (thus R is also com-
eff
where f andǫ arethevolumefractionanddielectricpermit- parable), but different monomer size a . Thus the effect of
i i 0
tivity of each composition, including vacuum. There is al- monomersize,ifthereisany,isdecoupledfromothereffects.
ways only one solution of ǫ that is physically meaningful. Wefirstconsiderthesameexampleusedinfigure8ofPaperI:
eff
Forgraphite,wemaketheusual 1- 2 approximation,andtake theN=32BAM1clusterrealizationBAM1.32.4(P=0.619,
3 3
ǫ=ǫ(Ekc)for f = 1f ,ǫ=ǫ(E⊥c)for f = 2f . R/aeff =1.379) with monomer size a0 =0.0504µm and the
3 graphite 3 graphite N=512BAM2clusterrealizationBAM2.512.14(P =0.613,
The EMT-Mie results are shown in Fig. 2 for the silicate-
R/a =1.372)with a =0.02µm. Both clusters have a =
graphite and the pure silicate cases, in parallel to Fig. 1. eff 0 eff
0.160µm and R = 0.220µm. The orientation-averaged re-
The EMT calculations at fixed wavelength show resonances
sultsare shown in Fig. 4a,b fortwo wavelengthsandforthe
that arise from the use of spheres, but these should be
silicate-graphitecompositiononly. Althoughthereareslight
smoothed out when modeling nonspherical particles, which
differences, the two cases have similar phase functions and
are randomly-oriented and will not show such well-defined
polarizations:atconstantRandP,varyingthemonomersize
resonances.ThereforewehavesmoothedtheEMTresultsus-
a hadlittleeffectonthephasefunctionandpolarization.
ingaGaussiankernel 0
Theaboveexampleemployedmoderate-sizedclusters(x=
S¯ = Rdlnaexph- (cid:2)ln(a/a¯)(cid:3)2/2σ2iSij(a) (3) 2FπigR./4λc,.d c3o.9m)pcaormespothseedscoaftstemrianllgmporonpoemrteierss(o2fπ2a0l/aλrg.e c0l.u9s)-.
ij
dlnaexp - ln(a/a¯) 2/2σ2 ters (R≈1.4µm, x=11.1 and 13.9) with similar porosities
R h (cid:2) (cid:3) i P ≈0.6butdifferentmonomersizes. Forλ=0.631µmand
whereσ=a¯/(a¯+λ/2π)anda¯=(1- f )- 1/3a istheradius 0.794µm,clusterswitha =0.10µmand0.16µmshowsim-
vac eff 0
of the Mie sphere. The phase function and polarization are ilar (though not identical because of the slight difference in
¯
thencomputedusingthesmoothedS . porosity of the two clusters; see §3.4) polarization p(θ), de-
ij
BydirectlycomparingFig.1andFig.2itisevidentthatthe spitethesubstantialdifferenceinmonomersize.
EMT-Mie results for S (θ) and linear polarization p(θ) do
11 3.3. DependenceonClusterSize
sharethesametrendswesee intheDDAcalculations. Nev-
ertheless,therearesubstantialdifferencesbetweentheEMT- As we have discussed in §3.1 for fixed-size clusters, the
MieresultsandourDDAresults. Oneobviousfeatureisthat dependenceof the phase function and linear polarization on
the EMT-Mie model tends to underestimate the backscatter- wavelength λ is likely caused by the changes in both the
ingforshortwavelength(λ.0.6µm,x&1.8),afeatureal- size parameteranddielectricfunction. To investigatethe ef-
ready revealed by the behavior of the asymmetry parameter fectsofclustersize atfixedincidentwavelength(i.e.,theef-
g≡hcosθidiscussedinPaperI(fig. 12). Forexample,con- fectsofsize parameteralone),we use N =256,BAM2 clus-
sider the forward-backwardasymmetryS (0)/S (180◦) for ters with monomer size a = 0.02,0.025,0.03µm, or R ∼
11 11 0
λ=0.168µm: the DDAcalculationsforthemixedgraphite- 0.169,0.212,0.254µm. Theseclustershavethesameporos-
silicateBAM2clustergive∼450,whiletheEMT-Miecalcu- ity, andas arguedin theprevioussection, monomersize has
lationgives∼7000. negligible effects, hence any difference must be caused by
To compare the EMT-Mie results and our DDA calcula- changesinR.TheresultsareshowninFig.5forbothcompo-
tions in detail we plot the relative differences in Fig. 3, for sitions. Wepresentresultsattwowavelengths: λ=0.126µm
the silicate plus graphite case (upper) and the pure silicate (< R) and λ = 0.631µm (> R). In both cases the back-
case (bottom). The differencecanbe substantialforspecific ward/forwardscatteringasymmetryincreaseswithincreasing
wavelengths or scattering angles. For example, for θ ≈90◦ thesizeparameter2πR/λ.
