The Microwave Thermal Emission from the Zodiacal Dust ...

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Astronomy& Astrophysicsmanuscript no. paper-con-aa-2 c©ESO 2015January 21, 2015

The Microwave Thermal Emissionfrom the Zodiacal Dust Cloud

Predicted with Contemporary Meteoroid ModelsValery V. Dikarev and Dominik J. Schwarz

Faculty of Physics, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germanye-mail:[email protected]

Received<date> / Accepted<date>

ABSTRACT

Predictions of the microwave thermal emission from the interplanetary dust cloud are made using several contemporary meteoroidmodels to construct the distributions of cross-section area of dust in space, and applying the Mie light-scattering theory to estimatethe temperatures and emissivities of dust particles in broad size and heliocentric distance ranges. In particular, themodel of theinterplanetary dust cloud by Kelsall et al. (1998, ApJ 508: 44–73), the five populations of interplanetary meteoroids ofDivine (1993,JGR 98(E9): 17,029–17,048) and the Interplanetary Meteoroid Engineering Model (IMEM) by Dikarev et al. (2004, EMP 95: 109–122) are used in combination with the optical properties of olivine, carbonaceous and iron spherical particles. The Kelsall model hasbeen widely accepted by the Cosmic Microwave Background (CMB) community. We show, however, that it predicts the microwaveemission from interplanetary dust remarkably different from the results of application of the meteoroid engineering models. We makemaps and spectra of the microwave emission predicted by the three models assuming variant composition of dust particles. Predictionscan be used to look for the emission from interplanetary dustin CMB experiments as well as to plan new observations.

Key words. Cosmology: cosmic background radiation - Zodiacal dust - Radiation mechanisms: thermal

1. Introduction

The unprecedented high-precision surveys of the mi-crowave sky performed recently by theWilkinson MicrowaveAnisotropy Probe (WMAP, Bennett et al. 2013) andPlanck(Planck Collaboration I 2014) observatories in search and char-acterisation of the large- and small-scale structure of theCosmicMicrowave Background (CMB) anisotropies pose new chal-lenges for simulation and subtraction of the foreground emissionsources, including the Solar-system dust. Previous templatesdesigned for this purpose were based on the Kelsall et al.(1998) model. They indicated little significance of the inter-planetary dust for theWMAP survey (Schlegel et al. 1998), yetremarkable contribution was detected at high frequencies ofPlanck (Planck Collaboration XIV 2014).

The angular power spectrum of CMB anisotropies is ingood agreement with the inflationaryΛ-cold dark matter model(Planck Collaboration I 2014). However, the reconstructedCMBmaps at the largest angular scales reveal some intriguing anoma-lies, among them unexpected alignments of multipole mo-ments, in particular with directions singled out by the So-lar system (Schwarz et al. 2004). The quadrupole and octopoleare found to be mutually aligned and they define axes thatare unusually perpendicular to the ecliptic pole and paral-lel to the direction of our motion with respect to the restframe of the CMB (the dipole direction). For reviews seeBennett et al. (2011), Copi et al. (2010) and Bennett et al. (2013)and Planck Collaboration XXIII (2014) as well as Copi et al.(2013) for a detailed analysis of final WMAP and first Planckdata. It has been suggested that an unaccounted observationbiasmay persist, e.g. yet another foreground source of the microwave

emission bound to the Solar system. Dikarev et al. (2009) ex-plored the possibility that the Solar-system dust emits more in-deed than it was previously thought, and found that the macro-scopic (> 0.1 mm in size) meteoroids can well contribute∼10 µK in the microwaves, i.e. comparable with the power of theCMB anomalies, without being detected in the infrared (IR) lightand included in the IR-based models like that of Kelsall et al.(1998). Babich et al. (2007) and Hansen et al. (2012) have alsostudied possible contribution of the Kuiper belt objects tothe mi-crowave foreground radiation. Dikarev et al. (2008) constructedand tested against theWMAP data some dust emission templates.

Here we improve and extend preliminary estimates ofDikarev et al. (2009). In addition to the Kelsall model, we usetwo meteoroid engineering models to make accurate and thor-ough maps of the thermal emission from the Zodiacal cloud inthe wavelength range from 30µm to 30 mm.

The paper is organized as follows. Section 2 introduces threemodels of the interplanetary meteoroid environment that weuseto predict the thermal emission from the zodiacal dust cloud. Adetailed description of the theory and observations incorporatedin each model is provided, it may be helpful not only for cosmol-ogists interested in understanding the solar-system microwaveforegrounds, but also for developers of new meteoroid modelswilling to comprehend earlier designs. Section 3 describesthethermal emission models. Empirical models as well as the Mielight-scattering theory are used to calculate the temperatures andemission intensities of spherical dust particles composedof sil-icate, carbonaceous and iron materials, for broad ranges ofsizeand heliocentric distance. The models of the spatial distributionof dust and thermal emission are combined in Section 4 in order

Article number, page 1 of 17

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Table 1. Frequenciesν, wavelengthsλ and bandwidths∆ν/ν ofthe CIB/CMB observations withCOBE DIRBE wavelength bands 4through 10,WMAP, andPlanck.

Instrument ν λ ∆ν/ν

COBE 61.2 THz 4.9µm 0.13DIRBE 25.0 THz 12µm 0.54

12.0 THz 25µm 0.345.00 THz 60µm 0.463.00 THz 100µm 0.322.14 THz 140µm 0.281.25 THz 240µm 0.40

Planck 857 GHz 350µm 0.25HFI 545 GHz 550µm 0.25

353 GHz 850µm 0.25217 GHz 1.4 mm 0.25143 GHz 2.1 mm 0.25100 GHz 3.0 mm 0.25

Planck 70 GHz 4.3 mm 0.2LFI 44 GHz 6.8 mm 0.2

30 GHz 10.0 mm 0.2WMAP W 94 GHz 3.2 mm 0.22

V 61 GHz 4.9 mm 0.23Q 41 GHz 7.3 mm 0.20Ka 33 GHz 9.1 mm 0.21K 22 GHz 13.6 mm 0.24

to make all-sky maps and spectra of the thermal emission fromthe interplanetary dust. Conclusions are made in Sect. 5.

Throughout this paper, we will be dealing with the wave-lengths mostly, since the dust optical properties are naturally de-scribed in terms of linear scales. The CMB community, however,is more accustomed to frequencies. A conversion table is there-fore useful for the observation wavebands of infrared detectorsand radiometers (Table 1).

2. Meteoroid Models

In this paper, we use three recent and contemporary modelsof the Zodiacal dust cloud due to Kelsall et al. (1998), Divine(1993) and Dikarev et al. (2004). The first model is constrainedby the infrared observations only from an Earth-orbiting satel-lite COBE. For brevity, it is referred to as the Kelsall modelhereafter. The Kelsall model is most familiar to and most of-ten used by the Cosmic Microwave Background (CMB) researchcommunity as it helps, and was designed to assess and mitigatethe contamination of the background radiation maps by the fore-ground radiation from interplanetary dust. The two other mod-els incorporate data obtained by diverse measurement techniquesand serve to predict meteoroid fluxes on spacecraft and to assessimpact hazards. Being applied in spacecraft design and analysismostly, they are often dubbed as meteoroid engineering models.We choose the Divine model and IMEM, with the latter abbre-viation standing for the Interplanetary Meteoroid EngineeringModel. We also check if the most recent NASA meteoroid en-gineering model, MEM (McNamara et al. 2004), can be used inour study, and explain why it cannot be.

