The Microwave Thermal Emission from the Zodiacal Dust Cloud Predicted with Contemporary Meteoroid Models

0.93 MB·English

by  Valery V. Dikarev

Astronomy&Astrophysicsmanuscriptno.paper-con-aa-2 (cid:13)cESO2015
January21,2015
The Microwave Thermal Emission
from the Zodiacal Dust Cloud
Predicted with Contemporary Meteoroid Models
ValeryV.DikarevandDominikJ.Schwarz
FacultyofPhysics,BielefeldUniversity,Postfach100131,33501Bielefeld,Germany
e-mail:[email protected]
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1 Received<date>/Accepted<date>
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2
ABSTRACT

n
a Predictionsofthemicrowavethermalemissionfromtheinterplanetarydustcloudaremadeusingseveralcontemporary meteoroid
J modelstoconstructthedistributionsofcross-sectionareaofdustinspace,andapplyingtheMielight-scatteringtheorytoestimate
the temperatures and emissivities of dust particles in broad size and heliocentric distance ranges. In particular, the model of the
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interplanetarydustcloudbyKelsalletal.(1998,ApJ508:44–73),thefivepopulationsofinterplanetarymeteoroidsofDivine(1993,
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JGR98(E9):17,029–17,048) andtheInterplanetaryMeteoroidEngineeringModel(IMEM)byDikarevetal.(2004,EMP95:109–

122)areusedincombinationwiththeopticalpropertiesofolivine,carbonaceousandironsphericalparticles.TheKelsallmodelhas
]
P beenwidelyacceptedbytheCosmicMicrowaveBackground(CMB)community.Weshow,however,thatitpredictsthemicrowave
E emissionfrominterplanetarydustremarkablydifferentfromtheresultsofapplicationofthemeteoroidengineeringmodels.Wemake
mapsandspectraofthemicrowaveemissionpredictedbythethreemodelsassumingvariantcompositionofdustparticles.Predictions
.
h canbeusedtolookfortheemissionfrominterplanetarydustinCMBexperimentsaswellastoplannewobservations.
p
- Keywords. Cosmology:cosmicbackgroundradiation-Zodiacaldust-Radiationmechanisms:thermal
o
r
t
s1. Introduction emission bound to the Solar system. Dikarevetal. (2009) ex-
a
[ ploredthe possibilitythattheSolar-system dustemitsmorein-
The unprecedented high-precision surveys of the mi- deedthanit waspreviouslythought,andfoundthatthemacro-

1crowave sky performed recently by the Wilkinson Microwave scopic (> 0.1 mm in size) meteoroids can well contribute ∼
vAnisotropy Probe (WMAP, Bennettetal. 2013) and Planck 10µKinthemicrowaves,i.e.comparablewiththepowerofthe
0(PlanckCollaborationI2014)observatoriesinsearchandchar- CMBanomalies,withoutbeingdetectedintheinfrared(IR)light
8
acterisationofthelarge-andsmall-scalestructureoftheCosmic and included in the IR-based models like that of Kelsalletal.
7
Microwave Background (CMB) anisotropies pose new chal- (1998). Babichetal. (2007) and Hansenetal. (2012) have also
4
0lengesforsimulationandsubtractionoftheforegroundemission studiedpossiblecontributionoftheKuiperbeltobjectstothemi-
.sources, including the Solar-system dust. Previous templates crowaveforegroundradiation.Dikarevetal.(2008)constructed
1designed for this purpose were based on the Kelsalletal. andtestedagainsttheWMAPdatasomedustemissiontemplates.
0
(1998) model. They indicated little significance of the inter-
5 Here we improve and extend preliminary estimates of
planetarydust for the WMAP survey(Schlegeletal. 1998), yet
1 Dikarevetal. (2009). In addition to the Kelsall model, we use
remarkable contribution was detected at high frequencies of
:
vPlanck(PlanckCollaborationXIV2014). two meteoroid engineering models to make accurate and thor-
i oughmapsof the thermalemission from the Zodiacalcloud in
X The angular power spectrum of CMB anisotropies is in
thewavelengthrangefrom30µmto30mm.
goodagreementwiththeinflationaryΛ-colddarkmattermodel
r
a(PlanckCollaborationI2014).However,thereconstructedCMB Thepaperisorganizedasfollows.Section2introducesthree
mapsatthelargestangularscalesrevealsomeintriguinganoma- modelsoftheinterplanetarymeteoroidenvironmentthatweuse
lies, among them unexpected alignments of multipole mo- topredictthethermalemissionfromthezodiacaldustcloud.A
ments, in particular with directions singled out by the So- detaileddescriptionofthetheoryandobservationsincorporated
lar system (Schwarzetal. 2004). The quadrupoleand octopole ineachmodelisprovided,itmaybehelpfulnotonlyforcosmol-
are found to be mutually aligned and they define axes that ogists interested in understanding the solar-system microwave
are unusually perpendicular to the ecliptic pole and paral- foregrounds,but also for developers of new meteoroid models
lel to the direction of our motion with respect to the rest willing to comprehend earlier designs. Section 3 describes the
frame of the CMB (the dipole direction). For reviews see thermalemission models. Empiricalmodelsas wellas the Mie
Bennettetal.(2011),Copietal.(2010)andBennettetal.(2013) light-scatteringtheoryareusedtocalculatethetemperaturesand
and PlanckCollaborationXXIII (2014) as well as Copietal. emissionintensitiesofsphericaldustparticlescomposedofsil-
(2013) for a detailed analysis of final WMAP and first Planck icate,carbonaceousandironmaterials,forbroadrangesofsize
data.Ithasbeensuggestedthatanunaccountedobservationbias andheliocentricdistance.Themodelsofthespatialdistribution
maypersist,e.g.yetanotherforegroundsourceofthemicrowave ofdustandthermalemissionarecombinedinSection4inorder
Articlenumber,page1of17
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Table 1. Frequencies ν, wavelengths λ and bandwidths ∆ν/ν of level-of-damagethresholds,hencetheyprovidethesizedistribu-
the CIB/CMB observations with COBE DIRBE wavelength bands 4
tionexplicitly.
through10,WMAP,andPlanck.
Profilesofparticlecross-sectionareadensityperunitvolume
of space are plotted for the Kelsall and Divine models as well
Instrument ν λ ∆ν/ν as IMEM in Fig. 1. Interestingly, the Kelsall model is sparser
COBE 61.2THz 4.9µm 0.13 thanboththeDivinemodelandIMEMatmostheliocentricdis-
DIRBE 25.0THz 12µm 0.54 tances.Themeteoroidengineeringmodelspredictsubstantially
12.0THz 25µm 0.34 more dust than Kelsall does, especially beyond 1 AU from the
5.00THz 60µm 0.46 Sun:Divine’sdensityisalmostflatintheeclipticplanebetween
3.00THz 100µm 0.32 1and2AU,exceedingKelsall’sdensitybyafactorof3intheas-
2.14THz 140µm 0.28 teroidbelt.TheIMEMdensityremainssimilartothatoftheKel-
1.25THz 240µm 0.40 sallmodelupto∼ 3AU,showingalocalmaximumbeyondthe
Planck 857GHz 350µm 0.25 asteroidbeltnear5AUfromtheSun,inthevicinityofJupiter’s
HFI 545GHz 550µm 0.25 orbit.IMEMpossessesthelatitudinaldistributionsomewhatnar-
353GHz 850µm 0.25 rowerthanthatoftheKelsallandDivinemodels.