scatteringatλ≈0.50µm,theEMT-Miecalculationunderes- In general, we expect p(90◦)→1 in the Rayleigh scatter-
timatesthepolarizationbyafactor∼2,forboththegraphite- inglimitR/λ≪1,withthepeakpolarizationdecreasingwith
silicateclustersandthepuresilicateclusters. AlthoughPaper increasingR/λ. Thisdecline with increasingR/λis seen in
4 SHEN,DRAINE&JOHNSON
Figure 5b,d. However, the results in Figure 5a,c show that featureswillbepresentedin§4.2.
the variation of p with increasing R is not monotonic: at
max
λ=0.126µm when R&λ and when the dielectric function 4. APPLICATIONSOFBALLISTICAGGREGATES
is very absorptive, for both the 100% silicate and 50% sili- The light scattering properties of our ballistic aggregates
cate+50%graphiteN=256BAM2clusters,thepolarization can be applied to various observations. Here we focus
isanincreasingfunctionofRovertherange1.3.R/λ.2. on debris disks and cometary dust, where single scattering
Thus for these cases the polarization at λ=0.126µm has a dominates in the optically thin regime. We show exam-
minimumatsomesizeR <1.3λ. ples of aggregates that can reproduce qualitative and quan-
crit
However, the dependenceof polarizationon R/λ depends titative features observed in the scattered light from the de-
onthedielectricfunction(andthereforeonbothcomposition bris disk around AU Mic (Grahametal. 2007) and from
andwavelength). Forλ=0.631µmthe100%silicateBAM2 cometary dust (e.g., Lumme&Rahola 1994; Petrovaetal.
clusterswithP≈0.6havethepolarizationdecliningwithin- 2000; Kimuraetal. 2006; Lasueetal. 2009). Due to com-
creasingRouttoR/λ=2.2(thelargestvaluecomputed,see putationallimits,wecannotprobeasufficientlylargeparam-
§4.2andFigs.7dand8)–withoutshowinga reversalin the eter space to claim that our models are unique; nor do we
polarizationbehavior.Thesituationisevenmorecomplicated attempt to fit a sophisticated model to the observations of a
forthesilicateplusgraphitecase,wherethereisnocoherent specificcomet. Nevertheless,ourexamples(inparticularthe
trend when R≈λ (see Figs. 1a, 7a and 7b). Based on the moderate-porosityBAM2 clusters) nicelyreproducemostof
casesinvestigatedthusfar,itappearsthatwhenthedielectric thefeaturesobservedinlightscatteredbydebris-diskdustand
function has only weak absorption (e.g., 100% astrosilicate cometarydust.
at λ=0.631µm), for fixed porosity P the polarization is a
monotonicallydecreasingfunctionofclustersizeRfromthe 4.1. DebrisdiskaroundAUMic
RayleighlimitR≪λuptoR/λ.2.Ontheotherhand,when
Polarizationmapsofthedebrisdisksurroundingthenearby
the dielectric function is strongly absorptive (e.g., materials
Mstar AUMicroscopiihavebeenobtainedbyGrahametal.