The Kelsall model does not consider the size distribution ofparticles in the Zodiacal cloud. The size distribution is implic-itly presented by the integral optical properties of the cloud. Incontrast, the meteoroid engineering models are obliged to spec-ify the fluxes of meteoroids for various mass, impulse, or other

level-of-damage thresholds, hence they provide the size distribu-tion explicitly.

Profiles of particle cross-section area density per unit volumeof space are plotted for the Kelsall and Divine models as wellas IMEM in Fig. 1. Interestingly, the Kelsall model is sparserthan both the Divine model and IMEM at most heliocentric dis-tances. The meteoroid engineering models predict substantiallymore dust than Kelsall does, especially beyond 1 AU from theSun: Divine’s density is almost flat in the ecliptic plane between1 and 2 AU, exceeding Kelsall’s density by a factor of 3 in the as-teroid belt. The IMEM density remains similar to that of the Kel-sall model up to∼ 3 AU, showing a local maximum beyond theasteroid belt near 5 AU from the Sun, in the vicinity of Jupiter’sorbit. IMEM possesses the latitudinal distribution somewhat nar-rower than that of the Kelsall and Divine models.

All these distinctions stem from different observations andphysics incorporated in different models. When describing themin subsequent subsections, we do not aim at selecting the bestmodel. We take all three of them instead, with the intention to“bracket” the more complex reality by three different perspec-tives from different “points of view”.

2.1. The Kelsall model

A concise analytical description of the infrared emission fromthe interplanetary dust captured by theCOBE Diffuse InfraRedBackground Experiment (DIRBE) was proposed by Kelsall et al.(1998). It recognizes five emission components, a broad andbright smooth cloud, three finer and dimmer dust bands, andcircumsolar dust ring along the Earth orbit with an embeddedEarth-trailing blob. Each component is described by a parame-terised empirical three-dimensional density function specifyingthe total cross-section area of the component’s dust particles perunit volume of space (Fig. 2), and by their collective light scatter-ing and emission properties such as albedo, absorption efficien-cies, etc. The particle size distributions of the components arenot considered. Most of the light scattering and emission prop-erties are neither adopted from laboratory studies of natural ma-terials nor predicted by theories of light scattering, theyare freeparameters of the model fit to the DIRBE observations instead.

The smooth cloud is the primary component of the Kelsallmodel. Its density function is traditionally separated into radialand vertical terms. The radial term is a power law 1/Rαc withα = 1.34 ± 0.022 being the slope of the density decay withdistance from the cloud’s centreRc. The slope is known to beequal to one for the dust particles in circular orbits migrating to-ward the Sun under the Poynting-Robertson drag (orbiting theSun, the particles absorb its light coming from a slightly for-ward direction due to aberration, and the radiation pressure forcehas a non-zero projection against the direction of their motion;that projection causes gradual loss of the orbital energy and de-crease of the semimajor axis of particle’s orbit). The slopeisgreater than one if the particle orbits are initially eccentric, or ifthe zodiacal cloud is fed from sources broadly distributed overradial distances, so that the inner circles of the cloud are sup-plied from more dust sources than the outer circles (Leinertet al.1983; Gor’kavyi et al. 1997).

The vertical term of smooth cloud’s density is representedby a widened, modified fan model. The smooth cloud’s symme-try plane is inclined off the ecliptic plane. This happens becausethe Earth has no influence on the orbital dynamics of the vastmajority of cloud’s particles. The giant planets, Jupiter primar-ily, perturb the orbits of dust particles as well as their sources,or parent bodies, such as comets and asteroids, and control the


V. V. Dikarev & D. J. Schwarz: Microwave Thermal Emission from the Zodiacal Dust Cloud

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Fig. 1. Profiles of particle cross-section area density predicted with selected meteoroid models and compared with each other:ecliptic radial (left)and latitudinal at 1 AU from the Sun (right, the vernal equinox is at latitude 0◦).

inclinations of the orbits of the particles. An offset of the centerof the cloud from the Sun is also allowed in the Kelsall model(of the order of 0.01 AU). The vertical motion of the Earth withrespect to the cloud’s symmetry plane leads to the modulationsof the infrared emission from the zodiacal cloud reaching 30%in the ecliptic polar regions.

The dust bands are the remnants of collisional disruptions ofasteroids that resulted in formation of the families of asteroidsin similar orbits and remarkable dust belts. The dust bands werediscovered first on the high-resolution maps made by theIRASsatellite (Low et al. 1984) and recognized later in theCOBEDIRBE data as well. Kelsall et al. (1998) introduced three dustbands in their model, attributed to the Themis and Koronis aster-oid families (±1◦.4 ecliptic latitude), the Eos family (±10◦), andthe Maria/Io family (±15◦). The bands have symmetry planesdifferent from that of the smooth background cloud. The bandsare all double, with the density peaking above and below therespective symmetry plane, since their constituent particles re-tain the inclination of the ancestor asteroid orbit but the longi-tudes of nodes get randomized by the planetary perturbations.The vertical motion of such particles is similar to that of anos-cillator, which moves slower and spends more time near its ex-tremities, thus the density at the latitudes of each family’s or-bital inclination is high. One would expect a similarly shaped“edge-brightened” radial density with the peaks at the perihe-lion and aphelion distances of the parent body orbit (e.g. Sect. 3in Gor’kavyi et al. 1997). However, the dust in asteroid bandsis proven to be migrating toward the Sun under the Poynting-Robertson drag: Reach (1992) has demonstrated that the tem-peratures of particles and parallaxes of the bands measuredfromthe moving Earth are both higher than those supposed to be inthe respective asteroid families of their origin, implyingthat thebands extend farther towards the Sun than their ancestor bodies.Unlike Reach (1992), Kelsall et al. (1998) allowed for a partialmigration only of dust, by introducing a cut-off at a minimalheliocentric distance either defined or inferred individually foreach band. They also fixed rather than inferred most of the dustband shape parameters. It is noteworthy that Kelsall et al. (1998)did not introduce an explicit outer cut-off for the densities of thesmooth cloud and dust bands. Instead, they integrated the densi-ties along the lines of sight from the Earth to the maximal helio-centric distance of 5.2 AU (close to the Jovian orbital radius).

The dust particles on nearly circular orbits migrating underthe Poyinting-Robertson drag toward the Sun are temporarilytrapped in a mean-motion resonance with the Earth near 1 AU(Dermott et al. 1984). The resulting enhancement of the zodia-cal cloud density is described in the Kelsall model by the solarring and Earth-trailing blob. The trailing blob is the only com-ponent of the model that is not static: it is orbiting the Sun alonga circular orbit with a period of one year, and as the name sug-gests, its density peak is located behind the moving Earth. Thesolar ring and trailing blob have density peak distances slightlyoutside the Earth orbit, their symmetry planes are inclinedoff theecliptic plane, and the trailing blob orbits the Sun at a constantvelocity whereas the Earth velocity is variable due to the eccen-tricity of our planet’s orbit. Consequently, the Earth moves withrespect to the solar ring and trailing blob over the course oftheyear much like the other components of the Kelsall model.

The DIRBE instrument performed a simultaneous survey ofthe sky in 10 wavelength bands at 1.25, 2.2, 3.5, 4.9, 12, 25,60, 100, 140, and 240µm. It was permitted to take measure-ments anywhere between the solar elongations of 64◦ and 124◦.As the Earth-boundCOBE observatory progressed around theSun, DIRBE took samples of the infrared background radiationfrom all over the sky. The foreground radiation due to interplane-tary dust was a variable ingredient of the sky brightness becauseit depends on the changing position of observatory with respectto the cloud. Thus the fitting technique was based on minimizingthe difference between the brightness variations in time observedalong independent lines of sight and those predicted by the Kel-sall model for the same lines of sight, ignoring the underlyingphotometric baselines contaminated or even dominated by thebackground sources.

