Modelling interstellar extinction and polarization with spheroidal grains

0.44 MB·English

by  N. V. Voshchinnikov

Modelling interstellar extinction and polarization
with spheroidal grains
N.V. Voshchinnikova,1 H.K. Dasb
8 aSobolev AstronomicalInstitute, St.PetersburgUniversity, St.Petersburg, 198504Russia
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bIUCAA,PostBag4,Ganeshkhind, Pune411007,India
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(Received..October2007)

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Abstract

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p We calculate the wavelength dependence of the ratio of the linear polarization degree
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o to extinction (polarizing efficiency) P(λ)/A(λ) from the ultraviolet to near-infrared. The
r prolate and oblate particles with aspect ratios from a/b = 1.1 up to 10 are assumed to
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s be rotating and partially aligned with the mechanism of paramagnetic relaxation (Davis–
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[ Greenstein).Size/shape/orientation effectsareanalyzed.Itisfoundthatthewavelengthde-

pendence of P(λ)/A(λ)ismainly determined bytheparticle composition andsizewhereas
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v thevaluesofP(λ)/A(λ)dependontheparticleshape,degreeanddirection ofalignment.
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4 Keywords: Lightscattering; Non-spherical particles; Extinction; Polarization
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v 1 Introduction
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ModellingofthewavelengthdependenciesofinterstellarextinctionA(ł)andlinear
polarization P(ł) allows one to obtain information about such properties of inter-
stellar grains as size, composition, shape, etc. Interstellar polarization also tells us
about the structure of magnetic fields because it arises due to dichroic extinction
of non-spherical grains aligned in large-scale Galactic magnetic fields [1], [2]. A
correlationbetweenobservedinterstellarextinctionandpolarizationshowsthatthe
sameparticlesare responsibleforbothphenomena.
Veryofteninterstellarextinctionandpolarizationaremodelledseparately(e.g.,[3],
[4]). Themodellingofthesephenomenausuallyincludesconsiderationofnormal-
1 Towhomallcorrespondenceshouldbeaddressed.Phone:[phone];Fax:(+7)
812/4287129;e-mail:[email protected]
PreprintsubmittedtoElsevier 2February2008
izedextinctionE(λ−V)/E(B−V)andnormalizedpolarizationgivenbySerkowski’s
curve(P is the maximumdegreeof polarizationand λ thewavelength corre-
max max
spondingtoit, K isthecoefficient) P(λ)/P = exp[−Kln2(λ /λ)](see,e.g., [5]
max max
and discussion in [6], [7]). In these cases, the important observational data (abso-
lute values of extinction and polarization) are ignored and normalized curves can
be fitted using particles of different composition and slightly different sizes (see,
e.g., discussion in [1], [7]). For example, the consideration of absolute extinction
gives the possibility to use the cosmic abundances to constrain grain models (see
[8]). In the case of interstellar polarization it is better to interpret the wavelength
dependence of polarizing efficiency per unit extinction P(λ)/A(λ) (ratio of the lin-
ear polarization degree to extinction). This allows one to estimate the degree and
directionofgrainalignmentandparticleshape.Notethatpreviousconsiderationof
polarizing efficiency was restricted by one wavelength band only (like P(V)/A(V)
or P(K)/A(K)).
Here,weanalyzetheextinctionandpolarizationproducedbyhomogeneousspheroidal
particleswithdifferentaspectratiosa/b(aandbaremajorandminoraxes,respec-
tively).Spheroidalparticleshavetheimportantapplicationsinvariousfieldsofsci-
ence(spheroidalantennas,bacteriaandmicroweeds,rainingdrops,etc.).Bychang-
ing a/b the particles with a shape varying from close to as sphere (a/b ≃ 1) up to
needles(prolatespheroids)ordisks(oblatespheroids)canbestudied.Theparticles
are assumed to be rotating and partially aligned with the Davis–Greenstein mech-
anism (mechanism of paramagnetic relaxation) [9]. We focus on the behaviour of
polarizingefficiencies andconsiderthenormalizedextinctionA(λ)/A (i.e.,ignore
V
the cosmic abundances which will be taken into account in next paper). The the-
ory iscompared withobservationsofstars seenapparently throughoneinterstellar
cloud.