217GHz 1.4mm 0.25 All these distinctions stem from different observations and
143GHz 2.1mm 0.25 physicsincorporatedindifferentmodels.Whendescribingthem
100GHz 3.0mm 0.25 in subsequent subsections, we do not aim at selecting the best
Planck 70GHz 4.3mm 0.2 model. We take all three of them instead, with the intention to
LFI 44GHz 6.8mm 0.2 “bracket” the more complex reality by three different perspec-
30GHz 10.0mm 0.2 tivesfromdifferent“pointsofview”.
WMAPW 94GHz 3.2mm 0.22
V 61GHz 4.9mm 0.23
2.1.TheKelsallmodel
Q 41GHz 7.3mm 0.20
Ka 33GHz 9.1mm 0.21 A concise analytical description of the infrared emission from
K 22GHz 13.6mm 0.24 theinterplanetarydustcapturedbytheCOBE DiffuseInfraRed
BackgroundExperiment(DIRBE)wasproposedbyKelsalletal.
(1998). It recognizes five emission components, a broad and
tomakeall-skymapsandspectraofthe thermalemissionfrom bright smooth cloud, three finer and dimmer dust bands, and
theinterplanetarydust.ConclusionsaremadeinSect.5. circumsolar dust ring along the Earth orbit with an embedded
Throughout this paper, we will be dealing with the wave- Earth-trailingblob.Each componentis describedby a parame-
lengthsmostly,sincethedustopticalpropertiesarenaturallyde- terised empiricalthree-dimensionaldensity functionspecifying
scribedintermsoflinearscales.TheCMBcommunity,however, thetotalcross-sectionareaofthecomponent’sdustparticlesper
ismoreaccustomedtofrequencies.Aconversiontableisthere- unitvolumeofspace(Fig.2),andbytheircollectivelightscatter-
fore usefulfor the observationwavebandsof infrareddetectors ingandemissionpropertiessuchasalbedo,absorptionefficien-
andradiometers(Table1). cies, etc. The particle size distributions of the components are
notconsidered.Mostofthe lightscatteringandemission prop-
ertiesareneitheradoptedfromlaboratorystudiesofnaturalma-
2. MeteoroidModels terialsnorpredictedbytheoriesoflightscattering,theyarefree
parametersofthemodelfittotheDIRBEobservationsinstead.
In this paper, we use three recent and contemporary models The smooth cloud is the primary componentof the Kelsall
of the Zodiacal dust cloud due to Kelsalletal. (1998), Divine model.Itsdensity functionis traditionallyseparatedinto radial
(1993)andDikarevetal.(2004).Thefirstmodelisconstrained and vertical terms. The radial term is a power law 1/Rα with
c
by the infrared observations only from an Earth-orbitingsatel- α = 1.34 ± 0.022 being the slope of the density decay with
lite COBE. For brevity, it is referred to as the Kelsall model distance from the cloud’s centre R . The slope is known to be
c
hereafter. The Kelsall model is most familiar to and most of- equaltooneforthedustparticlesincircularorbitsmigratingto-
tenusedbytheCosmicMicrowaveBackground(CMB)research ward the Sun under the Poynting-Robertson drag (orbiting the
communityasithelps,andwasdesignedtoassessandmitigate Sun, the particles absorb its light coming from a slightly for-
thecontaminationofthebackgroundradiationmapsbythefore- warddirectionduetoaberration,andtheradiationpressureforce
groundradiation from interplanetarydust. The two other mod- hasa non-zeroprojectionagainstthe directionof their motion;
elsincorporatedataobtainedbydiversemeasurementtechniques thatprojectioncausesgraduallossoftheorbitalenergyandde-
andservetopredictmeteoroidfluxesonspacecraftandtoassess crease of the semimajor axis of particle’s orbit). The slope is
impacthazards.Beingappliedinspacecraftdesignandanalysis greaterthanoneiftheparticleorbitsareinitiallyeccentric,orif
mostly,theyareoftendubbedasmeteoroidengineeringmodels. the zodiacalcloud is fed from sourcesbroadlydistributed over
We choosethe Divine modelandIMEM, with the latter abbre- radial distances, so that the inner circles of the cloud are sup-
viation standing for the Interplanetary Meteoroid Engineering pliedfrommoredustsourcesthantheoutercircles(Leinertetal.
Model. We also check if the most recent NASA meteoroid en- 1983;Gor’kavyietal.1997).
gineeringmodel,MEM(McNamaraetal.2004),canbeusedin The vertical term of smooth cloud’s density is represented
ourstudy,andexplainwhyitcannotbe. byawidened,modifiedfanmodel.Thesmoothcloud’ssymme-
TheKelsallmodeldoesnotconsiderthesizedistributionof tryplaneisinclinedofftheeclipticplane.Thishappensbecause
particles in the Zodiacal cloud. The size distribution is implic- the Earth has no influence on the orbital dynamics of the vast
itly presentedby the integralopticalpropertiesof the cloud.In majorityof cloud’sparticles. The giantplanets, Jupiter primar-
contrast,themeteoroidengineeringmodelsareobligedtospec- ily, perturb the orbits of dust particles as well as their sources,
ifythefluxesofmeteoroidsforvariousmass,impulse,orother or parentbodies, such as cometsand asteroids, and controlthe
Articlenumber,page2of17
V.V.Dikarev&D.J.Schwarz:MicrowaveThermalEmissionfromtheZodiacalDustCloud
1.8
Kelsall Kelsall
10-6 Divine 70 1.6 Divine
U 1
A IMEM U • 1.4 IMEM
er A 1.2
p -7 r
a 10 pe 1
re a
A e 0.8
n Ar
um 10-8 mn 0.6
ol 0.4
C u
ol
C 0.2
10-9 0
0.1 1 10 -20 0 20 40 60 80
Heliocentric Distance, AU Ecliptic Latitude, degree
Fig.1.Profilesofparticlecross-sectionareadensitypredictedwithselectedmeteoroidmodelsandcomparedwitheachother:eclipticradial(left)
andlatitudinalat1AUfromtheSun(right,thevernalequinoxisatlatitude0◦).