The accuracy of the Kelsall model in describing the infraredemission from interplanetary meteoroids in DIRBE’s wave-length bands and within the range of permitted solar elonga-tions is reported to be better than 2%. Interpolations of themodelbrightnesses between the instrument wavelengths and extrapola-tions beyond the wavelength and solar elongation ranges maybeprone to higher uncertainties. Indeed, the concise description ofthe zodiacal emission contribution to the DIRBE data providesno clue as to how the light scattering and emission properties ofits components depend on wavelength between and beyond theten DIRBE wavelength bands.



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Fig. 2. Radial and latitudinal profiles of particle cross-section area density in the Kelsall model, componentwise.

When applying the Kelsall model in the far infrared wave-lengths and microwaves, one should bear in mind that alreadyKelsall et al. (1998) found the 140 and 240µm bands nearlyuseless in constraining the density distribution parameters dueto relatively high detector noise and small contribution ofthe ra-diation from the interplanetary dust at these wavelengths.Con-sequently, the inferred parameters of the density distributions inthe Kelsall model are based on the observations at the wave-lengths up to 100µm. The dust particles are efficient in interact-ing with the electromagnetic radiation if their sizes are not toosmall with respect to the wavelength (2πs > λ, wheres is theparticle radius andλ is the wavelength). Dust grains with radiifrom∼ 16µm are therefore visible at the wavelength of 100µm.Dikarev et al. (2009) argue that a considerable amount of mete-oroids are present in the Solar system with the sizes bigger than∼ 100 µm which are outshined in the infrared light by moreabundant and ubiquitous smaller dust grains. The longest obser-vation wavelength of DIRBE at which Kelsall et al. (1998) couldstill constrain the density distributions of dust is short of beingcapable to resolve these particles unless they compose distinctivefeatures like dust bands.

We use the formulae for the density functions provided byPlanck Collaboration XIV (2014) since the original paper byKelsall et al. (1998) contains typographical misprints in the def-initions of the asteroid dust band and circumsolar ring densities.

2.2. The Divine model

A model of the interplanetary meteoroid environment (Divine1993) constructed to predict fluxes on spacecraft anywhere inthe Solar system from 0.05 to 40 AU from the Sun, is composedof five distinct populations, each of which has mathematicallyseparable distributions in particle mass and in orbital inclination,perihelion distance, and eccentricity (Fig. 3). Using the distribu-tions in orbital elements, Divine explicitly incorporatedKeple-rian dynamics of meteoroids in heliocentric orbits.

The Divine model is constrained by a large number of diversemeteoroid data sets. The mass distribution of meteoroids from10−18 to 102 g is fitted to the interplanetary meteoroid flux model(Grün et al. 1985), which in turn is based on the micro-cratercounts on lunar rock samples retrieved by theApollo missions,i.e. the natural surfaces exposed to the meteoroid flux near Earth,

and data from several spacecraft. The orbital distributions are de-termined using meteor radar data, observations of zodiacallightfrom the Earth as well as from theHelios andPioneer 10 inter-planetary probes, and meteoroid fluxes measured in-situ by im-pact detectors on boardPioneer 10 and 11,Helios 1, Galileo andUlysses spacecraft at the heliocentric distances ranging from 0.3to 18 AU. The logarithms of the model-to-data ratios were mini-mized in a root-mean-square sense. When modeling the zodiacallight, Divine assumed that the scattering function is independentof meteoroid mass. The albedos of dust particles were definedsomewhat arbitrarily in order to hide the populations necessaryto fit the in-situ measurements from the zodiacal light observa-tions. No infrared observations were used to constrain the model.

The core population is the backbone of the Divine model, itfits as much data as possible with a single set of distributions.The distribution in perihelion distancerπ of the core-populationparticles can be used as strictly proportional tor−1.3

π up to 2 AUfrom the Sun. This function resembles a spatial concentration,and its slope is close to Kelsall’sα = 1.34 for the radial densityterm: much like the infrared data fromCOBE, the zodiacal lightobservations from the Earth and interplanetary probes demandthat the slope be steeper than a unit exponent. The eccentricitiesare moderate (peaking at 0.1 and mostly below 0.4), inclinationsare small (mostly below 20◦).

The other populations fill in the gaps where one populationwith separable distributions is not sufficient to reproduce the ob-servations. Their names hint at distinctive features of their orbitaldistributions. The inclined population is characterized by incli-nations largely in the range 20◦–60◦, its eccentricities are all be-low 0.15. In contrast, the eccentric population is composedofparticles in highly eccentric orbits (the eccentricities are largelybetween 0.8 and 0.9), the inclinations are same as in the corepopulation. These two populations are added in order to com-pensate for a deficit of meteoroids from the core population withrespect to in-situ flux measurements on board theHelios space-craft. (The eccentric population is also used to match the inter-planetary flux model between 10−18 and 10−15 g.) The halo popu-lation has a uniform distribution of orbital plane orientations and“surrounds” the Sun as a halo between roughly 2 and 20 AU. Itpatches the meteoroid model where the above-listed populationsare not sufficient to reproduce theUlysses andPioneer in-situflux measurements. The eccentric, inclined and halo populations



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Fig. 3. Radial and latitudinal profiles of particle cross-section area density in the Divine model, componentwise.

are composed of meteoroids much smaller than those significantenough for predictions of the infrared and microwave observa-tions.

The asteroidal population is described last but it makes ahuge difference between Kelsall’s and Divine’s models. Alreadywhen fitting the first population to the meteor radar data, Divineobserved that the results for the distribution in perihelion dis-tance show a minimum nearrπ = 0.6 AU, suggesting that twocomponents are involved (i.e., inner and outer). The radialslopeof the inner component was consistent with that demanded bythe zodiacal light observations, and the inner and outer compo-nents contributed 45% and 55%, respectively, to the flux nearEarth. The outer component’s concentration peaked outsidetheEarth orbit, in the asteroid belt. Since the main contribution tothe zodiacal light comes from the particles with masses smallerthan 10−5 g and the median meteoroid mass of the meteor radarwas 10−4 g, it was reasonable to ascribe the inner fraction to apopulation dominated by smaller particles responsible forthe zo-diacal light, and the outer fraction to another one dominated bylarger particles detected as meteors only with the radar. The twofractions were divided into the core and asteroidal populations,accordingly. Their inclination and eccentricity distributions areidentical, but the distributions in perihelion distance and massare different. The asteroidal population is used to fit the inter-planetary flux model at large masses> 10−4 g.

Derived from data analysis rather than postulated theoreti-cally, the distinction between the core and asteroidal populationsin the Divine model separates – both on the mass and perihe-lion distance scales – the small dust particles migrating towardthe Sun under the Poynting-Robertson drag, the concentration ofwhich increases with the decrease of the heliocentric distance,and their immediate parent bodies, the larger particles swarmingfurther away from the Sun, i.e. in the asteroid belt according toDivine (1993).

2.3. IMEM

The Interplanetary Meteoroid Engineering Model (IMEM) isdeveloped for ESA by Dikarev et al. (2004). Like the Divinemodel, it uses the distributions in orbital elements and massrather than the spatial density functions of the Kelsall model, en-suring that the dust densities and fluxes are predicted in accord

with Keplerian dynamics of the constituent particles in heliocen-tric orbits. Expansion of computer memory has also enabled theauthors of IMEM to work with large arrays representing multi-dimensional distributions in orbital elements to replace Divine’sseparable distributions. The distribution in mass remainssepara-ble from the 3-D distribution in inclination, perihelion distanceand eccentricity.