atλ=0.126µmorsilicate-graphiteclustersatλ=0.631µm),
(2007) using HST ACS in the F606W optical band (λ =
c
forfixedporositythepolarizationdeclineswithincreasingR 0.590µm,∆λ = 0.230µm). The scattered light is polar-
from the Rayleigh limit until it reaches a local minimum at
ized perpendicular to the disk plane. Grahametal. (2007)
R≈λ (the transition is less distinct for the silicate-graphite
adopted the form for the phase function introduced by
case than for the pure silicate case), and then rises as R is
Henyey&Greenstein(1940):
furtherincreased, atleast outto R/λ≈2 (e.g., Figs. 5a and
7a,b). 1 1- g2
S = , (4)
11 4π(1+g2- 2gcosθ)3/2
3.4. DependenceonPorosity
assumedthatthepolarizationvs. scatteringanglevariesas
We nowinvestigatetheeffectofporosityonthescattering
phase function and linear polarization. Previous studies on S sin2θ
p(θ)=- 21 =p , (5)
theporosityeffectusingonlytheBPCAand/ortheevenmore S max1+cos2θ
11
porous“ballisticcluster-clusteragglomeration(BCCA)”clus-
and simultaneously fitted the phase function and linear po-
ters were quite limited in the dynamical range of porosity,
larizationasfunctionofscatteringangletotheobservational
and changes in porosity were coupled with changes in clus-
data, obtaining g ≈ 0.68 and p ≈ 0.53. Grahametal.
ter size. To decouplefrom the cluster size effect we choose max
(2007) suggestthat very porous(P ≈0.91- 0.94)µm-sized
clusters with comparable sizes, but different porosities. We
spherical grains or aggregates can produce these features
use N =256BA, BAM1 andBAM2 clusters, with monomer
based on Mie theory and DDA calculations for BA clusters
size a = 143,170,200Å respectively; hence these clusters
0 (Kimuraetal.2006).
have comparable size R ∼ 0.17µm, but different porosities
Here we will show that random aggregates with a much
P =0.85,0.74,0.58.
lower porosity (P ≈0.6) can, in fact, better fit the observa-
Weconsidertworegimes:λ<Randλ>R.Theresultsare
tionsof the AU Mic debrisdisk. To reproducethe observed
showninFig. 6fortwo examplewavelengths,λ=0.126µm
features,werequirethatthe phasefunctionandlinearpolar-
andλ=0.447µm,andforthetwocompositions.Itisevident
izationarebothclosetothefunctions(4)and(5)withthethe
thatinbothregimes,porosityhaslittleeffectontheshapeof
thephasefunction4. Ontheotherhand,higherporositytends best-fit values of g and pmax found by Grahametal. (2007).
In particular, the intensity of scattered lightat θ=0◦ should
toincreasethelinearpolarizationforbothR>λandR<λ.
be approximatelya factor of 150 larger than the intensity at
Most of the cases shown in this section have large
linear polarization fraction [p(90◦) & 50%]. Cometary θ=180◦,andthemaximumpolarizationshouldbe≈0.5,al-
dust typically has p(90◦) . 40% and a negative branch thoughp(θ)neednotnecessarilypeakatθ=90◦.
of polarization at scattering angle ∼ 160◦ - 180◦ (e.g., Dustgrainsindebrisdiskswillhaveadistributionofsizes.
Much of the interstellar grain mass can be approximatedby
Levasseur-Regourd&Hadancik 2001), observed at optical
a power-lawsize distributiondn/dR∝R- α forR.0.25µm
wavelengths. From the results of §3.1-§3.4 we expect that,
withα≈3.5(Mathisetal.1977). Sizedistributionswithα≈
in general, a reduced peak polarization and appearance of a
negativepolarization branch for 160.θ<180◦ can be ob- 3.5 can be obtained from models with coagulation and col-
lisional fragmentation (Dohnanyi 1969; Tanakaetal. 1996;
tainedby(1)increasingtheclustersizeRand(2)makingthe
Weidenschilling1997).
clustermorecompact(lowerporosityP). Examplesofballis-
For modelingcomets and debris disks, we will consider a
ticaggregatesthatareabletoreproducethesecometarydust
size distribution dn/dR∝R- 3.5. For a fixed porosity (i.e., a
om4etTryheofsltihgehtBdAif,fBerAenMc1eainndS1B1A/SM112(0c)luisstelriks.elycausedbythedifferentge- ptiaornticisuljaursttydpen/odfaa0gg∝reag0-a3t.e5,wwithhearefixae0disN)th,tehimsosinzoemdeisrtrsiibzue-.