inclinationsoftheorbitsoftheparticles.Anoffsetofthecenter The dustparticles on nearlycircular orbitsmigratingunder
of the cloud from the Sun is also allowed in the Kelsall model the Poyinting-Robertson drag toward the Sun are temporarily
(oftheorderof0.01AU).TheverticalmotionoftheEarthwith trapped in a mean-motionresonance with the Earth near 1 AU
respectto the cloud’ssymmetryplaneleadsto the modulations (Dermottetal. 1984). The resulting enhancementof the zodia-
of the infraredemission from the zodiacalcloud reaching30% calclouddensityisdescribedin the Kelsallmodelbythe solar
intheeclipticpolarregions. ring and Earth-trailingblob. The trailing blob is the only com-
ponentofthemodelthatisnotstatic:itisorbitingtheSunalong
Thedustbandsaretheremnantsofcollisionaldisruptionsof acircularorbitwithaperiodofoneyear,andasthenamesug-
asteroids that resulted in formation of the families of asteroids gests, its densitypeak is located behindthe movingEarth.The
insimilarorbitsandremarkabledustbelts.Thedustbandswere solarringandtrailingblobhavedensitypeakdistancesslightly
discoveredfirst on the high-resolutionmaps made by the IRAS outsidetheEarthorbit,theirsymmetryplanesareinclinedoffthe
satellite (Lowetal. 1984) and recognized later in the COBE eclipticplane,and thetrailing bloborbitsthe Sunat a constant
DIRBE data as well. Kelsalletal. (1998) introducedthree dust velocitywhereastheEarthvelocityisvariableduetotheeccen-
bandsintheirmodel,attributedtotheThemisandKoronisaster- tricityofourplanet’sorbit.Consequently,theEarthmoveswith
oidfamilies(±1◦.4eclipticlatitude),theEosfamily(±10◦),and respecttothesolarringandtrailingbloboverthecourseofthe
the Maria/Io family (±15◦). The bands have symmetry planes yearmuchliketheothercomponentsoftheKelsallmodel.
differentfromthatofthe smoothbackgroundcloud.Thebands
TheDIRBEinstrumentperformedasimultaneoussurveyof
are all double, with the density peaking above and below the
the sky in 10 wavelength bands at 1.25, 2.2, 3.5, 4.9, 12, 25,
respective symmetry plane, since their constituent particles re-
60, 100, 140, and 240 µm. It was permitted to take measure-
tain the inclination of the ancestor asteroid orbit but the longi-
mentsanywherebetweenthesolarelongationsof64◦ and124◦.
tudes of nodes get randomized by the planetary perturbations.
As the Earth-bound COBE observatory progressed around the
Theverticalmotionofsuchparticlesissimilartothatofanos-
Sun,DIRBEtooksamplesoftheinfraredbackgroundradiation
cillator,whichmovesslowerandspendsmoretime nearitsex-
fromalloverthesky.Theforegroundradiationduetointerplane-
tremities, thus the density at the latitudes of each family’s or-
tarydustwasavariableingredientoftheskybrightnessbecause
bital inclination is high. One would expect a similarly shaped
itdependsonthechangingpositionofobservatorywithrespect
“edge-brightened” radial density with the peaks at the perihe-
tothecloud.Thusthefittingtechniquewasbasedonminimizing
lionandapheliondistancesoftheparentbodyorbit(e.g.Sect.3
thedifferencebetweenthebrightnessvariationsintimeobserved
in Gor’kavyietal. 1997). However, the dust in asteroid bands
alongindependentlinesofsightandthosepredictedbytheKel-
is proven to be migrating toward the Sun under the Poynting-
sall model for the same lines of sight, ignoring the underlying
Robertson drag: Reach (1992) has demonstrated that the tem-
photometric baselines contaminated or even dominated by the
peraturesofparticlesandparallaxesofthebandsmeasuredfrom
backgroundsources.
the moving Earth are both higher than those supposed to be in
therespectiveasteroidfamiliesoftheirorigin,implyingthatthe TheaccuracyoftheKelsallmodelindescribingtheinfrared
bandsextendfarthertowardstheSunthantheirancestorbodies. emission from interplanetary meteoroids in DIRBE’s wave-
Unlike Reach (1992), Kelsalletal. (1998) allowed for a partial length bands and within the range of permitted solar elonga-
migration only of dust, by introducing a cut-off at a minimal tionsisreportedtobebetterthan2%.Interpolationsofthemodel
heliocentric distance either defined or inferred individually for brightnessesbetweentheinstrumentwavelengthsandextrapola-
eachband.Theyalsofixedratherthaninferredmostofthedust tionsbeyondthewavelengthandsolarelongationrangesmaybe
bandshapeparameters.ItisnoteworthythatKelsalletal.(1998) pronetohigheruncertainties.Indeed,theconcisedescriptionof
didnotintroduceanexplicitoutercut-offforthedensitiesofthe the zodiacalemission contributionto the DIRBE data provides
smoothcloudanddustbands.Instead,theyintegratedthedensi- noclueastohowthelightscatteringandemissionpropertiesof
tiesalongthelinesofsightfromtheEarthtothemaximalhelio- itscomponentsdependon wavelengthbetweenandbeyondthe
centricdistanceof5.2AU(closetotheJovianorbitalradius). tenDIRBEwavelengthbands.
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1
-U 1.8
-1U 10-6 A Smooth Cloud
A 7, 1.6 Band 1
sity, 10-7 y •10 1.4 BBaanndd 23
en -8 sit 1.2 Ring+Blob
D 10 n
a De 1 TOTAL
Are 10-9 ea 0.8
n Ar
ctio 10-10 on 0.6
Se cti 0.4
- -11 e
s 10 S
os s- 0.2
r s
C 10-12 o 0
r
0.1 1 10 C -20 0 20 40 60 80
Heliocentric Distance, AU Ecliptic Latitude, degree
Fig.2.Radialandlatitudinalprofilesofparticlecross-sectionareadensityintheKelsallmodel,componentwise.
When applying the Kelsall model in the far infrared wave- anddatafromseveralspacecraft.Theorbitaldistributionsarede-
lengths and microwaves, one should bear in mind that already terminedusingmeteorradardata,observationsofzodiacallight
Kelsalletal. (1998) found the 140 and 240 µm bands nearly fromtheEarthaswellasfromtheHeliosandPioneer10inter-
useless in constraining the density distribution parameters due planetaryprobes,andmeteoroidfluxesmeasuredin-situbyim-
torelativelyhighdetectornoiseandsmallcontributionofthera- pactdetectorsonboardPioneer10and11,Helios1,Galileoand
diationfromthe interplanetarydustatthese wavelengths.Con- Ulyssesspacecraftattheheliocentricdistancesrangingfrom0.3
sequently,theinferredparametersofthedensitydistributionsin to18AU.Thelogarithmsofthemodel-to-dataratiosweremini-
the Kelsall model are based on the observations at the wave- mizedinaroot-mean-squaresense.Whenmodelingthezodiacal
lengthsupto100µm.Thedustparticlesareefficientininteract- light,Divineassumedthatthescatteringfunctionisindependent
ing with the electromagneticradiationif their sizes are not too of meteoroid mass. The albedos of dust particles were defined
small with respect to the wavelength (2πs > λ, where s is the somewhatarbitrarilyin orderto hidethepopulationsnecessary
particle radiusand λ is the wavelength).Dust grains with radii tofit the in-situmeasurementsfromthe zodiacallightobserva-
from∼16µmarethereforevisibleatthewavelengthof100µm. tions.Noinfraredobservationswereusedtoconstrainthemodel.