2 Polarizing efficiency: observations
Interstellarextinctiongrowswitharadiationwavelengthdecreasewhileinterstellar
linear polarization has a maximum in the visual part of spectrum and declines at
shorterandlongerwavelengths(see [6], [7]andFig. 1, leftpanel).
The polarizing efficiency of the diffuse interstellar medium is defined as the ratio
ofthepercentagepolarization(P) totheextinction(A)observed atthesamewave-
length P(λ)/A(λ). There existsan empiricallyfoundupperlimitonthisratio
P /A(V) <∼ 3%/mag, (1)
max
where P isthemaximumdegreeoflinearpolarizationwhichisreachedonaver-
max
agenear thewavelengthλ ≈ 0.55µm(see[6], [7]formorediscussion).
max
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Fig.1.Interstellarextinction(starsandthinline)andpolarizationcurvesforstarHD24263
(left panel) and the polarizing efficiency of the interstellar medium in the direction of this
star (right panel; solid curve shows the power-law approximation P/A ∝ λ1.47). Observa-
tionaldataweretakenfrom[10](extinction) and[11](polarization).
We chose five stars located not far than 500 pc with measured ultraviolet (UV)
polarization [11], then found the extinction (data published in [10] were mainly
used) and calculated the ratio P(λ)/A(λ). It is suggested that these stars are seen
through one interstellar cloud that is supported by a weak rotation of positional
angle [11]. The obtained polarizing efficiencies are shown in Fig. 2. Apparently,
firstpresentationoftheobservationaldatainthesimilarformwasmadebyWhittet
([12], Fig. 9)whogavetheaveragenormalizeddependence P(λ)/A(λ)·A /P .
V max
Fig.2.Polarizing efficiency oftheinterstellar medium inthedirection offivestars. Obser-
vationaldataweretakenfrom[10](extinction) and[11](UVandvisualpolarization), [13]
(IRpolarization).
Notethat the polarizingefficiency generally increases witha wavelength growthif
λ <∼ 1µm. It may be approximatedusing the power-lawdependence P/A ∝ λǫ. For
stars presented in Fig. 2 the values of ǫ vary from 1.41 for HD 197770 to 2.06 for
HD 99264.
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3 Modelling
Let us consider non-polarized light passing through a cloud of rotating spheroidal
grains. Interstellar grains are usually partially aligned (see, e.g., [14]). Imperfect
alignment is also supported by the fact that values of the polarizing efficiencies
calculated for non-rotating or perfectly aligned particles are generally higher than
theempiricallyestimatedupperlimitgivenby Eq.(1) [7], [15].
Theextinctioninstellarmagnitudesandpolarizationinpercentagecanbefoundas
[7]
A(λ) = 1.086N hC i , P(λ) = N hC i 100%, (2)
d ext λ d pol λ
where N isthedustgraincolumndensityandhC i andhC i aretheextinction
d ext λ pol λ
andlinearpolarizationcrosssections,respectively,averagedoverthegrainorienta-
tions
π/2π/2π/2
2
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hC i = πr2 Q f(ξ,β)dϕdωdβ, (3)
ext λ π! Z Z Z V ext
0 0 0
π/2 π π/2
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hC i = πr2 Q f(ξ,β) cos2ψdϕdωdβ. (4)
pol λ π2Z Z Z V pol
0 0 0
Here, r is the radius of a sphere with the same volume as spheroidal grain, β is
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the precession-cone angle for the angular momentum J which spins around the
magnetic field direction B, ϕ the spin angle, ω the precession angle (see Fig. 1
in [15]). The quantities Q = (QTM + QTE)/2 and Q = (QTM − QTE)/2 are,
ext ext ext pol ext ext
respectively,theextinctionandpolarizationefficiencyfactorsforthenon-polarized
incidentradiation, f(ξ,r )isthecone-angledistribution.