AGGREGATESASPOROUSDUSTGRAINS.II. 5
Hence the averaged phase function S (θ) and polarization Levasseur-Regourdetal. 1996), although gas contam-
11
are: ination in polarimetric measurements with wide-band
amaxda (dn/da )S (a ,θ) filters might depolarize the observed scattered light
S¯ (θ)= Ramin 0 0 11 0 , (6) (Kiselevetal.2004).
11 amaxda (dn/da )
Ramin 0 0 • Within the 4000–7000Å window, the polarization
p¯(θ)= Raamminaxda0(dn/da0)S11(a0,θ)p(a0,θ) , (7) increases with wavelength, which is the so-called
amaxda (dn/da )S (a ,θ) polarization color effect (e.g., Chernovaetal. 1996;
Ramin 0 0 11 0 Levasseur-Regourd&Hadancik2001).
where a and a are the minimumand maximumvalues
min max • Manycometsshowanegativebranchofpolarizationat
ofmonomersizeinoursizedistribution.
scatteringanglelargerthan150- 160◦,withaminimum
WeconsiderN=256BAM2clusters(P =0.58),witha =
0 of&- 2%(e.g.,Dollfusetal.1988;Eatonetal.1992).
0.02,0.03,0.04,0.05,0.06,0.07,0.08µm,whichcorrespond
to R ≈ 0.169,0.254,0.339,0.424,0.508,0.593,0.678µm. Most of these features, in particular the negative polariza-
We show the calculated phase function and polarization for tion branch, have been successfully reproduced using vari-
each of these clusters (averaged over orientations and 3 re- ous aggregates which differ in geometry, composition and
alizations) in Fig. 7, at two wavelengths λ = 0.501 and porosity (e.g., Lumme&Rahola 1994; Petrovaetal. 2000;
0.631µm. For comparison with the scattering propertiesin- Kimuraetal. 2006; Bertinietal. 2007; Lasueetal. 2009).
¯
ferredforthedustaroundAUMic,wecalculateS11(θ)andpo- Theaggregatesstudiedherearecapableofproducingallthese
larizationp¯(θ)averagedoverasizedistributiondn/dR∝R- 3.5 features as well. In addition, we have demonstrated the ef-
withamin=0.015µmandamax=0.065µm(Rmin=0.127µm, fects of grain size and porosity, and pointed out that the
Rmax = 0.551µm), which are shown as dashed lines. The monomer size effect claimed by previous authors is in fact
best-fit Henyey-Greenstein phase function (4) and polariza- due to changes in cluster size R and/or porosity P. For ex-
tionfittingfunction(5)fromGrahametal.(2007)areshown ample, in Petrovaetal. (2000) and Bertinietal. (2007), the
as solid black lines. We plot the comparison for both the differenceoftheprominenceofthenegativebranchiscaused
silicate-graphitecomposition(upperpanels)andthepuresil- by the effect of grain size when they increase the monomer
icatecomposition(bottompanels). sizeforthesameconfiguration/porosity.
As we can see from Fig. 7, the size-averaged silicate- Thereasonthatthoseauthorsdidnotfindanegativebranch
graphite clusters produce close matches to the Henyey- of polarization for small monomer size (a .0.1 µm) and
0
Greenstein model at these optical wavelengths for both the moderatenumberofmonomers(afewtens)isthatcomputa-
phase function and linear polarization. These clusters have tional limits prevented them from using a sufficient number
porosityP ≈0.6andoverallsizeR∼0.2- 0.5µm,i.e.,they ofmonomers. Totest this, we havecomputeda few realiza-
aresub-µm-sizedclusterswithmoderateporosity. tions of BAM2 clusters with N =1024, composed of small
Asdiscussed in §3.4, increasingporositywillincrease the monomers (a = 0.08 µm). The results are shown in Fig.