Dikarevetal.(2009)arguethataconsiderableamountofmete-
ThecorepopulationisthebackboneoftheDivinemodel,it
oroidsarepresentintheSolarsystemwiththesizesbiggerthan
fits as much data as possible with a single set of distributions.
∼ 100 µm which are outshined in the infrared light by more
Thedistributioninperiheliondistancer ofthecore-population
π
abundantandubiquitoussmallerdustgrains.Thelongestobser- particlescanbeusedasstrictlyproportionaltor−1.3 upto2AU
vationwavelengthofDIRBEatwhichKelsalletal.(1998)could π
from the Sun. This function resembles a spatial concentration,
still constrainthe density distributionsof dustis shortof being
anditsslopeisclosetoKelsall’sα = 1.34fortheradialdensity
capabletoresolvetheseparticlesunlesstheycomposedistinctive
term:muchliketheinfrareddatafromCOBE,thezodiacallight
featureslikedustbands.
observations from the Earth and interplanetary probes demand
We use the formulae for the density functions provided by thattheslopebesteeperthanaunitexponent.Theeccentricities
PlanckCollaborationXIV (2014) since the original paper by aremoderate(peakingat0.1andmostlybelow0.4),inclinations
Kelsalletal.(1998)containstypographicalmisprintsinthedef- aresmall(mostlybelow20◦).
initionsoftheasteroiddustbandandcircumsolarringdensities.
Theotherpopulationsfillin thegapswhereonepopulation
withseparabledistributionsisnotsufficienttoreproducetheob-
2.2.TheDivinemodel servations.Theirnameshintatdistinctivefeaturesoftheirorbital
distributions.The inclined populationis characterizedby incli-
A model of the interplanetary meteoroid environment (Divine nationslargelyintherange20◦–60◦,itseccentricitiesareallbe-
1993) constructed to predict fluxes on spacecraft anywhere in low 0.15. In contrast, the eccentric population is composed of
theSolarsystemfrom0.05to40AUfromtheSun,iscomposed particlesinhighlyeccentricorbits(theeccentricitiesarelargely
of five distinct populations, each of which has mathematically between 0.8 and 0.9), the inclinations are same as in the core
separabledistributionsinparticlemassandinorbitalinclination, population. These two populations are added in order to com-
periheliondistance,andeccentricity(Fig.3).Usingthedistribu- pensateforadeficitofmeteoroidsfromthecorepopulationwith
tions in orbitalelements, Divine explicitly incorporatedKeple- respecttoin-situfluxmeasurementsonboardtheHeliosspace-
riandynamicsofmeteoroidsinheliocentricorbits. craft.(Theeccentricpopulationis also usedto match theinter-
TheDivinemodelisconstrainedbyalargenumberofdiverse planetaryfluxmodelbetween10−18and10−15g.)Thehalopopu-
meteoroid data sets. The mass distribution of meteoroids from lationhasauniformdistributionoforbitalplaneorientationsand
10−18to102gisfittedtotheinterplanetarymeteoroidfluxmodel “surrounds”theSunasahalobetweenroughly2and20AU.It
(Grünetal. 1985), which in turn is based on the micro-crater patchesthemeteoroidmodelwheretheabove-listedpopulations
countsonlunar rocksamplesretrievedbythe Apollomissions, are not sufficient to reproduce the Ulysses and Pioneer in-situ
i.e.thenaturalsurfacesexposedtothemeteoroidfluxnearEarth, fluxmeasurements.Theeccentric,inclinedandhalopopulations
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V.V.Dikarev&D.J.Schwarz:MicrowaveThermalEmissionfromtheZodiacalDustCloud
1
-U 1.8
-1U 10-6 A Core
A 7, 1.6 Asteroidal
sity, 10-7 y •10 1.4 EIcnccelinnteridc
en -8 sit 1.2 Halo
D 10 n
a De 1 TOTAL
Are 10-9 ea 0.8
n Ar
ctio 10-10 on 0.6
Se cti 0.4
- -11 e
s 10 S
os s- 0.2
r s
C 10-12 o 0
r
0.1 1 10 C -20 0 20 40 60 80
Heliocentric Distance, AU Ecliptic Latitude, degree
Fig.3.Radialandlatitudinalprofilesofparticlecross-sectionareadensityintheDivinemodel,componentwise.
arecomposedofmeteoroidsmuchsmallerthanthosesignificant withKepleriandynamicsoftheconstituentparticlesinheliocen-
enoughfor predictionsof the infraredand microwave observa- tricorbits.Expansionofcomputermemoryhasalsoenabledthe
tions. authorsofIMEMtoworkwithlargearraysrepresentingmulti-
The asteroidal population is described last but it makes a dimensionaldistributionsinorbitalelementstoreplaceDivine’s
hugedifferencebetweenKelsall’sandDivine’smodels.Already separabledistributions.Thedistributioninmassremainssepara-
whenfittingthefirstpopulationtothemeteorradardata,Divine blefromthe 3-Ddistributionininclination,periheliondistance
observed that the results for the distribution in perihelion dis- andeccentricity.
tance show a minimum near rπ = 0.6 AU, suggesting that two IMEMisconstrainedbythe micro-cratersize statisticscol-
componentsareinvolved(i.e.,innerandouter).Theradialslope lectedfromthelunarrocksretrievedbytheApollocrews,COBE
of the inner componentwas consistent with that demanded by DIRBE observations of the infrared emission from the inter-
thezodiacallightobservations,andtheinnerandoutercompo- planetarydustat4.9,12,25,60,and100µm wavelengths,and
nents contributed 45% and 55%, respectively, to the flux near GalileoandUlyssesin-situfluxmeasurements.Anattempttoin-
Earth.Theoutercomponent’sconcentrationpeakedoutsidethe corporatenewmeteorradardatafromtheAdvancedMeteorOr-
Earth orbit, in the asteroid belt. Since the main contributionto bitRadar(AMOR,seeGalligan&Baggaley2004)wasnotsuc-
thezodiacallightcomesfromtheparticleswithmassessmaller cessful (Dikarevetal. 2005), as the latitudinal number density
than10−5gandthemedianmeteoroidmassofthemeteorradar profile of meteoroidsderivedfrom the meteor radar data stood
was 10−4 g, it was reasonable to ascribe the inner fraction to a in stark contrast to that admissible by the COBE DIRBE data,
populationdominatedbysmallerparticlesresponsibleforthezo- duetoincompleteunderstandingoftheobservationbiases.
diacallight,andtheouterfractiontoanotheronedominatedby
The meteoroids in IMEM are distributed in orbital ele-
largerparticlesdetectedasmeteorsonlywiththeradar.Thetwo
mentsinaccordwithanapproximatephysicalmodeloftheori-
fractionswere dividedintothe coreand asteroidalpopulations,
gin and orbital evolution of particulate matter under the plan-
accordingly.Their inclination and eccentricity distributions are etary gravity, Poynting-Robertsoneffect and mutual collisions.
identical, but the distributions in perihelion distance and mass
Grün&Dikarev(2009)visualisetheIMEMdistributionsofme-
are different. The asteroidal population is used to fit the inter-
teoroidsinmassandinorbitalelementsingraphandcolorplots.
planetaryfluxmodelatlargemasses>10−4g.