V
Weconsiderso-calledimperfectDavis–Greenstein(IDG)orientation[9].TheIDG
mechanism is described by the function f(ξ,β) which depends on the alignment
parameterξ andtheangleβ
ξsinβ
f(ξ,β) = . (5)
(ξ2cos2β+sin2β)3/2
The parameter ξ depends on the particle size r , the imaginary part of the grain
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magnetic susceptibility χ′′ (= κω /T , where ω is the angular velocity of grain),
d d d
gas density n , the strength of magnetic field B and dust (T ) and gas (T ) temper-
g d g
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atures
r +δ (T /T ) κB2
ξ2 = V 0 d g , where δIDG = 8.231023 µm. (6)
r +δ 0 n T1/2T
V 0 g g d
TheangleψinEq.(4)isexpressedviatheanglesϕ,ω,βand Ω(anglebetweenthe
line of sight and the magnetic field, 0◦ ≤ Ω ≤ 90◦). Note that for the case of the
perfect DG orientation (PDG, perfect rotational or 2D orientation) the major axis
of a non-spherical particle always lies in the same plane. For PDG, integration in
Eqs.(3), (4)is performed overthespinangleϕonly.
As usual, the particles of different sizes are considered. In this case the cross-
sections hC i and hC i are obtained after the averaging over size distribution
ext λ pol λ
functionn(r )
V
rV,min
hCi (r )n(r )dr ,
λ V V V
Z
rV,min
wherer and r are thelowerand uppercutoffs,respectively.
V,min V,max
Wechoosethepower-lawsizedistributionfunction
n(r ) ∝ r−q,
V V
which was often used for the modelling of interstellar extinction [16], [17]. The
averageGalacticextinctioncurvecan bereproduced usingthemixtureofcarbona-
ceous (graphite) and silicate particles (in almost equal proportions) with parame-
ters: q = 3.5,r ≈ 0.001µmand r ≈ 0.25µm.
V,min V,max
4 Results and discussion
Our calculations have been performed for prolate and oblate rotating spheroids
of several sizes and aspect ratios. The particles consisting of astronomical silicate
(astrosil) and amorphous carbon (AC1) were considered. The optical properties
of spheroids were derived using the solution to the light scattering problem by
the separation of variables method [18] and modern treatment of spheroidal wave
functions[19].
Below webriefly discussextinctionby spheroidsand in sufficientdetail polarizing
efficiency.ThecomparisonwithobservationsoftwostarsisperformedinSect.4.3.
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4.1 Extinctioncurve
Extinction (and sometimes polarization) produced by spheroids was studied by
Rogers and Martin [20], Onaka [21], Vaidya et al. [22], Voshchinnikov [23] and
morerecentlybyGuptaetal.[24].Mostlytheparticleswiththeaspectratioa/b ≤ 2
were considered.
Theparametersofourmodel(seeSect.3)are:thetype(prolateoroblate)andcom-
position of spheroids, the size r or the parameters of the size distribution r ,
V V,min
r , q, the degree specified through parameter δ and direction of alignment Ω.
V,max 0
Figure3illustrateshowvariationsofparticleshapeandalignmentinfluenceonthe
normalized extinction A(λ)/A . The value of δ for IDG orientation is typical of
V 0
diffuse interstellar medium [15]. It is seen that the difference in curves are in evi-
dence only in the UV. Prolate spheroids produce larger extinction when the angle
of alignment reduces from Ω = 90◦ (magnetic field is perpendicular to the line of
site) to Ω = 0◦ (magnetic field is parallel to the line of site; Fig. 3, left panel).
ForoblateparticlesthedependenceonΩisopposite.However,onlyforPDGthese
changes are at a detectable rate. Extinction also grows when the particles (prolate
and oblate)becomemoreelongatedorflattened (see Fig.3, rightpanel).