0
polarization of scattered light. Our highest-porosityclusters 8 for optical wavelength λ=0.631 µm, where the negative
arethoseBAclusters,whicharecommonlyusedintheliter- branch is evident for both compositions (it is more promi-
ature, referred to as “ballistic particle-cluster agglomeration nent at the usually observed red band λ=0.55µm). In Fig.
(BPCA)”clusters(e.g.,West1991;Kozasaetal.1992,1993; 8, we have also computed for the N =1024 BAM2 clusters
Ossenkopf 1993; Kimuraetal. 2006; Bertinietal. 2007; using monomer size a = 0.1µm, shown as open squares;
0
Lasueetal. 2009). We found that if we replace the BAM2 the dashed lines are the average of the results of the two
clusters in Fig. 7 with BA clusters with the same number monomersizes,forasizedistributiondn/dR∝R- 3.5running
of monomers and monomer sizes, we can reproduce simi- fromR =0.98µmtoR =1.54µm.
min max
lar phase function features but over-predictthe polarization. Theseclustersareµm-sizegrains,whichisconsistentwith
This is already seen in Grahametal. (2007) (e.g., their fig. the values found by other authors (e.g., Lumme&Rahola
8). ThusweconcludethatcompactBAM2clustersfittheob- 1994; Petrovaetal. 2000; Kimuraetal. 2006), although the
servationsofAUMicbetterthanthemoreporousBAclusters exact values depend on composition as well as porosity.
thathavebeenconsideredpreviously. We find that the N = 1024, a = 0.08,0.1 µm BAM2 clus-
0
ters composed of silicate and graphite (Fig. 8a) with R ≈
4.2. CometaryDust 1.1- 1.4µmandP ≈0.6,reproduceabackscatteringalbedo
Thephasefunctionandlinearpolarizationofscatteredlight A=[S11(180◦)λ2]/(4π2R2)∼0.04andasmallnegativepolar-
have been observed in a variety of comets. Though differ- izationforscatteringangles∼155–180◦peakingat(∼- 1%),
entcometsshowquantitativedifferencesinthephasefunction representative of the typical values found in cometary ob-
andpolarization,therearesomecommonfeatures: servations, although the maximum polarization (∼45%) is
a little higher than observed. We may need somewhatmore
• The phase function shows strong forward scattering compactaggregates,ordifferentcomposition(e.g.,moresil-
withaweakenhancementinthebackscattering;thege- icate, see the right panel of Fig. 8) to lower the peak polar-
ometric albedo [defined as A ≡ (S [180◦]λ2)/(4πG) ization. Alternatively,onemayconsiderthemixtureoffluffy
11
where G ≈πR2 is the averaged geometric cross sec- aggregatesandcompactsolidgrains(e.g.,Lasueetal.2009).
tionofthegrain]ofbackscatteredlightislessthan0.06 In their study, a larger fraction of porous BCCA aggregates
(e.g.,Hanner&Newburn1989). is needed to produce a higher peak polarization for comet
C/1995 O1 Hale-Bopp than for comet 1P/Halley, consistent
• The linear polarization p(θ) is a bell-shaped curve withourargumentthathighporosityhelpsincreasethepolar-
as function of scattering angle, with typical max- ization. Since only very porous BCCA clusters were used
imum value of 10- 30% (Dobrovolskyetal. 1986; in their modeling, it will be interesting to see if our more
6 SHEN,DRAINE&JOHNSON
compact BAM1 and BAM2 clusters will provide better fits ing, increasing R/λ results in decreasing polarization,
forthesecometsinmodelingthemixtureoffluffyaggregates at least for R/λ.2 (e.g., Figure 7c,d, showing100%
andcompactsolidgrains. silicateBAM2clustersatλ=0.501µmand0.631µm);
however,atvacuumUVwavelengthswherethemateri-
5. CONCLUSIONS alsarestronglyabsorbing,increasingR/λcanincrease
We have studied the phase function and linear polariza- thepolarization(e.g.,Figure5a,c,showingscatteringat
tionpropertiesoflightscatteringbyballisticaggregates. We λ=0.126µm).
studiedthewavelengthdependence,clustersizedependence,
andporositydependenceofthelightscatteringpropertiesus- 5. Thedegreeofpolarizationdependsonthesizeparame-
ingthediscrete-dipole-approximation,andcomparedwiththe teraswellasporosity,buthighporosityhelpsincrease
EMT-Miemodel.Ourmainconclusionsare: polarizationinboththeR.λandR&λregimes.