IMEM is constrained by the micro-crater size statistics col-lected from the lunar rocks retrieved by theApollo crews,COBEDIRBE observations of the infrared emission from the inter-planetary dust at 4.9, 12, 25, 60, and 100µm wavelengths, andGalileo andUlysses in-situ flux measurements. An attempt to in-corporate new meteor radar data from the Advanced Meteor Or-bit Radar (AMOR, see Galligan & Baggaley 2004) was not suc-cessful (Dikarev et al. 2005), as the latitudinal number densityprofile of meteoroids derived from the meteor radar data stoodin stark contrast to that admissible by theCOBE DIRBE data,due to incomplete understanding of the observation biases.

The meteoroids in IMEM are distributed in orbital ele-ments in accord with an approximate physical model of the ori-gin and orbital evolution of particulate matter under the plan-etary gravity, Poynting-Robertson effect and mutual collisions.Grün & Dikarev (2009) visualise the IMEM distributions of me-teoroids in mass and in orbital elements in graph and color plots.Cross-section area density profiles are shown here in Fig. 4.

All interplanetary meteoroids are divided into populationsby origin/source and mass/dynamical regime. Grün et al. (1985)constructed a model of the mass distribution of interplane-tary meteoroids. Based on this model, they estimated particlelifetimes against two destructive processes, mutual collisionsand migration downward toward the Sun due to the Poynting-Robertson drag (terminated by particle evaporation or thermalbreak-up). Figure 5 shows that the lifetime against mutual colli-sions is shorter than the Poynting-Robertson time for the mete-oroids bigger than∼ 10−5 g in mass, located at 1 AU from theSun. The destruction of meteoroids is dominated by a factor often in rate by the respective process already one order of magni-tude away from this transition mass in either direction. Theratesvary with the distance from the Sun, however, the transitionmassremains nearly unchanged (Fig. 6 in Grün et al. 1985). That iswhy Dikarev et al. (2004) could simplify the problem by divid-ing the mass range into two subranges with distinct mass dis-



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Fig. 4. Radial and latitudinal profiles of particle cross-section area density in IMEM, componentwise. The density plots are not smooth since theorbital distributions in IMEM are approximated by step functions.

tribution slopes and dynamics. Meteoroids below the transitionmass of 10−5 g are treated as migrating from the sources towardthe Sun under the Poynting-Robertson drag, meteoroids abovethe transition mass are assumed to retain the orbits of theirul-timate ancestor bodies over their short lifetimes until collisionaldestruction.

The ultimate ancestor bodies of interplanetary dust in IMEMare comets and asteroids. A catalogue of asteroids is used toaccount for the latter source of meteoroids. Three distinctpop-ulations of asteroids are recognized and allowed to have inde-pendent total production rates: the Themis and Koronis families(semi-major axes 2.8 < a < 3.25 AU, eccentricities 0< e < 0.2,and inclinations 0< i < 3◦.5), Eos and Veritas families (2.95 <a < 3.05 AU, 0.05 < e < 0.15, 8◦.5 < i < 11◦.5), and the mainbelt (a < 2.8 AU). The orbital distributions of meteoroids morethan 10−5 g in mass are constructed by counting the numbersof catalogued asteroids per orbital space bins. The orbitaldistri-butions of meteoroids less than 10−5 g in mass are described asthe flow of particles along the Poynting-Robertson evolutionarypaths (Gor’kavyi et al. 1997) starting from the source distribu-tions defined earlier. The mass distributions are adopted piece-wise from the interplanetary meteoroid flux model by Grün et al.(1985) as shown in Fig. 5.

The orbital distributions of meteoroids from comets cannotbe defined as easy as those of meteoroids from asteroids. Be-cause of a number of loss mechanisms, such as comet nuclei de-cay, accretion, tidal disruption or ejection from the Solarsystemby planets, very few comets are active at a given epoch and listedin the catalogues. Moreover, the catalogues are prone to obser-vation biases since the comet nuclei are revealed by gas and dustshed at higher rates at lower perihelion distances.

In order to describe the orbital distributions of meteoroids ofcometary origin, Dikarev & Grün (2004) proposed an approx-imate analytical solution of the problem of a steady-state dis-tribution of particles in orbits with frequent close encounterswith a planet. This solution is applied to represent the orbitaldistributions of meteoroids from comets in Jupiter-crossing or-bits, i.e. the vast majority of catalogued comets. Big meteoroidswith masses greater than 10−5 g are confined to the region ofclose encounters with Jupiter. There is only one quantity that isapproximately conserved in the region of close encounters with

Jupiter. It is the Tisserand parameterT

T =aJ

a+ 2

√

aaJ

(1− e2) cosi,

with aJ being the semimajor axis of Jovian orbit. The num-bers of meteoroids inT -layers are therefore stable over a longperiod of time. They are free parameters to be determinedfrom the fit of IMEM to observations. The relative distribu-tions in orbital elements within eachT -layer are found theoreti-cally (Dikarev & Grün 2004).

Finer dust grains with masses less than 10−5 g leak fromthis region into the inner solar system and then migrate to-ward the Sun under the Poynting-Robertson drag. The Poynting-Robertson migration time is much longer than the close approachtime in the region of close encounters with Jupiter, consequently,the orbital distributions of all meteoroids are shaped there bythe gravitational scattering on Jupiter. It is only the dustleak-ing through the inner boundary of the region that is distributedas the flow of particles along the Poynting-Robertson evolution-ary paths (Gor’kavyi et al. 1997). Their distributions are also es-tablished theoretically in IMEM, with the normalisation factorsbeing inferred from the fit to observations.

An empirical finding by Divine of a helpful separation of thebulk of meteoroid cloud into the core and asteroidal populations– segregation of big and small dust particles – has become oneof the physical assumptions in IMEM.

The weights of populations of dust from comets and aster-oids in IMEM were fitted to in-situ flux measurements using thePoisson maximum-likelihood estimator and to infrared observa-tions using the Gaussian maxiumum-likelihood estimator.

IMEM was tested against those data not incorporated inthe model by Dikarev et al. (2005), confirming that the or-bital evolution approach allows for more reliable extrapola-tions of the observations and measurements incorporated inthemodel, it is compared with several other meteoroid models byDrolshagen et al. (2008).

2.4. MEM

The Meteoroid Engineering Model (MEM) described byMcNamara et al. (2004) is another recent development of the or-



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Fig. 5. Cumulative mass distribution of interplanetary meteoroids(Grün et al. 1985, top), meteoroid lifetimes against mutualcollisionsand Poynting-Robertson drag (middle), and cumulative massdistribu-tions of meteoroid populations in IMEM (Dikarev et al. 2005,bottom).The flux of meteoroids withm > 10−5 g is lower in IMEM than in theGrün model since the impact velocities of big meteoroids arehigher at1 AU from the Sun in IMEM and they produce larger craters on thelunarrocks than in the Grün model, which assumed a single impact velocityfor all meteoroid sizes.

bital distributions and software to predict fluxes on spacecraft inthe Solar system and near Earth. It is constrained by the Earth-based meteor radar dataCMOR (Canadian Meteor Orbit Radar)and zodiacal light observations from two interplanetary probes,Helios 1 and 2. The nominal heliocentric distance range at whichthe model is applicable, from 0.2 to 2 AU, is rather limited, how-ever. Even though the meteoroid mass range of 10−6 to 10 g cov-ered by the model is extremely interesting for the purposes ofour study, some assumptions made by the authors are rather ar-guable. In particular, the mutual collisions between meteoroidsare considered in the model ignoring dependence of the mete-oroid collision probability on its size, whereas Grün et al.(1985)calculated that the lifetime against collisional disruption of me-teoroids varies by a factor of 100 between the masses of 10−6 and1 g (their Fig. 6 and our Fig. 5 in the text above)! It is not onlythe mass distribution of particles that is determined by thecolli-sion probability, but also the orbital distributions: as the lifetime

against collisions becomes shorter than the Poynting-Robertsontime by orders of magnitude, the meteoroids can no longer sur-vive a travel from their sources toward the Sun and the orbitaldistributions of bigger particles are more similar with thedis-tributions of their sources than the distributions of smaller par-ticles. MEM is unfortunately unable to reveal and describe thisimportant feature of the interplanetary meteoroid cloud.