Cross-sectionareadensityprofilesareshownhereinFig.4.
Derived from data analysis rather than postulated theoreti-
All interplanetary meteoroids are divided into populations
cally,thedistinctionbetweenthecoreandasteroidalpopulations
byorigin/sourceandmass/dynamicalregime.Grünetal.(1985)
in the Divine model separates – both on the mass and perihe-
constructed a model of the mass distribution of interplane-
lion distance scales – the small dustparticles migratingtoward
tary meteoroids. Based on this model, they estimated particle
theSununderthePoynting-Robertsondrag,theconcentrationof
lifetimes against two destructive processes, mutual collisions
which increases with the decrease of the heliocentric distance,
and migration downward toward the Sun due to the Poynting-
andtheirimmediateparentbodies,thelargerparticlesswarming
Robertson drag (terminated by particle evaporation or thermal
furtherawayfromtheSun,i.e.intheasteroidbeltaccordingto
break-up).Figure5showsthatthelifetimeagainstmutualcolli-
Divine(1993).
sionsisshorterthanthePoynting-Robertsontimeforthe mete-
oroidsbiggerthan ∼ 10−5 g in mass, located at 1 AU fromthe
Sun.Thedestructionof meteoroidsisdominatedbya factorof
2.3.IMEM
teninratebytherespectiveprocessalreadyoneorderofmagni-
The Interplanetary Meteoroid Engineering Model (IMEM) is tudeawayfromthistransitionmassineitherdirection.Therates
developed for ESA by Dikarevetal. (2004). Like the Divine varywiththedistancefromtheSun,however,thetransitionmass
model, it uses the distributions in orbital elements and mass remains nearly unchanged (Fig. 6 in Grünetal. 1985). That is
ratherthanthespatialdensityfunctionsoftheKelsallmodel,en- whyDikarevetal. (2004)couldsimplifytheproblembydivid-
suringthatthe dustdensitiesandfluxesarepredictedin accord ing the mass range into two subranges with distinct mass dis-
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-1U 10-6 A
A 7, 1.6 Asteroidal,
sity, 10-7 y •10 1.4 Amste>r1o0id-5a gl,
en -8 sit 1.2 m<10-5 g
D 10 n
a De 1 Cometary,
Are 10-9 ea 0.8 m>10-5 g
n Ar Cometary,
ctio 10-10 on 0.6 m<10-5 g
Se cti 0.4 TOTAL
s- 10-11 Se
os s- 0.2
r s
C 10-12 o 0
r
0.1 1 10 C -20 0 20 40 60 80
Heliocentric Distance, AU Ecliptic Latitude, degree
Fig.4.Radialandlatitudinalprofilesofparticlecross-sectionareadensityinIMEM,componentwise.Thedensityplotsarenotsmoothsincethe
orbitaldistributionsinIMEMareapproximatedbystepfunctions.
tributionslopes anddynamics.Meteoroidsbelow the transition Jupiter.ItistheTisserandparameterT
massof10−5garetreatedasmigratingfromthesourcestoward
the Sun under the Poynting-Robertson drag, meteoroids above T = aJ +2 a(1−e2)cosi,
the transition mass are assumed to retain the orbits of their ul- a ra
J
timateancestorbodiesovertheirshortlifetimesuntilcollisional
with a being the semimajor axis of Jovian orbit. The num-
destruction. J
bers of meteoroids in T-layers are therefore stable over a long
TheultimateancestorbodiesofinterplanetarydustinIMEM
period of time. They are free parameters to be determined
are comets and asteroids. A catalogue of asteroids is used to
from the fit of IMEM to observations. The relative distribu-
accountfor the latter sourceof meteoroids.Threedistinct pop-
tionsinorbitalelementswithineachT-layerarefoundtheoreti-
ulations of asteroids are recognized and allowed to have inde-
cally(Dikarev&Grün2004).
pendenttotalproductionrates:theThemisandKoronisfamilies Finer dust grains with masses less than 10−5 g leak from
(semi-majoraxes2.8<a<3.25AU,eccentricities0<e<0.2,
this region into the inner solar system and then migrate to-
andinclinations0 < i < 3◦.5),EosandVeritasfamilies(2.95 <
wardtheSununderthePoynting-Robertsondrag.ThePoynting-
a < 3.05AU, 0.05 < e < 0.15,8◦.5 < i < 11◦.5),andthe main
Robertsonmigrationtimeismuchlongerthanthecloseapproach
belt(a < 2.8AU).Theorbitaldistributionsofmeteoroidsmore
timeintheregionofcloseencounterswithJupiter,consequently,
than 10−5 g in mass are constructed by counting the numbers
the orbital distributions of all meteoroids are shaped there by
ofcataloguedasteroidsperorbitalspacebins.Theorbitaldistri-
the gravitational scattering on Jupiter. It is only the dust leak-
butionsofmeteoroidslessthan10−5 ginmassaredescribedas
ing throughthe innerboundaryof the regionthat is distributed
theflowofparticlesalongthePoynting-Robertsonevolutionary
astheflowofparticlesalongthePoynting-Robertsonevolution-
paths (Gor’kavyietal. 1997) starting from the source distribu-
arypaths(Gor’kavyietal.1997).Theirdistributionsarealsoes-
tions defined earlier. The mass distributions are adopted piece-
tablishedtheoreticallyinIMEM,withthenormalisationfactors
wisefromtheinterplanetarymeteoroidfluxmodelbyGrünetal.
beinginferredfromthefittoobservations.
(1985)asshowninFig.5.