1. ItisshownthatthoughtheEMT-Miemodelreproduces
6. WepresentaggregateswithBAM2geometry,moderate
similar trends in these dependences, it differs quanti-
porosity P ≈0.6, and sub-µm sizes which can repro-
tatively from the DDA calculations. We recommend
ducethescatteredlightphasefunctionandpolarization
usingDDAcalculationsifaccurateresultsaredesired.
observedintheAUMicdebrisdisk.
2. Monomer size has negligible effects on the scattered
light properties as long as monomers are small com- 7. We present aggregate models with BAM2 geometry
pared with the incident wavelengthλ. Even when the andmoderateporosityP ≈0.6 thatcanreproducethe
monomers are no longer small (e.g., 2πa /λ∼ 1.6), albedo and polarization p(θ) observed for cometary
0
monomer size appears to be of secondary importance dust, including the negative polarization observed at
forthephasefunctionandpolarizationp(θ). scattering angles 160◦ .θ <180◦. These aggregates
arecomposedofsilicateandgraphite,andareof&µm
3. The phase function is mainly determined by R/λ; in- size. Suchmoderatelyporousaggregatesarepromising
creasing R/λ decreases the backscattering relative to candidatesforcometarydust.
forwardscattering.
4. WhenR/λ≪1(e.g.,intheRayleighlimit),increasing
R/λdecreasesthepolarization. ForR&λ, thedepen- This research was supported in part by NSF grant AST
dence of polarization on R/λ depends on the dielec- 04-06883. Computations were performed on the Della and
tricfunction:formaterialsthatarenotstronglyabsorb- ArtemiscomputerclustersatPrincetonUniversity.
REFERENCES
Bertini,I.,Thomas,N.,&Barbieri,C.2007,A&A,461,351 Krause,M.,&Blum,J.2004,Phys.Rev.Lett.,93,021103
Blum,J.,&Wurm,G.2000,Icarus,143,138 Lasue, J.,Levasseur-Regourd, A.C.,Hadamcik, E.,&Alcouffe, G.2009,
Bohren,C.F.,&Huffman,D.R.1983,AbsorptionandScatteringofLight Icarus,199,129
bySmallParticles(NewYork:Wiley) Levasseur-Regourd,A.C.,Hadamcik,E.,&Renard,J.B.1996,A&A,313,
Brownlee,D.E.1985,AnnualReviewofEarthandPlanetarySciences,13, 327
147 Levasseur-Regourd,A.C.,&Hadancik,E.2001,inESASP-495:Meteoroids
Chernova,G.P.,Jockers,K.,&Kiselev,N.N.1996,Icarus,121,38 2001Conference,ed.B.Warmbein,587–594
Dartois, E., Muñoz Caro, G. M., Deboffle, D., & d’Hendecourt, L. 2004, Lumme,K.,&Rahola,J.1994,ApJ,425,653
A&A,423,L33 Mackowski,D.W.1991,Proc.RoyalSoc.LondonSerA,433,599
Dobrovolsky,O.V.,Kiselev,N.N.,&Chernova,G.P.1986,EarthMoonand Mathis,J.S.,Rumpl,W.,&Nordsieck,K.H.1977,ApJ,217,425
Planets,34,189 Mathis,J.S.,&Whiffen,G.1989,ApJ,341,808
Dohnanyi,J.W.1969,J.Geophys.Res.,74,2531 Ossenkopf,V.1993,A&A,280,617
Dollfus, A., Bastien, P., Le Borgne, J.-F., Levasseur-Regourd, A. C., & Pendleton,Y.J.,&Allamandola,L.J.2002,ApJS,138,75
Mukai,T.1988,A&A,206,348 Petrova,E.V.,Jockers,K.,&Kiselev,N.N.2000,Icarus,148,526
Dorschner,J.,&Henning,T.1995,A&ARev.,6,271 Purcell,E.M.,&Pennypacker,C.R.1973,ApJ,186,705
Draine,B.T.2003,ApJ,598,1017 Rouleau,F.,&Martin,P.G.1991,ApJ,377,526
Draine,B.T.,&Flatau,P.1994,J.Opt.Soc.Am.A,11,1491 Shen,Y.,Draine,B.T.,&Johnson,E.T.2008,ApJ,689,260
—.2008,http://arXiv.org/abs/astro-ph/0809.0337 Tanaka,H.,Inaba,S.,&Nakazawa,K.1996,Icarus,123,450
Draine,B.T.,&Lee,H.M.1984,ApJ,285,89 Warren,J.L.,Barrett,R.A.,Dodson,A.L.,Watts,L.A.,&Zolensky,M.E.