2.5. The Cross-Section Area Distributions

Figure 6 maps the total cross-section area of dust particlesperunit volume of space, as a function of the position in the Solarsystem, for each component or population of every meteoroidmodel introduced in the previous Section. TheX andZ coordi-nates are measured from the Sun, with theX axis pointing tothe vernal equinox andZ axis being perpendicular to the eclip-tic plane. The integral of the cross-section area density along aline of sight gives the optical depth of the cloud along that lineof sight, assuming the geometric-optics approximation, sphericalparticles, and ignoring that the particle efficiencies in absorbingand scattering light can in fact be higher or lower than unity.

The Kelsall model is reproduced in the upper row of maps.Three dust bands are shown together in the middle plot. Thesmooth cloud is to the left, and the solar ring is to the right ofit. The dominant component of the Kelsall model, the smoothcloud is getting higher in density only toward the Sun and thesymmetry plane slightly inclined off the ecliptic plane. The dustbands are the only component of the model bulking beyond theEarth orbit, and they are by definition bound to the ecliptic lati-tudes of their ancestor asteroid families’ orbit inclinations. Boththe smooth cloud and the dust bands extend up to the heliocen-tric distance of 5.2 AU only, since Kelsall et al. (1998) stoppedintegrating the model densities there. A more rigorous model ofthe bands composed of dust migrating toward the Sun due tothe Poynting-Robertson effect would put their densities’ cut-offsat the outer boundaries of the corresponding asteroid families.Note, however, that most of the thermal infrared emission fromthe interplanetary dust observed byCOBE was due to the dustwithin 0.5–1 AU from the Earth orbit, sinceCOBE could not bepointed too close to the Sun and dust is too cold and inefficientat thermal emission far from the Sun (Dikarev et al. 2009, theirSect. 2.1 and Fig. 2). Thus for the purpose of modelling the in-frared thermal emission observed from the Earth, the behaviourof the density at multiples of an astronomical unit was not im-portant. This may not be the case for the microwave emissionthough.

The next two rows of maps depict the populations of the Di-vine model. The core and asteroidal populations are importantin the infrared and microwave emission ranges. They are dis-tributed remarkably different in space, with the core populationdensity growing higher toward the Sun and the asteroidal popula-tion density peaking beyond the orbit of the Earth. The two popu-lations are also composed of particles of different sizes: the corepopulation mostly smaller and the asteroidal population moslybigger than∼ 50 µm. The halo, inclined and eccentric popula-tions are provided for the sake of completeness. Their tiny,up to∼ 10 µm-sized particles are ignorable in the wavelength rangesof interest.

The bottom row of maps and the right map in the third rowexhibit the populations from IMEM. The asteroidal dust parti-cles bigger than 10−5 g in mass are confined to the asteroid belt,naturally. They are not migrating from their ancestor body orbitstoward the Sun nor extend beyond the outer edge of the asteroidbelt. The smaller dust particles move their ways toward the Sun



Fig. 6. Particle cross-section area densities (per unit volume of space, AU2/AU3) for each component or population of the meteoroid models byKelsall et al. (1998), Divine (1993) and IMEM (Dikarev et al.2004). Note that two maps for the cometary dust in IMEM are plotted in differentscale.

due to the Poyting-Robertson effect. The inclinations of their or-bits are intact, since IMEM does not take into account any plan-etary perturbations other than gravitational scattering by Jupiterthat require a close approach to the giant planet. Consequently,the latitudinal distribution of their density is independent on he-liocentric distance away from the source region, i.e. asteroid belt.

Big cometary particles have a density peak along the orbitof Jupiter. This occurs because all their orbits are required to

approach Jovian orbit within 0.5 AU or less. Small cometaryparticles leak through the inner boundary of the region of closeencounters with Jupiter and migrate toward the Sun under theac-tion of the Poynting-Robertson effect, they reach the highest den-sity at the shortest heliocentric distances. A cometary dust den-sity enhancement in the form of a spherical shell with a radius of5.2 AU is a negligibly small defect caused by the assumption ofa uniform distribution of particle orbits in longitudes of perihe-



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Fig. 7. Cumulative distribution functions of particle cross-section area in mass (size) for each population of the interplanetary meteoroid model byDivine (1993) and IMEM (Dikarev et al. 2004). The material density of particles is equal to 2.5 g cm−3 in both models except Divine’s eccentricpopulation, in which it is ten times lower. The particle sizescale should not be used in combination with the plot for the latter population.

lia in IMEM. Concentric spherical “shells” of different densitieson the map of cometary meteoroids withm > 10−5 g are due tofinite discretization of the distribution in perihelion distance.

Cumulative distribution functions of meteoroid cross-sectionarea in mass and size for each population of the meteoroid mod-els by Divine (1993) and Dikarev et al. (2004) are plotted inFig. 7 (normalized to unity). Note that one population of theDivine model has an assumed material density of 0.25 g cm−3,whereas the standard value assumed for all other populations aswell as in many other meteoroid models is 2.5 g cm−3.

The Kelsall model bets on essentially single-population rep-resentaion of the interplanetary meteoroid cloud, with thesolarring and dust bands being rather minor features (Fig. 2). Themeteoroid engineering models, however, state it clearly that thecloud is composed of different sorts of dust in major populationsthat are also distributed differently in space. We will see belowhow this affects the microwave emission predictions.

3. Thermal Emission Models

The thermal emission from dust particles is expressed with re-spect to the blackbody emissionBλ(T ) at the wavelengthλ andtemperatureT , using an emissivity modification factor whichmatches the absorption efficiency factorQabs(s, λ) for the parti-cle sizes (Bohren & Huffman 1983).Qabs is defined as the ratioof the effective absorption cross-section area of the particle to itsgeometric cross-section area.

3.1. The Kelsall model

The light scattering and emission properties of dust are notde-fined in the Kelsall model for the wavelengths between and be-yond those of theCOBE DIRBE instrument. Figure 8 showsKelsall’s emissivity modification factors, or absorption efficien-cies to preserve the uniformity of terms, for the DIRBE wave-length bands. Planck Collaboration XIV (2014) used the cross-section area density of the Kelsall model and found the absorp-tion efficiencies for its components to describe approximatelythe thermal emission from interplanetary dust at the wavelengthsof Planck’s High Frequency Instrument (HFI). Fixsen & Dwek

(2002) used the FIRAS (Far-Infrared Absolute Spectrometer)data fromCOBE and found that the annually averaged spec-trum of the zodiacal cloud can be fitted with a single blackbodyat a temperature of 240 K with an absorption efficiency beingflat at wavelengths shorter than 150µm andQabs ∝ λ

−2 beyond150µm.