Fig. 3. Normalized extinction for ensembles of aligned prolate spheroids from as-
trosil. Spheroids have power-law size distribution with parameters: r = 0.001µm,
V,min
r = 0.25µm and q = 3.5; δ = 0.196µm. The effect of variations of particle degree
V,max 0
anddirection ofalignment (leftpanel)andparticleshape(rightpanel)isillustrated.
Variationsofparameters ofsizedistribution(r , r , q)on extinctionare sim-
V,min V,max
ilartothoseforspheres are notdiscussedhere(see, e.g., [25]).
4.2 Polarizingefficiency: theory
Theoretical dependence of thepolarizing efficiency of theinterstellarmedium was
discussed several times for particles of constant refractive index (see, e.g., [20],
[21]). The wavelength dependence of P/A was considered in [26] for infinite and
finite cylinders and in [7] for perfectly aligned spheroids of single size. Here, we
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analyse firstly the dependence P(λ)/A(λ) for single size particles and then for en-
sembleswithasizedistribution.
4.2.1 Singlesize
Some results for prolate particles are plotted in Figs. 4 and 5. They show the po-
larizing efficiency in the wavelength range from the infrared to far ultraviolet. The
dependence P(λ)/A(λ) observed for two stars is given for comparison. We made
calculationsforcompactgrainsandforporousgrainsusingtheBruggemanmixing
ruleforrefractiveindexandparticlesofsamemassascompactones.Notethatcal-
culated efficiencies can be considered as upperlimitsbecause somepopulationsof
grains (spherical, non-oriented) may contribute into extinction but not to polariza-
tion.
Fig. 4. Wavelength dependence of polarizing efficiency for homogeneous rotating
spheroidal particles of astronomical silicate and amorphous carbon. The effect of varia-
tionsofparticlecomposition anddirectionofalignmentisillustrated. Theopencirclesand
squaresshowtheobservational dataforstarsHD24263andHD99264, respectively.
From Figs. 4, 5 one can conclude that the wavelength dependence of polarizing
efficiency is mainly determined by the particle composition and size. Variations
of other parameters influence on the value of efficiency (the dependence of P/A
is scaled). A growth of the efficiencies P(λ)/A(λ) takes the place if we increase
angle Ω (direction of alignment deviates widely from the line of site), parameter
δ (degree of alignment) or aspect ratio a/b (consider more elongated or flattened
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particles) and decrease particleporosity (volumefraction of vacuum)P or particle
size r . It is also evident that the contribution of particles with size r ≈ 0.1µm
V V
toobservedpolarizationmustberatherimportantandthepolarizationproducedby
perfectly alignedparticles significantlyexceeds theobservedone.
4.2.2 Sizedistribution
We also calculated the wavelength dependencies of P/A separately for spheroids
fromAC1andastrosilwithstandardpower-lawsizedistribution(r = 0.001µm,
V,min
r = 0.25µmandq = 3.5)andconsideredtheeffectofvariationsdifferentmodel
V,max
parameters.TheresultsareshowninFigs.6,and7.NoteagainthatthevalueofP/A
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Fig. 5. Wavelength dependence of polarizing efficiency for homogeneous rotating
spheroidalparticlesofastronomicalsilicate.Theeffectofvariationsofparticlesize,shape,
porosity and degree of alignment isillustrated. Theopen circles and squares show the ob-
servational dataforstarsHD24263andHD99264,respectively.
depends on the degree and direction of alignment and the shape of particles while
theparticlecompositionandsizestronglyinfluencesonthewavelengthdependence
ofpolarizingefficiency.
Fig.6. Polarizing efficiency ensembles ofaligned prolate spheroids witha/b = 2from as-
trosil (left panel) and amorphous carbon (right panel). Spheroids have power-law size dis-
tributionwithparameters:r = 0.001µm,r = 0.25µmandq = 3.5;δ = 0.196µm.