AnempiricalfindingbyDivineofahelpfulseparationofthe
The orbital distributionsof meteoroidsfrom comets cannot bulkofmeteoroidcloudintothecoreandasteroidalpopulations
be defined as easy as those of meteoroids from asteroids. Be- –segregationofbig andsmalldustparticles– hasbecomeone
causeofanumberoflossmechanisms,suchascometnucleide- ofthephysicalassumptionsinIMEM.
cay,accretion,tidaldisruptionorejectionfromtheSolarsystem The weights of populations of dust from comets and aster-
byplanets,veryfewcometsareactiveatagivenepochandlisted oidsinIMEMwerefittedtoin-situfluxmeasurementsusingthe
in the catalogues.Moreover,the cataloguesare proneto obser- Poissonmaximum-likelihoodestimatorandtoinfraredobserva-
vationbiasessincethecometnucleiarerevealedbygasanddust tionsusingtheGaussianmaxiumum-likelihoodestimator.
shedathigherratesatlowerperiheliondistances. IMEM was tested against those data not incorporated in
Inordertodescribetheorbitaldistributionsofmeteoroidsof the model by Dikarevetal. (2005), confirming that the or-
cometary origin, Dikarev&Grün (2004) proposed an approx- bital evolution approach allows for more reliable extrapola-
imate analytical solution of the problem of a steady-state dis- tionsoftheobservationsandmeasurementsincorporatedinthe
tribution of particles in orbits with frequent close encounters model, it is compared with several other meteoroid models by
with a planet. This solution is applied to represent the orbital Drolshagenetal.(2008).
distributions of meteoroidsfrom comets in Jupiter-crossing or-
bits,i.e.thevastmajorityofcataloguedcomets.Bigmeteoroids
2.4.MEM
with masses greater than 10−5 g are confined to the region of
closeencounterswithJupiter.Thereisonlyonequantitythatis The Meteoroid Engineering Model (MEM) described by
approximatelyconservedintheregionofcloseencounterswith McNamaraetal.(2004)isanotherrecentdevelopmentoftheor-
Articlenumber,page6of17
V.V.Dikarev&D.J.Schwarz:MicrowaveThermalEmissionfromtheZodiacalDustCloud
againstcollisionsbecomesshorterthanthePoynting-Robertson
-6 timebyordersofmagnitude,themeteoroidscannolongersur-
-1s] 10 vive a travel from their sources toward the Sun and the orbital
-2m 10-8 distributions of bigger particles are more similar with the dis-
tributionsof their sourcesthan the distributionsof smaller par-
AU [ 10-10 ticles. MEMis unfortunatelyunableto revealanddescribethis
at 1 10-12 Cum(uGlarutievne eDti satlr.i b1u9t8io5n) importantfeatureoftheinterplanetarymeteoroidcloud.
x
Flu 10-14 2.5.TheCross-SectionAreaDistributions
Figure 6 maps the total cross-section area of dust particles per
-12 -10 -8 -6 -4 -2 0
10 10 10 10 10 10 10 unitvolumeof space, as a functionof the position in the Solar
system, for each component or population of every meteoroid
ars Poynting-Roberson Drag modelintroducedinthepreviousSection.The X andZ coordi-
ye106 Collisional Destruction nates are measured from the Sun, with the X axis pointing to
dU, thevernalequinoxandZ axisbeingperpendicularto the eclip-
oiA
or1 105 tic plane. The integralof the cross-sectionarea densityalong a
eteat lineofsightgivestheopticaldepthofthecloudalongthatline
Me ofsight,assumingthegeometric-opticsapproximation,spherical
m 4
eti10 particles,andignoringthattheparticleefficienciesinabsorbing
Lif andscatteringlightcaninfactbehigherorlowerthanunity.
The Kelsallmodelisreproducedin the upperrow of maps.
10-12 10-10 10-8 10-6 10-4 10-2 100 Three dust bands are shown together in the middle plot. The
smooth cloud is to the left, and the solar ring is to the right of
10-6 it. The dominant component of the Kelsall model, the smooth
-1s] cloud is getting higher in density only toward the Sun and the
2 10-8 symmetryplaneslightlyinclinedofftheeclipticplane.Thedust
-m
bandsaretheonlycomponentofthemodelbulkingbeyondthe
AU [ 10-10 Earthorbit,andtheyarebydefinitionboundtotheeclipticlati-
1 Meteoroid Populations tudesoftheirancestorasteroidfamilies’orbitinclinations.Both
x at 10-12 CoDlliosimoninsa, tmed> 1b0y-:5g tthriecsdmisotaonthcecloofu5d.2anAdUthoendlyu,sstibnacnedKseelxstaelnldetuapl.t(o19th9e8)hesltioopcpeend-
u -5
Fl -14 P-R Drag, m<10 g integratingthemodeldensitiesthere.Amorerigorousmodelof
10 Total
the bands composed of dust migrating toward the Sun due to
-12 -10 -8 -6 -4 -2 0 thePoynting-Robertsoneffectwouldputtheirdensities’cut-offs
10 10 10 10 10 10 10
at the outer boundaries of the corresponding asteroid families.
Meteoroid Mass [g]
Note,however,thatmostofthethermalinfraredemissionfrom
Fig. 5. Cumulative mass distribution of interplanetary meteoroids the interplanetary dust observed by COBE was due to the dust
(Grünetal. 1985, top), meteoroid lifetimes against mutual collisions within0.5–1AUfromtheEarthorbit,sinceCOBEcouldnotbe
andPoynting-Robertson drag(middle),andcumulative massdistribu- pointedtooclosetotheSunanddustistoocoldandinefficient
tionsofmeteoroidpopulationsinIMEM(Dikarevetal.2005,bottom).
atthermalemissionfarfromtheSun(Dikarevetal.2009,their
Thefluxofmeteoroidswithm > 10−5 gislowerinIMEMthaninthe
Sect.2.1andFig.2).Thusforthepurposeofmodellingthein-
Grünmodelsincetheimpactvelocitiesofbigmeteoroidsarehigherat
fraredthermalemissionobservedfromtheEarth,thebehaviour
1AUfromtheSuninIMEMandtheyproducelargercratersonthelunar
of the density at multiplesof an astronomicalunit was notim-
rocksthanintheGrünmodel,whichassumedasingleimpactvelocity
forallmeteoroidsizes. portant. This may not be the case for the microwave emission
though.
ThenexttworowsofmapsdepictthepopulationsoftheDi-
bitaldistributionsandsoftwaretopredictfluxesonspacecraftin vine model. The core and asteroidal populations are important
theSolar systemandnearEarth.Itis constrainedbythe Earth- in the infrared and microwave emission ranges. They are dis-
basedmeteorradardataCMOR(CanadianMeteorOrbitRadar) tributedremarkablydifferentinspace,withthecorepopulation
andzodiacallightobservationsfromtwointerplanetaryprobes, densitygrowinghighertowardtheSunandtheasteroidalpopula-
Helios1and2.Thenominalheliocentricdistancerangeatwhich tiondensitypeakingbeyondtheorbitoftheEarth.Thetwopopu-
themodelisapplicable,from0.2to2AU,isratherlimited,how- lationsarealsocomposedofparticlesofdifferentsizes:thecore
ever.Eventhoughthemeteoroidmassrangeof10−6to10gcov- population mostly smaller and the asteroidal population mosly
ered by the model is extremely interesting for the purposes of biggerthan∼ 50 µm. The halo,inclined andeccentric popula-
ourstudy,someassumptionsmadebytheauthorsareratherar- tionsareprovidedforthesakeofcompleteness.Theirtiny,upto
guable.In particular,the mutualcollisionsbetween meteoroids ∼ 10 µm-sizedparticlesareignorableinthewavelengthranges
are considered in the model ignoring dependence of the mete- ofinterest.