The absorption efficiencies of dust from all three bands co-incide in Kelsall et al. (1998) for the wavelengths up to 240µm.Planck Collaboration XIV (2014) have removed this constraintand allowed for individual weights of the band contributions inthe microwaves. They found that the bands keep highQabs ∼ 1up to λ ∼ 3 mm, implying that their constituent particles aremacroscopic. The absorption efficiency of the smooth cloud de-cays in the microwaves, however, not exactly as sharp as a simpleapproximation of Fixsen & Dwek (2002) suggests.

The inverse problem solution sometimes ledPlanck Collaboration XIV (2014) to negative absorptionefficiencies of the smooth cloud or circumsolar dust ring andEarth-trailing blob. (Those negative efficiencies are simplymissing from Fig. 8 at certain wavelengths, as the logarithmicscale does not permit negative ordinates.) Obviously, somecomponents of the Kelsall model were used by the fittingprocedure to compensate for excessive contributions from theother components in such cases. This is a strong indication ofinsufficiency of the Kelsall model in the microwaves.

Our study requires the absorption factors even further inthe microwaves. We use the numbers inferred by Kelsall et al.(1998) and Planck Collaboration XIV (2014) whenever pos-sible, i.e. when they are available and positive. A nega-tive absorption efficiency found for the smooth cloud byPlanck Collaboration XIV (2014) atλ = 2.1 mm is replacedwith the result of interpolation of the positive efficiencies foundat λ = 1.4 and 3 mm, whereas negative efficiencies for the cir-cumsolar ring atλ = 0.85, 1.4 and 2.1 mm are simply nullified.The absorption efficiency is extrapolated beyond thePlanck/HFIwavelengths (λ > 3 mm, i.e. in theWMAP range) using theapproximation due to Fixsen & Dwek (2002) for the smoothcloud Qabs ∝ λ

−2, similar to Maris et al. (2006) who assessedthe level of contamination of thePlanck data by the Zodiacalmicrowave emission based on the Kelsall model, and using flat



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Fig. 8. Absorption efficiencies of dust in the Kelsall model determined for theCOBE DIRBE wavelengths by Kelsall et al. (1998), gray area on theplot, and for thePlanck/HFI wavelengths by Planck Collaboration XIV (2014). Fixsen& Dwek (2002) have also used the data fromCOBE/FIRASinstrument to determine the annually averaged spectrum of the Zodiacal cloud plotted here with a dotted line.

Qabs= 1, 0.5, 1, and 0.1 for the asteroid dust bands 1, 2, 3, andthe circumsolar ring with the trailing blob, respectively.

3.2. Selected Materials and Absorption Efficiencies

Predictions of the thermal emission from interplanetary dustclouds using the meteoroid engineering models require opti-cal properties of constituting particles. Following Dikarev et al.(2009), we use the Mie light-scattering theory and optical con-stants of silicate and amorphous carbonaceous spherical particlesin order to estimate the absorption of solar radiation and thermalemission by meteoroids. Dikarev et al. (2009) banned iron fromsubstances for a hypothetical meteoroid cloud that could bere-sponsible for anomalous CMB multipoles. In this paper, how-ever, we take it back into account since iron is known to com-pose some meteorites as well as asteroid surfaces, it is presentin the interplanetary dust particles and therefore it must be con-sidered when discussing the real, not hypothetical Solar-systemmedium. Water and other ices are still ignored as extremely in-efficient emitters of the microwaves.

Dealing with the microwave thermal emission from dust withan assumed temperature, Dikarev et al. (2009) did not need opti-cal constants for the visual and near-infrared wavelengths. Herewe calculate the temperatures of dust in thermal equilibrium andthe absorption efficiencies are required for the brightest part ofthe Solar spectrum. Figure 9 plots the data for the wavelengthsbetween 0.1 and 100µm.

3.3. The Dust Temperatures

Let us now calculate and discuss the temperatures of dust par-ticles of different sizes at different distances from the Sun forthe substances selected in the previous section. Figure 10 plotsthe equilibrium temperatures of spherical homogeneous particlescomposed of silicate, carbonaceous and iron materials as well asthe dust temperature used in the Kelsall model.

The equilibrium temperature is found from the thermal bal-ance equation

πs2∫ ∞

0Qabs(s, λ)F⊙(λ) dλ = 4πs2

∫ ∞

0Qabs(s, λ)Bλ(TD) dλ,

(1)

where s is the radius of a spherical dust grain,λ is the wave-length,F⊙ is the incident Solar radiance flux,Bλ(TD) is theblackbody radiance at the dust particle’s temperatureTD. As theleft-hand side provides the total energy absorbed from a mono-directional incident flux, the right-hand side gives the total en-ergy emitted, omni-directionally. By denoting the absorption ef-ficiency averaged over the Solar spectrum withQ̄⊙, and the samequantity averaged over a blackbody spectrum at temperatureTDwith Q̄(TD), then using the Stefan-Boltzmann law and the So-lar constant, one can rewrite Eq. (1) in a more concise form (cf.Reach 1988):

TD = 279 K[

Q̄⊙/Q̄(TD)]1/4

( R1 AU

)−1/2

, (2)

whereR is the distance from the Sun. A perfect black bodywith Qabs = 1 throughout the spectrum has therefore a temper-ature of 279 K at 1 AU from the Sun, inversely proportional tothe square root of the distance. This inverse-square-root trend isoften closely followed by the real dust particles.

The Kelsall model uses the temperature given by equation

T = 286 K

(

1 AUR

)0.467

. (3)

Kelsall et al. (1998) emphasize that the dust temperature at1 AUand absorption efficiencies could not be determined indepen-dently, so that the temperature was found by assuming thesmooth cloud to be the dominant component with its spectrum inthe mid-IR being that of a pure blackbody (i.e., unit absorptionefficiencies atλ = 4.9, 12, and 25µm).

The equilibrium temperatures of dust particles are shown inFig. 10 for a broad range of heliocentric distances. Temperaturesused in the Kelsall model are plotted for comparison as well.Temperatures of the micrometer-sized particles are in mostob-vious disaccord with Kelsall’s model, but they tend to be higherthan temperatures of the larger particles of their composition aswell. The reason is simple: their absorption efficiencies are toolow at the wavelengths of ten and more micrometers, at whichtheir larger counterparts emit the energy absorbed from theSo-lar radiation flux, thus they warm to higher temperatures in order



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Fig. 9. Optical constants and absorption efficiencies for olivine and iron of Pollack et al. (1994), and for amorphous carbonaceous grains from the‘BE’ sample of Zubko et al. (1996) used in this paper, in addition to the data introduced by Dikarev et al. (2009). The absorption efficiencies arecalculated for the sizes of spherical dust grains indicatedin the bottom row of plots.

to shift the maximum of the Planck function to shorter wave-lengths. Iron is an exception from this rule, all particles madeof this substance are considerably warmer, being unable to emitthe radiation thermally at longer wavelengths regardless of size.(It does not contradict to the fact that communications in ra-diowaves are assisted by iron and other metallic antennas, as ironis very good at scattering the electromagnetic emission at longerwavelengths simultaneously!)