Eaton,N.,Scarrott,S.M.,&Gledhill,T.M.1992,MNRAS,258,384 1994,CosmicDustCatalog,14
Graham,J.R.,Kalas,P.G.,&Matthews,B.C.2007,ApJ,654,595 Weidenschilling,S.J.1997,Icarus,127,290
Hanner,M.S.,&Newburn,R.L.1989,AJ,97,254 West,R.A.1991,Appl.Opt.,30,5316
Henyey,L.G.,&Greenstein,J.L.1940,Annalesd’Astrophysique,3,117 Wurm,G.,&Blum,J.1998,Icarus,132,125
Kimura,H.,Kolokolova,L.,&Mann,I.2006,A&A,449,1243 Xu,Y.-L.1997,Appl.Opt.,36,9496
Kiselev,N.N.,Jockers,K.,&Bonev,T.2004,Icarus,168,385
Kozasa,T.,Blum,J.,&Mukai,T.1992,A&A,263,423
Kozasa,T.,Blum,J.,Okamoto,H.,&Mukai,T.1993,A&A,276,278
AGGREGATESASPOROUSDUSTGRAINS.II. 7
FIG. 1.—WavelengthdependenceofS11[normalizedusingS11(0)]andpolarizationfortheN=256anda0=0.02µmBAM2clustersfortwocompositions,
averagedover3realizations(BAM2.256.1-3)and54randomorientationsforeachrealization. Upper:thesilicate-graphitecase;bottom:the100%silicatecase.
8 SHEN,DRAINE&JOHNSON
FIG. 2.—SameasFig.1,butfortheEMT-Mieresults(smoothedasineq.3)computedfor fvac=0.55andsameamountofsolidmaterialasintheN=256
BAM2clustersinFig.1.
AGGREGATESASPOROUSDUSTGRAINS.II. 9
FIG.3.—DifferenceinthephasefunctionandpolarizationfortheDDAresultsfortheN=256BAM2clustersfromFig.1andEMTresultsfromFig.2.
10 SHEN,DRAINE&JOHNSON
FIG.4.—Testsfortheeffectofmonomersize.(a)Comparablephasefunctionandpolarizationfortwosilicate-graphiteclusters–BAM1.32.4andBAM2.512.14
–withsimilarporositiesP∼0.619,0.613andclustersizeR∼0.221,0.220µm,butdifferentmonomersizea0=0.0504,0.02µm,forλ=0.355µm(x≡2πR/λ=
3.9).(b)Sameas(a),butforλ=0.631µm(x=2.2).(c)Comparablephasefunctionandpolarizationatλ=0.631µmfortwolargesilicateclusterswithsimilar
porosity P ∼0.57and0.64, andsizeR∼1.4µm (x=13.9), butdifferent monomersizea0 =0.10,0.16µm (2πa0/λ=1.0,1.6). (d)Sameas (c), butfor
λ=0.794µm.Theclustersin(c)and(d)areeachrepresentedby3realizations(BAM2.256.1-3andBAM2.1024.1-3)and54orientationsperrealization. These
fourexamplesshowthatforfixedsizeRandporosityP,themonomersizea0 isunimportantif2πa0/λ<1,andofonlysecondaryimportanceevenwhen
2πa0/λ≈2.