oroidcollisionprobabilityonitssize,whereasGrünetal.(1985) Thebottomrowofmapsandtherightmapinthethirdrow
calculatedthatthelifetimeagainstcollisionaldisruptionofme- exhibit the populations from IMEM. The asteroidal dust parti-
teoroidsvariesbyafactorof100betweenthemassesof10−6and clesbiggerthan10−5ginmassareconfinedtotheasteroidbelt,
1g(theirFig.6andourFig.5inthetextabove)!Itisnotonly naturally.Theyarenotmigratingfromtheirancestorbodyorbits
themassdistributionofparticlesthatisdeterminedbythecolli- towardtheSunnorextendbeyondtheouteredgeoftheasteroid
sionprobability,butalsotheorbitaldistributions:asthelifetime belt.ThesmallerdustparticlesmovetheirwaystowardtheSun
Articlenumber,page7of17
A&Aproofs:manuscriptno.paper-con-aa-2
Fig.6.Particlecross-sectionareadensities(perunitvolumeofspace,AU2/AU3)foreachcomponentorpopulationofthemeteoroidmodelsby
Kelsalletal.(1998),Divine(1993)andIMEM(Dikarevetal.2004).NotethattwomapsforthecometarydustinIMEMareplottedindifferent
scale.
duetothePoyting-Robertsoneffect.Theinclinationsoftheiror- approach Jovian orbit within 0.5 AU or less. Small cometary
bitsareintact,sinceIMEMdoesnottakeintoaccountanyplan- particlesleakthroughtheinnerboundaryoftheregionofclose
etaryperturbationsotherthangravitationalscatteringbyJupiter encounterswithJupiterandmigratetowardtheSunundertheac-
thatrequirea close approachtothe giantplanet.Consequently, tionofthePoynting-Robertsoneffect,theyreachthehighestden-
thelatitudinaldistributionoftheirdensityisindependentonhe- sityattheshortestheliocentricdistances.Acometarydustden-
liocentricdistanceawayfromthesourceregion,i.e.asteroidbelt. sityenhancementintheformofasphericalshellwitharadiusof
Big cometary particles have a density peak along the orbit 5.2AUisanegligiblysmalldefectcausedbytheassumptionof
of Jupiter. This occurs because all their orbits are required to auniformdistributionofparticleorbitsinlongitudesofperihe-
Articlenumber,page8of17
V.V.Dikarev&D.J.Schwarz:MicrowaveThermalEmissionfromtheZodiacalDustCloud
Size, µm Size, µm
10-2 10-1 1 10 102 103 10-1 1 10 102 103
1 1
a
e
Ar Divine’s
n 0.75 Asteroidal 0.75
o
cti Core
e Halo
S
- 0.5 Inclined 0.5
s
s Eccentric
o
Cr IMEM
of 0.25 0.25 m>10-5g
F m<10-5g
D
C 0 0
-18 -15 -12 -9 -6 -3 0 -15 -12 -9 -6 -3 0
10 10 10 10 10 10 10 10 10 10 10 10 10
Mass, g Mass, g
Fig.7.Cumulativedistributionfunctionsofparticlecross-sectionareainmass(size)foreachpopulationoftheinterplanetarymeteoroidmodelby
Divine(1993)andIMEM(Dikarevetal.2004).Thematerialdensityofparticlesisequalto2.5gcm−3inbothmodelsexceptDivine’seccentric
population,inwhichitistentimeslower.Theparticlesizescaleshouldnotbeusedincombinationwiththeplotforthelatterpopulation.
liainIMEM.Concentricspherical“shells”ofdifferentdensities (2002) used the FIRAS (Far-Infrared Absolute Spectrometer)
onthemapofcometarymeteoroidswithm > 10−5 garedueto data from COBE and found that the annually averaged spec-
finitediscretizationofthedistributioninperiheliondistance. trumofthezodiacalcloudcanbefittedwithasingleblackbody
Cumulativedistributionfunctionsofmeteoroidcross-section at a temperature of 240 K with an absorption efficiency being
areainmassandsizeforeachpopulationofthemeteoroidmod- flatatwavelengthsshorterthan150µmand Q ∝ λ−2 beyond
abs
els by Divine (1993) and Dikarevetal. (2004) are plotted in 150µm.
Fig. 7 (normalized to unity). Note that one population of the The absorptionefficienciesof dustfromallthree bandsco-
Divine modelhas an assumed materialdensity of 0.25g cm−3, incideinKelsalletal.(1998)forthewavelengthsupto240µm.
whereasthestandardvalueassumedforallotherpopulationsas PlanckCollaborationXIV (2014) have removed this constraint
wellasinmanyothermeteoroidmodelsis2.5gcm−3. andallowedforindividualweightsofthe bandcontributionsin
TheKelsallmodelbetsonessentiallysingle-populationrep- themicrowaves.Theyfoundthatthebandskeephigh Q ∼ 1
abs
resentaionof theinterplanetarymeteoroidcloud,with the solar up to λ ∼ 3 mm, implying that their constituent particles are
ring and dust bands being rather minor features (Fig. 2). The macroscopic.Theabsorptionefficiencyofthesmoothcloudde-
meteoroidengineeringmodels,however,stateitclearlythatthe caysinthemicrowaves,however,notexactlyassharpasasimple
cloudiscomposedofdifferentsortsofdustinmajorpopulations approximationofFixsen&Dwek(2002)suggests.
thatare also distributeddifferentlyin space.We will see below
The inverse problem solution sometimes led
howthisaffectsthemicrowaveemissionpredictions.
PlanckCollaborationXIV (2014) to negative absorption
efficiencies of the smooth cloud or circumsolar dust ring and
Earth-trailing blob. (Those negative efficiencies are simply
3. ThermalEmissionModels
missing from Fig. 8 at certain wavelengths, as the logarithmic
The thermal emission from dust particles is expressed with re- scale does not permit negative ordinates.) Obviously, some
specttotheblackbodyemission B (T)atthewavelengthλand components of the Kelsall model were used by the fitting
λ
temperature T, using an emissivity modification factor which procedure to compensate for excessive contributions from the
matchestheabsorptionefficiencyfactor Q (s,λ)fortheparti- other components in such cases. This is a strong indication of
abs
clesize s(Bohren&Huffman1983).Q isdefinedastheratio insufficiencyoftheKelsallmodelinthemicrowaves.
abs
oftheeffectiveabsorptioncross-sectionareaoftheparticletoits Our study requires the absorption factors even further in
geometriccross-sectionarea. the microwaves. We use the numbers inferred by Kelsalletal.