Amorphous carbonaceous particles (labeled “Zubko BE” inour plots) from∼ 10 µm in size are in the best agreement withthe Kelsall model temperatures, among the real species con-sidered here. This is a good indication for the Kelsall model,since most interplanetary meteoroids are indeed expected to becomposed of carbonaceous material (Reach 1988; Dikarev et al.2009). The temperatures of the silicaceous particles are some-what lower, as olivine is rather transparent in the visual light thatit absorbs, remaining opaque in the infrared light that it emits,but this difference is quite negligible starting from particle radiiof 100µm, especially at the heliocentric distances beyond 1 AU(Fig. 11).

In subsequent sections, however, we assume a single temper-ature for all dust particles of each substance, regardless of size.Our results are therefore less precise for dust particles smallerthan∼ 100 µm in radii. Fortunately for achievement of this pa-per’s goal, the microwave emission from large particles is moreimportant to predict accurately, since small particles arenot effi-cient emitters at that long wavelengths.

4. The Microwave Thermal Emission from Dust

We are ready now to make maps of the thermal emission fromthe Zodiacal dust for the broadest range of wavelengths and ob-servation points. Indeed, the surface brightness of the skydue

to the Zodiacal cloud, measured in units of power per unit solidangle and unit wavelength, observed from the locationr in thedirection specified by a unit vectorp̂, is given by∫ ∞

0Bλ(TD(|r + lp̂)|)Cabs(λ, r + lp̂)dl, (4)

whereCabs(λ, r) is the absorption cross-section of dust per unitvolume of space. In the Kelsall model,Cabs is a sum of the prod-ucts of the total cross-section area of dust by the emissivity mod-ification factors for the wavelengthλ, over all model compo-nents. Meteoroid engineering models require an evaluationof anested integral over the particle sizes:

Cabs(λ, r) = −∫

πs2 Qabs(λ, s)∂N(s, r)∂s

ds, (5)

where N(s, r) is the number density of meteoroids with radiigreater thans at the locationr. The minus sign in Eq. (5) is nec-essary since∂N(s, r)/∂s ≤ 0. Figure 11 shows that the equilib-rium temperatureTD depends on particle size as well, however,this dependence is important only for the small dust grains withs < 100µm, which can be safely neglected in the microwaves.

Note that no color correction factor is applied, so that a dif-ference of at least several percent is to be expected betweenour intensities and Kelsall’s Zodiacal Atlas for the wavelengthswhere a direct comparison is possible.

4.1. All-Sky Maps

All-sky maps of the thermal emission from the interplanetarydust are made in Fig. 12 using the Kelsall model as well as me-teoroid engineering models assuming that their particles are car-bonaceous. Five observation wavebands ofCOBE, Planck and



Fig. 12. All-sky maps of the thermal emission from the Zodiacal cloud, according to the Kelsall model and two meteoroid engineering mod-els (Divine 1993; Dikarev et al. 2004) populated by the carbonaceous particles, as seen from Earth at the fall equinox time, for selected wavebandsof theCOBE, Planck andWMAP observatories. The reference frame is ecliptic, each map’scenter points to the antisolar direction (i.e., the vernalequinox), a 30◦-wide band around the Sun is masked. The grey scale of the upper row of maps is in MJy sterad−1, the other rows are inµK of atemperature in excess of the CMB. Each map has its own brightness scale.

WMAP are selected to demonstrate the transformation of thebrightness distribution from the far infrared light (240µm) tomicrowaves (13.6 mm).

At short wavelengths, emission from fine dust grains is dom-inant, and all models of the Zodiacal cloud show the maximumemission near the Sun. The smooth background cloud is re-sponsible for this brightness distribution in the Kelsall model,the core population in the Divine model and the meteoroidswith massesm < 10−5 g migrating toward the Sun under thePoynting-Robertson drag in IMEM are also concentrated near

the Sun. As the wavelength grows, the fine dust grains fade out,but bigger meteoroids remain bright. The components of the Kel-sall model possessing spectra of macroscopic particles, i.e. theasteroid dust bands, Divine’s asteroidal population and the me-teoroids with massesm > 10−5 g in IMEM are located mostlybeyond the orbit of the Earth. Instead of the brightness peakofthe fine dust at small solar elongations, the big meteoroids ap-pear as a band, or bands, along the ecliptic, broadening in theantisolar direction: the surface brightness of diffuse objects does



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Fig. 13. Brightness profiles of the thermal emission from the Zodiacal cloud along the great circles in the ecliptic plane (ecliptic latitudel = 0) andperpendicular to the ecliptic plane at solar elongationε = 90◦, according to the Kelsall model and two meteoroid engineering models populated bythe carbonaceous particles, as seen from Earth, for selected wavebands of theCOBE, Planck andWMAP observatories. The excess temperaturesare calculated w.r.t. the CMB.

not depend on the distance from the observer, but their angularsize increases due to their proximity in the antisolar direction.

Figure 13 shows the brightness profiles built with the modelsfrom Fig. 12 for Earth-bound observatories scanning the celes-tial sphere in two important great circles, one in the ecliptic plane(zero ecliptic latitude) starting from the vernal equinox counter-clockwise, and another perpendicular to the ecliptic plane, at thesolar elongation of 90◦, from the ecliptic north pole toward theapex of Earth’s motion about the Sun. The scan in the eclipticplane is only unmasked, however, when the target area on thesky was visible withWMAP (solar elongations from 90◦ to 135◦)

or COBE DIRBE (solar elongations from 64◦ to 124◦). Note thatboth profiles match at scan angles of 90◦ and 270◦ since the greatcircles are crossing there.

The Divine model and IMEM with the carbonaceous me-teoroids appear to be substantially brighter than the Kelsallmodel. Note that IMEM was fitted to the DIRBE data up toλ = 100 µm only, using a composition of dust different fromthe triple used here, and Divine used no infrared data at all.Thus no exact match with the Kelsall model is expected, espe-cially at the wavelengthsλ > 100 µm for which Kelsall et al.(1998) themselves found theCOBE DIRBE data insufficiently



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Fig. 10. Equilibrium temperatures of homogeneous spherical dust par-ticles composed of different substances.

constraining the model (Sect. 2.1). The difference is substantiallylower (25–30%) at shorter wavelengths (12–60µm). However,Planck Collaboration XIV (2014) inferred the absorption effi-ciencies of the Kelsall model components for thePlanck wave-bands as well, and reported a lower emission from the Zodiacalcloud as shown in Fig. 12 and 13 on the plots forλ = 850 µmand 2.1 mm.

To pave the way for an explanation of the discrepancy, weremind that the interplanetary dust is not entirely carbonaceous,and meteoroids of other chemical composition yield lower in-frared and microwave thermal emission. Besides, meteoroiden-gineering models are fitted to many other data sets, typically withan accuracy substantially lower than the models of the electro-magnetic emission from dust. Finally, the fitting of the Kelsallmodel to thePlanck data by Planck Collaboration XIV (2014)resulted in confusingly negative absorption efficiencies of dustsometimes, implying that the model may be incomplete, withsome model components used by the fitting procedure to com-pensate for excessive emission from the other components. Onecan also see that the Kelsall model is deficient in the eclipticplane, with the asteroid dust bands allowed to shine just a lit-tle above and below that plane only, and not e.g. in the ecliptic

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plane. A new fit to thePlanck data using different meteoroidmodels would therefore be of great value, since these modelsmay be better in reproducing the relative brightness distributionof the microwave emission from dust, but they also may cur-rently overestimate the number of big meteoroids in the inter-planetary space, so they also may be improved by such a fit.

Maps of the thermal emission of silicate particles are qualita-tively similar with those of the carbonaceous meteoroids and arenot shown here. However, the emission spectra of the zodiacalcloud for the silicate particles are discussed in the next Section.