V,min V,max 0
Theeffectofvariationsofparticlecompositionanddirectionofalignmentisillustrated.The
open circles and squares show the observational data for stars HD 24263 and HD 99264,
respectively.
AsfollowsfromtheseFigures,theoreticalpolarizingefficiencymoreorlessresem-
bles the observational dependencies only if the contribution of small particles into
polarizationisdepressed(r > 0.05µmorq < 3,seeFig.7).Itiswellknownfor
V,min
a long time that the polarization of forward transmitted radiation is determined by
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Fig.7.Polarizingefficiencyensemblesofalignedprolatespheroidsfromastrosil.Spheroids
have power-law size distribution, Ω = 90◦. The effect of variations of parameters of size
distribution, degree of alignment and particle shape is illustrated. The open circles and
squaresshowtheobservational dataforstarsHD24263andHD99264, respectively.
particles of small sizes while large particles produce no polarization independent
oftheirshape(see discussionin[1]and [27]).
The influence of small particles on P/A is reduced if: a) the particles are absent in
theinterstellarcloud,b)thesmallparticlesarelessorientedorc)theshapeofsmall
particlesisclosertospherical.Thereasona)mustmanifestsitselfbythedepressed
UV extinctionwhilethereasonsb)and c)wouldbedifficultto distinguish.
The relationships found can be used for more detailed comparison of the theory
withobservations.
4.3 Comparisonwithobservations
We calculated thenormalized extinction A(λ)/A and thepolarizingefficiency and
V
compared the results with observations of stars mentioned in Sect. 2. In order to
fit the UV bump on the extinction curve a small amount of graphite spheres with
radiusr ≈ 0.015µmwas added tothemixtureofsilicateand carbon spheroids.
V
According to the model presented in Sect. 3, we adopted that the same particles
produce extinction and polarization and the alignment of silicateand carbon parti-
cles is identical. At the same time, parameters of size distribution for silicate and
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carbon can differ.
Theresultsofcomparisonareshownfortwostars inFigs.8 and 9.
Fig. 8. Normalized extinction (left panel) and the polarizing efficiency (right panel) in
the direction of the star HD 147933. The error bars show 1σ uncertainty. Dashed curves
show the results of calculations for model with prolate spheroids with a/b = 5 from
amorphous carbon (24%) and astrosil (73%) and graphite spheres (3%). Spheroids have
a power-law size distribution with the parameters: amorphous carbon, r = 0.03µm,
V,min
r = 0.15µm,q = 1.5;astrosil, r = 0.08µm,r = 0.25µm,q = 1.5.Alignment
V,max V,min V,max
parameters are:δ = 0.5µm,Ω = 35◦.
0
The coincidence of the theory with observations is rather good. The better agree-
ment can be achieved if we assume that the alignment parameters for silicate and
carbonaceous grains differ. Forexample,it ispossibleto considernon-alignedcar-
bonaceous grains (as it was made in [28]). Evidently, more attention should be
givento modernalignmentmechanisms.
Fig. 9. Normalized extinction (left panel) and the polarizing efficiency (right panel) in the
directionofthestarHD62542.Theerrorbarsshow1σuncertainty.Dashedcurvesshowthe
resultsofcalculations formodelwithprolatespheroids witha/b = 2fromamorphous car-
bon(85%)andastrosil (10%)andgraphite spheres (5%).Spheroids haveapower-lawsize
distribution with the parameters: amorphous carbon, r = 0.07µm, r = 0.10µm,
V,min V,max
q = 2.7; astrosil, r = 0.005µm, r = 0.25µm, q = 2.7. Alignment parameters are:
V,min V,max
δ = 0.5µm,Ω= 40◦.
0
These improvements of the model require to invoke additional observational data
such as dust-phase abundances in the directions of considered stars, information
about magnetic fields, etc. Otherwise, the modelling cannot give non-ambiguous
results. In particular, the important feature is the wavelength where the ratio P/A
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