(1998) and PlanckCollaborationXIV (2014) whenever pos-
sible, i.e. when they are available and positive. A nega-
3.1.TheKelsallmodel
tive absorption efficiency found for the smooth cloud by
The lightscatteringand emissionpropertiesofdustare notde- PlanckCollaborationXIV (2014) at λ = 2.1 mm is replaced
finedintheKelsallmodelforthewavelengthsbetweenandbe- withtheresultofinterpolationofthepositiveefficienciesfound
yond those of the COBE DIRBE instrument. Figure 8 shows atλ = 1.4 and3 mm,whereasnegativeefficienciesfor the cir-
Kelsall’semissivitymodificationfactors,orabsorptionefficien- cumsolarringatλ = 0.85,1.4and2.1mmaresimplynullified.
cies to preserve the uniformity of terms, for the DIRBE wave- TheabsorptionefficiencyisextrapolatedbeyondthePlanck/HFI
length bands. PlanckCollaborationXIV (2014) used the cross- wavelengths (λ > 3 mm, i.e. in the WMAP range) using the
sectionareadensityoftheKelsallmodelandfoundtheabsorp- approximation due to Fixsen&Dwek (2002) for the smooth
tion efficiencies for its components to describe approximately cloud Q ∝ λ−2, similar to Marisetal. (2006) who assessed
abs
thethermalemissionfrominterplanetarydustatthewavelengths the level of contamination of the Planck data by the Zodiacal
of Planck’sHigh FrequencyInstrument(HFI). Fixsen&Dwek microwaveemissionbasedon theKelsall model,andusingflat
Articlenumber,page9of17
A&Aproofs:manuscriptno.paper-con-aa-2
ss
bb
aa
QQ 11

yy
cc
nn
ee
ficifici 1100--11 Cloud
EfEf Ring

nn Blob
oo
ptipti 1100--22 Band 1
rr Band 2
oo
ss Band 3
bb
AA Fixsen & Dwek
--33
1100
1100 µµmm 110000 µµmm 11 mmmm 1100 mmmm
WWaavveelleennggtthh
Fig.8.AbsorptionefficienciesofdustintheKelsallmodeldeterminedfortheCOBEDIRBEwavelengthsbyKelsalletal.(1998),grayareaonthe
plot,andforthePlanck/HFIwavelengthsbyPlanckCollaborationXIV(2014).Fixsen&Dwek(2002)havealsousedthedatafromCOBE/FIRAS
instrumenttodeterminetheannuallyaveragedspectrumoftheZodiacalcloudplottedherewithadottedline.
Q = 1,0.5,1,and0.1fortheasteroiddustbands1,2,3,and (1)
abs
thecircumsolarringwiththetrailingblob,respectively.
where s is the radius of a spherical dust grain, λ is the wave-
length, F is the incident Solar radiance flux, B (T ) is the
⊙ λ D
3.2.SelectedMaterialsandAbsorptionEfficiencies blackbodyradianceatthedustparticle’stemperatureT .Asthe
D
left-handsideprovidesthetotalenergyabsorbedfromamono-
Predictions of the thermal emission from interplanetary dust
directionalincident flux, the right-handside gives the total en-
clouds using the meteoroid engineering models require opti-
ergyemitted,omni-directionally.Bydenotingtheabsorptionef-
calpropertiesofconstitutingparticles.FollowingDikarevetal. ficiencyaveragedovertheSolarspectrumwithQ¯ ,andthesame
(2009), we use the Mie light-scatteringtheoryand opticalcon- ⊙
quantityaveragedoverablackbodyspectrumattemperatureT
stantsofsilicateandamorphouscarbonaceoussphericalparticles with Q¯(T ), then using the Stefan-Boltzmann law and the SoD-
inordertoestimatetheabsorptionofsolarradiationandthermal D
larconstant,onecanrewriteEq.(1)inamoreconciseform(cf.
emissionbymeteoroids.Dikarevetal.(2009)bannedironfrom
Reach1988):
substancesfora hypotheticalmeteoroidcloudthatcouldbe re-
sponsible for anomalous CMB multipoles. In this paper, how- T =279K Q¯ /Q¯(T ) 1/4 R −1/2, (2)
epvoesre,swoemteakmeeitteobraicteksinastowaeclcloausnatsstienrcoeidirsounrfiasckesn,oiwtnistporceosemn-t D h ⊙ D i (cid:18)1AU(cid:19)
where R is the distance from the Sun. A perfect black body
intheinterplanetarydustparticlesandthereforeitmustbecon-
with Q = 1 throughoutthe spectrumhasthereforea temper-
sideredwhendiscussingthereal,nothypotheticalSolar-system abs
atureof279K at1 AU fromtheSun,inverselyproportionalto
medium.Waterandothericesarestill ignoredasextremelyin-
thesquarerootofthedistance.Thisinverse-square-roottrendis
efficientemittersofthemicrowaves.
oftencloselyfollowedbytherealdustparticles.
Dealingwiththemicrowavethermalemissionfromdustwith
TheKelsallmodelusesthetemperaturegivenbyequation
anassumedtemperature,Dikarevetal.(2009)didnotneedopti-
calconstantsforthevisualandnear-infraredwavelengths.Here 1AU 0.467
wecalculatethetemperaturesofdustinthermalequilibriumand T =286K . (3)
R !
the absorptionefficienciesare requiredfor the brightestpartof
the Solar spectrum.Figure 9 plotsthe data forthe wavelengths Kelsalletal.(1998)emphasizethatthedusttemperatureat1AU
between0.1and100µm. and absorption efficiencies could not be determined indepen-
dently, so that the temperature was found by assuming the
smoothcloudtobethedominantcomponentwithitsspectrumin
3.3.TheDustTemperatures
themid-IRbeingthatofa pureblackbody(i.e.,unitabsorption
Let us now calculate and discuss the temperaturesof dust par- efficienciesatλ=4.9,12,and25µm).
ticles of different sizes at different distances from the Sun for Theequilibriumtemperaturesofdustparticlesareshownin
the substancesselected in the previoussection. Figure10 plots Fig.10forabroadrangeofheliocentricdistances.Temperatures
theequilibriumtemperaturesofsphericalhomogeneousparticles used in the Kelsall model are plotted for comparison as well.
composedofsilicate,carbonaceousandironmaterialsaswellas Temperaturesof the micrometer-sizedparticlesare in mostob-
thedusttemperatureusedintheKelsallmodel. viousdisaccordwithKelsall’smodel,buttheytendtobehigher
Theequilibriumtemperatureisfoundfromthethermalbal- thantemperaturesofthelargerparticlesoftheircompositionas
anceequation well. The reason is simple: their absorptionefficienciesare too
low at the wavelengthsof ten and more micrometers, at which
∞ ∞ theirlargercounterpartsemittheenergyabsorbedfromtheSo-
πs2 Q (s,λ)F (λ)dλ=4πs2 Q (s,λ)B (T )dλ,
abs ⊙ abs λ D larradiationflux,thustheywarmtohighertemperaturesinorder
Z Z
0 0
Articlenumber,page10of17