We make maps of the thermal emission from the iron me-teoroids since they are remarkably different from the two ma-terials discussed above (Fig. 14). Fine dust grains bulkingnearthe Sun remain the brightest source regardless of the wavelengthof observation. This stems from a very weak dependence of theabsorption efficiency of iron particles on their size (within therange considered in this paper, see Fig. 9).

The difference in morphology of the maps for carbonaceousand iron particles suggests the way to discriminate betweendif-ferent sorts of interplanetary meteoroids from the microwave ob-servations, when their accuracy permits.



Fig. 14. All-sky maps of the thermal emission from the interplanetary dust predicted by two meteoroid engineering models, assuming that theirconstituent particles are composed of iron. The reference frame is as in Fig. 12. The brightness is provided in units of MJy sterad−1 for theCOBEwaveband centered at 240µm, and in units ofµK for the Planck andWMAP wavebands.

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Fig. 15. Thermal emission spectra of the Zodiacal cloud predicted bythe Kelsall model for the apex of Earth’s motion about the Sunandecliptic pole. Large squares are the excess temperatures w.r.t. the black-body emission of the CMB (right scale), small squares are theabsoluteintensities of emission (left scale).

4.2. Thermal Emission Spectra

Figures 15, 16 and 17 show the thermal emission spectra ofthe Zodiacal cloud predicted by Kelsall et al., pointed towardthe apex of Earth’s motion about the Sun and ecliptic pole.The graphs are provided with two brightness scales: the abso-lute intensities are more natural and easier to read in the in-frared wavelengths, whereas the excess temperatures are instan-taneously comparable with other foreground emission sourcesin the CMB studies. The excess temperatures∆T are definedin accord with Dikarev et al. (2009) so thatBλ(TCMB + ∆T ) =Bλ(TCMB) + ǫ, whereǫ is the intensity of emission in excess ofthe CMB radiation at the temperatureTCMB = 2.725 K, withthe blackbody intensityBλ being calculated precisely with thePlanck formula rather than in the Rayleigh-Jeans approximation.

The Kelsall model is evaluated at the wavelengths of obser-vations withCOBE DIRBE andPlanck HFI, for which the ab-sorption efficiencies (“emissivity modification factors”) of thedust particles in the model components have been determined,and using the extrapolations described in Sect. 3.1 for theWMAPwavelengths. The model predicts rather low emission in the mi-crowaves, e.g. below 1µK in the apex and below 0.1µK in theecliptic poles forλ > 3 µm. The emission in the apex is almostentirely due to the asteroid dust bands presumably having flatabsorption efficiencies at these wavelengths, while a continuingdecrease of the excess temperature with the wavelength increasefor the polar line of sight forλ > 3 mm indicates that the smoothbackground cloud is contributing withQabs ∝ λ

−2. Accordingto the Kelsall model, the foreground emission due to the Zodia-cal dust could indeed be disregarded in theWMAP survey of themicrowave sky with its accuracy of 20µK per 0.3◦ square pixelover one year.

The Divine model in Fig. 16 turns out to be a lot brighter,however. It predicts an excess temperature of up to∼ 20 µK inWMAP’s W Band atλ = 3 mm, and∼ 5 µK in the K Bandat λ = 13.6 mm, assuming that the meteoroids are all carbona-ceous. Silicate meteoroids would add much less than 10µK inthe entireWMAP wavelength range. If the interplanetary dustparticles were composed exclusively of iron, the Divine modelwould not let them shine brighter than the Kelsall model.

Predictions made with IMEM for the same compositions asabove are shown in Fig. 17. IMEM is dimmer than the Divinemodel, it is still brighter than that by Kelsall.

We have also estimated the total mass and cross-section ofmeteoroids in the Divine model and IMEM within 5.2 AU fromthe Sun, i.e. inside a sphere considered by Kelsall et al. (1998)as the outer boundary of the Zodiacal cloud. The Divine modelcontains∼ 2 · 1020 g mass and 6· 10−6 AU2 cross-section areaof meteoroids. Assuming a density of 2.5 g cm−3, the total massof meteoroids could be contained in a single spherical body ofthe radius∼ 30 km, the total cross-section area correspons to asingle sphere of the radius∼ 2 · 105 km. The last value of radiuswould be possessed by a planet three times bigger than Jupiter.



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Fig. 16. Thermal emission spectra of the Zodiacal cloud predicted bythe Divine model for the apex of Earth’s motion about the Sun andecliptic pole, assuming that the dust particles are composed of amor-phous carbon (top), silicate (middle) and iron (bottom) material. Thickcurves are the excess temperatures w.r.t. the blackbody emission of theCMB (right scale), dashed curves are the absolute intensities of emis-sion (left scale).

IMEM has only 6· 1019 g mass and 6· 10−6 AU2 cross-sectionarea of meteoroids within 5.2 AU from the Sun.

Fixsen & Dwek (2002) assumed a certain size distributionof interplanetary meteoroids and used the optical properties forsilicate, graphite and amorphous carbonaceous particles in or-der to reproduce the annually-averaged spectrum of the Zodiacalcloud measured with the FIRAS instrument on boardCOBE. Thesmooth cloud of the Kelsall model was used to describe the spa-tial number density distribution of meteoroids. They foundthetotal mass of meteoroids in the range 2–11·1018 g, i.e. remark-ably lower than in the Divine model and IMEM.

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Fig. 17. Thermal emission spectra of the Zodiacal cloud predicted byIMEM for the apex of Earth’s motion about the Sun and eclipticpole,assuming that the dust particles are composed of amorphous carbon(top), silicate (middle) and iron (bottom) material. Thickcurves are theexcess temperatures w.r.t. the blackbody emission of the CMB (rightscale), dashed curves are the absolute intensities of emission (left scale).

5. Conclusion

We have made predictions of the microwave thermal emis-sion from the Zodiacal dust cloud in the wavelength rangeof COBE DIRBE, WMAP and Planck observations. Weused the Kelsall et al. (1998) model extrapolated to the mi-crowaves using the optical properties of interplanetary dust in-ferred by Planck Collaboration XIV (2014) and Fixsen & Dwek(2002), and two engineering meteoroid models (Divine 1993;Dikarev et al. 2004) in combination with the Mie light-scatteringtheory to simulate the thermal emission of silicate, carbonaceousand iron spherical dust particles.

We have found that the meteoroid engineering models depictthe thermal emission substantially brighter and distributed dif-ferently across the sky and wavelengths than the Kelsall model



does, due to the presence of large populations of macroscopicparticles in the engineering meteroid models that are only avail-able in the asteroid dust bands of the Kelsall model. Both theDi-vine model and IMEM confirm an earlier estimate of a∼ 10 µKthermal emission from interplanetary dust (Dikarev et al. 2009)for theWMAP observations, provided that the dominant particlecomposition is carbonaceous. At smaller solar elongations, in-terplanetary dust can be orders of magnitude brighter, naturally.

More detailed search for and account of interplanetary dustare therefore worthwhile in the CMB experiment results. Mapsof the microwave sky should also be scrutinized to constrainthenumber of macroscopic particles in the engineering meteoroidmodels, as they are important to assess the impact hazard forlong-term manned missions with large spaceships to the Moonor Mars.

Acknowledgements. The authors thank Carlo Burigano, Craig Copi, KennethGanga, Dragan Huterer, Pavel Naselsky and Glenn Starkman for insightfuldiscussions. This work was supported by the German ResearchFoundation(Deutsche Forschungsgemeinschaft), grant reference SCHW1344/3 – 1, and viathe Research Training Group 1620 ‘Models of Gravity’.

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